Millimeter-Wave Radar Targets and Clutter (Artech House Radar Library)

Millimeter-Wave Radar Targets and Clutter

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Millimeter-Wave Radar Targets and Clutter

Gennadiy P. Kulemin Technical Editor David K. Barton

Artech House Boston • London www.artechhouse.com

Library of Congress Cataloging-in-Publication Data Kulemin, G. P. (Gennadii Petrovich) Millimeter-wave radar targets and clutter / Gennadiy P. Kulemin. p. cm. — (Artech House radar library) Includes bibliographical references and index. ISBN 1-58053-540-2 (alk. paper) 1. Radar targets. 2. Radar—Interference. 3. Millimeter waves. I. Title. II. Series. TK6580.K68 2003 621.3848—dc22 2003060064

British Library Cataloguing in Publication Data Kulemin, Gennadiy P. Millimeter-wave radar targets and clutter. — (Artech House radar library) 1. Radar—Interference 2. Backscattering 3. Radar targets 4. Millimeter wave devices I. Title 621.3’848 ISBN 1-58053-540-2

Cover design by Yekaterina Ratner

 2003 ARTECH HOUSE, INC. 685 Canton Street Norwood, MA 02062 All rights reserved. Printed and bound in the United States of America. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher. All terms mentioned in this book that are known to be trademarks or service marks have been appropriately capitalized. Artech House cannot attest to the accuracy of this information. Use of a term in this book should not be regarded as affecting the validity of any trademark or service mark. International Standard Book Number: 1-58053-540-2 Library of Congress Catalog Card Number: 2003060064 10 9 8 7 6 5 4 3 2 1

Contents Preface

ix

Acknowledgments

xi

CHAPTER 1 Radar Characteristics of Targets 1.1 Introduction 1.2 Target RCS 1.2.1 RCS Models 1.2.2 RCSs of Real Targets 1.3 Radar Reflections from Explosions and Gas Wakes of Operating Engines 1.3.1 Analysis of Radar Reflection Mechanisms 1.3.2 Spatial-Temporal Characteristics of Explosion and Fuel Combustion 1.3.3 Radar Reflections from Explosion and Gas Wake 1.3.4 Centimeter Wave and MMW Attenuation in Explosions 1.3.5 Radar Backscattering from Sonic Perturbations Caused by Aerodynamic Object Flight 1.4 Statistical Characteristics of Targets 1.4.1 Target Statistical Models 1.4.2 Real Target Statistical Characteristics 1.4.3 Echo Power Spectra 1.5 Surface Influence on the Statistical Characteristics of Radar Targets 1.5.1 Diffuse Scattering Surface Influence on the Statistical Characteristics 1.5.2 Multiple Surface Reflection Influence References CHAPTER 2 Land Backscattering 2.1 Classification and Physical Characteristics of Land

1 1 3 3 7 18 18 23 28 34 41 55 55 58 62 72 72 78 84

89 89

v

vi

Contents

2.2 State of the Theory 2.2.1 RCS Models 2.2.2 Power Spectrum Model 2.3 Normalized RCS 2.3.1 Normalized RCS of a Quasi-Smooth Surface 2.3.2 Normalized RCS for Rough Surfaces Without Vegetation 2.3.3 Backscattering from Snow 2.3.4 Backscattering from Vegetation 2.3.5 Normalized RCS Models 2.4 Depolarization of Scattered Signals 2.5 Statistical Characteristics of the Scattered Signals 2.6 Power Spectra of Scattered Signals References

95 95 101 108 108 109 114 118 120 123 126 128 133

CHAPTER 3 Estimation of Land Parameters by Multichannel Radar Methods

137

3.1 Estimation of Soil Parameters 3.1.1 Introduction 3.1.2 Soil Backscattering Modeling 3.1.3 Efficiency of Multichannel Methods 3.2 Soil Erosion Experimental Determination 3.2.1 Set and Technique of Measurement 3.2.2 Statistical and Agrophysical Characteristics of Fields 3.2.3 On-Land Radar Measurement Results 3.2.4 Aircraft Remote Sensing 3.3 Methods of Multichannel Radar Image Processing 3.3.1 Image Superimposing 3.3.2 Methods of Multichannel Radar Image Filtering 3.4 Soil Erosion Determination from Ratio Images: Experimental Results References

137 137 138 145 150 150 151 155 157 159 159 163 166 168

CHAPTER 4 Sea Backscattering at Low Grazing Angles

171

4.1 Sea Roughness Features for Small Grazing Angles 4.1.1 Sea Roughness Characteristics 4.1.2 Shadowing and Peaks in Heavy Sea 4.2 Sea Backscattering Models 4.3 Sea Normalized RCS 4.4 Depolarization of Scattered Signals 4.5 Sea Clutter RCS Model 4.6 Sea Clutter Statistics

171 171 184 189 193 197 202 206

Contents

vii

4.7 Radar Spike Characteristics of Sea Backscattering 4.8 Backscattering Spectra References

209 213 222

CHAPTER 5 Microwave and MMW Backscattering by Precipitation and Other Meteorological Formations

227

5.1 5.2 5.3 5.4

Structure of Meteorological Formations Atmospheric Attenuation Backscattering Theory Experimental Results Review 5.4.1 Precipitation Backscattering 5.4.2 Cloud Backscattering 5.5 The Statistical Characteristics of Scattered Signals 5.6 Radar Reflections from ‘‘Clear’’ Sky (Angel-Echo) 5.6.1 Point Reflections 5.6.2 Backscattering from the Turbulent Atmosphere References

227 233 236 239 239 242 242 250 250 254 256

CHAPTER 6 Sea and Land Radar Clutter Modeling

259

6.1 Land Clutter Modeling 6.1.1 Initial Data 6.1.2 Peculiarities of Land Clutter Simulation 6.2 Sea Clutter Modeling 6.2.1 Peculiarities of Sea Clutter Simulation 6.2.2 Algorithm of Sea Clutter Simulation 6.3 Clutter Map Development 6.3.1 Initial Data for Modeling 6.3.2 Software Input and Processing Components 6.3.3 Raster Image Processing Module 6.3.4 Automatic Highlighting of Contours on the Raster 6.3.5 Steady Algorithm of Surface Recovery from Contours 6.3.6 Simulation of the Absolute Reflectivity References

259 259 262 267 267 268 276 276 277 278 280 282 283 284

CHAPTER 7 Clutter Rejection in MMW Radar

287

7.1 Influence of Propagation Effects on MMW Radar Operation 7.1.1 Introduction 7.1.2 Multipath Attenuation

287 287 288

viii

Contents

7.2 Influence of Rain and Multipath Attenuation on Radar Range 7.3 Influence of Land and Rain Clutter on Radar Detection Range 7.4 Land and Rain Clutter Rejection in Millimeter Band Radar 7.4.1 General Notes 7.4.2 Land and Sea Clutter Rejection 7.4.3 Rain Clutter Rejection References

290 292 297 297 298 305 311

About the Author

313

Index

315

Preface

For the last 40 to 50 years, the intensive development of the millimeter band of radiowaves has taken place to address communication, radar, remote sensing, and many other problems. The interest in this band is due to a number of millimeter wave (MMW) advantages in comparison to longer wave bands. With this interest comes the possibility of developing super wide bandpass communication paths between on-land points. The possible development of narrow beam formations for acceptable antenna sizes would enable better tracking, detection, and surveillance in modern radar. In addition, the reserve and stability to countermeasures would be higher. The successful solving of problems for low-altitude, on-land, and maritime target detection and tracking has determined the propagation effects of MMWs near land and sea surfaces and in the troposphere. Among them, we can note the multipath propagation attenuation and the attenuation in precipitation (e.g., rain, fog, and snow) limiting the maximal range of detection. The small influence of multipath attenuation in comparison with radar of the centimeter or longer wave band is the essential advantage of MMW radar. The precipitation influence does not show itself in microwaves, and it is necessary to take this limiting factor into consideration in the millimeter band at ranges of more than few kilometers. The second problem limiting the application of MMW radar is the land and sea clutter conditioned by backscattering from distributed scatterers and the volume clutter from such scatterers as precipitations; the latter role increases in the millimeter band and results in limitations in radar frequency. The investigations in propagation of MMWs have been carried out in the Institute for Radiophysics and Electronics of the National Academy of Science of the Ukraine for more than 50 years, and great experimental data were collected during this time. Part of these results obtained by the author or with his participation has been included in works presented to the reader. Millimeter Wave Scattering by Earth’s Surface at Small Grazing Angles by G. P. Kulemin and V. B. Razskazovsky published in 1987 in Russian was the first monograph in the former Soviet Union in which the problems of forward scattering and backscattering of MMWs by land and sea surfaces were discussed and the theoretical and experimental results were

ix

x

Preface

presented, including the statistical characteristics of arrival angles due to multipath propagation over the surface. This book remained unknown for a wide circle of readers. Since then, many new results have been obtained, and no other new books were published.

Acknowledgments The preparation and publishing of this work was made possible by the enthusiastic support of David K. Barton. He made significant technical and scientific contributions, as well as providing valuable editorial suggestions. It has been a pleasure to work with personnel of Artech House Publishers. Special thanks are due to Tiina Ruonamaa for support and attention.

xi

CHAPTER 1

Radar Characteristics of Targets 1.1 Introduction The knowledge of radar statistical characteristics for targets to be detected is the usual starting point for radar system designers. The approach to statistical characterization differs significantly depending upon the radar system functions to be investigated. For instance, the tasks of target recognition, target identification in noise and clutter, and simple target detection require different amounts of information on the target scattering properties. In the first case, the signal must be presented as a multidimensional random vector in sine space; in the last situation, it is enough to know the average signal power or energy. The knowledge of the latter characteristic is the starting point for any radar system design and analysis of predetermining feasibility and nature of all further radar signal-processing techniques. Therefore, the first and the most important target characteristic is the radar cross section (RCS). It is also necessary to know the probability density function (pdf) of RCS fluctuation for given conditions of target observation for derivation of radar energy requirements; the minimally needed characteristic is the average RCS value, which is contained in the expression for the target echo power. A sufficient number of theoretical and experimental papers are devoted to the investigations of the RCS for different targets, but this material is mainly presented in the periodical references and requires analysis and generalization. Besides, experimental data on target statistical characteristics in the shortwave part of microwave and MMW bands are limited, and this obstacle stimulated the author’s interest in target characteristics at these wave bands. Discussion of the results of target modeling using simple shapes and experimental investigations of radar characteristics for cone-cylinder bodies in the resonance area (ka ∼ 1, where k = 2␲ /␭ is the wave number, ␭ is the signal wavelength, and a is the object diameter) seem to be most interesting. Chemical explosions and gas exhausts of operating engines are rather complex media involving some mechanisms for microwave backscattering [1, 2]. These mechanisms are competing for different stages of the explosions, and for long-life reflections the effects of microwave interaction with turbulent media possessing

1

2

Radar Characteristics of Targets

time-variable parameters are of great interest for physics. Moreover, there are possibilities of applications dealing with radar observation of explosions under battlefield conditions and with the detection of air targets with small RCS when backscattering is observed from gas wakes. The theoretical analysis of the radio wave scattering by a slightly ionized turbulent wake was performed in [3], and the radar method of measurements for the turbulent wake was considered in [4]. While considering the problem of a microwave reflection from an explosion area, attention is focused on the primary stage when the reflection from a shock wave ionized front (SWIF) is observed, and the reflection coefficient has the value approximately equal to unity (i.e., it is possible to consider the reflection from an ideally conducting body) [5]. Chemical explosions are the main object of investigations because the mechanism of backscattering at later stages of an explosion and that produced by a gas wake is the same. For this reason, the results concerning radar reflection from an explosion are significant. The operation of some types of radar and communication systems must be reliable when chemical explosions occur on propagation paths as found, for instance, in quarries or operation under battlefield conditions. In such situations, the spatial volume important for wave propagation becomes fully or partially blocked by the cloud formed by the explosion products and by accompanying particles of soil. This results in wave absorption. Further, the space surrounding the charge of explosives (where the explosion products—both solid and gas—and particles of soil are present) will be called simply the explosion volume. The main focus of investigators in studies of radio wave propagation through the explosion volume (as can be seen, for example, from survey [6]) was on the influence of the sand-dust cloud formed by soil particles and the influence of artificial smoke (e.g., phosphorus, hexachlorine, ethane, and oil fog). At the same time, obviously, the physical processes of microwave attenuation in the explosion volume of chemical explosives when there is no involvement of other substances are also interesting because even in this case rather large values of attenuation are observed experimentally. Detection of low-RCS airborne vehicles using the secondary effects observed during their flight through the atmosphere is also a significant issue [7]. Such effects may involve the following phenomena: the forming of an atmosphere shock wave due to object flight at ultra- and supersonic speeds; the presence of strong sound perturbation resulting in modulation of the atmosphere parameters; and the variation of turbulent troposphere parameters caused by intensive sound fields. Microwave scattering from atmosphere inhomogenities, which arise due to sound and shock wave propagation, can increase the total air-target cross section and improve radar efficiency. The statistical models of radar targets are considered using the common approach. It is shown that for derivation of detection characteristics, the standard

1.2 Target RCS

3

Swerling models can be extended to include chi-square distributions with small numbers of degrees of freedom. Special attention is paid to the results of experimental investigations of the echo power spectra for different classes of targets. The statistical characteristics of the echo from low-altitude targets are changed because of two circumstances [8]. First, the electromagnetic field from the target in the presence of multipath propagation is the sum of the direct wave and one reflected by the rough surface (sea or land). As it is known [9, 10], the statistical characteristics of a point nonfluctuating target placed over the surface are described by the Rician distribution. This is due to the influence of the statistical propagation factor, which introduces a diffuse component of electromagnetic field for the long paths with many random scatterers. In the shortwave part of the centimeter band and, particularly, in the millimeter band, the diffuse component of the electromagnetic field scattered by the surface increases along with the destruction of the specular reflection. In this situation, the spatial correlation radii of the field diffuse components over the target are often greater than the geometric dimensions of the target [10]. Then the received signal is a product of two terms: the first describes the target signal fluctuations in free space, and the second describes the fluctuations of the propagation factor. In addition, the statistical characteristics of the target echoes are changed because of the signal reflection from the target to the radar via the surface (multiple reflections). This effect, conditioned by the multiple reflections, can be significant for comparatively small ranges from the target to the surface when it is possible to neglect the propagation losses. Such interaction of the target and surface was first considered in [11] for a plane plate placed at an angle of 45° to the surface and the possibility of the RCS growth was shown. The analysis results of these mechanisms of target and surface interaction are presented and their influence on the radar target statistical characteristics is shown.

1.2 Target RCS 1.2.1 RCS Models

The choice of a radar target mathematical model is a rather complicated problem in the majority of situations, and it does not yield to exact analysis. While deriving diffraction from complex objects, we often use the results obtained for simple cases. The complex surface of the object is divided into several simple areas for which the reflection can be easily determined and described, and then the summing of partial contributions is performed according to techniques proposed in [12, 13]. Usually, the following geometrical shapes are used for approximation of the real targets and their parts: segments of spheres, ellipsoids and ogive objects, cone segments, cylinders or wires, plane plates, and dihedrals. Only the illuminated parts of such bodies must be taken into account. For derivation of

4

Radar Characteristics of Targets

reflection from these typical surfaces, Kirchhoff’s technique is the most widely applied because of its simplicity. The main difficulty for this approach lies in determining of the angle sector (target aspects) for which the derived expressions are valid and in the formula transformation for aspect changes. According to this approach, one supposes the independence of the bright points that contribute mainly into the total reflected signal (i.e., the effects of multiple reflection are not taken into consideration). This can result in significant errors in the determination of the object RCS because shapes such as the corner reflector are not taken into account. A separate problem is the technique of combining the bright point reflections. The calculation of the phase for every component makes sense only when there is an accurate description of the surface and knowledge of the operational frequency, such that the errors of relative phase predictions between the different target parts do not exceed a fraction of one wavelength. In this case, the technique permits us to estimate sufficiently accurately the complex target scattering pattern with the errors less than 1–2 dB [14, 15]. Only rough estimation of RCS is possible by means of signal noncoherent addition from all bright points if the accuracy of surface and frequency description is insufficiently high. It was proposed in [16] to use random relative phase with uniform distribution in the interval of [0, 2␲ ] for RCS estimation. This approach is valid if the number of bright points is rather large and the linear distances between them exceed the wavelength. Commonly, the use of the random phase model provides RCS estimation and permits us to determine its most important statistical characteristics. The technique of scattering pattern calculation for complex objects based on the geometrical optics approximation was proposed in [17, 18]. For example, the results of the scattering pattern derivation for a Convair-990 aircraft and a comparison with experimental data at a frequency of 10.0 GHz from these papers are presented in Figure 1.1; the experimental RCS values are integrated in the sector of aspect angles of 10°. The techniques of RCS derivation for comparatively simple objects are elaborated carefully for two cases: •

•

The wavelength is significantly greater than the target dimensions (Rayleigh scattering); The wavelength is significantly less than the target dimensions, corresponding to surface and edge scattering.

For the intermediate resonance region (wavelength comparable with the target dimensions), the establishment of some RCS relationships is a rather complicated problem, but there exist some techniques for approximation presented, in particular, in [16].

5

1.2 Target RCS

Figure 1.1 The scattering pattern of the Convair-990 aircraft at X-band: (a) horizontal plane and (b) vertical plane. (After: [17].)

Let us consider the techniques of the RCS evaluation for rather simple aerodynamic objects, which can be considered as combinations of round cones and round cylinders, both having limited dimensions. Much attention has been paid to derivation for the round finite cone [19, 20]. The scattering from its tip is given by

␴t =

␭2 ⭈ ␲ tan4 ␥ 16

(1.1)

where ␥ is the half angle at the cone tip and ␭ is the wavelength. This value is quite small in comparison with the scattering from the edge (cone base). For a cone with dimensions significantly greater than ␭ the RCS along the symmetry axis (␸ = 0) is determined as [20]

冉

␴ 1 ka sin ␲ /n = n ␭2 ␲

冊冉 2

cos

␲ 3␲ − cos n n

where n = 3/2 − ␥ /2, and a is the cone base radius.

冊

(1.2)

6

Radar Characteristics of Targets

Equation (1.2) shows that the RCS of a cone for the forward aspects does not depend on the wavelength and is determined by the base diameter strongly. This dependence is approximately ␴ ∼ a 2. For other angles, the RCS depends upon the electromagnetic wave polarization. In particular, the RCS for the vertical polarization is determined as [20]

冉 冋冉

冊 | 冋冉 冊 冉 冋冉 冊册 冋冉

ka sin ␲ /n ␴ = 2 n ␭ 4␲ 2 ×

␲ cos − 1 n

2

冊册 冊册 冊 冉

1 ␲ exp −i 2ka sin ␸ − sin ␸ 4

−1

␲ 3 ␲ − 2␸ − cos − cos n n

␲ + exp i 2ka sin ␸ − 4

×

␲ cos − 1 n

−1

−1

(1.3)

冊 册| −1

2 ␲ 3 ␲ − 2␸ − cos − cos n n for 0 < ␸ < ␥

and

冉

␴ ka sin ␲ /n = 2 n ␭ 4␲ 2

冊

2

1 × sin ␸

冋冉

␲ cos − 1 n

冊 冉 −1

␲ 3 ␲ − 2␸ − cos − cos n n

冊册

−1 2

for ␥ < ␸ <

␲ 2

Derivation of the cone RCS using (1.2) and (1.3), as a rule, gives estimates that are greater than those ones obtained experimentally. For example, for a finite round cone (␥ = 15°, 2a = 150 mm, ␸ = 0° ) the RCS derivation gives the value of ␴ = −15.5 dB(m2 ), while the experimental RCS is about −(20 − 25) dB(m2 ). A better match to the experimental results for a cone-cylindrical body, for the axial direction (␸ = 0° ), is defined by expression [20]

␴ = ␲a2

4␲ 2 sin2 (␲ + ␥ )

2

冋

␲2 ␲+␥

2␲ 2 ␲2 cos − cos ␲+␥ ␲+␥

册

(1.4)

RCS estimation for the same cone (␥ = 15°, 2a = 150 mm) according (1.4) gives ␴ = −18 dB(m2 ), closer to experimental results. For the second simple body—the cylinder of rather small length—the RCS can be determined as

7

1.2 Target RCS

3 1 (1 − q 2 ) sin 2qx ␴ = ⭈ ⭈ ␭ 2 8␲ (␲ /2)2 + {ln [cka (1 − q 2 )1/2/2]}2 4q 2

(1.5)

where q = cos ␸ , x = kl , l is the half-length of the cylinder, and c = 0.5772 is Euler’s constant. The analysis of (1.5) shows that the maximal RCS for cylinder takes place at angles of incidence close to normal with respect to the cylinder axis. Reducing the wavelength reduces the maximal RCS. The application of (1.1)–(1.5) for targets having dimensions comparable with the wavelength (i.e., in the resonance region) provides estimates only of the order of expected RCS in the most favorable situations. In particular, we performed a comparison of the predicted and experimental RCS for cone-cylinder objects using (1.2)–(1.5). The experiments were carried out at wavelengths of 3.0 and 0.8 cm, and the results are presented in Figure 1.2. It is worth noting that the RCS is higher by 10–15 dB for objects with diameters of 15.0 mm in comparison with diameters of 7.5 mm; this is significantly greater than that predicted by the geometrical optics approximation. The experimental RCS decreases by 14–20 dB for the smaller wavelength, while according to the derivations this should be only 4–12 dB. Thus, the modeling of the real radar target RCS, including objects with rather simple geometrical shapes, provides only the expected order of the RCS. More detailed data on the RCS and the scattering patterns can be obtained by natural experiments, especially for X- and Ka-bands, where the small-dimension constructive and technological target peculiarities are important. 1.2.2 RCSs of Real Targets

The RCSs for different classes of targets including marine, land, and air objects have been thoroughly investigated at X-band and longer wave bands, but less accurate and complete data are available for millimeter bands. RCS dependence on the wavelength, as a rule, is not evident or is totally absent for most radar targets having dimensions many times greater than the wavelength. It can be observed only for the objects for which the reflection is mainly caused by the corner reflectors on their surface. Such constructive elements are especially typical of ships and other marine vessels. In connection with this phenomenon, their RCS significantly exceeds their projected area in the plane perpendicular to the illumination direction. The RCS of objects usually approximately equals such projected area if the scattering is mainly caused by quasi-flat or curved surface elements. These effects allow us with some carefulness to use the quantitative data obtained in X-band for RCS estimation in millimeter bands. Let us more thoroughly consider the results of RCS measurements for targets of different classes. The RCSs of marine vessels are rather high, and their mean values are presented in Table 1.1 [21].

8

Radar Characteristics of Targets

Figure 1.2 The scattering patterns of cone-cylinder bodies at X- and Ka-bands: (a) diameter 7.5 mm and (b) diameter 15 mm.

Table 1.1 RCS of Large Marine Vessels Mean RCS (m 2 ) > 2 ⭈ 104 3 ⭈ 103–104 50–250 35–140 0.3–0.4

Type Ships with over 104 tons displacement Middle-class vessels with 103–3 ⭈ 103 tons displacement Small vessels with 60–200 tons displacement Submarine in above water state Submarine periscope (height is 0.5m over water surface) Source: [22].

For practically all microwave bands (1–10 GHz), the median value of RCS from side aspects can be determined using the empirical expression from [22]

␴ 0.5 = 52 ⭈ f

1/2

D 3/2

(1.6)

1.2 Target RCS

9

where f is the operational frequency in GHz, and D is the ship displacement in kilotons. The mean RCS of these objects decreases with increasing range, as a result of the ship’s structure falling into the shadowing zone, and this dependence is presented in Figure 1.3 for ships of three classes [23]. For MMW bands, the mean RCS of large ships increases with frequency more quickly than suggested by (1.6). For ships with displacements less than 200 tons, this increase is 3–5 dB, and for ships such as motor vessels, it is 15–20 dB. This confirms the assumption that in millimeter bands, the corner reflector shapes of ship superstructures influence the mean RCS. As an illustration, the mean RCS dependence on range for three types of ships is presented in Figure 1.4. The RCS of the small marine targets presented in Table 1.2 are significantly less [24]. Such objects as marine buoys have a special place among small marine targets because they are characterized by rolling motions with height oscillations due to rough sea and the presence of an anchor. Their mean RCS decreases with increasing sea states due to the shadowing effect by sea waves. For instance, such change of RCS for a small marine buoy is about 7 dB for sea state changing from 1 to 5, while for the same change of sea state, the RCS change is 18 dB for a buoy of medium size and 9 dB for a large one. RCS values for various marine objects reported in [25] are presented in Table 1.3. The RCS of land objects also varies within rather wide limits depending on the object type. The mean forward aspect RCS for some land targets obtained while moving along a dirt road are presented in Table 1.4 [26]. The measurements are carried out at a 3-cm wavelength.

Figure 1.3 RCS dependence on the range for (1) trawler, (2) dry cargo ship, and (3, 4) tankers.

10

Radar Characteristics of Targets

Figure 1.4 RCS dependences on range at wavelengths of 3.0 and 0.8 cm for (1) patrol boat, (2) tanker, and (3) motor ship.

The RCS of air targets at microwave have also been investigated. As shown in [27], the mean RCS for the piston-engine B-26 aircraft at forward aspects in the ±10° sector is 20–25 dB(m2 ), and a similar mean RCS is typical for the C-54 at the 3-cm wavelength. As was shown in [28], the mean RCS values are 8–15 dB(m2 ) for large jet aircraft, about 1 m2 for light aircraft of the L-200 type, and about −(0.9–3.3) dB(m2 ) for a Russian Mi-4 helicopter [23]. One of the main trends in modern military airplane construction is the design of low-observable vehicles, decreasing their detection probability by air defense

11

1.2 Target RCS Table 1.2 RCS of the Small Marine Targets Object Yacht, sailboat Scull boat Gum boat Large marine buoy with radar reflector Medium marine buoy with radar reflector Small marine buoy Channel cone buoy Man on windsurf Source: [24].

Mean RCS (m 2 ) ␭ = 3 cm ␭ = 8 mm 10–20 12–14 2–4 0.8–5.0 1.0–2.0 1.2–2.5 20–20 — 7–10 — 10 — 10 — 2.5–3.0 2.5–3.5

Table 1.3 RCS of Some Small Marine Objects

Source: [25].

Table 1.4 Mean RCS of Land Targets for Forward Aspects Object Tank Armored car Heavy artillery tractor Light artillery tractor Truck Source: [26].

Mean RCS (m 2 ) 6.0–9.0 8.9–30.0 15.0–20.0 10.0–15.0 6.0–10.0

radar systems. The efforts of the airplane designers led to RCS reduction over the past decades as illustrated in Table 1.5. The contributions of reflections from the different elements of the aircraft structure to the total RCS are determined by the aspect relative to the radar. For side aspects, reflections from the fuselage and vertical stabilizer are predominant, along with reflections from the leading edges of the wing and stabilizer. The contributions to the total RCS from different aircraft structures are illustrated in Figure 1.5.

12

Radar Characteristics of Targets Table 1.5 Airplane RCS (m2 ) Decreasing over Past Decades Airplane Type Bomber Fighter

1970s 50–100 5–15

1980s 5–10 1–3

1990s 0.5–1.0 0.1–0.3

Figure 1.5 Contributions of different structures to total aircraft RCS.

The main directions and trends of aircraft design with decreased RCS are presented in Table 1.6. Taking into consideration these trends, it is possible to predict that one can expect light and medium-weight aircraft with RCS of order 10−2–10−3 or less. This significantly decreases their detection range. Hence, it is necessary to find new characteristics of targets to ensure their reliable detection at great ranges. Data on the scattering properties of biological objects have significant interest for short-range radar designers. In some cases they are the desired targets, and in other cases they are false targets. The backscattering from a human body is determined by its mass and the radar operational frequency. The connection between frequency and RCS of a man is seen from data presented in Table 1.7.

13

1.2 Target RCS Table 1.6 Main Methods of Decreasing RCS Direction Flying apparatus aerodynamic shape and structural element improvements

Radio transparent and radio-absorbing material applications

Decreasing visibility of on-board antenna systems Ionized absorbing cloud (IAC) creation

Technical Realization Possibilities The removal of the sharp selvages, corner forms, gaps in aerodynamic surfaces. Decreasing the vertical stabilizer area. Use of the aerodynamic shape of the ‘‘flying wing’’ type. Integration of the glider-engine and gliderarmament systems. Use of composite material. Application of radio-absorbing coatings. Application of conducting material for gap removal. Antenna scattering in directions other than specular reflection to the radar. Decreasing numbers of antennas. Electronic gun use. Application of coatings using radioactive isotopes.

Table 1.7 Man RCS Dependence on the Frequency Frequency (GHz) 0.4 1.1 2.9 4.8 9.4 Source: [28].

Mean RCS (m 2 ) 0.033–2.33 0.1–1.0 0.14–1.05 0.37–1.88 0.5–1.22

It is seen that RCS of a man does not practically depend on the operational frequency in microwave bands. The scattering pattern presented in Figure 1.6 [29] shows that the RCS at 3-cm wavelength is maximal for frontal aspect and minimal for side aspect. The detailed investigation of backscattering from birds and insects is a rather hard problem for several reasons. The ratio of object dimension to wavelength can change over several orders, while the difference of the shape from spherical (even without considering the wings) leads to strong dependence of RCS on the observation aspect and the polarization of radiation. Besides, the target is not rigid, and its shape periodically changes with the wing flaps and respiration. The problem is more complex for determination of temporal RCS dependence because this is determined by target behavior (i.e., migration, local food extraction). The RCS dependences on the aspect for three bird species at 3-cm wavelength are presented in Figure 1.7. The RCS dependence on bird mass obtained in [30] is presented in Figure 1.8. The predicted values of the water sphere RCS at wavelengths of 3 and 0.8 cm are

14

Radar Characteristics of Targets

Figure 1.6 Scattering pattern of a man. (From: [29].  1984 Radio and Communication.)

Figure 1.7 The scattering pattern for three bird species at X-band: (1) pigeon, (2) starling, and (3) crow.

shown by the lines, the experimental data at the 3-cm wavelength are presented by the points. The simplest model for RCS estimation of biological objects is the equivalent water sphere for which mass is equal to the object mass. However the length-to-diameter ratio for bird body parts, containing water, equals to 2:1 or 3:1 [30].

1.2 Target RCS

15

Figure 1.8 RCS dependences on the bird mass at X-band (solid line) and Ka-band (dotted line). Points are the experimental data. (After: [30].)

The nonspherical shape of scatterers leads to the appearance of a cross-polarized component of the echo. This component value equals to −12 to −13 dB in comparison to the main one for the objects with RCS greater than 5 ⭈ 10−3 cm2. The dependences on wavelength of bird and insect RCS are presented in Figure 1.9. At wavelengths of more than 10 cm, the bird RCS can be approximated by the relation ␴ ∼ ␭ −4 where ␭ is the radar wavelength (i.e., Rayleigh scattering takes place). The maximal RCS value of birds at this band equals to 0.1–20 cm2, this value decreasing by 10 dB at a 3-cm wavelength and by 15 dB at a 30-cm wavelength. The insect RCS is 10−1–10−4 cm2 up to the wavelength of 8 mm. RCS values for some bird and insect species are presented in Tables 1.8 and 1.9 [30, 31]. In the period from spring to autumn, the majority of birds are above most of the land surface at heights of 1 to 2 km. The migrating birds of some species regularly fly at heights more than 4 km and appear at distances from the nearest land of more than 1,000 km. At the heights from 0 to 2 km, the volume density of the bird distribution is often from 10−7 to 10−6 m−3, and in regions of flock accumulations, densities of order 10−5 can be found for durations of days. If the radar resolution cell volume equals to 106–107 m3 (as typical of many radars at ranges less than 20 km), many such cells will contain at least one bird. Table 1.10 shows the order of bird density in the some regions of flocking (the bird number is summed in height); the data is averaged over the considerable geographical area [32]. It is necessary to take into consideration that factors of social behavior can raise the local density of the flocks, especially in approaches to places of night rest, bird colonies, or flock nutrition. In the last column of Table

16

Radar Characteristics of Targets

Figure 1.9 Bird and insect RCS dependences on the wavelength.

Table 1.8 RCS of Birds at 10-cm Wavelength Type Pigeon Starling Sparrow Seagull Source: [30].

From Side 1.0 × 10−2 2.5 × 10−3 7.0 × 10−4 1.5 × 10−2

RCS (m 2 ) From Front 1.1 × 10−4 1.8 × 10−4 2.5 × 10−5 2.0 × 10−3

From Behind 1.0 × 10−4 1.3 × 10−4 1.8 × 10−4 —

Table 1.9 RCS of Insects at 10-cm Wavelength Type Butterfly Butterfly Bee Dragonfly Source: [30].

Wing Span (cm) 10.0 3.0 1.0 —

RCS (cm 2 ) 1.0 5.0–10−3 2–10−3 10−3

1.10, the estimates of the top averaged RCS are given. RCS bounds are wide because the target aspect, radiation polarization, and wavelength are not taken into consideration. These estimates are acceptable for wavelengths from X- to S-bands. The bird-averaged distribution by altitudes is presented in Table 1.11 [33].

17

1.2 Target RCS Table 1.10 Bird Density in Flocking Places Accumulation Type Winter refuges for crows, seagulls, geese, ducks in the littoral waters Stormy petrel migration by California coast Coastal and sea birds in the reproduction period Source: [32].

Area (km 2 )

Bird Number

Density (m −3 )

RCS (cm 2 )

<103

104–106

10−9–10−6

10–500

<103

>106

10−5

50–500

<105

109

10−7

50–500

Table 1.11 The Bird Distribution by Altitude Height (m) 250 500 1,000 1,500 2,000 Source: [33].

Percentage of Birds at Heights Lower Than Shown 45 65 80 90 100

The distributions of insect density in the air often can be greater by many orders than for birds. Some data from the works of Rainy and Johnson [30] are presented in Table 1.12; the RCS are given for X-band, and they represent the most probable values and are close to the real data. It appears that the insect distribution density often can exceed 10−5–10−4 m−3, and densities of order 10−3–10−2 apparently are found regularly for cases when the converging winds concentrate the insects. The greatest densities have most probably local behavior, and the typical concentration area for these cases does not exceed 100 km2. Thus, considerable data permit us to estimate the RCS of objects that can be either targets or interference to radar systems designed for the detection of small RCS objects.

Table 1.12 Distribution Density for Some Insects Object Middle butterfly during intensive migration Night butterfly All insects for 1 hour Source: [30].

Height (m) Near surface Near surface Near surface

Distribution Density (m −3 ) −2

10 10−2 101

RCS (cm 2 ) 10−2–100 10−1–100 10−3–100

18

Radar Characteristics of Targets

1.3 Radar Reflections from Explosions and Gas Wakes of Operating Engines 1.3.1 Analysis of Radar Reflection Mechanisms

Let us first consider the possible mechanisms of microwave backscattering from an explosion volume. This point is the most complex for the analysis, and it includes the models of backscattering from a turbulent gas wake as a particular case. During the primary stage of an explosion, a dominant factor is the reflection from the SWIF. Depending on the ratio between the pressure p o of an undisturbed gas and the SWIF pressure p = p o + ⌬p , it is possible to classify three typical stages of the shock wave: • • •

Strong shock waves (p >> p o ); Shock waves of a middle range intensity (⌬p ∼ p o ); Weak shock waves (⌬p << p o ).

For strong and weak shock waves, it is possible to derive the analytical solution for the shock wave propagation equation. Its accuracy is determined only by the accuracy of some primary assumptions. The problem of the strong point-source explosion is solved in [34]. The author of the paper ignored the contra pressure of the medium. Taking contra pressure into account, this problem was solved on the basis of numerical integration in [35]. For a weak shock wave, the asymptotic expressions were also obtained (primarily in [36] and those in generalized form in [37]). For a strong shock wave, the gas density decreases sharply from the SWIF to the center; practically all of the gas mass is concentrated in the thin layer near the front surface. With increasing distance from the SWIF to the center, the pressure reduces by two to three times and remains practically stable for the entire sphere. The temperature increases from the front to the center—especially very quickly in the area of constant pressure, which is determined by the presence of particles heated by the shock wave, which in turn have high entropy and are located near the center. With the passage of time, the shock wave amplitude decreases and the front pressure diminishes asymptotically to the atmospheric level. Correspondingly, the SWIF gas compression decreases, and the shock wave propagation speed asymptotically diminishes approaching the sound velocity. The SWIF propagation law transforms from r f ∼ t 2/5 to r f ∼ a o t where r f is the SWIF radius, t denotes time, and a o is the sound speed (i.e., the shock wave transforms gradually to a sound wave). The zone with low pressure follows the compression area in such a wave, after which the air resumes its original state.

19

1.3 Radar Reflections from Explosions and Gas Wakes of Operating Engines

The high temperature of the SWIF, reaching 105K for the initial explosion stage, causes ionization of the air and creation of a plasma layer. In this case, the electromagnetic field reflection coefficient is described [38] in the following way:

|⌫|

2

冋

册

2 1 − (1 − n e /n*e )1/2 = 1 + (1 − n e /n*e )1/2

(1.7)

where n e is the plasma electron density in the SWIF; n*e = (␲ m /e 2 ) f 2 is the electron density corresponding to the plasma resonance frequency, f ; and m and e are the mass and the charge of the electron, respectively. According to [39], the critical electron concentration corresponding to the plasma resonance frequency in the SWIF can be reached in the air at temperatures of 3,000K for the 3-cm wavelength and 4,000K for the 8-mm wavelength. The presence of admixtures can decrease the temperature at which the critical concentration appears. Such temperatures in the SWIF exist for 80–100 ms after the explosion for the chemical explosives (like trotyl) that have a weight of 1 kg. In later stages, the ionization becomes lower than critical, resulting in a rapid decrease of the refractive index. Table 1.13 presents the RCS and the SWIF radius values derived from (1.7) and taking data [35] into consideration. The results are obtained for the explosion of a 1-kg trotyl charge at 3-cm wavelength; it was assumed that | ⌫ | was constant for the first Fresnel zone on the spherical SWIF. It is worth mentioning that for t > 200 ␮ s (corresponding to r f ≈ 1m), the rapid decreasing of RCS begins, and for t > 1 ms, the contribution of the shock wave ionization to a total reflected signal becomes insignificant. Microwave reflection from the discontinuity in concentration at the shock wave front is the second possible mechanism. For this case, the reflection coefficient can be determined as

冉 冊 n−1

| ⌫ | 2 = 2␥ p o

2

⭈ ⌬p 2

(1.8)

for an infinitely thin layer on the shock wave front and

Table 1.13 The Temporal Dependence of the SWIF Radius and RCS of an Explosion at 3-cm Wavelength Time from the Explosion Start ( ␮ s) SWIF radius (m) RCS (m2 ) Source: [1].

4.7 0.15 0.071

6.2 0.164 0.085

20 0.264 0.226

50 0.383 0.46

84 0.47 0.69

140 0.64 0.13

180 0.7 0.031

20

Radar Characteristics of Targets

|⌫|2 =

冉 冊 ⌬n 2

2

⭈

1 1 + 16␲ 2⌬2/␭ 2

(1.9)

for a layer of finite effective width ⌬. In these expressions (n − 1) = 320 ⭈ 10−6 for a standard atmosphere when the altitude is 500m, ␥ is the adiabatic constant equal to 1.4 for air, ⌬p is the pressure discontinuity in SWIF, ␭ is the radar wavelength, and p o is the pressure of an undisturbed atmosphere. Taking into consideration the data on the SWIF pressure discontinuity derived from [17], we obtain | ⌫ | 2 ≈ 8 ⭈ 10−8 for the explosion of a 3-kg trotyl charge at 0.5s after the explosion start and | ⌫ | 2 ≈ 1.5 ⭈ 10−8 for 30 ms after the start of the explosion (i.e., the reflection coefficient is not large and it decreases rather slowly). The SWIF expansion, which results from a finite viscosity and the presence of turbulent pulsations of temperature, pressure, and speed, gives a considerable reflection coefficient decrease in comparison to values derived from (1.8). The estimates of the explosion RCS stipulated by this scattering mechanism have values comparable to the background reflection from the troposphere, and they are much less than the experimental data. Moreover, both mechanisms can cause reflections existing only for short intervals of time for which the SWIF does not exceed the limits of the radar resolution volume. But the reflections exist for a rather great time interval for area volume of relatively small dimensions. The analogous estimations of microwave reflections from the Mach cone taking place for an air vehicle flight with supersonic and ultrasonic speeds show that the ionization of a shock wave front becomes considerable for speeds greater than 2.5 km/s. For other speeds, the shock wave RCS is approximately equal to 10 (i.e., it is comparable to the troposphere reflective ability). Therefore, the reflections from the front of an explosion or from the Mach cone can be the important mechanism for the detection of these objects only for short time periods after the explosion or during the air vehicle flight at supersonic speeds. One more mechanism of reflected radar signal formation is conditioned by the perturbations arising in the explosion area after passage of the shock wave front. The refractive index pulsation intensity increases due to medium turbulence as well as to the chemical content changing in this area. For aerodynamic object detection, the reflections from the turbulent gas wake of an operating engine can be used. For real explosions, a volume occupied by explosion products is formed. Their compositions can be easily specified. The explosion products composition for 1 kg of trotyl C6H2(NO2)3CH3 is presented in Table 1.14. It is evident that the main components of explosion products are CO, CO2 , H2O, and N2 gases and amorphous carbon (soot); the contribution of the other ingredients is negligible. During the later explosion stages, the partial combustion of CO and C, with some additional carbon oxide formation, takes place. The noncombusted particles of carbon with dimensions of about 10−6–10−7 mm [40]

21

1.3 Radar Reflections from Explosions and Gas Wakes of Operating Engines Table 1.14 The Composition of 1-kg Trotyl Explosion Products Products

Mole

Weight Gram

CO2 CO H2O C N2 NH3 Source: [1].

1.92 11.64 10.96 17.32 6.6 0.04

84.5 326 192 208 185 0.6

Dipole Moment (Debye)

Polarizationability

0.1 ± 0.05 0.112 ± 0.15 1.65 ± 0.25 — — —

2.6 ⭈ 10−24 2.02 ⭈ 10−24 1.5 ⭈ 10−24 — 1.84 ⭈ 10−24 —

according to these derivations cannot explain the RCS values observed for the explosion volume. For the total refractive index estimation in the explosion products volume, let us use the ratio presented in [41]: N = (n − 1) ⭈ 106 = 2␲ A o

冉

␳ ␮2 ␣o + M 3kT

冊

⭈ 106

(1.10)

where n is the refractive index; A o is the Avogadro number; ␳ denotes the density; M is the gas molecular weight; ␣ o and ␣ are the polarization ability and the dipole moment of the gas molecule, respectively; k is the Boltzmann’s constant; and T is the temperature in K. The contribution of different components and the total reflection coefficient of the formed mixture are presented in Table 1.15. It is seen that the maximum contribution to the refractivity is made by water in a molecular phase, while among the other components the influence of the carbon oxide is the most significant. It is worth noting that at high temperatures (typical for explosion products) the contribution caused by the molecule polarization can increase by several orders. Analogous phenomena occur for fuel combustion in turbojet and turboprop engines. In Table 1.16, we present the data on gas volumes for 1-kg kerosene combustion (kerosene is now the main type of fuel for modern aircraft) exhausting

Table 1.15 Gas-Like Explosion Products Refractivity for a 1-kg Trotyl Explosion Gas Type CO2 CO N2 H2O ⌺N Source: [1].

Percentage in Mixture 4.0 24 13.6 22

Contribution in Mixture N, N-Units Due to ␣ 0 , t = 20°C Due to ␮ 26 0.82 140 5.2 66 — 90 358.0 322 404.0

22

Radar Characteristics of Targets Table 1.16 The Chemical Composition and Refractivity of Combustion Products for 1 kg of Kerosene Gas Type Volume m 3 ⭈ kg −1 CO2 1.6 H2O 1.8 N2 9.1 Total refractivity Source: [1].

N for Pure Gas 700 1760 410

Percentage of Content 12.8 14.5 72.7

Refractivity 90.0 255.0 300.0 645.0

to the atmosphere under normal pressure. The results of refractivity derivation from (1.10) are also presented. According to the obtained estimates in the volume occupied by explosion products and fuel combustion products, the mixing of gas-like products with the surrounding air takes place under the influence of atmosphere turbulence and gravity. The sharp margins of areas with different refractive indices remain intact because of turbulent diffusion, the speed of which is greater than the speed of the molecular gas diffusion. Later, blurring of the turbulent product wake margins occurs, resulting from intermolecular diffusion. The dimensions of the volume occupied by the explosion products are limited. For an air explosion, the shock wave front moves more rapidly than the explosion products, so from the very beginning of the expansion process the pressure decreases in the area occupied by the explosion products. A short time later, after the explosion, its products will occupy the maximum volume V∞ , which is described for a spherical charge by radius r∞ [42] 3

r∞ = (20 − 30)r o = (20 − 30) ␤ √C

(1.11)

where C is the charge weight in kg; ␤ is the coefficient depending on the explosion substance density (for pressed trotyl it is equal to 0.053), and r o is the spherical charge radius in meters. The dimensions of the gas wake of an operating engine in the cross direction are also limited; the wake diameter is four to six times greater than that of the nozzle. Using the simplifying assumption on a turbulent isotropy for the area occupied by the explosion products and combustion products, it is possible to estimate a specific volume RCS of this area using [43]

␩=

␲ ⭈ 〈 ⌬n 2 〉 ⭈ k 2 ⭈ F n (k ) 8

(1.12)

where k = 2␲ /␭ is the wave number; 〈 ⌬n 2 〉 denotes the refractive index fluctuation variance; and F n (k ) is the one-dimensional spatial spectrum of refractive index fluctuations.

23

1.3 Radar Reflections from Explosions and Gas Wakes of Operating Engines

Consequently, for the estimation of RCS of the explosion and engine fuel combustion volumes, it is necessary to know two local turbulence characteristics: the variance and the spatial spectrum of the refractive index. It is possible to expect that the normalized one-dimensional spectrum of the refractive index pulsations can be described as a spectrum of a random telegraph signal with a Poisson distribution of refractive index steps F n (k ) = [1 + (kL )2 ]−1

(1.13)

where L is some typical effective turbulence scale. For kL << 1, we have F n (k ) ≈ 1 and the volumetric normalized RCS ␩ ≈ ␭ −2; for kL >> 1, the specific RCS has a very insignificant dependence on radar wavelength. Therefore, from the point of view of radar detection, the latter mechanism seems to be the most important among those considered because it provides the greatest duration of the reflected signal. The next section is devoted to the estimations of the variance and of the spatial spectrum of the refractive index fluctuations for a disturbed volume. 1.3.2 Spatial-Temporal Characteristics of Explosion and Fuel Combustion

Experimental investigations into the spatial and temporal characteristics of the refractive index of the explosion volume were carried out using a refractometer and a thermoanemometer. The first instrument directly obtained the temporal fluctuations of refractive index differences at two different points of space; the second determined the speed pulsations in the air flow. The necessity of the use of two instruments was conditioned by considerable dimensions of the microwave refractometer open resonators, which prevented the estimation of the spectrum with wave numbers more than 1 cm (i.e., with linear dimensions less than 10–20 cm). The use of a thermoanemometer with a time constant about 0.01 second permitted us to investigate the inhomogeneous media with scales about 1 to 2 mm. Taking into consideration the similarity of the spatial spectra for a velocity field and the refraction index [43]. It was possible to combine the data obtained by these instruments. The data were recorded by a high-speed photoelectric recorder. For data processing, a sample with a duration of about 10 seconds was divided into segments, each having a duration of about 1 second. Two segments preceded the beginning of the explosion. For each segment, statistical and spectral processing was carried out. For the spatial spectra determination from the temporal spectra, the hypothesis of frozen turbulence was used [43], according to which the whole spatial stochastic field moves with a mean velocity of an air flow. This allowed us to obtain the spatial spectra of the refraction index fluctuations for spatial dimensions from 2 to 100 cm.

24

Radar Characteristics of Targets

The experimental investigations were conducted on an open flat surface. The trotyl charges with weights of 1–3 kg were placed at a height of about 1.5m above the surface. The refractometer and the thermoanemometer sensors were placed at a 10-m range from the explosion center. About 40 explosions were carried out for different wind speeds. Refractometrical investigations into the turbulence local characteristics for the explosion volume showed the following: 1. For 2–4 seconds at 10m from the center of the explosion of a 3-kg trotyl explosive, the root mean square (rms) value of pulsations exceeded (2–3) N -units, compared to (0.1–0.5) N -units for the undisturbed atmosphere. This phenomenon was observed both for calm weather and for a wind direction toward sensors. For the cross-wind direction, there was a 3–5 dB increase in the refractive index pulsation in comparison to the undisturbed atmosphere. The time interval when the effective value of fluctuations was more than 1 N -unit was equal to 3–5 seconds. The illustration in Figure 1.10 presents the refractive index pulsation values as the temporal functions for wind absence—Figure 1.10(a)—and for an explosion product moving toward the sensors—Figure 1.10(b). 2. The refractive index temporal fluctuation spectra retained their shape. Besides, for the frequency band from 5 to 30 Hz, the slope of the disturbed area spectra did not change in comparison to the undisturbed atmosphere spectra corresponding closely to the phenomena predicted theoretically for a homogeneous turbulent atmosphere. Figure 1.11 presents the refractive index temporal fluctuation spectra for the different moments of time after the explosion for the same experiment. Using the hypothesis of frozen turbulence [16], the transformation to spatial fluctuation spectrum was carried out (the lower horizontal axis). For the frequency region F < 5 Hz, a modification of the spectrum slope was observed for several experiments, probably resulting from the finite dimensions of the refractometer baseline (0.7m) acting as a spatial lowpass filter. Its influence also resulted in the structural functions that had a tendency to saturation for the baseline dimensions of 0.5–1m. 3. For approximately 70% of the experiments carried out in conditions of an explosion product movement toward the sensors, the difference of the correlation intervals of the flow velocity fluctuations was observed before and after the explosion. The decrease of the correlation interval at 2–5 seconds after the explosion start was typical (Figure 1.12) in comparison to the correlation interval for the undisturbed atmosphere. It was the evidence that the typical dimensions of the explosion product turbulence diminished for the case of movement toward the sensors in comparison to the undisturbed atmosphere.

1.3 Radar Reflections from Explosions and Gas Wakes of Operating Engines

Figure 1.10

25

Temporal dependence of the refractive index fluctuation rms values in explosion (a) without wind and (b) with a longitudinal wind. (From: [1].  1997 IEEE. Reprinted with permission.)

26

Radar Characteristics of Targets

Figure 1.11

Instantaneous power spectra of the refractive index fluctuations in explosion. (From: [1].  1997 IEEE. Reprinted with permission.)

4. For a distance between the sensors and the explosion center equal to 17m, the fluctuation intensity before and after the explosion remained almost the same, excluding the cases of movement toward the sensors of the expanding volume occupied by the explosion product. Analogous results were obtained when the experimental study of local spatialtemporal characteristics was carried out for a gas wake of an operating jet engine. The investigations were made both for a jet engine model with fuel expenditure 2 g/s and for a MIG-21 aircraft engine. For the model experiments when the distance from the nozzle was of about 2–3m, the refractive index fluctuations were 10–20 dB more than ones for the undisturbed atmosphere reaching 〈 ⌬n 2 〉 = 100 (N -units)2. Reduction of the fluctua-

1.3 Radar Reflections from Explosions and Gas Wakes of Operating Engines

Figure 1.12

27

Temporal dependence of an air velocity decorrelation time after a 1-kg trotyl explosion. (From: [1].  1997 IEEE. Reprinted with permission.)

tion intensity occurred with increase of the distance from the nozzle; when the distance exceeded 8–9m, they decreased to the level of the undisturbed atmosphere. The shape of the spectrum of fluctuations for the gas wake was practically identical to the spectrum of undisturbed atmosphere; for their description the functional dependence ∼F −5/3 could be used. The spatial spectra obtained using the freezing approach with taking the local speed of the wake into account were characterized by the shift to the area of the large spatial wavelengths ⌳ for an increase of the distance from the nozzle. This resulted from the greater degree of generating small perturbations. The refractive index fluctuation intensity for a gas wake of the MIG-21 operating engine for the different distances from the nozzle and the regimes of operation is presented in Table 1.17. It is seen that with fuel expenditure increase, the same fluctuation intensity is observed for the greater distances. The frequency and spatial fluctuation spectra of the refractive index are analogous to those typical for the model of a gas wake, and they can be described using ∼F −5/3 and ∼ (1/⌬)−5/3 dependencies. Therefore, the refractive index fluctuations for the area occupied by the explosion products and for the operating engine gas wake possess the following features:

28

Radar Characteristics of Targets Table 1.17 The rms Values of Refractivity in the Gas Wake of an MIG-21 Aircraft Engine Operating Regime Minimal Nominal (normal) Maximal Source: [1].

•

•

•

Distance Along the Axis (m) from Nozzle 20 25 50 65

冠 ⌬N 2 冡1/2 ⭈ N-units 5.5 4.5 4.4 3.7

In the disturbed volume, the refractive index fluctuation intensity increases greatly in comparison to that of the undisturbed atmosphere, proving the applicability of the model proposed earlier for this region; The refractive index fluctuation spatial-temporal spectra shape of the disturbed areas is similar to the undisturbed atmosphere spectra; The dimensions of the disturbed volume are limited by the volume of explosion products propagation and by the nozzle gas flow.

1.3.3 Radar Reflections from Explosion and Gas Wake

The experimental study of the radar characteristics for the explosion area of a trotyl explosive was carried out at wavelengths from 10 cm to 4.1 mm. The parameters of the pulsed and continuous-wave (CW) radars used for experimental investigations are presented in Table 1.18. The data from the pulsed radars were recorded by a high-speed photoelectric recorder and by a 10-channel spectral analyzer of parallel type covering the frequency band from 10 to 500 Hz. With use of the spectral analyzer of this type, it

Table 1.18 The Parameters of the Pulsed and CW Radars Parameters Type Central frequency (GHz) Transmitter power: –Pulsed (kW) –Average (W) Polarization Pulse duration ( ␮ s) Repetition frequency (Hz) Antenna pattern width: –Azimuthal –Elevation Threshold sensitivity (Wt) Frequency band of analysis (Hz)

1 Pulsed

2 Pulsed

3 CW

4 CW

5 CW

3.0

10.0

10.0

10.0

74.0

250 — VV, HH 0.5

250 — VV, HH 0.7

— 10 VV —

— 4 VV —

— 0.6 VV —

1,750

1,100

—

—

—

2° 2.3°

0.5° 0.75°

2° 2°

1.5° 1.5°

0.6° 1.0°

10−12

0.5 ⭈ 10−12

2 ⭈ 10−18

10−17

5 ⭈ 10−17

500

500

0.10–40,000

0.10–40,000

0.10–40,000

1.3 Radar Reflections from Explosions and Gas Wakes of Operating Engines

29

was possible to obtain the instantaneous power spectra of the reflected signals. Moreover, the multichannel gate unit was used, which permitted us to obtain the explosion volume spatial dimensions. The data from the CW radars were recorded by the same recorder and by a 10-channel spectral analyzer of parallel type covering the frequency band of analysis from 0.01 to 40 kHz. When the position of the explosions was chosen, great attention was paid to the selection of the surface area with a clutter minimum level. The explosions of the trotyl charges with weights of 1 and 3 kg were carried out at a range of about 2 km from the pulsed radars and over 50m from the CW radars. Calibration of the radars was carried out by a set of the corner reflectors. The rms error of the RCS estimation was equal to 2 dB. We should like to note the following peculiarities of experiments: 1. The long-life reflections from the explosion products volume were the subjects of the study, but not the reflections from the short-time high-temperature nucleus; 2. Coherent processing techniques and Doppler frequency filtering in the frequency domain F < 10 Hz (in some experiments, 5 Hz) were used for removal of the obstructing reflections from environment. The RCS of the explosion in this case was determined as Fu

␴=

冕

G (F ) dF

FL

where G (F ) is the power spectrum of the reflected signal, and F L and F U are the low and the upper bandpass filter frequencies. Obviously, for strong dependencies of spectral density on frequency (which according to the results presented later did take place for reflections from the explosion), the RCS value depended significantly upon F L and F U for this method of data processing. The experiments showed that the maximum reflection level is observed for a radar antenna beam directed to the explosion center, and for wavelengths of 10 cm and 3 cm, the total RCS was equal to several square meters, reaching 10m in some cases. Besides, the RCS did not depend on the wavelength and polarization of the signal; only a dependence on wind direction was observed. The RCS for the crosswind was less than that for radial wind direction because of an explosion product drift from the explosion area by a cross-direction wind (i.e., the linear azimuth resolution of the radar was better than the radial one).

30

Radar Characteristics of Targets

The mean RCS values obtained in the frequency band from 10 to 500 Hz and 0.6s after the explosion are presented in Table 1.19. Table 1.20 presents the average RCS values measured for wavelengths 3 cm, 0.8 cm, and 0.4 cm in the band of analysis 5–200 kHz. It is seen, in particular, that a significant decrease in total RCS occurs for 3.2-cm wavelength due to the growth of the lower bound of frequency analysis. At the same time, the RCS for the surface explosion of 1-kg trotyl reached 0.2–0.3 m2 for a 3-cm wavelength and the same frequency band. The measurements of the dimensions of the explosion volume that formed the echo were carried out by means of the estimation of the azimuthal cross sections of this volume using narrow-beam antennas with main lobe widths less than 1°. They showed that the volume diameter was about 6–7m for the explosion of the trotyl charge (its weight was 1 kg) without the envelope and reached 8–10m for the explosion of charge with a metal envelope. Demonstrating this effect, Figure 1.13 presents the dependencies of the echo power when the antenna axis rotates by some angle with respect to the explosion center (curve 1 corresponds to the charge without the envelope, curve 2 corresponds to an enveloped charge). If we take into account that, according to (1.11), the limit diameter of the area occupied by the explosion product of a 1-kg trotyl explosion is equal to 2.2–3.2m, it is possible to suppose that the reflected signal is partially formed by a turbulent atmosphere created by the passing of the shock wave front. It is worth noting that the dimensions of the reflecting volume are determined in sufficient degree by the band of analysis of the echo and increase with decreasing low-bound frequency. For instance, for the wavelength of 3 cm, the reflecting volume effective dimensions are equal to 2.5–6m for a frequency band of 30–60 kHz and to 8.5–11m for a frequency band of 8–27 kHz. This phenomenon, in our opinion, is explained by the fact that the power spectrum of the reflected signal becomes poor in the high frequency area as the explosion products volume expands.

Table 1.19 The RCS of the Explosion for Band from 10 Hz to 500 Hz Wavelength (cm) 3.2 0.8 0.4 Source: [1].

␴ ⌬F ⭈ m 2 0.017 0.02 0.0035

Table 1.20 The RCS of the 1-kg Trotyl Explosion for the Frequency Band of 5–200 kHz Charge Mass (kg) Trotyl, 3 kg Trotyl, 3 kg Source: [1].

Wavelength (cm) 3.0 10.0

RCS (m 2 ) Cross Wind Longitude Wind 16.3 1.75 4.2 2.3

1.3 Radar Reflections from Explosions and Gas Wakes of Operating Engines

Figure 1.13

31

The angular dependencies of the echo at wavelengths of (a) 3 cm and (b) 8 mm: (1) trotyl explosion without metallic envelope, and (2) trotyl explosion with metallic envelope. (From: [1].  1997 IEEE. Reprinted with permission.)

The band of analysis essentially determines the duration of the signal reflected from the explosion. This is illustrated by the dependence of the signal duration on the lower bound frequency of the analysis band presented in Figure 1.14. If for F L = 10 Hz, the total duration was equal to 1–3 seconds; it decreased to 0.5–0.7 second when F L = 350 Hz. The power spectrum analysis of echoes has shown that in the frequency band of 10–500 Hz, the spectra are described by the relationship G (F ) ∼ F −5/3 (see

32

Radar Characteristics of Targets

Figure 1.14

Echo duration as a function of filter low-band frequency. (From: [1].  1997 IEEE. Reprinted with permission.)

Figure 1.15). As the explosion evolves, the spectrum becomes poorer in the highfrequency region. When the wind had the direction from the explosion center toward the radar, the dependence could have the shape G (F ) ∼ F −1 − F 0 at frequencies less than 30 Hz due to the Doppler shift. The shape of the echo power spectrum does not remain the same in the high frequency region. The rapid decrease of the spectral intensity occurs during the explosion products volume expansion at F > 5 kHz. From the analysis of the instantaneous power spectra presented in Figure 1.16, which were obtained at a 3-cm wavelength for the 1-kg trotyl explosion, it is seen that during 6 ms after the explosion, the spectrum shape looks like G (F ) ∼ F −1; later, with the process evolution, it approaches ∼F −4 − F −5. Finally, it is worth mentioning that the use of circular polarization does not result in a change in the radar characteristics of the signals scattered by the explosion, in particular, reduction in the RCS. This obstacle excludes the use of scattering by ground particles as a possible model because in this case circular polarization will attenuate the intensity of the reflected signal. The results of radar observation of gas wake of MIG-21 and AN-24 aircraft are quite similar to those described earlier. These investigations were carried out with pulsed Doppler radar at a 10-cm wavelength. The experiments showed that these aircraft, moving both with sonic and ultrasonic speeds, have tails in the echo detectable up to distances 1000m. The existence or absence of the inverse optically visible track did not influence the intensity of radar tail essentially. These facts

1.3 Radar Reflections from Explosions and Gas Wakes of Operating Engines

33

Figure 1.15

Echo power spectra at the different moments of time after the explosion start obtained for the radar wavelength of 10 cm. (From: [1].  1997 IEEE. Reprinted with permission.)

Figure 1.16

Instantaneous echo power spectra from the explosion at a 3-cm wavelength. (From: [1].  1997 IEEE. Reprinted with permission.)

34

Radar Characteristics of Targets

permitted us to conclude that presence of this phenomenon was caused by scattering from the track created by the gas-like fuel combustion products. The estimates obtained showed that the RCS of the gas wake was about 10−4–10−2 m2, reaching values of 0.1 m2 in some cases, and the track length reached 500–1,000m. These RCS values corresponded well to ones predicted using the model considered in the first section of this chapter. If we suppose that the gas wake can be approximated by a cylinder with the diameter four to six times that of the engine nozzle and the refractive index fluctuation variance does not change along and across the axis, from (1.12) one can obtain the specific RCS ␴ = 3 ⭈ 10−7 m2/m (RCS for 1m of the track). We suppose here that ⌬N 2 = 25 (N -units)2 (see Table 1.5) and F (k ) = 0.1 for the nozzle diameter 1m at a 10-cm wavelength. Then the total RCS is equal to 10−4–10−3 m2, conforming enough well with the RCS of the track obtained experimentally. Experimental investigations into the microwave radar reflections from chemical explosions enable us to conclude that the most important mechanisms of scattering are the following: • •

The reflection from the SWIF for the initial explosion stage; The reflection from the gas-like explosion products for further evolution of the explosion.

The refractive index fluctuation intensity of the explosion volume is significantly greater than the fluctuation intensity in the undisturbed atmosphere. The spatialtemporal fluctuation spectra of the refractive index do not differ practically from the spectra of the turbulent atmosphere. The RCS of the explosion volume for chemical explosives with the weight about several kilograms reaches 10 m2, and the intensive reflections exist during intervals less than 1 second or several seconds. These characteristics do not practically depend on radar wavelength in the frequency band of 10–75 Hz. The reflected signal spectrum is rather wide, especially during the initial stage of the explosion products cloud formation. These peculiarities permit us to realize effective radar detection with clutter filtering in some situations. The backscattering from the turbulent gas wake of jet engines results in an increase in the RCS of aerodynamic objects. This phenomenon can be used for their detection, especially in the case of a premeditated RCS decrease of the object itself. 1.3.4 Centimeter Wave and MMW Attenuation in Explosions

The first important characteristic of the explosion volume that influences the total attenuation is the volume of explosion products flying away at the final stage. It is determined in [42] by

1.3 Radar Reflections from Explosions and Gas Wakes of Operating Engines

V∞ =

4 ␲ ⭈ r ∞3 = (1.5 − 6.5) ⭈ 103 ⭈ ␲ ⭈ ␤ 3 ⭈ C 3

35

(1.14)

where C is the charge weight (kg) and ␤ is the factor depending on explosion substance density (e.g., its value for compressed trotyl is equal to 0.053). The content of gas-like and solid explosion products can be easily determined for all known explosives and is presented in Table 1.14. It is seen that the main part of the explosion products is formed by gases CO2 , CO, N2 , H2O, and amorphous carbon C; the contribution of other substances is negligible. During the final stage of explosion evolution, partial burning of CO and C takes place forming carbon dioxide; this leaves part of the carbon sediments as a dust. Among gas-like products, only water vapor and oxygen, concentrated under high pressure at the shock wave front of the explosion, possess comparatively large dipole moments and attenuation spectra in the microwave band. The attenuation in carbon dioxide is significant only in the wavelength band 12.9–17.1m [44] (i.e., far from the microwave band). In the microwave band, there exist weak absorption lines (frequencies) of CO and NO [45], but their dipole moments have values about 0.1 Debye (approximately 20 times less than the dipole moment of water vapor) and the CO concentrations only 1.5 times higher than that of H2O. Therefore, these gases cannot play important roles in microwave absorption. For nonpolar molecules (N2 ), the dipole moment can appear as the result of collisions, but for usual conditions the absorption factor ␥ resulting from this phenomenon is much less than that for water vapor (␥ N 2 /␥ H2O = 10−6 ). The derivation of the oxygen absorption factor for a pressure of 10 atmospheres (this value corresponds to the shock wave front pressure at 0.5 ms after a 1-kg trotyl explosion) has shown that for wavelengths from 0.4 to 3 cm, ␥ had the values 1.27 ⭈ 10−3–2.3 ⭈ 10−2 m−1. Thus, the attenuation caused by this phenomenon is very small (taking into account that the width of its layer following the shock wave front equals several centimeters). Estimation of the water vapor absorption factor has shown that for the explosion of 1-kg trotyl, this factor does not exceed 4 ⭈ 10−3 – 6 ⭈ 10−2 m−1 in the same waveband (i.e., it has the same level as the absorption in oxygen). The second cause of microwave attenuation is the temperature ionization of air at the shock wave front and its heating due the burning of nonreacted remainders, which generates temperatures of about (2–3) ⭈ 103K. This effect can result in the longtime existence of plasma in the explosion products volume. The electron concentration in this situation is much less than critical, so in the microwave band the condition ␻ >> ␯ is satisfied and the absorption factor can be expressed by the following expression [46]

冉

n ␯⭈n ␥ = 0.1 2 e 2 1 − 0.3 2 e 2 ␻ +␯ ␻ +␯

冊

−1/2

(1.15)

36

Radar Characteristics of Targets

where ␯ is the number of efficient collisions of electrons with molecules, ␻ is the frequency of the radar, and n e denotes the electron concentration. For typical plasma parameters of burning and for atmospheric pressure we use ␯ = 1011 s−1, n e = 108 cm−3. For f = 10 GHz, the absorption factor derived from (1.13) is equal to ␥ ≈ 7.5 ⭈ 105 cm−1 (i.e., the attenuation is very small). Furthermore, the absorption factor in plasma should decrease with decreasing wavelength, although experiments showed its growth. Finally, the third cause of microwave attenuation in explosions is absorption by solid explosion products. As is seen from Table 1.12, a large amount of carbon is given off during the detonation process. Carbon particle dimensions have the most probable radius ␮ 0 = 0.05–0.15 ␮ m [47] (i.e., usually they are much less than the wavelength). For derivation of absorption in such particles, the theory of Mie [48] can be used. For particles having ␮ << ␭ , the attenuation cross section is determined as

␴a ≈

4␲ 2 ␮ 2 6⑀ ″ ⭈ ␭ (⑀ ′ + 2)2 + (⑀ ″ )2

(1.16)

where ⑀ ′ and ⑀ ″ are the real and imaginary components of the dielectric constant, respectively. Taking into consideration that according to [49], the particle dimension distribution can be approximated by an exponential law with sufficient accuracy N ( ␮ ) = No ⭈

冉

␮ ␮2 ⭈ exp − ␮ o3 2␮ o2

冊

(1.17)

where N o is the total number of particles in a volume equal to 1 m3, the attenuation factor (specific attenuation per 1m) is

␥=

冕

N ( ␮ ) ␴ a ( ␮ , ␭ ) d␮ ≈

冋

450N o ␮ o 6⑀ ″ ␭ (⑀ ′ + 2)2 + (⑀ ″ )2

册

(1.18)

It is seen from (1.18) that the attenuation factor grows inverse proportionally to the wavelength ␭ . However, it is necessary to take into account the frequency dependence of the dielectric constant, which can results in a weaker dependence of ␥ with wavelength. The results of the carbon complex dielectric constant derivation for burning temperatures using the data of [50] are presented in Table 1.21. It is seen from Table 1.21 that the real component of dielectric constant does not depend on wavelength, but the imaginary component quickly reduces with decreasing ␭ , especially in the millimeter band. Consequently, for frequencies higher

1.3 Radar Reflections from Explosions and Gas Wakes of Operating Engines

37

Table 1.21 Complex Dielectric Constant of Carbon for Burning Temperatures Wavelength (mm) 30 8.0 4.0 Source: [2].

⑀′ 2.02 2.02 2.02

⑀″ 10.8 2.7 1.35

than 30–40 GHz, the influence of this effect causes some decrease of the attenuation factor wavelength dependence in comparison with the ␥ ∼ ␭ −1 dependence. The derivation according to (1.18) for an explosion of 1-kg trotyl, which was done under the assumption that the probable radius of the particles was ␮ = 0.1 ␮ m and the total number of particles per specific volume occupied by the explosion products was equal to N ≈ 4 ⭈ 1019 m3, gave the results presented in Table 1.22. Let us estimate the total microwave attenuation for propagation through the explosion products volume. The length of the propagation path can be easily determined from the simplest geometry analysis (see Figure 1.17).

Table 1.22 Dependence of Attenuation Factor in Carbon Derived According to (1.16) Wavelength (mm) ␥ , m −1 Source: [2].

Figure 1.17

30.0 0.29

8.0 1.68

4.0 1.8

The geometry for derivation of microwave total attenuation in explosion area (T = the transmitter position, Rec. = the receiver position, O = the explosion center).

38

Radar Characteristics of Targets

Assuming r 1 >> ␭ , R >> ␭ , r 2 >> ␭ , and supposing r 1 = r 2 = r for simplicity, it is easy to calculate the path length of the transmitter-receiver line for angle ␪ with respect to the explosion center, l = 2r √q 2 − sin2 ␪ , tan ␪ << q

(1.19)

where q = R /r. If we assume that the particle distribution along this path is described by the expression N=

冉

No R ⭈ tan ␪ exp − ⌬ ⌬

冊

(1.20)

where ⌬ is the effective radius of the volume occupied by the explosion products (i.e., that the particle density exponentially decreases with increasing distance from area center), we get S=␥⭈l=

冋

册

冉

冊

450N o ␮ o3 R 6⑀ ″ 2r √q 2 − sin2 ␪ ⭈ exp − tan ␪ 2 2 ␭⭈⌬ ⌬ (⑀ ′ + 2) + (⑀ ″ ) (1.21)

Refractive focusing effects can exert influence on the signal intensity when the signal propagates through the explosion volume. Using the results of [43], it is easily shown that the relative growth of signal intensity ⌬ p /p beyond the spherical area is determined as ⌬ p /p ≈ 2

r (n − n o ) R

(1.22)

where r is the distance from the area center to the reception point, and R is the radius of dielectric sphere with refractive index n (usually n − n o << 1, where n o is the refractive index of the atmosphere). Substituting the experimental values r = 25m, R = 2.5 m, and n − n o = 300 ⭈ 10−6 (this value corresponds to maximal predicted ones) into (1.22), we obtain ⌬ p /p ≈ 6 ⭈ 10−3 (i.e., the influence of refraction is negligible). These estimates of the influence of diffraction on the dielectric sphere show that the inhomogeneous sphere with dimensions R << r , where r is the distance from the explosion center to reception point (see Figure 1.17), does not create significant focusing or defocusing effects [43] and, consequently, cannot change the intensity of the received signal significantly. As mentioned earlier, the experimental investigation into microwave attenuation in the explosion area was carried out at wavelengths of 3 cm, 0.8 cm, and

1.3 Radar Reflections from Explosions and Gas Wakes of Operating Engines

39

0.4 cm. The propagation path was of length 50m; the transmitter and receiver were placed at the ends and the explosions took place in the middle of the path. Trotyl explosives were used; their weights were 1 kg. To eliminate the influence of ground debris, the charges were placed at heights 1–2m over the land surface (on slabs or poles). Magnetron oscillators were used as transmitters, operating in the continuous regime with output power 1–10W. For measuring system calibration, polarization attenuators were used. They provided calibration accuracy of 0.2–0.5 dB; the worst value corresponded to the system with wavelength 4 mm. A typical temporal diagram of total explosion signal attenuation is presented in Figure 1.18. At the moment of charge explosion, rapid drop in the received signal intensity is observed. Maximal attenuation occurs for 30–50 ms for all investigated operating frequencies, increasing slightly (up to 10%) with increasing frequency. The sharp splash in the curve labeled ‘‘4’’ is the time of explosion. The recovery of the signal level takes place more slowly than its decay at the start of the explosion. The total duration of significant attenuation is about 80–100 ms. The results of the entire radar signal attenuation study for charge detonation on the transmitter-receiver line of sight are presented in Table 1.23, and the results of the attenuation factor measurements are given in Table 1.24. The significant differences between results obtained for the different experiments can be explained by the nonidentity of charges and conditions of their explosion (technical trotyl charges were used for the experiments). A comparison of Tables 1.23 and 1.24 shows that for 3-cm wavelength, satisfactory agreement of predicted and experimental attenuation coefficient values is observed, but for shorter wavelengths the difference increases.

Figure 1.18

Samples of temporal diagram for received power at wavelengths of (4) 3 cm and (3) 8 mm. (From: [2].  1997 IEEE. Reprinted with permission.)

40

Radar Characteristics of Targets Table 1.23 Maximal Total Attenuation (dB) for 1-kg Trotyl Explosion Wavelength (mm) 30 8.0 4.0 Source: [2].

Attenuation (dB) 3.2; 4.3; 6.5; 7.5; 11.0; 7.2; 7.1; 4.3; 3.2 13.8; 14.6; 11.7; 14.2; 11.3; 14.3 11.8; 8.7; 11.0; 9.0; 12.0

Average Attenuation (dB) 6.03 ± 2.4 13.3 ± 1.71 10.5 ± 1.94

Table 1.24 Experimental Attenuation Factors for Explosions of a 1-kg Trotyl Explosive Wavelength (mm) 30.0 8.0 4.0 Source: [2].

␥ , m −1 0.15; 0.3; 0.2; 0.35 3.3; 3.6; 2.8 3.8; 2.9; 2.8; 3.6

␥ , m −1 0.25 ± 0.075 3.2 ± 0.19 3.3 ± 0.22

When the explosion center is shifted by some angle with respect to the transmitter-receiver line of sight, less signal attenuation occurs. The observed experimental dependence S = f (␪ ) and the results of derivations performed according to (1.22) are presented in Figure 1.19. Satisfactory agreement of prediction and experimental data is observed. For a deviation angle ␪ ≈ 6° from the explosion center (for experiment geometry, this

Figure 1.19

The angular dependencies of total attenuation for wavelengths of 3 cm, 0.8 cm, and 0.4 cm. The solid line is the result of derivation from (1.21). (From: [2].  1997 IEEE. Reprinted with permission.)

1.3 Radar Reflections from Explosions and Gas Wakes of Operating Engines

41

corresponds to 5–6-m shifting), attenuation is practically absent. This deviation corresponds to a distance between the line of transmitter-receiver and the charge position equal to 40r o (the lower axis of abscissas in Figure 1.19), where r o is determined by (1.12) and characterizes the dimensions of area occupied by the explosion products. The lack of knowledge of dissociation processes occurring in the explosion makes it difficult to explain the attenuation temporal dependence. One of the probable mechanisms is connected with the fact that after primary explosion products dispersal, burning of the small dispersed carbon particle takes the place due to the presence of a high temperature that is sufficient for burning (higher than 2,500K). In these conditions, the carbon particle concentration quickly decreases, resulting in recovery of the received signal level. The probability of this hypothesis is supported by data presented in [1], the author of which gives the following results: using brisant explosive (the weight was 36 kg), the dust was present for 5–10s after start of the explosion, and the attenuation for frequencies 35–140 GHz was not higher than 0.4 dB. Dust particle concentration had a value of about 108 m−3 (i.e., it was less than the primary carbon particle concentration that appeared as the result of detonation). In the explosion area occupied by trotyl explosive explosion products, significant microwave attenuation occurs, and it increases with increased radar frequency. The most probable mechanism for explaining the experimental data is the absorption by solid products of transformation, in particular, by amorphous carbon. 1.3.5 Radar Backscattering from Sonic Perturbations Caused by Aerodynamic Object Flight 1.3.5.1 Reflection from the Shock Wave Front Caused by Supersonic Flight

For straight flight of an object when its speed is V, a shock wave is formed in the homogeneous atmosphere. The shock wave possesses axial symmetry, and its shape is close to conical (Mach cone) with the object at its apex. For a cone angle ␸ 0 , the following expression is valid

␸ 0 = arcsin (a 0 /V ) = arcsin M −1

(1.23)

where a 0 is the sound speed and M = V /a 0 is the Mach number. This wave spreading to the sides of the trajectory includes increasing air mass, simultaneously with which its intensity diminishes, and transformation into a sound wave is observed. The pressure jump ⌬ p at the shock wave front (SWF) for flight of a body of revolution is determined as [51] ⌬ p (M 2 − 1)1/8 D = ⭈ kS ⭈ 3/4 p0 L (H /L )

(1.24)

42

Radar Characteristics of Targets

where L is the body length, H is the normal distance to the body flight axis, D is the equivalent maximal body diameter, p 0 is the pressure of surrounding gas, and k S is the body shape coefficient equal to ␶o

kS =

1.19 (␥ + 1)

3/4

⭈

冢冕

F (␶ ) ⭈ L ⭈ dL

0

冣

⭈

L D

(1.25)

Here ␥ denotes the adiabatic constant, which for air is 1.4; F (␶ ) is the effective area distribution function [7, 52] ␶

F (␶ ) =

√ 冕 L ⭈ 2␲

A eff (t ) (␶ − t )1/2

dt

0

in which A eff (t ) is the body cross section. As a rule, when the body of revolution configuration is simple the value ⌬ p can be calculated analytically, but for more complicated bodies (like aircraft) reliable data relating p to the SWF are usually obtained experimentally. According to [53], the SWF pressure difference is 50–200 newton m−2 (the larger value corresponding to a bomber) when aircraft speed equals (M1.5–M2) for distances 6–8 km from the aircraft trajectory. In a homogeneous atmosphere, ⌬ p decreases with the increase of distance to object according to the law ⌬ p ∼ r −3/4 and the rate of pressure variation in the interval between jumps is inversely proportional to r [54]. The pressure impulse, being the important effect of the shock wave on the medium, is expressed as t1

I=

冕

⌬ p (t ) dt ≈

1 ⌬p ⭈ T 2

(1.26)

to

where T = t 1 − t o and the parameter T is calculated on basis of the following expression for rate of pressure variation as a function of time d (⌬ p ) M ⭈ r −1 = 0.2 ⭈ dt 2 √M − 1

(1.27)

The considered dependences permit us to extrapolate the experimental data given in references into the area of comparatively small distances from the object.

1.3 Radar Reflections from Explosions and Gas Wakes of Operating Engines

43

Figure 1.20 presents the SWF parameter values for distances less than 1 km from the airplane (see shaded areas; the lower bound corresponds to a fighter and the upper one corresponds to a transport aircraft). Point 1 in Figure 1.20 is obtained for an F-4C moving at an altitude of 30m [55]. Satisfactory agreement is observed between predicted data presented in Figure 1.20 and the experimental data given in [1, 51, 55] and extrapolated to small distances. In Figure 1.20, the dependences ⌬ p (r ) and I (r ) are also presented for flight with the speed of M2 of a cone-cylinder body with diameter of 120 mm (curves 2). While pressure jump values for aircraft and cone-cylinder body have almost the same level, the pressure impulse for the second is one or two orders less. From the point of view of radar observation of the SWF and the interaction of the SWF with a turbulent atmosphere, the most interesting items are the spectra of pressure jump (see Figure 1.21). The duration of the N -profile, depending on the object shape and its speed, can vary within the limits 0.01–0.5 second. The theoretical spectrum of such an impulse decreases with increased frequency at the rate of 6 dB/octave (line 1). The experimental spectra (curves 2 and 3) are obtained in [51] for different atmospheric conditions. With increased frequency, the experimental spectra decrease with the slope close to that predicted. The spectra maxima are observed at frequencies of 10–20 Hz. Expansion of the front, due to atmosphere turbulence, results in reduction of the spectrum high frequency components. Electromagnetic field reflection from an infinitely thin SWF is determined by the refractivity jump, the reflection coefficient ⌫ given by

冉

n −n

| ⌫ | 2 = n1 + no 1 o

aaaaaaa aaaaaaa aaaaaaa aaaaaaa aaaaaaa Figure 1.20

冊

2

≈

⌬n 2 4

(1.28)

aaaaaaaa aaaaaaaa aaaaaaaa aaaaaaaa

(a) Pressure jump and (b) pressure impulse as functions of distance from an aircraft and a cone-cylinder body. (From: [7].  1997 SPIE. Reprinted with permission.)

44

Radar Characteristics of Targets

Figure 1.21

Power spectra of the pressure jump. (From: [51].  1966 Acoustical Society of America.)

where n 1 and n o are the refractivity coefficients for the peak and atmosphere, respectively. Taking into account the adiabatic link between density and pressure for weak shock waves and the proportionality of refractivity coefficient to density, we get ⌬(n − 1) 1 ⌬ p = ⭈ n−1 ␥ p

(1.29)

冉 冊

(1.30)

and n−1

| ⌫ | 2 = 2␥ p

2

⭈ ⌬p2

where (n − 1) = 320 ⭈ 10−6 for a standard atmosphere at height about 500m above sea level. Because the SWF expands as it propagates in the atmosphere, the reflection coefficient is less than its value determined by (1.30) for an infinitely thin jump. Let us assume that the refractivity coefficient varies n0 =

再

n0

for x < 0

n 0 + ⌬n [1 − exp (−x /⌬)]

for x ≥ 0

(1.31)

1.3 Radar Reflections from Explosions and Gas Wakes of Operating Engines

45

where ⌬ is the effective front width and x denotes the direction normal to the front. It is easy to show that for this case

|⌫|

2

冉 冊

⌬n = 2

2

1

⭈ 1+

16␲ 2⌬2

(1.32)

␭2

Two obstacles cause SWF expansion in the turbulent atmosphere. The first is that the presence of turbulent temperature pulsations results in sound speed fluctuations. The second is that the sound waves are carried by moving air, and therefore the presence of turbulent motions, described by turbulent pulsations of speed U ′, contributes to additional random distortions of front shape. In this case, the effective SWF width can be determined by the [56] ⌬ = 16

␥ p ⭈ ⑀ 2 ( ␮ + ␺ )2 L o o ␥+1 ⌬p

(1.33)

where L o is the outer scale of the atmosphere turbulence and ⑀ , ␮ , and ␺ are the parameters characterizing the atmospheric state. The value of L o increases linearly with altitude, changing from 1–2m near the land surface to 300–400m at the upper boundary of atmosphere. According to data presented in [57], for altitudes 100–200m, the typical value is L o ≅ 10m. The value of ⑀ 2 lies within the limits 10−7–10−5 and decreases with the altitude. The coefficient ␮ characterizes the turbulent pulsations of the sound speed a T , according to ␮ = a T /⑀ a o , and is usually close to unity. The coefficient ␺ characterizes the turbulent component of the flow speed U x′, which is parallel to the SWF normal and is equal to ␺ = U x′ /U o , where U o is the mean speed of the flow. When the wave propagates in the vertical direction the typical value of ␺ is 0.1–0.2. The results of deriving the effective width of the SWF for several parameters of turbulence ⑀ 2 ( ␮ + ␺ )2—carried out using (1.33)—are represented in Figure 1.22 by the solid lines. In the same figure, the experimental data [58] are shown for shock wave resulting from supersonic airplane flights. The dashed curve corresponds to dependence ⌬ = f (⌬ p ) for dry air when the SWF width is determined by intermolecular bonds and viscosity only. It is seen from Figure 1.22 that the real SWF width is two orders greater than the theoretical value for dry air, leading to reduction of the reflection coefficient. The dependences of the reflection coefficient upon the distance to the front derived according to (1.30) and (1.32) are presented in Figure 1.22(a). For distances about 10m from the object, the reflection coefficient is approximately equal to 10−10— almost the same in comparison with cases considered in Figure 1.22(b) and diminishing for greater distances.

46

Radar Characteristics of Targets

Figure 1.22

(a) The effective front width and (b) the reflection coefficient as functions of distance to pressure peak ⑀ 2 <( ␮ + ␺ )2> equals to (1) 3 ⭈ 10−3, (2) 3 ⭈ 10−4, and (3) 3 ⭈ 10−5. (From: [7].  1997 SPIE. Reprinted with permission.)

Taking into consideration that in the turbulent medium, besides SWF expansion, the SWF refractivity varies irregularly and reflection is possible in directions other than normal to the front. In this case, backscattering occurs due to the front roughness. This permits observation of reflections from the shock wave front in an angular region determined by the scattering pattern. For preliminary estimation, it is possible to use the assumption of fully isotropic scattering (meaning that the front distortions have amplitude less than the wavelength and a rather small correlation radius). This assumption results in higher estimates in comparison to the real data over much of space excepting the narrow region of angles near the direction of specular reflection. The effective radar cross section ␴ in this case is equal to

␴=

冕

⌫2(r ) dS

S il

where the integration must be carried out over an all-illuminated area. The estimates show that for 10-cm wavelength, when the antenna pattern width in both planes is about 0.01 radian and the distance to the object is 10 km, the value of the RCS of the SWF front at the distances from the object less than 10m is about 10−5–10−6 m2. This value is comparable with background reflections

1.3 Radar Reflections from Explosions and Gas Wakes of Operating Engines

47

from the turbulent troposphere. Taking into account that the spectrum maximum of the SWF pressure is observed at the frequencies 10–20 Hz (see Figure 1.21), the RCS of the SWF front can be greater at decimeter and meter wavelength in comparison with the estimates obtained for the centimeter waveband.

1.3.5.2 Microwave Scattering from Sound Perturbations in the Atmosphere

The airplane flights with near-sonic and supersonic speeds result in the appearance of a powerful sound wave field. For instance, the sound radiation power of ‘‘Phantom’’ airplane is about 10 millimeter waves. The sound wave propagation is followed by a dielectric constant variation of the medium and, as result, by electromagnetic wave scattering from these dielectric constant perturbations. Let us evaluate the scattered field intensity for this case. The dielectric constant of the medium in the sound wave field can be presented as e ( r›, t ) = ⑀ o + ⑀ 1 ( r›, t )

(1.34)

where ⑀ o = const denotes the dielectric constant of undisturbed medium, ⑀ 1 ( r›, t ) is the sound perturbation of the dielectric constant, usually ⑀ 1 ( r›, t ) << 1. The value of ⑀ 1 can be expressed from the Lorentz-Lorenz relationship [59] by the following

⑀ 1 ( r›, t ) ≈

n −1 ⑀o − 1 ⭈ ␦␳ = 2 o ⭈ ␦␳ ␳ ␳

(1.35)

where ␦␳ is the air density variation caused by the pressure changing due to the sound or shock wave propagation. It is expressed as

␦␳ 1 ␦ p = ⭈ ␳ ␥ p where ␳ is the air density. Taking into account that the sound speed in an undisturbed medium is equal to ao =

√

␥ ⭈ po ␳o

after simple derivations we get from (1.13)

48

Radar Characteristics of Targets

⑀ 1 ( r›, t ) ≈ 2

no − 1 ⌬ p ⭈ 2 ␳ ao

(1.36)

The pressure variation in atmosphere during sound wave propagation can be presented as [60]

冋冉

›

冊册

R o √2␳ a o I o G a exp [−␣ (r − R o )] V ⌬p(r , t) = ⭈ exp −i ⍀t − k a r + k a r ao (1.37) ›

where G a is the normalized sound source directivity pattern, ⍀ denotes the frequency of the sound wave, k a = 2␲ /␭ a is the acoustic wavenumber, V is a wind speed (later it is supposed that it is constant in the scattered area), I o denotes the sound field intensity measured for some sample range R o from source along the axis of radiation pattern, ␣ is an atmosphere absorption coefficient, and r is the current range from the sound source (see Figure 1.23). Substituting (1.37) into (1.36), we get

⑀ 1 ( r›, t ) =

冉

›

⑀1 V G cos ⍀t − k a r + k a r a ao

冊

(1.38)

where

⑀ 1 = 2√2 ␳ −1/2 a o−3/2 I o1/2 R o (n o − 1) exp [−␣ (r − R o )]

aaaaa aaaaa aaaaa aaaaa

Figure 1.23

Geometry for derivation.

(1.39)

49

1.3 Radar Reflections from Explosions and Gas Wakes of Operating Engines

The electromagnetic field reflected from the sound wave package can be described in the Born approach as k 2 exp (−i␻ t ) E 1 (R 1 , t ) = 4␲ ⑀ o ›

冕冕 冕 |

—›

–›

e ik | R 1 − r

(V )

›

›

R1 − r

|

|

⑀ 1 ( r›, t )[ n›(⑀ o n›)] dV (1.40)

›

Here R 1 is the radius vector in the point of electromagnetic wave reception and ␻ denotes the frequency of electromagnetic field. Omitting the rather complex intermediate derivations, let us write the ratio of received-to-transmitted electromagnetic field power as

冉 冊

Pr Ro = 2I o Pt R1

2

(n o − 1)2 −2␣ (R 1 − R o ) k 2a 32 e 2 ␳ a o3 ⌬4R 14 | C 3 − C 2 /2R 1 |

冉

冤

V C1 + C2 x ao Vx × exp −4 2 2 − ⌬ C ⌬ ao 2 C3 − 2 2R 1

冉

冊 冊

2

冥

sin ⭈

冋

(1.41)

冉

冊册 冊册

␲ a3 V ␭ 2⫿ 1− z ␭ ␭o ao

冋

冉

V ␲ a3 ␭ 2⫿ 1− z ␭ ␭o ao

Here a 3 is the radial length of the sound package, k = 2␲ /␭ is the wavenumber, ⌬ is the radar directivity pattern width, Vx and Vz are the speed vector projections onto the coordinate axes,

冋

C 2 = ik 2 ⫿

冉

V ␭ 1− z ␭a ao

冊册

: C 3 = ik

冉

2 ⫿ ␭ /␭ o 1 1 1 − 2 2+ 2 2R 1 R 1 ⌬a ⌬

冊

and ⌬a denotes the width of the acoustic source directivity pattern. It follows from (1.41) that the scattering of the electromagnetic field has a resonant character; its maximum occurring for ␭ /␭ a determined by the expression 2−

冉

V ␭ 1− x ␭a ao

冊

=0

(1.42)

The resonant maximum value is proportional to the number of wavelengths in the sound package (i.e., to the sound package length). Besides, the resonant length of the radiowave depends upon longitudinal component of wind speed Vz because the sound wave length is functionally connected with Vz . The wind speed cross component Vx leads to two effects, decreasing the level of reflected signal. The first phenomenon consists of sound wave package expanding

50

Radar Characteristics of Targets

out of the volume of the radar resolution cell; it is taken into consideration by the factor exp (−4Vx2 /⌬2a o2 ). The second effect deals with the distortion of the spherical shape of the sound wave phase front. In natural conditions, one more factor appears diminishing the backscattering efficiency. It is connected with turbulent pulsations of the atmosphere refractive index, which result in additional distortions of the sound wave phase front. The results of numerical derivation of the reflection coefficient for resonant scattering without wind, which were obtained for two radiowave bands (S- and X-bands), are presented in Figure 1.24. The derivations are carried out for the following values of parameters included in (1.39): a S = 200␭ a , R o = 10m, I o = 10 Wm−2, ⌬2 = 0.37 ⭈ 10−2, ⌬a2 = 0.03, T = 20°C, n o − 1 = 2.68 ⭈ 10−4, a o = 340 m/sec, ␳ = 1.29 kg/m3, ␣ = 2.5 ⭈ 10−2. It is seen that for the most favorable geometry and for distances between the radar and the sound wave source less than 100m, the ratio P r /P t is about 10−8–10−12, and it decreases with further distance (to −200 dB when the distance is about 500m). This means that the observation of radar reflections from sound waves is practically impossible at ranges typical for radar operation, especially for directions that do not coincide with normal to sound wave front. It is also worth noting that the main energy component in the jet propulsion engine sound radiation spectrum is concentrated in a frequency interval having a

Figure 1.24

Reflected signal ratio as a function of distance between the radar and sound wave source. (From: [7].  1997 SPIE. Reprinted with permission.)

51

1.3 Radar Reflections from Explosions and Gas Wakes of Operating Engines

width less than 10 Hz [61]; the maximum of the weak shock wave spectrum also lies in the low frequency area (see Figure 1.21). This means that for radar detection of signals scattered by sound waves, it is more expedient to use the UHF or L-bands as in the case of detection of SWF reflections for supersonic flights. 1.3.5.3 Microwave Backscattering from Atmosphere Turbulent Perturbations Caused by the Interaction with Sound Field

Microwave signal backscattering from atmosphere turbulent perturbations is the result of their refractive index variations caused by temperature gradients or local humidity variations. The volume RCS in this case is determined as [62]

␩=

␲ 〈 ⌬n 2 〉 ⭈ k 2 ⭈ F n (k ) 8

(1.43)

where k = 2␲ /␭ is the wave number, F n (k ) denotes one-dimension refractive index spectrum, and 〈 ⌬n 2 〉 is the variance of refractive index fluctuations. It is known [43] that the RCS is connected with the speed of energy dissipation ⑀ by the dependence ␩ ∼ ⑀ 2/3 and, consequently, ⑀ increase caused by any reasons leads to increased RCS. Shock wave or sound wave propagation in the atmosphere is accompanied by energy losses; part of the power is transformed into the power of turbulent motion, which can result in variation of energy dissipation speed and, as a result, in changing of the intensity and fluctuation spectrum of the refractive index. The sound intensity of the source of power P S at some range r is expressed as I (r ) =

PS 4␲ r 2

⭈ exp (−2␣ r )

(1.44)

It is easy to show that the sound energy loss of the sound wave is

␤ (r ) =

I (r ) ⭈ m; ␳

m = 2␣

(1.45)

or

␤ (r ) =

mP S 4␲␳ r 2

exp (−mr )

(1.46)

The sound absorption coefficient in a homogeneous atmosphere depends on humidity and temperature, and for a standard atmosphere, it is equal to m = 4 ⭈ 10−5 at a frequency of 10 kHz. The presence of turbulent viscosity of ␯ T, which

52

Radar Characteristics of Targets

exceeds the cinematic viscosity of ␯ as ␯ T /␯ ∼ R e [62], results in significant increase in the absorption coefficient; here R e is the Reynolds number. Taking into account that in the near-ground layer of the atmosphere R e ≥ 30–60 × 109, the sound energy loss rate is 10 times greater than that determined by (1.46). For example, for a sound source power P S = 5 MW and for range from the source equal to 10m the value of ␤ , derived using (1.46), is ␤ ≈ 0.01 m2sec−3, but taking into consideration the turbulent viscosity, it increases to ␤ ≈ 0.2 m2sec−3. At the same time, the turbulent energy dissipation rate in lower atmosphere layers does not exceed 0.01–0.1 m2sec−3 [57] (i.e., it is approximately 10 times less than the additional energy obtained by atmosphere and due to the sound absorption). In several papers devoted to the concept of turbulence, it is shown that in some conditions the sound oscillations, even having relatively low power, can result in significant change of the turbulence characteristics. Omitting the theory, let us consider some conclusions and experimental results of such interaction investigations on the basis of papers [63–66]. 1. For rather great Reynolds numbers, there exist two-dimension speed fluctuations of a sinusoidal type (in atmosphere, their appearance is stipulated by Helmholtz instability). Under the influence of sound wave, having the frequency equal to the frequency of autooscillations of sinusoidal fluctuations, a sharp increase of fluctuations occurs. Sometimes several maxima at several frequencies occur, but always the main frequency can be distinguished. According to [65], for initial turbulence intensity approximately equal to 16%, the acoustic influence leads to the increasing of turbulent energy by 97%. 2. The spectra of the speed pulsations in the flow after acoustic influence have a resonant form. From the point of view of instability theory, the general tendency of resonant frequency behavior seems to be interesting. First of all, the resonant frequency is a continuous function of the flow speed, and for speeds less than 20 m/s it lies in the interval from 10 Hz to 1,000 Hz. 3. If the sound oscillations affect turbulent flow rather strongly, increase in the turbulence intensity occurs at the sonic frequency while its decrease takes the place at other frequencies [i.e., a transformation of F (k ) is observed]. 4. The influence of acoustic waves shows itself even when their intensity is not high. According to [67], the sound field amplitude must have the value about 0.01–0.1% of intensity of the main flow, and the sound field intensity does not exceed 0.001–1 W/m2. The forecasting estimates show that such a level of intensities can be expected at ranges less than some tens of meters from the object (e.g., airplane or missile), which produces this acoustic field. Experimental investigations were carried out, taking into account the complexity of the theoretical estimates of such sound wave interaction with atmospheric

1.3 Radar Reflections from Explosions and Gas Wakes of Operating Engines

53

turbulence and their influence on characteristics of the microwave backscattered field. An operating jet airplane engine was used as the source of intensive sound field. The parameters of the turbulent atmosphere were analyzed by two techniques. In the first, the measurements were made of flow speed turbulent pulsations for an engine operating in different regimes by means of thermoanemometer. The measured parameter in this case was the variance of the flow speed differences at several spatially separated points. The distance between sensors was 5 cm, and this provided a decrease in the influence of the direct sound field on the results of measurements. Because the air oscillations for frequencies of hundreds of Hertz, where the main part of the engine noise power is concentrated, were practically identical for both sensors, the signals received by them were mutually canceled. The experiment scheme is presented in Figure 1.25(a). The directions between the sensors and the axis of jet wake formed the angle 40°–60°, which excluded the entry of the combustion products into the displacement zone of the sensors. A sample of the process for a shutdown engine and for one operating in its nominal regime is demonstrated in Figure 1.25(b). It is seen that the speed fluctuation difference variance for the operating engine is larger than that with an undisturbed atmosphere. A forced engine operation regime did not lead to significant change in fluctuation variance in comparison with the nominal regime. The cumulative distributions of different variance are represented in Figure 1.26 (abscissa axis has the normalized values). It is seen that the influence of the sound field results in an increase in the speed difference variance in comparison

Figure 1.25

(a) The scheme of experiment and (b) the sample of flow speed turbulent pulsations for muffled and operating engine. (From: [7].  1997 SPIE. Reprinted with permission.)

54

Radar Characteristics of Targets

Figure 1.26

Cumulative distributions of flow speed different variance. (From: [7].  1997 SPIE. Reprinted with permission.)

with the experiment when the engine is shut down. For the 50-m range, this difference is less evident and shows itself in the area of large variances only. Consequently, in the area of spatial scales of some centimeters, the sound influence on the turbulent atmosphere is observed only for ranges less than 10–20m. The second direction of investigations consisted of direct estimation of the microwave backscattering electromagnetic field level in the area of an operating engine. These measurements were performed with X-band continuous-wave radar. The illuminated spot was raised to the height of 100m over the aircraft to remove the reflections from its vibrating surface and from the area behind the engine. The minimal detectable RCS of this radar in the area of supposed backscattering was about 10−4–10−5 m2 in the filter bandwidth of 10 Hz. The observations have shown that the reflections from the atmosphere zone surrounding the aircraft did not exceed the minimal detectable level of RCS for all operation regimes of the jet engine. Therefore, further radar measurements are expedient using more sensitive radars. The results of investigation show that among three considered mechanisms of possible radar observation of aerodynamic objects using atmospheric sound perturbations caused by their flight, the most promising is the backscattering from

55

1.4 Statistical Characteristics of Targets

the shock wave front occurring during supersonic flight. The microwave backscattering from perturbations of the dielectric constant of the air, resulting from sound wave influence, turns out to be comparable with backscattering from natural turbulence of the atmosphere. The backscattering from the turbulent atmosphere due to variations in its spatial and temporal characteristics, caused by sound energy transformation into the power of turbulence, seems to be promising for radar detection and requires the further investigations with more sensitive radars. The general requirement is the use of L- or UHF-band radars for which these mechanisms appear to work more efficiently.

1.4 Statistical Characteristics of Targets 1.4.1 Target Statistical Models

The statistical model of a target, as in general the description of its properties, must be developed with clear orientation to the scope of the problems being considered. Here the model analysis is chosen for applicability to the task of the detection and radar range estimation. The target model for this case has to correctly reflect only the fluctuation statistics of the signal amplitude or power (proportional to RCS) at a given moment of time. These properties are fully determined by the amplitude and phase relations of the elementary electromagnetic waves created by the target scattering elements in the radar receiver. For rather general assumptions about the characteristics of the bright points forming the radar target, it is possible to form the statistical description of the echo. The diffraction field (or signal amplitude) can be presented in the following way [68] S (␪ ) =

∑ A j e j␸

j

= X + iY

(1.47)

j

where A j is the echo amplitude of j th bright point. The phase ␸ j of every elementary signal depends on the range d j from the j th point to the plane of signal registration and the character of this point.

␸ j = 2kd j + ␸ jo

(1.48)

The coordinate of d j can depend on the target aspect according to d j = d jo cos ␪ or in a more complicated manner for continuous change of the surface radius of curvature. For an unknown type of target motion and bright point distribution, the values d j can be considered as the random values with some distribution law p (d ). The pdf of phase does not practically depend on the distribution p (d ) and approaches a uniform distribution in the interval [−␲ , ␲ ] if d /␭ >> 2␲ .

56

Radar Characteristics of Targets

For slow change of amplitude A j in comparison with phase variation, the signal components in X and Y have asymptotically Gaussian distributions with equal variances (i.e., the echo is a narrowband Gaussian process). Generally, for symmetrical pdf p (␸ ), the mean value of the sine-component is equal to zero; the mean of the cosine-component is a ; and the variances of each component s 1 , s 2 are equal to a = X = nA 2 s 1 = nA 2 s 2 = nA 2

冕 冕

冕

p (␸ ) cos ␸ d␸

p (␸ ) cos2 ␸ d␸ −

a 2

(1.49)

p (␸ ) sin2 ␸ d␸

The pdf of amplitude A is expressed as p (A ) =

冋

A d 2 (s 1 + s 2 )2 exp − − 2s 1 4s 1 s 2 √s 1 s 2

册

⭈

∑ (−1)m I m m

冉

冊 冉冊

s1 − s2 2 a A I 2m 4s 1 s 2 s1

(1.50)

For uniform phase distribution, we have a = 0, s 1 = s 2 and the amplitude pdf is Rayleigh: p (A ) =

冉 冊

2A A2 exp − n n

(1.51)

Such targets are called the Rayleigh targets. If among the elementary scatterers there exists a bright point nonfluctuating both in amplitude and phase, then the signal amplitude is described by the Rician pdf (a = c , s 1 = s 2 ) p (A ) =

冉

2A A2 + c2 exp − n n

冊 冉 冊 ⭈ I0

2c n

(1.52)

For more generalized case with s 1 ≠ s 2 one can find p (A ) =

冉

冤

k2 + 1 k2 + 1 k2 + 1 2 B2 + A A exp − k 2 2k 2 ×

∑ (−1)m I m m

where k =

2 √s 1 /s 2 ≠ 1, B = c /(s 1 + s 2 ) ≠ 0.

冉

k2 − 1 4k

2

冊

冊

A 2 I 2m (k 2 + 1) BA

(1.53)

冥

57

1.4 Statistical Characteristics of Targets

The set of parameters B = 0, k = 1 corresponds to the Rayleigh distribution, and B ≠ 0, k = 1 corresponds to the Rician distribution. For the objects with reflecting area dimensions many times larger than the wavelength, the parameter k differs from unity very little because the phase fluctuations in this case exceed ␲ . It permits wide use of the Rayleigh target model. It is worth noting that it describes well the scattering characteristics of complex targets if accuracy is not required in the pdf tails. To improve agreement between the model and experimental pdf, some authors made attempts to use more complicated concepts of real target reflection models. For example, there was the model in which the reflector forming the Rician distribution was augmented by high directivity reflectors with large RCS (comparable with total contribution from the other elements). Other models used strong specular reflections appearing only for narrow angle regions, resulting in RCS spikes for specific target aspects. However, these models did not receive the wide application in practice because of complexity or impossibility of multiple parameter estimations from the real data of these distributions. In the practice of statistical theory of radar, priority was given to the chi-square distribution, which for appropriate selection of parameters coincides rather well with the Rayleigh or produces enough accurate approximations of most empirical pdfs. The Swerling models [69] are widely used in radar theory as variants of the models with RCS variation according to chi-square distributions. The chi-square pdf takes the form [69] p (␴ , ␴ ) = =

冉 冊

1 ␯ ␯␴ ⭈ ⭈ ⌫(␯ /2) 2␴ 2␴

(v − 2)/2

冉 冊 冉 冊

exp −

␯␴ 2␴

(1.54)

␯ ␹2 2 (v − 2)/2 ( ␹ ) exp − 2 ␴ 2v /2 ⌫(␯ /2)

where ␹ 2 = ␯␴ /␴ , ⌫(␯ /2) is the gamma-function, ␴ is the mean RCS value, and ␯ is the number of degrees of freedom determining the ratio of the square of the mean RCS to RCS variance. Widely applied in radar, Swerling’s models are shown in Table 1.25. It is supposed in the first and third Swerling models that the received signals are nonfluctuating ones for one scan (slowly fluctuating pulse train) and uncorrelated between two scans. There are the varying signal fluctuations within one scan duration for the second and fourth models (independently fluctuating pulse train). For the Rayleigh case (1 and 2), the RCS pdf is described by the exponential law p (␴ , ␴ ) =

冉 冊

1 ␴ exp − ␴ ␴

(1.55)

58

Radar Characteristics of Targets Table 1.25 The Swerling Models Type of Model Model 0 (Marcum’s model) Model 1 Model 2 Model 3 Model 4

RCS pdf

Echo Fluctuations

Stable target Rayleigh Rayleigh Four degrees of freedom Four degrees of freedom

No Slow Fast Slow Fast

Number of Freedom Degrees ∞ 2 2 4 4

For the two other models, the RCS pdf has the form p (␴ , ␴ ) =

冉 冊

4␴ 2␴ exp − ␴ ␴

(1.56)

They are similar to the RCS pdf of a target having one dominant stable scatterer together with many randomly distributed scatterers. In this sense, the expressions (1.55) and (1.56) give close results in the region of large probabilities. For the majority of complex targets, the Swerling models 1 and 2 are applicable, especially for observations over the angle range of about 360°. The application of the standard Swerling models, as a rule, leads to decreased estimations of RCS probabilities for the area of small values of probability, precluding reliable estimates of detection probability for a single signal at the area of correct detection probabilities of about PD ∼ 1. Besides, the Swerling models are poorly applicable to rather simple targets with small numbers of bright points. These difficulties were overcome by Weinstock [70, 71]. He extended the area of application of the chi-square distribution to degrees of freedom less than two. The number of degrees of freedom for some bodies of simple shape evaluated by Weinstock is presented in Table 1.26. This approach permits us to use the chi-square distribution for a wide class of approximations from Rayleigh and Rician ones to the popular log-normal distribution by adjusting of the numbers of degrees of freedom. 1.4.2 Real Target Statistical Characteristics

The most extensive experimental data are obtained for air target RCS distributions [72–75]. These results permit us to confirm that for air targets, the amplitude Table 1.26 The Number of Degrees of Freedom for Simple Bodies Type Randomly oriented cylinder Stabilized cylinder Randomly oriented bodies of spheroid shape Source: [70].

Freedom Degrees Number 0.6–1.4 4.0 4.0

1.4 Statistical Characteristics of Targets

59

distributions can be approximated rather well by the Rayleigh law up to amplitude values six times greater than the mean. As illustration, the cumulative RCS distributions for piston-engine and jet airplanes are presented in Figure 1.27 obtained in [72] at the 3-cm wavelength; the exponential distribution is showed by the dashed line. It is seen that the experimental results agree with the Rayleigh model in the area of most probable RCS values, although the RCS mean value is somewhat greater than in this model (within limits from 1.1 to 1.5) (i.e., the distribution does not predict the largest observed signal intensities). For jet airplanes, the agreement of the experimental results with the Rayleigh model is more satisfactory than for piston-engine aircraft. These results are obtained for long-period observation of targets or for averaging of signals over a wide aspect sector. If the observation time is small relative to the echo fluctuation period, a different chi-square law describes the experimental data. For long observation time and large airplane aspect change, the data approach the Rayleigh statistics (␯ = 2), and for small time of observation the degree of freedom number increases up to ␯ = 10. The results of

Figure 1.27

Cumulative RCS distributions of (a) piston-engine and (b) jet airplanes at the X-band. (After: [72].)

60

Radar Characteristics of Targets

freedom degree number determination for three types of airplanes and for a helicopter obtained at the wavelength of 10 cm are presented in Figure 1.28 [73]. It is seen that for a spectrum width of 20 Hz, good agreement between the experimental data and the Rayleigh model is observed for observation time more than 5 seconds. For air target observation over a sector of 10°, the freedom degree number decreases to values from 0.9 to 2.4. The RCS distributions for large land and marine targets, as a rule, coincide well with the Rayleigh model in the area of the most probable RCS values [23, 26]. The distributions tails, characterizing the probability of large RCS values, are usually higher than for the model. However, chi-square distributions with freedom degree number ␯ ≥ 2 can approximate the experimental results. As an illustration, the distributions of the instantaneous values of signal from the output of a phase detector are presented in the Figure 1.29(a) for onshore and inflatable boats. These data were obtained at the 3-cm wavelength and are plotted on a scale linearizing the Gaussian distribution. The amplitude distributions for an inflatable are presented in Figure 1.29(b) on a scale linearizing the Rayleigh law. Similar results are obtained for small marine targets. The amplitude distributions for marine buoys in X-band are closer to the Rician distribution, while for the majority of targets like motor boats, rowboats, spheres, and corner reflectors placed on the sea surface, the amplitude distributions are approximated by Rayleigh and Rician laws in a satisfactory way. As an illustration, the RCS cumulative distribution for an anchored sphere obtained at a 3-cm wavelength for sea state about of two is presented in Figure 1.30 [27]. The solid curves correspond to chisquare distributions with different number of degrees of freedom. It is seen that

Figure 1.28

Freedom degree number for airplanes and helicopter at S-band. (After: [73].)

1.4 Statistical Characteristics of Targets

Figure 1.29

61

(a) The cumulative instantaneous value distributions for inshore and inflatable boats and (b) amplitude distribution for inflatable boats at X-band.

the experimental data satisfactory coincide with the chi-square distribution for ␯ = 5–6. Some RCS distribution peculiarities are observed for simple cone-cylinder bodies. In the majority of situations during experimental investigations, their difference from Rayleigh model was noticed; in particular, the ratio of the power of the fluctuation component to the total power was equal to 0.03–0.19, which is less than the corresponding value for a Rayleigh target [23, 74]. RCS cumulative distributions for a cone-cylinder body with 150-mm diameter and about 500-mm length obtained at X-band are presented in Figure 1.31 [23, 26] (solid lines are the chi-square distributions with different number of freedom degrees). The comparison of the experimental data with the model permits us to assert that with reasonable confidence, the approximation of these data is possible by chi-square distributions with 0.8 ≤ ␯ ≤ 4.

62

Radar Characteristics of Targets

Figure 1.29

(Continued.)

In spite of comparatively poor data for the statistical characteristics of biologic objects, apparently, it is possible to establish as a working hypothesis that the RCS distribution for man, large animals, and birds is lognormal with a standard deviation of 8–10 dB [29, 30, 75] and that the amplitude distribution of insect backscattering is Rayleigh. Consequently, the analysis of RCS experimental distributions for practically all types of targets confirms the validity of their approximation by chi-square distribution with different number of degrees of freedom for solving detection problems. For targets of complex shape, it is possible the use of the standard Swerling models for practically all cases.

1.4.3 Echo Power Spectra

The radar signals scattered from moving objects have Doppler frequencies determined by target velocity and aspect, and these frequencies can reach rather high values. Besides a linear motion, the motion of object can have angular and linear

1.4 Statistical Characteristics of Targets

Figure 1.30

63

Cumulative RCS distributions for anchored sphere at X-band and comparison with chi-square distribution.

oscillations in three planes. As a result of object vibration and oscillatory motion, the individual parts of the object can have different radial velocities with respect to the radar. This phenomenon creates the echo spectrum. For a complex object motion, including linear motion with some velocity V and an oscillating component mV cos (⍀ + ␸ ), where m is the modulation coefficient, the echo is frequency modulated with the spectral components that are equally spaced with respect to the Doppler frequency determined by target velocity and spatial orientation. Acceleration of the object results in spreading of the spectrum. As D. K. Barton showed [75, p. 82], the power spectrum of the echo from a target of complex shape can be presented as the spectrum of a Markov’s process with intensity reduced at frequencies above some defined ⌬F

64

Radar Characteristics of Targets

Figure 1.31

Cumulative RCS distributions for cone-cylinder targets with diameter of 150 mm at X-band.

冋 冉

F − F0 G (F ) = G 0 1 + ⌬F

冊册

2 −1

(1.57)

where G 0 is the spectral density at frequency F 0 determined as F 0 = 2V /␭ , V is the object radial velocity, and ⌬F is the half-width of the spectrum at the halfpower level. Then the spectral density decreases at the rate of 6 dB per octave for | F − F 0 | >> ⌬F. The signal power spectra at the X- and Ka-bands for all air targets including airplanes and helicopters are characterized by comparatively small spectrum width and large value of F 0 due to linear motion. So, for the L-200 aircraft, the effective

65

1.4 Statistical Characteristics of Targets

spectral width at X-band is 20–30 Hz at the −6 dB level, almost independent of the polarization This corresponds to the measured spectra results for piston-engine and jet airplanes of [72], the author of which expressed the normalized autocorrelation function as

␳ (␶ ) = exp (−␶ /␶ 0 )

(1.58)

where ␶ 0 is the correlation interval having a value close to 0.05-second 3-cm wavelength. For all piston-engine and jet airplanes for forward aspects, pronounced peaks in the spectra are observed that result from propeller or compressor vane rotation, as illustrated in Figure 1.32. For lower levels (less than −30 dB), one can expect spectral components with frequencies corresponding to fuselage vibrations. As is shown in [76], the fuselage vibrations for piston-engine and jet airplanes are described by the dependence G (F ) ∼ F −2 and have frequency components of 500– 600 Hz. Those components with vibration amplitude of I max ≥ 2.5 are found in the frequency band less than 40 Hz. In this situation at a wavelength of 8 mm, the phase modulation index is approximately equal to two, which is why the spectral components reach 150–200 Hz (i.e., noticeable spectrum spreading is observed).

Figure 1.32

Power spectrum for L-200 airplane at X-band. (From: [26].  1995 SPIE. Reprinted with permission.)

66

Radar Characteristics of Targets

The echo spectra of helicopters have evident propeller modulation. To demonstrate this phenomenon, the power spectra of echoes from the Russian helicopter Mi-4 is presented in Figure 1.33, obtained at a wavelength of 3 cm for different aspects. For flight directions toward the radar, the power spectra are quite narrow; their width at the −10-dB level does not exceed 300–400 Hz. For increasing range between helicopter and radar, the spectrum spreads significantly. This can be caused by modulation of the echo by the rotating rear propeller. For a hovering helicopter, the spectrum central frequency F 0 shifts into the zero frequency area, and significant spectrum spreading at the level of (1–2) ⭈ 10−6 m2/Hz is observed up to frequencies of 10 kHz. Besides, there are the peaks of propeller modulation that are at 6–10 dB higher than the surrounding average level. The propeller modulation peaks were not observed at a wavelength of 8 mm, and the spectra were still wider, reaching 1 kHz at the −10-dB level. The echo power spectra for land objects are also comparatively narrowband, their width at the −3-dB level lying within the limits of ⌬F /F 0 = 0.06–0.23. The spectrum width values for some land objects at the 3-cm wavelength are presented in Table 1.27.

Figure 1.33

Power spectra of helicopter Mi-4 at X-band. (From: [26].  1995 SPIE. Reprinted with permission.)

1.4 Statistical Characteristics of Targets

67

Table 1.27 Land Target Spectrum Width at Wavelength of 3 cm Object Tank Armored car Heavy artillery tractor Light artillery tractor Truck

Spectrum Width (Hz) at Level −10 dB −20 dB 50 200 50 190 115 400–600 100–300 300–550 50–135 200–450

Examples of power spectra for tank and truck at the wavelength of 8 mm are presented in Figures 1.34 and 1.35. It is seen from Table 1.27 that at low levels of intensity the spectrum width for targets of the first two types (caterpillar objects) is significantly less than for objects having a large number of independently moving parts (wheeled objects). The central frequency F 0 and spectral width ⌬F increase in inverse proportional to the wavelength.

Figure 1.34

Power spectra for tank at X-band.

68

Radar Characteristics of Targets

Figure 1.35

Power spectra of GAZ-63 truck at X-band.

The power spectra for large marine targets caused by pitch, roll, and heave in rough sea are characterized by comparatively small width, not exceeding a few hertz in X-band, and rather large central frequencies. Let us determine the power spectra shape, taking into account that the spectrum of echo G (␻ ) is connected with the spectrum of target heave G z (␻ ) by the following relation [77] G (␻ ) = ␻ 2 ⭈ G z (␻ )

(1.59)

The spectrum of heave is determined by the spectrum of the waves G m (␻ ) through the transfer function K (␻ ) of the linear dynamic system. It is known that a dynamic system like a ship has a rather narrow passband, and its transfer function can be presented as [77]

69

1.4 Statistical Characteristics of Targets

| K (␻ ) | Z =

kZ 2 2 2 2 2 √(1 − ␻ /␻ Z ) + 4␮ Z ␻ /␻ Z

(1.60)

where ␻ Z is the natural frequency of ship oscillations, ␮ Z is the heave decrement, and k Z is the reduction coefficient. Then one can get the following expression for the power spectra of heave G Z (␻ ) =

| K (␻ ) | Z2

⭈ G m (␻ ) =

2 kZ G m (␻ )

(1 − ␻ 2/␻ Z2 )2 + 4␮ Z ␻ 2/␻ Z2

(1.61)

For the majority of ships and other marine objects, especially small ones, the natural frequencies of oscillations lie in the area ␻ Z >> ␻ , then 2 G m (␻ ) G Z (␻ ) ≈ k Z

(1.62)

(i.e., the spectrum of heave is proportional to the wave power spectrum). Taking into consideration that the spectrum of fully developed waves in the gravitational domain can be approximated as

冋 冉 冊册

G m (␻ ) = ␤ g 2␻ −5 exp −0.74

g ␻U

2

(1.63)

where U is the wind velocity, g is the gravitational constant, and ␤ is the parameter, which weakly depends on the frequency varying from 2 ⭈ 10−3 in the low-frequency region to 10−2 in the high frequency region of the gravitational spectrum. Then it is possible to predict from (1.60) and (1.61) that the echo spectrum in the high-frequency region is described by the dependence G (␻ ) ∼ ␻ −3. Taking this into account, we derive the spectrum approximations for marine objects as [23, 26]

冋 |

G (F ) = G 0 1 +

F − F0 ⌬F

|册

n −1

(1.64)

where G 0 is the spectral density at frequency F 0 determined as F 0 = 2V /␭ , ⌬F is the spectrum half-width at the −3 dB level, and V is the radial component of the target velocity in respect to the radar. The spectrum width and power index experimentally obtained for some marine targets at the 3-cm wavelength are presented in Table 1.28 and the spectrum width at wavelengths of 8 mm and 4 mm are shown in Table 1.29 (data were obtained for sea state 1–2). It can be noticed

70

Radar Characteristics of Targets Table 1.28 Spectrum Width and Power Index of Marine Targets at Wavelength of 3 cm

Object Onshore motor boat –Anchored –Moving Message cutter Cruiser yacht Sport yacht Motor boat Rowboat Inflatable boat Windsurf Anchored barrel

Spectrum Width (Hz) at Level (dB) −3 −10 2.0 5.0 7.0 4.0 6.0 3.0 2.0 8.5 7.0 5.0

4.6 10.0 15.0 10.0 12.0 11.0 5.5 20.0 19.0 11.5

Mean Index n 2.6 3.2 2.75 2.6 3.7 1.9 2.6 2.5 2.4 3.0

Table 1.29 Spectrum Width of Marine Targets at Millimeter Band Object Motor boat Boat Inflatable boat Navigation buoy Spherical buoy Spherical buoy

Wavelength (mm) 8 8 8 4 8 4

Spectrum Width (Hz) at Level −3 dB 13.0 8.5 8.0 5.5 0.6–0.8 0.65

that the power degree indices in the expression for target power spectrum usually decrease with decreasing object size or increasing sea state. The power spectra of biological objects have some rather typical peculiarities. The spectra of human echoes are narrowband, their width for the −3-dB level equal to 20–30 Hz and mean Doppler frequency F 0 for the motion is 80–100 Hz at the 3-cm wavelength. The echo amplitude and frequency modulation take a form controlled by motion of arms and legs and is expressed in the appearance of step modulation peaks. As an illustration, the current spectra of moving man and swimming man obtained at a 3-cm wavelength are presented in Figures 1.36 and 1.37. The signals scattered from birds are, as a rule, amplitude modulated ones. The frequency of modulation is inversely proportional to bird size. In [63], it was experimentally determined that the modulation frequency F m (Hz) is connected with the wing length l (mm) as F m l 0.897 = 572

(1.65)

The frequency of wing strokes for the white heron equals to 2–4 Hz, and is about 10 Hz for the swallow, 3–4 Hz for the seagull, and 6–7 Hz for ducks. The wing

71

1.4 Statistical Characteristics of Targets

Figure 1.36

Power spectrum of moving man.

oscillations are nonsinusoidal, and there are a number of harmonic components in the echo. The velocity of birds can be about 15 m/s and more for migration and local flights, corresponding to Doppler frequencies of about 1 kHz at a 3 cm wavelength. The Doppler spectrum width can be determined as [63] ⌬F = 10.9Ad 0.21␭ −1

(1.66)

where A is the angular motion amplitude of wing for stroke and d is the distance from body to forearm end (m) excluding the length of flapping feathers. It is seen from (1.66) that a bird size change by 10 times leads to only 60% spectrum widening. As a rule, these spectra are rather narrowband; their width being not greater than 10–20 Hz at a 3-cm wavelength. The spectrum of the echo from a seagull obtained at X-band is presented in Figure 1.38. Thus, we have presented the experimentally obtained power spectra at microwave and millimeter bands for practically all types of targets and a convenient approximation of spectrum shape by expression of (1.64).

72

Radar Characteristics of Targets

Figure 1.37

Power spectra of swimming man at X-band.

1.5 Surface Influence on the Statistical Characteristics of Radar Targets 1.5.1 Diffuse Scattering Surface Influence on the Statistical Characteristics

In the shortwave part of the centimeter band and, particularly, in the millimeter band, the diffuse component of the electromagnetic field scattered by the surface grows significantly together with the destruction of the specular reflection. In this situation the spatial correlation radii of the reflected diffuse component over the target is larger in most cases than the geometric dimensions of the target [78]. Then the received signal is a product of two terms: the first describes the radar target signal fluctuations in free space and second one describes the fluctuations of the propagation factor. This division was used in some papers [79, 80]. Later we will consider the point target placed over the rough surface that is statistically equivalent (in its RCS distribution functions and power spectra) to the real target. In this case, the surface influence reduces to modulation of the target echo. The signal amplitude and RCS in the point of reception can be presented as

1.5 Surface Influence on the Statistical Characteristics of Radar Targets

Figure 1.38

73

Power spectrum of seagull at X-band.

A*t = A t ⭈ F 2

(1.67)

␴ *t = ␴ t ⭈ F 4

(1.68)

where A t , ␴ t are the amplitude and RCS of the target in free space, respectively, and F is the surface propagation factor. Using the relations for the pdf the product of two random values [80] and assuming that the RCS probability function in free space is described by the standard Swerling models [69, 81], we obtain the expression for the probability function of the normalized RCS as ∞

p I (␰ ) = A 0

冕 冋冉 exp −

0

␰ +␤⭈x x

冊册

dx ⭈ I 0 ( ␥ ⭈ x 1/2 ) x

(1.69a)

74

Radar Characteristics of Targets

for Swerling models 1 and 2, and ∞

p II (␰ ) = 4␰ A 0

冕 冋冉 exp −

2␰ +␤⭈x x

0

冊册

dx ⭈ I 0 ( ␥ ⭈ x 1/2 ) 2 x

(1.69b)

for Swerling models 3 and 4. Here

␰=

␴*i is the target RCS normalized to mean value. ␴* i

A0 =

␤=

冉 冊

1 f2 exp − 02 . 2␳ d 2␳ d 1 2

2␳ d

f0 =

;

␥=

f 02 2

2␳ d

.

√1 + ␳ 0 − 2␳ 0 cos ␪ is the specular reflection coefficient. 2

␳ 0 , ␳ d are the specular and diffuse reflection factors depending on surface roughness, radar wavelength and determined, for example, in [11]. I 0 is the Bessel function of zero order. Calculating (1.69) we obtain finally p I (␰ ) = A 0

p II (␰ ) = 8␰ A 0

冤∑ ∞

␯ −1

∞

∑

␯ =0

冉冊

␥ 2␯ ␰ ⭈ 2␯ 2 ␤ 2 (␯ !)

冉冊

␥ 2␯ ␰ ⭈ 2␯ 2 ␤ 2 (␯ !)

(␯ − 1)/2

␯ /2

K ␯ 冠2 √␰␤ 冡

K ␯ − 1 冠2 √␰␤ 冡 +

冉冊 ␰ ␤

1/2

(1.70a)

⭈ K 1 冠2 √␰␤ 冡

冥

(1.70b) where K (⭈) is the modified Bessel function. Equations (1.70a) and (1.70b) permit us to obtain the moments for the RCS probability functions in the first and second Swerling models as ∞

m l (I ) = A 0 l !

冕 0

exp (−␤ ⭈ x ) ⭈ I 0 (␥ x 1/2 ) x l ⭈ dx

(1.71a)

75

1.5 Surface Influence on the Statistical Characteristics of Radar Targets ∞

−l

m l (II ) = 2 A 0 (l + 1)!

冕

exp (−␤ ⭈ x ) ⭈ I 0 (␥ ⭈ x 1/2 ) x l ⭈ dx

(1.71b)

0

Comparing (1.71a) and (1.71b), it is easily to obtain the dependence connecting the moments for two models in the form m l (II ) = 2−l ⭈ (l + 1) ⭈ m l (I )

(1.72)

Equations (1.71a), (171b), and (1.72) permit us to obtain all l-moments of the normalized target RCS including the mean value, rms, and the skewness and asymmetry for most models. Let us consider some particular cases of the rough surface effect on the radar target statistical characteristics. For a weakly rough surface in the maximum of the interference lobe, when the condition f 0 / ␳ d > 1, is fulfilled, and for a surface with considerable roughness at great ranges, when the diffuse scattering coefficient ␳ d decreases more quickly than the specular reflection coefficient ␳ 0 , the propagation factor density function is close to the Gaussian distribution [78] p (F ) =

冋

1 (F − f 0 )2 exp − 2 2␳ d √2␲ ⭈ ␳ d

册

(1.73)

and for f 0 / ␳ d >> 1 p (F ) = ␦ (F − f 0 )

(1.74)

where ␦ (⭈) is the Dirac function. Then the RCS distribution of the target can be presented as

冉 冊 冉 冊

2␴* 1 p I (␴*t ) = ⭈ exp − t * ␴t ␴ t* p II (␴*t ) =

2␴* 4␴*t exp − t 2 ␴ t* (␴ t* )

(1.75a)

(1.75b)

2 for the first and second Swerling models, respectively, where ␴*t = ␴ t /f 0 . The comparison of (1.75) with the RCS distributions in the standard Swerling model shows that for a weak diffuse surface scattering, the common shape of the RCS distributions is retained and a scale transformation takes the place.

76

Radar Characteristics of Targets

For small values of f 0 / ␳ d << 1, typical for pure diffuse scattering and also in the interference minima of the electromagnetic field (f 0 → 0), the density function of the surface propagation factor has the form

冉 冊

(1.76)

p I (␰ ) = 2B 0 K 0 冠2 √ ␤␰ 冡

(1.77a)

p II (␰ ) = 8␰ B 0 K 0 冠2 √ ␤␰ 冡

(1.77b)

p (F ) =

F

exp − 2

␳d

F2

2

2␳ d

and the RCS distributions equal

The comparison of (1.75) and (1.76) shows the significant difference of the RCS distributions from the Swerling’s models at the last case. For great values of ␰ (i.e., in the band of the intensive spikes of the echo), using the asymptotic form [82] K v (z ) = z→∞

√

␲ ⭈ e −z 2z

(1.78)

one can show that the rate of decrease of the distribution for large values of target RCS in (1.77) is determined by the term exp (−2␤ 1/2␰ 1/2 ) and is less than in the Swerling models for all values of ␤ ≠ 0. An examination of these models was carried out using experimental data on the statistical characteristics of the RCS for some types of the cone-cylindrical targets with a diameter of 100 mm moving over a land surface with a velocity of about 180 m/s. The cumulative functions of the RCS distribution obtained experimentally (curve 1) and derived with use of (1.77) for two values of the diffuse scattering coefficient (curve 2 for ␳ d = 0.2 and curve 3 for ␳ d = 0.4), and the RCS distribution for the first Swerling model (line 4) are presented in Figure 1.39; line 4 describes in the best way the target RCS distribution in free space [83, 84]. It is seen that except in the regions of high confidence, the data coincide with the derived model. The difference between the derived and experimental results in the initial part of the cumulative function can be explained by the use of the approximate method of derivation based on the inclusion of only diffuse echo (i.e., the application of the Gaussian distribution of the diffuse scattering coefficient instead of the Rician distribution) and the large errors of the experimental results in this part of the curve. The RCS distributions for targets over a land surface smoother decrease for the large values of the RCS in comparison with the standard Swerling distributions.

1.5 Surface Influence on the Statistical Characteristics of Radar Targets

Figure 1.39

77

The RCS cumulative functions for cone-cylindrical targets. (From: [8].  2001 SPIE. Reprinted with permission.)

Besides, the increasing of the RCS fluctuations is seen at small ranges when the specular component of the electromagnetic field is absent because of the increasing grazing angle and, as consequence, the increase of the roughness parameter. At large ranges, the specular component of the field increases, and, as result, the RCS distribution is near that for free space. The results of derivation of the RCS distribution quantiles for experimental data and distributions of (1.75) for two values of the diffuse scattering coefficients are presented in Table 1.30. It is seen from Table 1.30 that the surface effect leads to an increase in the intensive spikes of the RCS. Increasing the diffuse scattering coefficient by a factor

78

Radar Characteristics of Targets Table 1.30 The Comparison of the Experimental and Derived Quantiles of RCS Distributions

Conditions Experimental data for free space Derivation for ␳ d = 0.2 Derivation for ␳ d = 0.4 Source: [8].

Distribution Quantiles, dB (m 2 ) for Confidence Level 0.1 0.5 0.9 −13 −15 −24.5 −12 −18 −26 −8 −16 −26

of two leads to RCS growth by 4 dB for confidence level of 0.1. At the same time, the RCS values practically coincide for large confidence levels. Therefore, a diffuse scattering surface leads to RCS distribution transformation of radar targets, especially for regions of large RCS. 1.5.2 Multiple Surface Reflection Influence

Let us estimate the RCS change for low-altitude targets with very small altitudes. The geometry of this problem is presented in Figure 1.40. For echo power reflected by the target in the direction of the radar (the solid lines 1), the effect of the rough surface is absent. For this case, the target is characterized by RCS ␴ t ; for most of the targets, these characteristics are known rather well. Power is also reflected to the surface and back to the radar (dotted lines 2), increasing the target total RCS

␴ ⌺ = ␴ t + ␴ *t

(1.79)

where ␴ * t is the component created by the interaction of the target with the surface (bistatic scattering). One can estimate the power reflection from the target to surface direction using the bistatic RCS of the target in the direction of the surface. The power density at the surface can be determined as

Figure 1.40

The geometry of the problem: (a) the scattering directions and (b) the lighted area.

1.5 Surface Influence on the Statistical Characteristics of Radar Targets

⌸2 =

Pt G t ␴ b (4␲ )2r 2h 2

79

(1.80)

where P t , G t are the transmitter power and antenna gain, respectively. r is the range from the radar to the target. h is the target altitude.

␴ b is the target bistatic RCS. According to Crispin’s theorem [84, 85], the mean value of the bistatic RCS for all target aspects can be determined through the monostatic RCS. The power density at the target equals

⌸3 =

⌸2 ␴ 0 S eff 4␲ h 2

(1.81)

where ␴ 0 = ␴ 0(␥ ) is the surface normalized RCS depending on the incidence angle ␥ and S ef−f is the area of the illuminated surface that effectively reflects in the target direction. The normalized RCS of the sea surface for small incidence angles ␥ strongly depends on the value of this angle. The normalized RCS for incidence angles from 0° to 15°, obtained in [86] at the wavelengths of 3.2 cm and 0.86 cm for different wind velocities, are presented in Table 1.31. The value of the effective incidence angle ␥ eff , determining the incidence angle interval for which the surface effectively reflects the field energy to the target direction can be determined as

Table 1.31 Dependence of the Sea Normalized RCS on the Incidence Angle Wavelength (cm) 0.86

3.2

Source: [8].

Wind Velocity (ms −1 ) 2.5–5.0 5.0–7.5 7.5–10.0 10.0–12.0 2.5–5.0 5.0–7.5 7.5–10.0

0 16 14 13 12 5 3 −1

␴ 0 (dB) for Incidence Angles (degree) 2.5 5.0 7.5 10.0 12.5 11 7 2.5 −5 −13 13 12 11 9 6 12.5 12 11 9.5 8 11.5 10.5 9.7 8 7 3 0.5 −2.0 −5.0 −7.5 2.5 1.5 0.5 −0.5 −2.5 −1.5 −3.0 −4.0 −5.0 −6.5

15.0 — 2 6.5 6.0 −10.5 −4.0 −8.0

80

Radar Characteristics of Targets ␲ /2

冕

␥ eff =

␴ 0(␥ ) ⭈ d␥

0

(1.82)

␴ 0(0)

The derivation results of ␥ eff from the data of Table 1.2 are presented in Table 1.32. It is seen from Table 1.32 that for increasing wind velocity, a broadening of the angle cone takes place, determining the effective reflection of the signal from the surface to the target direction. This area equals to S eff = ␲ (␥ eff ⭈ h )2

(1.83)

The reflected energy density at the radar is also determined by the target bistatic RCS ⌸4 =

⌸3 ␴ b

(1.84)

4␲ ⭈ r 2

Then, taking into consideration (1.80)–(1.84), the received power can be presented as 2

0 P G 2␭ 2 ␴ S eff ␴ b Pr = t 3 4 ⭈ (4␲ ) r (4␲ h 2 )2

(1.85)

This expression is a radar equation for which the mean RCS for reflections from the surface equals to 2

␴ *t =

␴ 0␴ b S eff (4␲ h 2 )2

2

=

2

␴ 0␴ b ␥ eff 16␲ h 2

(1.86)

It is seen from (1.86) that ␴ * t is proportional to the square of the target bistatic RCS. It is inversely proportional to the square of target altitude, and it increases Table 1.32 Dependence of ␥ eff on Wind Velocity Wavelength (cm) 0.86 3.2 Source: [8].

␥ eff (degree), for Wind Velocity (ms −1 ) 2.5–5.0 5.0–7.5 7.5–10.0 10.0–12.5 3.75 8.75 11.1 11.1 6.0 10.5 10.5 —

81

1.5 Surface Influence on the Statistical Characteristics of Radar Targets

for increasing wind velocity because of the increasing effective angle. The derivation results of ␴ * t for some objects at the altitude h = 1m are presented in Table 1.33 (␥ eff values are taken from the Table 1.32). It is seen from Table 1.33 that significant growth of ␴*t is observed for targets having a bistatic RCS considerably exceeding, in the surface direction, the monostatic RCS. For a nonfluctuating target moving over the surface, as shown in Figure 1.40, the reflected signal becomes a fluctuating one because of the surface effect. Let us consider this phenomenon for the movement over the sea. For known dependencies of ␴ 0(␥ ), one can determine the probability function of ␴ 0 and, consequently, ␴ * t , taking in consideration the amplitude modulation of the normalized RCS of the sea, because of the slope angle change of the scattering area caused by large waves. As is known [87], the probability function of the slope angles ␪ of the sea surface can be expressed as p (␪ ) =

冉

1 ␪2 exp − 2 2␴ ␪ √2␲ ␴ ␪

冊

(1.87)

where ␴ ␪ is the rms value of the surface slope angles, which do not exceed ␴ ␪ ≤ 0.1 for sea [87]. Then we obtain using the results of [80] p (␴ * t ) = p [␪ 1 (␴* t )] ⭈

| |

| |

d␪ 1 d␪ 2 + p [ ␪ 2 (␴ * t )] ⭈ d␴ * d␴*t t

(1.88)

Taking into consideration that the ␴ 0(␪ ) dependence at the angular band of interest can be approximated with sufficient accuracy by

␴ 0(␪ ) = ␴ 0 exp (−␣␪ ),

␪≥0

we obtain

Table 1.33 Values of ␴ *t for Different Objects

Type of Target Bullet of caliber 22 Sphere-cone-sphere with top half-angle 15° Cone with top half-angle 15° Source: [8].

Wavelength (cm) 3.2 0.86

␴ b /␴ b (dB) 1.25 7.5

3.2 3.2

6.0 −15.0

␴ t* (dB) for Wind Velocity (ms −1 ) 2.5–5.0 7.5–10.0 −29.2 −30.3 −9.6 −4.3 −12.6 −54.6

−7.3 −49.3

82

Radar Characteristics of Targets

冋 册

1 ␴ 0(␪ ) ␪ = − ln 2 ␴0

(1.89)

Using (1.86) for ␴*t determination, we present it as

␴*t = ␴ ⭈ ␴ 0(␪ ) =

再

␴ ⭈ ␴ 0 exp (−␣␪ 1 ),

␪1 ≥ 0

␴ ⭈ ␴ 0 exp (␣␪ 2 ),

␪2 < 0

(1.90)

where 2

␴=

2

␴ b ⭈ ␥ eff 16␲ h 2

Then

冉 冊 冉 冊

1 ␴*t ␪ 1 = − ln 2 ␴ ⭈ ␴0

␪2 =

(1.91a)

1 ␴*t ln 2 ␴ ⭈ ␴0

(1.91b)

and

| | | |

1 d␪ 1 d␪ 2 = = d␴*t d␴*t ␣␴*t

(1.92)

Taking in consideration (1.91), (1.92), (1.88) can be transformed to the form

p (␴ * t ) =

冤

2 exp − √2␲ ⭈ ␣␴ ␪ ␴ *t

冉

ln2 −

␴ *t ␴ ⭈ ␴0

2␴ ␪ ␣ 2

冊

冥

for 0 ≤ ␴ * t ≤ ␴ ⭈ ␴0 (1.93)

It is seen from (1.93) that the probability density of ␴ * t conforms to a truncated logarithmic Gaussian law. Introducing y = ␴ * t /␴␴ 0 and ␤ = ␣␴ ␪ , we have finally p(y) =

冉

冊

ln2 y 2 exp − , 2␤ 2 √2␲ ␤ y

0≤y≤1

(1.94)

1.5 Surface Influence on the Statistical Characteristics of Radar Targets

83

and the cumulative function

F(y) =

冦

2⌽ 0,

冉 冊

ln y , ␤

y≥0

(1.95)

y<0

where ⌽(⭈) is the Laplace function. The obtained distribution of ␴*t has higher level of tails in comparison with the standard Swerling models. The probabilities of exceeding the median value by an amount y at wavelengths of 3.2 cm and 0.86 cm are presented in Figure 1.41. The analysis of these results shows that the increase of sea echo fluctuations in the millimeter band in comparison with the centimeter band leads to increasing the probability of the ␴ * t value exceeding the median value. For a simple shape target at small altitude the over the surface of a rough sea, the increase in the RCS caused by the reflection from the surface exceeds the monostatic RCS of these targets with 10% confidence.

Figure 1.41

Probability of median value exceeding by y. (From: [8].  2001 SPIE. Reprinted with permission.)

84

Radar Characteristics of Targets

Thus, it is possible to arrive at the following conclusions: •

•

•

The flight of low-altitude targets over a diffuse scattering surface leads to a change in their distribution laws in the region of large values of RCS, in comparison with Swerling’s models. An analogous result gives the electromagnetic field reflection for the path target to surface. The RCS increase and higher level of tails in the RCS distributions are most visible for targets when the bistatic RCS in the surface direction exceeds the monostatic RCS. The effect of RCS increase is observed most strongly for target altitudes of meters over the surface. The differences of the RCS distributions from the Swerling models increase with the radar wavelength reduction.

It is necessary to note that these conclusions can be similarly applied to the land surface.

References [1]

[2]

[3] [4] [5]

[6] [7]

[8] [9] [10]

Kulemin, G. P., and V. B. Razskazovsky, ‘‘Radar Reflections from Explosion and Gas Wake of Operating Engine,’’ IEEE Trans. on Antennas and Propagation, Vol. 45, No. 4, 1997, pp. 731–739. Kulemin, G. P., and V. B. Razskazovsky, ‘‘Centimeter and Millimeter Radio Wave Signal Attenuation in Explosion,’’ IEEE Trans. on Antennas and Propagation, Vol. 45, No. 4, 1997, pp. 740–743. Neuringer, J. L., ‘‘Derivation of the Spectral Density Function of the Energy Scattering from Underdense Turbulent Wake,’’ AIAA Journal, Vol. 7, No. 4, 1969, pp. 728–730. Fox, J., ‘‘Space Correlation Measurements in the Fluctuating Turbulent Wakes Behind Projectiles,’’ AIAA Journal, Vol. 5, No. 2, 1969, pp. 362–368. Velmin, V. A., Yu. A. Medvedev, and B. M. Stepanov, ‘‘Radar Echoes from Explosion Area,’’ Journal of Experimental Techniques and Physics, Vol. 7, December 1968, pp. 455–457 (in Russian). Boynton, F. P., C. B. Ludwig, and A. Thomson, ‘‘Spectral Emissivity of Carbon Particles Clouds in Rocket Exhausts,’’ AIAA Journal, Vol. 6, No. 5, 1968, pp. 116–124. Kulemin, G. P. and V. B. Razskazovsky, ‘‘Radar Backscattering from Sonic Perturbations Caused by Aerodynamic Object Flight,’’ Proc. SPIE Radar Sensor Technology II, Vol. 3, No. 066, June 1997, pp. 194–202. Kulemin, G. P., ‘‘Sea and Land Surface Influence on the Statistical Characteristics of the Low-Altitude Radar Targets,’’ SPIE Proc., Vol. 4, No. 374, 2001, pp. 156–164. Shtager, E. A., Radiowave Scattering on the Complex Objects, Moscow, Russia: Radio i svyaz, 1986 (in Russian). Pavelyev, A. G., ‘‘About Scattering of Electromagnetic Waves by the Rough Surface and Frequency Spectrum of the Scattered Signal,’’ Radiotechnics and Electronics, Vol. 14, No. 11, 1969, pp. 1923–1931 (in Russian).

85

References [11] [12]

[13]

Kulemin, G. P., and V. B. Razskazovsky, Millimeter Radiowave Scattering by Earth’s Surface at Small Angles, Kiev, Russia: Naukova Dumka, 1987 (in Russian). Blore, W. E., ‘‘The Radar Cross-Section of Ogives, Double-Backed Cones, DoubleRounded Cones and Cone-Sphere,’’ IEEE Trans. on Antennas and Propagation, Vol. AP-12, No. 5, 1964, pp. 582–590. Ruck, G. T., Radar Cross-Section Handbook, Vol. 1, New York: Plenum Press, 1970.

[14]

Crispin, J. W., and A. L. Maffett, ‘‘Radar Cross Section Estimation of Complex Shapes,’’ Proc. IEEE, Vol. 53, No. 8, 1965, pp. 972–981.

[15]

Beckmann, P., and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces, London, England: Pergamon Press, 1963; Norwood, MA: Artech House, 1987.

[16]

Crispin, J. W., and K. M. Siegel, Methods of Radar Cross-Section Analysis, New York: Academic Press, 1968.

[17]

Weiss, M. R., ‘‘Numerical Evolution of Geometrical Optics Radar Cross Section,’’ IEEE Trans. on Antennas and Propagation, Vol. AP-17, No. 2, 1969, pp. 229–231.

[18]

Howell, N. A., ‘‘Computerized Ray Optics Method of Calculating Average Value of Radar Cross Section,’’ IEEE Trans. on Antennas and Propagation, Vol. AP-16, No. 5, 1968, pp. 569–574. Crispin, J. W., and A.L. Maffett, ‘‘Radar Cross-Section Estimation of Simple Shapes,’’ Proc. IEEE, Vol. 53, No. 8, 1965, pp. 972–981. Keller, J. B., ‘‘Backscattering from a Finite Cone,’’ IRE Trans. on Antennas and Propagation, Vol. AP-8, No. 2, 1960, pp. 175–182. Dimova, A. I., M. E. Albats, and A. M. Bonch-Bruevich, Radiotechnical Systems, Moscow, Russia: Soviet Radio, 1975, p. 438 (in Russian).

[19] [20] [21] [22]

Skolnik, M. I., ‘‘An Empirical Formula for the Radar Cross Section of Ships at Grazing Angles,’’ IEEE Trans. on Acoustics, Speech, and Signal Processing, Vol. AES-10, No. 2, 1974, p. 292.

[23]

Kulemin, G. P., and V. B. Razskazovsky, The Statistical Characteristics of Radar Targets, Inst. Radiophysics Electr., Kharkov, Ukraine, report ‘‘Shore,’’ 1993, p. 185 (in Russian).

[24]

Kulemin, G. P., et al., ‘‘About Selection Small Marine Targets from Sea Backscattering by Coherent Radar,’’ Signal Processing in Radiotechnic Systems, Aviation Institute, Kharkov, Ukraine, 1988, pp. 88–98 (in Russian).

[25]

Williams, P. D. L., H. D. Cramp, and K. Curtis, ‘‘Experimental Study of the Radar CrossSection of Maritime Targets,’’ Adv. Radar Techn., 1985, pp. 69–83.

[26]

Kulemin, G. P., and V. B. Razskazovsky, ‘‘The Radar Characteristics of Targets at X- and Ka-Bands,’’ Proc. SPIE, Vol. 2,469, 1995, pp. 132–141.

[27]

Kulemin, G. P. and V. B. Razskazovsky, ‘‘The Statistical Characteristics of Radar Targets,’’ Preprint IRE NASU, No. 92-2, Kharkov, Ukraine, 1992, p. 32.

[28]

Skolnik, M. I., Radar Handbook, New York: McGraw-Hill, 1970, pp. 13–27.

[29]

Nebabin, V. G., and V. V. Sergeev, Methods and Technique of Radar Recognition, Moscow: Radio and Communication, 1984, p. 152 (in Russian); trans., Norwood, MA: Artech House, 1995.

[30]

Von, T. P., ‘‘Birds and Insects As Radar Targets: A Review,’’ Proc. IEEE, Vol. 75, No. 2, 1985, pp. 35–63.

[31]

Mueller, E. A., ‘‘Differential Reflectivity of Birds and Insects,’’ Proc. 21st Conf. Radar Meteor., 1983, pp. 465–466.

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[33]

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[53]

Haykin, S. S., C. R. Carter, and M. V. Patriarche, ‘‘Identification of Areas of Bird Clutter and Weather Clutter Using Air Traffic Control Radar,’’ IEE Conf. and Exhib., 1975, pp. 172–173. Edgar, A. K., E. J. Dodsworth, and W. P. Warden, ‘‘The Design of a Modern Surveillance Radar,’’ Int. Conf. Radar: Present and Future, London, England, October 1973, pp. 8–13. Sedov, L. I., Methods of Similarity and Dimensionality in Mechanics, Moscow, Russia: Ed. of Phys.-Math. Lit., 1967 (in Russian). Ohocimsky, D. Y., ‘‘The Derivation of Point Explosion Taking Contrapressure into Account,’’ Proc. of Math. Institute (named after Steklov), Vol. 87, April 1966 (in Russian). Landau, L. D., ‘‘On Shock Waves,’’ USSR Ac. of Sc. Proc., Physics Ser., Vol. 6, January– February 1942, pp. 64–69 (in Russian). Korotkov, P. F., ‘‘On Nonlinear Geometrical Acoustics: Weak Short Waves,’’ Journal of Appl. Math. and Phys., No. 5, 1964, pp. 30–35 (in Russian). Lin, S. C., and J. D. Teare, ‘‘Rate of Ionization Behind Shock Waves in Air: Theoretical Interpretations,’’ Phys. Fluids, Vol. 6, 1963, pp. 355–375. Krinberg, I. A., ‘‘Air Electric Conductivity for Admixtures Presence,’’ Journal of Appl. Math. and Phys., No. 1, 1965, pp. 78–87 (in Russian). Baum, F. A., K. P. Stanyukovich, and B. I. Shehter, Explosion Physics, 1959 (in Russian). Nasilov, D. N., Radiometeorology, Moscow, Russia: Ed. Nauka, 1966 (in Russian). Adushkin, V. V., ‘‘On Shock Wave Forming and Explosion Products Flying Away,’’ J. Appl. Match. and Phys., No. 5, 1963, pp. 107–120 (in Russian). Tatarski, V. I., Wave Propagation in a Turbulent Medium, New York: McGraw-Hill, 1961. Kondratiev, K. L., Ray Thermal Exchange in Atmosphere, Leningrad: Gidrometeoizdat Ed., 1956 (in Russian). Britt, C. O., C. W. Tolbert, and A. W. Straiton, ‘‘Radio Wave Absorption of Several Gases in the 100 to 117 kMc/s Frequency Range,’’ J. Res. NBS., Vol. 6-D, January 1961, pp. 15–18. Bazhenova, T. V., Shock Waves in Real Gases, Moscow, Russia: Nauka Ed., 1968 (in Russian). Boynton, F. P., C. B. Ludwig, and A. Thomson, ‘‘Spectral Emissivity of Carbon Particles Clouds in Rocket Exhausts,’’ AIAA Journal, Vol. 6, No. 5, 1968, pp. 116–124. Van de Hulst, H. C., Light Scattering by Small Particles, New York: John Wiley, 1957. Aleksandrov, Ye. N., et al., ‘‘Solid Explosion Substances Crushing in Shock Wave,’’ Physics of Burning and Explosion, No. 3, 1968, pp. 84–89 (in Russian). Still, V. R. and G. N. Gloss, ‘‘Emissivity of Dispersed Carbon Particles,’’ J. Opt. Soc., Vol. 50, February 1960, pp. 186–191. Von Gierre, H. E., ‘‘Effects of Sonic Boom on People: Review and Outlook,’’ J. Acoustic Soc. Amer., Part 2, Vol. 39, No. 5, 1966, pp. S43–S50. Rizer, Y. P., ‘‘Sonic-Boom Propagation in Inhomogeneous Atmosphere to Side of Density Decreasing,’’ J. Applied Mechanics and Technical Physics, No. 4, 1964, pp. 269–274 (in Russian). Hubbard, H. H., ‘‘Nature of the Sonic-Boom Problem,’’ J. Acoustic Soc. Amer., Part 2, Vol. 39, 1966, pp. S1–S9.

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[66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76]

87 Witham, G. B., ‘‘On the Propagation of Weak Shock Waves,’’ J. Fluid. Mech., Vol. 1, No. 9, 1952, pp. 290–318. Kane, E. J., ‘‘Some Effects on the Nonuniform Atmosphere on the Propagation of Sonic Booms,’’ J. Acoust. Soc. Amer., Part 2, Vol. 39, No. 5, 1966, pp. S26–S30. Plotkin, K. J., and A. R. George, ‘‘Propagation of Weak Shock Waves Through Turbulence,’’ J. Fluid Mech., Part 3, Vol. 54, 1972, pp. 449–467. Lamley, J., and H. Panofsky, The Structure of Atmospheric Turbulence, New York: Interscience Publishers, 1964. Maglieri, D. J., ‘‘Some Effects of Airplane Operations and the Atmosphere on the SonicBoom Signatures,’’ 2nd Conf. on Sonic Boom Res., NASA, 1968, pp. 19–23. Landau, L. D., and Ye. M. Livshits, Mechanics of Continious Media, Moscow, Russia: Gostehizdat, 1954 (in Russian). Blohintsev, D. N., Acoustics of Inhomogeneous Moving Medium, Moscow, Russia: Fizmatgiz Ed., 1981 (in Russian). Melnikov, B. P., ‘‘Noise Caused by Tu-124 Airplane Take-Off and Landing,’’ Acoustic J., Vol. 1, No. 2, 1965, pp. 138–146 (in Russian). Lane, J. A., ‘‘Radar Echoes from Tropospheric Layers by Incoherent Backscatter,’’ Electronic Letters, Vol. 3, No. 4, 1967, pp. 173–174. Shlihting, G., The Origin of Turbulence, Moscow, Russia: Foreign Literature Edit., 1962 (in Russian). Sato, H., ‘‘The Stability and Transition of a Two-Dimensional Jet,’’ J. Fluid Mech., Part 3, Vol. 7, 1960, pp. 321–329. Isatajev, S. I., and S. B. Tarasov, ‘‘On Acoustic Field Influence on Wake Directed Along Stream Axis,’’ USSR Acad. Sci. Proc., Fluid and Gas Mechanics, No. 2, 1971, pp. 164–167 (in Russian). Crow, S. C., and F. H. Champagne, ‘‘Orderly Structure in Jet Turbulence,’’ J. Fluid Mech., Part 3, Vol. 48, 1971, pp. 547–591. Shade, H., ‘‘Contribution to the Nonlinear Stability Theory of Inviscid Shear Layers,’’ Phys. Fluids, Vol. 7, No. 5, 1964, pp. 623–628. Delano, R. H., ‘‘Theory of Target Glint or Angular Scintillation in Radar Tracking,’’ Proc. IRE, Vol. 41, No. 3, 1953, pp. 61–67. Swerling, P., ‘‘Probability of Detection for Fluctuating Targets,’’ IEEE Trans. Inf. Theory, IT-6, No. 4, 1960, pp. 269–308. Mayer, H. A., and D. P. Meyer, ‘‘Chi-Square Target Models of Low Degrees of Freedom,’’ IEEE Trans. Aerosp. Electr. Syst., Vol. AES-11, No. 5, 1975, pp. 694–707. Mayer, H. A. and D. P. Meyer, Radar Target Detection, New York: Academic Press, 1973. Moll, P. L., ‘‘On the Radar Echo from Aircraft,’’ IEEE Trans. Aerosp. Electr. Syst., Vol. AES-3, No. 3, 1967, pp. 574–577. Muchmore, R. B., ‘‘Aircraft Scintillation Spectra,’’ IEEE Trans. Antennas and Propagation, AP-8, No. 2, 1960, pp. 201–212. Wilson, I. D., ‘‘Probability of Detecting Aircraft Targets,’’ IEEE Trans. Aerosp. Electr. Syst., AES-8, No. 6, 1972, pp. 757–762. Barton, D. K., Radar System Analysis, Dedham, MA: Artech House, 1979. Krendell, C., (Translat. Ed.), Random Oscillations, Moscow, Russia: Mir, 1967, p. 361 (in Russian).

88

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[84] [85] [86] [87]

Nogid, L. M., Ship Stability and Its Behaviour at Rough Sea, Leningrad, Russia: Sudostroenie, 1967, p. 240 (in Russian). Barton, D. K. and H. R. Ward, Handbook of Radar Measurement, Englewood Cliffs, NJ: Prentice Hall, 1969; Dedham, MA: Artech House, 1984. Eaves, J. L., and E. K. Reedy, Principles of Modern Radar, New York: Van Nostrand Reinhold, 1987, p. 712. Levin, B. R., Theoretical Foundations of the Statistic Radioelectronics, Moscow, Russia: Soviet Radio, 1969 (in Russian). Berkowitz, R. S., (ed.), Modern Radar, New York: John Wiley, 1968. Gradshtein, I. S., and I. M. Ridzik, Tables of Integrals, Sums, and Products, Moscow: Fizmatgiz, 1963 (in Russian); trans. New York: Academic Press, 1965. Kulemin, G. P. and V. B. Razskazovsky, ‘‘Complex Effects of Clutter, Weather and Battlefield Conditions on the Target Detection in Millimeter-Wave Radars,’’ Proc. SPIE, Vol. 2,222, 1994, pp. 862–871. Kelly, R. E., ‘‘On the Derivation of Bistatic RCS from Monostatic Measurements,’’ Proc. IEEE, Vol. 53, No. 8, 1965, pp. 983–988. Chernyak, V. S., Fundamentals of Multisite Radar Systems, Amsterdam, the Netherlands: Gordon and Breach, 1998. Skolnik, M. I., Radar Handbook, New York: McGraw-Hill, 1970. Cox, C., and W. Munk, ‘‘Statistics of the Sea Surface Derived from Sun Glitter,’’ J. Mar. Res., Vol. 13, 1954, pp. 198–227.

CHAPTER 2

Land Backscattering 2.1 Classification and Physical Characteristics of Land Classification of land for backscattering prediction must take into account the presence or absence of vegetation; for a surface without vegetation, the roughness is also an essential additional parameter, depending on the wavelength and grazing angle. According the Rayleigh criterion, the surface can be considered rough if the rms height deviation ␴ h meets the criterion ␴ h > (␭ /8) sin ␺ [1], where ␭ is the wavelength and ␺ is the grazing angle. Among surfaces without vegetation, quasismooth surfaces (e.g., a road with concrete or asphalt paving, rocks, salt-marshes, or dry salt lakes) are distinguished, to which a simple mathematical model can be applied. For estimating scattering properties of such surfaces, the essential parameters are the statistical characteristics of the roughness and the complex dielectric constant. The real and imaginary components ⑀ 1 and ⑀ 2 of the dielectric constant for concrete and asphalt in the MMW band are shown in Table 2.1. The statistical characteristics (rms height, correlation functions, spatial correlation radii, and angle slope deviation) of various surfaces are shown in Table 2.2. Surfaces without vegetation are in a separate group among rough surfaces. Their scattering properties are also determined by the complex dielectric constant and height deviation. The complex dielectric constant data for various surface types in the MMW band are shown in Table 2.3. The numerous experimental data sets on the dielectric constant of agricultural fields obtained in [4, 5] support the following conclusions. The frequency behavior of the dielectric constant for wet soils has the following tendencies: ⑀ ′ decreases and ⑀ ″ increases with increasing frequency from 4 GHz to Table 2.1 Dielectric Constants of Concrete and Asphalt Wavelength (mm) 30.0 8.6 2.2 Source: [2, 3].

Concrete ⑀′ ⑀″ 6.5 1.5 5.5 0.5 5.55 0.36

Asphalt ⑀′ ⑀″ 4.3 0.1 2.5 0.6 2.25 0.18

89

90

Land Backscattering Table 2.2 The Statistical Characteristics of Various Surfaces Cover Type Concrete Asphalt Sand Snow Soil harrowed Source: [2–4].

␴ h , cm

␳ (r )

␳ 0 , cm

1.6 ⭈ 10−2 0.4 ⭈ 10−1 0.2–0.6 0.1–0.3 2.0

√␥ 2

exp (−7r ) (1 + 2r 2 )−3/2 — — —

0.14 0.22 0.6–2.5 0.2–200 10.0–80.0

0.14 0.3 0.1–0.4 <0.2 0.5

Table 2.3 Complex Dielectric Constant for Some Covers at Wavelengths of 8 and 2 mm Cover Soil Soil (clay) Sand Sand Snow Ice Brick (red) Brick Pine board Source: [3].

Wavelength of 8 mm ⑀′ ⑀″ 3.0 0.4 7.0 3.5 3.0 0.56 7.0 4.4 2.0 0.4 ⭈ 10−2 3.1 0.7 ⭈ 10−2 — — — — — —

Wavelength of 2 mm ⑀′ ⑀″ — — 2.5 9.4 ⭈ 10−2 2.5 6.2 ⭈ 10−2 — — 1.4 10−2 — — 3.2 1.1 ⭈ 10−1 3.3 1.4 ⭈ 10−1 2.0 8 ⭈ 10−2

18 GHz. These dependences for a loam field, obtained for soil moisture from 0.02 g/m3 to 0.37 g/m3, are shown in Figure 2.1. The dependence of the dielectric constant on the volumetric moisture, measured at 10 GHz and 18 GHz, show that ⑀ ′ and ⑀ ″ depend very weakly on the soil type. As an illustration, Figure 2.2 shows the soil moisture dependence for five fields with different contents of sand, clay, and silt. It is seen that the soil content has insignificant influence. The roughness parameters of agricultural fields are determined by the means of tillage and change over wide limits. As an illustration, the summary of roughness parameters is presented in Table 2.4 for four fields with different tillage. The dielectric properties of snow cover play a significant role in backscattering model development and explanation of results. Much experimental data have been gathered for snow dielectric characteristics covering the frequency band of 0.8–37 GHz [6–9]. Models have been developed in which snow is represented as a two-component or three-component mixture [10, 11]. The model replaces the inhomogeneous medium by a homogeneous one with an effective dielectric constant ⑀ . It is possible to determine the dielectric constant of the mixture with rather simple models if the inclusions in the heterogeneous medium are small in comparison with the wavelength (permitting consideration only of the absorption while neglecting the scattering losses) and their shape is known.

2.1 Classification and Physical Characteristics of Land

91

Figure 2.1 Soil dielectric constant dependence on frequency. (From: [4].  1985 IEEE. Reprinted with permission.)

Figure 2.2 Soil dielectric constant dependence on soil moisture for five fields at frequencies of 10.0 GHz and 18.0 GHz. (From: [4].  1985 IEEE. Reprinted with permission.)

92

Land Backscattering Table 2.4 Summary of Roughness Parameters Type rms Height (cm) Field 1 0.40 Field 2 0.32 Field 3 1.12 Field 4 3.02 Source: [5].

Correlation Length (cm) 8.4 9.9 8.4 8.8

rms Slope 0.048 0.032 0.133 0.485

For dry snow (i.e., a mixture of air and ice) the real part of the dielectric constant in the microwave band does not depend on temperature or frequency but is a function of snow density. This dependence can be represented as [11]

⑀ ds ′ =

再

1 + 1.9␳ s

for ␳ s ≤ 0.5 g/cm3

0.51 + 2.88␳ s

for ␳ s > 0.5 g/cm3

(2.1)

where ␳ s is the snow density, which under natural conditions rarely exceeds 0.5 g/cm3. The experimental results obtained in the frequency band 0.8–37 GHz are described by (2.1) very well, as is shown in Figure 2.3. The imaginary part of the snow dielectric constant depends on frequency and temperature in a rather complex way. The dependence of the loss tangent on frequency is shown in Figure 2.4 [12] for snow density of 0.45 g/cm3 [9]. The dielectric constant of wet snow increases sharply because of the influence of water in its liquid phase. The content of liquid water in snow is usually described by a volumetric moisture m v that is the ratio of water volume to total snow volume. There are several models of wet snow dielectric properties. For the millimeter band, the modified Debye model gives the highest accuracy for the dielectric constant [13]. Numerous experiments for wet snow [12, 14] support the following conclusions. The real part ⑀ ′ of the dielectric constant increases with an increase in the volumetric moisture from 1.4 for dry snow to 3.3 for m v = 12% at a frequency of 2 GHz. It decreases with an increase in frequency to 1.35 for the millimeter band. With increasing frequency the real part of dielectric constant decreases. This is illustrated by Figure 2.5(a) obtained at a frequency of 10.0 GHz for moisture change from zero to 12%. The imaginary part ⑀ ″ for wet snow increases with increasing volumetric moisture—Figure 2.5(b). The maximal values are found at approximately the relaxation frequency, beyond which it decreases with increasing frequency. Land territories with vegetation form the third group. The complex dielectric constants of vegetation media are determined by their biometrical indices (e.g., vegetation volumetric density and square cover degree) and by the relative moisture content in the cellular tissue. The experimental data on the dielectric constant for corn leaves in the band 1.5–8.0 GHz, for volumetric moisture from zero to 0.8 g/cm3, are shown in Figure

2.1 Classification and Physical Characteristics of Land

93

Figure 2.3 Dry snow dielectric constant versus snow density. (From: [12].  1986 Artech House, Inc. Reprinted with permission.)

2.6 [12]. The dielectric constant for wheat grains as a function of the weighted moisture is shown in Figure 2.7. The vegetation biometrical indices change continuously as a function of vegetation growth. The vegetation moisture can change as a function of season and weather conditions. The biometrical indices of some crops and grass for different periods of growth are shown in Table 2.5 [15]. In [3], the notion of the effective complex dielectric constant of a medium is introduced for investigations of propagation peculiarities in vegetation

⑀ eff = ⑀ eff (1 − i tan ␦ eff )

(2.2)

94

Land Backscattering

Figure 2.4 Dry snow loss tangent versus frequency. (From: [9].  1986 IEEE. Reprinted with permission.)

where ⑀ eff and tan ␦ eff are the dielectric constant and loss tangent, respectively. The use of this parameter permits derivation of the reflection coefficients from the parameters of homogeneous dielectric media [16]. In [17], a simple expression for effective complex dielectric constant is obtained for homogeneous vegetation covers

⑀ eff = 1 + Bd ⑀ w M

(2.3)

where d is the weight of scattering elements in a unit volume, B is an empirical parameter equal to 0.3 for coniferous forest and to 0.6 for deciduous forest, ⑀ w is the dielectric constant of water at the radar frequency, and M is the relative weighted moisture content in the vegetation. It has been established that huge tracts of forest are homogeneous scatterers, just as are grass, cereals, and some types of bushes—all having an inhomogeneity of different degree. The averaged parameter values for coniferous and deciduous forests are shown in Table 2.6 [18]. Data on dielectric properties of vegetation are insufficient for general model development of land surfaces with vegetation.

2.2 State of the Theory

95

Figure 2.5 Frequency dependence (a) real and (b) imaginary parts of dielectric constant of wet snow for different water contents. (From: [12].  1986 Artech House, Inc. Reprinted with permission.)

2.2 State of the Theory 2.2.1 RCS Models

Radars for detection of on-land and low-altitude targets are subject to land backscattering, characteristics of which depend on the land surface type, the vegetation state, the season, the radar operating frequency, and the wind velocity.

96

Land Backscattering

Figure 2.6 Dielectric constant versus volumetric water content for corn leaves. (From: [12].  1986 Artech House, Inc. Reprinted with permission.)

There are significant difficulties in the development of theoretical models for land backscattering because of the variety of land types. The major theoretical works can be divided into two groups. In the first, the land surface is considered to consist of elementary scatterers (e.g., spheres and cylinders), for which the scattering properties are known and for which parameters are chosen to secure satisfactory agreement with experimental results. These models usually predict quite well the angular dependence of the normalized RCS, which is a measure of the scattered signal intensity, and the scattered signal statistical characteristics. The second group of works [1–3, 19–21] requires the choice of individual, exactly formulated backscattering parameters; however, these choices give large errors in derivations of scattered signals because of the idealization of the surface. The simplified models of first group are shown in Table 2.7 [22–24]. Here the grazing angle, related to the incidence angle ␪ by ␺ = ␲ /2 − ␪ , is used as a parameter. In the model proposed in [2] for grassy terrain, it is assumed that the scattering comes from long thin cylinders with considerable loss, permitting multiple scattering

97

2.2 State of the Theory

Figure 2.7 Dielectric constant versus water content for grain: (a) real and (b) imaginary parts. (From: [12].  1986 Artech House, Inc. Reprinted with permission.)

Table 2.5 Biometrical Indices of Crops

Type Grass Grass Alfalfa Sugar beet Wheat Source: [15].

Weight (g/cm 2 ) 0.12 0.16 0.077 — 0.25

Height (cm) 10 17 18 55 110

Cover Degree (%) 100 95 90 95 25

Moisture (%) 86 85 87 — 36

Steam Amount (cm −2 ) 1.5 0.6 0.64 0.02 0.043

Steam Section (cm 2 ) 0.008 0.016 — — 0.096

98

Land Backscattering Table 2.6 Experimental Values of Model Parameters for Forests Forest Type Coniferous forest, thick Coniferous forest, thin Deciduous forest, thick Deciduous forest, thin Source: [18].

Parameters d M 10−4 0.4 5 ⭈ 10−3 0.6 10−4 0.6 5 ⭈ 10−3 0.8

Table 2.7 Simplified Models of Land Backscattering Model Diffuse scattering Diffuse scattering (Lambert law) Ament’s model Facet model Model of half-cylinders Model of half-spheres Vegetation model as cylinders

Angular Dependence of Normalized RCS sin2 ␺ sin ␺ ⌿ exp [−cot2 (␺ /2s 2 )] ␺ 4, ␺ 2 ␺ 4, ␺ 2 sin2 ␺

RCS Dependence on Wavelength — — ␭ −2 ␭ −2, ␭ −6 ␭ −2, ␭ 0, ␭ −3, ␭ −1 ␭ −2, ␭ 0, ␭ −4, ␭ −1 —

Reference — — — [22] [23] [24] [2]

effects to be neglected. For small incidence angles close to normal incidence (grazing angle ␺ → 90° ), models have been developed to satisfactorily explain the experimental data. For example, in [25] a cover cloud model is proposed in which a cloud of identical particles with volumetric homogeneous distribution represents the vegetation. For simplicity in the cover cloud model, the approach of single scattering is used and the contributions of all particles are summed, taking into consideration the double attenuation in the cover layer between its surface and the particle. This model satisfactorily describes the experimental results in the band 8.6–17.0 GHz for incidence angles less than 70°. In [26], a dielectric layer model is proposed, in which the vegetation is represented by a homogeneous dielectric medium. This model is satisfactory at wavelengths from 1.65–3.75 cm for incidence angles close to nadir. A great number of discs with different spatial distribution is used as a model of forest foliage in [27]. Attempts have been made to formulate models of the second group via exact solutions to the electrodynamic backscattering problem, using measured surface parameters as initial data for derivation of the scattered signal. These models assume that all land surfaces can be divided into two classes: those with and those without vegetation. The surface without vegetation has a significant additional dependence on the degree of roughness. According to the Rayleigh criterion, the surface is rough if the rms deviation from the average surface is ␴ h > (␭ /8) sin ␺ . Otherwise it can be described as quasi-smooth.

99

2.2 State of the Theory

Theoretical models are based on methods of solving the problem of scattering by a statistically rough surface. Rather accurate models of backscattering based on Kirchhoff’s or small perturbation methods are strictly applicable only for quasismooth surfaces without vegetation (e.g., concrete or asphalt). For these surface models, it is sufficient for estimation of RCS to know the surface electrophysical characteristics such as the dielectric constant and surface roughness. For Kirchhoff’s method, it is assumed that for sufficiently gently sloping surfaces with the radii of curvature considerably greater than the wavelength, limiting conditions can be represented as those of the plane facets of which this surface consists. The problem is solved by introduction of local Fresnel coefficients followed by integration along the surface currents. The vector form of Green’s theorem is used to obtain the full scattering matrix [28]. For a strongly rough surface, the normalized RCS can be represented as in [19]

冉

2

␴0 =

Rf0

⭈ cos2 ␤ 0 ⭈ exp −

sin4 ␺

cot2 ␺ tan2 ␤ 0

冊

(2.4)

where tan ␤ 0 = 2␴ h l and l is the spatial correlation radius of surface roughness. Consequently, the value of ␤ 0 can be interpreted as the mean slope of the rough surface. The value of R f 0 is the Fresnel reflection coefficient of the plane surface, a function of its electrophysical characteristics. The use of reflection coefficients for the plane surface does not permit us to take fully into account the polarization effects on the scattered signal. The determination of the scattered electromagnetic field in the small perturbation method is based on the following assumptions, applicable to quasi-smooth surfaces. Let the surface height deviation be set as z = f (x , y )

(2.5)

Let us choose the plane z = 0 such that (2.5) describes the deviation from mean height z = 0. Then the small perturbation method is applied for small gently sloping roughness

| k␴ h sin ␺ | << 1

|

|

df (x , y ) << 1; dx

|

(2.6a)

|

df (x , y ) << 1 dy

(2.6b)

The normalized RCS of a rough surface with complex dielectric constant ⑀˙ can be expressed as in [28]

100

Land Backscattering 0 ␴ HH = 4␲ k 4 sin4 ␺ | ␣ HH | S (␬ 0 ) 2

(2.7)

0 ␴ VV = 4␲ k 4 sin4 ␺ | ␣ VV | S (␬ 0 ) 2

where k = 2␲ /␭ is the wavenumber; and S(␬ 0 ) is the spectral density of surface roughness with a wavenumber

␬0 =

冉 冊 4␲ ␭

cos ␺

(2.8)

The physical sense of (2.8) is that the backscattering takes place for the spectral component with a period one-half the radar wavelength. The Fresnel coefficients in (2.7) are determined by the surface dielectric properties and have the form

␣ HH =

␣ VV =

⑀˙ − 1

冠sin ␺ + √⑀˙ − cos2 ␺ 冡2 (⑀˙ − 1) [⑀˙ (1 + cos2 ␺ ) − cos2 ␺ ]

冠⑀˙ sin ␺ + √⑀˙ − cos2 ␺ 冡2

␣ HV = ␣ VH = 0

(2.9a)

(2.9b) (2.9c)

The absence of scattered signal depolarization is inherent in estimation of the normalized RCS in the small perturbation method characterized by (2.9c). This contradicts experimental results even for quasi-smooth surfaces. The coefficients of ␣ HH , ␣ VV depend very weakly on wavelength for small grazing angles, their change being not greater than 1–3 dB in the band 10–140 GHz. From (2.7) we see a strong dependence of normalized RCS on wavelength (inversely proportional to the fourth power of wavelength) and on grazing angle (␴ 0 ∼ sin4 ␺ ). The normalized RCS for vertical polarization is greater than that for horizontal by 8–10 dB. A comparison of derived and experimental results for quasi-smooth concrete or asphalt surfaces shows that the model (2.7) does not apply for these surfaces in the shortwave part of millimeter band because the surface becomes strongly rough. A comparative analysis of the two latter models leads to the conclusion that normalized RCS estimations using Kirchhoff’s method can be used for rather small incidence angles ␪ near nadir, while estimations using the small perturbation method can be used for incidence angles ␪ > 30–40°, but neither applies at extremely small grazing angles when the influence of shadowing is significant.

101

2.2 State of the Theory

This is why these models are not applicable for such surfaces as rough ones without vegetation or those with vegetation. The main way to derive general models in these cases is the development of empirical models based on experimental results.

2.2.2 Power Spectrum Model

The backscattering power spectra of land surfaces with vegetation are determined by the wind-induced motion of scatterers as the radar scans its volume and/or by radar platform motion. For identical radar transmitting and receiving antennas or for a single combined antenna, the signal at the receiver input from the j th scatterer in the far zone can be represented as u j (t ) ⬀

√P t ␴ j 2

rj

∞

G (␪ ) =

冕

⭈ G (␪ j ) ⭈ exp [i (␻ 0 t − kr j − ␸ j )]

(2.10)

f (x ) ⭈ exp (ikx sin ␪ ) dx

−∞

where P t is the transmitter power, ␴ j is the RCS of the j th scatterer, ␸ j is the phase of the signal from the j th scatterer, ␻ 0 is the radar frequency, k = 2␲ /␭ is the wavenumber, r j is the range to j th scatterer, G (␪ ) is the antenna voltage pattern, and f (x ) is the electromagnetic field distribution at antenna aperture. The variables in (2.10) can be divided into two groups. The first group consists of the values r j and ␪ j , which change with radar platform motion and antenna scanning, and which can be functionally expressed through the current antenna position and time. The variables of the second group, ␴ j and ␸ j , characterizing the scatterer state at the present moment of time, are random functions of time in most cases and depend on the scatterer number. If the extent of each scatterer is small in comparison to its range, (2.10) can be written as the product of two factors, each containing the variables of one of these groups. Consequently, the scattered signal spectrum is the convolution of two spectra: The first describes the signal fluctuations resulting from scatterer motions and the second results from antenna scan and radar platform motion. The conclusions for a single scatterer can be generalized for the sum of scatterers. Assuming that all scatterers are in the far zone, the signal at the receiver input is U (t ) ⬀

∞

∑

j =1

1 2

rj

G (␪ j , t ) √␴ j (t ) ⭈ exp {i [␻ 0 t − ␸ j (t )]}

(2.11)

102

Land Backscattering

Then the autocorrelation function can be found as 〈 U (t ) U* (t ) 〉 ⬀

∑∑〈 j′

j

1 2 2 rj rj′

√␴ j (t ) ␴ j ′ (t ′ ) ⭈ G (␪ j , t ) G (␪ j ′ , t ′ )

× exp {i [␻ 0 (t − t ′ ) − ␸ j (t ) + ␸ j*′ (t ′ )]}

(2.12)

〉

Here the sign * denotes the complex conjugate value. Assuming that the scatterers are statistically independent and the phases of scattering are distributed uniformly in [0, 2␲ ], (2.12) can be written in form 〈 U (t ) U* (t ′ ) 〉 ⬀

∞

∑

j =1

1 4

rj

√␴ j (t ) ␴ j (t ′ )

(2.13)

⭈ G (␪ j , t ) G (␪ j , t ′ ) exp {i [␻ 0 (t − t ′ ) − ␸ j (t ) + ␸ j (t ′ )]} Taking into consideration the independence of scatterer fluctuations and antenna scanning, this expression can be transformed to 〈 U (t ) U* (t ′ ) 〉 ⬀

∑ 〈 A j (t ) A j (t ′ ) exp {−[␸ j (t ) − ␸ j (t ′ )]} exp [i␻ 0 (t − t ″ )]〉 ∞

(2.14)

j =1

× 〈 G (␪ j , t ) G (␪ j , t ′ ) 〉 where A j (t ) ⬀

2 √␴ j (t ) ⁄ r j is the amplitude of the j th scatterer signal.

Let us note that R sj (␶ ) = 〈 A j (t ) A j (t ′ ) exp {i [␻ 0 (t − t ′ ) − ␸ j (t ) + ␸ j (t ′ )]} 〉

(2.15)

R Gj (␶ ) = 〈 G (␪ j , t ) G (␪ j , t ′ ) 〉

(2.16)

The expression (2.15) represents the autocorrelation function of the signal from the j th scatterer and (2.16) characterizes the antenna scanning law. Substituting (2.14) and (2.15) into (2.14), we obtain 〈 U (t ) U * (t ′ ) 〉 ⬀

∑ R sj (␶ ) R Gj (␶ )

(2.17)

j

The power spectrum is the Fourier transformation of the autocorrelation function, and hence for our case it is the convolution of the scatterer fluctuation spectrum and antenna-scanning spectrum. This leads to the conclusion that for

103

2.2 State of the Theory

determination of the spectrum of land backscattering caused by antenna scanning and radar platform motion, it is sufficient to find separately the spectra of scatterer fluctuation and of antenna motion (or the electromagnetic field modulation across its aperture). Let us consider the spectrum model formed only by the motions of scatterers under the influence of wind or other reasons. The scattering surface can be represented as a linear system with a transient response K (t ) = K 0 + ⌬K (t )

(2.18)

where K 0 characterizes the reflectors that are stable in time (e.g., buildings, bare soil, or rocks) and the second term characterizes the fluctuating scatterers (e.g., grass, branches, or leaves of trees). Then the autocorrelation function of this system can be represented as 2

g ( ␶ ) = K 0 + ␳ (␶ )

(2.19)

If the continuous signal is u (t ) = A cos ␻ 0 t at the system input, the autocorrelation function at the system output (the scattered signal) will have the form A2 2 R out (␶ ) = R in (␶ ) ⭈ g (␶ ) = [K 0 + ␳ (␶ )] cos ␻ 0 ␶ 2

(2.20)

The power spectrum is the Fourier-transform of autocorrelation function and can be written as ∞

G 1 (␻ ) =

冕

R out (␶ ) ⭈ e −j␻␶ d␶ = a 2␦ (␻ − ␻ 0 ) + G (␻ )

(2.21)

−∞

where ␦ (␻ − ␻ 0 ) is the Dirac function, and a 2 is the ratio of stable to fluctuating components of scattered power (RCS). It is seen that the total power spectrum of the scattered signal consists of one component that is produced by scattering from stable reflectors and another produced by moving scatterers. The first term can considerably exceed the second for terrains without vegetation. Let us consider the power spectra of the surface model, for which the scatterers of branch, leaf, and grass types move under the influence of wind. The basis of this model is the derivation technique developed by G. S. Gorelik [29, 30] for radiowave scattering problems with moving inhomogeneities. Assume that the

104

Land Backscattering

scatterer velocity autocorrelation function reproduces with some scale correction that of the wind velocity. Assuming for simplicity that the signal amplitude from the j th scatterer is unity, one can obtain from (2.9) the expression for signal phase of j th scatterer

␸ j = 2kr j (t ) + ␸ 0

(2.22)

where r j (t ) is the range from the radar to the j th scatterer, and ␸ 0 is the initial phase. From Figure 2.8 we have r j = r + x j sin ␪

(2.23)

where x j is the scatterer displacement relative to its initial position. Without violating generality, one can choose the initial phase ␸ 0 such that

␸ j = 2kx j (t ) sin ␪

(2.24)

Assume that the velocities of the scatterers are a stationary random process, mutually independent, and have identical statistical properties and zero mean values. For a displacement velocity of the j th scatterer u j (t ), the displacement during time ␶ is t +␶

(⌬x j )␶ = x j (t + ␶ ) − x j (t ) =

冕 t

Figure 2.8 Geometry of derivation.

u j (s ) ds

(2.25)

105

2.2 State of the Theory

This value is stationary by the previous assumptions, and the phase difference has the analogous properties. The autocorrelation function is R (␶ ) = 〈 u (t ) ⭈ u (t + ␶ ) 〉 =

∑ ∑ 〈 cos [␻ 0 t − ␸ j (t )] cos [␻ 0 t ′ − ␸ j ′ (t ′ )] 〉 j

j′

(2.26)

where t ′ = t + ␶ . Because of statistical independence of ␸ j (t ) and ␸ j ′ (t ′ ) for j ≠ j ′ (2.26) can be written as R (␶ ) =

∑ 〈 cos [␻ 0 t − ␸ j (t )] 〉 + ∑ ∑ 〈 cos [␻ 0 t − ␸ j (t )] cos [␻ 0 t − ␸ j ′ (t ′ )] 〉 j

j

j′

(2.27)

where terms having j = j ′ in the second term are omitted. Taking into consideration the independence of the statistical properties of separate scatterers, (2.27) can be written in the form R (␶ ) = n 〈 cos [␻ 0 t − ␸ (t )] cos [␻ 0 t ′ − ␸ (t ′ )] 〉

(2.28)

+ (n 2 − n ) 〈 cos [␻ 0 t − ␸ (t )] cos [␻ 0 t ′ − ␸ (t ′ )] 〉 Under the assumption that the mean scatterer velocities are zero, 〈 sin ⌬␸ 〉 = 〈 sin ␸ 〉 = 〈 sin ␸ ′ 〉 = sin (␸ + ␸ ′ ) = 0

(2.29)

and for a uniform phase distribution in [0, 2␲ ] one can approximately assume [31] 〈 cos ␸ 〉 = 〈 cos ␸ ′ 〉 = 〈 cos (␸ + ␸ ′ ) 〉 = 0

(2.30)

Then (2.28) has the form R (␶ ) =

n cos ␻ 0 ␶ ⭈ 〈 cos ⌬␸ 〉 2

(2.31)

The envelope of the autocorrelation function

␳ (␶ ) = 〈 cos ⌬␸ 〉

(2.32)

106

Land Backscattering

is determined by the distribution of scatterer displacement probabilities because from (2.24) and (2.25) we have ⌬␸ = 2k ⌬x (t ) sin ␪

(2.33)

Letting m = 2k sin ␪ , we obtain ∞

␳ (␶ ) = 〈 cos m ⌬x (t ) 〉 =

冕

cos m ⌬x ⭈ p (⌬x ) ⭈ d (⌬x )

(2.34)

−∞

If the scatterer displacements are described by a Gaussian distribution p (⌬x ) =

1

√2␲ (⌬x )

冋

exp − 2

(⌬x )2 2(⌬x )2

册

(2.35)

we obtain [32]

冋

␳ (␶ ) = exp −

m 2(⌬x )2 2

册

(2.36)

Consequently, the envelope of the autocorrelation function is determined by the mean square value of scatterer displacement, which is easily expressed through the autocorrelation function of scatterer velocities [30] R u (␶ ) = 〈 u (t ) ⭈ u (t ′ ) 〉 = (⌬u )2 ⭈ ␳ (␶ )

(2.37a)

where (⌬u )2 is the scatterer velocity variance. Actually [30], ␶

2

冕

(⌬x ) = 2 (␶ − ␰ ) R u (␰ ) d␰ 0

and

冤

n R (␶ ) = cos (␻ 0 ␶ ) ⭈ exp −m 2 (⌬u )2 2

␶

冕 0

(␶ − ␰ ) ␳ u (␰ ) d␰

冥

(2.37b)

107

2.2 State of the Theory

The spectral density is found as Fourier-transform of the autocorrelation function

2 G (␻ ) = ␲

∞

冕

R (␶ ) cos (␻␶ ) d␶

(2.38)

0

n = ␲

∞

冕

␶

冤

2

cos (␻ − ␻ 0 ) ␶ ⭈ exp −m (⌬u )

0

2

冕

冥

(␶ − ␰ ) ␳ u (␰ )

0

d␶

Taking into account that the autocorrelation function of the horizontal component of wind velocity at small heights above the surface is determined as [33, 34]

␳ u (␶ ) = e −␣ | ␶ | cos ␤␶

(2.39)

(2.38) can be written in the form

n G (␻ ) = ␲

∞

冕

␶

冤

2

cos (␻ − ␻ 0 ) ␶ ⭈ exp −m (⌬u )

0

2

冕

(␶ − ␰ ) e ␣ | ␶ | cos ␤␶

0

冥

d␶ (2.40)

The solution of this integral has the form [35]

冉

冊∑

冢冋冉

m 2 (⌬u )2

m 2 (⌬u )2 n G (␻ ) = exp ␲ k2

⭈

where k = (␣ 2 + ␤ 2 )/2␣ .

∞

m =0

k2

m 2 (⌬u )2 k

2

(−1)m ⭈ m!

k2

+ km

冊

2

+ km

冉

m 2 (⌬u )2

册

+ (␻ − ␻ 0 ) 2

冣

冊

m

(2.41)

108

Land Backscattering 2

Letting m 2 (⌬u )2 = ␴ f , we obtain

冉 冊∑ 2 ␴f 2

∞

n G (␻ ) = exp ␲ k

m =0

(−1m ) ⭈ m!

冉冊

2 ␴f m 2

k

2

␴f ⭈

冉

k2 k2

冊

2

2

␴f

+ km

+ km

+ (␻ − ␻ 0 ) 2 (2.42)

The analysis of this expression shows that the power spectrum of land scattering 2 2 has different forms as a function of ␴ f /k 2. For ␴ f /k 2 >> 1, the expression for spectral density has the form [35] G (␻ ) ≅

冋

n (␻ − ␻ 0 )2 2 (2␲␴ f )1/2 exp − 2 ␲ 2␴ f

册

(2.43)

Thus, for slow scatterer oscillations (small Doppler frequencies less than the −3 dB spectrum bandwidth), the power spectrum is practically Gaussian. The spectrum width is ⌬F =

␴f ␲ √2

⭈

√ (⌬u )2 =

2√2 ␭

√ (⌬u )2

(2.44)

From the analysis of (2.44), it can be seen that the land backscattering spectrum width is inversely proportional to wavelength and proportional to the rms wind velocity fluctuations. 2 In the region of higher frequencies when ␴ f /k 2 << 1, the power spectrum has the form [35] G (␻ ) =

n ␲

冋

2

␴ f /2k 2

(␴ f /2k )2 + (␻ − ␻ 0 )

2

−

␴ f /2k 4k 2 + (␻ − ␻ 0 )2

册

+...

(2.45)

where only the first terms are significant. Thus, for the frequency region greater than the spectrum width, the change of spectral density is described better by a power function with an exponent approaching two.

2.3 Normalized RCS 2.3.1 Normalized RCS of a Quasi-Smooth Surface

The normalized RCS is one of the most important and universal parameters characterizing the land backscattering, and it depends on such variables as the surface

2.3 Normalized RCS

109

type, wavelength (or frequency), grazing angle, transmitting and receiving polarizations, the season of year, and weather conditions. Although the normalized RCS concept is strictly applied only to homogeneous rough surfaces or flat land areas, it is also used in practice for a description of backscattering from nonuniform terrains (e.g., for towns or land areas with sparse trees). In these cases, it is necessary to use the normalized RCS concept with caution. The influence of the factors mentioned earlier has been investigated in the wide band from meter to short millimeter wavelengths and was generalized in [20]. The main conclusions for quasi-smooth surfaces are the following. Smooth surfaces are characterized by weak scattering (except at angles near nadir), and the angular dependence of normalized RCS for these surfaces is ␴ 0 ∼ sin3 ␺ , where ␺ is the grazing angle. As was noted in Section 2.2.1, theoretical models have been developed most completely for quasi-smooth surfaces, and the derivations of normalized RCS within the framework of these models are satisfactorily adjusted to experimental results. The rms error does not exceed 2–3 dB, and maximal error is 10 dB [12]. The dependence of ␴ 0 on wavelength can be represented as ␴ 0 ∼ ␭ −4. For grazing angles less than 30°, the RCS is from 6 dB to 10 dB higher for vertical polarization than for horizontal. As an example, the angular dependences of normalized RCS for concrete at wavelengths of 3.2 cm and 8.6 mm for vertical and horizontal polarizations of transmitting and receiving are shown in Figure 2.9, and the dependence of ␴ 0 on wavelength is shown in Figure 2.10 for concrete, asphalt, and gravel. The dependence of experimental normalized RCS on frequency for different types of road surfaces for horizontal polarization is presented in Table 2.8, confirming the conclusions on strong frequency dependence of the normalized RCS. The presence of a water film on the surface decreases the normalized RCS by 10 dB for small grazing angles, as shown in Figure 2.11; this is explained by the smoothing of surface roughness by the water. This same smoothing increases the normalized RCS for nadir radiation. For nadir radiation, the rms roughness height strongly influences the normalized RCS, leading to its decrease with increasing ␴ h . The values of normalized RCS for nadir operation in the frequency band 40–135 GHz are shown in Table 2.9. 2.3.2 Normalized RCS for Rough Surfaces Without Vegetation

Rough surfaces without vegetation have greater values of normalized RCS (by 15–25) dB and other angular and frequency characteristics in comparison with quasi-smooth surfaces In Figure 2.12, the angular dependences of the normalized RCS are shown for rough surfaces without vegetation at wavelengths of 1–3 cm. The normalized RCS of a rough surface without vegetation is a function of the surface roughness ␴ h and volumetric soil moisture m v [i.e., ␴ 0 = f (␴ h , m v )].

110

Land Backscattering

Figure 2.9 The normalized RCS of concrete versus grazing angle for (a) vertical and (b) horizontal polarizations. (After: [36].)

The growth of surface roughness changes the angular dependence of normalized RCS to ␴ 0 ∼ sin ␺ for highly rough terrains and the dependence of ␴ 0 on wavelength to ␴ 0 ∼ ␭ −1. As an illustration, the angular dependences of the normalized RCS for ploughed fields with different types of cultivation (degrees of roughness) are shown in Figure 2.13. It is seen that the angular dependence of the normalized RCS for k␴ h ≥ 1.2 can be approximated as ␴ 0 ∼ sin ␺ . However, right up to these values of k␴ h , the experimental angular dependences are satisfactorily adjusted to derivations according Kirhhoff’s method. For greater values of k␴ h the surface is practically diffuse, as shown in Figure 2.13. All of these surfaces scatter diffusely

111

2.3 Normalized RCS

Figure 2.10

Dependence of normalized RCS for quasi-smooth surfaces on radar wavelength. (After: [36].)

Table 2.8 Normalized RCS (dB) Frequency Dependence of Different Surfaces at the Grazing Angle of 10° Frequency (GHz) 10.0 15.5 35.0 Source: [36].

Concrete −(30–54) −(29–45) −(20–43)

Asphalt −(26–46) −(25–39) −(18–33)

Asphalt-Gravel −(25–41) −(20–33) −(15–29)

Slag-Gravel −(25–44) −(18–34) −(18–28)

in the millimeter bands, and the angular dependence of the normalized RCS can be approximated as ␴ 0 ∼ sin ␺ . The second factor determining the normalized RCS is the dielectric constant, which depends on frequency, soil type, and its volumetric moisture. Increasing the volumetric soil moisture leads to increasing ␴ 0 and changes the frequency dependence of ␴ 0 because of increasing soil dielectric constant. However, the normalized RCS change of bare soil is less than 10 dB over the entire range of field moisture change; this is considerably less than the change caused by surface roughness.

112

Land Backscattering

Figure 2.11

Angular dependences of normalized RCS for dry and wet asphalt at wavelength of 8 mm and vertical polarization (VV). (After: [36].)

Table 2.9 Average Values of Normalized RCS for Nadir Radiation Surface Type Lake (smooth surface) Asphalt Concrete Sand, gravel Brick Veneer, thickness 5 mm Source: [36–39].

10.0 GHz 11.4 — — 6.5 — —

Normalized RCS (dB) at Frequencies, 40–90 GHz 70.0 GHz 20.0 15.2 16.0 — 15.2 11.5 −7.4 −1.2 — — — —

135.0 GHz — — — — 4.0 −10.0

As was shown in Section 2.1, the influence of soil type on its dielectric constant is rather small. The Fresnel coefficients in (2.7) depend rather weakly on the frequency. In this case, the dependence of normalized RCS on relative moisture can be represented by following empirical expression [11]

␴ 0 = 0.148m f − 15.96 (dB)

(2.46)

where m f = (m v /C v ) ⭈ 100%, m v is the volumetric soil moisture in grams per cubed centimeter, and C v is the field moisture capacity. The dependences of the

113

2.3 Normalized RCS

Figure 2.12

Angular dependences of the normalized RCS for three types of land.

normalized RCS on volumetric soil moisture are shown in Figure 2.14. It is seen that a volumetric moisture change from 0.05 g/m3 to 0.3 g/m3 leads to an increase in normalized RCS of 7–8 dB. Vegetation masks the soil surface, and in this case the total normalized RCS is determined as [11] 0 ␴ 0⌺ (␺ , ␭ ) = ␴ veg (␺ , ␭ ) +

0 ␴ soil (␺ , ␭ ) 2 L (␺ )

(2.47)

0 0 where ␴ veg and ␴ soil are the normalized RCS of vegetation and soil, respectively, and L (␺ ) is the attenuation coefficient in the vegetation layer. At frequencies above 8 GHz and for grazing angles less than 60°, the contribution of the first term in (2.47) is predominant, because one can neglect the soil influence, and the contribution of the second term is significant at lower frequencies and for ␺ ≥ 60°. Figure 2.15 shows the attenuation in a vegetation layer for different types of plants as a function of the penetration depth at the 3-cm wavelength (X-band). Under conditions of incomplete surface screening, the normalized RCS dependence on the soil moisture takes the form [11]

114

Land Backscattering

Figure 2.13

Angular dependence of normalized RCS for different ␴ h . (From: [40].)

␴ 0 = 0.113m f − 13.84 (dB)

(2.48)

Thus, it is not as steep as for soil without vegetation, for which the normalized RCS is given by (2.46). The ␴ 0 dependences on soil moisture for bare soil and for soil with vegetation are shown in Figure 2.16.

2.3.3 Backscattering from Snow

Microwave backscattering from snow involves surface scattering by the air-snow and snow-soil boundaries and volumetric scattering by ice crystals within the snow layer. Multiple scattering also takes place, conditioned by the upper and lower boundaries. The normalized RCS of snow is determined by several factors: the radar frequency, the transmitted and received polarizations, and the grazing angle, as well as the electrophysical and geometrical properties of the snowpack [41]. The first snow parameter influencing the normalized RCS is its water equivalent

2.3 Normalized RCS

115

Figure 2.14

Soil normalized RCS dependences on moisture at 3–4.5 GHz. (After: [12].)

Figure 2.15

Attenuation dependences on the penetration depth for three types of vegetation.

116

Land Backscattering

Figure 2.16

Normalized RCS dependences on soil moisture for bare and vegetated surfaces. (From: [12].  1986 Artech House, Inc. Reprinted with permission.)

W = ␳s h

(2.49)

where ␳ s is the snow density and h is the snowpack height. The second parameter is the volumetric water content m v . The surface roughness parameters and ice crystal size distribution are also significant characteristics of snow. In most cases, the normalized snow RCS (if one does not take into consideration multiple scattering) can be expressed as [42] 2

0

␴ =

␴ ss0 (␭ ,

␪) +

␴ s0(␪ ′ )

+

␥ sa (␪ ′ ) L 2( ␪ ′ )

0 (␪ ′ ) ⭈ ␴ soil

(2.50)

where ␴ ss0 is the normalized surface scattering RCS of the air-snow boundary, 2 ␴ s0(␪ ′ ) is the normalized volumetric scattering RCS of the snow, ␥ sa (␪ ′ ) is the 0 is the normalized power displacement factor of the air-snow boundary, and ␴ soil RCS of the underlying soil. The angle ␪ ′ is related to the incidence angle ␪ by

117

2.3 Normalized RCS

sin ␪ =

√⑀ s ⭈ sin ␪ ′

(2.51)

where ⑀ s is the dielectric constant of snow. For dry snow and small grazing angles (i.e., ␴ s0 ≈ 0), one can neglect the scattering from the air-snow boundary. In this case, only volumetric scattering and reflection from the snow-soil boundary determine the normalized RCS in (2.50), and the expression for the normalized RCS becomes 0 ␴ 0(W ) = A 0 − (A 0 − ␴ soil ) ⭈ exp (−C 0 W sec ␪ ′ )

(2.52)

where A 0 and C 0 are coefficients depending on radar frequency, incidence angle and, polarization [43]. The dependences of the normalized RCS on the water equivalent of snow are shown in Figure 2.17. It is seen that the saturation of the normalized RCS is observed for snowpack height and its water equivalent growth, assuming the absence of soil influence on the scattering signal intensity. For wet snow, one cannot neglect the surface scattering of the air-snow boundary because of the large value of the snow dielectric constant resulting from water

Figure 2.17

The normalized RCS of snow versus depth of snow cover. (From: [43].)

118

Land Backscattering

in the liquid phase. In some cases, this surface scattering predominates in forming the scattered signal. A rapid decrease of the normalized RCS takes place with increase in liquid water content, as is shown in Figure 2.18. For m v ≥ 4–5% the rate of change of ␴ 0 decreases as a result of the considerable attenuation of the free water. The result is a predominant influence of the surface scattering. The frequency dependence of the normalized RCS for dry and wet snow is shown in Figure 2.19 [41]. First of all, one can see the interesting dependence of ␴ 0 on the liquid water content. A change of m v from 0% to 1.25% gives a decrease in the normalized RCS of about 1 dB in the decimeter band; the difference grows for increasing frequency, and it reaches the maximal value of 10–12 dB at 35.0 GHz. This difference decreases with further increase in frequency and is 2–3 dB at 140.0 GHz.

2.3.4 Backscattering from Vegetation

Vegetation leads to changes in RCS angular and frequency dependences. The smallest value of the normalized RCS is observed for thin vegetation covers. This value varies within limits of −(20 to 25) dB for grazing angles less than 10°. The back-

Figure 2.18

The normalized RCS of snow versus the volumetric moisture. (From: [43].)

2.3 Normalized RCS

Figure 2.19

119

The frequency dependence of snow normalized RCS. (After: [12].)

scattering is essentially diffuse, with an angular dependence of form ␴ 0 ∼ sin ␺ and very weak frequency dependence for all grazing angles less than 10°. Dense forests have the largest RCS in spring and summer as a result of the correlation of the scattering intensity with vegetation biomass. A high correlation between the normalized RCS and biomass is observed in the shortwave part of centimeter and in millimeter bands for grazing angles less than 60°–70°, where the soil influence is practically negligible. Thus, at 11 GHz for ␺ ≤ 60°, the correlation between normalized RCS and biomass exceeds 0.9 for both copolarization and cross-polarization [41]. An increase in biomass to 0.4 kg/m2 leads to increasing the normalized RCS by 3 dB (i.e., RCS change at a rate of 7.5 dB/kg/m2 ). There is considerable seasonal dependence of normalized RCS for vegetation covers. In the spring–summer period, the scattering intensity increases by 10–20 dB in comparison with the autumn–winter period, and the largest values are observed in June–July. This corresponds to an increase in biomass and vegetable water content during this time period. As an illustration, the seasonal changes of the normalized RCS and leaf cover coefficient are shown in Figure 2.20(a), and the seasonal changes of the normalized RCS and the vegetable water content are shown in Figure 2.20(b). There is no great difference in the normalized RCS for deciduous and coniferous forests in summer and autumn [44]. As noted in [20], no differences in the normalized RCS for coniferous forest in summer and winter are observed, while for deciduous forest this difference is 20–22 dB. There is a very significant influence of weather conditions on the normalized RCS. After a rain, the backscattering of grass increases by about 3 dB for all

120

Land Backscattering

Figure 2.20

(a) Cover coefficient and normalized RCS and (b) water content and normalized RCS dependence on season. (After: [40].)

grazing angles, and for forest this increase is 5 dB in comparison with dry weather [42, 43]. The dependence of normalized RCS for vegetation surfaces on the wavelength is very weak in centimeter and millimeter bands, having the form ␴ 0 ∼ f n where 0 ≤ n ≤ 1. The best approximation for frequency dependence for both forest and fields covered by the grass and agricultural plants has the form ␴ 0 ∼ f 0.6, giving satisfactory results from 3 GHz to 100 GHz. For grazing angles less than 1°, the experiments do not disclose even this much frequency dependence, as illustrated by Table 2.10 [20, 23, 38, 45]. 2.3.5 Normalized RCS Models

The features of land backscattering derived earlier permit us to develop several empirical models for normalized RCS of different terrain types. Table 2.10 The Vegetable Surfaces Normalized RCS for ␺ < 1° Vegetation Type Meadow (flat surface) Steppe (roughness heights up to 0.5m) Dry meadow Sparse mixed forest, bush Dense foliage forest

32 −27 −23 −20 −22 −12

Wavelength 12.5 −30 −23 −20 −20 −14.5

(mm) 8.6 −15 −23 −27 −23 −11

8.15 −25.5 −24 −21 −21 −8

4.1 −30 −2 −22.5 −21.5 −9.5

121

2.3 Normalized RCS

For the land surfaces with vegetation, the angular and frequency dependences for incidence angles from 0° to 60° (grazing angles greater than 30° ) and in the band 1–18 GHz can be represented by the empirical expression [46]

␴ 0 (dB) = a 0 + a 1 e a 2 ␪ + (a 3 + a 4 e −a 5 ␪ ) ⭈ exp [−(a 6 + a 7 ␪ ) f ]

(2.53)

where ␪ = 90° − ␺ is the incidence angle and a 1 − a 7 are coefficients determined by the transmitted and received polarizations; their values are shown in Table 2.11. The frequency dependence of the normalized RCS given by (2.53) is very weak at frequencies above 4 GHz, and at frequencies less than 4 GHz, ␴ 0 decreases very rapidly with decreasing frequency (i.e., frequency dependence appears in the decimeter band). At the same time, the angular dependence is that of diffuse backscattering. A simpler empirical expression (with fewer number of coefficients) for the normalized RCS of these surfaces was proposed in [47]

␴ 0 = D + 10␣ ⭈ log f + 8.6␤ ⭈ f ␣ − M␪ (dB)

(2.54)

where the coefficient values are as shown in Table 2.12. Another model applies to the band 3–100 GHz, for grazing angles less than 30°, and is suitable for RCS description of various surfaces including quasi-smooth, rough with and without vegetation, snow, and city and country areas. The normalized RCS in this model has the form [21, 48]

␴ 0 (dB) = A 1 + A 2 log ␺ /20 + A 3 log f /10

(2.55)

Table 2.11 Coefficient Values for Use in (2.53) Polarization Horizontal polarization (HH) VV Cross polarization (HV) Source: [46].

a0

a1

a2

a3

a4

a5

a6

a7

2.69 3.49

−5.35 −5.35

0.014 0.014

−23.4 −14.8

33.14 23.69

0.048 0.066

0.053 0.048

5.1 ⭈ 10−3 2.8 ⭈ 10−3

3.91

−5.35

0.013

−25.5

14.65

0.098

0.258

2.1 ⭈ 10−3

Table 2.12 Coefficient Values for Use in (2.54)

␣ ␤ 0.8 0.04 Source: [47].

D −15.5

M 0.1

122

Land Backscattering

Here f is the frequency in gigahertz, and ␺ is the grazing angle in degrees. The coefficients A 1 − A 3 for various types of terrain are shown in Table 2.13. The significant feature of this model lies in the replacing of the real terrain variety by a limited number of land surface types, which include all of the main terrain types. This is sufficient for land clutter estimations in typical conditions of radar operation. As an illustration, the angular and frequency dependences of the normalized RCS for deciduous forest are shown in Figures 2.21 and 2.22, obtained for a grazing angle of 10° by various authors, where the solid and dotted lines correspond to derivations according to (2.55). The variance of experimental data amounts to 10–15 dB, and this is associated with differences both in various land types and in the methods of experimental data processing. Table 2.13 The Coefficients A 1–A 3 in Land Clutter Model Terrain Type Concrete Arable land Snow Deciduous and coniferous forests, summer Deciduous forest, winter Grass with height more than 0.5m Grass with height less than 0.5m Urban territories (town and country buildings) Source: [48].

Figure 2.21

A1 −49 −37 −34 −20 −40 −21 −(25−30) −8.5

A2 32 18 25 10 10 10 10 5

A3 20 15 15 6 6 6 6 3

Normalized RCS ␴ 0 versus grazing angle for forest at wavelengths of 8 mm (curve 1) and 3 mm (curve 2).

2.4 Depolarization of Scattered Signals

Figure 2.22

123

Normalized RCS versus frequency for forest at ␺ = 10°.

Consequently, we have normalized RCS estimations for various types of land for practically all microwave and millimeter-wave bands.

2.4 Depolarization of Scattered Signals The polarization differences of backscattered signals are determined, to a major degree, by the ratio of surface roughness to wavelength. For quasi-smooth terrain (concrete, asphalt), the normalized RCS at grazing angles less than 30° is greater by 8–10 dB for VV than for HH, and HV components are practically absent [20]. This is in good agreement with the land surface model obtained by the small perturbation method for which the depolarization compo0 0 nents are zero (␴ HV = ␴ VH = 0). For bare rough surfaces, the difference between RCS values for HH and VV polarizations disappears as the rms roughness ␴ h increases, and the difference does not exceed 2–3 dB for grazing angles less than 60°. The normalized RCS ratio for co- and cross-polarizations for these surfaces is 7–15 dB [1] decreasing with reduced

124

Land Backscattering

wavelength and grazing angle and with increasing ␴ h . For example, in Figure 2.23, the dependence of ␴ 0 on incidence angle is shown for three soils with different ␴ h measured at the 3-cm wavelength, and in Figure 2.24 the copolarization to crosspolarization ratios are shown for wavelengths of 3 cm and 8 mm for soils with different erosion state [49]. The polarization ratios are small for land surfaces with vegetation. The ratio 0 0 ␴ VV /␴ HH decreases with vegetation biomass growth. For agricultural plantings (e.g., potatoes, alfalfa, or sugar beets), this ratio is 2–3 dB at 10 GHz [50] and 3–4 dB between 10 and 100 GHz [51]. The copolarization to cross-polarization ratio varies over wide limits. As shown in [51], this ratio is 7–12 dB at the 3-cm wavelength, decreasing with grazing angle decrease, and is 10–12 dB at 9-mm wavelength (Figure 2.25). Thus, for various land surfaces, the polarization ratios at a 3-cm wavelength are shown in Table 2.14. The polarization features of signals scattered by snow are not pronounced. There is little difference between RCS for VV and HH polarizations for snow at frequencies above 10 GHz, and copolarization to cross-polarization ratio is about 10 dB for dry snow and 10–15 dB for wet snow. This ratio decreases with further increase in frequency, to 5 dB at 95 GHz and 3 dB at 140 GHz [41]. 0 0 For sea ice, the ratio of ␴ VV /␴ HH is practically unity, and the copolarization to cross-polarization ratios are 3–10 dB in the band 10–40 GHz [52], depending on surface roughness. These data permit us to use the following hypotheses for description of land backscattering: •

0 0 The ratio ␴ VV /␴ HH is 10 dB for quasi-smooth surfaces at 10 GHz, and its dependence on frequency is rather weak.

Figure 2.23

(a–c) The normalized RCS versus incidence angle for three types of soil. (After: [12].)

2.4 Depolarization of Scattered Signals

Figure 2.24

• •

125

0 0 Ratios ␴ cop /␴ cross versus incidence angle at wavelengths of (a) 3 cm and (b) 8 mm for soils with different erosion states (curves 1–3). (From: [49].  1995 SPIE. Reprinted with permission.)

This ratio is zero for all types of rough surfaces. The copolarization to cross-polarization ratio is 10 dB with the exception of quasi-smooth surfaces for which the cross-polarization components of the backscattered signal are very small.

Thus, the following expressions can be used for normalized RCS of land clutter

126

Land Backscattering

Figure 2.25

(a–c) Polarization ratios for vegetation at a 9-mm wavelength for different crops.

0 0 0 Table 2.14 Ratios of ␴ VV /␴ 0HH and ␴ copal /␴ cross , in Decibels, for 3-cm Wavelength

Surface Type Quasi-smooth Rough bare soil Rough surface with vegetation Snow Ice Town and country areas

0 ␴ VV ≅

冦

0 ␴ HH + 10 0 ␴ HH

0 0 ␴ VV /␴ HH 8.0–10.0 2.0–3.0 2.0–3.0 0 1.0–2.0 0

冉 冊 f 10

0 0 ␴ copal /␴ cross — 7.0–15.0 7.0–12.0 10.0–15.0 3.0–10.0 2.0–3.0

−1/2

for quasi-smooth surfaces

(2.56a)

for other surfaces

0 Here ␴ HH is the value of ␴ 0 from (2.55), and f is the frequency in gigahertz. The cross-polarized component of the normalized RCS is 0 0 0 0 ␴ cross = ␴ HV = ␴ VH ≈ ␴ HH − 10 (dB)

(2.56b)

for all types of surfaces other than quasi-smooth. For quasi-smooth surfaces, the cross-polarized component of the normalized RCS in model is zero (−∞ in decibels).

2.5 Statistical Characteristics of the Scattered Signals The fluctuations of backscattered signals from land are associated with the motion of scatterers within the resolution cell and the shift of the resolution cells due to

127

2.5 Statistical Characteristics of the Scattered Signals

radar platform motion and antenna scanning. This determines the normalized RCS temporal and spatial probability distributions ∞

0

p (␴ ) =

冕

p 冠␴ 0 | m 冡 ⭈ p (m ) dm

(2.57)

−∞

Here p (m ) is the pdf that characterizes the spatial distribution of the fluctuations, p 冠␴ 0 | m 冡 is the conditional pdf that describes the temporal fluctuations in a single resolution cell, and m is the mean value of the normalized RCS. The pdf of RCS can depend in various ways on the distribution of mean or median values p (m ) of ␴ 0 within the scan volume. This is because the RCS distribution in a single resolution cell is transformed, and the resulting distribution has larger tails (i.e., greater probability of occurrence of large RCS values). The temporal statistics of the scattered signal follows the RCS distribution within the limits of a single resolution cell p 冠␴ 0 | m 冡. Typical approximations of experimental distributions of normalized RCS are the Rayleigh and Rician [1]. For these cases, the probability function of the in-phase (I ) and quadrature (Q ) components of the scattered signal are Gaussian with variance equal to the fluctuating component power and with nonzero mean value (the mathematical expectation) equal to the amplitude of the stable component. The stable component of the scattered signal is formed by reflectors that are stable in time (e.g., rocks, buildings, or bare surfaces). In some cases, its value is considerably larger than the fluctuating component. As result, the normalized RCS can be represented as a sum of two terms

␴ 0 = ␴ st0 + ␴ fl0 = ␴ fl0(1 + a 2 )

(2.58)

where ␴ st0 and ␴ fl0 are the normalized RCS of the stable and fluctuating components, respectively, and a 2 is the ratio of stable-to-fluctuating RCS. For various land types, the values of a 2 are determined by the mean wind velocity and the wavelength and are 30.7␭ 2 ⭈ V −2.9

a2 =

冦

2

63␭ ⭈ V 2

−3

100␭ ⭈ V

−3

for forests for short vegetation

(2.59)

for bare land

For town and country areas, the a 2 values are approximately 104–105. In these expressions, ␭ is the wavelength in centimeters and V is the wind velocity in meters per second.

128

Land Backscattering

Two models are applied rather widely for modeling the probability distribution functions of land backscattering. In the first (the Rayleigh model), an ensemble of independent reflectors with random amplitude and phase produces the scattered signal. This is a typical model for land surfaces with dense vegetation. The pdf of the normalized RCS in this model is p (␴ 0 /m ) =

冉 冊

1 ␴0 exp − m m

(2.60)

For terrains with the stable scatterers, a model using the Rician pdf is applied: p 冠␴ 0 | m 冡 =

冋

1 + a2 ␴0 ⭈ exp −a 2 − (1 + a 2 ) m m

册

冉√

⭈ I 0 2a

0

␴ (1 + a 2 ) m

冊

(2.61)

where I 0 is the Bessel function of the first kind and zero order. The normalized RCS differs from the Gaussian model only for small sizes of the radar resolution cell, and this function can be represented as p (␴ ) =

冉 冊

b␴ b − 1 −␴ b exp ␣ ␣

(2.62)

This is the Weibull distribution of RCS. Here ␣ is the shape parameter and b is the slope parameter, b = 1 for the Rayleigh model (2.10). The experimental results of [53] showed that the b parameter for forest and grass in summer and in winter and for the wind velocities of 5–10 m/s is 10–15 (i.e., the Weibull distribution is rather similar to the Gaussian). Examples of these distributions for scattering from grass at 3.2-cm wavelength are shown in Figure 2.26. Therefore, it is possible to use the Gaussian pdf to describe the statistics of two quadrature components of the scattered signals from the land surfaces.

2.6 Power Spectra of Scattered Signals The power spectra of land backscattering are determined, as a rule, by fluctuations of scatterers that move under wind conditions and by volume scanning and radar platform motion. The influence of volume scan for various scanning methods that lead to power spectrum broadening is analyzed in detail in [10, 20], and radar platform motion is considered in [11]. The scatterer motions induced by wind fluctuations in the troposphere control the power spectrum of backscattering from terrain with vegetation. In the frame-

129

2.6 Power Spectra of Scattered Signals

Figure 2.26

Histograms of relative amplitude distributions for backscattering from grass at a 3-cm wavelength and for wind velocity (a) 5 m/s and (b) 7 m/s. (From: [53].)

work of this model, the power spectra are described as functions of the fractal type in which the spectral width and the power exponent are the functions of the wind velocity. The power spectrum can be represented as [20, 54] G ′ (F ) = a 2␦ (F ) + G (F )

(2.63)

where ␦ (F ) is Dirac’s function, which characterizes the scattering from the stable reflectors and G (F ) is the power spectrum of the fluctuating component. The experimental investigations of land clutter power spectra carried out by the author in 1966–1969 and later ones [20, 54, 55] showed that the power spectrum of the fluctuating component can be represented by the following empirical expressions

冋 冉 冊册

F G (F ) = G 0 1 + ⌬F 2(U + 2)

n −1

冉 冊

100 n= ⭈ f U +1

(2.64)

0.2

(2.65)

where F is the Doppler frequency; ⌬F is the −3 dB width of the power spectrum, which is the function of wind velocity, the operating frequency, and the land surface type; and n is the power exponent, which depends on wind velocity and vegetation type. For formal description of these values, one can use the following expressions for surfaces with vegetation

130

Land Backscattering

⌬F = 1.23 ⭈

冉 冊 3.2 ␭

⭈ U 1.3

(2.66)

where ␭ is the wavelength in centimeters, f is the frequency in gigahertz, and U is the average wind velocity in meters per second. The latter expression does not take into consideration the seasonal change and variations for various vegetation covers. As an example, the dependence of ⌬F on wind velocity is shown in Figure 2.27, where the dots are the experimental results and the solid line is from (2.66). The presence of vegetation and increase in biomass leads to an increase in the scattering intensity [i.e., increase of 10–15 dB in the spectral components for all frequencies in summer as compared with autumn and winter, with some decrease in the power exponent of (2.65)]. As an example, the backscattering power spectra for a swamp with grass and bushes are presented in Figure 2.28 for the 3-cm wavelength (the straight line 1 for summer, the line 2 for winter). The power spectra of backscattering from forest and grass at 3 cm in summer and winter are shown in Figure 2.29 [20, 53]. It is seen that the power exponent in (2.65) decreases in winter for forest and does not change for grass. It has been determined experimentally [20] that the power spectra of backscattering from land with vegetation extend over Doppler frequency bands in the order

Figure 2.27

The −3-dB spectrum bandwidth versus wind velocity for a 3-cm wavelength.

2.6 Power Spectra of Scattered Signals

Figure 2.28

131

Doppler spectra of backscattering from swamp with bushes at the 3-cm wavelength in summer (1) and in winter (2).

of 10–20 kHz. Examples of such spectra obtained at wavelengths of 3 cm, 8 mm, and 4 mm for forest and grass are shown in Figure 2.30. The power spectra of intensity (the amplitude spectra) have practically the same properties as the power spectra of coherent signals. First of all, square-law detection of signals with power spectra of the form G (F ) ∼ F −n does not broaden the amplitude spectrum in comparison with the power spectrum if the signal-tonoise ratio is chosen correctly. The amplitude spectral shape can be described by (2.64) and the spectral width by (2.66). For a light wind (about 1 m/s), the correlation interval decreases with reduced wavelength more rapidly than for moderate wind. This can be explained by the greater efficiency of the phase modulation of the scattered signal because the influence of small radial movements of the scatterers is more visible. There may be flatter spectra at low-frequency F for further reduction in wavelength.

132

Land Backscattering

Figure 2.29

Doppler spectra of backscattering from forest and grass at the 3-cm wavelength in (a) summer and (b) winter. (From: [53].)

Figure 2.30

The power spectra of backscattering from forest and grass at wavelengths of 3 cm, 8 mm, and 4 mm. (From: [48].  1994 SPIE. Reprinted with permission.)

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[28] [29]

[30] [31] [32] [33] [34] [35] [36] [37]

Redkin, B. A., and V. V. Klochko, ‘‘Derivation of Averaged Tensor of Vegetation Complex Dielectric Constant,’’ Radiotechnics and Electronics, No. 8, 1975, pp. 1596–1603 (in Russian). Beckmann, P., and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces, Pergamon Press, 1963; Norwood, MA: Artech House, 1987, p. 303. Kulemin, G. P., and V. B. Razskazovsky, Scattering of Millimeter Radiowaves by the Earth’s Surface for Small Grazing Angles, Kiev, Ukraine: Naukova Dumka, 1987 (in Russian). Ishimaru, A., Wave Propagation and Scattering in Random Media, New York: Academic Press, 1978. Ament, W. A., F. C. MacDonald, and R. D. Shewbridge, ‘‘Radar Terrain Reflections for Several Polarizations and Frequencies,’’ Proc. Symp. Radar Return, Arlington, NY, October 1959, pp. 346–349. Spetner, L. J., and I. Katz, ‘‘Two Statistical Models for Radar Terrain Return,’’ IRE Trans. Antennas and Propagation, Vol. AP-8, No. 5, 1960, pp. 242–246. Twersky, V., ‘‘On Scattering and Reflection of Electromagnetic Waves by Round Surfaces,’’ IRE Trans. Antennas and Propagation, Vol. AP-8, No. 1, 1960, pp. 81–89. Attema, E., and F. T. Ulaby, ‘‘A Radar Backscatter for Vegetation Targets,’’ Proc. Open Int. Symp. EM Propag. Nonionized Media, La Baule, France, May 1977, pp. 579–584. Bush, T., and F. Ulaby, ‘‘Radar Return from a Continuous Vegetation Canopy,’’ IEEE Trans. Antennas and Propagation, Vol. AP-24, No. 3, 1976, pp. 269–276. Zamaraev, B. D., Y. F. Vasilyev, and V. G. Kolesnikov, ‘‘Foliage Forest Normalized RCS Estimation at Millimeter Band of Radiowaves,’’ Spatial-Temporal Signal Processing in Radiotechnical Systems, Aviation Institute, Kharkov, Ukraine, 1985, pp. 50–53 (in Russian). Bass, F. G., and I. M. Fuks, Wave Scattering from Statistical Rough Surfaces, Moscow, Russia: Soviet Radio, 1972, p. 424 (in Russian). Gorelik, G. S., ‘‘On Scatterer Velocity Correlation Influence on the Statistical Properties of Scattered Radiation,’’ Radiotechnics and Electronics, Vol. 2, No. 10, 1957, pp. 1227–1233 (in Russian). Gorelik, G. S., ‘‘To Radiowaves Scattering Theory on the Wondering Inhomogeneities,’’ Radiotechnics and Electronics, Vol. 1, No. 6, 1956, pp. 695–703 (in Russian). Tikhonov, V. I., Statistical Radiotechnics, Moscow, Russia: Soviet Radio, 1966, p. 678 (in Russian). Gradshtein, I. S., and I. M. Rizhik, Tables of Integrals, Sums, Series and Products, Fizmatgiz, Moscow, 1963, p. 1100 (in Russian). Pinus, N. Z., and S. M. Shmeter, Aerology, Moscow, Russia: Gidrometeoizdat, 1965, p. 351 (in Russian). Zubkovsky, S. L., ‘‘Fluctuation Spectra of Wind Velocity Horizontal Component at Height of 4m,’’ Izv. AS USSR, Physics of Atmosphere and Ocean, No. 10, 1962, pp. 1425–1428. Malakhov, A. N., ‘‘About Generator Spectral Line Shape for its Frequency Fluctuations,’’ Journal of Experimentental and Theoretical Physics, Vol. 30, No. 5, 1956, pp. 884–889. Taylor, R., ‘‘Terrain Return Measurements at X-, Ku , and Ka Bands,’’ IRE Nat. Conv. Rec., Part 1, 1959, pp. 19–30. Long, M. W., Radar Reflectivity of Land and Sea, 3rd ed., Norwood, MA: Artech House, 2001.

References [38] [39] [40] [41] [42] [43]

[44] [45] [46] [47]

[48]

[49]

[50]

[51] [52]

[53]

[54]

[55]

135 Trebits, R. N., R. D. Hayes, and L. C. Bomar, ‘‘MM-Wave Reflectivity of Land and Sea,’’ Microwave J., Vol. 21, No. 8, 1978, pp. 49–53. King, H. E., et al., ‘‘Terrain Backscatter Measurements at 40 to 90 GHz,’’ IEEE Trans. Antennas and Propagation, Vol. AP-18, No. 6, 1970, pp. 780–784. LeToan, T., ‘‘Active Microwave Signatures of Soil and Crops,’’ Proc. IGARSS’82, Digest, Vol. 1, 1982, pp. Tp 2.3/1–Tp 2.3/5. Kulemin, G. P., T. N. Kharchenko, and S.E. Yatsevich, ‘‘Snow Remote Sensing by the Radar Techniques,’’ Preprint of IRE NASU, No. 92-8, Kharkov, Ukraine, 1992 (in Russian). Ulaby, F. T., W. H. Stiles, and M. Abdelrazik, ‘‘Snowcover Influence on Backscattering from Terrain,’’ IEEE Trans. Geosc. Remote Sens., Vol. GE-22, No. 2, 1984, pp. 126–132. Ulaby, F. T., and W. H. Stiles, ‘‘The Active and Passive Microwave Response to Snow Parameters: Part II—Water Equivalent of Dry Snow,’’ J. Geophys. Res., Vol. 85, No. C2, 1980, pp. 116–122. Kim, Y. S., et al., ‘‘Surfaces-Based Radar Scatterometer Study of Kansas Rangeland,’’ Remote Sens. Environ., Vol. 11, 1980, pp. 253–265. Grant, C. R., and B. S. Yaplee, ‘‘Backscattering from Water and Land of Centimeter and Millimeter Wavelengths,’’ Proc. IRE, No. 6, 1957, pp. 976–982. Ulaby, F. T., ‘‘Vegetation Clutter Model,’’ IEEE Trans. Antennas and Propagation, Vol. AP-28, 1980, pp. 538–545. Vasilyev, Y. F., B. D. Zamaraev, and G. P. Kulemin, ‘‘The Angular and Season Backscattering Dependences of Millimeter Radiowaves by Vegetation Cover,’’ Preprint of IRE NASU No 91-3, Kharkov, Ukraine, 1991, p. 26, (in Russian). Kulemin, G. P., and V. B. Razskazovsky, ‘‘Complex Effects of Clutter, Weather and Battlefield Conditions on the Target Detection in Millimeter-Wave Radars,’’ Proc. SPIE, Vol. 2,222, 1994, pp. 862–871. Kulemin, G. P., et al., ‘‘Soil Moisture and Erosion Degree Estimation from Multichannel Microwave Remote Sensing Data,’’ Proc. Europ. Symp. SPIE on Satellite Remote Sensing, Paris, Vol. 2,585, September 1995, pp. 144–155. Moore, R. K., K. A. Soofy, and S. M. Purduski, ‘‘A Radar Clutter Model: Average Scattering Coefficients of Land, Snow and Ice,’’ IEEE Trans Aerosp. Electr. Syst., Vol. AES-16, 1980, pp. 783–799. Skolnik, M. I., (ed.), Radar Handbook, New York: McGraw-Hill, 1970. Kulemin, G. P., ‘‘Growler Detection Method in Sea Clutter by Coherent Radar,’’ Proc. Sixth Int. Conf. Remote Sensing for Marine and Coastal Environment, May 2000, Charleston, SC; Veridan ERIM Int., Vol. 2, pp. II.47–II.54. Savchenko, A. K., S. J. Haimov, and G. P. Kulemin, ‘‘On the Experimental Study of Radar Backscattering from Land,’’ XVII Europ. Microwave Conf., Stockholm, October 1988, pp. 705–709. Kulemin, G. P. and V. B. Razkazovsky, ‘‘Land Backscattering Spectra of Centimeter and Millimeter Radiowaves for Small Grazing Angles,’’ Preprint of IRE NASU, No. 195, Kharkov, Ukraine, 1982, p. 39 (in Russian). Brukhovetskiy, A. S., and A. A. Puzenko, ‘‘About Signal Spectrum for Transversal Movement of Shadowing Reflectors,’’ Radiotechnics and Electronics, Vol. 15, No. 12, 1970, pp. 2533–2538 (in Russian).

CHAPTER 3

Estimation of Land Parameters by Multichannel Radar Methods 3.1 Estimation of Soil Parameters 3.1.1 Introduction

Efficient use of agricultural fields requires application of modern remote sensing techniques for soil characteristic determination because the traditional methods of in situ measurements do not provide data detailed enough for practice and are too labor consuming. During recent decades, besides optical and infrared methods, radar methods of soil characteristic study have been intensively developed—in particular, nearsurface soil moisture estimation, determination of humus content, and degree of soil erosion. The high resolution and the weather independence of radars permit their practical application to remote sensing of large terrain areas through the use of airborne and spaceborne radars—synthetic aperture radar (SAR) or side-looking radar (SLAR). In radar remote sensing, the electromagnetic field scattered by the objects serves as a source of information about the physical and chemical properties of the surface. For this reason, the processes of microwave scattering from bare soil have been the subject of theoretical and experimental investigations for many years. This research activity was directed at finding a correlation between the parameters of the scattered electromagnetic field and the statistical and agrophysical characteristics of soil. The capability of active microwave techniques to sense near-surface soil moisture has a considerable research interest. The basis for microwave remote sensing of soil moisture is the dependence of the soil’s dielectric properties on its moisture content due to a large contrast between the dielectric constant of water and that of dry soil. Boundary conditions affecting soil characteristics include the smallscale random surface roughness generated by cultivation, soil erosion processes, azimuthally dependent ridge/furrow patterns, and the slope of a terrain element affecting the local angle of incidence. As is well known, the scattered signal intensity is determined by the statistical characteristics of the surface and the dielectric constant of the medium. However,

137

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Estimation of Land Parameters by Multichannel Radar Methods

the range of the soil specific RCS variations due to moisture does not exceed 8–10 dB, while the roughness variations can lead to 15–25 dB of RCS variations. Consequently, the effects of surface roughness create obstacles to correct estimation of soil moisture. The use of the dual-polarization ratio of normalized RCS for obtaining unbiased estimation of the real part of the complex dielectric constant was first proposed in [1], and this ratio was also used in [2]. The application of multifrequency radar technique to determination of surface roughness was proposed in [3], where this approach was used for small incidence angles, and joint processing was not applied. In papers [4–8], the application of multichannel techniques to separate determination of near-surface soil moisture and surface roughness was proposed, and joint processing of data was used. Taking into account that roughness parameters depend mainly on mechanical and aggregate soil structure, the successful retrieval of this information permits determining soil areas with different degrees of erosion and analyzing the dynamics of their evolution. This chapter presents the results of theoretical and experimental investigations into multichannel remote sensing techniques applied to estimation of soil parameters in the microwave band. 3.1.2 Soil Backscattering Modeling

In modeling wave scattering from natural terrains, one generally expects a combination of surface and volume scattering, especially when a dry soil medium is inhomogeneous. The soil medium can be treated as a volume consisting of variable fractions of soil solids, aqueous fluids, and air. Soil solid material is a mixture of sand, silt, and clay, and it is characterized by a distribution of particle sizes (texture) and mineralogy of their consistent particles (particularly, a clay fraction). The particle diameter of clay is less than 2 ␮ m, the particle diameter of silt is between 2 ␮ m and 50 ␮ m, and the particle diameter of sand is between 50 ␮ m and 200 ␮ m. The water in soil is classified as free or bound water. Modeling of scattering from bare natural and agricultural soils starts from a characterization of the surface roughness and dielectric behavior of materials. The latter depends on several parameters, including the bulk density, the particle size distribution, the mineralogical composition, the content of organic matter, the content of bound and free water, the soil salinity, and the temperature. The problem of electromagnetic wave scattering from rough surfaces has been studied extensively for many years. However, because of the complexity of the problem, satisfactory solutions are available only when very stringent restrictions are imposed on the rough surface profile, the electromagnetic parameters of the irregular boundary, or the frequency. Models can be divided into two groups: field-approach models and intensity-approach models [9]. The disadvantage of the existing field-approach models is that they cannot in practice include multiple incoherent scattering beyond the second order. On the other hand, the intensity-

3.1 Estimation of Soil Parameters

139

approach models can include more multiple scattering terms and the interaction between surface and volume scattering, but it assumes a far field interaction between scatterers. We restrict ourselves to rough surface scattering models only and neglect volume scattering, although in practice these two effects are sometimes difficult to separate. For bare soil, the surface scattering is dominant only if the terrain can be considered homogeneous. Two rough surface scattering models are widely applied because of their simplicity: Kirchhoff’s model and the first-order small perturbation model [10]. In most backscattering applications, Kirchhoff’s model is used over the incidence angular region 0 < ␪ < 20°. This model is restricted to high frequencies and indicates that for perfectly conducting surfaces the backscattered fields do not depend on polarization. The small perturbation model is used over the angular range 20° < ␪ < 60°. Larger incidence angles are not considered because the scattering mechanisms for grazing incidence are likely to be different from a purely surface scattering phenomenon. Using a perturbational approach derived for surfaces with small gradients (k␴ h < 0.3, where k = 2␲ /␭ is the wavenumber and ␴ h is the rms roughness height), the backscattering field is shown to be strongly dependent on the polarization of the incident and scattered waves. Other scattering models are, perhaps, potentially more powerful than the previous ones, but they have not yet been extensively applied to the interpretation of experimental measurements. Without going into detail, we mention three methods that seem to be the most promising: the full-wave method of Bahar [11] bridging the wide gap existing between the perturbational solutions for rough surfaces with small slopes and the quasi-optics solutions, the diagram method [12], and the stochastic Fourier method [10]. These methods are not yet fully developed, but the results obtained appear to be very encouraging. In all of the models of rough surface scattering, the normalized RCS of surface characterizing the intensity of the scattered field is a product of two functions 0 ␴ pp ( f , ␪ i ) = D pp [⑀ s ( f ), ␪ i ] ⭈ S ( f , ␪ i )

(3.1)

The first function D (⭈), or so-called dielectric function, characterizes the dielectric properties of the scattering medium and depends upon the polarization pp = HH or VV, the angle of incidence ␪ i , and the dielectric constant ⑀ s . For these models, the dielectric functions are equal or proportional to the Fresnel reflectivity (i.e., they present equally the RCS dependence as a function of dielectric constant of the medium). Their main limitation is an inadequate estimation of the scattering coefficients for cross-polarized components of the scattered signals. The second function S (⭈) takes into account the surface roughness influence on normalized RCS, where f denotes the radar frequency.

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Estimation of Land Parameters by Multichannel Radar Methods

For analysis of the capabilities of the multichannel method, we use the simple surface scattering models, particularly the small perturbation model described in the previous chapter. There are several reasons for this. First of all, the condition k␴ h > 1 is usually fulfilled for agricultural soils with different methods of cultivation in the microwave band. Then, the small perturbation model can be used only for approximated estimation of the normalized RCS. However, the model derivations of normalized RCS dependences as functions of incidence angles, rms surface roughness, and soil moisture coincide rather accurately with the experimental results up to and beyond k␴ h > 2.5–3.0 [12]. The most significant differences between the model and experimental results are observed for the cross-polarized components of the scattered signals. Besides, for the considered multichannel methods, we have used the ratios of the normalized RCS at different frequencies and polarizations, where the absolute error of RCS estimation from the perturbational model do not greatly influence the ratio values. For the case of backscattering we obtain D pp [⑀ s ( f ), ␪ i ] = | ␣ pp (␪ i ) |

2

2

S ( f , ␪ i ) = 8(k cos ␪ i )4 ⭈ ␴ h ⭈ W (2k cos ␪ i )

␣ VV =

(⑀ s − 1) [sin2 ␪ i − ⑀ s (1 + sin2 ␪ i )]

(⑀ s cos ␪ i + √⑀ s − sin2 ␪ i )2

␣ HH =

⑀s − 1

(cos ␪ i + √⑀ s − sin2 ␪ i )2

(3.2) (3.3) (3.4)

(3.5)

where W (⭈) is the surface roughness spectrum. The moisture content determination is based on the correlation between the dielectric function (3.2) and the soil dielectric constant, as well as on the dielectric function dependence on frequency and soil moisture. The results of simulation [5–7] have shown that the dielectric functions had a weak dependence upon frequency in the microwave band, while the soil dielectric constant differed significantly for different frequencies. This is illustrated in Figure 3.1. Analysis shows that the maximal differences of dielectric functions do not exceed 1 dB if the ratio of two frequencies satisfies the condition 1 < f2 /f1 < 2 − 3

(3.6)

(i.e., for the microwave band, the dielectric functions are practically frequency independent). The most obvious dependence of D on f takes place for moisture

3.1 Estimation of Soil Parameters

141

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Figure 3.1 The dielectric functions D versus soil moisture for horizontal polarization, frequencies 1–18 GHz, and incidence angle 60°. (From: [5].)

0.05–0.2 g ⭈ cm−3. The lower bound of this interval deals with the appearance of free water molecules in the soil, and when the moisture content exceeds 0.2–0.25 g ⭈ cm−3, saturation of dielectric functions occurs. This phenomenon leads to two important conclusions: •

•

The potential moisture content sensitivity is equal for all frequencies in the microwave band. The weak frequency dependence of the dielectric functions permits soil moisture estimation for the upper soil layer and measurement of other statistical characteristics on the basis of joint analysis of multichannel remote sensing data.

As shown in [5], in the framework of the small perturbation approach, it is theoretically possible to separate and to accurately estimate the roughness and the moisture parameters on the basis of multichannel measurements without a priori knowledge of surface statistical characteristics. In the most simple case of twofrequency remote sensing, it is possible to select the frequencies f 1 , f 2 and the

142

Estimation of Land Parameters by Multichannel Radar Methods

angles of incidence ␪ 1 , ␪ 2 in such a way that the ratio of corresponding functions of roughness does not depend on surface statistical characteristics. The condition of such independence is described by the equation k 1 sin ␪ 1 = k 2 sin ␪ 2

(3.7)

Then the ratio of the RCS for two polarizations is 0 ␴ pp ,1 0 ␴ qq ,2

=

0 ␴ pp ,1( f1, ␪1) 0 ␴ qq ,2( f2, ␪2)

=

冉

冊

4 D pp (⑀ 1 , ␪ 1 ) k 1 cos ␪ 1 ⭈ D qq (⑀ 2 , ␪ 2 ) k 2 cos ␪ 2

(3.8)

Here pp , qq = HH or VV, ⑀ 1 = ⑀ ( f 1 ), and ⑀ 2 = ⑀ ( f 2 ). The ratio (3.8) is a function of dielectric constants, wavelengths, and angles of incidence, and it does not depend on surface statistical characteristics. Remote moisture determination can be performed with (3.8), taking into account that the dielectric constants are functions of the volume moisture content of the soil upper layer. Here we can consider five polarization ratios R1 =

0 ␴ HH, 1 0 ␴ HH, 2

; R2 =

0 ␴ HH, 1 0 ␴ VV, 2

; R3 =

0 ␴ VV, 1 0 ␴ HH, 2

; R4 =

0 ␴ VV, 1 0 ␴ VV, 2

; R5 =

0 ␴ HH 0 ␴ VV

(3.9)

The polarization ratio R 5 is the particular case when f 1 = f 2 = f , pp = HH, and qq = VV. It was proposed for the first time in [1], where it was shown that this ratio served for obtaining unbiased estimation of the real part of the complex dielectric constant; this ratio is also used in [5]. Moisture dependence R 5 for f = 10 GHz and for different angles of incidence is shown in Figure 3.2. The nonlinear behavior of this dependence is evident. The variation range of R 5 increases with an increase in the angle of incidence and for ␪ = 50°–60°, it is approximately equal to 6 dB. The maximal moisture sensitivity is observed for volume moisture content less than 0.25 g/cm−3 and is approximately equal to 0.24 dB/0.01 g/cm−3. For wetter soil, saturation is observed, and the sensitivity decreases by a factor more than 3. Analysis of ratios R 1 –R 4 as functions of the moisture is done in [5], where it is shown that forming of these estimates provides a maximum moisture content sensitivity greater than 0.1 dB/0.01 g/cm−3 when frequency ratio f 1 /f 2 = 1.2–2.5; this is practically available. Multichannel techniques for estimation of statistical characteristics for some surfaces in the framework of the small perturbation method is based on the use of the following relationship [4] SR i =

0 ␴ pp (␪ i , f 1 ) 0 ␴ pp (␪ i , f 2 )

=

D pp [␪ i , ⑀ s ( f 1 )] ⭈ S (␪ i , f 1 ) D pp [␪ i , ⑀ s ( f 2 )] ⭈ S (␪ i , f 2 )

(3.10)

143

3.1 Estimation of Soil Parameters

Figure 3.2 The R 5 ratio versus soil moisture at 10 GHz for incidence angles 20°–60° (From: [5].)

The weak frequency dependence of the dielectric function in the microwave band permits us to assume that D pp [␪ i , ⑀ s ( f 1 )] ≅ D pp [␪ i , ⑀ s ( f 2 )]

(3.11)

This assumption is permissible taking into account the fact that the instrumental accuracy of normalized RCS estimation is 1.5 dB, especially if the operation frequencies f 1 and f 2 are comparable [i.e., when ( f 2 − f 1 ) << ( f 2 + f 1 )/2 and f 2 / f 1 < 1.5]. In such conditions (3.10) is a function of surface parameters only and does not depend on ⑀ s ( f ). That is, SR i =

冉 冊 k1 k2

4

⭈

W (2k 1 sin ␪ i , 0) W (2k 2 sin ␪ i , 0)

(3.12)

where, as noted earlier, W is the two-dimensional surface roughness spectrum. It is possible to assume for simplicity that

冉 冊

4

SK i = SR i ⭈

k2 ; ⌳ 1, i = 2k 1 ⭈ sin ␪ i ; ⌳ 2, i = 2k 2 ⭈ sin ␪ i k1

(3.13)

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Estimation of Land Parameters by Multichannel Radar Methods

The model of surface roughness spatial spectrum determines the further estimation of the statistical characteristics. Most often, surfaces with a Gaussian surface height distribution and any surface autocorrelation function are used. For surfaces with this height distribution and a Gaussian autocorrelation function, analytical expressions are available for cross checking. In addition, many surface scattering theories have been reported in the literature using the assumption of Gaussian height distribution, although the roughness spectrum of naturally occurring surfaces generally has more high frequency components than the Gaussian. For a Gaussian surface, the autocorrelation function is

⁄

2

␳ (␰ ) = ␴ h ⭈ exp (−␰ 2 l 2 )

(3.14)

and the spatial power spectrum of roughness is W (⌳) =

2 ␲␴ h l 2

冋 冉 冊册

⌳l exp − 2

2

(3.15)

Here l is the spatial correlation radius, ⌳ is the spatial wavenumber. The exponential autocorrelation function of the surface is also used

冉 | |冊

2

␳ (␰ ) = ␴ h exp −

␰ l

(3.16)

with a spatial power spectrum 2

⁄

W (⌳) = 4␴ h l 2 (1 + ⌳2l 2 )

(3.17)

The rapidly increasing number of applications of fractal models and fractal geometry in physics deserves close attention in studies of various areas, and particularly in remote sensing of land. It was first demonstrated by M. Berry [13] that for Gaussian statistics the surfaces with a simple power spectrum of the type W 1 (⌳) = C 1 ⌳−␣,

1<␣<3

(3.18)

are fractal rough surfaces. We note that the spectrum W is defined for any real ⌳, and therefore, the corresponding surface has no characteristic scales and the rms height is not defined (␴ h → ∞). For the values of the spectral exponent ␣ given by (3.18), the Fourier transform of W is a divergent integral; thus, the correlation function does not exist.

145

3.1 Estimation of Soil Parameters

The second surface type has the same expression for the roughness spectra but with an outer (or large scale) cutoff ⌳0 W 2 (⌳) =

再

C ⌳−␣

for ⌳ ≥ ⌳0

0

for ⌳ ≤ ⌳0

(3.19)

In view of the experimental data, type II surfaces appear to be of particular importance in the practice of remote sensing (here the usual case is ␣ = 2). We should like to note that for large ⌳ the spectra (3.17) and (3.18) are identical, but type II spectra have finite rms height ␴ h and infinite rms slope. For a Gaussian surface autocorrelation function, we find the correlation radius estimate as ˆl g =

√4 ln (SK i ) ⁄ 冠⌳2, i − ⌳1, i 冡 2

2

(3.20)

For surfaces with an exponential correlation function, the estimate of the correlation radius can be easily obtained as ˆl e = 冋(SK i − 1)

⁄ 冠⌳2,2 i − SK i ⌳1,2 i 冡册1/2

(3.21)

The estimate of the fractal spectrum exponent is

⁄

␣ˆ = ln (SK i ) ln (⌳2, i /⌳1, i )

(3.22)

The constant C is linked with ␴ h by the expression

⁄

␴ h = 冠√C ⭈ ⌳1o − ␣ /2 冡 [a ⭈ sin (␲ /␣ )]

(3.23)

The value of ⌳0 is not greater than 2 when ␣ = (1–3⌳); for ␣ = 2, the value of C = ␴ h . Therefore, it is obvious that in the multichannel approach, we can obtain only one surface characteristic from the radar measurements: its correlation radius or the fractal spectrum exponent. 3.1.3 Efficiency of Multichannel Methods

The efficiency of considered multichannel approaches was investigated on the basis of the experimental results obtained in [3, 14]. As an example, the first surface

146

Estimation of Land Parameters by Multichannel Radar Methods

type is asphalt, for which the small perturbation method can be applied without limitations. The angular dependences of the normalized RCS for asphalt are obtained in [14] at frequencies of 8.6 GHz, 17.0 GHz, and 35.6 GHz for two polarizations (horizontal and vertical) and for the incidence angle region of 20–80°. Before analysis of asphalt dielectric and statistical characteristics, it is worth noting that the corresponding values are presented in some references (e.g., in [15, 16]). At frequencies of 10 GHz and 35 GHz, the averaged dielectric constants of asphalt are ⑀ a = 4.3 ± 0.5 and 2.5 ± 0.3, respectively. The surface roughness correlation radius and rms height are l = 0.22 cm and ␴ h = 0.04 cm. The derivation of the dielectric function D for the small perturbation model shows that, for asphalt, its frequency dependence should be taken into consideration. The difference between dielectric functions for frequencies of 10 GHz and 35 GHz is approximately 2–4 dB for incidence angle variation from 20° to 60°. That is why the use of (3.10) for asphalt statistical characteristics estimation at frequencies of 17 GHz and 35.6 GHz is impossible, as (3.11) is not valid for these frequencies. The dielectric constant of asphalt was estimated using the polarization ratio R 5 that is a monotonic function of ⑀ a . The averaged values obtained for all three frequencies are given in Table 3.1. These results match well with other reference data, and they confirm the practical applicability of polarization ratio R 5 for dielectric constant estimation. Using (3.10), estimation of asphalt surface roughness correlation radius and fractal spectrum exponent was performed. In this case (3.10) is a function only of l and ␣ . The data for every frequency and polarization were considered statistically independent. The estimates of surface parameters obtained for different channels and polarizations as well as their averages are presented in Table 3.2. It is seen from Table 3.2 that from the point of view of statistical reliability, the averaged estimate of the fractal spectra exponent is the best in comparison to similar estimates of correlation radius. Its confidence interval is approximately Table 3.1 Averaged Values of ⑀ a f (GHz) ⑀a

8.6 3.56 ± 1.22

17 3.34 ± 0.38

35.6 1.77 ± 0.51

Table 3.2 Estimates of Asphalt Surface Roughness Correlation Radius, Height rms Values, and Fractal Spectra Exponent f (GHz) 8.6 17.0 35.6 Averaged values

l g (cm) 0.71 0.41 0.20

l ex (cm) 0.69 0.50 0.26

␴ hg (cm) 0.12 0.23 0.20

␴ hex (cm) 0.14 0.25 0.21

0.44 ± 0.2

0.48 ± 0.24

0.20 ± 0.05

0.20 ± 0.05

␣ 1.34 1.66 1.61

␴ C (cm) 0.107 0.111 0.093

1.39 ± 0.16 0.103 ± 0.008

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3.1 Estimation of Soil Parameters

20% in respect to ␣ , while the range of respective confidence intervals for estimates of l is in the limit 50%. Estimates of the last unknown parameter ␴ h (or C ) were formed using (3.1) 0 by means of minimizing the difference between experimental data of ␴ pp and 0 theoretically derived approximate values of ␴ˆ pp . Because the amplitude of fluctuations depends on (␣ − C ) value, it is easier to analyze not C but the value ␴ C , taking into account that ␴ C = C /M 4 − ␣ [m] (i.e., expressed in meters as well as ␴ h ). The estimates obtained for different frequencies are shown in Table 3.2. As seen from Table 3.2, the estimate of ␴ C is the best from the point of view of its statistical reliability. Its confidence interval is less than 5% with respect to its average value, while the analogous intervals for ␴ h lie in the limits 20%–30% of the estimates for Gaussian and exponential surfaces. As illustration of fractal approximation possibilities, Table 3.3 shows the derived values for the surface roughness spectra W (⌳i ) obtained using C estimates at different frequencies (here ⌳i = k 1 , k 2 , k 3 and ␪ i = 30°). These generalized data for all frequencies with application to average values of asphalt dielectric and statistical characteristics show that the Gaussian approximation is not appropriate. It is illustrated well by Figure 3.3, where the experimentally obtained data at 35.6 GHz for horizontal and vertical polarizations and the theoretical curves for Gaussian surface are compared. Figure 3.4 represents the angular dependence of ␴ 0 of asphalt at 35.6 GHz and model representations for exponential and fractal surfaces. The absolute average discrepancies of these curves from obtained values are 1.15 and 1.3 dB for exponential and fractal surfaces, respectively. In practice, the two approximations are equally valid. Analogous estimates of asphalt dielectric constant and its surface statistical characteristics were obtained for 8.6 GHz and 17 GHz. They coincide well with l and ␣ estimates obtained earlier, on the basis of which the dielectric constant estimates for asphalt were derived. Because for practical purposes we were interested in applicability of fractal spectra for description of surface roughness, this approximation is a subject of further consideration. The asphalt dielectric constant values obtained at 8.6 GHz and 17 GHz (␣ = 1.38) were equal to 3 ± 1.9 and 4.5 ± 2.0, respectively. The confidence intervals of these estimates were rather wide but they overlapped. This could result from the accuracy of measurements and the validity of assumption (3.10) for asphalt at these frequencies. It is worth noting Table 3.3 The Spectral Density of Asphalt Roughness (in Decibels) at Frequencies f 1 = 8.6, f 2 = 17.0, and f 3 = 35.6 GHz for ␪ i = 30° f (GHz) 8.6 17.0 35.6

W (k 1 ) −97.8 −97.4 −99.4

W (k 2 ) −101.9 −101.5 −101.0

W (k 3 ) −106.4 −106.0 −105.5

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Estimation of Land Parameters by Multichannel Radar Methods

Figure 3.3 Angular dependence of ␴ 0 for VV and HH polarizations and their comparison with Gaussian models. (Solid and dotted curves are derived from models; signs are the experimental data.)

the very close values obtained for the latter model parameter C (␴ C ). We got ␴ C = 0.146 cm and 0.148 cm for 8.6 GHz and 17 GHz, respectively. The second type surface is a bare field. The experimental angular dependence of the normalized RCS for this field obtained at 4.7 GHz for two polarizations is presented in [3]. The soil is characterized by its high clay content (about 40%). The measurements have been carried out for dry soil (gravimetric moisture 4.3%) and for wet soil with gravimetric moisture (about 30.2%). The high clay content results in a temporally invariant rms surface roughness of about 2.5 cm. The estimates of exponent ␣ were formed from data of ␴ 0 using (3.22) and the regression analysis method, where the second frequency was 8.6 GHz. The | ⑀ | and ␣ data for dry and wet soils are shown in Table 3.4. The ␣ values for dry soil are practically identical for both polarizations, and they differ significantly for wet soil. Using C estimations, we derived the values of surface roughness spectral density, also presented in Table 3.4. The comparison of W (k ) for soil with W (k ) for asphalt at 17.0 GHz shows that the roughness influence on the normalized RCS of a field is greater than that for asphalt by 15–20 dB (see data in Table 3.3). The application of Gaussian and exponential models for the autocorrelation function permits us to estimate the correlation radius of field roughness. The surface roughness for this field does not satisfy the small perturbation model. In these conditions, the derived values of l are approximately 1 cm and do not correspond

3.1 Estimation of Soil Parameters

149

Figure 3.4 Angular dependences of ␴ 0 for VV and HH polarizations and their comparison with (a) exponential and (b) fractal models.

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Estimation of Land Parameters by Multichannel Radar Methods Table 3.4 The Values of | ⑀ |, ␣ , and W (k 1 ) for Dry and Wet Soil Polarization Gravimetric volume, % |⑀ | ␣ W (k 1 ) (dB)

HH 4.3 3 0.8–1.1 −(76.7–78.5)

HH 30.2 35 0.8–1.1 −(76.7–78.5)

VV 4.3 3 1.0–1.3 −(79.8–80.5)

VV 30.2 35 2.3–2.5 −(87.4–88.1)

to real values of l for agricultural fields. At the same time, the fractal approximation of surface spectra seems to be more suitable because the spectral exponent is in a good agreement with the value range (␣ = 1–3). Thus, the application of multichannel methods permits us to obtain the soil dielectric constant and, consequently, the soil moisture and the soil statistical characteristics using the fractal model for the surface spectra—even for rough surfaces when the limitations of the small perturbation model are inapplicable.

3.2 Soil Erosion Experimental Determination 3.2.1 Set and Technique of Measurement

One of the purposes of multichannel methods of remote determination of soil parameters is estimation of the degree of erosion, because this determines the soil fertility. The erosion influences the radar characteristics of the soil, and, consequently, one can find the correlation between the scattered signal and the soil statistical characteristics. An experimental study of the multichannel method of soil characteristics estimation has been carried out [17]. The agricultural field was located on a hill, and its different areas had average slopes between 2° and 8°. The cultivation performed 3 weeks earlier across the slope included plowing, disking, and harrowing. The soil type was chernozem (black soil) with four stages of soil erosion. Visual inspection indicated that the periodic row structure was almost destroyed by many rainfalls. A dual-frequency CW radar was used for experimental investigation with characteristics as shown in Table 3.5. The radar system included two radars with antennas placed on a common platform, the antennas receiving backscattered signals from the same area of surface. A peculiarity of this radar was the possibility of signal radiation with any linear polarization and simultaneous reception of two orthogonal components of the scattered signal. Radar measurements were followed by simultaneous in situ measurements of soil characteristics, including obtaining the surface profiles along and across the direction of cultivation, the moisture of the near-surface soil layer (from 2 cm to 5 cm in depth), and the agrophysical characteristics of the soil. Determination of the normalized RCS was made using the method of comparison with the RCS of a corner reflector for calibration. The latter was installed at

151

3.2 Soil Erosion Experimental Determination Table 3.5 The Main Characteristics of the Radar Wavelength (cm) Modulation Transmitted power (mW) Sensitivity (dBW) Antenna pattern width (degree) Polarization —Radiation —Reception Polarization isolation (dB) Maximal bandwidth of receiver (kHz)

3.2 CW 50.0 −165 10.0

0.8 CW 50.0 −155 10.0

H or V Two orthogonal components 35 2.0

H or V Two orthogonal components 30 2.0

a height of 3m over the land, providing a negligible influence of background surface clutter on the calibration accuracy. The main radar characteristics were the normalized RCS dependence on the incidence angles for two polarizations (horizontal and vertical) and for two wavelengths (3 cm and 0.8 cm), (i.e., four angular dependences—for copolarized and cross-polarized components of the scattered signal—for every polarization of the radiated signal). The measurements were carried out for an incidence angular interval 35°–70° with step of 5°. For the normalized RCS estimates, the averaging of angular dependence for several azimuthally spaced angles was used. This resulted in decreasing of the normalized RCS fluctuations to 1.0–1.5 dB. 3.2.2 Statistical and Agrophysical Characteristics of Fields

The erosion state of the soil areas was determined by the average-weighted diameter of water-stable aggregates and by the aggregation coefficient Ka [18]. This depends on the quantity and quality of humus, granulometer, and mineral content of the soil and characterizes the genetic particularity of the soil. The moisture and aggregate content characteristics are shown in Table 3.6. As seen from Table 3.6, the agrophysical characteristics of the soil in areas 1–3 are rather similar, and the most different properties are observed for area 4. The moisture measurements at the reference points were carried out by the thermostat-weight technique and showed that the moisture for the near-surface layer of the soil was rather homogeneous for all areas and the gravimetric soil moisture was near the field capacity. This was caused by weak soil drying because

Table 3.6 Weighted Soil Moisture and Aggregate Content Characteristics Terrain area Erosion degree Soil moisture (%) d (mm) Ka

1 Noneroded 16.2 0.85 0.57

2 Weakly eroded 12.9 0.46 0.39

3 Middle eroded 17.0 0.38 0.37

4 Heavy eroded 11.8 0.25 0.23

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Estimation of Land Parameters by Multichannel Radar Methods

of the low air temperature (it was about 5°–6° ). Only the moisture in area 3 was higher than for the other areas because it was obtained for the day after a rain. For this reason, it was possible to neglect the moisture influence on the normalized RCS for different areas of soil. The surface roughness was measured with a profile meter with density of 1 measurement per 10 mm, which ensured errors not greater than 5 mm. The common length of every surface profile was 6–7.5m. The profile processing included the obtaining of autocorrelation functions and spatial spectra along and across the directions of cultivation. The method of cultivation can result in a significantly different degree of roughness for the same agricultural region. It is worth noting that for fields with a periodic structure, the rms surface height depends greatly on the profile orientation in respect to the plowing furrows. However, for fresh-plowed fields, the periodicity is often significantly distorted by presence of clods. The small and rather large clods appearing for breast plowing are oriented randomly. Their dimensions depend on the moisture and the soil type. In general, the surface normalized autocorrelation function can be approximated by

␳ (␰ ) = exp (−␥ ⭈ ␰ ) ⭈ cos (2␲␰ /␤ )

(3.24)

where ␰ is the spatial coordinate. No statistical dependence between parameters ␥ and ␤ was detected. A more general approximation can be used for dual-scale roughness (the typical cases are the plowed and harrowed fields), taking into account the peculiarities of the cultivation. For this case

␳ (␰ ) = (1 − A ) ⭈ exp (−␥ ␰ ) ⭈ cos (2␲␰ /L p ) + A ⭈ exp (−␩ ␰ ) ⭈ cos (2␲ ␰ /L h ) (3.25) where the coefficient A is determined by the ratio of harrowing to plowing depths, L p denotes the distance between the furrows of plowing, and L h defines the distance between the furrows of harrowing. The observations show that the dynamics of change of field roughness (if the fields are not subjected to additional tillage) are fully determined by the atmosphere humidity conditions—the aggregate and mechanical soil contents. Initially, the roughness smoothing caused by rainfall can result in an increase of periodicity because the process of furrow destruction is more prolonged in comparison to the process of clod destruction. Table 3.7 shows the approximate range of ␴ h variations for different types of cultivation (the measurements were performed for along and across directions). It is worth noting that the percent content of clay in the soil is a basic factor determining the degree of field roughness after natural furrow and clod destruction. The mechanical and aggregate soil contents are subjected to slow permanent change

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3.2 Soil Erosion Experimental Determination Table 3.7 Roughness rms Values ␴ h (cm) for Some Fields Measurements Along direction of cultivation Across direction of cultivation

Fresh Plowed >4–5 3–4

Plowed and Washed Off 3–5 2–3.5

Harrowed 1.5–3 1–2

Harrowed and Rolled <1.5 <1.5

and, as the result, to erosion. The field slope of about 5°–10° is the reason for the formation of heavily and medium-eroded areas. The mechanical and hydrophysical properties of such soils differ from those of noneroded soils. Therefore, we expect that in the borders of the same terrain, different degrees of erosion can be found as the result of the same cultivation. For such fields, successful separation of roughness and moisture effects and their accurate estimation offers the ability to retrieve information about the erosion process and its evolution. Figure 3.5 presents the correlation functions of roughness across the direction of cultivation just after plowing (curve 1) and 20 days later (curve 2) when the field became smoothed and rolled. The process of furrow destruction manifests itself in the removal of the periodic structure of the autocorrelation function. The autocorrelation functions of surface roughness along the direction of cultivation hardly differ from the autocorrelation functions of isotropic surfaces. Examples of spatial spectra for investigated land areas are shown in Figure 3.6. The spectra were obtained using the standard fast Fourier transform (FFT) and the maximum entropy methods (Berg’s algorithm). It is seen from Figure 3.6(a)

Figure 3.5 The spatial autocorrelation functions of plowed field (1) right after plowing and (2) 20 days after plowing.

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Estimation of Land Parameters by Multichannel Radar Methods

Figure 3.6 The spatial spectra of investigated land lot obtained by (a) FFT and (b) maximal entropy method.

that harrowing leads to a second maximum appearing in the soil spatial spectrum, while the maximal entropy method smoothes this maximum. The rms values of surface roughness ␴ h for the cross direction are greater than for along furrow direction. In the spatial spectra, the second maximum appeared rather often; it is

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3.2 Soil Erosion Experimental Determination

connected with the cultivation type. However, in our measurements, it is not observed due to washing off of the soil. For determination of roughness characteristics, the fractal approximation of spatial spectra was used. The average values of ␴ h and fractal exponent ␣ are shown in Table 3.8. The averaged exponent in the experimental spectra was equal to ␣ = 1.61 ± 0.89. For profiles with a strongly marked periodic row structure, the average value of ␣ of the fractal spectra does not change significantly; its values are ␣ = 1.64 ± 1.02. The data analysis showed that the value of ␴ h along the direction of cultivation did not practically depend on the degree of soil erosion, but we observed a decrease of ␴ h with an increase of erosion degree across the direction of cultivation. At the same time, a decrease of ␣ was observed along the direction of cultivation when the erosion degree increased. This shows an increase of the spatial correlation radius for noneroded soils in comparison to eroded ones. 3.2.3 On-Land Radar Measurement Results

The results of the angular normalized RCS dependence study have shown the following [6, 19, 20]: •

•

•

The normalized RCS for an 8-mm wavelength is greater by 3–6 dB than for a 3-cm wavelength for all selected surface areas and for both polarizations with reception of copolarized components. The normalized RCS values for vertical polarization exceed those for horizontal polarization by 0.2–8 dB, depending upon the degree of surface erosion. The speed of the angular RCS variations depends on the degree of erosion.

The longitudinal furrow structure of the field is a dominating factor in forming the scattered signal for horizontal polarization. The washing off of roughness for heavily eroded lots results in decreasing RCS in comparison with noneroded regions. At the same time, for vertical polarization, the RCS values for strongly eroded lots are greater than for noneroded ones. This is probably explained by the different degree of cross-furrow microstructure destruction, which is larger for strongly eroded soil. For example, Figure 3.7 shows the angular dependence of normalized

Table 3.8 Roughness Characteristics of Investigated Lots Erosion Degree Along furrows ␴ h (cm) ␣ Across furrows ␴ h (cm) ␣

Noneroded 1.7 1.8 2.1 1.9

Weakly Eroded 1.54 1.6 2.45 1.75

Middle Eroded 1.46 1.35 1.9 1.85

Heavy Eroded 1.28 1.3 1.8 1.7

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Estimation of Land Parameters by Multichannel Radar Methods

Figure 3.7 Specific RCS vs incidence angle for areas with different erosion degree at 0.8 cm for (a) vertical and (b) horizontal polarizations; curve 1 is obtained for noneroded areas, the curve 2 for weakly and middle-eroded areas, the curve 3 for heavily eroded areas.

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3.2 Soil Erosion Experimental Determination

RCS for areas 3 (middle eroded) and 4 (heavily eroded) obtained at 3 cm and 0.8 cm for horizontal polarization. The mean rate of angular variation of the normalized RCS is approximately 0.4–0.5 dB/degree for area 3 and about 0.2–0.3 dB/degree for area 4. This angular dependence can be understood because the exponent of the surface roughness fractal spectrum decreases with increasing degree of erosion, as seen from Table 3.8. Dual-channel polarization processing (using ratio R 5 ) has permitted us to emphasize its sensitivity to variations of soil agrophysical parameters. This is seen 0 0 from Table 3.9 where the polarization ratios ␴ HH /␴ VV are shown averaged for all regions of incidence angles. The maximal value of this ratio at 3 cm is −6.6 dB for heavy eroded areas and 7.7 dB for non-eroded areas. Good correlation between the ratio R 5 and the agrophysical parameters of soil are observed. For incidence angles ␪ = 35°–45°, the maximum sensitivity of R 5 to agrophysical parameter variations is observed at 3 cm, but at 0.8 cm the maximum sensitivity is observed for the incidence angles greater than 60°. The angular dependence of the normalized RCS ratios for cross-polarized reception seems to have no correlation with soil erosion degree. The depolarization increases with wavelength decrease from −(10–13) dB for 3 cm to −(5–8) dB for 8 mm, and correlation of these values with erosion degree is not observed. In Figure 3.8, this dependency is shown for 3 cm and 0.8 cm and for the horizontal polarization. 3.2.4 Aircraft Remote Sensing

Sections 3.2 and 3.3 give the radiophysical model an experimental basis for determination of bare soil erosion from multichannel remote sensing data. According to this approach, one needs to have available a few radar images of the same terrain area (bare soil region) formed for VV and HH polarizations of transmitted and received sensing signals in the microwave band. But the availability of such images is not the only prerequisite for successfully measuring soil erosion using multichannel radar remote sensing data. There are also some normalized requirements on the imaging systems used, along with the algorithms of multichannel data processing. First, radar remote sensing systems should provide appropriate spatial resolution in order to be sensitive to local variations of normalized radar cross sections in the terrain region of interest. The desirable resolution is about 20 × 20 m2 or better, and this can be provided by either SLAR or SAR systems.

Table 3.9 The Ratio R 5 of Investigated Areas at 0.8 cm and 3 cm Erosion Degree 0 ␴ HH 0.8 cm (dB) 0 3 cm ␴ VV

Noneroded

Weakly Eroded

Middle Eroded

Heavy Eroded

11.4

6.0

3.4

−4.5

7.7

0.2

−3.1

−6.6

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Estimation of Land Parameters by Multichannel Radar Methods

Figure 3.8 The ratios ␴ 0HV /␴ 0HH versus the incidence angle at (a) 3 cm and (b) 0.8 cm (curves 1–3 are labeled as in Figure 3.7).

Second, RCS estimates for each pixel obtained from the observed radar images should be sufficiently accurate. This means that the imaging systems have to be properly calibrated [21], and it is also desirable to use either images with low levels of noise or to properly smooth them. At the same time, the presence of multiplicative noise is inherent in radar images, and this noise, also referred as speckle, is especially intense in images formed by SARs. Therefore, multichannel images are to be

3.3 Methods of Multichannel Radar Image Processing

159

processed in such a manner that the multiplicative noise is effectively suppressed, while the edges and fine details corresponding to local variations of RCS (and, respectively, different degree of soil erosion) are well preserved. Finally, the component images of multichannel remote sensing data should be properly registered (i.e., represented in a common spatial coordinate system with correspondence of the multichannel image cell to the element of the sensed terrain). In practice, the resolutions of different radars used for remote sensing are various, the linear resolution depending on range for all radars. Besides, the sensed terrain strips are sometimes obtained by radar during different flights of the radar platform, and, certainly, the images do not fully coincide. For example, we obtained 8-mm images from two approximately opposite directions of platform trajectory for radar operation with different polarizations while we were conducting the experiment. Terrain relief and Earth’s surface curvature are additional sources of geometric distortions of the radar image in multichannel systems for remote sensing. The images were obtained simultaneously by two radar systems at 10 GHz and 35 GHz installed onboard an IL-18D (Russian) aircraft [22]. The spatial resolution of the Ka-band SLAR was about 20 × 20 m2; for the X-band SLAR, it was approximately 30 × 30 m2; the average altitude was about 7,000m; and the agricultural field under investigation was observed at an incidence angle about 50°. The field dimensions were 1.2 × 1.2 km2, and the field occupied about 50 × 50 pixels.

3.3 Methods of Multichannel Radar Image Processing 3.3.1 Image Superimposing

The radar images are to be superimposed, using both linear and nonlinear techniques. For a case when we deal with analyzed terrain areas having relatively small sizes, it is possible to use the linear aphine transformation; otherwise, it becomes reasonable to apply a preliminary geometrical correction of images and then more complicated methods of nonlinear transformations of data arrays. In practice, several factors obstruct getting the image-to-topology map and image-to-image registration without errors. These factors are the fluctuations of the imaging system platform trajectory (for airborne systems), the curvature of the sensed terrain relief, and the nonlinear relationship between slant and ground range. So the task of accurate registration, transformation, and interpolation of multichannel remote sensing data is also important because these registration errors can result in errors in further interpretation for soil erosion determination, especially in the neighborhood of edges and details [23]. The procedure for image-to-image and image-to-topology map superimposing include the following stages:

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Estimation of Land Parameters by Multichannel Radar Methods

1. Selection of common reference points in all images. This operation is performed by qualified experts using the corners of agricultural fields, or in more general cases any bright quasi-point objects (e.g., bridges, buildings, or highway crossroads) easily recognizable in all images can be selected as reference points and found interactively. Contour (edge) features of images can be used for reference point selection as well; sometimes image preliminary filtering is expedient. 2. Derivation of coefficients describing the aphine transform of the superimposed image to the coordinate system of the reference image or topology map. Being represented in a matrix form this aphine transform is [8] F=A⭈H+D where operator D =

(3.26)

再 冎 再 冎 再 冎 再 冎 Dx Dy

image H and matrix A =

denotes the transformation of superimposed

A 11 A 12

defines the scaling and rotation operaA 21 A 22 tions to be done over the same image H. Denoting the reference point F xi G xi coordinates as F xyi = and G xyi = , i = 1, 2, . . . , l for the F yi G yi superimposed and reference images (maps), respectively, the superimposing integral errors are determined by horizontal and vertical components ␦ h and ␦ v l

␦h = ␦v =

∑ (F xi − A 11 G xi − A 12 G yi − D x )2

(3.27)

i =1 l

∑ (F yi − A 21 G xi − A 22 G yi − D y )2

i =1

Equation (3.27) produces an equation system that can be uniquely solved with respect to A and D parameters for l = 3. But for l ≥ 4 (the greater l is, the higher the accuracy of superimposing), some optimization procedure should be applied; obviously, the requirement of total square mean error minimization is typical and reasonable. This leads to a solution of the equation system obtained after calculation of partial derivatives on matrix transform coefficients. These operations are rather easy; at the first stage, the matrix D parameters are estimated, and then the matrix A coefficients have to be derived. We also analyze the problem of how many reference points it is reasonable to select and what their optimal spatial locations are.

3.3 Methods of Multichannel Radar Image Processing

161

Usually the selection of 7–12 reference points forming an equilateral polygon is applicable to many practical situations. They should be placed as sparsely as possible in the sense of maximal total distance between them. This conclusion is quite trivial [24] and intuitively understandable. 3. Interpolation of superimposed image H to the coordinate grid nodes of the reference image G. The necessity of this operation is explained by the fact that both images are sampled, and after spatial transformation the majority of grid nodes for images do not coincide. Several methods can be used for transformed image interpolation after superimposition: nearest neighbor, bilinear, and bicubic [25]. The first method happened to be more expedient because it resulted in minimal dynamic errors in the neighborhood of edges and fine details. Besides, this method is very simple and requires minimal computational efforts. Bilinear and bicubic interpolation methods provide better visual perception but are more complicated. Because of these advantages, the first technique was applied in data processing. It is based on substitution by the nearest sample value or on linear (or median) approximation taking into account the four nearest neighbor pixel values, and it provides appropriate interpolation accuracy. The errors and distortions caused by interpolation are usually smaller than multiplicative noise existing in real images. Completing the discussion on image superimposing, it is worth mentioning that the errors of this operation can exceed one or two pixels; therefore, it is desirable to decrease them if possible during further stages of data processing. Let us demonstrate these effects for real multichannel radar remote sensing data. The airborne SLAR image of an agricultural region in the Ukraine is shown in Figure 3.9 (␭ = 8 mm, HH polarization). The multiplicative noise is not very intensive, its relative variance ␴ ␮2 ≈ 0.005. The SLAR image of the same region for VV polarization is shown in Figure 3.10. The multiplicative noise in this image has the same relative variance ␴ ␮2 ≈ 0.005. The general appearance of this image is in a sense similar to Figure 3.9 (because the same terrain region is sensed) and in another sense different, as Figure 3.10 is formed for another polarization. Just this simultaneous similarity and difference between component images of multichannel remote sensing data are exploited for retrieval of useful information from them and separation of factors affecting the RCS for different wavelengths and polarizations. Figure 3.11 shows the airborne SLAR image formed by another radar (␭ = 3 cm, HH polarization). The multiplicative noise in this image is more intensive (␴ ␮2 ≈ 0.011), although it is still Gaussian with a unity mean, as with both 8-mm images considered earlier. This image is also similar to those ones shown in Figures 3.9 and 3.10, but a small difference of resolution cell size between the images is seen.

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Estimation of Land Parameters by Multichannel Radar Methods

Figure 3.9 Radar image of agricultural region of Ukraine at 8 mm for HH polarization.

Due to similarity of component images of multichannel remote sensing data, it is quite easy to select the set of ground control points (GCPs) in all images and, if needed, the topology map of the sensed region. An example of such set selection is shown in Figures 3.9 and 3.11 (10 GCPs are indicated by flags). This approach to multicomponent image joint registration is rather traditional (see [26] for details) and, in our opinion, does not require too many additional explanations. The only thing worth mentioning is the following: For joint registration of small-size component images (or when the component images have a common fragment of size tens to tens pixels) even linear (affined) transforms provide appropriate accuracy—the residual errors are usually about one pixel. In case of larger size of the fragment common for all component images, the nonlinear spatial transforms are worth applying [26] to minimize the residual superimposing errors. The application of nonlinear transforms at the image registration stage permits us to keep these errors at the level about one resolution element. An example of a three-channel jointly registered radar image obtained for component images in Figures 3.9–3.11 and represented in the monochrome mode is given in Figure 3.12. The image in Figure 3.9 was chosen as a strong point, providing the greatest sizes of surface fragments common for three images. Only the fragment common for all component images is shown in Figure 3.12, which is why this image does not have a rectangular shape. Noise in this image is observed, and the edges are a little bit smoothed (not sharp) due to residual registration errors.

3.3 Methods of Multichannel Radar Image Processing

Figure 3.10

163

Radar image of agricultural region of Ukraine at 8 mm for VV polarization.

The total accuracy of Ka-band image-to-image superimposition was characterized by an rms value of 1.97; for superimposing the X-band image to Ka-band HH, it was 3.06. Thus, the tasks of multichannel radar preprocessing resulting from the previous analysis are the following: The noise in these images should be suppressed while the edges and small size objects are to be preserved. Besides, if the residual image registration errors are rather large, it is desirable to sharpen these edges to avoid possible data interpretation errors in their neighborhoods. 3.3.2 Methods of Multichannel Radar Image Filtering

Noise in multichannel radar images or in ratio images obtained as the ratios (3.9) calculated for each pixel can be suppressed in several different ways. First, the required ratio image can be calculated as the pixel-by-pixel ratio of the corresponding jointly registered component images without prefiltering. Then, the resulting

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Estimation of Land Parameters by Multichannel Radar Methods

Figure 3.11

Radar image of agricultural region of Ukraine at 3 cm for HH polarization.

ratio image can be postprocessed by a filter [27, 28]. However, according to recent experience, this is not the best method. The reasons are the following: 2

2

2

1. The relative variance of noise in the ratio image is ␴ ␮rat ≥ ␴ ␮ 1 + ␴ ␮ 2 , where 2 2 ␴ ␮ 1 and ␴ ␮ 2 are the relative variances in component radar images used for obtaining the ratio image; thus, efficient filtering of the ratio image can be problematic because of rather intensive noise. 2. The residual image-to-image registration errors cannot be removed due to postprocessing of the ratio image. Another way is to perform multichannel radar image filtering before getting the required ratio image(s). In this case, there are two methods: 1. To perform component processing of multichannel data using filters well suited for each component image (the proper filter selection depends on the properties of noise in the corresponding component image [27]); 2. To use vector filtering of multichannel radar remote sensing data. The comparison of these two approaches shows that the latter one (i.e., the vector filtering of multichannel remote sensing data) is more useful [29]. This is

3.3 Methods of Multichannel Radar Image Processing

Figure 3.12

165

The joint radar image in monochrome regime.

because the vector filtering methods take into account the correlation that practically always exists between the component images of multichannel data. Among vector filtering methods, we designed two techniques to best suit the tasks that should be solved for the situation at hand. The first method is the use of adaptive nonlinear vector filter [28]. This filter exploits the advantages of component and vector filtering. Due to application of the vector median for several subapertures at the final stage of data processing, it provides sharpening of the smeared edges and the fine details arising because of residual registration errors. This is clearly seen in Figure 3.13, where the output multichannel image appears in monochrome representation. Comparing the images in Figures 3.12 and 3.13, it is seen that the adaptive nonlinear vector filter [16] also effectively suppressed noise in homogeneous regions of the image. The benefits provided considerably improve the interpretation of multichannel data (see the results for test data in our paper [26]). Another vector filter that performs appropriately well in the situation considered is the modified vector sigma filter [27]. This filter is unable to remove the

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Estimation of Land Parameters by Multichannel Radar Methods

Figure 3.13

Multichannel radar image after adaptive nonlinear vector filtering.

residual registration errors, which is why its application is recommended for cases when these errors are negligibly small. The noise-suppressing efficiency of the modified vector sigma filter is approximately the same as for the adaptive nonlinear vector filter. The obvious advantage of the modified vector sigma filter is that it is able to preserve the low-contrast edges in component images. For the application considered, this is a very important property, as the contrasts between the areas with different erosion degree are usually not too large.

3.4 Soil Erosion Determination from Ratio Images: Experimental Results The use of vector filtering of multichannel radar remote sensing data, as shown earlier, leads to preservation of valuable information and simultaneous removal of noise and errors. These results offer ratio images with appropriately high quality. Keep in mind that the ratio images are needed only for estimation of soilerosion degree from multichannel radar remote sensing data, according to the radiophysical model approach (Chapter 2). For other types of surfaces, like ice, water surface, forest, and agricultural fields with vegetation, other models of backscattered signals and RCS can be valid; therefore, ratio image forming can be of

3.4 Soil Erosion Determination from Ratio Images: Experimental Results

167

no use. In other words, this means that one should know a priori what pixels of jointly registered multichannel remote sensing data and ratio images correspond to bare soil lots. In our case, it was an agricultural field of size approximately 1 × 1 km2. In all images presented in Figures 3.9–3.13, this field is placed approximately in the center (up and to the left from flags 5 and 7 in Figures 3.9 and 3.11).

Two ratio images 10 lg 冠I ij /I ij 冡 that characterize the RCS ratio have been obtained using the images presented in Figures 3.9 and 3.10. Before getting these ratio images, the component 8-mm images with different polarizations have been jointly registered using the nonlinear transform and nearest neighbor interpolation. Then they have been processed by the adaptive nonlinear vector filter (the first ratio image) and by the modified vector sigma filter (the second ratio image). The difference between the RCS ratios estimated from the two ratio images was almost always less than 0.3 dB, and thus the considered methods of vector filtering produce similar results. Erosion state classification has been performed by means of setting the threshold values 8 dB, 4.5 dB, and 0 dB. Then, if the ratio image value is larger than 8 dB, the corresponding pixel is considered a noneroded area and indicated by white color in the erosion classification map given in Figure 3.14. The values in the ratio image that are within the limits 4.5–8 dB are indicated by light gray color in this figure (weakly eroded areas). The pixels with values within the limits 0–4.5 dB HH

Figure 3.14

Erosion state classification.

VV

168

Estimation of Land Parameters by Multichannel Radar Methods

correspond to middle-erosion degree (dark gray color), and the pixels with values below 0 dB correspond to heavily eroded areas of soil (black color). In situ measurements of erosion state have been performed for several control points—locations of five of them are shown in Figure 3.14. Due to this, it has become possible to compare the remote sensing classification results with data obtained from in situ measurements. This comparison has demonstrated the following: 1. For all five control points, the remote sensing classification results agree with conclusions based on in situ measurements. HH

VV

0 0 /␴ VV 2. The estimates of the ratios I ij /I ij differ from the ratios ␴ HH defined for the centers of the classes (6 dB for weakly eroded lots and 2.3 dB for middle-eroded ones) in the points of in situ measurement by less than 1.3 dB.

Such coincidence and accuracy of the obtained remote sensing and in situ measurement results, to our mind, can be considered appropriate for practical applications. Thus, it is shown theoretically that the multichannel methods permit us to separate and to estimate the roughness parameters and the dielectric constant of the near-surface layer in the soil. The obtained ratios for multichannel approaches can be used when the small perturbation model conditions are not fully satisfied. The fractal spectra are the best approximations of the surface roughness spectra. The correlation between the radar and agrophysical characteristics of soil was proved experimentally. In particular, the use of dual-polarization processing permits us to increase the sensitivity of radar remote sensing techniques to the soil erosion change.

References [1]

[2] [3]

[4]

Yakovlev, V. P., ‘‘On Radiometer and Radar Abilities for Surface Remote Sensing,’’ Proc. of State Scientific Center of Natural Resources Study, No. 26, Gidrometeoizdat, Leningrad, 1986, pp. 27–31 (in Russian). Shi, I., et al., ‘‘SAR-Derived Soil Moisture Measurements for Bare Fields,’’ Proc. of IGARSS 91, 1991, pp. 393–396. Mo, T., J. R. Wang, and T. J. Schmuggle, ‘‘Estimation of Surface Roughness Parameters from Dual Frequency Measurements of Radar Backscattering Coefficients,’’ IEEE Trans. Geoscience and Remote Sensing, Vol. GE-26, 1988, pp. 574–579. Zerdev, N. G., and G. P. Kulemin, ‘‘Surface Statistical Characteristics Determination by Multichannel Radar Methods,’’ Proc. Scientific Apparatus Design for MM and subMM Radiowave Bands, Institute of Radiophysics and Electronics of Ukrainian Academy of Science, Kharkov, Ukraine, 1992, pp. 90–98 (in Russian).

References [5] [6]

[7]

[8]

[9] [10]

[11] [12]

[13] [14] [15] [16] [17]

[18] [19] [20]

[21]

[22] [23]

169 Zerdev, N. G., and G. P. Kulemin, ‘‘Soil Moisture Determination by Multichannel Radar Techniques,’’ Soviet Journal of Remote Sensing, 1993, Vol. 1, No. 11, pp. 139–152. Kulemin, G. P., et al., ‘‘Soil Moisture and Erosion Degree Estimation from Multichannel Microwave Remote Sensing Data,’’ Proc. Europ. Symp. SPIE on Satellite Remote Sensing, Paris, France, September 1995, Vol. 2585, pp. 144–155. Kulemin, G. P., et al., ‘‘Soil Erosion Characteristics Estimation Techniques Using Multipolarization MM-Band Remote Sensing Radar Systems,’’ Proc. URSI Open Symp., Ahmedabad, India, November 1995, pp. 185–188. Kulemin, G. P., et al., ‘‘Radar Dual-Polarization Remote Sensing of Soil Erosion,’’ Proc. European Symp. Aerospace Remote Sensing, London, England, September 1997, Vol. 3222, pp. 89–100. Ishimaru, A., Wave Propagation and Scattering in Random Media, New York: Academic Press, 1978. Bahar, E., ‘‘Full Wave Solutions for the Scattered Radiation Fields from Rough Surfaces with Arbitrary Slopes and Frequency,’’ IEEE Trans. Antennas and Propagation, Vol. AP-28, 1980, pp. 11–21. Zipfel, C. C., and J. A. DeSanto, ‘‘Scattering of a Scalar Wave from a Random Rough Surface: A Diagrammatic Approach,’’ J. Math. Phys., Vol. 13, 1972, pp. 1903–1911. Brown, G. S., ‘‘A Stochastic Fourier Transform Approach to Scattering from Perfectly Conducting Randomly Rough Surfaces,’’ IEEE Trans. Antennas and Propagation, Vol. AP-30, 1982, pp. 1135–1144. Feder, J., Fractals, New York: John Wiley, 1988. Ulaby, F. T., R. K. Moore, and A. K. Fung, Microwave Remote Sensing: Active and Passive, Vol. III: From Theory to Applications, Norwood, MA: Artech House, 1986. Ulaby, F., ‘‘Radar Measurement of Soil Moisture Content,’’ IEEE Trans. Antennas and Propagation, Vol. AP-22, 1974, pp. 257–265. Kulemin, G. P., and V. B. Razskazovsky, The Scattering of Millimeter Radiowaves by the Earth Surface at Low Angles, Kiev: Naukova Dumka, 1987 (in Russian). Zerdev, N. G., and G. P. Kulemin, ‘‘Soil Erosion Effects in Microwave Backscattering from Bare Fields,’’ 24th Europ. Microwave Conf. Proc., Vol. 1, Cannes, France, September 5–8, 1994, pp. 431–436. Bulygin, S. J., and F. N. Lisitsky, ‘‘Microaggregation as the Indicator of Anti-Erosion Soil Stability,’’ Pochvovedenie, No. 12, 1991, pp. 98–104 (in Russian). Kulemin, G. P., ‘‘Soil Erosion Estimation by Dual-Polarization Radar Remote Sensing,’’ IGARSS’99 Proc., Vol. 2, 1999, pp. 846–848. Kulemin, G. P., ‘‘Radar Estimation of Soil Parameters by Multichannel Methods,’’ Physics and Engineering of Millimeter and Submillimeter Waves, MSMW Third Int. Kharkov Symp., Vol. 1, 1998, pp. 53–59. Zelensky, A. A., et al., ‘‘Locally Adaptive Robust Algorithm of Image Processing,’’ Inst. Radiophysics and Electr. NANU, Preprint 93-8, Vol. 39, Kharkov, Ukraine, 1993 (in Russian). Kalmikov, A. I., et al., ‘‘Multipurpose Aircraft Radar System for Earth’s Investigations,’’ IRE NANU, Preprint 90-21, Kharkov, Ukraine, 1990 (in Russian). Kulemin, G. P., et al., ‘‘MM-Wave Multichannel Remote Sensing Systems and Algorithms of Image Processing,’’ Conf. Digest of Int. Conf. on MM-waves and Infrared Science and Technology, Guanzhou, China, August 1994, pp. 359–362.

170

Estimation of Land Parameters by Multichannel Radar Methods [24] [25] [26] [27]

[28]

[29]

Richard, J. A., Remote Sensing Digital Image Analysis: An Introduction, Berlin, Germany: Springer-Verlag, 1986. Achmetyanov, V. P., and A. Y. Pasmurov, ‘‘Processing of Radar Images in Tasks of Earth’s Remote Sensing,’’ Foreign Electronics, No. 1, 1987, pp. 70–81. Kulemin, G. P., et al., ‘‘Soil Erosion State Interpretation Using Pre- and Postprocessing of Multichannel Radar Images,’’ Proc. Europ. Symp. SPIE, Vol. 3499, 1998, pp. 142–151. Kurekin, A. A., et al., ‘‘Processing Multichannel Radar Images by Modified Vector Sigma Filter for Soil Erosion Degree Determination,’’ Proc. Europ. Symp. SPIE, Vol. 3868, 1999, pp. 162–173. Lukin, V. V., et al., ‘‘Data Fusion and Processing for Airborne Multichannel System of Radar Remote Sensing: Methodology, Stages, and Algorithms,’’ Proc. SPIE, Vol. 4051, 2000, pp. 188–197. Lukin V. V., et al., ‘‘Digital Adaptive Robust Algorithms for Radar Image Filtering,’’ Electronic Imaging, Vol. 5, No. 3, 1996, pp. 410–421.

CHAPTER 4

Sea Backscattering at Low Grazing Angles

4.1 Sea Roughness Features for Small Grazing Angles 4.1.1 Sea Roughness Characteristics

Sea roughness is determined by different atmospheric factors, the most significant of which is the wind. The roughness state can be determined by the significant wave height: the mean (peak-to-trough) height of the highest one-third of the waves. There are two numerical scales for roughness description. One of them is the Douglas scale for description of surface roughness and swell, shown in Table 4.1. The second is the Beaufort scale, characterizing the surface wind velocity, shown in Table 4.2. Measurement of the probability distribution of wave heights shows that 45% of oceanic waves have heights less than 1.2m, and 80% have heights less than 3.6m. The sea surface is a complex natural formation on which wind speed, sharp crests, breaking waves, water spray, and foam are all observed to affect the scattered signal and, along with radar frequency and grazing angle, to determine the radar clutter level.

Table 4.1 Douglas Scale Roughness and Swell State 0 1 2 3 4 5 6 7 8 9

Significant Height (m) 0 0.3 0.3–0.9 0.9–1.5 1.5–2.4 2.4–3.6 3.6–6.0 6.0–12.0 12.0 —

State Characteristics Calm Smooth Slight Moderate Rough Very rough High Very high Precipitous Confused

171

172

Sea Backscattering at Low Grazing Angles Table 4.2 Beaufort Scale Wind State 0 1 2 3 4 5 6 7 8 9 10 11 12

Wind Velocity (m/s) 0.5 0.5–1.5 2.0–3.0 3.5–5.0 5.5–8.0 8.5–10.0 11.0–13.5 14.0–16.5 17.0–20.0 20.5–23.5 24.0–27.5 28.0–31.5 32.0

Wind Characteristics Swell Light air Light breeze Gentle breeze Moderate breeze Fresh breeze Strong breeze Moderate gale Fresh gale Strong gale Whole gale Storm —

The origin of these phenomena on the surface is the near-water wind, whose duration T is related to the fetch X S by X S = 1/2 ⭈ c (w ) ⭈ T

(4.1)

where c (w ) is the phase velocity of sea waves. For different wind velocities, the fetch changes within bounds of one to a few hundred kilometers (i.e., the normalized fetch for fully developed roughness x S = X S g /u *2 equals 107 to 108, where g is the acceleration constant and u * is the friction velocity usually introduced for wave structure analysis of the near-water atmospheric layer and independent of the height of wind velocity measurement. Usually, this value is related to the mean wind velocity V at height of z above the surface by

冉

gz u V (z ) = * ln ϖ 0.035u *2

冊

(4.2)

where ϖ is Karman’s constant, equal to 0.4 according to results of many measurements. The approximate friction velocity dependence on the mean wind velocity can be presented as [1] u * ≈ 0.05V

(4.3)

As seen from the experimental data obtained in wave tanks, the dependence of X S , the wave rms height ␴ w and its slope variation ␴ ⵜ␰ on the wind velocity V for fixed fetch has the form shown in Figure 4.1. At first, the rms height and wave slope angles increase slowly for increasing wind velocity, and after a certain velocity is reached, depending on the fetch, a more rapid growth is observed. The

173

4.1 Sea Roughness Features for Small Grazing Angles

Figure 4.1 Sea wave height dependence on wind velocity for different fetches. (After: [2].)

wind velocity for which this change takes the place is called as the critical velocity Vc . The dependence ␴ w = f (V ) is described rather well by the function [2]

冉

V − Vc ␴W = A Vc

冊

2

(4.4)

and the slope angle variation by 2

冋冉

␴ ⵜ␰ = ␥ 2 ≅ ln

␳ a ␯ u* ␣

冊册

2

(4.5)

where ␳ a is the air density, ␣ is the surface tension coefficient, and ␯ is the water viscosity. The critical wind velocity varies inversely with the fetch. The wave height and slope angle increase for further increase in wind velocity right up to loss of stability when the wave top angle decreases. As shown by Stokes [3], the angular stream with top angle ␸ is stable if ␸ = ␲ /n (n = 3/2). However, for n ≠ 3/2, the system is unsteady and boundary deformation takes place. It is

174

Sea Backscattering at Low Grazing Angles

shown that this deformation occurs with small vertical acceleration, as the water jet is ejected from the top. This eruption of a water jet from the wave top is called wave breaking. According to the accepted classification [4], waves break by spilling, plunging, or surging, depending on the wave steepness. Spilling occurs when the wave crest becomes unstable at the top and the crest flows down the front face on the wave, producing an irregular, foamy water surface that eventually takes the aspect of a bore. Plunging occurs when the wave crest curls over the front face and falls into the base of the wave, resulting in a high splash and the development of a bore-like wave front. Surging occurs when the wave crest remains unbroken while the base of the front face of the wave, with minor breaking, advances up a beach. It is necessary to note that breakings of the plunging type are 40%–45% of the total. Such a description of sea waves applies for any arbitrary small but final values of vertical acceleration. Wave top breaking occurs when the local inertial forces exceed the gravitational ones. From Stoke’s theoretical papers, the breaking criterion was obtained and introduced to practice in the form [5] H C = 0.027gT 2

(4.6)

which connects the sea wave maximal height H C and period T. Experimental investigations confirm this relationship, giving coefficient values between 0.02 and 0.022. In the period preceding breaking, the wave parameters are changed; first, the wavelength ⌳ decreases, leading to increasing steepness ␥ = H /⌳. Then, horizontal and vertical asymmetry appears, determined by the coefficients of horizontal ␮ and vertical ␭ asymmetry, which are (see Figure 4.2)

Figure 4.2 Asymmetry of sea wave.

175

4.1 Sea Roughness Features for Small Grazing Angles

␮ = h /H ;

␭ = F2 / F1

(4.7)

Initially, the coefficient of horizontal asymmetry is close to 0.5, quickly increasing to values close to 0.9, after which vertical asymmetry appears. The experimental data of ␮ and ␭ obtained in [5] are shown in Table 4.3. The breaking process can be divided into three stages: the start of overturning, the appearance of one or a few splashes, and a chaotic movement. As the front slope steepness increases to the stability threshold, a water jet is ejected in the direction of wave movement. When the jet falls onto the undisturbed water surface, the splash appears, after which the process becomes chaotic (i.e., turbulent movement of the air-water mixture occurs). After splashing, the inertial forces quickly decrease and the water particles move in parabolic trajectories. Consider the characteristics of foam formations, because these permit us to indirectly judge the breaking process. The important quantitative characteristics of foam activity are the dependences of isolated regions for different type formations (crests and striped structures) and their relative areas on the surface wind velocity and the sea surface state. Investigations of these formations and their division into two types were carried out in [6]: the crests (dynamic foam with typical lifetime of some seconds corresponding to wave crest breaking) and striped structures (static foam with lifetime of minutes). We first analyze the wind dependence of the crest areas and relative areas of surface cover (and wind at a height of 20m) in a spatial window of 100 × 100 m2 for wind velocities of 5–10.5 m/s. The experimental pdf of relative areas of foam structures [6, 7] differs strongly from the Gaussian model, having positive values of skewness and kurtosis, and the gamma distribution is most suitable for their description p (S ) =

␭ (␭ S )␩ − 1 exp (−␭ S ) ⌫(␩ )

(4.8)

where ␭ = (␩ /␴ 2 )1/2; ␩ = 6/␥ 1 = 4/␥ 2 ; ⌫(␩ ) is the gamma-function; ␥ 1 is the coefficient of skewness; and ␥ 2 is the kurtosis coefficient. The parameters of foam characteristics are shown in Table 4.4.

Table 4.3 The Asymmetry Coefficients of Plunging Asymmetry Coefficients ␮ ␭ Source: [5].

Symmetrical Wave 0.5 1.0

Asymmetrical Coefficients Minimal Maximal Average 0.63 0.93 0.76 0.9 2.70 1.85

176

Sea Backscattering at Low Grazing Angles Table 4.4 Distribution Parameters of Foam Characteristics Wind Velocity (m/s) 5.7 9.5 10.5 Source: [6].

Mean Value (m) 0.0146 0.04 0.0695

␴ /m 1.24 0.71 1.08

␥1 2.31 1.67 4.36

␥2 6.97 3.22 16.6

The probability of a match of the experimental histograms to the gamma distribution was estimated according Pierson’s goodness-of-fit test and was found to be greater than 0.5–0.6. The mean value m and variance ␴ 2 have a tendency to grow with increasing wind velocity. The relationships between mean values of foam cover and single formation areas and wind velocity show that the threshold wind velocities for foam structure formation are as noted earlier in papers [7–9]. For the Black Sea, this threshold velocity is 4.5–5.5 m/s. An analogous threshold appeared for mean values of relative area of foam cover, approximated by a step function of the form [6] S⌺ =

再

0,

V < 5 m/s

0.015[1 + 2.2 ⭈ 10−2 (V − 5)3 ],

V ≥ 5 m/s

(4.9)

The dependence of S ⌺ = f (V ) is shown in Figure 4.3(a). The wind velocity dependence of single foam formations is approximated by S (m 2 ) =

再

0,

V < 5 m/s

0.4 + 0.0384(V − 5)2,

V ≥ 5 m/s

(4.10)

Dependence S = f (V ) for single foam regions are shown in Figure 4.3(b).

Figure 4.3 Dependences of (a) foam cover–relative sea area and (b) single foam formations on wind velocity. (After: [6].)

4.1 Sea Roughness Features for Small Grazing Angles

177

One of the conclusions of power spectral theory for developed sea roughness [7] is the relationship between the dissipation energy of the breaking wave and the crest foam cover area. From this, there follows a dependence of the dissipation energy threshold for breaking on the cube of wind velocity. The S ⌺ /S ratio permits us to evaluate the threshold density of dissipation energy regions, equal to about 300 per km2 [6], above which it varies with the cube of velocity to about 580 per km2 for V = 10.5 m/s. From [10], the mean number of crest foam formations depends on the wind velocity; this dependence has a very sharp slope from a few units for wind velocities less 6 m/s to 10–15 for wind velocities greater than 7 m/s over a sea surface area of 104 m2. For further wind velocity increases up to 16 m/s, the mean number of such formations remains practically constant. During breaking and boiling surf formation, an intensive drop-spray phase arises, with a duration of a few seconds. The foam formation occurs after the large drops enter the water. The foam spot achieves its maximal size in 4–5 seconds after breaking, and after that a collapse takes place. The foam spot size is 10–15m; the foam structure has a velocity of about 1.0–1.5 m/s at birth, and then it stops moving. The spray component originates with foam crest formation as a result of the aerodynamic jet gap at the water surface, accompanied by steepening and destruction of the wave profile. The water particle horizontal velocities are 0.7–0.9 times the wind velocity; the average diameter of drops is 10−2 cm, and their maximal diameter is about 0.2 cm. The lifetime of the spray approximately equals the surf lifetime up to transition of the foam structure to striped foam. The foam bubble structure is the source of the drop component. The burst bubble ejects water drops into air. Their size is rather small, as a rule, because they form from film material. The drop diameter in surf zone foam is determined by the film thickness h and can be up to 1.5 ␮ m [11]. The large drops arise, as a rule, at the end of their injection and mostly do not remain long in the air, with the exception of the turbulent regions of the surf zone. The experimental diameters of drops from [11] are shown in Table 4.5. Some of these drops move almost horizontally along the sea surface; vertically moving drops appear for bubble diameters greater than 1 mm. The vertical velocities

Table 4.5 The Drop Diameters Formed for Bubble Bursting Drop Number 2 4 6 8 10 Source: [11].

Drop Diameter 8.4h 13.6h 19.4h 24.3h 29.5h

178

Sea Backscattering at Low Grazing Angles

of so-called reactive drops can reach 10–50 m/s. The height of drop ejection also depends on the bubble diameter d . From data of Figure 4.4 [12], the maximal ejection heights are 15–20 cm for foam bubble diameters of about 1.0–1.5 mm. It is noted that with the exception of the first few seconds, the drop ejection height is practically constant and is 4–5 cm over the sea surface. The Phillips-Miles model for sea wave generation [13] is the most developed and is accepted by many authors as a working hypothesis for wave evolution with limited wind duration and large fetch. This evolution is observed at low frequencies, and spectral component saturation takes place in the high frequency band, where the slope can be approximated by a dependence proportional to ␻ −n where n = 4. The Phillips mechanism explains the rise and temporal linear growth of wave spectral components only at the earliest stages of their development. For later stages, the spectral component energy grows exponentially according to the Miles mechanism of instability. The main factors limiting the wave growth are their breaking and the nonlinear interaction of spectral components. The quantitative estimations of energy transfer rate for real spectra at the expense of instability are given in [14, 15]. From these papers, one can reach the conclusion that for the frequency band lower than the frequency of the sea wave spectral maximum, the energy transfer from wind to waves contributes the main increase in the wave energy. As the main parameter for sea surface description in oceanography, the wave spectrum is used, containing general information about the sea surface state and

Figure 4.4 Dependences of drop heights on the bubble diameter (curves A–F: the first drop, curve G: the second drop, curve H: the third drop, curve I: the fourth drop; temperature A: 4°; temperature B: 16°; and temperatures C–I: 22–26°C). (From: [12].)

179

4.1 Sea Roughness Features for Small Grazing Angles

not taking into consideration the earlier noted features. It is also the main parameter used in Bragg’s hypothesis for microwave scattering from sea surface. The cause of sea roughness is the wind, but this does not mean that the local wind fully determines the sea roughness structure. When the wind influences the sea surface over a rather long time and for large fetches, we obtain the fully developed roughness regime. For such a regime, the main expressions are obtained describing the sea roughness spectrum. The frequency spectrum of sea roughness for the simplest conditions of wave formation, when one wave system is observed, can be presented as S (␻ ) = A ⭈ ␻ −S exp (−B␻ −n )

(4.11)

The Pierson-Moskovitch spectrum is often used for fully developed roughness described by (4.11), for which the constants are equal to A = 8.1 ⭈ 10−3g, B = 0.74 ⭈ (g/V)4, S = 5, n = 4, where V is the mean wind velocity at a height of 10m, g is the gravitational constant. This spectrum has its maximum at frequency

␻ 0 ≈ 0.9(g /V )

(4.12)

and decreases with increasing frequency ␻ ≥ ␻ 0 . It is seen from (4.12) that the spectral maximum shifts to a lower frequency band with increased wind velocity. The Pierson-Moscovitch spectrum width at the half power level is ⌬F eff ≈ 0.133 ⭈

g V

(4.13)

The roughness power spectrum can distinguish the energy transfer interval with its maximum at the frequency ␻ 0 in which the energy transfer from the air flow to the sea waves takes place. The energy decreases quickly at lower frequencies because the wave energy transfer at large temporal scales from nonlinear effects is negligible. Phillips [16] advanced a hypothesis about the equilibrium region at frequencies above the frequency of the spectral maximum. He proceeded from the assumption that the wave dissipation process (crest breaking to achieve limited stable configuration and the transfer of wave energy to turbulence, capillary wave formation at face fronts of the primary waves, and secondarily nonlinear interaction) determines the spectral shape in this region. In this spectral interval, there is a consistent mechanism of energy transfer from larger to smaller turbulence, with dissipation of mechanical energy to thermal. Some 70%–80% of the energy is in the energy transfer interval and 20%–30% is in the equilibrium interval. Neglecting the surface tension and molecular viscosity in the equilibrium interval, he obtained from similar considerations the expression for the frequency spectrum

180

Sea Backscattering at Low Grazing Angles

S (␻ ) = a p g 2␻ −5, ␻ 0 ≤ ␻ ≤ ␻ m

(4.14)

where a p is the universal dimensionless constant defined in (4.16). The upper boundary of the equilibrium region is determined by the frequency for which the influence of surface tension becomes significant:

␻m =

冉 冊 ␳g3 ␣

1/4

(4.15)

where ␳ is the water density. The constant of a p varies with the fetch function X S [17] a p = 0.076

冉 冊 gX S

−0.22

u *2

(4.16)

(i.e., the value of a p increases rather slowly with increasing fetch). Often this constant is equal to (6.9 ± 2.4)10−3 for developed roughness and to (12.9 ± 2.5)10−3 for limited fetches. The equilibrium interval ends with a sharp cut off at the frequency ␻ m , above which the spectral density decreases more rapidly than for Phillips’s spectrum. The physical mechanism of this cut off is the mechanical energy dissipation to thermal in the frequency band ␻ > ␻ m for breaking waves. From (4.11), it is easy to obtain the approximate expression for wave height variance ∞

2 ␴W

=

冕

− S (␻ ) d␻ = const ⭈ g 2 ⭈ ␻ 04

(4.17)

0

This result was first obtained by Hicks [18], with the constant equal to about 0.04. Besides the frequency spectrum, the wave spectrum or the wavenumber spectrum is often used. The frequency and the wavenumber for gravitational waves are connected by the dispersion relation

␬=

2␲ ␻ 2 = ⌳ g

(4.18)

where ⌳ is the sea wavelength. The wave and frequency spectra are related by

181

4.1 Sea Roughness Features for Small Grazing Angles

S ( ␬ ) = S (␻ ) ⭈

d␻ d␬

(4.19)

Transforming the frequency spectrum to an isotropic spectrum of wavenumbers, Phillips obtained the spectrum in the form

S (␬ ) =

再

0.005/␬ 4,

␬ > g /V 4

0,

␬ < g /V 4

(4.20)

where k determines by (4.18) and equal to ␬ 0 ≤ ␬ ≤ k m . Here k 0 is the wavenumber corresponding to the wavelength of the spectral maximum and k m is the wavenumber corresponding to the upper limit of the equilibrium interval determined as k m = (g /␣ )1/2. According to analytical estimates in the spectral model JONSWAP [19], the microscale value ⌳m = ⌳0 /k m can be represented as ⌳m = ⌳ 0 /65 and is approximately 0.5m. This result agrees with data of [20], according to which the transfer from the equilibrium interval to the interval of gravitational-capillary waves takes place for wavelengths of about 0.2m. The period and wavelength are related to the wind velocity by the simple expressions T = 2␲ /␻ = 0.64V

(4.21a)

⌳ = 2␲ /k

(4.21b)

where V is the wind velocity. The sea surface statistical description [21] permits us to confirm that the wave height distribution is rather close to a Gaussian distribution, with a variance that can be obtained from (4.17). The rms wave height dependence on the wind velocity can be approximately represented as

␴ w = 0.005V 2 [m]

(4.22)

It should be noted that the sea roughness autocorrelation function can be approximated with good accuracy by [13] 2 R (␶ ) = ␴ W ⭈ e −␤␶ cos ␻ 0 ␶

(4.23)

where ␤ is related with the spectral width by ⌬F eff = ␤ /4

(4.24)

182

Sea Backscattering at Low Grazing Angles

In conclusion, note that the sea wave slope variance can be determined from the wave spectrum. For Phillips’s spectrum (4.20), one can obtain ␬m 2

␴ ⵜ␰ =

冕

␬ 2S (␬ ) d␬ = B ⭈ ln (␬ m /␬ 0 )

(4.25)

␬0

where B ≈ 0.005. 2 Using for ␴ ⵜ␰ a value determined from (4.6) with k 0 from (4.18) and for k m = 2␲ /⌳m , the minimal wavelength of equilibrium interval ⌳m ≈ 2 cm, we obtain the rms slope value for wind velocity V = 10 m/s

冋

␴ ⵜ␰ = 0.005 ln

冉

2␲ 9.8 0.81 2 0.02 10

冊册

1/2

≈ 0.203

(4.26)

The experimental data of Cox and Munk [10] show that the slope variance lies in the interval 0.02 to 0.07—that is, the derivations from (4.26) give results coinciding with experimental data. The sea roughness characteristics described earlier reflect the main features of roughness from its birth through the wave breaking process. Beginning with 1958, Phillips’s spectrum for the equilibrium interval of the sea roughness spectrum was used as the universal law for a description of developed roughness. However, the spectral density decrease with increasing k depends on the environment, particularly on wave development [22–24]. This led to a generalization of Phillips’s law [22, 25] and to presentation of the wavenumber spectrum in fractal form. The main wind wave feature for open sea, unlike in lakes, shallows, and wave tanks, is the large width of the wave spectrum. This means that there is great number of secondary waves in the roughness spectrum, besides those determined by the frequency of the spectral maximum. This leads to validity of the assumption about Gaussian statistics of sea roughness as the traditional first approach. Then the sea roughness spectrum for the equilibrium interval in terms of wavenumbers can be presented as [25]

冉 冊

V2 S (k , ␺ ) ≈ ␤ (x ) g

2␮

k −(4 − 2 ␮ m) Y (␺ )

(4.27)

where Y (␺ ) describes the angular distribution of wave energy, and ␤ and ␮ are dimensionless functions of normalized fetch x s .

183

4.1 Sea Roughness Features for Small Grazing Angles

The exponent (4 − 2 ␮ m) of k , characterizing the equilibrium interval, is a monotonic function of the degree of roughness development, as observed in [26, 27]. The fractal surface scale for spectrum (4.27) is D = 2 + ␮ , and therefore ␮ is the fractal surface codimension. The particular consequence of the condition ␮ > 0 is that it represents the large number of secondary waves forming the wave packet on the sea surface between two main waves with wavelength of ⌳ = 2␲ /k 0 . This leads to increasing numbers of brilliant points and single facet curvature. For a rather wide equilibrium interval or well-developed roughness, the spectrum falls with increasing k . So, for a relative fetch x s > 105, the maximal value is ␮ = 1/3 in the low frequency part of the equilibrium interval, decreasing to ␮ = 1/4 in the high frequency part. The value of ␮ is zero in the dissipative region and for small fetch 103 < x s < 4 10 . Because of the rather small change in the fractal codimension ␮ = ␮ (x ), it can be considered as independent of k over the entire equilibrium interval, conforming to the conclusion of similarity theory [22]. The coefficient ␤ is a nondimensional parameter called Phillips’s generalized constant. It depends slightly on the fetch and is [17]

␤ ≈ 0.0331x −0.2

(4.28)

It is emphasized that rapid decrease of spectral density in the dissipative interval, proportional to k −4, is a property inherent in developed roughness. The upper boundary of the equilibrium interval is conditioned by the hydrodynamic instability of high gravitational waves that have the tendency to break for rms wave angular slopes ␥ reaching some threshold value ⌫ ≈ 0.4. As shown in [28], for microscale k m , the variance of slope angles can be determined as 2

␥ =

2 ␴ ⵜ␰

冋

册冉 冊

␤ ␦ 2␮ V2 ≈ ⌫( ␮ ) − ⭈ 2 ␮ g ⌳m

2␮

(4.29)

where ␦ = k 0 ⌳m . For well developed roughness when ␮ ≥ 0.2 and ␦ << 1, the wave slope is practically independent of ⌳0 . Particularly, for ␮ = 1/4 (the Zacharov-PhilonenkoToba spectrum), we have

␥2 ≅

␤ ⌫(1/4) V ⭈ 2 √g ⌳m

(4.30)

For fully developed roughness (␮ → 0), the influence of the main wavelength ⌳0 is significant and

184

Sea Backscattering at Low Grazing Angles

␥ 2 ≈ ␤ ln

冉 冊 c ⌳0 h

(4.31)

where c = 0.1193. It is seen from these expressions that the wind dependence of slope angle variance is significantly changed as a function of the roughness regime. 4.1.2 Shadowing and Peaks in Heavy Sea

Microwave backscattering from the sea surface for small and extremely small grazing angles has some peculiarities. First of all, the larger part of surface is in the shadowing zone for low grazing angles, and the scattered signal is formed only by areas that rise above the shadowing zone. Consequently, the signal changes from being spatially continuous to discrete. In these conditions, the term normalized radar cross section loses its sense because this term assumes a homogeneous surface with uniform illumination, while in our case the scattered signal is formed by local areas of the sea surface. Besides, spikes of the scattered signal with specific statistical features are observed even for large grazing angles in addition to the range continuous signal. A strict physical model of the spikes is unavailable, preventing us from explaining their statistics. It is noted only that the spatial statistics of spikes are associated with breaking sea waves and presence of foam on the sea surface. In these situations, the rough sea is not a stationary random process, and one can identify two phases in the backscattered signal: spikes for which the mean intensity is much greater than the mean level, and gaps in which the scattered signal level is considerably less than the mean level or in which the signal is practically absent for extremely small grazing angles because of the surface shadowing. Recently, the interest in investigations of sea backscattering for small grazing angles has increased considerably. This deals with attempts to explain the physical nature of the scattered signal spikes because of the necessity to take spikes into consideration in sea radar clutter models [29–39]. The total RCS of sea clutter for a single resolution cell is determined by the following. In the illuminated area, the total RCS is

␴ ⌺ = ␴ sea + ␴ spray

(4.32)

Here ␴ sea is the RCS of the sea surface, as determined by the general laws of sea backscattering without shadowing and spikes. The second term in the righthand part, ␴ spray , takes into account the spatial statistics of spikes. The spatial statistics are determined by the statistics of random sea surface peaks above some threshold. The sea surface is a complex natural formation that includes wind waves, spray, and foam. Each component takes part in forming the backscattered signal

185

4.1 Sea Roughness Features for Small Grazing Angles

as a function of the radar frequency and the grazing angle and determines the radar sea clutter level. The main reason for these phenomena on the sea surface is the surface wind. First of all, let us consider the shadowing function of the sea surface. The surface can be presented as a stationary random process ␰ (x , y ) with a Gaussian 2 pdf of surface heights, zero mean and variance ␴ W p (␰ ) =

1

␴ W √2␲

冉

exp −

␰2 2

2␴ W

冊

(4.33)

The incident field vector lies in the plane XZ, the incidence angle is denoted by ␪ , and the grazing angle is ␺ = ␲ /2 − ␪ (see Figure 4.5). The simplest technique to describe surface shadowing is to introduce a shadowing function S (x , y ) that is unity for the illuminated part of surface and zero for the surface in the shadowing zone. Then, a mean shadowing function S (␪ ) = 〈S (x , y , z )〉 is determined. The mean power of the backscattered signal is proportional to S (␪ ). In our case S (x , y , z ) is a function only of the coordinate x. Using the results of [40, 41], the shadowing function can be represented as S (␪ ) =

⌽( ␮ /␴ ⵜ␰ ) ⌳( ␮ ) + 1

where ⌽(⭈) is the probability integral, ␮ = tan ␺ , and

Figure 4.5 Geometry for derivation.

(4.34)

186

Sea Backscattering at Low Grazing Angles

冉

␴ ␮2 ⌳( ␮ ) = (2␲ )−1/2 ⵜ␰ exp − ␮ 2␴ ⵜ␰

冊

− erfc

冉

␮ ␴ ⵜ␰ √2

冊

(4.35)

2

The variance of the surface slopes ␴ ⵜ␰ is determined by the autocorrelation 2 function of the surface because ␴ ⵜ␰ = | −␳ ″(0) | and ␳ is the normalized autocorrelation function. For the two most common spatial surface autocorrelation functions (Gaussian and exponential), one can obtain 2 ␴ ⵜ␰

=

再

2

2␴ w / T 2 2

␴w /T 2

for Gaussian function for exponential function

(4.36)

2

where ␴ w is the wind velocity variance and T = 1/4⌬F eff ; ⌬F eff is the effective spectral width of the surface determined by (4.13). The variance of the sea surface slope is 2 ␴ ⵜ␰

冉冊

g ≅ 0.533 V

2

(4.37)

Introducing the normalized height ␨ = ␰ /␴ w and slope ␩ = ␮ /␴ ⵜ␰ , one can finally obtain S (␨ , ␩ ) = [⌽(␨ )]⌳; ⌳(␩ ) =

(4.38)

2 1 1 ⭈ ⭈ e ␩ /2 + ⌽(␩ ) − 1 ␩ √2␲

For small ␩ << 1 ⌳(␩ ) ≈

冋

1 1 − 2 ␲ ␩ (1 + ␩ 2/2) √

√2(1 − ␩ )

册

(4.39)

The dependence of the shadowing function on the grazing angle ␺ for ␨ = 1 is shown in Figure 4.6 (for rms values of slope equal to 0.14 and 0.26, which characterize the boundary values of the sea surface slope variance [24]). The transition from the illuminated zone to the strongly shadowing zone takes place in the narrow range of the grazing angles from 0.5° to 5° and the sea surface is illuminated almost fully for grazing angles larger than 5°. Let us consider the problems of sea wave peaks (i.e., the case in which a sea wave rises above some boundary), using the first approximation of rough sea as

187

4.1 Sea Roughness Features for Small Grazing Angles

Figure 4.6 The shadowing function versus grazing angle. (After: [24].)

a stationary random process with finite second spectral moment and continuous spectrum, at all stages up to wave breaking. For such a random process, there is a sufficiently developed theory of peaks [42, 43]. In the framework of this theory and for the condition of high threshold when the number of crossings by the stationary random process is small, the peak distribution can be approximately described by Poisson’s law P (k , T ) =

[N (␰ 0 )]k ⭈ exp [−N (k , T )], k!

k = 0, 1, 2, . . .

(4.40)

where N (␰ 0 ) is the mean number of peaks. For a Gaussian random process, the mean number of peaks is determined by Rice’s formula [42] N (␰ 0 ) =

√−␳ ″ (0) 2␲

冉 冊

␰ ⭈ exp − 0 2

(4.41)

where ␳ ″ (0) is the second derivative of the autocorrelation function of the process for ␶ = 0, and ␰ 0 is the threshold normalized to the rms wave height. According to evaluation of N (␰ 0 ) from (4.41), the mean number of peaks decreases with increasing wind velocity. This contradicts the experimental data and is the result of the use in the derivation of the Pierson-Moscovitch spectrum

188

Sea Backscattering at Low Grazing Angles

of sea waves. This spectrum is not applicable to a heavy sea because it does not take into consideration the breaking of large numbers of waves. For the fractal model of the sea wave spectrum, the mean number of peaks is a cubic function of the wind velocity or friction velocity [16] N (␰ 0 ) ⬀ g −1 ⭈ k m ⭈ u *3

(4.42)

The dependence of the peak frequency on friction velocity experimentally obtained in [36] for a unimodal sea wave spectrum is shown in Figure 4.7, and the analogous results are given in [35]. The dependence of the mean number of peaks on range for two wind velocities is shown in Figure 4.8. One can see a decreasing mean number of peaks with increasing range. The distribution of the mean peak duration for the normal random process can be approximately represented as P␶ (␶ = k␶ 0 ) = [1 − ⌽(␰ 0 )]k − 1 ⭈ ⌽(␰ 0 )

(4.43)

where ␶ 0 is a discretization interval that can be determined from the sample theorem as ␶ 0 ≅ 1/⌬F eff ; k = 0, 1, 2, . . . ; ⌬F eff is the effective Pierson’s sea wave spectral width. From this expression, it is seen that for increasing threshold ␰ 0 the probability of short-time spikes increases and the probability of long-time spikes decreases. The mean spike and gap duration can be determined as

Figure 4.7 The mean number of peaks versus the friction velocity for (a) all conditions of sea roughness and (b) sea wave with unimodal spectrum. (From: [36].)

189

4.2 Sea Backscattering Models

Figure 4.8 The spike mean number dependence on range for wind velocities (1) 7 m/s and (2) 10 m/s.

␶ s (␰ 0 ) = 2␲ [1 − ⌽(␰ 0 )]

␶p = 1

⁄ 冋√

⁄ 冋√

冉 冊册

␳ ″ (0) ⭈ exp −

冉 冊册

␰ −␳ ″ (0) ⭈ exp − 0 2

␰0 2

(4.44)

(4.45)

The mean duration of the spikes and gaps as functions of the threshold for two wind velocities are shown in Figure 4.9. Thus, the results presented permit us to estimate the main characteristics of shadowing and peaks of the sea surface above some boundary under the assumption that the sea surface can be described by a differential random process (i.e., wave breaking is absent). If breaking is taken into account, the results must inevitably change.

4.2 Sea Backscattering Models Theoretical and experimental investigations of backscattering from the sea surface have provided a basic understanding of the phenomenon. However, the development of theoretical models of backscattering has proven to be a challenging task.

190

Sea Backscattering at Low Grazing Angles

Figure 4.9 The mean duration of spikes and gaps as the functions of relative threshold.

Among the sea backscattering models, two models are best known. The first is the facet model [44, 45], in which geometrical optics techniques are applied for backscattering description, and the surface is represented by the totality of small, flat plates. It is supposed that the radar reflection is formed by the facets that are normal to the direction of radiation. If the facet slope distribution is known, one can determine the fraction that is perpendicular to a given radiated beam and, consequently, the scattered signal intensity. The facet model proved to be useful for qualitative analysis of clutter phenomena and was modified [46] on the basis of the actual scattering of finite-length radiowaves by facets of finite sizes, including the influence of wavelength on the effective number of facets contributing to the scattered signal. In those cases when the facets extend in height to at least one interference lobe width, the radiation conditions are equivalent to a homogeneous field, and the normalized RCS does not depend on the incidence angle. If the facets occupy only part of the interference lobe, the radiation intensity decreases very quickly with increasing range, and the normalized RCS varies as r −4 where r is the range from the radar. For small grazing angles, this model leads to the following conclusions: •

Facets whose circumference is near ␭ /2 scatter most intensively;

191

4.2 Sea Backscattering Models

•

•

The scattering varies with the square of crest slope because the facets at the wave crest scatter most intensively; The normalized RCS dependence on the frequency is related to the facet size distribution, leading to its varying inversely with wavelength.

Surface shadowing for small grazing angles and heavy sea and scattering from spray and white caps formed for breaking waves are not considered in the facet model. In addition, the variation of normalized RCS dependence with wavelength does not conform with experimental investigation results; this dependence is considerably stronger in the model than in experiments. A more detailed description of the observed effects [47, 48] is given in the twoscale scattering model, in which the sea surface is represented as a superposition of irregularities with pronounced differences in scale length. The model provides a relatively good explanation of the experimental results, especially at decimeter wavelengths and the longer wavelength part of the centimeter band for grazing angles larger than several degrees. The two-scale model is more suitable for theoretical interpretation of the phenomenon. In the framework of this model, the normalized RCS is [47] 0 ␴ VV = 16␲ k 4 | ⑀ | sin4 ␺ ⭈ f (⑀ , ␺ ) ⭈ S (ϖ 0 ) 2

(4.46)

0 ␴ HH = 16␲ k 4 sin4 ␺ ⭈ S (ϖ 0 )

where

⑀ is the dielectric constant of sea water; f (⑀ , ␺ ) = [(1 + ␩ 1 sin ␺ )2 + ␩ 2 sin2 ␺ ]−2;

␩ 1 + ␩ 2 = √⑀ ; 4␲ ϖ 0 = 2␲ /⌳0 = cos ␺ ; ␭ S (ϖ ) is the roughness spectral density for sea wavenumber ϖ ; ⌳ is the sea wavelength;

␺ is the grazing angle. The sea roughness spectrum for grazing angles 20° to 60° can be represented by the expression [49] S (ϖ ) ≅

B −4 ϖ ; ␲

B = (2 − 6) ⭈ 10−3 ⭈ [. . .]

192

Sea Backscattering at Low Grazing Angles

Analysis of these expressions shows that the normalized RCS does not practically depend on the frequency in the entire UHF band, and the normalized RCS is smaller for horizontal polarization than for vertical. For small grazing angles, the two-scale model fails to account for a number of features in the backscattered signal. An attempt was made [50] to make use of the model to gain an insight into such experimentally noted phenomena as the large normalized RCS and the increase in the central frequency in the spectrum for horizontal polarization as compared with vertical. Yet it was not possible to give an adequate explanation for all of the features observed. One effect is the lower rate of change (with respect to that predicted by the model) of the normalized RCS as a function of grazing angle for small ␺ . The problem of signal characteristics observed for breaking waves and subsequent generation of sprays has not been addressed so far. Some authors [51, 52] describe experiments that were performed in an attempt to distinguish between two model backscattering mechanisms: facet reflection and two-scale scattering. It was pointed out that in X-band, the measured normalized RCS exceeded that predicted (according to the two-scale model) by as much as 12 dB. Besides, the normalized RCS was equal for horizontal and vertical polarizations, and a more significant central frequency shift was detected. Estimates showed that, along with Bragg scattering, reflection from facets with sufficiently great radii of curvature r could occur (it should be noted that for wind speed greater than 4 m/s, the contribution from the facets for 10 ≤ kr ≤ 100 proves to be significant). As a result, with wind speed from 0 to 7 m ⭈ s−1, the contribution from the facets to the total scattered signal is 30%–40%, while the lifetime of these reflections is 20% of the total observation period. Moreover, as the wave crests steepen prior to breaking, it becomes necessary to take into account wedge diffraction that (as reported in [50]) can result in an increase of the normalized RCS for horizontally polarized radio waves in comparison with the vertical by 10–20 dB. Thus, ␴ 0 can be equal for two polarizations and can even give a result that is inconsistent with the standard 0 theory, for which ␴ H ≤ ␴ V0 . As far as the shorter wavelength part of X- and Ka-bands is concerned, the contribution to the scattered signals due to spray blown from the wave crests during wave breaking is of great importance. As a consequence, wave breaking caused by the wind is accompanied by the appearance of a variable density layer at the air-water interface, which is a mixture of finite volumes of these components. The contribution of the spray to the total scattered signal can be calculated as an integral over the volume V [32],

␦ spray =

冕 V

␩ (h, u ) dV

(4.47)

193

4.3 Sea Normalized RCS

where ␩ (h, u ) is the specific volume RCS of spray, which is a the function of the height h above the sea surface and the wind speed u . Assuming the specific volume RCS to be uniform in the cross-sectional plane h = const, (4.47) can be simplified and if a pulsed signal is used it becomes c␶ ␦ spray = u r␪ 0 2

∞

冕

␩ (h, u ) dV

(4.48)

0

where ␶ u is the transmitted pulse duration, ␪ 0 is the beamwidth in the azimuthal plane, and r is the range. The volume normalized RCS of spray in (4.48) can be defined as [32] ∞

␩ (h, u ) =

冕

␴ (D, ␭ ) N (D, h, u ) dD

(4.49)

0

where ␴ (D, ␭ ) is the RCS of a water drop of radius D and N (D, h, u ) is the number of drops with radii from D to D + dD in a unit volume. The RCS of an individual particle can be calculated using the well-known Mie theory, which is valid up to the shorter wavelength part of the millimeter band. The size distribution of water drops can be described by the Marshall-Palmer formula [53] N (D ) = N 0 exp (−⌳D )

(4.50)

where N 0 = 8 × 104 m−3, ⌳ = 26.7␻ −0.24 [32], and ␻ is the water content in g/m3. Analysis of experimental data for water content at small altitudes above the sea surface permits derivation of the following empirical formula

冉

␻ = A 0 exp 0.83u −

h h0

冊

(4.51)

where A 0 = 2.4 × 10−11 g ⭈ cm−3 and h 0 = 11 cm. Taking the effect of spray into account, it becomes possible to explain both the angular dependence of the normalized RCS in the region of small grazing angles (which cannot be done with the two-scale model) and some other features of the power spectra and polarization characteristics of the scattered signals.

4.3 Sea Normalized RCS The experimental investigations of the normalized RCS variations that have been carried out for a variety of radar system parameters and external factors over the

194

Sea Backscattering at Low Grazing Angles

frequency range from 3 GHz to 100 GHz at small grazing angles [32, 54–57] reveal several phenomena inherent in behavior of ␴ 0 at millimeter wavelengths. These involve the effect of sprays generated during wave breaking on the angular dependence and magnitude of ␴ 0 and the quicker saturation of the normalized RCS with increasing wind speed in the millimeter band as compared with the X-band. Three regions can be distinguished in the angular dependence of the normalized RCS of the sea surface: the quasi-specular region, the plateau, and the interference region—the latter being of greatest interest for our applications. The magnitude of the normalized RCS at grazing angles smaller than few degrees shows a sudden drop as the angle decreases, ␴ 0 varying with ␺ . The transition from the plateau region to the interference region takes the place at the critical angle ␺ cr , whose value depends on the wavelength and the sea surface state:

␺ cr =

␭ ␭ ≈ 5H 0.015 u 2.5

(4.52)

where u is the mean wind speed in m ⭈ s−1 and H is the effective sea wave height, which is lower than the peak-to-trough height by a factor about three. In the decimeter and the longer-wavelength part of the centimeter band, the relationship ␴ 0 ∼ ␺ 4 applies for ␺ < ␺ cr . However, concerning the shorter wavelength part of the centimeter and the MMW band, the boundary between the plateau and interference regions should be set with care. As was noted in [55], the ␴ 0 ∼ ␺ 4 dependence holds in the X-band under standard refraction conditions, whereas superrefraction at grazing angles less than 0.1° leads to a ␴ 0 ∼ ␺ relationship. The results of measurements performed over the frequency range 10 GHz to 75 GHz [32, 54] identified some additional features in the angular dependence of the normalized RCS. Between the plateau and the interference region, there exists a transition zone. In this transition zone, the normalized RCS is proportional to ∼␺ 2, which corresponds to a range dependence P range ∼ r −5 for received power. At shorter wavelengths, the transition zone shifts towards smaller grazing angles. To explain the presence of this region in the angular dependence, two mechanisms might be conceived [32]. One is the presence of increased refractivity with steep gradients in the near-surface layer of troposphere. Under such circumstances, the real grazing angle is larger than that determined from geometric considerations; this may result in increased normalized RCS. For the interference region we can use the relationship

␴ 0 = k␺ 4 where

(4.53)

195

4.3 Sea Normalized RCS

k = const;

␺ = ␺g + ␺r ; ␺ g ≈ h /r is the geometric grazing angle; h is the height of the radar antenna;

␺ r = 1/2 ⭈ r ⭈ grad n is the grazing angle due to refraction; grad n is the refractivity index gradient. Then, assuming a constant grad n in the near-surface layer, (4.53) transforms to

冋

h ␴ = k ␺r + grad n 2␺ r 0

册

4

(4.54)

(i.e., the normalized RCS magnitude at small grazing angles is seen to be gradient dependent). The meteorological measurements that were performed together with the radar measurements showed that the gradient, which was highly variable throughout the day, was equal to 2–2.5 N ⭈ m−1, and the value was normally larger in the layer 0m to 5m than 0.157 N ⭈ m−1. In other words, there existed super-refraction that could even give rise to atmospheric ducting. Within the layer from 5m to 10m, such conditions existed during 50% of the observation period. However, the measured gradients on the average appeared to be lower than those required to account for the normalized RCS angular dependence (i.e., the superrefraction mechanism cannot provide a complete explanation of the results obtained). Under wave-breaking conditions, a variable density layer appears at the airwater boundary, representing an air-water mixture. Accounting for the microwave scattering caused by spray blown from the wave crests and generated as the waves break, the total RCS can be written as (4.32) where ␴ spray is given by (4.47)–(4.51). Then, as follows from [32], ∞

冕 0 ␴ 0 = ␴M +

␩ (h, u ) dh

0

0 ␴M

(4.55)

The analysis of [32] shows that the rate at which ␴ 0 decreases with decreasing grazing angle slows down at higher wind speeds (i.e., in MMW band at small grazing angles the normalized RCS increases), mainly due to the contribution of reflections from spray. The decrease in the normalized RCS at still lower grazing

196

Sea Backscattering at Low Grazing Angles

angles can be attributed to the shadowing effect. As a major part of the surface is located in the shadow zone, and the contribution to scattering is only made by partial areas (crests of the larger waves), the very concept of a normalized RCS becomes meaningless. Owing to quick changes in the fine surface structure produced by the wind, the normalized RCS of the sea in the shorter wavelength part of the centimeter band and at millimeter wavelengths becomes dependent on wind speed. The intensity of reflections starts to increase at wind speeds of more than 2.5–3 m ⭈ s−1 and reaches its maximum values at 3–10 m ⭈ s−1. With further increase of the wind speed, the rate of growth of the RCS drops down and amounts to 0.4–0.6 dB/ms−1 [58]. At millimeter wavelengths, the normalized RCS begins to be saturated more quickly than in the shorter wavelength part of the centimeter band [32] starting at wind speeds of 5–7 m ⭈ s−1. Many authors make special reference to the dependence of the normalized RCS upon the wind direction with respect to the beam. Under moderate sea conditions and at small grazing angles, the normalized RCS normally is maximum in the upwind direction and minimum crosswind. According to the data of [59, 60] the ratio of upwind to downwind measured at wavelengths of 1–3 cm for HH polarization is 8 dB at 10° grazing angle and varies with angle. For VV polarization, it did not exceed 3–4 dB. The upwind-to-crosswind ratio at ␭ = 3 cm reaches 5–6 dB. At millimeter wavelengths the upwind-to-crosswind normalized RCS ratio is 5 dB or 6 dB, rising to 10–15 dB for a calm sea [32]. The normalized RCS can be greatly influenced by the transmitting and receiving polarizations. In the X-band, the normalized RCS for HH polarization exceeds that of VV by 8–12 dB for grazing angles less than 1–2° [60]. The normalized RCS for HH polarization is 1–2 dB higher than for VV both in the X-band and for MMWs, over high sea and for grazing angles less than 0.5° [32]. When the sea was calm, the normalized RCS for VV polarization was greater than for HH by 5–7 dB. Analysis of the cross-correlation function for scattered signals at the outputs of synchronous and amplitude detectors and of the sliding-average amplitudes shows the following: •

•

The correlation of cross-polarized signal components from the outputs of synchronous and amplitude detectors is relatively small (the correlation factors never exceed 0.5). The sliding-average amplitudes of two orthogonally polarized signals are better correlated; the correlation factor increases with the volume of the pulse packet integrated and with the wind speed. It varies from 0.45 at low sea states up to 0.95 for a stormy sea with pronounced periodic structure of roughness.

The correlation factors for ␭ = 8 mm and ␭ = 4 mm are shown in Table 4.6.

197

4.4 Depolarization of Scattered Signals Table 4.6 Correlation Factors for Cross-Polarized Components

Wavelength (mm) 8.0

Polarization Linear, at 45°

8.0

Linear, 45°

4.0

Linear, 45°

4.0

—

Wind Speed and Sea State u w < 3 m ⭈ s−1, calm u w < 16 m ⭈ s−1, Sea state 5 u w < 11 m ⭈ s−1, Sea state 4 u w < 11 m ⭈ s−1, Sea state 4

Correlation Factors for Cross-Polarized Components Sliding Channels Amplitudes Average −0.15 0.13 0.49

Number of Experiments 6

0.04

0.22

0.6

4

0.19

0.42

—

7

0.19

0.21

0.71

12

4.4 Depolarization of Scattered Signals Measurements at frequencies of 10–75 GHz and wind speed up to 14–15 m ⭈ s−1 show strong depolarization of the signals, with depolarization factors of −3 dB to −6 dB. Investigations of the signal depolarization at 10 GHz and 35 GHz, in which two states of the scattering process could be clearly identified (i.e., spikes caused by reflections from breaking waves and gaps in the absence of wave breaking) were reported in [61–63]. The most widely used criterion of depolarization degree is the depolarization coefficient D i , j = 10 log (I i , j / I i , i )

(4.56)

where D i , j is the depolarization coefficient (in decibels), i and j are the polarizations of transmitted and received signal, respectively, and I is the corresponding signal component intensity. It is easy to recognize intervals of comparatively high and low intensity (spikes and gaps) in temporal dependences of the copolarized component intensity and the scattered signal depolarization coefficient for both wavelengths and both polarizations. In the future, the term spike means the intervals for which signal intensity is 3 rms values greater than its average value. It is seen from the analysis of temporal dependences that for vertical polarization, the scattered signal depolarization level increases in spikes for both wavelengths. Meanwhile, for horizontal polarization and wavelength of 3 cm, it is practically impossible to determine the difference between the depolarization coefficients in spikes and gaps, corresponding closely with results of [60]. Averaged depolarization coefficients in spikes and gaps for different beam directions with respect to wave direction are given in Table 4.7. The analysis of that data shows the following:

198

Sea Backscattering at Low Grazing Angles Table 4.7 The Mean Depolarization Coefficients in Spikes and Gaps Direction of Radiation Upwind

Crosswind

Surf zone

Wavelength (cm) 3.0 3.0 0.8 0.8 3.0 3.0 0.8 0.8 3.0 3.0 0.8 0.8

Polarization H V H V H V H V H V H V

D i , j spike (dB) −8.0 −15.1 −5.3 −11.7 −8.0 −15.5 −8.0 −10.0 −9.0 −16.0 −3.5 −11.5

D i , j gap (dB) −9.0 −17.9 −8.1 −15.0 −8.0 −16.5 −8.0 −15.5 −9.0 −16.5 −8.5 −17.0

Source: [34].

•

•

•

•

•

•

•

Depolarization coefficients do not practically depend on the beam and wave angles. Depolarization coefficient values for horizontal polarization are usually higher by 2–8 dB in spikes and 7–9 dB in gaps for both wavelengths than corresponding values for vertical polarization. In the millimeter band, for both polarizations, the difference of depolarization coefficients in spikes and gaps does not depend upon beam direction and is 3–5 dB. For a 3-cm wavelength, there is no difference in depolarization coefficients in spikes and gaps for horizontal polarization. For vertical polarization, the difference varies from 0 dB for surf illumination to 2.5 dB or 3 dB for upwind illumination. Increasing depolarization coefficient is observed in approximately 40% (X-band) and 60% (millimeter band) of cases when spikes appear in samplings having a duration of 3 minutes. In gaps, comparatively weak depolarization change is observed with radar frequency change (approximately D i , j ∼ f 0 − f 0.2 ) for both polarizations, while in spikes more rapid depolarization coefficient increasing occurs (D i , j ∼ f 0.5 − f 0.9 ).

Experimental investigations conducted by author [63] at 6.3 GHz showed a similar difference between depolarization coefficients for horizontal and vertical 0 0 polarizations. Supposing that ␴ HV = ␴ VH , where in HV and VH, the first letter denotes the transmitted polarization and the second denotes the received one (this assumption results from the reciprocity theorem [62]), the author explained this 0 0 phenomena by the supposition that ␴ VV > ␴ HH . But experiments at X- and Ka-bands and generalized in [32, 33] showed that specific RCS values for horizontal 0 0 and vertical polarizations are practically equal (i.e., ␴ VV ≈ ␴ HH , excluding the

199

4.4 Depolarization of Scattered Signals

mechanism presented in [63] as an explanation of sea clutter polarization differences). Consequently, we must search for other mechanisms dealing with the presence of small-structure elements on the sea surface (created by wind and resulting in partial depolarization of the scattered signal). One of these elements is the contribution to scattered field by the sharp crest of a breaking wave. In particular, the larger average specific RCS for horizontal polarization over vertical was explained 0 0 by this fact in [63]. Taking [64] into account, let us estimate the ratio ␴ HH /␴ VV for an ideally conducting wedge having the geometry shown in Figure 4.10. The derivation uses

0 ␴ HH 0 ␴ VV

=

|

冋 冉 冊册 冋 冉 冊册

␲ ␺ 1 + cos ␲+ ␣ 2

␲2 ␲ ␺ 2 cos + cos + ␲+ ␣ ␣ 2

−1

|

2

(4.57)

where ␺ is the grazing angle and ␣ is the angle of the wedge top. The results of this derivation are shown in Table 4.8. They show that only for the angle range from 100° to 120°, exactly corresponding to wave crest breaking, the scattered signal for horizontal polarization can be greater than for vertical, 0 resulting in increase in the depolarization component ␴ HV and D i , j . But this mechanism cannot explain difference in gaps where sea wave breaking is absent.

Figure 4.10

Geometry of wedge.

200

Sea Backscattering at Low Grazing Angles Table 4.8 The Ratio of ␴ HH /␴ VV , dB for Different Angles in Wedge Top Grazing Angle 1 2 3 4 5 6 Source: [34].

100° 2.05 2.04 2.03 2.02 2.0 1.99

110° 14.8 14.5 14.16 13.86 13.57 13.3

120° 0.23 0.45 0.68 0.9 1.14 1.36

130° −9.9 −9.7 −9.5 −9.3 −9.1 −8.9

140° −17.4 −17.2 −17.0 −16.7 −16.5 −16.3

150° −24.8 −24.5 −24.2 −23.9 −23.7 −23.4

The second mechanism results from signal scattering by spray formed during wave breaking. According to [65] data, during crest breaking and boiling, an intensive drop-spray phase appears, lasting for about 1s, after which foam forms. During crest breaking, drops of large size fall into the water and stimulate the forming of small spray, the lifetime of which can exceed several seconds [11]. These drops are moved by the wind, forming a layer saturated by water in different phases over the sea surface. Greater depolarization for horizontal polarization in comparison with vertical can be explained by the flattening of the spheroids (drops) in the vertical plane. The higher intensity for horizontal polarization results from the horizontal axis a being larger than the vertical one b. According to [16] data, when b /a = 0.2 the ratio ␴ HH /␴ VV ≈ 8.3 dB for a /␭ ≈ 0.1 to 0.15 (i.e., when the drop diameter is significantly less than the wavelength). A spheroid of this type can be approximately considered as a dipole randomly oriented in space. If the dipole is oriented along axis OA and the plane wave is propagating along the axis OZ (Figure 4.11), the dipole cross section ␴ xx (␪ , ␸ ) for a polarization plane a coinciding with axis OX is [63]

␴ XX (␪ , ␸ ) = ␴ max ⭈ sin4 ␪ ⭈ cos4 ␸

(4.58)

where ␴ max = ␴ xx (␲ /2, 0) is the cross section of a dipole oriented along axis OX. Correspondingly, the cross section ␴ XY of this dipole when received on the polarization coinciding with OY is

␴ XY (␪ , ␸ ) = ␴ max ⭈ sin4 ␪ ⭈ sin2 ␸ ⭈ cos2 ␸

(4.59)

For the ensemble of randomly oriented dipoles 2␲

␲

冕 冕 d␸

␴ XX 0 = ␴ XY 2␲

0 ␲

冕 冕 d␸

0

␴ XX (␪ , ␸ ) ⭈ sin ␪ ⭈ d␪ (4.60)

␴ XY (␪ , ␸ ) ⭈ sin ␪ ⭈ d␪

0

(i.e., significant depolarization occurs).

201

4.4 Depolarization of Scattered Signals

Figure 4.11

Geometry of the spheroid.

Depending on the phase of breaking and the wavelength, the contribution of drops to the total scattered signal differs for copolarizations and cross-polarizations, explaining both the increase in depolarization coefficient for horizontal polarization in comparison with vertical and the increasing depolarization in spikes relative to gaps. Using the model of normalized RCS proposed in [32, 62] and assuming significant reflection from spray during the spikes, we may write: 0 0 0 ␴ spike = ␴ sea + ␴ drop 0 0 0 ␴ spike cross = ␴ sea cross + ␴ drop cross

D i , j spike =

where

0 0 ␴ sea cross + ␴ drop cross 0 0 ␴ sea + ␴ drop

(4.61)

= D i , j gap ⭈

冢

0 D drop ␴ drop 1+ ⭈ D i , j gap ␴ 0 sea

1+

0 ␴ drop 0 ␴ sea

冣

202

Sea Backscattering at Low Grazing Angles

D i , j spike and D i , j gap denote depolarization coefficients for spikes and gaps, respectively. 0 0 0 0 ␴ sea , ␴ sea cross , ␴ drop , and ␴ drop cross are the specific sea and spray RCS for copolarized and cross polarized components, respectively. 0 0 D drop = ␴drop cross /␴drop is the spray depolarization coefficient.

Derivation of d = D i , j spike /D i , j gap is carried out under the assumption that the maximal spray depolarization coefficient is D drop = 4.8 dB, resulting from (4.59), and that the ratio 0 ␴ drop 0 ␴ sea

=

再

−(10 to 15) dB

for ␭ = 3 cm

−(3 to 5) dB

for ␭ = 8 mm

is determined using data presented in [66]. The results of this derivation are shown in Table 4.9, which also contains the experimental values of difference between depolarization coefficients in spikes and gaps. Analysis of data in Table 4.9 shows that spray-drop fraction depolarization mechanism (the fraction formed due to sea wave collapsing) permits us to explain the experimental results for the excess depolarization coefficients in spikes compared to gaps.

4.5 Sea Clutter RCS Model The investigations that have been carried out permitted us to develop an empirical model of backscattering at centimeter and millimeter wavelengths and low grazing

Table 4.9 The Derived and Experimental Values of d The Direction of Radiation Upwind

Crosswind

Surf zone

Source: [62].

Wavelength (cm) 3.0 3.0 0.8 0.8 3.0 3.0 0.8 0.8 3.0 3.0 0.8 0.8

Polarization of Radiation H V H V H V H V H V H V

d derived (dB) 0.26 4.1 1.5 6.2 0.32 3.5 1.34 6.3 0.6 3.5 1.6 7.9

d experim. (dB) 1.0 2.8 2.8 3.3 0 1.0 0 5.5 0 0.5 5.0 5.5

203

4.5 Sea Clutter RCS Model

angles [66]. The presence of super-refraction in the near-surface troposphere is taken into account for this model in that the real grazing angle appears to be greater than one determined from geometric considerations, and is

␺=

h 1 + r | grad n | r 2

(4.62)

The transition from the plateau region to the interference region occurs at a critical angle

␺c ≈

3.6 ⭈ 103

␲ f u 2.5

(4.63)

where u is the average wind speed in m ⭈ s−1. f is the operating frequency in gigahertz.

␭ is the wavelength in meters. The existence of a transition zone near ␺ ≅ ␺ c , where the angular dependence of the normalized RCS changes in character, is described by the coefficient

A␺ =

冉 冊 冉 冊 ␺ ␺c

4

␺ 1+ ␺c

4

(4.64)

The factor taking into consideration the wind-dependent saturation of the normalized RCS can be written as

Au =

冉 冊 冉 冊 u u0

1+

n

u u0

n

(4.65)

where u 0 is the critical wind speed equal to u 0 = 7 m ⭈ s−1 and n the power index depending upon the operating frequency and defined as n = 1.25f 0.5

(4.66)

204

Sea Backscattering at Low Grazing Angles

The coefficient permitting the variations of the normalized RCS with a varying angle between the beam and the general sea wave run can be taken in the form suggested in [59], A ␣ = exp [0.375 cos ␺ (1 − 2.8␣ ) f 0.33 ]

(4.67)

The operating frequency dependence of the normalized RCS in the saturation region and at grazing angles greater than the critical value is relatively weak and can be approximated as ␴ 0 ∼ f 0.4. The contribution due to sea spray can be taken into account through (4.48). Making use of the relations of [32] and assuming the volume RCS to be constant within the plane h = const, it is then not difficult to obtain the value 0 ␴ spray ≈ 1.36 ⭈ 10−18 f 4 exp (1.4 u )

(4.68)

Taking into consideration that the maximum normalized RCS of the sea surface at 10 GHz is of the order of 35 dB to 37 dB in the plateau region, its dependencies on these parameters can be written as 0

␴ = 7 ⭈ 10

−4

冉 冊冉 冊 f 10

0.4

␺ 10

0.5

A ␺ A u A ␣ + 136 ⭈ 10−18 f 4 exp (1.4u ) (4.69)

The comparison of the experimental results with the normalized RCS calculated according to (4.69) have shown good agreement. This is illustrated in Figure 4.12(a), representing the angular dependences of the normalized RCS at ␭ = 3 cm: 8 mm and 4 mm for mean wind velocities of 8–10 m ⭈ s−1. The similar dependence of the normalized RCS on wind speed for various grazing angles is given in Figure 4.12(b). The lines represent computations according to (4.69), while the signs show the measured results. Figure 4.12(c) shows the normalized RCS as a function of the operating frequency for grazing angles ␺ > ␺ cr , where it can be seen to agree with the frequency dependence of (4.69), and for grazing angles 0.3° and 0.15°. It is seen that the RCS dependence on the operating frequency is stronger at small grazing angles (␺ < ␺ cr ). The effect is due to the decrease in ␺ cr , with increased operating frequency as seen from (4.55), and to the growth of ␴ 0 for the same frequencies [see (4.64)]. Once again, the experimental results are in satisfactory agreement with the data calculated from (4.69). At frequencies less than 10 GHz, the normalized RCS depends on the polarization of transmission and reception. This dependence can be presented as 0 ␴ VV 0 ␴ HH

≅ e 0.25(10 − f )

205

4.5 Sea Clutter RCS Model

Figure 4.12

Dependence of the normalized RCS: (a) upon the grazing angle; (b) upon the wind speed for ␭ = 3 cm (1), ␭ = 8 mm (2), and ␭ = 4 mm (3); (c) upon the operating frequency for the grazing angle ␺ > ␺ cr (1), ␺ = 0.3° (2) and ␺ = 0.15 (3); here the experimental data show by circles at 3 cm, by crosses at 8 mm, and by squares at 4 mm.

or 0 0 (␴ VV − ␴ HH ) dB ≈

再

1.08(10 − f )

for f < 10 GHz

0

for f > 10 GHz

Thus, the normalized RCS dependence on the sea wave parameters in the frequency band 1–100 GHz can be represented by (4.69). In the frequency band

206

Sea Backscattering at Low Grazing Angles

1–10.0 GHz, the polarization dependence of the normalized RCS is taken into consideration in the form 0 ␴ HH

≅ 7 ⭈ 10

−4

冉 冊 f 10

0.5

A␺ AV A␣

0 0 ␴ VV (dB) = ␴ HH + 1.08(10 − f )

(4.70a) (4.70b)

The depolarization of the scattered signals can be taken into account by 0 0 0 0 ␴ cross (dB) = ␴ VH = ␴ HV ≅ ␴ HH − 10

(4.71)

Thus, the empirical model estimates the normalized RCS of the sea surface with both superrefraction and scattering from spray in a frequency range 10 GHz to 100 GHz at grazing angles ␺ < 30° and wind speeds u < 15 m ⭈ s−1. It gives satisfactory agreement with experimental results and can be used to evaluate the contribution of sea clutter during operation of radar detection systems.

4.6 Sea Clutter Statistics The signals returned from the air-water boundary are fluctuating due to scatterer motions within a single resolution cell (a surface area limited by the pulse length and the antenna azimuth beamwidth), as well as to shift of the surface areas viewed by a moving radar. Therefore, the pdf of the normalized RCS is a function of space and time. Measurements at X-band at small grazing angles (1° to 5° ) and with small illuminated cells [67, 68] show that the best approximation to the amplitude probability density is provided by the log-normal distribution

p (␴ ) =

1

␴␴ s √2␲

冤

exp −

ln

冉 冊 ␴ ␴m 2

2␴ s

冥

(4.72)

where ␴ m is the median RCS and ␴ s the rms deviation of ln (␴ ). This distribution is characterized by longer tails in comparison with the Rayleigh distribution, which takes into account the higher probabilities of larger signal amplitudes than in Rayleigh statistics. The experimental distributions are markedly different from the Rayleigh for horizontal polarization, although some differences can be noticed for vertical polarization as well. These differences become sharper for shorter pulses, longer ranges, and greater wind speeds. One can also notice that the distribution of instantaneous signal voltages obeys the composite Gaussian law,

207

4.6 Sea Clutter Statistics

p (x ) = (1 − ␥ )

冉 冊

1 x2 exp − 2 ␴ √2␲ 2␴

+

␥ k␴ √2␲

冉

exp −

x2 2k 2␴ 2

冊

(4.73)

where ␥ is the weight coefficient and k is the ratio of variances of two Gaussian probability densities. It should be noted that these parameters depend only weakly on the angle between the wind direction and the beam. Similar results were obtained in [32, 69] for the shorter wavelength part of the centimeter band and for MMWs. The best approximations of the amplitude probability functions were Johnson’s S B and lognormal distributions for horizontal polarization and the Rayleigh distribution for vertical. At shorter wavelengths and higher sea waves, the standard deviation in (4.72) decreases, as illustrated by the data of Table 4.10 [70]. The differences between the measured distributions and the Rayleigh model are less significant for VV polarization. Therefore, the amplitude distributions are often represented for VV polarization by the Weibull distribution [30] P (A ) =

冉 冊

B B −1 AB exp − A C C

(4.74)

where C is the shape parameter and B the slope parameter. It should be emphasized that for a distribution characterized by two independent parameters, it is relatively easy to select them in such a manner that they fit the experimental distributions. With B = 2 (4.74) becomes the Rayleigh amplitude distribution, while with B = 1 it becomes an exponential. The values of B and C for (4.74) that were derived at ␭ = 3 cm for HH and VV polarizations are listed in Table 4.11 [30]. The measurements in the shorter wavelength part of the centimeter band and for MMWs [32, 69] have shown that instantaneous signal strengths (outputs of a synchronous detector) differed from the standard Gaussian model, the major distinctions being observed, as in [68], for HH polarization. In the case of vertical or circular polarization, the measured data can be approximated reasonably well by the Gaussian distribution. However, the probabilities of signal large values are Table 4.10 ␴ S Dependences on Frequency and Wave Height Frequency (GHz) 10.0

35.0 Source: [70].

Wave Height (m) 0.24 0.48 1.10 0.24 0.48

Wind Speed (m ⭈ s−1 ) 0 6.1 5.6 0 6.1

␴ S (dB) HH VV 9.0 5.2 7.7 4.4 5.7 5.4 7.2 5.2 5.7 4.3

208

Sea Backscattering at Low Grazing Angles Table 4.11 Dependences of B and C on Sea State VV Sea State Sea state 2 Sea state 5 Source: [30].

B 0.622 0.495

HH C 0.065 0.228

B 0.833 0.625

C 0.006 0.034

seen to be higher than predicted by this distribution. Figure 4.13 presents the cumulative probability functions of the signal instantaneous values at ␭ = 3 cm and ␭ = 8 cm using a scale linearly representing the Gaussian law for a range of 0.6 km and wind speed of 6 ms−1. The best approximation of these distributions for HH polarization is provided up to the probability level of 10−4 to 10−5 by the composite Gaussian law, with a ratio of component variances k = 10 to 20 and the weight coefficient ␥ < 0.1. At small grazing angles, the signal returned by the sea surface shows a set of specific

Figure 4.13

Cumulative probability functions of signal instantaneous values.

4.7 Radar Spike Characteristics of Sea Backscattering

209

features. Along with the continuous noise-like signal, spikes at 10 to 15 dB above the average level are observed at short ranges (1–1.5 km). As the grazing angle decreases (range increases), the returned signal acquires a pulsed character, which can be explained by the shadowing of a large part of the surface by such large waves that only the crests of the larger waves extend above the shadow zone. Under these circumstances, the normalized RCS is no longer a characteristic that would give a complete description of the intensity of echoes from the sea surface. To estimate the radar immunity against clutter, the statistical characteristics of the signal spikes become of particular importance.

4.7 Radar Spike Characteristics of Sea Backscattering As can be seen, the sea normalized RCS is a sum of two components for practically all models (the idea of discrete scatterers is discussed in the introduction). Here we will discuss the radar characteristics of scattered signal spikes using the results obtained in [71–73]. For spikes [16] 0 ␴ spike ≅ F 2 (␺ , ␣ ) ⭈ [u *2 /g ]3/2

(4.75)

where F 2 (␺ , ␣ ) is a function of the grazing angle ␺ and the angle ␣ between the wind and beam directions. This function characterizes the scattering of a single spike. Let us consider the ratio between the components in (4.69). In the normalized RCS model [66], the second term is determined by scattering from spray, giving a large contribution to the total signal in X-band and more especially in the millimeter band. In the framework of this model, the ratio is 0 ␴ spike

␴0

= [1 + 0.65 ⭈ 1014 A ␺ ⭈ A V ⭈ A ␣ ⭈ ␺ 0.5 ⭈ f −3.6 ⭈ exp (−1.4V )]−1 (4.76)

where f is the frequency in gigahertz. A ␺ , A V , and A ␣ are the coefficients determining the dependence of the normalized RCS on the grazing angle, the wind velocity, and the angle between the wind and radiation directions. V is the wind velocity. Strong dependencies of this ratio on the wind velocity and the frequency are observed. For decreasing grazing angle, the contribution of the first term in the total

210

Sea Backscattering at Low Grazing Angles

scattered signal decreases because of shadowing, and the backscattering assumes a more discrete shape. The experimental data permit us to confirm that the probability of spikes obtained in a single resolution cell at a certain time can be approximated by the Poisson distribution P (n ) =

Nn ⭈ exp (−N ), n!

N > 0, n > 0

(4.77)

where N is the mean number of spikes. This probability depends on the grazing angle, the threshold, and the sea state. From our data [32], the mean intensity of spikes at 3 cm is 10–12 min−1 for a grazing angle of 0.4° and decreases to 6–8 min−1 for grazing angles of 0.1°–0.15°. This agrees rather well with the data on the mean number of crossings of some boundary by the Gaussian random process shown in Figure 4.7 for wind velocities of 7–10 m/s. Our measurements carried out at 140 GHz for wind velocities less than 5 m/s and zero threshold showed that for grazing angles less than 3.5°, the mean number of spikes was 20 to 24 per minute. The dependence of the mean number of spikes on the friction velocity obtained in [30] at 15 GHz for two polarizations shows that it can be approximated by N = C 1 ⭈ u˜ *␣ 1

(4.78)

where log C 1 = 1.1 and ␣ 1 = 2.9 ± 0.6. 0 This agrees with Philips’ model [16]. The ratio ␴ sp /N = const, and with increasing wind velocity, the mean RCS of the spikes increases. As the result, the spike normalized RCS distributions are independent of the friction velocity for velocities from 25–45 cm/s. The spike probability decreases with increased shadowing. The spike statistics at the output of the phase detector of a coherent radar measured at 3.0 cm and 0.8 cm for the grazing angles less than 0.3° is shown in Table 4.12. Here the polarization dependence of spike probability is not significant, but its great dependence on the threshold is seen rather clearly. Besides, the spike

Table 4.12 The Spike Probabilities at 3.0 and 0.8 cm for Different Polarizations Relative Threshold 1.5 1.5 2.0 2.0

Polarization of Radiation and Reception HH VV HH VV

Spike Probability at 3.0 cm 0.27 0.22 0.09 0.09

Spike Probability at 0.8 cm 0.26 0.26 0.10 0.19

Coincidence Probability of Spikes 0.06 0.05 0.02 0.01

211

4.7 Radar Spike Characteristics of Sea Backscattering

probability as a function of the threshold coincides with conclusions for sea wave peaks over some boundary. The weak coincidence of spikes for the instantaneous values of backscattering at two frequencies can be expected from the different mechanisms of the scattered signal formation at these frequencies. The spike and gap probabilities are considerably greater for the sliding mean amplitudes of the scattered signals at 3 cm and 0.4 cm. The joint probability data for orthogonally polarized scattered signals and for horizontal, vertical, and tilted polarizations are shown in Table 4.13 (here the symbols 0 and 1 refer to gaps and spikes, respectively; the first argument corresponds to vertical polarization and the second to horizontal received polarization). It is seen that for the sliding mean amplitudes, coincidence of orthogonally polarized components for spikes and gaps is observed 30%–40% of the total time and the absence of coincidence only 10%–20%. At the same time, the spikes of the instantaneous signal are observed 10% of time at both polarizations. In [32, 51, 71], the results for maximal RCS distributions of spikes for a threshold level of 6 dB above the mean RCS at frequencies of 10 GHz and 37 GHz and for horizontal, vertical, and circular polarizations of radiation are presented. The following conclusions are made: •

•

•

•

For identical weather conditions, the maximal spike RCS increases with decreasing wavelength, and at 8 mm it is larger than at 3 cm by 3–7 dB. The maximal spike RCS increases for decreasing grazing angle, explained by the presence of more intense scatterers in the shadowing zone. Polarization dependence of maximal spike RCS is not seen clearly. The probability of the spike RCS larger than 0.1 m2 is higher for horizontal polarization than for vertical polarization, while the probability of spikes with large RCS at 8 mm is larger for vertical polarization. The spike RCS decreases for circular polarization in comparison with linear by about 10 dB. The most probable duration of spikes is 0.4–0.6 second, and the maximal duration is 4–5 seconds. These results agree with the sea surface peak duration shown in Figure 4.9.

Table 4.13 Statistics of Spike Amplitudes at the Wavelengths of 3.0 cm and 0.8 cm Wavelength (cm) 3.0 3.0 3.0 0.4 0.4

Sea State Sea state 5, wind velocity 15 m/s

Transmitted Polarization V H Tilted V V

Spike Probabilities PV PH 0.53 0.50 0.51 0.50 0.51 0.46 0.45 0.15 0.42 0.49

P(0,0) 0.34 0.32 0.46 0.49 0.37

Joint Probabilities P(0,1) P(1,0) 0.13 0.17 0.17 0.19 0.40 0.10 0.05 0.36 0.21 0.14

P(1,1) 0.37 0.32 0.14 0.10 0.28

212

Sea Backscattering at Low Grazing Angles

•

•

•

The mean duration of gaps increases for increasing threshold, especially for sea state 2, where the results of the peak theory are clearly applicable. These data also agree with the results in Figure 4.9. In our investigations, the mean duration of spikes and gaps depends on the wind velocity. The mean spike duration is 0.1–0.3 second for the wind velocities of 2–4 m/s and 0.2–0.5 second for the wind velocities greater than 6 m/s. The mean gap duration is 0.5–5 seconds for the wind velocities of 4–8 m/s. The mean duration of spikes increases and the gap duration decreases with decreased wavelength.

As an example, the histograms of spike duration distribution at 3.0 cm and 0.8 cm for vertical and horizontal polarization are shown in Figure 4.14, and the distributions of spike and gap duration at 2.0 mm are shown in Figure 4.15. As shown in [62], a rather intense depolarization in spikes is observed for vertical polarization, while the difference of depolarized components for spikes and gaps is small for horizontal polarization. The depolarization coefficients for gaps depend on frequency rather weakly, and the spike depolarization coefficients increase with increasing frequency. The total power spectra of the scattered signals are determined by the scattering from sea and spray. There is an increase in Doppler frequency for spikes (by two

Figure 4.14

Spike duration distribution at 3 cm and 8 mm for (a) horizontal and (b) vertical polarizations.

4.8 Backscattering Spectra

Figure 4.15

213

(a) Spike and (b) gap duration distributions at 2 mm.

to three times) in comparison with gaps. In 15%–20% of the spectra, a second maximum appears that is caused, in our opinion, by the scattering from spray blown by the wind.

4.8 Backscattering Spectra The power spectra of the signals backscattered from the sea are generally determined by fluctuations of scatterers driven by the wind as well as by antenna scanning and radar platform motion. The effects of various scanning techniques and platform motion on the spectra are not discussed here. The following discussion addresses the power spectra of scattered signals with a fixed radar antenna.

214

Sea Backscattering at Low Grazing Angles

The power spectra of X-band and MMW signals scattered by the sea surface show a marked dependence on the wind speed and antenna polarizations. As the wind speed increases, the central frequency F 0 and the spectral width ⌬F tend to increase. In this case, the spectra for vertical and circular polarizations have lower magnitudes at the central frequency F 0 , as compared with horizontal polarization, while all of the spectral widths are the same. The difference in the central frequency decreases with decreasing grazing angle. For instance, for ␭ = 3 cm the ratio F 0HH /F 0VV was 1.2 to 1.5 with a grazing angle about 1°, whereas the difference of frequency shifts vanished at grazing angles less than 0.3°. To approximate the scattered signal spectra over the wide band of radiowaves (from 1 GHz to 140 GHz) and small grazing angles, the authors suggest the relationship [32, 74]

冉 |

F −F G (F ) = G 0 1 + 0 ⌬F

|冊

n −1

(4.79)

where G 0 is the maximum value of the spectral density at F 0 ; ⌬F is the spectral half-width at the −3-dB level, and F is the current frequency. The rate of spectral density decrease characterized by the power exponent n is dependent on the wind speed and sea surface state; decreasing with an increase in the wind speed. Figure 4.16 shows, as an illustration, the spectra measured at ␭ = 3 cm with horizontal polarization for the frequency range F < F 0 . Triangles show measurements at wind speeds less than 4 ms−1, and circles and squares represent the experimental data obtained at wind speeds of 5–8 ms−1. The

Figure 4.16

(a) Power spectra of radar returns from the sea surface at ␭ = 3 cm for horizontal polarization: curve 1: n = 2; curve 2: n = 3; curve 3: n = 4 and curve 4: n = 5. The dots correspond to measured results. (b) Power spectra of intensity at ␭ = 3 cm with the wind speed of 10–12 ms−1; 1: exponential function; 2: power law with n = 2; 3: power law with n = 3; the dots correspond to measured results.

215

4.8 Backscattering Spectra

power exponent is seen to decrease with increasing wind speed. Similar magnitudes of the exponent were obtained in [32] for the high-frequency spectral region (F > F 0 ), which leads to the conclusion that the spectrum is symmetrical with respect to the central frequency shift F 0 . In order to describe the spectral density dependence on the radar operating frequency (wavelength) and wind speed, the following empirical relationship can be used n = 8.9␭ 0.1V −0.5

(4.80)

where ␭ is the wavelength in centimeters and V is the mean wind speed (ms−1 ) (see [6]). The central frequency and the spectral width vary proportionally to the operating frequency. The measured values happen to be larger than predicted ones, apparently due to the contributions from wave crests and from breaking waves that move at considerably greater speeds. Let us consider the change of central frequency F 0 because of the Bragg’s scattering and the phase speed of sea waves in the framework of this mechanism, the orbital movement of sea waves, and the wind drift. Within the framework of the two-scale model, the account of these factors permits the reception of the following expression for determination of the spectrum central frequency F0 =

√

g ⭈ cos ␸ 2V WD ␲␮ 2V0 + 16 3 + cos ␣ + cos ␣ ␲␭ ␭ ␭ ␳␭

(4.81)

where ␣ is the angle between directions of radiation and sea wave movement; V0 is the orbital speed of large gravitational sea waves; V WD is the speed of a wind drift; ␮ is the factor of the surface tension; ␳ is the density of water; and ␭ is the wavelength of radiowave. The first item in the right part (4.81) is determined by the phase speed of movement of the great sea waves and defined from the dispersive equation [50]. The expression under the root square caused by the surface tension is small in comparison with item of g /␲␭ in practically all range of frequencies except for a shortwave part of a millimeter range. Besides, the point on sea surface moves in close to a circular orbit. For a case of simple harmonious movement, orbital speed is determined as V0 =

␲H T

where H and T are the height and the period of sea wave accordingly.

(4.82)

216

Sea Backscattering at Low Grazing Angles

For small grazing angles when scattering elements are near to sea wave crests (i.e., the other part of a wave is in a zone of shadowing), the orbital speed may be submitted as V0 =

冉

␲H H 1 + 2␲ 2 T gT 2

冊

(4.83)

The last expression is the sum of the classical orbital speed (4.82) and the contribution of Stocks drift. The second item, usually rather small, nevertheless brings the contribution of a spectrum of the reflected signal to the formation. For rough seas, the height and the period of a sea wave are unequivocally connected to speed of wind by the dependences H = 7.2 ⭈ 10−3 ⭈ V 2.5

(4.84a)

T = 0.556 ⭈ V

(4.84b)

Therefore, (4.82) and (4.83) may be presented as V0 ≅ 1.3 ⭈ 10−2 ⭈ ␲ V 1.5

冉

V0 = 1.3 ⭈ 10−2 ⭈ V 2.5 1 + 4.66 ⭈ 10−2

␲ 2 0.5 V g

冊

(4.85)

= 1.3 ⭈ 10−2 ⭈ V 2.5 (1 + 4.69 ⭈ 10−2 ⭈ V 0.5 ) Here V is the mean speed of a wind at height of 10m above the sea surface. Due to the interaction of a wind and sea surface, one more movement of a sea surface named a wind drift is observed. Available experimental results specify that speed of a wind drift does not exceed 3% from wind speed at height of 10m [11, 12] and often the following ratio is used: V WD ≈ 0.02V

(4.86)

For the estimation of a role considered above three factors, forming the central frequency in a spectrum of scattered signals, we shall consider dependence of F 0 on wavelength presented in Figure 4.17. The experimental data of different authors obtained in the frequency band of 1–140 GHz is marked by the different signs; the straight lines 1–4 are some approximations of F 0 dependence, presented under the figure. A better agreement of the dependence of F 0 on the wavelength with the experiment is provided by the following empirical expressions [74]

217

4.8 Backscattering Spectra

Figure 4.17

The spectrum central frequency dependences on the wavelength 1:

√

g 2V WD 2V 0 + + ; 2: F 0 = ␲␭ ␭ ␭ signs show the experimental data. F0 =

√

g V ; 3: F 0 = 44.4 ; 4: F 0 = ␲␭ ␭

F 0 = 44.4

V ␭

√

g 2V WD + ; ␲␭ ␭

(4.87)

As marked earlier, the F 0 value is determined by the polarization of radiation and reception. Its value is smaller for the vertical polarization in comparison with horizontal, and this difference decreases for the grazing angle decreasing. The experimental data is marked as in Figure 4.17. The ⌬F = f (␭ ) dependence is presented in Figure 4.18. Here the different signs are the same as in the Figure 4.17. The experimental results are satisfactory described by ⌬F = 30.7

V 0.75 ␭

(4.88)

These are valid for wind speeds of 2–15 m ⭈ s−1 (local wind velocities achieve 20 m ⭈ s−1 and more) and were checked at frequencies of 10–140 GHz and grazing

218

Sea Backscattering at Low Grazing Angles

Figure 4.18

The spectral width dependence on the wavelength.

angles ␺ < 5°. As the grazing angle is changed, the power spectra of the backscattered signals undergo changes that are especially noticeable for vertical and circular polarizations. As has been noted, at lower grazing angles F 0VV → F 0HH , whereas F 0HH is virtually independent of the grazing angle. Moreover, as ␺ decreases, the spectra become somewhat broader, as described by the empirical expression [74]: ⌬F ⌬F

= 0.63 + 0.064

h c␺

(4.89)

where h is the radar height. This expression holds for ranges r ∈ (0.3 to 3.0) km. Varying the angle ␣ between the illumination direction and the general wave run (which coincides with the wind direction in the case of developed roughness) leads to variations of the central frequency in the power spectrum for all polarizations. The central frequency is maximal if the surface is illuminated normally to the wave front and is practically equal to zero for illumination along the wave. The azimuthal dependence can be approximated by F0 = cos2.5 ␣ , F 0max

| ␣ | ≤ 30°

(4.90)

219

4.8 Backscattering Spectra

Neither the spectral width nor the exponent of power depends on the angle ␣ . Besides, the spectral width does not depend on the polarization of radiation and reception. Finally, the spectral parameters can be represented in the form F 0HH = F 0HV = 44.4 F 0VV = F 0VH =

再

V ␭

(4.91)

F 0HH (1 − 0.4␺ )

for ␺ ≤ 2°

0.5F 0HH

for ␺ > 2°

⌬F VV = ⌬F HH = ⌬F VH = ⌬F HV ≅ 30.7

V 0.75 ␭

(4.92)

(4.93)

Basically, the power spectra of the scattered signal intensity (i.e., the spectra at the amplitude detector output) have properties similar to those of the power spectra of the signals themselves. Figure 4.16(b) presents the measured power spectra of intensity and the curves corresponding to an exponential spectrum (curve 1) and to the power law

冋 冉 冊册

F G (F ) = G 0 1 + ⌬F

n −1

(4.94)

The comparison of Figure 4.16(a, b) shows the spectrum of (4.94) to retain the general form of those obtained from (4.79), but with a lower power index. In other words, the rate of the drop off in spectral density slows down in comparison with the power spectrum of the signal. As shown in [75], the intensity spectra (amplitude spectra) are characterized by a small width in their energy-carrying part, not exceeding 1.5–2 Hz at the −20-dB level. With the antenna pattern oriented parallel to the sea wave, their spectral peaks lie in the frequency range 0.1–0.15 Hz, coinciding with the maximum of the sea spectrum. The spectra are characterized by the absence of peaks for backscattering from the surf zone. At frequencies above 0.5–0.15 Hz, the spectrum can be approximated by an expression similar to (4.88), G (F ) ∼ F −n. The probable values of the exponent n lie between 1.9 and 2.6 for ␭ = 3 cm and 2.6 to 3.4 for ␭ = 8 mm. Regression analysis shows that in the X-band, the spectra in their energycarrying part are, on the average, 1.5 to 1.8 times wider than at ␭ = 8 mm. No stable differences have been observed in the spectra of cross-polarized components. Here, the analysis revealed a high correlation of the orthogonal scattered components on both operating frequencies. The correlation factors were 0.7 to 0.95, the lower values being observed for illumination perpendicular to the sea wave and the higher noted in the surf zone. This tendency is even more pronounced in

220

Sea Backscattering at Low Grazing Angles

cross-correlation factors of the signals at 10 GHz and 35 GHz. The lowest values are 0.3–0.4 for the calm sea; they tend to increase with the appearance of wave breaking and reach 0.8 when the breaking is intense. Because the sea state is far from being steady under the wave-breaking conditions (which result in spikes and pauses in the scattered signals), the current power spectra reveal some distinctive features never shown by the average power spectra. These effects have been studied experimentally in the 3-cm, 8-mm, and 2-mm bands [33, 76] and can be summarized as follows. In breaking conditions the current spectrum of sea backscattering can be presented as G (F ) = G sea (F ) + G sp (F )

(4.95)

where G sea (F ) and G sp (F ) are the spectral densities of sea and spray backscattering, respectively. The first component is described by (4.79), and dependences of its parameters on the sea state and wind velocity have been discussed earlier. Some features of second component were considered in [33]. The spectral width characteristic of the spike period of the returned signal is considerably greater (up to a factor of two) than that one shown in gaps. The central frequency also increases. Table 4.14 presents the averaged data on the spectral widths and central frequency at two wavelengths, with different transmitted and received polarizations. The results in Table 4.14 show that the central frequencies during the spike and gaps are higher for horizontal polarization than for vertical polarization. This is in agreement with the conclusions drawn from the averaged spectra. The forms of the power spectra for the cross-polarization components are similar to those obtained in the copolar channels. A relatively high correlation between the Doppler shift and the intensity of the return signal is observed under the heavy sea, for all polarizations, whereas similar correlation is practically absent for the calm sea. Instantaneous power spectra of the scattered signal spikes often show, along with

Table 4.14 The Central Frequency and Spectral Width in Spikes and Gaps Frequency (GHz) 9.6

35.0

Source: [33].

Polarization HH HV VV VH HH HV VV VH

Spikes F 0 (Hz) ⌬F (Hz) 256 ± 56 56 ± 20 255 ± 66 65 ± 22 222 ± 82 78 ± 26 233 ± 83 84 ± 19 890 ± 145 575 ± 153 969 ± 190 629 ± 52 606 ± 230 356 ± 118 720 ± 200 301 ± 70

Gaps F 0 (Hz) 141 ± 25 141 ± 30 62 ± 31 63 ± 32 521 ± 85 471 ± 11 222 ± 63 260 ± 96

⌬F (Hz) 38 ± 18 34 ± 22 45 ± 23 59 ± 21 219 ± 44 271 ± 53 157 ± 52 224 ± 50

221

4.8 Backscattering Spectra

the effect of power concentration at higher frequencies than in the gaps, a second maximum in their high-frequency part (around 220–360 Hz or 900–1,300 Hz for the 3-cm and 8-mm wavebands, respectively). The second maximum frequency can be derived from the empirical relationship F sp ≅

冉 冊

2U 0 h ␭ h0

0.25

cos ␣

(4.96)

where U 0 is the mean wind velocity at height of h 0 and h is the height of wave breaking. The experimental results of [34] show that the second spectral width can be from 50 Hz to 2.0 kHz. The mean spectral width is presented in Table 4.15. The second peak was observed in 15% and 21% of the processed spectra at ␭ = 3 cm and ␭ = 8 mm, respectively. The relative frequency of appearance of the second peak reported in [32] was lower and was 1.5% to 6%. Similar multimode spectra from breaking waves were observed at ␭ = 8 mm in [33]. The presence of the second peak can be explained in two ways. First, at the moment of wave spiking, just before breaking, its orbital velocity increases. Second, the sprays generated after wave breaking are carried away by the wind, their drift velocity reaching 60% of the wind speed. The predominant effect of the second mechanism is confirmed by the increased depolarization factor, which can be attributed to the emergence of fast-moving nonspherical droplets. The degree of correlation between the copolarized and cross-polarized components of the scattered signal can be found from the coherence function. Analysis of its behavior indicates that in the X-band, the major contribution to the scattered signal during the gaps was made by scatterers having relatively low speeds (Doppler frequencies below 100 Hz). At the moments when spikes occur, additional highspeed scatterers appear (the corresponding Doppler frequencies are 200–400 Hz), which are characterized by a low degree of coherence between the orthogonally polarized components. At ␭ = 8 mm the coherence of cross-polarized components is low over the entire range of Doppler frequencies, both in the spikes and in the gaps.

Table 4.15 Mean Spectral Width (kHz) Wavelength (mm) ⌬F sp (kHz)

30 0.05–0.08

8.0 0.36–0.58

4.0 0.62–0.9

2.0 1.25–2.0

222

Sea Backscattering at Low Grazing Angles

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[75]

[76]

225 Ufimtsev, P. Y., Edge Wave Technique in Physical Theory of Diffraction, Moscow, Russia: Soviet Radio, 1962. Cherny, I. V., and F. A. Sharkov, ‘‘Remote Radiometry of the Sea Wave Breaking Cycle,’’ Earth Research from Space, No. 2, 1988, pp. 17–28 (in Russian). Kulemin, G. P., ‘‘Sea Backscattering Model for Millimeter Band of Radiowaves,’’ Proc. 8th URSI com. F Triennial Open Symp., Session 3, Aveiro, Portugal, September 1998, pp. 128–131. Croney, J., ‘‘Clutter and Its Reduction in Shipborne Radars,’’ Proc. Int. Conf. Ant. and Propag., London, England, Vol. 105, No. 2, 1973, pp. 213–219. Trunk, G. V., ‘‘Radar Properties of Non-Rayleigh Sea Clutter,’’ IEEE Trans. on Aerosp. Electr. Syst., Vol. AES-8, No. 2, March 1972, pp. 196–204. Kulemin, G. P., and V. I. Lutsenko, ‘‘On the Distribution Laws of Millimeter Wave Signals Scattered by the Sea Surface at Small Grazing Angles,’’ II Soviet Symp. on Millimeter and Submillimeter Waves: Conf. Digest., Gorky, 1980, Vol. 1, pp. 293–294 (in Russian). Dyer, F. R., N. C. Currie, and M. S. Applegate, ‘‘Radar Backscatter from Land, Sea, Rain and Snow at Millimeter Wavelengths,’’ Adv. Radar Techn., London, England, 1985, pp. 250–253. Kulemin, G. P., ‘‘Spike Characteristics of Radar Sea Clutter for Extremely Small Grazing Angles,’’ SPIE Int. Symp. Radar Sensor Technology V, Vol. 4033, Orlando, FL, April 2000, pp. 129–138. Kulemin, G. P., ‘‘Spike Characteristics of Radar Sea Clutter for Extremely Small Grazing Angles (Part 2),’’ SPIE Int. Symp. Radar Sensor Technology VI, Vol. 4374, Orlando, FL, April 2001. Kulemin, G. P., ‘‘Microwave Sea Backscattering Features for Extremely Small Grazing Angles,’’ Modern Radioelectronics Progress, No. 12, 1998, pp. 17–47 (in Russian). Kulemin, G. P., and V. I. Lutsenko, ‘‘Special Features of Centimeter and Millimeter Radio Wave Backscattering by the Sea Surface at Small Grazing Angles,’’ Preprint No. 237, Institute of Radio Physics and Electronics, Academy of Sciences of the Ukrainian SSR, Kharkov, 1984 (in Russian). Balan, M. G., et al., ‘‘Statistics of Envelopes of Microwave and Millimeter Wave Signals Scattered by Nonstationary Sea Waves,’’ Conf. Digest Representations and Processing of Random Signals and Films: II Soviet Conference, Kharkov, 1991, p. 199 (in Russian). Balan, M. G., et al., ‘‘Nonstationary Radar Reflections from the Sea in the Millimeter Band,’’ Conf. Digest Representations and Processing of Random Signals and Films: II Soviet Conference, Kharkov, 1991, pp. 82–83 (in Russian).

CHAPTER 5

Microwave and MMW Backscattering by Precipitation and Other Meteorological Formations

5.1 Structure of Meteorological Formations Millimeter-band radar systems offer several advantages over electrooptical systems for operation in the battlefield and for weapon guidance. Among these are allweather operation, operation in smoke, and operation in dusted atmosphere. To determine the radar characteristics of meteorological formations, we must classify them, in order to present the complex meteorological conditions of the propagation path in terms of simpler phenomena. A useful classification is based on grouping meteorological formations by size and the physical properties of their constituent particles, including the particle sphericity. Such a classification corresponds to definitions issued by the International Meteorological Organization in 1956. According to it, all precipitations are divided into liquid and solid ones, and in turn all liquid precipitations are divided into rain and drizzle. Rain is water precipitation formed by drops with radius greater than 0.25 mm. Observations show that drops with radius greater than 3.5 mm become flat and break into smaller drops. The raindrop terminal velocity reaches 8–10 m/s. Drizzle is rather homogeneous precipitation consisting mostly of drops with radii less than 0.25 mm. Drizzle intensity is not greater than I = 0.25 mm per hour, and terminal velocity through fixed air is less than 0.3 m/s. Consequently, all spherical water drops with diameters of 0.5–5.5 mm can be considered rain. The geometrical characteristics of rain zones depend on the rain intensity and climatical conditions in the local area, connected with geographical coordinates of this area. The horizontal and vertical extents of rain zones with different intensities are shown in Table 5.1 [1]. Rains in their precipitation zones are distributed nonuniformly, especially for rain intensities greater than 40 mm/hr.

227

228

Microwave and MMW Backscattering by Precipitation and Other Meteorological Formations Table 5.1 Rain Geometrical Characteristics Rain Intensity (mm/hr) 2 4 8 16 32 64 Source: [1].

Diameter (km) 300 45 35 20 8 1

Height (m) 4,000 4,000 8,000 8,000 8,000 8,000

Showers and heavy rains are distinguished. Showers fall, as a rule, from nimbostratus clouds and are characterized by small to moderate intensities (less than 20 mm/hr), insignificant temporal changes, and small drop sizes. Heavy rains fall from cumulonimbus clouds. They are characterized by large intensities (I ≥ 40 mm/hr), temporally changeable intensity, and comparatively large drop sizes. The determination of rain intensity corresponding to a certain percentage of observation time is an important problem. Different precipitation rates are found in different climatic regions. The probability (or percentage of time during which rain of given intensity or more falls) is approximately described by the gamma distribution of form [2] p=

␤␯ ⭈ ␯ ⭈ I ␯ − 1 e −␤ I ⌫(␯ + 1)

(5.1)

where ⌫(␯ + 1) is the gamma function, and ␤ and ␥ are parameters taking into account the climatic features of the region. It is possible to use global rain models [3] for Earth’s regions for which the statistical distributions of rain intensity are unknown. For the most simple RiceHolmberg model [3], the percentage of mean yearly time p (I ), percentage during which the rain intensity at a given station exceeds I mm/hr, is determined by p (I ) =

M0 [0.03␤ e −0.03l + 0.2(1 − ␤ ) (e −0.258l + 1.86e −1.63l )] 87.6

(5.2)

where ␤ = M ⌫ /M 0 ; M 0 and M ⌫ are the mean annual total rainfall and the mean rainfall per storm, respectively, in millimeters. Heavy rains usually are 20% and moderate rains are 40%–60% of all rain types. Taking into consideration that the mean rain duration usually is not less than 2 hours, the rain mean intensity over the ocean will be equal to 0.5–5.0 mm/hr as a function of region. The probability of rain at 100 mm/hr for seaside regions does not exceed 5 ⭈ 10−4 and the probability of rain at 10 mm/hr is 10−2; therefore, rain of moderate

5.1 Structure of Meteorological Formations

229

intensity (5 mm/hr and less) makes the main contribution to total annual precipitation. One can find the rain distributions of certain intensity for different regions in many papers [2, 4, 5]. The mean number of days with precipitations equivalent to rain intensities more than 0.1 mm/hr and the total annual precipitation for oceanic regions are shown in Table 5.2. Note that even for a large number of rainy days, the mean rain probability usually does not exceed 10% because of the small rain duration. Considering this, the requirement on maritime radar to detect the objects in rain with intensity of 1–5 mm/hr is justified. As a practical matter, maritime radar must be designed for rather extreme conditions, as one collision per 10 years is still too many, and most ships must operate in heavy snowfall as well as in rain. Hail is formed, mainly in summer, in powerful convective clouds. The large sizes of cloud drops and high liquid water content in clouds assist in formation of solid ice layers on the ice particles falling through a supercooling part of the cloud. Usually, the hail particles have the form of ice balls, but hemispherical, cone, and lentil-like shapes are also found. The particle sizes do not exceed some millimeters but cases of hail with diameter up to 10–12 cm have been observed. The hail density varies from 0.5 g/cm3 to 0.9 g/cm3, and density of small hail and ice grains is approximately 0.3 g/cm3. The particle size and precipitation intensity for hail formation and hail falling change as determined by rising airflow features. For continuous rising flow with a velocity of about 5 m/s, the hail diameter is constant some time after hail begins to fall, and then a quick decrease is observed. If rising airflow is absent, the hail diameter quickly decreases in the first minutes of falling. The hail surface is heated and melted for hail falling into air with positive temperature. Formed water spreads around the hail surface, either as a thin water film or as a water and ice mixture. We must recall that hail can form for internal cloud temperatures less than −(10 to 12)°C, and hail surface temperature for melting

Table 5.2 Precipitation Distributions for World Ocean Regions Region Norway Sea Barents Sea Newfoundland North Sea Equatorial part of Atlantic Ocean Indian Ocean Equatorial part of Pacific Ocean Philippines Arabian Sea Antarctic region Source: [2].

Number of Days with Precipitation 150 150 150 200 100–200 100–200 100–150 200 50 250–300

Total Annual Precipitation 75 100 75 — — — 100 430 — —

230

Microwave and MMW Backscattering by Precipitation and Other Meteorological Formations

is 0°C. The water film thickness on most hail surfaces at landing is fractions of a millimeter. Usually, hail falls in narrow zones with width from fractions of a kilometer to 10–15 km and length from units to tens of kilometers. The period of hail falling is small (typically 5–10 minutes). The annual number of days with hail is 2–3 days in plain regions. Cumulonimbus clouds with thunderstorm activity are very significant radar clutter sources. The thunderstorm centers depend on physical and geographical conditions, weather, and season. Their maximal number is observed in June–July. The mean number of days with thunderstorm for some towns in the former USSR is shown in Table 5.3 [6]. Radar pulses reflect from the lightning channel boundaries in thunderstorms because of the large air refractivity factor gradients arising through intensive air heating in the channel and the high concentration of gas ions and free electrons in the discharge channel. A cloud is the visual accumulation of condensation or sublimation products of evaporated water at some height in the free atmosphere. In meteorology, sublimation is the transference process of evaporated water steam to its solid phase, skipping the liquid phase. Clouds are characterized by great variety of form and physical structure. The bases of their classification are the forming conditions and morphological sign (i.e., the outer shape of the cloud). For forming conditions, all clouds are divided into three classes: •

•

•

Cumuliform clouds are strongly developed in vertical planes and comparatively small horizontally sized ones. They are formed as a result of airintensive rising (convectional movement). Undulating clouds are the great, horizontal layers that have the shape of fleecy clouds, rollers, or banks. Stratus clouds are the layers in the shape of compact cover; their horizontal sizes exceed the vertical ones by some hundred times. They form as a result of air that slows smooth rising movement.

For the cloud height, the clouds are divided into four classes: upper, middle, and lower tiers, and the cloud family of vertical development. The clouds of the Table 5.3 Mean Number of Thunderstorm Days in June (Numerator) and July (Denominator) Town Moscow Kiev Odessa Simpheropol Source: [6].

Median Value 6/7 7/6 7/5 6/6

Maximal Value 13/16 14/16 15/14 14/16

5.1 Structure of Meteorological Formations

231

upper tier are disposed at heights more than 6,000m; the clouds of the middle tier are at heights 2,000–6,000m, and the clouds of the lower tier are at heights less than 2,000m. The vertical-development cloud foundations are placed at heights of lower tier clouds, and the tops are at the height of middle- or high-tier clouds. For the external shape, all clouds are divided into 10 forms (or families) having the following names and abbreviations: cirrus (Ci), cirrocumulus (Cc), cirrostratus (Cs), altocumulus (Ac), altostratus (As), nimbostratus (Ns), stratocumulus (Sc), stratus (St), cumulus (Cu), and cumulonimbus (Cb). For content, the clouds are divided into three groups: water (liquid-drop), consisting of water drops and supercooled drops at negative temperatures; freezing (crystal), consisting of ice crystals; and mixed, consisting of a mixture of supercooled water drops and ice crystal. The primary particles for cloud formation are, as a rule, liquid drops. Ice crystals form in a cloud if the cloud’s high part has rather lower temperature. Usually the cloud crystallization begins near the isotherm of −10°C and then can propagate to the entire supercooled part of cloud. Water clouds are found most often in summer, and ice clouds are found most often in winter. The clouds of mixed structure are not clearly seasonal. The cloud vertical extent, generally, can reach 10,000m, and the horizontal extent can be up to 1,000 km for stratus clouds and only 10 km for cumulus clouds. The water drop size distribution depends on cloud height; the drop mean size grows with altitude. The drop size distribution can be represented as

f (a ) =

1 a␮ ␮ ␮ − 1 ⭈ ␮ + 1 e −␮ (a /r ) ⌫( ␮ + 1) r

(5.3)

where ␮ is the half-width parameter and a is the drop radius. The microphysical cloud characteristics and their typical thickness are shown in Table 5.4. The mean value of maximal water content depends on cloud types and their power, and it varies from 0.1 g/m3 to 0.6 g/m3. The water content of powerful cumulus clouds can reach considerably greater values: in the European part of the former USSR, it can be 1.4–1.55 g/m3; for temperate and tropical latitudes, it can be up to 4 g/m3 or more; and in the United States, the cumulonimbus cloud water content has been observed up to 20 g/m3 at heights of 5,500–7,500m. Fog is the aggregate of water drops or ice crystals balanced in air, which decreases the visibility to 1 km or less. As a function of visibility range, one can describe heavy fog corresponding to visibility less than 50m, moderate fog with visibility from 50m to 500m, and light fog with visibility from 500m to 1,000m. The fog water content changes within wide limits from thousandths to 1.5–2.0 g/m3. With cooling, the fog water content increases, while the water content

232

Microwave and MMW Backscattering by Precipitation and Other Meteorological Formations Table 5.4 Average Physical and Microphysical Characteristics of Clouds

Type of Cloud Cirrus Cirrocumulus Cirrostratus Altocumulus Altostratus Stratocumulus Stratus Nimbostratus Cumulus Cumulus congestus Cumulonimbus Source: [5].

Thickness (km) 3 0.3 1 0.5 1 0.5 0.5 1 2.5 3 7

Height of Lower Boundary (km) 8.5 7 7 4 4 1 0.4 0.5 1.1 1 0.7

Water Content (g/m 3 ) 0.005 0.005 0.005 0.1 0.2 0.1 0.1 0.2 0.2 1.2 1.0

GammaDistribution Parameters ␮ r 5 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2 6 3 6 3 6 1

of fog formed by evaporation from a water steam decreases with increasing temperature. Experimental data show that the fog water content and visibility change significantly with height only at the upper and lower boundaries. Fogs are otherwise rather uniform in the vertical plane. The horizontal extents of fog can reach some hundred kilometers, with vertical extents up to 1,000m. The average vertical extent in Arctic Regions is 250m. The horizontal sizes of fog areas extend to 130–180 km, and 20% of cases cover an entire sea (e.g., the Black Sea). It is established that the number of drops per 1 cm3 changes from 0.5 to 93 in advective fogs, from 50 to 860 for radiative fogs, and from 70 to 500 for evaporation fogs of medium intensity. The drop sizes change from fractions of a micron to some tens of microns, and average radius ranges from 2 microns to 18 microns. The average number of days with fogs in world oceans is 50 days per year. Fogs are often observed along coasts of the North Sea, Baltic Sea, Sea of Ochotsk, and in regions of Florida and California. In the region of Newfoundland, Canada, fogs take place up to 250 days per year, and they are also typical for the entire North Sea. They appear 30%–70% of the year in the Kara Sea, and 40% in Laptevs Sea. The average duration of fog is 6–8 hours, and this extends up to two days for 1%–3% of all cases. A special type of sea fog is sea vapor accompanied by strong wind, the duration of which can reach 80 hours. Such fog puffs occupy the vertical layer about 10m, impeding the detection of marine objects. Sandstorm and dust-storm formations are distinguished by spatial extents of 10–500 km2 and time durations of 3 hours to a few days, and by their chemical content, geometrical parameters, and water content in aerosol particles. The use of the theoretical relations for radiowave attenuation in sand and dust clouds demands knowledge of particle shapes and sizes and their dielectric constant value

233

5.2 Atmospheric Attenuation

and size distributions. For practically used bands, including Q- and K-bands, one can use the Rayleigh approximation, in which the particle shape does not affect the wave attenuation. The estimates show that the Rayleigh approximation can be applied in frequency bands up to 100 GHz. Dust storms are characterized by the following particle sizes: a large fraction consists of particles with diameters of about 0.01 mm, while a smaller fraction has diameters of 0.001 mm or less [7, 8]. This aerosol density can be very large. There is about 10 ␮ g/m3 of dust in the clear atmosphere, about 120 ␮ g/m3 for moderately dusty atmosphere, and more than 200 g/m3 for a dust storm (i.e., dustiness is greater than for clear air by as much as 107 ).

5.2 Atmospheric Attenuation As it is well known, the received power Pr of a signal scattered by a target depends on both radar system parameters and terrain and environmental parameters. Radar system parameters include the transmitted power P t , the frequency f (or wavelength ␭ ), and the antenna gain G. The other parameters are the two-way propagation factor V 4, determined by terrain type, roughness, reflectivity and location with respect to the radar, and atmospheric attenuation ␥ (decibels per kilometer, one way). Thus, P G 2␭ 2␴ t Pr = t ⭈ V 4 ⭈ exp (−0.46␥ r ) (4␲ )3r 4

(5.4)

where r is the range to target and ␴ t is the target RCS. Radar signal attenuation in the atmosphere is determined by ionosphere absorption as well as molecular absorption in the tropospheric gases (i.e., water vapor and oxygen) and by attenuation in meteorological formations (i.e., rain, fog, clouds, and smog). The absorption factor in the ionosphere decreases quickly with increasing radar frequency, and absorption is significant at L-band and longer wave bands. For radar of X-band and shorter wave bands, the ionospheric attenuation is negligible and only tropospheric effects need to be analyzed. Atmosphere gas attenuation is most significant at the resonance absorption lines: 22 GHz, 182 GHz, and 340 GHz for water vapor and 60 GHz and 120 GHz for oxygen. The most intense absorption takes place at frequencies above 60 GHz, where for standard conditions at sea level the attenuation coefficient reaches 10–20 dB/km. Accordingly, radar frequencies for obtaining long ranges are chosen in transmission windows between the resonance absorption lines. In the millimeter band, these windows are near 35 GHz and 95 GHz. The averaged data for atmospheric absorption by gases in the frequency bands from 10–95 GHz are shown in Table 5.5 [9].

234

Microwave and MMW Backscattering by Precipitation and Other Meteorological Formations Table 5.5 Atmosphere Gas Specific Absorption

Frequency (GHz) 10.0 35.0 95.0 Source: [9].

Water Vapor with Density (g/m 3 ) 7.5 20.0 5.3 ⭈ 10−3 7 ⭈ 10−3 0.15 0.13 0.2 0.52

Oxygen 8 ⭈ 10−3 0.05 0.045

Total Absorption for Water Vapor Density 7.5 20.0 0.013 0.015 0.1 0.18 0.25 0.57

A water vapor density of 7.5 g/m3 is accepted as the standard for continental areas with moderate climate, and a density of 20.0 g/m3 is normal for lower layers of the troposphere above oceans and seas of equatorial, tropical, and subtropical zones and for coastal regions in these areas. The absorption in atmospheric gases can be predicted, and for some cases one can find seasonal maximum humidity values for total attenuation derivation with adequate accuracy. Microwave attenuation in precipitation can be predicted only in a probabilistic sense because the duration and intensity of precipitation are the random processes. Estimation techniques have been developed for paths of different ranges, taking into consideration the spatial and temporal inhomogeneity of precipitation. Precipitation is the limiting factor for millimeter-band radar systems at ranges of more than several kilometers. The most serious source of attenuation is rain. The attenuation coefficient in decibels per kilometer for homogeneous rain can be determined as [10]

␥ = k ⭈ In

(5.5)

where k , n are parameters depending on wavelength and temperature k (␭ ) = a 0 + a 1 ␭ + a 2 ␭ 2;

(5.6a)

n (␭ ) = b 0 + b 1 ␭ + b 2 ␭ 2;

(5.6b)

a 0 = 2.026, a 1 = −3.759 ⭈ 10−1, a 2 = 1.949 ⭈ 10−2, b 0 = 4.721 ⭈ 10−1, b 1 = 8.084 ⭈ 10−2, b 2 = 3.761 ⭈ 10−3. Based on the Marshall-Palmer raindrop size distribution model, Olsen et al. [11] tabulated a rain attenuation coefficient formula for the frequency band from 1 GHz to 1,000 GHz in simpler forms that for temperature of 20°C are given by

235

5.2 Atmospheric Attenuation

␥ (dB/km) =

冦

0.256I 0.9

for f = 30 GHz

0.412I 0.841

for f = 40 GHz

0.792

for f = 50 GHz

0.572I

(5.7)

In order to estimate the rain attenuation, a moderate rainfall rate of 4 mm/hr was chosen as a reference because, as seen from [12], the rainfall duration for this or more intense precipitation in Europe is less than 0.5% of total radar operation time. In some cases, the radar efficiency estimate has been obtained for rainfall rate of 1 mm/hr, for which the rainfall time was not greater than 1.5% of total operation time in Europe (for areas with mean precipitation from 300 mm to 500 mm per year). Attenuation data in decibels per kilometer for MMWs in rains of these intensities are shown in Table 5.6. Important information for rain attenuation influence on radar operation is data on the degree of rain homogeneity within the radar range because this determines the required value of radar power. The cell sizes for different rain intensities are shown in Table 5.1. These data permit us to make an assumption about rain homogeneity at all radar ranges for low-RCS target detection. The attenuation in fog depends on the liquid water quantity per unit volume (water content of this formation). The attenuation coefficient is

␥ f = kf ⭈ Mf

(5.8)

where k f is the specific factor of attenuation in dB(m3/gkm) and M f is the water content, g/m3. The values of k f factor for a temperature of 18°C are shown in Table 5.7. The fog water content can be approximately determined using the data in Table 5.8 [13]. Attenuation by dust storms, as with rain attenuation, is determined by the dust particle dimension distributions and the dielectric constant of particles. Theoretical and experimental investigations of dust parameters showed that:

Table 5.6 Attenuation Coefficient (dB/km) in Rain Frequency (GHz) 10.0 37.5 95.0 140.0 Source: [12].

Rainfall Rate of 1 mm/hr 0.02 0.25 0.6 0.7

Rainfall Rate of 4 mm/hr 0.086 1.0 3.0 3.2

236

Microwave and MMW Backscattering by Precipitation and Other Meteorological Formations Table 5.7 Values of k f in Fog for Temperature of 18°C Frequency (GHz) 10.0 15.0 40.0 140.0 Source: [4].

k f dB(m 3/gkm) 0.05 0.112 0.876 7.14

Table 5.8 Dependence of Optical Visibility on Fog Water Content Water Content (g/m 3 ) 2 1 0.5 0.2 0.1 Source: [13].

•

•

•

Optical Visibility (m) 30 50 80 200 300

The real part of the dielectric constant of dust aerosols does not practically depend on the water content of aerosols from 0% to 20–30% in the frequency band of 3–37 GHz for all soil types. The imaginary part of the dielectric constant increases rapidly with increasing water content up to 0.4 for water content of 4.3%. The approximate functions for dimension distributions of dust aerosols are very different for different examples and can be presented as power, exponential, Gaussian, and lognormal functions. It is difficult to determine the exact form of this distribution.

Data from experimental investigations of the microwave attenuation in dust storms are extremely limited. One can expect that for regions of Africa, Arabia, and Sudan, there are dry dust storms and the microwave attenuation in them is rather small. Greatest attenuation is observed for rains, and this should be taken into consideration.

5.3 Backscattering Theory The theory of microwave scattering by precipitation and clouds is developed using the assumption on noncoherent volumetric scattering. The mean power of the scattered signal for this case is proportional to the effective backscattering area of a unit of volume (or normalized volumetric RCS ␩ ): ∞

␩=

冕 0

␴ (␭ , D ) ⭈ N (D ) dD

(5.9)

237

5.3 Backscattering Theory

where ␴ (␭ , D ) is the RCS of a drop with diameter D, N (D ) is the number of drops with diameter from D to D + dD in the unit volume, and ␭ is the wavelength. The RCS of a single particle of spherical shape is derived from Mie’s formula [14]

␴ 1 = 2 2 ␲ D /4 ␳

|∑ ∞

n =1

2

2

(−1)n (2n + 1) (a n − b n )

|

2

(5.10)

where a n and b n are coefficients determined by complex spherical Bessel and Hankel functions, ␳ = 2␲ D /␭ . Taking into consideration that the drop diameter is not greater than 0.01 cm for clouds and fogs without precipitation and changes within limits from 0.01 cm to 0.6 cm with most probable value of 0.1 cm [15] for rain, the expression for RCS of a single drop in the millimeter band can be considerably simplified. For ␳ = 2␲ D /␭ << 1, it is easy to obtain

␴ (␭ , D ) ≈

|

␲ 5D 6 m 2 − 1 ␭4 m2 + 1

|

2

(5.11)

⑀−1 ; ⑀ is the water dielectric constant. ⑀+2 Taking into account the data on drop sizes, we reach the conclusion that (5.6) is applicable to drop RCS of fog, clouds, and light precipitations up to frequencies of 300 GHz. For heavy rain, when the drop sizes are large, the use of (5.11) at frequencies above 30 GHz leads to considerable errors, and it is necessary to use the general formula (5.10). The second factor necessary to estimate normalized volumetric RCS is the drop size distribution. Theoretical investigations of precipitation do not permit us to propose a universal analytical formula for drop size distribution, and therefore we use empirical distributions. The Marshall-Palmer distribution is most applicable where m =

N (D ) = N 0 ⭈ exp (−⌳D )

(5.12)

where N 0 = 0.08 cm−4; ⌳ = 41.1I −0,21 cm−1; and I is the precipitation intensity in mm/hour. As the derivation shows, this distribution gives satisfactory results at frequencies less than 100 GHz, while for higher frequency bands it is necessary to take into account the contribution of small drops. Often the Laws-Parsons distribution is applied

238

Microwave and MMW Backscattering by Precipitation and Other Meteorological Formations

N (D ) =

6Cn n − 9 −n r − n␤ n − 4 10 ␣ I D exp (−␣ −n 10n I − n␤ D n ) ␲

(5.13)

C = 72; ␣ = 1.25; ␤ = 0.199; n = 2.29; r = 0.867. It is not possible to take into account the drop size distribution for cloud scattering because of the absence of experimental data on change in the distribution with cloud height. In radio meteorology, the integral characteristic of clouds and precipitations is called the reflectivity Z=

∑ N (D i ) D i6 ⌬D i [mm6/m3 ]

(5.14)

i

There are numerous data about its value for clouds and precipitation (e.g., in [4]). The normalized RCS is connected with reflectivity by

␩=

|

|

␲5 m2 − 1 ⭈Z ␭4 m2 + 2

(5.15)

and can be used for estimations of different cloud types. Snow microwave backscattering is a more complex process. It is possible to use for normalized RCS derivation of small snow crystals the relation from [14]

␩ = 0.11

Wm ′k

␳ 2␭ 4

冉

m2 − 1 m2 + 2

冊

2

(5.16)

where W is the quantity of ice in grams per m3, m ′ is the crystal mass in grams, ␳ is the density, and k is the shape coefficient depending on the crystal shape; its value is near unity. The crystals making the largest contribution require accounting of scattering particle shape. Multiplier

|

m2 − 1

|

is about 0.9 for water and 0.2 for ice for microwaves and m2 + 2 the longwave part of the millimeter band, changing slightly with air temperature change. In the shortwave part of the millimeter band ( f > 40–60 GHz), it is necessary to take into consideration the quick decrease of the real part of the water dielectric constant with increasing frequency that takes place up to frequencies of about 100 GHz. This leads to considerable differences in the frequency dependence of the normalized RCS from the law ␩ ∼ ␭ −4.

5.4 Experimental Results Review

239

5.4 Experimental Results Review 5.4.1 Precipitation Backscattering

Let us analyze the experimental data for microwave radar backscattering from the precipitation and clouds [4, 14–17]. For the centimeter band, it is necessary first of all to note the satisfactory coincidence of experimental data with the theoretical derivations using typical size distributions. The rain normalized volumetric RCS dependence on the precipitation intensity at wavelengths of 3.0 cm and 10.0 cm are shown in Figure 5.1. It is seen that for increasing rain intensity from weak (I = 2.5 mm/hr) to heavy (I = 20 mm/hr), the normalized RCS grows by about 20 dB. In [2], it is shown that the normalized RCS can change by about 7–8 dB as a function of changeability of the size distribution law N (D ) in rains of three types for I = const and ␭ = const. The temperature change from 0° to 40°C does not lead to significant change of normalized RCS, and the value varied by less than 20%–25% for I = const. The normalized RCS dependences on the radar wavelength for different precipitation intensity are shown in Figure 5.2. It is seen that rain of 10 mm/hr at 3 cm for resolution cell volume of about 107 m3 produces clutter of about 1 m2, and this value is comparable with fighter RCS. Dry falling snow and rain of equal rate have practically equal backscattering intensity. So, for snow crystals with weight of 1–2 mg and for rate of I = 10 mm/hr,

Figure 5.1 Volumetric normalized RCS dependence on precipitation intensity. (From: [12].)

240

Microwave and MMW Backscattering by Precipitation and Other Meteorological Formations

Figure 5.2 Volumetric normalized RCS of different precipitation intensity dependence on the wavelength.

the backscattering is comparable with backscattering from the rain of the same intensity (i.e., at X-band, the normalized RCS of snow and rain do not differ). This is illustrated by Figure 5.1. In fact, the clutter from falling snow is considerably less than that of rain. This is because the rainfall rate of 1 mm/hr corresponds to shower, and the same rate of snowfall correspondents to heavy snowfall, which happens considerably less infrequently. The snow reflectivity values increase when snow was mixed with rain. For the normalized RCS of rain in microwave band (1 cm ≤ ␭ ≤ 10 cm), the empirical expression is [1]

␩ = 7 ⭈ 10−12 f 04 I 1.6

[m−1 ]

(5.17)

where f 0 is the frequency in gigahertz. The theoretical and experimental results in millimeter bands show worse agreement with theory. The greater dependence of normalized RCS on the drop size distribution than in microwave bands is the reason for this, and this dependence can greatly change within short temporal intervals. The dependences of normalized RCS on rain intensity at wavelengths of 8.0 mm (crosses) and 4.0 mm (points) and derivative dependences are shown in Figure 5.3(a) [16]. The experimental results are rather exactly described by the expression

241

5.4 Experimental Results Review

Figure 5.3 Specific RCS of rain via (a) rain intensity and (b) frequency dependences of A and ␤ coefficients. (After: [12, 16].)

␩ = A ⭈ I␤

(5.18)

where A = 0.18 ⭈ 10−4, ␤ = 1.2 at wavelength of 8.0 mm and A = 0.53 ⭈ 10−4, ␤ = 1.06 at wavelength of 4.0 mm. Results of rain backscattering investigations presented in [17] are less than our data and data of Russian authors [2, 4, 16] by 5–10 dB. The possible reason of these differences is the different techniques of experimental investigations and different rain distributions in different regions of Earth. The analysis of experimental data from [2, 4, 16, 17] showed that for the rainnormalized RCS at the millimeter band, one can use (5.18); A and ␤ coefficients dependences on the frequency are shown in Figure 5.3(b). For precipitation clutter within a radar resolution cell, one can use the relation

␴≈␩

c␶ 0 ␪ ␸ r2 2 0 0

(5.19)

where ␶ 0 is the radiation pulse duration; r is the range; and ␪ 0 , ␸ 0 are the antenna pattern half-power widths in azimuthal and elevation planes. The results of rain clutter RCS at wavelengths of 2 mm, 4 mm, 8 mm, and 32 mm for typical radar

242

Microwave and MMW Backscattering by Precipitation and Other Meteorological Formations

parameters of ␶ 0 = 1 ␮ s, ␪ 0 = ␸ 0 = 0.01 rad are shown in Figures 5.4 and 5.5. The estimation of rain clutter RCS for other radar parameters is found by addition to the RCS obtained from these figures of the factor

冉

K (dB) = 10 lg

␶ ␪ ␸ + lg + lg ␶0 ␪0 ␸0

冊

(5.20)

The results of total attenuation derivations are shown in these figures by the dotted lines; the scale for attenuation estimation is on the right coordinate axis. It is known that the shape of large raindrops differs from spherical. As a result, the RCS of heavy rains is greater for horizontal polarization than for vertical by a factor 1.4. The use of circular polarization leads to reduced scattering. At X-band, this decrease is approximately 18 dB for heavy rains (I = 15 mm/hr), 20–23 dB for moderate rains, and 30–35 dB for light rains (I = 3 mm/hr). 5.4.2 Cloud Backscattering

The microwave normalized RCS for clouds is lower by about 40 dB than for rain. According to [4], for the European part of Russia, the normalized RCS for cumulonimbus clouds with precipitation can be about 10−5 m−1 at the X-band; the data is shown in Table 5.9. It is seen from Table 5.9 that the meteorological formations that give the most powerful scattered signal are the clouds of the last three types. As a rule, the averaged duration of backscattering from them is 1–2.5 hours, and it reaches 3–6 hours for stratus. The scattering characteristics from hail and rain clouds differ visibly. Typically, the temperature at the upper boundary of hail clouds is, as a rule, less than −30°C, and the normalized RCS is ␩ = 5 ⭈ 10−7 cm−1. The typical profile of hail clouds ␩ (h ) differs from the analogous profile for heavy rain (see Figure 5.6). The maximal values of ␩ for hail zones are observed near the isotherm 0°C, where the hailstones reach the maximal sizes and supply with water. In the supercooled part of hail clouds up to heights of some kilometers above the zero isotherm, ␩ (h ) is about constant, and it then very quickly decreases with height. The value of ␩ near the Earth’s surface is less than at heights near the zero isotherm by about two orders on account of hailstone melting.

5.5 The Statistical Characteristics of Scattered Signals The backscattered signals from meteorological formations are normally distributed because they are the superposition of large numbers of independent (or weakly

5.5 The Statistical Characteristics of Scattered Signals

243

Figure 5.4 Rain clutter RCS and attenuation at wavelengths of 2 mm and 4 mm. (From: [12].)

244

Microwave and MMW Backscattering by Precipitation and Other Meteorological Formations

Figure 5.5 Rain clutter RCS and attenuation at wavelengths of 8 mm and 32 mm. (From: [12].)

245

5.5 The Statistical Characteristics of Scattered Signals Table 5.9 The Normalized RCS of Clouds

␩ (m −1 ) Cloud Type Cirrus Altocumulus Altostratus Stratocumulus Stratus Cumulus congestus Stratonimbus Cumulonimbus Cumulonimbus with thunderstorm Source: [4].

10.0 cm 2.5 ⭈ 10−12 3.7 ⭈ 10−12 2 ⭈ 10−12 5 ⭈ 10−12 2 ⭈ 10−12 1.5 ⭈ 10−10 10−9 7 ⭈ 10−9 5.5 ⭈ 10−8

3.0 cm 3 ⭈ 10−10 4.5 ⭈ 10−10 3 ⭈ 10−10 6 ⭈ 10−9 3 ⭈ 10−10 2 ⭈ 10−8 10−7 8.5 ⭈ 10−7 7 ⭈ 10−6

Figure 5.6 The normalized RCS of rain and hail on the height. (After: [4].)

correlated) components. This is why the amplitude fluctuation distributions are Rayleigh and the RCS distributions are exponential p (␩ ) =

1 2␴ ␩2

冉 冊

⭈ exp −

␩2

2␴ ␩2

(5.21)

where ␴ ␩2 is the normalized RCS fluctuation variance. The generation of experimental results for signal amplitude fluctuation distributions applicable to backscattering from meteorological formations [4, 14], was carried out using dipole scatterer ‘‘clouds’’ having identical shapes and sizes and, consequently, equal radar reflectivity. For dipole clouds, the experimental distributions matched the theoretical model satisfactorily. It is shown in [4] that analogous

246

Microwave and MMW Backscattering by Precipitation and Other Meteorological Formations

results are obtained in precipitation. It was shown in [17] that the amplitude fluctuation distributions in the millimeter band conformed to the lognormal law, the rms does not depend on the rain intensity, and it grows with decreasing resolution cell sizes, especially for high-resolution radar. Power spectral analysis of the scattered signals was carried out in some papers [4, 14, 18, 19]. Most authors, particularly [18, 19], take the position that the backscattering spectrum shape is described by a Gaussian shape to the level of −(30–40) dB. The precipitation power spectrum width at −3 dB reaches 140–150 Hz at a 20-cm wavelength for wind velocity of 36 km/hr and increases in inverse proportion to the radar wavelength. It is shown in [18] that the spectrum shape of backscattering intensity from dipole scatterer clouds is approximated rather well by the Gaussian law and does not practically depend on the wavelength; the spectrum width increases in inverse proportion to the wavelength. That is,

冉 冊

F2 G (F ) = G 0 exp −a 2 f0

(5.22)

where G 0 is the maximum spectral density, f 0 is the radar frequency, F is the Doppler frequency, and a is the nondimensional coefficient characterizing the spectrum width (≈ 3 ⭈ 1015 for rain clouds). As an illustration, the power spectra at the S- and X-bands are shown in Figure 5.7 [14]. In [19], it is confirmed that for small elevation angles (less than 6° ) at the X-band, the precipitation backscattering spectrum is mainly determined by the

Figure 5.7 Rain backscattering power spectra. (From: [14].  1951 McGraw-Hill, Inc.)

5.5 The Statistical Characteristics of Scattered Signals

247

turbulence and the radial components of wind velocity having a Gaussian power spectrum. Therefore it has, as a rule, a Gaussian shape in its intensive part, down to −(15 dB or 20 dB), and then falls down according to G (F ) ∼ F −6. For rain of moderate intensity, the spectral width in this band is 20–35 Hz at −3 dB. The spectral shape does not change for wet snow; permitting us to repudiate the concept of drop vibration for explanation of long tails in the spectra (in rains of moderate intensity, the frequencies are 80–100 Hz, and they are less by about one order for snow). Authors explored the significant variations of the spectrum shape from Gaussian at levels below −40 dB, explaining this by drop fall-velocity variations; the variations at higher levels were explained by drop motion in the turbulent air. The measurements of [19, 20] at centimeter and decimeter bands (wavelengths of 4 cm and 35 cm) showed that the intensity (incoherent) power spectra of scattered signals from most meteorological formations are also rather well described by the Gaussian curve (5.16). The analogous conclusion about the precipitation intensity power spectral shape is made in [21] based on measurements carried out in the millimeter band of radiowaves. In some cases, bimodal spectra are observed, usually explained by particles with different laws of fall velocities (e.g., drop-snowflake or drop-hailstone). The general data about the spectrum width of signal intensity fluctuations for different meteorological objects obtained in [18] are shown in Table 5.10. The minimal, mean, and maximal values of spectral width ⌬F are given for objects observed at an elevation angle of 30°. The spectral width is inversely proportional to wavelength ␭ ; therefore, the experimental data are presented in scale of (␭ /2) ⭈ ⌬F. This permits us to determine the spectral width at any wavelength. The spectral width distribution for different meteorological objects is shown in Figure 5.8 [18]. These data show that the intensity fluctuations spectral width for thunderstorms can exceed the data for mean spectra obtained for the same days and heights for dipole scatterer clouds. There is no monotonous dependence between the spectral width and precipitation intensity. So, the spectra in rains with intensity of 10–15 mm/hr are narrower than the spectra in rains with intensity of 2–4 mm/hr.

Table 5.10 Intensity Spectral Width for Some Meteorological Formations Type of Formation Rain Zero isotherm and wet snow Clouds Clouds Cumulus clouds with positive temperature Thunderstorms and heavy rains Source: [18].

Spectrum Width (␭ /2) ⭈ ⌬F (cm-Hz) Minimal Mean Maximal 30 45–60 110 25 40 50 20 35 65 30 45 75 35 75 125 60 125 400

248

Microwave and MMW Backscattering by Precipitation and Other Meteorological Formations

Figure 5.8 Spectrum width histograms. (After: [18].)

In powerful cumulus clouds, the spectrum of fluctuations is considerably wider than in clouds of other types. As was shown in [19, 20], the intensive part of backscattering intensity spectra from clouds and precipitation in coherent radar can be approximated by a Gaussian function

249

5.5 The Statistical Characteristics of Scattered Signals

冋

G (F ) = G 0 exp −a

(F − F 0 )2 f 02

册

(5.23)

where F 0 is the Doppler frequency shift that determined the wind drift; the rest of the parameters coincide with those in (5.16). For small elevation angles, the value of F 0 is determined as F0 =

2U H cos ␪ ␭

where U H is the horizontal component of wind velocity, and ␪ is the angle between the wind and the radar antenna pattern axis. The Doppler shift varies over wide limits. In Table 5.11 some characteristics of clouds with precipitation are shown, including the Doppler frequency shift F 0 and the parameter a. The empirical formula from [22] for rain clouds determines the spectral width at −6 dB as ⌬F =

435 − 940 , Hz ␭

(5.24)

This gives results coinciding with experimental data of other authors. The results of investigations carried out by the Institute for Radiophysics and Electronics of NAS of the Ukraine [12] show that the backscattering spectral shape from cumulonimbus clouds and from rain changes considerably with elevation angle and time. There are both unimodal and bimodal spectra. The variations of spectral density permit us to approximate the spectral shape as a Gaussian curve (5.22) modified by a power function, for which G (F ) ∼ F −6. The maximal spectral width is observed at the lower cloud edge, where the turbulent diffusion is maximal; the width is decreased higher in the cloud.

Table 5.11 Spectral Parameters of Cloud Backscattering Type of Spectrum Unimodal ″ ″ ″ ″ Bimodal ″ Source: [12].

␭ (cm) 35 35 35 4 4 4 4

Spectral Width (Hz) at Level 10 dB 20 dB 30 dB 75 130 150 78 100 140 50 100 130 55 100 250 400 500 — 150 400 500 400 600 800

F 0 (Hz) 45 40 100 240 100 260 50

4.6 1.9 2.5 1.8 5.6

a ⭈ 1014 ⭈ 1014 ⭈ 1014 ⭈ 1014 ⭈ 1014 — —

250

Microwave and MMW Backscattering by Precipitation and Other Meteorological Formations

The spectral shape for nonrain clouds and fogs of small intensity is determined by atmospheric turbulence and is very close to G (F ) ∼ F −5/3. Thus, the conclusions can be drawn that the most applicable model for a description of backscattering signal fluctuations is the Gaussian distribution; the amplitude fluctuations are described rather well by a Rayleigh distribution; and for estimation of levels in the tails level, the lognormal distribution can be used. The power spectral shape of backscattering signals is described by the Gaussian curve (5.22) and for spectrum description at levels less than −40 dB, it is possible to use a power function.

5.6 Radar Reflections from ‘‘Clear’’ Sky (Angel-Echo) 5.6.1 Point Reflections

The origin of radar point reflections in the atmosphere can be reflections from socalled dot angels (the term angel is used in Russian literature) (i.e., closed areas with large refractivity index gradients and with higher humidity and temperature at their boundaries) that preserve these characteristics for long periods. The stability of these formations is determined by annular vortexes that are the air circulation inside angel, with velocity up to 7 m/s. The angels exist as the bubbles and streams forming in a layer with unstable stratification; they are formed near the sea or land surface and then detach from it and rise. Stratification is an air temperature vertical distribution determining the equilibrium conditions in the atmosphere that reflects or prevents vertical movement of the air. For unstable stratification, the temperature decreases with height and prevents atmospheric convection. The mixed model of angel forming has been recently developed [23], in which an angel is a rising stream with a bubble cap at the top; it is this bubble, mainly, that generates the scattered signal. Inside the unstable atmospheric layer, the angel arises with a small rising velocity of about 0.8 m/s. The vertical velocity decreases upon reaching a stable layer, and it fluctuates in height with an average period of 10–20 minutes. Temperature and humidity measurements show that their diameters are, mainly, 30–80m and can reach 300–500m. For estimation of the total number of angel-echoes, it was proposed in [24] for marine radar and for moderate latitudes to use N ⌺ ≈ 5(C − 160) ␪ H ␪ V

(5.25)

where ␪ H , ␪ V are the antenna pattern width in horizontal and vertical planes, respectively, in degrees; C=

P t G 2␭ 2␩ (4␲ )3 P min q 2 L p

(5.26)

251

5.6 Radar Reflections from ‘‘Clear’’ Sky (Angel-Echo)

where P t is the transmitter power, G is the antenna gain, ␩ is the microwave transmission line efficiency factor, L p is the power loss factor for processing in radar microwave sets, P min is the receiver threshold power, and q 2 is the signal-tonoise ratio for target detection with requisite detection and false alarm probabilities. It is seen that the number of angel-echoes increases with increasing radar energy where angels with smaller RCS will be observed. Their mean concentration over land is 750 km−3 in layers of 1–3 km. The concentration, as a rule, decreases with increasing height. Seasonal and diurnal variation is observed for the appearance of these reflectors. In the former USSR territory, angels are detected from May to September with maximal concentration in June–July [4]. The probability of their appearance grows with increasing air temperature and humidity. The diurnal variation of such reflections is characterized by maxima of radar backscattering concentration, intensity, and height at 13–14 hours. As an illustration, in Tables 5.12 and 5.13, the data obtained from [24] on angel concentration in height are shown as a function of temperature and wind velocity. It is seen that the concentration increases and the lower boundary of angel layer decreases with increasing temperature; the concentration and lower boundary increase with overland wind also increasing. Over sea, the angel-echo number is greater at night in comparison with day. Two models of angel backscattering are known: volumetric scattering from turbulence and specular reflection from angel surfaces. For the volumetric scattering model from air turbulence, it is necessary to keep in mind that the maximal probable normalized volumetric RCS of such turbulence is 10−10 cm−1 for clear sky. Consequently, for angel diameters of 30–80m, their RCS does not exceed to 3 ⭈ 10−7 – 6 ⭈ 10−6 m2, which is lower than experimentally observed values. For the specular backscattering mechanism, the angel RCS—more exactly, the bright point at its surface—depends on the surface curvature and the reflection coefficient; an approximation using the geometrical optics model giving Table 5.12 Angel Concentration and Lower Height Dependence on Air Temperature Temperature °C Mean concentration in layer of 150m Mean lower boundary of angel layer (m) Observation number Source: [24].

12–14 2.6 325 33

15–17 6.2 230 155

18–20 12.4 200 201

21–23 14.1 180 115

24–26 10.3 150 20

Table 5.13 Angel Concentration and Lower Height Dependence on Wind Velocity Wind velocity (m/s) Mean concentration in layer of 150m Mean lower boundary of angel layer (m) Observation number Source: [24].

0–5.5 24.7 265 109

7.5–13 36.3 283 96

15–20 51.9 318 68

22–28 43.7 417 39

30–35 45.5 375 5

252

Microwave and MMW Backscattering by Precipitation and Other Meteorological Formations

␴ = ␲␳ 2 | R x R y |

(5.27)

where ␳ 2 is the reflection coefficient and R x , R y are the main radii of curvature at the point of specular reflection. The point reflections from angels are explained by the reflections from semispherical thin layer zones with downwards-directed concave surfaces. The refractive index drop amounts to a few N-units and occurs in a layer with depth of about ␭ /4. The reflection coefficient depends on the refractive index variance gradient. For intermittent variance of refractive index at the boundary, the reflection coefficient is 2

␳ =

冉

n1 − n0 n1 + n0

冊

2

=

⌬N 2 ⭈ 10−12 4

(5.28)

where n 1 , n 0 are the refractive indexes on both sides of the angel boundary; and ⌬N = (n 1 − n 0 ) ⭈ 106 is the refractive index gradient. For angel diameters of 30–80m and ⌬N = 10 (such value can take place in the atmosphere), the derivation for the specular reflection model gives an RCS of order 2 ⭈ 10−8 – 1.6 ⭈10−7 m2. Measurements over the sea at the X-band [25, 26] show that the real values of angel RCS are greater by 5–6 orders than the derivations from (5.27), indicating the necessity for model elaboration. For moderate latitudes and at wavelength of 5 cm for horizontal polarization, the RCS of the angels lay within limits of 10−6 – 10−2 m2, as illustrated by Figure 5.9. Measurements show the angel RCS independence on wavelength at S- and C-bands. For the X-band and millimeter bands, one can expect the RCS to decrease.

Figure 5.9 Angel RCS cumulative distribution. (After: [25].)

5.6 Radar Reflections from ‘‘Clear’’ Sky (Angel-Echo)

253

The observations of angels carried out by the author at the S-band permitted us to find angels with maximal RCS up to 1 m2. It was noted that visible variance of the RCS does not take place during an observation time of about one second. The reflected signals slowly fluctuate, sometimes disappearing in noise for 10–30 seconds and rising again. The signal from angel at the S-band (on an A-type display) is shown in Figure 5.10(a); in Figure 5.10(b), the amplitude and Doppler frequency variance for signals from angels are shown at the C-band for a wind velocity of 2.9 m/s. The RCS distributions of the angel were approximated by a lognormal law with mean value of −42.5 dB/m2 and rms value of 5 dB. Angel backscattering is distinguished by its small amplitude variance and practically fixed position of its signal spectral line—the radial velocity variance did not exceed 1.5 m/s for 1 minute (i.e., it is coherent). In Table 5.14, the spectral width and the correlation intervals are shown, as obtained at the X-band for incoherent radar operation. In Figure 5.11, the power spectra at the S-band [Figure 5.11(a)] and C-band [Figure 5.11(b)] are shown [12]. It is seen from these data that angels have the narrowest amplitude and power spectra of all the meteorological formations considered here. The dependence of

Figure 5.10

(a) Angel backscattering at the S band (A-type display) and (b) temporal dependences of amplitude and Doppler frequency of scattered signal.

254

Microwave and MMW Backscattering by Precipitation and Other Meteorological Formations Table 5.14 The Correlation Intervals and Spectrum Width for Angel at X-Band

Object Angel: Clear sky Cloudy sky Source: [26].

Figure 5.11

Correlation Interval (ms) Spectrum Width (Hz) Level 0.5 0.1 0.5 0.1 20 12

45 26

6 12

22 41

Power spectra of angel backscattering at (a) the S-band and (b) the C-band.

spectral width on radar wavelength has the form ⌬F ∼ ␭ −1 (i.e., spectral broadening is inversely proportional to the wavelength). 5.6.2 Backscattering from the Turbulent Atmosphere

The majority of works from the theory of radar signal scattering in a turbulent atmosphere is based on the works of V. I. Tatarsky [27]. According to these, zones of refractive index microscale turbulent pulsations can be the reason for radar reflections in a clear sky. It is known that only spectral components of turbulence forming spatial grids with a size of l = ␭ /2 can take part in generating backscattering. The normalized volumetric RCS can be determined as

␩=

␲ 4 k ⭈ ⌽n (k ) 2

(5.29)

where k = 4␲ /␭ is the wavenumber, ␭ is the wavelength, and ⌽n (k ) is the threedimensional spectrum of refractive index fluctuations.

255

5.6 Radar Reflections from ‘‘Clear’’ Sky (Angel-Echo)

According to Kolmogorov-Obukhov theory, for the inertial interval of turbulence limited by the outer L 0 and inside l 0 scales, the spectrum of pulsations can be presented in form ⌽n (k ) = 0.033C n2 ⭈ k −1/3 ;

k0 ≤ k ≤ km

(5.30)

where C n2 is the value of refractive index fluctuation intensities and k 0 = 2␲ /L 0 ; k m = 2␲ /l 0 . The outer scale of turbulence inertial interval L 0 is approximately 10m [28] and the value of l 0 in an overland atmosphere layer is some millimeters or units of centimeters at a height of 10,000m. For C n2 , the usual expression is [27]

冉 冊

4/3 dn C n2 = a 2 L 0 dh

2

(5.31)

where a 2 is the nondimensional parameter and dn /dh is the vertical gradient of refractive index. The outer scale of the inertial interval can be determined as [28] L0 =

冉冊 ⑀ ␤

1/2

(5.32)

where ␤ is the vertical gradient of the average wind; and ⑀ is the velocity of turbulent energy dissipation. Then the expression (5.29) for the normalized RCS can be presented as

␩ = 0.38C n2 ␭ −1/3 = 0.38a 2 ⑀ 2/3 ␤ −2

冉 冊 dn dh

2

␭ −1/3

(5.33)

It is seen from this expression that the radar reflections from atmospheric turbulence weakly increase with decreased wavelength. The clear air normalized RCS variances as a function of wavelength, as obtained by the other authors and assembled (the points in Figure 5.12) are shown in Figure 5.12(a). The dependence ␩ (␭ ), obtained in a very dense layer of insects with concentration of about 5 ⭈ 106 m−3, is shown by the solid line. The values of C n2 are shown on the right ordinate axis and are 10−15–10−14 cm−2/3, but for some rare cases they can achieve values up to 10−12–10−11 cm−2/3. These data are obtained at great heights above 300–700m.

256

Microwave and MMW Backscattering by Precipitation and Other Meteorological Formations

Figure 5.12

(a) The normalized RCS of turbulent atmosphere and (b) histogram of spectral width distribution. (From: [12].)

At low altitudes over the surface, one can expect considerable increase of the normalized RCS for turbulent formation in the atmosphere for two reasons. First, the turbulent energy dissipation velocity quickly grows with decreasing height. As seen from [29], for height decreasing from 100m to 1m, the turbulent energy dissipation rate increases by more than two orders (from 6–8 cm2s−3 to 2 ⭈ 103 cm2s−3 ). Second, the vertical gradients of the refractive index for overland layers, as a rule, exceed the gradients in the free atmosphere not less than one order [30]. The backscattering power spectra from atmospheric turbulence are rather wide and are represented by G (F ) ∼ F −5/3 as was verified experimentally, particularly in [26]. A histogram for spectra of different width of scattered signals in the turbulent atmosphere is shown in Figure 5.12(b). In conclusion, we would like to note that the radiowave backscattering in the free atmosphere turbulent formations forms the minimal level of radar clutter for high-energy radar systems.

References [1] [2]

Edgar, A. K., E. J. Dodsworth, and W. P. Warden, ‘‘The Design of a Modern Surveillance Radar,’’ Int. Conf. Radar: Present and Future, London, 1973, pp. 8–13. Krasuk, N. P., V. L. Koblov, and V.N. Krasuk, Influence of Troposphere and Surface on Radar Operation, Moscow, Russia: Radio and Svyaz, 1988 (in Russian).

References [3] [4] [5]

[6] [7]

[8] [9] [10]

[11] [12]

[13] [14] [15] [16]

[17]

[18]

[19]

[20]

[21]

257 Ippolitto, L. J., ‘‘Atmospheric Propagation Condition Influence at Space Communication Systems,’’ Proc. IEEE, Vol. 69, No. 6, 1981, pp. 29–58 (in Russian). Stepanenko, V. D., Radar in Meteorology, Leningrad: Gidrometeoizdat, 1983 (in Russian). Sokolov, A.V., and E.V. Sukhonin, ‘‘Millimeter Wave Attenuation in Atmosphere,’’ in Science and Technics Results: ser. Radiotechnics, Moscow, VINITI, Vol. 20, 1980, p. 107–202 (in Russian). Stepanenko, V. G., and S. M. Galperin, Radiotechnical Methods of Hail Investigations, Leningrad: Gidrometeoizdat, 1983 (in Russian). Semenov, A. A., and T. I. Arsenyan, Microwave Attenuation in Sandy-Dusted Atmosphere, Preprint No. 4 (505), Inst. of Radiotechnology and Electronics, Academy of Science USSR, Moscow, 1989, p. 33 (in Russian). Arsenyan, T. I., and A. A. Semenov, ‘‘Attenuation of Microwaves in Sandy-Dusted Aerosol, Zarubeznaya Radioelktronika, No. 1, 1995, pp. 16–26 (in Russian). Kulemin, G. P., ‘‘Influence of Propagation Effects on Millimeter Wave Radar Operation,’’ Proc. SPIE Radar Sensor Technology IV, Vol. 3704, April 1999, pp. 170–178. Malinkin, V. G., ‘‘Engineering Formula for MMW Attenuation in Precipitations,’’ III AllUnion Symp. On Physics and Techn. MMW and subMMW, Gorky, September 1980, Thesis Reports, Moscow, Nauka, 1980 (in Russian). Olsen, R. L., D. V. Rogers, and D. B. Hodge, ‘‘The aR b Relation in the Calculation of Rain Attenuation,’’ IEEE Trans. Ant. Propag., Vol. AP-26, 1978, pp. 318–329. Kulemin, G. P., Backscattering of Microwaves and Millimeter Waves by Precipitations and Other Atmospheric Formations, Preprint No. 287, Inst. Radiophysics and Electr., AS Ukr.SSR, Kharkov, 1985, p. 34 (in Russian). Kulemin, G. P., and V. B. Razskazovsky, Millimeter Wave Scattering by Earth’s Surface for Small Grazing Angles, Kiev: Naukova Dumka, 1987 (in Russian). Kerr, D. E., Propagation of Short Radio Waves, Massachusetts Institute of Technology, Radiation Laboratory Series, Vol. 13, New York: McGraw-Hill, 1951. Borovikov, A. M., ‘‘Some Totals of Radar Observations for Powerful Cumulonimbus Clouds,’’ Trans. Central Aerological Observatory, Vol. 7, 1964, pp. 68–73 (in Russian). Vakser, I. X., ‘‘Rain Radar Reflection Measurements at Wavelengths of 4.1 and 8.15 mm,’’ X All-Union Conf. on Radiowave Propag., Irkutsk, June 1972, Report Thesis, Part IV, Moscow, Nauka, 1972, pp. 76–79 (in Russian). Currie, N. C., F. B. Dyer, and R. D. Hayes, ‘‘Some Properties of Radar Returns from Rain at 9.375, 35, 70 and 95 GHz,’’ Rec. IEEE Int. Radar Conf., Arlington, VA, 1975, pp. 215–220. Gorelik, A. G., Y. V. Melnichuk, and A. A. Chernikov, ‘‘Correlation of Radar Signal Statistical Characteristic with Dynamic Processes and Micro-Structure of Objects,’’ Trans. Central Aerological Observatory, Vol. 48, 1963, pp. 38–47 (in Russian). Kapitanov, V. A., Y. V. Melnichuk, and A. A. Chernikov, ‘‘About Spectrum Shape of Precipitation Radar Signals,’’ X All-Union Conf. Radio Wave Propag., Irkutsk, July 1972, Report Thesis, Moscow, Nauka, Part II, 1972, pp. 373–376 (in Russian). Kivva, F. V., et al., ‘‘Spectral Characteristics of Meteorological Formation Backscattering,’’ XII All-Union Conf. Radio Wave Propag., Tomsk, July 1978, Report Thesis, Moscow, Nauka, Part II, 1978, pp. 225–227 (in Russian). Sharapov, L. I., ‘‘Precipitation Radar Scattering Statistical Characteristics at Millimeter Band of Radiowaves,’’ Trans. 4th All-Union Meeting for Radiometeorology, Moscow, 1975, pp. 21–23 (in Russian).

258

Microwave and MMW Backscattering by Precipitation and Other Meteorological Formations [22] [23] [24] [25] [26] [27] [28] [29] [30]

Barton, D. K., Modern Radar System Analysis, Norwood, MA: Artech House, 1988. Skorer, R., Aerodynamica of Environment, transl. from English, A. Y. Presman, (ed.), Moscow, Russia: Mir, 1980 (in Russian). Edinger, J. G., and G. C. Holworth, ‘‘Angel Observations with AN/TPQ-6 at Santa Monica,’’ Proc. 8th Weather Radar Conf., San Francisco, CA, 1960, pp. 132–142. Gatkin, N. G., et al., Clutter Rejection in Typical Set of Signal Detection, Kiev, Russia: Technics, 1971 (in Russian). Gorelik, A. G., and L. N. Uglova, ‘‘Radar Characteristics of Clear Air Backscattering,’’ Izv. AS USSR, Physics Atmosph. Ocean, Vol. 4, No. 12, 1968, pp. 132–136 (in Russian). Tatarsky, V. I., Wave Propagation in Turbulent Atmosphere, New York: Dover Publications, 1961. Atlas, D., et al., ‘‘Optimizing the Radar Detection of Clear Air Turbulence,’’ J. Appl. Meteor., No. 5, 1966, pp. 450–461. Lumley, J. L., and H. A. Panofsky, The Structure of Atmospheric Turbulence, New York: John Wiley, 1964. Dorfman, N. A., et al., ‘‘Statistical Characteristics of Refractive Index in Over-Sea Layer of Atmosphere,’’ Izv. AS USSR, Physics Atmosph. Ocean, Vol. 14, No. 5, 1978, pp. 549–553 (in Russian).

CHAPTER 6

Sea and Land Radar Clutter Modeling 6.1 Land Clutter Modeling 6.1.1 Initial Data

The purpose of this chapter is the development of the mathematical models (algorithms and software) of land and sea radar clutter over a wide band of frequencies for different types of surface and polarizations. In the first stage, the main directions of modeling, as well as the model characteristics and principles of its development, are determined. The common principles for the formal model design are considered, taking into consideration the influence of additive clutter on radar operation. The most serious attention is paid to formal description of the modeling processes. There is a large volume of experimental data, and, in fact, the physical model of microwave land and sea backscattering is absent due to complexity of the backscattering processes and the variety of surface types. This requires the development of empirical models for land and sea clutter on the basis of experimental results obtained by the author and other authors, and the estimation of their accuracy. There are some principles for the computer modeling of external factor influence on radar operation. The first approach is the most simple and consists of generation of stochastic number group sequences that are the digital equivalent of clutter. In the simplest case, this digital sequence imitates the fluctuating amplitude or the quadrature components of the signal scattered from the land or sea surface. Its rms value is functionally determined using the normalized RCS of clutter, the transmitter power, the antenna gain, and the range to the target. The normalized distribution of fluctuations and their spectra (or autocorrelation functions) correspond to the chosen clutter model. The problem of modeling the external factor influence for such an approach reduces to an appropriate choice of the clutter statistical parameters and the generation of digital sequence with required statistical properties. Such a model is the most simple and economical in exploitation. It permits modeling the radar operation in real time. It can be augmented by the target model.

259

260

Sea and Land Radar Clutter Modeling

The second approach is distinguished from the previous one by the modeling of the physical mechanism of radiowave backscattering from the Earth’s surface. The first level of this approach is the input data including the data bank, describing the relief and the vegetation cover of the region and the meteorological conditions. The next level forms the scattering surface model for surface facets using geometrical optics to form the backscattering signal with its corresponding normalized RCS. Here the relief of scattering facets can be represented by smooth approximating functions. The third level consists of models that consider the influence of radar clutter on radar system operation. Having this type of data, one can model the processes of detection, automatic tracking, and classification of targets. The realization of such a model requires very large operative memory and high speed of computer operation. As a compromise, the model can be used in which the formal description includes propagation conditions and radar response to external factors. For example, radar clutter modeling can be carried out according to the following scheme: selection of the scattering surface facet, estimation of its RCS with the use of microrelief and vegetation data, and estimation of spectral parameters with the use of wind velocity data. The radar detection range is mainly determined by the land, sea, and precipitation clutter. As a basis for clutter map creation, digital topographical maps can be applied that permit, at the first stage, modeling the illuminated and shaded areas of the surface. One has to take into consideration the radar height, the curvature of the surface, and the vegetation height. For small grazing angles, the incorrect estimation of the heights can lead to significant errors in detection range prediction. The shadowed facets do not practically contribute to the scattered signal, because the diffraction field is very weak. As result, one can obtain the illuminated zones of land (or sea) by using digital topographical maps. The basis of this solution is the development of digital map database that is a part of geoinformatic system of a country. The main principles of database development include: • • • •

The hierarchy of the informational base; The ability for its continuous development and updating; Quick access to separate data and any data subset; The ability to automatically reorganize data storage.

The data banks for clutter modeling have to contain: • • •

Surface relief data; Climatical data; Vegetation cover data.

6.1 Land Clutter Modeling

261

The surface topographical data are used jointly with meteorological and climatical data. They are transformed to the radar clutter map at the next stage, using land and sea clutter models. The following problems are solved in this chapter: • •

The development of models for different types of the land and the sea clutter for different wind velocities (sea wave heights); The elaboration of the principles of topographical and vegetation map construction for estimation of radar clutter intensity.

The input data for simulation are set at the stage of clutter signal forming. They characterize the basic factors determining the parameters of clutter from different types of terrain and sea surface. Because clutter properties are determined by a large number of factors having different physical sense and value, it is expedient to divide the data for simulation into functional groups that integrate the parameters according to their semantic contents. While setting the input data, it is also necessary to define the ranges of their acceptable values. The value control of the entered parameters enables the software product reliability to increase, drawing attention to the inapplicability of mathematical models of radar clutter for some frequency bands, observation angles, and other parameters, as well as avoiding mistakes caused by incorrectness of entered values. Input data for radar clutter simulation according to their functional groups, including acceptable values of assigned parameters, are given next. Radar parameters comprise the first group, the initial parameters describing the radar position are in the second group, and data describing the land and sea surface parameters and weather conditions are in the third group. Mathematical models of clutter are developed for sea surface and basic types of land surface. The surface types for which the simulation is possible are given in Figure 6.1 with their classification. While solving the task of computer simulation of clutter signal and designing the appropriate methods and algorithms of simulation, it is important to satisfy the following requirements: 1. To apply simulation algorithms and methods that ensure a high degree of correspondence to clutter statistical characteristics in accordance with the model data; 2. To provide relative simplicity of simulation algorithm implementation; 3. To provide acceptable computation costs in computer simulations of clutters; 4. To minimize the degree of user intrusiveness in the process of simulation; 5. To provide the ability to visually and numerically control the parameters of the generated sequences (samples) of clutter signal.

262

Sea and Land Radar Clutter Modeling

Figure 6.1 Classification of land surface types.

Apart from these requirements, the selection of simulation methods and algorithms is essentially determined by the properties of the clutter signals and by the peculiarities of simulation tasks. The material presented in this chapter is the result of model development at the Institute for Radiophysics and Electronics of the National Academy of Science of Ukraine. Part of the results was published in [1–4].

6.1.2 Peculiarities of Land Clutter Simulation

Let us consider the basic peculiarities and properties of clutter signals backscattered from land that follow from a mathematical model of backscattering given in Chapter 2 and determine the selection of simulation methods and algorithms. For such peculiarities, it is first necessary to consider the possibility of representing the clutter as the sum of fluctuating and stable components. The stable component for each realization in azimuth can be represented as a vector with random phase and amplitude dependent on the RCS section of reflectors steady in time

263

6.1 Land Clutter Modeling

x˙ st (i, R m , k ) = A st (R m ) ⭈ e i␸ st (R m , k )

(6.1)

where A st (R m ) and ␸ st (R m , k ) denote the amplitude and phase of stable component, respectively. As follows from (6.1), A st depends only on range to the surface element. The values ␸ st for different ranges R m and the k th clutter realization are the independent random values characterized by uniform pdf within the interval [0, 2␲ ]. This assumption is stipulated by the fact that the stable component of clutter is formed as the result of coherent superposition of some number of reflectors stable in time and belonging to the same radar resolution cell. As a result, the vector corresponding to the signal x˙ st has random directions with equal probabilities. The clutter stable component x st does not change for the signal sample sequence in azimuth and, therefore, does not depend on the radiation pulse index i. The fluctuating component of clutter signal backscattered from land can be represented as a complex stochastic process having the property of stationarity [5]. Let’s consider the basic statistical and power characteristics of the fluctuating component, describing it as a stochastic process. The real and imaginary parts of fl fl complex signal (x Re (i, R m , k ) and x Im (i, R m , k )) are statistically independent and are characterized by the normal distribution with zero mean and identical 2 variances ␴ fl proportional to the RCS of reflectors unstable in time. Within the framework of the mathematical model of land clutter, it is also possible to consider the sample sequences of clutter corresponding to different ranges R m statistically independent, as are also the samples of fluctuating components obtained for different k th realizations. The correlation function ␳ (␶ ) describes important power characteristics of clutter signal and describes the relationship between random samples of fluctuating components of clutter for fixed resolution element

␳ (␶ ) = ␳ (i − j ) = m {x fl (i, R m , k ) ⭈ x fl ( j , R m , k )}

(6.2)

where m{⭈} denotes the average for an ensemble of realizations. Another power characteristic, strictly connected with correlation function according to WienerKhinchin theorem [5], is the power spectrum of clutter signal. Within the framework of the mathematical model of reflections from land, the power spectrum of the fluctuating component is characterized by the parameters from (2.64), such as the maximal value of spectral density, the spectrum width, and the power exponent. Formation of the sample of the clutter stochastic process from a given resolution cell at range R m consists of the following stages: 1. Initialization of the pseudorandom number generator by the value corresponding to the kth realization the modeled stochastic process; 2. Formation of fluctuating components of clutter signal for the range R m and the k th realization:

264

Sea and Land Radar Clutter Modeling fl

fl

x Re (i, R m , k ), x Im (i, R m , k )

i = 1 . . N az

3. Formation of the stable component of the clutter signal according to (6.1); 4. Obtaining total clutter at the output of phase detector as a result of adding the fluctuating and stable components. Simulation of the fluctuating component of clutter consists of the formation of two realizations of the stochastic process with given statistical and powerful properties (see Chapter 2) that correspond to signal real and imaginary components. For this purpose, we use the standard method—at the first step, independent pseudorandom values with uniform distribution are generated. Then they are subjected to linear and nonlinear transformations with the purpose of obtaining the required statistical properties [6]. The generation of samples of stochastic process is realized in several stages (see Figure 6.2): 1. The determination of the length of pseudorandom value sequence N prs (the number of stochastic process samples) used for formation of clutter signal with the given power spectrum; ud

ud

2. The generation of two sequences of uncorrelated samples x Re , x Im (white noise) with uniform pdf in the interval [0. . .1]; nd

nd

3. The obtaining of uncorrelated samples of stochastic process x Re , x Im with normal (Gaussian) pdf, zero mean, and variance equal to unity ␴ 2 = 1 as nd

nd

a result of the nonlinear transformation of the samples x Re , x Im of the initial process with uniform pdf; kn

kn

4. The formation of stochastic process samples x Re , x Im with the given power spectrum and normal distribution; 5. The truncation of the realization length of clutter signal to N az samples corresponding to the required sample size in azimuth); fl

fl

6. The formation of clutter x Re , x Im , with given intensity as a result of the cn cn transformation of samples x Re , x Im . For generation of uncorrelated samples with uniform pdf as well as for simulation of the clutter stable component, a pseudorandom number generator is used. nd

nd

The forming of sequence of uncorrelated samples X Re , X Im with normal distribution law can be carried out in two ways. The first simulation method is based on use of the central limit theorem, according to which the sum of a large number of independent random variables has approximately Gaussian distribution [5, 6]. Most standard mathematical programs contain built-in generators for random numbers with normal distribution. An advantage of this method is its comparative simplicity.

265

6.1 Land Clutter Modeling

Figure 6.2 Algorithms for simulation of the fluctuating component of land clutter.

The drawback is the need to use a certain number (as a rule ≥ 12) of pseudorandom values (usually with uniform distribution) for generation of one value with normal distribution. Besides, the maximum/minimum value of pseudorandom values obtained by this method is limited to the sum of maximum/minimum values of initial pseudorandom values. For generation of the correlated samples of clutter with a given power spectrum, 2 zero mean and variance ␴ nd = 1, we use a linear transformation of the initial nd

nd

(uncorrelated) sequence of samples X Re , X Im (see Figure 6.1, block 4). In this case, according to the central limit theorem, the distribution law of samples remains

266

Sea and Land Radar Clutter Modeling

normal [5]. For solving the tasks of practical simulation, two ways are widely applied: one based on application of methods of linear filtering and a method of canonical decompositions [7]. The method of a linear filtering is based on expression determining the power spectrum of clutter at the filter output where this filter has the transfer function K ( j␻ ) F␨ (␻ ) = | K ( j␻ ) | ⭈ F␰ (␻ ) 2

(6.3)

where | K ( j␻ ) | is the frequency characteristic of the filter, and F␰ (␻ ) and F␨ (␻ ) denote the power spectrums of stochastic process at the input and output of linear filter, respectively. In cases where the stationary stochastic process with F␰ (␻ ) = const (white noise) and normal distribution enters the filter input, the power spectrum at its output according to (6.3) has the power spectrum described by the 2 square of its frequency characteristic | K ( j␻ ) | . While using this method of linear filtering for generation of discrete sample sequence, the simulation task consists of the creation of a digital filter with the given frequency response. To obtain the sequences of correlated samples, digital filters with finite impulse response and filters with infinite impulse response can be used. For their design, the standard methods of digital filter design with given frequency response [8] can be used. One serious drawback of the linear filtering method is the impossibility in some cases to ensure precise conformity of filter frequency response square to the given power spectrum; therefore, the given method can be considered an approximation. For example, the power spectrum (2.64) can be formed precisely for values of spectrum parameter n = 2, 4, 8, . . . using the Batterworth filter of the order p = 1, 2, 3, . . . , respectively [8]. For other values of spectrum parameter n , the precise conformity of power spectrum to the desired spectrum cannot be provided. Taking this drawback into account, for simulation of stochastic process with power spectrum (2.64), we have selected the method of canonical decompositions. The essence of this method consists of representation of the simulated stochastic process ␰ (t ) by canonical decomposition [7]. One drawback of the KarhunenLoeve transform is the considerable difficulty of solving the equation for stochastic processes with a power spectrum that is not rational (and the power spectra of clutter from land and sea surfaces are in this class). Because of this we used a simpler approach for realization of the canonical decomposition method based on decomposition of stochastic process into Fourier series [7, 9]. The advantage of the method of simulation that uses Fourier-series expansion is the benefit in computation efficiency in contrast to the linear filtering method. A considerable reduction of computational load in execution of the transformation can be obtained using FFT algorithms [8, 9]. An additional benefit of this method is the simultaneous

267

6.2 Sea Clutter Modeling

formation of two mutually uncorrelated sequences of pseudorandom values kn

kn

x Re (n ) and x Im (n ) and the economical use of initial sequences of uncorrelated nd nd samples x Re (n ), x Im (n ). This method of clutter correlated sample formation is realized in block 4 of Figure 6.1. Along with the initial sequence of uncorrelated samples for realization of this method, it is also necessary to set the size of the sample array N prs and the 2

factors h k that determine the power spectrum form. 2 The algorithm for calculation of power spectrum factors h k is realized in blocks 7 and 8, and it consists of two stages. The first stage presumes the obtaining of power spectrum samples for a clutter signal sampled in time. Thus, the spectrum of the continuous signal is set by (2.64). The spectrum parameter G o determines the spectral density value for zero frequency, and it can be set arbitrarily. Conditionally, let us consider G o = 1. As follows from the Shannon sampling theorem, if there is discretization of the continuous signal, spectrum aliasing takes place for a period determined by the sampling rate. As the result, the spectrum of the sampled signal is distorted by multiple superposition of the continuous signal spectrum displaced in the frequency domain. As a rule, these distortions show themselves by increasing the level of high-frequency spectrum components. In [10], it is shown that in case of stochastic process discretization, the power spectrum behaves in a similar manner. The second stage of power spectrum factor calculation (Figure 6.2, block 8) consists of normalizing the values hk in such a manner that for the correlated stochastic process (obtained by the decomposition method for Fourier series), one 2

has to ensure the variance value ␴ kn = 1. fl The final simulation stage is the formation of the clutter signal x Re (n ), fl x Im (n ) with the required intensity.

6.2 Sea Clutter Modeling 6.2.1 Peculiarities of Sea Clutter Simulation

In contrast to clutter backscattered from land surfaces, sea clutter is characterized by a number of peculiarities and properties that lead to the necessity of introducing additional stages and complicating the clutter simulation algorithm. For instance, the backscattering from crests of breaking sea waves and spray results in the appearance of spikes in the scattered signal. Because of this, the distribution law of clutter differs from Gaussian. As shown in Chapter 4, the most satisfactory approximation of clutter distributions at the output of a quadrature detector is the compound normal distribution

268

Sea and Land Radar Clutter Modeling

w cn (x ) = (1 − ␥ )

1

√2␲␴ cn 2

冉

exp −

x2 2 2␴ cn

冊

+␥

1

√2␲ k

2 2 ␴ cn

冉

exp −

x2 2 2k 2␴ cn

冊 (6.4)

where ␥ and k 2 are the parameters describing the properties of spikes (see Chapter 2

4), and ␴ cn characterizes the variance of fluctuations. To obtain samples of a clutter signal that has a compound normal distribution, the simulation algorithm can use the method of nonlinear transformation of stochastic processes with Gaussian distribution [7]. Another distinctive peculiarity of sea clutter is the absence of reflectors that are steady in time and, consequently, the absence of stable clutter component. The calculation of RCS values for the sea is also executed in an essentially different manner. The power spectrum of real and imaginary components of clutter signal from sea differs from the power spectrum of land clutter by the presence of a mean Doppler frequency caused by the motion of particular surface scatterers. This value depends on wind speed, wave direction, and radar operation frequency, and it is characterized by parameter F 0 . The spectrum shift results in the appearance of a signal fluctuation correlation function for azimuthal samples with Doppler frequency of F 0 . In this case, the condition of stochastic process stationarity for the sequence of azimuthal sample is not valid as well; this obstacle requires additionally a correction to the parameters of the compound normal distribution (6.4), while simulating the clutter. 6.2.2 Algorithm of Sea Clutter Simulation

As in the case of land clutter, the simulation of real and imaginary components of sea clutter is executed in several stages. Let us consider the algorithm for generating the kth realization of clutter signal samples for sea clutter case in the range R m . As noted earlier, the real and imaginary parts of the clutter signal backscattered from the sea do not contain stable components. This enables us to eliminate from the algorithm of simulation the stage of stable component formation, as well as the stage of summing the stable and fluctuation components. Thus, the algorithm of clutter signal simulation is simplified, and it can be represented by the generalized diagram presented in Figure 6.3. The simulation process includes the following stages: 1. Initialization of the pseudorandom number generator by the value corresponding to the kth realization of the modeled stochastic process; 2. Formation of the clutter signal for the k th realization range R m : x Re (i, R m , k ), x Im (i, R m , k ), i = 1 . . . N az .

6.2 Sea Clutter Modeling

Figure 6.3 Algorithm of the simulation of sea clutter.

269

270

Sea and Land Radar Clutter Modeling

As follows from a comparison of Figures 6.2 and 6.3, the differences between this algorithm and that for the fluctuation component of land clutter consist in the presence of additional stages of simulation represented in Figure 6.3 by blocks 9–12. Common blocks for both diagrams of simulation are the blocks 1–8. Therefore, only the blocks 9–12 will be considered in detail. The algorithm of sea clutter simulation according to the scheme in Figure 6.3 consists of the following stages: 1. Defining the length of the pseudorandom sequence of samples N prs (the number of stochastic process samples) used for formation of the clutter signal with the power spectrum (4.79); 2. Finding the function y = f (x ) of nonlinear transformation of the stochastic process with Gaussian distribution law for given parameters ␥ and k of compound normal distribution (6.4); 2

3. Calculating the factors h k of the power spectrum (4.79) at F 0 = 0 and their normalization (realized in units 7 and 8); 2

4. Correcting the factors h k of the given power spectrum of the stochastic process according to the nonlinear transformation y = f (x ) and obtaining n2

power spectrum factors h k for initial normal stochastic process; kn kn 5. Generating two sequences of samples of the stochastic process x Re , x Im n2

with normal pdf and the power spectrum determined by the factors h k (realized in blocks 2–4); kn

kn

6. Truncating the realization length of the clutter signal x Re , x Im up to N az samples (i.e., to the number of samples corresponding to clutter signal in azimuth); ks

ks

7. Forming the samples x Re , x Im of the stochastic process with the power spectrum having the central frequency F 0 = 0 and compounding the normal distribution as the result of nonlinear noninertial transformation y = f (x ) kn

kn

of the samples x Re , x Im ; ks ks 8. Transforming the power spectrum for the stochastic process x Re , x Im by ds ds Doppler frequency F 0 and forming the samples x Re , x Im ; 9. Forming the clutter signal x Re , x Im with given intensity as the result of ds

ds

transformation of the samples x Re , x Im . At the initial stage of simulation, the calculation of the number N prs of samples of the stochastic process generated at subsequent stages with given power spectrum and normal distribution law is executed in block 1. The value N prs is determined according to peculiarities of the simulation of correlated stochastic processes with normal distribution using the Fourier series expansion method. In this case, the

6.2 Sea Clutter Modeling

271

number of samples N prs is larger or equal to the number N az of samples of the modeled signal in azimuth. The selection of the value N prs is realized according to the algorithm used for simulation of land clutter. The parameter F 0 at this stage of simulation is supposed equal to zero. The next simulation stage is the search for nonlinear transformation y = f (x ), permitting transformation of the Gaussian distribution of the initial stochastic process into the compound normal distribution. The modulating transformation resulting in the required change of the distribution law of the stochastic process can be applied for formation of the stochastic process with power spectrum (4.79) with central frequency F 0 ≠ 0 at one stage of simulation. To provide a high accuracy of conformity of the distribution law of the modeled stochastic process to the compound normal distribution law, the preliminary correction of given parameters of ␥ and k of the compound normal distribution is realized. As a result, the parameters ␥ F , k F of the distribution of the samples of the stochastic process ds

ds

x Re , x Im (see Figure 6.2), ensuring minimum value of goal function, are assumed. As seen, the algorithms of clutter simulation from land cannot be directly used for clutter formation for the sea clutter case. For example, the method of canonical decompositions [7, 9, 11] allows getting the stochastic process with a power spectrum of practically any kind—in particular, the power spectrum defined by (4.79). At the same time, the method of canonical decompositions is based on linear transformation of initial (uncorrelated) sample sequences, and by virtue of the central limit theorem it does not allow us to form the pseudorandom sample with compound normal distribution (6.4). For the method of canonical decompositions, the sequence of samples with given correlation properties is formed as a result of summation of a large number of pseudorandom numbers [9], and, in the case of a normal distribution law of the initial sequence samples, the obtained signal samples also have Gaussian distribution. If the distribution law of the initial sequence differs from normal, by virtue of the central limit theorem the obtained samples will have a distribution slightly different from a Gaussian one [9]. To form the sea clutter signal, the methods of simulation of non-Gaussian stationary stochastic processes [7] can be used. In this case, the stochastic process can be described by either a multidimensional distribution or structurally as a transformation from random variables and determined functions. In the first case, the problem of simulation can be solved as a problem of forming the realization of a random vector with given multidimensional distribution. For this purpose, we can apply either the multidimensional method of Neumann or the method based on use of the conditional probability density. In the second case, the probabilistic process is set parametrically. Its simulation consists of forming the realizations of random variables and their subsequent transformation. The practical use of these two simulation methods is significantly limited by problems arising in generation of stochastic process realizations with large lengths.

272

Sea and Land Radar Clutter Modeling

Besides, while carrying out the experimental observations, it is rather difficult (and in some cases impossible) to get multidimensional laws of random vector distributions or to set the stochastic process parametrically. In the case where the one-dimensional (marginal) probability law and power spectrum are known, for non-Gaussian stochastic process generation, it is possible to use a nonlinear transformation method. At the first stage of this method, the formation of stationary stochastic process ␰ x (t ), which has some specific power spectrum and the normal distribution, is made. Then, the stochastic process samples are subjected to nonlinear transformation without inertia y = f (x ), which transforms the initially normal pdf w x (x ) of the process ␰ x (t ) into the given pdf w y (y ) of the obtained process ␰ y (t ). As known, such a transformation always exists [5, 7]. Beside pdf transformation of the initial stochastic process, the nonlinear transformation also results in the power spectrum changing. Consequently, the correlation function corresponding to it also changes. Let us denote ␳ x (␶ ) the correlation function of initial stochastic process ␰ x (t ). Then, as the result of nonlinear transformation, the stochastic process ␰ y (t ) will have the correlation function ␳ y (␶ ), differing from ␳ x (␶ ) and connected with it by

␳y = ␸(␳x )

(6.5)

The form of relationship ␸ ( ␳ x ) is determined by the nonlinear transformation y = f (x ). The correlation functions of the initial process ␰ x (t ) is selected so that after transformation (6.5), we get the stochastic process with the given correlation function (power spectrum). For finding ␳ x (␶ ), it is necessary to execute the inverse transformation

␳ x = ␸ −1 ( ␳ y )

(6.6)

where ␸ −1 ( ␳ y ) is the function inverse to the function ␸ ( ␳ x ). Thus, the simulation of the stochastic process using the method of nonlinear transformations consists of the following stages: 1. Finding the nonlinear transformation y = f (x ) using the given pdf w y (y ); 2. Obtaining the dependence y = f (x ) for given function ␳ y = ␸ ( ␳ x ); 3. Finding the inverse function ␸ −1 ( ␳ y ) and correlation function ␳ x (␶ ) of initial process ␰ x (t ); 4. Forming the normal stochastic process ␰ x (t ) with correlation function ␳ x (␶ ); 5. Obtaining the stochastic process with required characteristics as the result of nonlinear transformation of the initial stochastic process ␰ y = f (␰ x ). Next, we suppose that the nonlinear transformation y = f (x ) is a monotonically increasing function. Obviously, this requirement is satisfied for the case of transfor-

273

6.2 Sea Clutter Modeling

mation of the normal distribution law into the compound normal law (6.4). Also consider the modeled and initial stochastic processes as having zero mean and variance ␴ 2 = 1. The assumption that the clutter signal mean equals zero follows from the mathematical model of sea backscattering. The condition ␴ 2 = 1 is accepted as a matter of convenience for simulations. The required variance of the clutter signal is set in the final simulation stage as a result of multiplying the sample values of the modeled sequence by a derived constant. For finding the transformation y = f (x ), let us use the condition of equality of cumulative density functions of random samples of the initial and modeled processes, Wy (y 0 ) = Wx (x 0 )

(6.7)

where y 0 = f (x 0), Wy (⭈), and Wx (⭈) are the cumulative distribution functions of stochastic processes ␰ x (t ) and ␰ y (t ). As the stochastic process ␰ x (t ) is Gaussian, Wx (⭈) is determined from the expression Wx (x ) = 0.5 ⭈ 冋⌽冠x / √2 冡 + 1册

(6.8)

where ⌽(⭈) is the error function x

2 ⌽(x ) = √␲

冕

e −z dz 2

(6.9)

0

If the stochastic process is preset by pdf w y (y ), (6.7) can be reduced to [5, 7] w y [ f (x )] ⭈

df (x ) = w x (x ) dx

(6.10)

where w x (x ) =

2 1 e −x /2 √2␲

(6.11)

The dependence y = f (x ) is found by solving (6.7) or (6.10). If it is not possible to find the function y = f (x ) analytically, the solution can be obtained by computer numerical methods as a table of values of y = f (x ). In this case, an interval of possible values of argument x is restricted by the limits for which the probability of x exceeding them is negligible. When the transformation y = f (x ) is found, the relation (6.5) between correlation functions of initial ␰ x (t ) and transformed ␰ y (t ) processes can be determined.

274

Sea and Land Radar Clutter Modeling

According to its definition, the correlation function can be derived as an expectation of product ␰ y (t ) ⭈ ␰ y (t + ␶ ) = f [␰ x (t )] ⭈ f [␰ x (t + ␶ )]. Then the correlation function ␳ y (␶ ) of the transformed process is determined by ∞ ∞

␳y = ␸(␳x ) =

冕冕

f (x 1 ) ⭈ f (x 2 ) ⭈ w x (x 1 , x 2 , ␳ x ) dx 1 dx 2

(6.12)

−∞ −∞

=

1 2␲ √1 − ␳ x2

2

∞ ∞

冕冕

f (x 1 ) ⭈ f (x 2 ) ⭈

2

x 1 − 2␳ x x 1 x 2 + x 2 2(1 − ␳ 2v ) e

dx 1 dx 2

−∞ −∞

where w x (x 1 , x 2 , ␳ x ) is the two-dimensional pdf corresponding to the Gaussian distribution, and x 1 and x 2 are the values of the initial stochastic process at time instants displaced from each other by the value ␶ . P x is the coefficient of correlation between x 1 and x 2 . Direct use of (6.12) for obtaining the dependence ␳ y = ␸ ( ␳ x ) is, as a rule, problematic because the integral cannot always be calculated in a closed form. Besides, if using the computer to solve (6.7) and (6.10), the function y = f (x ) is set as a table; this does not allow us to get the correlation function ␳ y (t ) analytically on the basis of (6.12). As shown in [5], the solution of (6.12) can be simply obtained if one represents the function ␸ ( ␳ x ) as power series. To find the coefficients of the power series, it is proposed to use the decomposition of the two-dimensional pdf w x (x 1 , x 2 , ␳ x ) into series using orthogonal Hermittian polynomials [12]. Then, the required dependence can be obtained as

␳y =

∞

∑

m =0

2 Cm

␳ xm m!

(6.13)

The coefficients C m can be found as [5] 1 Cm = √2␲

∞

冕

2 f (x ) ⭈ H m (x ) ⭈ e −x /2 dx

(6.14)

−∞

where H m (⭈) are Hermittian polynomials. Note that because initial and transformed stochastic processes have variance ␴ 2 = ␳ x (0) = ␳ y (0) = 1 and zero mean, the coefficient C 0 = 0. Consequently, the following equality is valid ∞

∑

m =0

2 Cm =1 m!

(6.15)

275

6.2 Sea Clutter Modeling

After getting the power series coefficients C m , for finding the correlation function ␳ x of the initial stochastic process, it is necessary to solve (6.13) with respect to ␳ x . The inverse function ␳ x = ␸ −1( ␳ y ), as a rule, cannot be found analytically, and it is expedient to get the solution of (6.13) by numerical computer methods [7]. In this case, because we get a numerical solution to this equation, the function ␸ −1( ␳ y ) is set by a table. We would like to note the peculiarities of practical use of the nonlinear transformation method for simulation of sea clutter. As it is mentioned in [7], this method cannot be used for generation of stochastic processes with the correlation function of definite type because the solution of (6.13) with respect to ␳ x (␶ ) does not always exist. However, when the correlation function is nonnegative, the solution of this equation always exists [7]. Let us consider the power spectrum of sea surface clutter defined by (4.79). Its difference from the spectrum of land clutter (2.64) consists in the presence of F 0 , the center frequency. In this case, the value of power spectral density is maximal at center frequency F 0 . The correlation function corresponding to the power spectrum (4.79) can be represented as

␳ y (␶ ) = ␳ y (␶ )* ⭈ cos (2␲ ⭈ F 0 ⭈ ␶ )

(6.16)

where ␳ y (␶ )* is the correlation function corresponding to spectrum (2.64). From (6.16), it follows that the function ␳ y (␶ ) has a fluctuating character with frequency F 0 , equal to the center frequency of the spectrum. Obviously, no stochastic process with spectrum (4.79) can be obtained as a result of initial process nonlinear transformation. As shown in [5], the nonlinear transformation results in spectrum widening and the appearance of the local extreme in frequency being a multiple of the center frequency F 0 . At the same time, the method of nonlinear transformation can be used for the formation of a stochastic process with power spectrum (2.64), which corresponds to spectrum (4.79) with the center frequency F 0 = 0. In this connection, the formation of a sea clutter signal with spectrum (4.79) can be done in two stages: 1. Generation by the nonlinear transformation method of a sequence of stochastic process samples with a compound normal distribution and power spectrum (2.64); 2. Transformation of power spectrum (2.64) of stochastic processes by the modulation method [5] to the form (4.79). If approximating the experimental distributions of clutter signal by the compound normal distribution (6.4), the weighting coefficient ␥ does not exceed the value 0.1, and the ratio of variances of distribution component k 2 is within the interval 10–20. Obviously, for typical values of parameters ␥ and k obtained for experimental data, the distribution does not practically depend on time and differs

276

Sea and Land Radar Clutter Modeling

only slightly from the compound normal distribution (6.4). The degree of conformity of the sample distribution law of the modulated stochastic process ␰ F (t ) to the compound normal distribution can be additionally increased as the result of a precorrection of the pdf parameters of initial stochastic process ␰ 0 (t ). The values of parameters ␥ F and k F ensuring minimum difference of the distribution law of process ␰ F (t ) from the compound normal law with parameters ␥ and k are determined by criterion ∞

Ew =

冕

2 2 2 [w cn (x , ␥ , k , ␴ cn ) − w F (x , ␥ F , k F , ␴ cn , F )] dx

(6.17)

−∞

As unknown values, the parameters ␥ F and k F , E w are accepted to provide the minimum of goal function (␥ F , k F ). In Table 6.1, for different ␥ and k of the compound normal distribution (6.4), the estimated values of total square error E w (6.17) are presented. The corrected values ␥ F and k F ensuring goal function minimum are also given. From the presented data, it follows that the deviation of the distribution law of samples of the formed stochastic process from a compound normal distribution is increased with increase of values ␥ and k . At the same time, for maximum values ␥ = 0.1 and k 2 = 20, the general quadratic errors are rather small (of order 10−6 ) and the distribution law deviation from the given one can be neglected. It is also necessary to mention that the difference between the corrected distribution parameters ␥ F and k F and the given ones is not great.

6.3 Clutter Map Development 6.3.1 Initial Data for Modeling

As shown in Chapter 2, the land clutter intensity characterized by its normalized RCS depends on a number of factors. The surface relief and vegetation type exert the primary influence. The alternation of the different vegetation types causes a

Table 6.1 Corrected Parameters of Compound Normal Distribution and the Values of Goal Function Corresponding to Them Typical Parameters of pdf ␥ k2 0.01 10 0.01 20 0.1 10 0.1 20

Corrected Parameters of pdf ␥F k 2F 0.00705 14.01 0.00660 30.06 0.07220 13.91 0.06768 29.82

Total Square Error Ew 4.49 ⭈ 10−9 1.34 ⭈ 10−8 5.62 ⭈ 10−7 1.99 ⭈ 10−6

6.3 Clutter Map Development

277

mixed character of land areas with clutter. As result, the conditions for target detection are changed. Analysis shows that, for most tasks, it is necessary to model surface areas with dimensions not less than 10 km. As a rule, a plain surface of such dimensions has a height variation greater than 100m and is described by contour intervals of 10–20m. These contours can have gaps or be absent. As long as the surface area has not less than 5–10 contours, it is possible to apply the simple and quick technique of mean heights for restoration, characterized by the simplicity of realization and small restoration time. The minimal initial data for modeling are: • •

• •

The set of surface contours; Radar characteristics (e.g., operation frequency, antenna height, and coordinates); The map of the land surface (e.g., grass, forest, and concrete); Wind velocity and direction.

As additional data, a map of land surface heights, a map of atmospheric precipitations, or a soil map can be used. 6.3.2 Software Input and Processing Components

The necessity of a digital relief model for compiling masking maps, in its turn, conditions the need for input, processing, and (perhaps) storing data on heights and vegetation. The software in question provides for inputting and processing such information from both ready electronic maps (the data export function from the exchange format of the MapInfo package) and printed maps. In order to prepare a digital relief model (raster image of a given locality on a given scale, with the brightness of every pixel designating the height of the relief in it), the information found on topographic maps about the lines of equal heights (isolines) is used. These data are exported into the internal data format from ready electronic maps or recognized from a scanned image of a topographic map. In the latter case, the information on equal height lines is highlighted in the image through stipulated color, brightness, and area; vectored; saved in the internal format; finished manually (removing, for example, breaks and wrongly recognized lines); and digitized. Data exported from ready electronic maps may also be edited manually (correction of mistakes). While rendering a digital relief model, three alternative algorithms can be used— the iterative algorithm based on discrete cosine conversion [13], the algorithm based on the Delone triangulation [14], and the one based on smoothing filters. The first algorithm ensures a more exact relief rendering with a more natural view, but it’s unstable towards error-containing incoming data (line breaks and incorrect

278

Sea and Land Radar Clutter Modeling

digitizing). The second algorithm shapes a less exact relief rendition; nevertheless, it is highly stable for erroneous data (able to function correctly even with fragments and dotty data). The third algorithm outruns the other two in terms of speed and can be used for quick approximate digital relief renditions and, as with the second algorithm, when processing erroneous data. For creating maps of vegetation, the information found on topographic maps about the area outlines and their types is used. Data can be exported from ready electronic maps (the exchange format of the MapInfo package) and subsequently edited manually (correction of downloaded mistakes). This software possesses mechanisms for storing data in separate sheets of topographic maps (equal height lines, vegetation areas, raster image of the sheet) and synthesizing data for a given locality on their basis (lacing and highlighting algorithms). Figure 6.4 presents an enlarged structural scheme of the software being discussed depicting in greater detail the modules responsible for input and processing data on heights and coating maps. The interface is arranged in a way that allows us at any time to gain access to any of the shown modules so the user does not have to stick to a rigid sequence of actions when inputting and processing data. This enables us, when setting a task, to easily distribute bits of work between various users specializing, for instance, only in scanning and interlacing topographic map sheets or only in correcting data on contour lines, and to carry out processing of data on an incomplete package (e.g., only scanning of topographic map fragments and their storage for future processing). In the following sections, we will get down to a more detailed description of the modules requiring attention. 6.3.3 Raster Image Processing Module

In the process of scanning topographic map fragments (as a rule, a scanner will not entirely accommodate map sheets), pieces of images are obtained having nonlinear distortions conditioned by paper folds and paper unevenly resting on the scanner working surface and distortions of deviation of the vertical axis of the map sheet from the vertical axis of the scanned image due to the user’s inaccuracy in placing the map sheet onto the scanner. These distortions must be removed prior to sheet interlacing and starting to highlight information on contour lines. If cheaper scanner models are used, an image brightness correction might also be needed to enhance its subjective visual quality. The procedures of interlacing images, highlighting an image fragment into a separate file, and rotating present no difficulty from the algorithmic point of view; still, the large size of images in processing—dozens and hundreds of megabytes— creates certain technological impediments. As a rule, the image size is far bigger

6.3 Clutter Map Development

279

Figure 6.4 Structural scheme of inputting and processing data on heights and vegetation maps as well as topographic map raster images.

280

Sea and Land Radar Clutter Modeling

than the size of the random access memory (RAM) available, and downloading the whole image into the RAM leads to the operational system creating a virtual memory on the hard drive, which increases the processing period by hundreds and thousands of times. In order to solve that problem, this software buffers the image to be processed, so at each moment of time only part of it is found in the RAM. In most cases, engaging this approach enables us to minimize spending nonscheduled time, though it complicates the application scripts to some extent and inflates the size of the execute files. The geometrical image correction procedure is nonlinear conversion on image pixel coordinates aiming at eliminating nonlinear distortions of scanning and distortions of the rotation angle. In this procedure, initial and resulting coordinates of four user-designated reference points of the image are employed.

6.3.4 Automatic Highlighting of Contours on the Raster

The procedure of automatic highlighting of contours on the raster consists of a rigid sequence of steps, though selection of parameters is needed at every step and, consequently, this step will be repeated several times, which leads to the module of automatic highlighting of contours on the raster. The initial data for the module of automatic highlighting of contours on the raster will be raster full-color (24-bit true color) BMP-format images. As resulting data, files of passports (this file contains detailed data contents, its sequence, and its size in bytes for each line), and vectors of contour lines are compiled. After submodule initiation, which is part of the module of automatic highlighting of contours on rasters, intermediate data may be both raster images saved in BMP files (modules from the linear filter to the removal of large objects with line color) and vector images saved in text files (modules from the vectoring line raster image to the offshoots removal). The key module here is the one of vectoring line raster image, which ensures the transition from a raster image to vector and, thus, unlike many other modules, cannot be deleted in the process of line highlighting. Another key module is the one of line narrowing, the algorithm of which will be examined later. The linear filter and the vector border underline modules, as a rule, are necessary when working with most scanners and printed topographic maps, although there are cases when they will not be needed (the image quality will meet the recognition requirements without resorting to them), so they might be dropped. The line fragments, cycles, and offshoots removal modules actually serve the purpose of saving subsequent manual work of correcting automatic recognition mistakes and might also be dropped if those mistakes are few in number. Now let us move to a more detailed description of submodules of the module of automatic highlighting of contours on the raster.

6.3 Clutter Map Development

281

The linear filter module, together with the module of vector underlining of borders, are designed to eliminate the effect of black line lamination into several lines of various colors, due to which part of them might erroneously be color classified as contour lines. The functions of the vector filter are described in [15]. It is seen that the line lamination into several lines of various colors has been neutralized, and now it can easily and fully be color classified. In the submodule of contour color and brightness highlighting, image points not belonging to contours are sifted off. We can use all sorts of point color interrelations R , B, and G as the operator-designated sifting conditions (e.g., B > 0.65R ) as well as the summary brightness value of a point R + G + B (e.g., R + G + B < 10). The median filters submodule sifts off separate small-sized and shapeless congestions of points mistakenly color selected and left there. Upon the operator’s choice, up to three median filters can be employed simultaneously, with the window size adjustable and the sequence statistics selectable. This will filter the image the number of times defined one by one. Activating the second and third median filters is not obligatory, and it is defined by the operator. Practically, good results are obtained, for example, by engaging in series two median filters with the window sizes 5 and 3, sequence statistics 10 and 4, and filtering repetition in the course of four iterations. As an alternative to recognizing lines on scanned topographic maps and their subsequent manual editing, this software enables data export on contour lines from ready electronic maps. The exchange format of the well-known MapInfo package has been chosen as a data format to be exported from. When data is exported from the exchange format of the MapInfo package, it is automatically bound to geocoordinates. When highlighting contour lines on raster and editing them manually, geocoordinates binding must be done manually through a special module of this software. The geocoordinates binding file is a text file where all the data is placed in one line and parted by spaces. The data contains the coordinates in degrees of North latitude and East longitude of the map’s top left-hand corner (the coordinates will be real numbers) and the number of pixels in one latitude degree and one longitude degree. The number of pixels per one degree of latitude or longitude depends on the chosen scanner resolution and on further scale changes of the scanned raster image. This system of binding data to geocoordinates possesses two essential advantages. First, the point coordinates are not directly bound to geocoordinates, so when the data about the geographic coordinates is unavailable, no fictitious geocoordinates for points need to be registered. Second, adding a point to or removing one from the isolines data file will not affect the file of geocoordinates binding, which could not be avoided if, for instance, the coordinates for two corners of the sheet (the top left-hand and the bottom right-hand) were recorded there.

282

Sea and Land Radar Clutter Modeling

Editing data on coating maps is fully identical to editing data on contour lines and is carried out by the same module. The only difference is that instead of the height value for a contour line, the coating-type value is recorded in the case of a coating map. 6.3.5 Steady Algorithm of Surface Recovery from Contours

Initial data for steady algorithms of surface recovery is the list of contours in a vector form. The coordinates of segment tops making contours are set as geographic coordinates of latitude and longitude. The contours can have gaps, passing or partially to miss. A scale of repaired surface is arbitrary one. The recovery has three levels: • • •

Recovery of normals (perpendiculars) to contours; Filling of interspace by mean altitudes; Smoothing of irregularities and discontinuities.

Recovery of normals to the contour segments is necessary for decreasing an error of surface recovery. Recovery of altitudes is made as follows: 1. Coordinates of a normal to a section are evaluated. 2. Three interceptions of normal with other contours (their altitude) are searched. 3. If not less than three altitudes have different values, surface altitudes between contours restore them by the two-dimensional spline for four points. The intervals between contours are filled by reference points, which are taken into account at the following stages of the surface recovery. The altitude recovery by this method has a great error rate but allows us to restore the surface with initial data of different quality, in view of neighboring contours. At the filling, the mean altitude takes into account both the altitude of contours and the altitude of points obtained earlier. The filling is made until three points with a different altitude will cover. A filling depth is the mean value between the first and second covered points. After filling, the surface looks like the domains of filling have a staircase, and the altitude of the steps is proportional to the quality of the initial contours. The stepwise surface is completely unacceptable for the calculation of shaded zones and simulation; therefore, it is necessary to receive a regular surface with the help of filtering. The analysis has shown that the best results for the given method of surface recovery reach the Gaussian pyramid-shaped filter. It is necessary to set the depth of smoothing manually, depending on the features of particular relief.

6.3 Clutter Map Development

283

6.3.6 Simulation of the Absolute Reflectivity

For calculation of the absolute value of normalized RCS, three parameters are needed: the grazing angle, the radar frequency, and the surface type. The general simulation algorithm permits us to evaluate the values of the absolute reflectivity for all facets of the modeled area. Having the values of the absolute reflectivity at any point of the land surface, radar parameters, and weather conditions makes it possible to calculate the fluctuating and steady components of the absolute reflectivity. To decrease the effect of the recovery relief errors, the grazing angle is calculated for a surface segment whose size is M × N pixels, and the radar altitude is set as absolute altitude at sea level. The size M and N are set depending on the quality of relief recovery, the model for minimum segment 7 × 7 facets is presented in Figure 6.5. In this figure, the brightness determines the normalized RCS for different areas of this fragment. The dark areas in the figure correspond to shaded terrains. The increasing of radar height leads, as a rule, to decreasing of shaded areas. As result of the sea and land clutter mathematical model development, the following results are obtained:

Figure 6.5 Reflectivity for radar height 20m, M = N = 2.

284

Sea and Land Radar Clutter Modeling

•

•

•

The algorithms are developed, and the land clutter mathematical model is carried out for a wide variety of surface types with and without the vegetation and for the different wind velocities on the basis of experimental investigations of the clutter statistical characteristics. The database has been developed for land clutter signals for real clutter signals obtained at on-land radars. Algorithms have been developed and the sea clutter mathematical model has been carried out for different wind velocities and sea states and for motionless radar.

References [1]

Kulemin, G. P., A. A. Kurekin, and E. A. Goroshko, Radar Clutter Modeling, Collected Articles, Kharkov Military University, Kharkov, Ukraine, Vol. 2, No. 28, 2000, pp. 59–65 (in Russian).

[2]

Kulemin, G. P., A. A. Kurekin, and E. A. Goroshko, ‘‘Radar Clutter with Non-Gaussian Distribution Modeling,’’ Radiophysics and Electronics, Vol. 7, No. 1, 2002, pp. 56–67 (in Russian).

[3]

Kulemin, G. P., and E. A. Goroshko, ‘‘Land Clutter Estimation in Airplane Pulsed Doppler Radar,’’ 2nd Int. Conf., CD Trans., Kiev, Ukraine, National Aerospace Academy, October 2000 (in Russian).

[4]

Kulemin, G. P., and E. V. Tarnavsky, ‘‘Modeling of Radar Land Clutter Map for Small Grazing Angles,’’ URSI General Assembly, Amsterdam, August 2002, to be published.

[5]

Levin, B. R., Theoretical Basics of Statistical Radio Engineering, Moscow, Russia: Soviet Radio, 1969 (in Russian)

[6]

Knut, D., Art of Programming for Computers, Moscow, Russia: Mir, 1977 (in Russian).

[7]

Bikov, V. V., Digital Modeling in Statistical Radio Engineering, Moscow, Russia: Soviet Radio, 1971 (in Russian).

[8]

Rabiner, L., and B. Gold, Digital Signal Processing Theory and Applications, Moscow, Russia: Mir, 1978 (in Russian).

[9]

Yaroslavsky, L. P., Digital Signal Processing in Optics and Holography: Introduction to Digital Optics, Moscow, Russia: Radio and Communications, 1987 (in Russian).

[10]

Gribanov, Y. I., and V. L. Malkov, Spectral Analysis of Stochastic Processes, Moscow, Russia: Energia, 1974 (in Russian).

[11]

Ermakov, S. M., and G. A. Mihailov, Statistical Modelling, Moscow, Russia: Science, 1982 (in Russian)

[12]

Ango, A., Mathematics for Electrical and Radio Engineers, Edition of K. S. Shifrin, Moscow, Russia: Science, 1965 (in Russian).

[13]

Ponomarenko, N. N., V. V. Lukin, and A. A. Zelensky, ‘‘The Iterative Procedure of Rendering Digital Relief Model on Isogram Map Using Discrete Cosine Conversion and Histogram Filtering,’’ Aviation and Space Techniques and Technologies, Kharkov, Russia: Kharkov Aviation Institute, 2000 (in Russian).

References [14] [15]

285 Shikin E. V., A. V. Boreskov, and A. A. Zaytsev, The Basics of Computer Graphics, Moscow, Russia: Dialog-MIFI, 1993 (in Russian). Kurekin, A. A., et al., ‘‘Adaptive Nonlinear Vector Filtering of Multichannel Radar Images,’’ Proc. of SPIE Conference on Multispectral Imaging for Terrestrial Applications II, Vol. 3119, San Diego, CA, July 1997, pp. 25–36.

CHAPTER 7

Clutter Rejection in MMW Radar 7.1 Influence of Propagation Effects on MMW Radar Operation 7.1.1 Introduction

MMW land-based and maritime radar systems are applied widely for weapon and missile control, and this chapter will be concerned with short-range, very-lowaltitude radar applications such as battlefield radars. This can be explained by the fact that the range, angle, and velocity resolution of MMW radar systems is better than for analogous systems in the centimeter band and that the reserve and stability to radio countermeasures are higher. The success in solving low-altitude target detection and tracking problems is determined, mainly, by propagation effects. Among these are multipath propagation attenuation and attenuation due to precipitation (i.e., rain, fog, or snow) that limit the maximum detection range. The problems of attenuation and backscattering of MMW in precipitation are considered in detail in Chapter 5. The precipitation influence on land-based radar operation is less important in the microwave band, and it is necessary to take into consideration this limiting factor in the MMW band at ranges more than few kilometers. The essential advantage of MMW-band radars is the small influence of multipath attenuation in comparison with radars in the centimeter band. Multipath propagation is the propagation of a wave from one point to another by more than one path. For radar, it usually consists of a direct path and one or more indirect paths by reflection from the land or sea surface or from large manmade structures. In this situation, there is simultaneous or near-simultaneous reception of waves that have reached the receiving antenna by direct and reflected paths. Depending on the relative phases and amplitudes of the several simultaneously received components, the result is a composite electromagnetic field that can be near zero or as much as twice that received by the direct path only. Consequently, multipath propagation can lead to attenuation of the electromagnetic field in comparison with free space propagation. There is also a second problem limiting the application of MMW radar systems. This is the clutter from the land or sea surface and volume clutter from such scatterers as precipitation, the latter increasing in the MMW band and limiting

287

288

Clutter Rejection in MMW Radar

the use of these radar frequencies. There are many papers and books [1–4] in which the influence of these propagation effects on radar operation is discussed separately, but the joint estimation of terrain and precipitation clutter and precipitation attenuation leads to more accurate determination of radar target detectability and available detection range. The joint influence of these effects on MMW radar operation, in particular on maximal attainable detection range and target detectability, is considered in this chapter. The parameters of MMW radar are compared with parameters of analogous X-band radar, and the comparison is carried out for two situations: an antenna aperture that is constant with frequency change and an antenna gain that is constant with frequency change (i.e., a change of antenna aperture area proportional to the square of wavelength takes place).

7.1.2 Multipath Attenuation

There are two reasons determining the total attenuation of signal in microwaves and millimeter bands. One of them is the multipath propagation over the Earth’s surface. For estimation of the multipath effect, the propagation factor V is used, as a rule, as a function of the heights of radar h r and target h t and the rms roughness height ␴ h [4]. For propagation over sea and land without vegetation for small grazing angles, the propagation factor can be presented in a form

|V | =

√

2␲ 1 + ␳ 2 − 2␳ ⭈ cos ␦≈ ␭ h + ht ; sin ␺ ≈ ␺ = r r

√

2␲ 1 + ␳ s2 − 2␳ s cos ␦; ␭

␦≈

(7.1)

2h r h t r

Here ␳ = ␳ 0 ␳ s ␳ v where ␳ 0 is the Fresnel reflection coefficient, ␳ s is the specular scattering factor, and ␳ v is a vegetation factor, depending on the presence of vegetation on the land surface. The Fresnel reflection coefficient ␳ 0 does approach −1 at low grazing angles. The minus sign in (7.1) implies that the phase angle of the reflection coefficient is exactly ␲ , which is true only for very low grazing angles. The vegetation greatly weakens the specular reflection even for microwave frequencies. So, at the S-band, the vegetation factor for grass with height about 10 cm and for grazing angle of 2° equals about 0.8, and for height of grass about 50 cm, it did not depends on the grazing angle and equaled 0.1–0.4 [4]. At a frequency of 35 GHz for field with short grass, the vegetation factor ␳ v was about 0.5, and at frequencies of 98 GHz and 140 GHz, its value was 0.17–0.24 [4]. The specular scattering coefficient ␳ s is

289

7.1 Influence of Propagation Effects on MMW Radar Operation

冋冉

␳ s2 = exp −

4␲␴ h sin ␺ ␭

冊册 2

(7.2)

The derivation of | V | as a function of range shows that a multipath structure in field strength as a function of height appears at relatively long ranges because of increase in ␳ s resulting from the reduced grazing angle ␺ . This structure is reduced for ␳ s ≤ 0.3—that is, for ranges less than rs ≅

␴ h ⭈ (h r + h t ) 0.12␭

(7.3)

where r s values for radar and target heights of 6m and 2m and ␴ h = 0.25m and 0.1m, shown in Table 7.1. It is seen that for minimal target heights, the multipath structure of the electromagnetic field is practically destroyed in the shortwave part of the MMW band for ranges less than 1.5–3.0 km. This permits us to neglect the multipath losses at these ranges, while at the X- and Ka-bands the multipath attenuation must be taken into consideration. In the dual-path propagation assumption, one can determine propagation factor values for available values of the specular scattering coefficient. The probability that the propagation factor is less than some value V is determined as 1 + | ␳s | − V 2 1 T (V ) = arccos x 2| ␳s | 2

(7.4)

For dual-path propagation, we obtain greater probabilities of deep multipath attenuations than for natural terrain. This is because for the natural terrain the amplitude pdf is closer to Rician, and, besides, it is necessary to take into consideration the scattered electromagnetic field attenuation by the antenna pattern. The multipath attenuation for real terrain paths is different for smooth and broken terrain. For broken terrain, the experimental data are the following [4]: at Ka-band, the electromagnetic field in the interference minimum is more than 7 dB below that for free space, while at W-band the difference is less than 6 dB. These

Table 7.1 The Ranges in Kilometers Within Which Specular Reflection Is Destroyed Frequency (GHz) 10.0 37.5 95.0 140.0

h r = h t = 6m ␴ h = 0.1m ␴ h = 0.25m 0.33 0.83 1.24 3.12 3.51 8.8 5.18 12.9

h r = h t = 2m ␴ h = 0.1m ␴ h = 0.25m 0.11 0.27 0.41 1.03 1.17 2.92 1.72 4.31

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Clutter Rejection in MMW Radar

values are smaller than ones derived from (7.1). The smooth surface is rather close to a plane over comparatively small areas (dimensions less than several hundred meters). For larger areas, a very gently sloping roughness influence becomes noticeable. This leads to an increase in grazing angle in comparison with that derived using the assumption of a plane surface for all paths, resulting in decreased ␳ s . The specular scattering coefficients and derived multipath attenuation factors that are less than these values 90% of the time are presented in Table 7.2. It is necessary to note that the data of V in Table 7.2 are, on average, less than those obtained from (7.1). Therefore, the derivative data with use of (7.1) for radar detection range estimations leads to some increase of multipath attenuation.

7.2 Influence of Rain and Multipath Attenuation on Radar Range Let us evaluate the influence of the effects discussed earlier on use of MMW radar for low-altitude target detection by land-based systems with antennas a few meters above the surface. It is worthwhile to compare radars in different bands for the same two conditions used in Section 7.1: the antenna aperture (and antenna area S A ) is constant with frequency change, and the antenna gain G A is constant with frequency change (i.e., an inversely proportional change of antenna aperture dimensions takes place). For S A = const we consider the dependence of the path loss coefficient A on frequency [5] A=

Pr

= 2

P t ␴ t SA

V4 4␲ 4

冉冊 c f

2

⭈ 10−0.2␥ r

(7.5)

where ␥ is the attenuation factor in precipitation. Here P r is the receiver power, P t is the transmitter power, ␴ t is the target RCS. Practically, this coefficient determines the energy potential of the radar in conditions of multipath and rain attenuation for constant S A . The dependence of A on frequency in rain and multipath conditions is presented in Figure 7.1. While for light 2

Table 7.2 The Scattering Coefficient and Multipath Attenuation V0.9 for Different Paths and Frequencies (h r = 4m, h t = 4–6m) Path Type Frequency (GHz) ␳s

Paths with Vegetation and Arable Land 35 95 140 0.6 0.2 0.1

Paths with Smooth Surface (Snow, Ice, and Sand) 35 95 140 0.8 0.6 0.4

V0.9 , dB

−6.6

−9.3

2

−1.8

−0.9

−6.6

−4.0

7.2 Influence of Rain and Multipath Attenuation on Radar Range

291

Figure 7.1 Coefficient A versus frequency for h r = h t = 6m, and for rainfall rates of 1 mm/hr (solid lines) and 4 mm/hr (dashed lines); ␴ h = 0.1m (curves 1, 2, 5, and 6), ␴ h = 0.25m (curves 3, 4, 7, and 8).

rains (I = 1 mm/hr), the shortwave part of the MMW band is preferable, for moderate rain the use of frequencies above 50 GHz is not beneficial. The performance of the MMW band compared to the X-band for constant antenna aperture can be done using the factor Cs =

冉 冊 冉 冊 Vf V 10

4

⭈

10 f

−2

⭈ 10−0.2r (␥ f − ␥ 10 )

(7.6)

here Vf , V 10 are the multipath attenuation factors and ␥ f , ␥ 10 are the attenuation coefficients at frequencies f and 10 GHz. The examples of derived values of C s as a function of range r are shown in Figure 7.2 (for ␴ h = 0.1m and h r = h t = 6m). It is seen that for ranges of 2–3 km, the MMW band has visible advantages over the X-band, especially for smooth paths. For ranges of about 5 km, rain attenuation is the prevailing factor, which is why the advantage of the shortwave part of the MMW band is seen only for light rain, while for moderate rain the Ka-band has insignificant advantage with respect to the X-band and shortwave part of MMW band. For light rains, the MMW band is more effective than the X-band, and for moderate rains only (I ≥ 4 mm/hr) the advantages of MMW appear for ranges less than 3.0–3.5 km. The estimation of MMW efficiency the for second case (when G A = const) can be expressed as

292

Clutter Rejection in MMW Radar

Figure 7.2 Factor C S versus range at frequencies of 35 GHz (solid lines) and 95 GHz (dashed lines) for rainfall rates of 1 mm/hr (curves 1 and 3) and 4 mm/hr (curves 2 and 4).

B=

V

Pr 2

P t GA ␴ t

=

4

冉冊 c f

2

(4␲ )3r 4

⭈ 10−0.2␥ r

(7.7)

The derivation results for ␴ h = 0.1m and h r = h t = 6m are shown in Figure 7.3(a). The comparative estimation of MMW-band advantages with respect to the X-band for this case are given by the factor CG =

冉 冊冉 冊 Vf V 10

4

10 f

−2

⭈ 10−0.2r (␥ f − ␥ 10 )

(7.8)

The derived dependences C G = f (r ) for ␴ h = 0.1m and h r = h t = 6m are shown in Figure 7.3(b). As seen from Figure 7.3, the longwave part of the MMW band has advantages in comparison to the X-band radar at ranges less than 1.5 km. In all conditions, the MMW band is less effective than the X-band at ranges greater than 3–3.5 km, and at smaller ranges only the frequency band 20.0–50.0 GHz is somewhat more effective than the X-band.

7.3 Influence of Land and Rain Clutter on Radar Detection Range The second basis for frequency choice in radars for low-altitude, land-based target detection is the land clutter. As is well known, the total land clutter RCS for pulsed

293

7.3 Influence of Land and Rain Clutter on Radar Detection Range

Figure 7.3 Factors (a) B versus frequency for rainfall rates 1 mm/hr (solid lines) and 4 mm/hr (dashed lines) and for ␴ h = 0.1m (curves 1, 2, 5, and 6) and ␴ h = 0.25m (curves 3, 4, 7, and 8), and (b) C G versus range at frequencies of 35 GHz (solid lines) and 95 GHz (dashed lines) for rainfall rates of 1 mm/hr (curves 1 and 3) and 4 mm/hr (curves 2 and 4).

radar (the term pulsed radar includes systems using pulse compression) can be determined as

␴ cl ≅

c␶ 0 r␪ 0 ␴ 0 ( ␺ , f ) 2

(7.9)

where ␶ 0 is the processed pulse duration, ␪ 0 is the azimuth beamwidth, and ␴ 0 is the normalized RCS of land. The ␴ 0 values can be determined, for example, from the model for different terrain types presented in Chapter 2. According to that model, the normalized RCS is a function of grazing angle and radar frequency only. All various land territories are classified into eight general terrain types. The coefficient values for different terrain types are shown in Table 2.13. For two limiting cases discussed earlier, we obtain the total RCS of land clutter as

冉冊 冉冊

␴ cl ≅ 0.03A 1 9 ␴ cl = A 1 ␲

9 ␲

A2

A2

⭈

冉 冊 冉 冊

f c␶ 0 h A 2 ⭈ ⭈ 2 L␪ 10

c␶ f ⭈ 0 ⭈ ␪ 0 ⭈ h A2 ⭈ 2 10

A2 − 1

A3

⭈ r 1 − A2

⭈ r 1 − A2

for S A = const (7.10) for G A = const

294

Clutter Rejection in MMW Radar

where L ␪ is the antenna aperture size in horizontal plane in meters, f is the frequency in gigahertz, ␪ 0 is the beamwidth in radians, and h r and r are the radar antenna height and range in meters, respectively. For different terrain types we have different dependences of total RCS on range: •

•

•

For quasi-smooth surfaces (e.g., concrete or surfaces with snow), RCS decreases rapidly with increasing range, so, for range change from 0.2 km to 5.0 km, the clutter RCS decreases by about 30 dB for concrete and 20–22 dB for surfaces with snow; For surfaces with vegetation (e.g., forest and grass), the clutter RCS does not practically depend on range (i.e., ␴ ≈ const); For country and town areas, a small increase of clutter RCS takes place for increasing range; its growth is about 7 dB for range increasing from 0.2 km to 5.0 km.

The frequency dependences of land clutter total RCS are different for two limiting cases of antenna size choice: •

•

For G A = const and at frequencies from 10.0 GHz to 100 GHz, the clutter total RCS increases with increasing frequency by up to 20.0 dB for quasismooth surfaces and 3.0 dB for urban terrain; For S A = const, the total RCS of quasi-smooth surfaces increases with increasing frequency by up to 10.0 dB for concrete; at the same time the RCS decreases for rough surfaces. Its decrease is 7.0 dB for urban areas and 4.0 dB for terrain with vegetation.

As an example, the RCS dependences on frequency are presented in Figure 7.4 for G A = const (curves 1 and 2) and for S A = const (curves 3 and 4) and for two radar heights (h r = 2.0m and 6.0m). At the right side of Figure 7.4, the minimal detectable target RCSs are presented for conditions that detection takes the place for single pulse with detection probability D = 0.9 and false alarm probability F = 10−3. Besides land clutter, the volumetric precipitation clutter is important in the MMW band. It is characterized by a normalized volumetric RCS ␩ that is a function of radar operation frequency and rainfall rate I given in millimeters per hour. As seen from Chapter 5, its value can be presented as

␩ = A ⭈ I␤

(7.11)

The A and ␤ dependences on frequency in the band 10.0–200.0 GHz are shown, for example, in Figure 5.3(b).

295

7.3 Influence of Land and Rain Clutter on Radar Detection Range

Figure 7.4 The clutter total RCS and minimal detectable target RCS (right axis) of land terrain with vegetation versus frequency for G A = const and S A = const and for h r = 2m and 6m.

The total RCS of rain clutter can be represented as

␴ cr =

冦

c␶ 0 2 r ⭈ ␪0 ⭈ ␾0 ⭈ ␩( f, I ) 2 0.09

c␶ 0 1 r 2 ⭈ f −2 ⭈ ␩ ( f , I ) ⭈ 2 L␪ L ␾

for G A = const (7.12) for S A = const

Here ␸ 0 is the beamwidth in the vertical plane (in radians), L ␸ is the antenna aperture size in the vertical plane (in meters), while ␪ 0 and L ␪ are the corresponding values in the horizontal plane. As we can see from (7.12), the total RCS of rain clutter rises quickly with increasing range. The dependences ␴ cr versus range are shown in Figure 7.5. They are derived for rainfall rates of 1.0 mm/hr, 4.0 mm/hr, and 10.0 mm/hr and for two frequencies—Figure 7.5(a) for G A = const and Figure 7.5(b) for S A = const. At 40.0 GHz and a rainfall rate of 4 mm/hr, the total RCS can be 1 m2 for a range of 5.0 km, exceeding the rain RCS at 10.0 GHz by 17–18 dB. The frequency dependence of the rain RCS has different forms for the two limiting cases. For G A = const, the RCS rises with increasing frequency, but this growth is slowed at frequencies above 40.0–50.0 GHz, as is seen from Figure 7.6(a). For S A = const, the total RCS is reduced at frequencies above 20.0–30.0 GHz, and in the short part of the MMW band (above 100 GHz), its values are

296

Clutter Rejection in MMW Radar

Figure 7.5 The clutter total RCS and minimal detectable target RCS (right axis) versus range at frequencies of 10.0 (solid lines) and 40.0 GHz (dashed lines) for rainfall rates of 1 mm/hr, 4 mm/hr, and 10 mm/hr for (a) G A = const and (b) S A = const.

greater by 4–10 dB than at 10.0 GHz, as seen in Figure 7.6(b). The aggregate effect of land and rain clutter leads to increasing the total clutter RCS in comparison with that at the X-band at ranges beyond 1.0 km for moderate rain (with intensity of I ≤ 4 mm/hr), even for cases of constant antenna aperture. This makes worse the clutter input power in mobile target indication (MTI) systems of MMW radars. Thus, for fine-weather conditions, the MMW radars with S A = const have smaller levels of land clutter than X-band radars. However, rain clutter is dominant for light rains (with intensity of about 1 mm/hr) at ranges beyond 1.0–2.0 km. This makes noticeably worse the clutter problem in MMW band radar systems in comparison with analogous systems at the X-band. Taking into consideration the multipath and rain attenuation and the land and rain clutter, MMW radar is better when applied to land-based and low-altitude

7.4 Land and Rain Clutter Rejection in Millimeter Band Radar

297

Figure 7.6 The clutter total RCS and minimal detectable target RCS (right axis) versus frequency (a) at ranges of 5.0 km (solid lines) and 2.0 km (dashed lines) for G A = const; (b) at range of 5.0 km for S A = const.

target detection at ranges less than 2.0–3.0 km. The application of microwave radar is preferable for all-weather conditions at greater ranges.

7.4 Land and Rain Clutter Rejection in Millimeter Band Radar 7.4.1 General Notes

One of the demands on radar for detection and tracking of low-altitude targets is high land clutter rejection. The main source of interference for the majority of radars operating over the land or sea surface is the clutter caused by backscattering from the surface, because the clutter power is significantly greater than the receiver noise power. The clutter echoes from a rough sea and from some types of terrain have many characteristics like those of thermal noise; they are randomly fluctuating in both amplitude and phase. But the spectrum is often much narrower than that of white or quasi-white noise (i.e., the clutter can be correlated, either partially or nearly totally, for times of the order of the typical period of signal integration).

298

Clutter Rejection in MMW Radar

The use of the millimeter band for such radars leads to a need to obtain precipitation clutter rejection, too. The use of stationary random processes with Gaussian pdf and the power spectra of white noise type as clutter models is a significant limitation for most papers in which the statistical theory of target detection has been developed [6–8]. This model is properly applied, as a rule, only for target detection in receiver noise, and the calculations of detection characteristics for this case are developed in detail [9]. As is well known [10], the qualitative indexes of optimal detection in the Gaussian noise don’t depend on the signal waveform but only on its energy relative to the spectral density of noise power. For extended correlated clutter, the signalto-noise ratio is determined by the ambiguity function of the radiated signal. It is possible to develop different techniques for modulation spectrum design; the signalto-clutter ratios (SCRs) are different for different signals. In this chapter, the land and rain clutter rejection is estimated for some types of radiated signals that are widely used in radar systems in centimeter and millimeter bands [11–13]. Among the most often used signals, one can note the periodic uncoded pulse sequence, ensuring range resolution and velocity indication of moving targets, pulsed-compression signals exemplified by pulsed signals with linear frequency modulated or phase-coded pulsed sequences, unmodulated continuous signals, and continuous signals with sinusoidal frequency modulation. 7.4.2 Land and Sea Clutter Rejection

Let us estimate the radar signal modulation required for high rejection of land and sea clutter. Clutter rejection is determined, first of all, by the characteristics of land and sea clutter. An empirical land model for normalized RCS is developed in Chapter 2, obtained from the experimental investigations in bands from 3 GHz to 100 GHz for grazing angles less than 45°. The empirical model for sea clutter is developed in Chapter 4; this model takes into account the scattering from spray formed by sea wave breaking and propagation in the boundary layer of the atmosphere with the enhanced refractivity over the sea. These models are used for clutter rejection estimation. It was noted earlier that the SCR depends on the transmitted signal ambiguity function. For ambiguity functions produced by a short individual pulse, with wide spectrum, the shift in Doppler frequency between target and clutter scattered signal is insignificant. Then the target velocities are usually ambiguous, and only the ranges of interest are unambiguous. The SCR for pulsed signals of this type without intrapulse phase modulation is q=

2␴ t ␴t ≈ ␴ c c␶ ␴ 0(r ) r␪ 0

(7.13)

7.4 Land and Rain Clutter Rejection in Millimeter Band Radar

299

where ␴ t is the target RCS, ␴ c is the clutter RCS, ␴ 0(r ) is the normalized clutter RCS as function of range r, ␶ is the pulse duration, and ␪ 0 is the azimuthal antenna pattern width at the level of −3 dB. For typical radar parameters of ␶ = 0.05–0.2 ␮ s, ␪ 0 = 1°, target RCS ␴ t = 1 m2, ␴ 0 = −30 dB, and r = 5 km, the SCR determined from (7.1) is equal to −(8 to 14) dB. High target detection probability in these conditions cannot be ensured. Velocity and high range resolution can be obtained using transmitted signals with line spectra. These periodic signals with arbitrary interperiod modulation have range resolution, and in the most of cases it is impossible to obtain unambiguous velocity and range indication simultaneously. One can form a signal with wide separation between spectral lines to provide unambiguous range determination [10]. Such signals, formed by n continuous sinusoidal oscillations, suffer from clutter accumulation from all of detection ranges. For pulsed sequences, the estimates of land and sea clutter rejection were done in [11, 12]. The main conclusions of this analysis are the following. For uncoded periodic pulsed signals, the SCR is determined as q=

␴t c␶ ␪ r 2 0

∞

∑

l =0

␴ 0 (r ) [1 − (l 0 − l ) cTr /2]

(7.14)

where Tr is the pulse repetition period and l 0 = [2r /cTr ] is the integer part of the range to unambiguous interval ratio, l = 0, 1, 2, . . . It is enough to limit oneself by value of l max = [2r max /cT ] for summing in the denominator of (7.14) because the clutter power contribution from the surface cells at ranges exceeding the maximal range of the target r max is significantly less than the backscattering from the surface cell under the target and from cells at shorter ranges. For unambiguous target range determination (r max ≤ cTr /2 and l 0 = 0), the clutter power is a sum of backscattering from the surface cell under the target and from the cells situated at ranges greater than the radar maximal range. Taking into account that the clutter power decreases with the range increase proportionally to r −3 for land and to r −7 for the sea, the calculation of clutter from ranges of r > r max is usually not necessary. Range ambiguities appear for high pulse repetition frequencies (PRFs) that are often necessary for effective clutter rejection by MTI systems. In this case, the clutter power is a sum of backscattering from the radar cell containing the target and from the closer cells. As a result, the SCR degrades and for ranges close to r = lcTr /2 the target observation is impossible due to transmitter leakage and receiver saturation by transmitted pulses. Upon first consideration, it may appear that shortening the pulse duration would increase SCR; however, it is necessary to consider spatial clutter spikes for

300

Clutter Rejection in MMW Radar

which the RCS can reach large amplitudes. For this spatial-temporal structure, the SCR is better than predicted by (7.14) at some ranges, while for other ranges this ratio is worse. The use of wideband signals permits an increase in clutter rejection. In particular, for pulses with linear frequency modulated or phase-coded signals, the gain in SCR compared with an uncoded pulsed sequence using the same transmitted pulse width is K = q w /q , where K = ␶ ⌬f is the compression ratio, ⌬f is the spectrum bandwidth, and q w is the SCR for wideband signals. If the medium contains a large number of scatterers, the resulting signal has random noiselike characteristics. In addition, if the range to target is large, it is possible to ignore the dependence of clutter level on range when calculating the performance of the pulse compression waveform. Then the resulting signal can be presented as a stationary and Gaussian random process for which the results of works [8–11] can be applied. In this case, the resulting clutter power density per resolution cell is decreased when the range resolution of radar increases. The advantage of range resolution is limited by the target dimensions as well as the change of the clutter statistical characteristics that leads to an increasing of the false alarm level. For signals with equal range resolution, the opposite situation is observed. The normalized RCS of clutter is practically constant in the radar resolution cell at long ranges for uncoded pulsed signals of short duration. For a wideband signal with equal range resolution ⌬r 0 but longer transmitted pulse width ␶ the cell sizes corresponding to the transmitted and processed pulse widths are determined by

⌬r =

Kc␶ 0 = K ⌬r 0 2

(7.15)

where ⌬r is the cell size corresponding to the transmitted pulse width ␶ and ⌬r 0 is the cell size of the processed pulse width ␶ 0 . Because K = ␶ ⌬f = ␶ /␶ 0 , it is clear that ␶ 0 and ⌬r 0 must refer to the resolution of the processed output, while ␶ = K␶ 0 and ⌬r = K ⌬r must refer to the longer transmitted pulse. When the radar energy potential is increased by increasing the transmitted pulse width without decreasing the range resolution (i.e., when the wideband transmitted pulse width is equal to ␶ = K␶ 0 , where ␶ 0 is the processed pulse width), and when ␶ c /2 becomes a significant fraction of the target range r, the clutter normalized RCS will not be constant in the bounds of the transmitted pulse. In this case the use of wideband, long-pulse signals leads to decreased SCR. For two dependencies of the normalized RCS on range, typical for grazing angles [␴ 0 ∼ r −4 for sea because for sea clutter the propagation factor starts to vary directly with grazing angle if the grazing angle is below the critical one, and ␴ 0 ∼ r −1 for land according to (7.10)], the clutter power is [12]

301

7.4 Land and Rain Clutter Rejection in Millimeter Band Radar

Pc =

冦

16 12 + ␤ 2 AK 3 (4 − ␤ 2 )3

256AK

冉

for land

冊

4 + ␤ 2 (12 + ␤ 2 ) 3 (4 + ␤ 2 )6

(7.16) for sea

Here A = Pt

G 2␭ 2␪ 0 ␴ 0(r )⌬r 0

(7.17)

(4␲ r )3

is the clutter power in a radar resolution cell at range of r for an uncoded pulsed signal with duration of ␶ 0 , ␤ = ⌬r /r = K ⌬r 0 /r is the ratio of transmitted pulse width to range delay. For the thumbtack ambiguity function with residue level ␦ , the clutter is distributed uniformly on the ‘‘range-velocity’’ plane. This reduces the signal-clutter ratio in comparison to that for an uncoded pulsed sequence. The losses for this case are

L=

q pc = q

冦

16 12 + ␤ 2 AK 3 (4 − ␤ 2 )3

256AK

冉

冊

4 + ␤ 2 (12 + ␤ 2 ) 3 (4 + ␤ 2 )6

for land (7.18) for sea

The dependence of SCR losses for pulse-compression signals as a function of ␤ , the relative transmitted pulse width, is shown in Figure 7.7. It is apparent from these results that the transmitted pulse width should be restricted to a small fraction of the target time delay, ␤ << 1. For residue level determined as ␦ = K −1 (this is typical for noise and noise-like signals), the losses are significant when the radar cell dimensions are compared with the range to target r 0 . If the residue level of is ␦ = K −1/2, then the SCR is considerably worse than that for uncoded pulsed sequences at all values of ␤ . Thus, the use of extended transmitter pulses with pulse compression in radar operating over land and sea surfaces does not improve the clutter rejection in comparison with an uncoded pulsed sequence of equal range resolution, and it may significantly increase the input clutter, especially for short-range radar. This obstacle limits the use of long pulses with pulse compression for increasing of the target detection range in clutter when the transmitted power is limited.

302

Clutter Rejection in MMW Radar

Figure 7.7 SCR losses as function of relative pulse width.

Let us discuss the clutter rejection for a CW signal without modulation. The clutter power at the receiver input is determined for CW radar by integration along the entire surface within the beam. As is shown in [14], for conditions of the normalized RCS constancy (␴ 0 = const) and the radar antenna pattern approximated by G (␸ , ␪ ) = G 0

冉

sin ␲␪ /␪ 0 ␲␪ /␪ 0

冊冉 2

sin ␲␸ /␸ 0 ␲␸ /␸ 0

冊

2

(7.19)

where ␸ 0 , ␪ 0 are the −3-dB beamwidths in the elevation and azimuth. The SCR at the output of a Doppler bandpass filter with rectangular frequency response, lower cutoff frequency F 1 , and upper cutoff frequency F 2 , and for horizontal elevation beam axis is 2

q con =

F1 F2 ␴ t h r 2␲ 2 ln 2 ⭈ ⭈ 0 4 2 ⌬F (F 2 − F 1 ) ␴ r ␪0␸0

(7.20)

where h r is the radar antenna height and ⌬F is the clutter spectrum width. As an illustration, the dependence of q con on the range r is shown in Figure 7.8 for h r = 3m, ␴ t = 0.02 m2, ␴ 0 = −25 dB, and ␸ 0 = ␪ 0 = 1°. The lower cutoff

303

7.4 Land and Rain Clutter Rejection in Millimeter Band Radar

Figure 7.8 The dependence of SCR on range for CW, frequency-modulated CW, and pulsed signals.

frequency of the filter corresponds to the target velocity of 150 m/s and the filter bandpass width is equal to F 2 − F 1 = 1 kHz. For comparison, the dependence of q (r ) is presented in the same figure (curve 3) for coherent pulsed radar with pulse width of ␶ 0 = 0.2 ␮ s and an MTI filter. The target and clutter parameters are the same as for the CW radar. It is seen that the SCR exceeds unity only at ranges less than several hundred meters for the CW signal, and the reliable detection of targets with small RCS can be ensured only at ranges less than 500m. This ratio equals to −(40–60) dB at ranges of 3–4 km, which determines the requirements on land and sea clutter rejection in MTI systems. When we use CW radar, the surface areas near the radar create the main contribution to clutter. A considerable decrease of this contribution is provided by application of a sinusoidal frequency modulated continuous signal. This results from the appearance of a number of harmonics at the radar input because of sinusoidal modulation. Each of them is modulated by a signal of Doppler frequency proportional to [15]

冉

2␲ r ⌬f I n (r ) = I n sin Fm ␭m where ⌬f is the deviation of the modulation frequency. F m is the modulation frequency. ␭ m = c /F m is the modulation wavelength. I n is a Bessel function of order n.

冊

(7.21)

304

Clutter Rejection in MMW Radar

For use of third harmonics of the modulation frequency and for the antenna pattern of (7.19), it is easy to obtain the clutter power at the receiver input

冉

P G 2␭ 2␴ 0␪ 0 2␲ r 2 ⌬f dPrc = t ⭈ I3 sin 3 4 F ␭m m (4␲ ) r

冊

⭈ r dr d␪

(7.22)

One can determine the full power of clutter using the condition for small h grazing angles ␸ ≈ r r ∞

2 2

0

P G ␭ ␴ ␪0 Prc = t ⭈ 2(4␲ )3

sin4

␲ hr ␸0r

2

I3

冮冉 冊 hr

␲ hr ␸0r

⭈ 4

冉

2␲ r ⌬f sin Fm ␭m r3

冊

dr

(7.23)

Let us divide the integrand in (7.23) into two parts, choosing some range R 0 as a boundary so that the maximal power of clutter from the ranges r < R 0 , where R 0 = h r /␸ 0 , is P G 2␭ 2␴ 0␪ 0 ␲ 6 max Prc = t ⭈ ⭈ 12 (4␲ )3

冉 冊冉 冊 6

⌬f Fm

5

␸0 ␲ hr

8

R0 6

␭m

if r < R 0

(7.24)

and P G 2␭ 2␴ 0␪ 0 0.32 Prc ≅ t ⭈ ␭m R0 (4␲ )3

if r > R 0

(7.25)

The ratio of the clutter power from the near and far zones is determined from (7.24) and (7.25) as Prcnear ␲6 ≅ ⭈ Prcfar 3.84

冉 冊冉 冊 ⌬f Fm

4

␸0 ␲ hr

4

9

R0 5

␭m

(7.26)

The estimations show that for antennas with narrow elevation beamwidths, this ratio is considerably less than one. For ␭ m = 6 ⭈ 104 m, ⌬f /F m = 4, ␸ 0 = 10−2, and h r = 10m, it is approximately 3 ⭈ 10−4. Therefore, the clutter from the near zone is considerably attenuated. For this case the SCR is

305

7.4 Land and Rain Clutter Rejection in Millimeter Band Radar

q FM =

3.12␴ t ␭ m h r

冉

2␲ r 2 ⌬f ⭈ I3 sin Fm ␭m ␴ r ␪0␸0 0 4

冊

(7.27)

The dependence of q FM on the range for the same radar parameters as for the continuous signal and for ␭ m = 20 km and ⌬f /F m = 4 is shown in Figure 7.8 (curve 2). The analysis of these results shows that, first of all, frequency modulated CW radar obtains higher SCRs in comparison with unmodulated CW at ranges greater than 1.5–3.0 km. The gain reaches 15–17 dB and is greater for broader beamwidths and for lower radar heights. Really, PrcFM 5h r ≅ Prc ␭m ␸0

(7.28)

The use of frequency modulated CW reduces the dynamic range of target signals at the expense of their attenuation by the modulating function when the target range is decreased. 7.4.3 Rain Clutter Rejection

Let us estimate the precipitation clutter rejection for transmitted signals of the types considered earlier. The SCR for an uncoded pulsed sequence and for c␶ /2 << r can be represented, as for land and sea clutter (7.14), by q=

␴ t exp (−2␥r ) ∞ c␶ exp (−2␥r ) ␩␪0␸0r2 ∑ 2 [1 − (l 0 − l )cT r /2r ]

(7.29)

l =0

where ␥ is the precipitation attenuation coefficient and ␩ is the precipitation normalized volumetric RCS. It is necessary to have in mind from the summation in the denominator of (7.29) that SCR decreases quickly as target range increases. For the determination of unambiguous target range (r max ≤ cT /2, l 0 = 0) the contribution of the reflecting areas at ranges r > r max is commensurable with the clutter power from the pulsed volume containing the target. With range ambiguity, the backscattering from the areas closer to the target and the backscattering from the volume containing the target add. This results in a significant reduction in the SCR. The quantitative presentation of the dependence q on range for rain intensities of 1 mm/hr and 4 mm/hr is given in Figure 7.9. Curves 1 and 2 correspond to the regime of unambiguous target range determination for r max = 3 km, curve 3 corresponds to the regime of ambiguous target range determination with additional scattered power from the volume closer to the target

306

Clutter Rejection in MMW Radar

Figure 7.9 The SCR dependence on range for pulsed signal and for (1) I = 1 mm/hr and (2–4) I = 4 mm/hr (␴ t = 1 m2, ␶ 0 = 0.1 ␮ s, ␸ 0 = ␪ 0 = 10 mrad, V = 1).

(l = 1), and curve 4 takes into account the backscattering from volumes closer than the target as well as from volumes beyond the target. The dependences of q on the radar frequency for pulsed signals, shown in Figure 7.10, permit us to reach some conclusions. First, transition to the millimeter band is accompanied by considerable decrease of the SCR in comparison with values obtained in the X-band, amounting to 20 dB at 35 GHz. This decrease is most visible in rains with intensity less than 4 mm/hr. The value of q decreases insignificantly in the shortwave part of the millimeter band ( f = 95 GHz). For rain intensities less than 4 mm/hr, these SCR decreases are less than 6 dB in comparison with values at 35 GHz. For rain intensities more than 16 mm/hr, the SCR does not change over all MMW bands. For the unmodulated CW, the main contribution to the clutter power at the receiver input is from areas near the radar. Let us discuss this problem in more detail, taking for the boundary between the near and far zones the range

r=

ka 2 2

(7.30)

where k = 2␲ /␭ is the wave number and a is the radius of the antenna aperture.

307

7.4 Land and Rain Clutter Rejection in Millimeter Band Radar

Figure 7.10

The SCR dependence on radar frequency for pulsed signal (␴ t = 1 m2, ␶ 0 = 0.1 ␮ s, ␸ 0 = ␪ 0 = 10 mrad, r = 5 km, V = 1): (1) I = 1 mm/hr, (2) I = 4 mm/hr, (3) I = 16 mm/hr.

ka 2 ≥ 1, the antenna 2r gain G depends on the coordinates of the observation point and, in to a first approximation, is determined by For reflector antennas in the near zone, defined by ␤ =

G≈

冋

2(ka )2 (ka )2 sin2 ␪ 1− 1+␤ 4

册

(7.31)

where ␪ is the angle between the antenna electrical axis and the direction to the observation point. The expression (7.31) is valid only near the antenna electrical axis where the condition

冉

ka sin ␪ 2

冊

2

<< 1

is fulfilled, for an exponential illumination of the aperture, and for ka >> 1. Then the rain clutter power from the near zone with spatial homogeneity is P ␭ 2␩ Pcnear = t 3 (4␲ )

冕 V

P t ␭ 2␩ dV = r4 (4␲ )3

G

冕 V

2(ka )2

冋 冉

1− 1 + ␤2

ka sin ␪ 2

冊册 2

dV r4

where the volume element dV in the spherical system of coordinates is

(7.32)

308

Clutter Rejection in MMW Radar

dV = r 2 sin ␪ ⭈ dr ⭈ d␪ ⭈ d␸ Let us divide the space on two areas of integration (see Figure 7.11). The clutter power from area 1 is Pcnear 1 ≈

5 P t ␭ 2␩ k (ka )2 ⭈ 64 (4␲ )2

(7.33)

␲ P t ␭ 2␩ k 2a ⭈ 10 (4␲ )2

(7.34)

and from area 2 is Pcnear 2 ≈

The clutter power from the far zone for identical antenna patterns in both planes is 2

Pcfar =

P t ␭ 2G 0 ␩ ␪ 0 (4␲ )3

ka 2

␪0 = ␸0

,

(7.35)

and the ratio of clutter from the near zone to the total power is

␣=

Pcnear = Pc ⌺

1

冉冊

␭ 1 + 0.48 a

2

=

1 2

1 + 0.48␪ 0

(7.36)

(i.e., it is inversely proportional to the square of beamwidth). Then for narrow directional antennas (␣ ≈ 1), the total scattered power is practically determined by the near zone of the antenna (for ␪ 0 = 30 mrad ␣ = 99.9%).

Figure 7.11

The areas of integration.

7.4 Land and Rain Clutter Rejection in Millimeter Band Radar

309

The ratio of signal to rain clutter is P 8G 0 ␭␴ t 25.6a 2␴ t q cont = s ≅ = P c 5 ␲ 2␩ r 4 ␲␭␩ r 4

(7.37)

It is seen from analysis of (7.37) that in the millimeter band, values of q cont do not practically change with wavelength shortening for the condition a = const (constant aperture area) because the wavelength decrease is balanced by increase in the normalized volumetric RCS. For proportional decrease of the antenna aperture with wavelength shortening and for the condition of G o = const, the SCR decreases in conformity with the character of ␩ (␭ ) dependence. The dependences of q on range for different rain intensities and for the condition D A = 2a = 0.6m are shown in Figure 7.12. It is seen that the SCR differs by less than 3–5 dB between 35 (curve 2) and 75 GHz (curve 3). One method of rejecting rain clutter from the near zone is the use of frequency modulated CW. When using the first harmonic of the modulating frequency we find 4 2 2 P ␭ 2␩ (ka ) a ␹ PnearFM ≈ t 2 ⭈ 512 (4␲ )

(7.38)

where ␹ = m␻ m /c, ␻ m = 2␲ F m , m is the modulation index.

Figure 7.12

The continuous signal SCR dependence on range for rain intensity of 4 mm/hr at wavelengths of (1) 3 cm, (2) 8 mm, (3) 4 mm, and of 1 mm/hr at wavelengths of (4) 3 cm and (5) 8 mm.

310

Clutter Rejection in MMW Radar

It is seen from comparing (7.38) with (7.33) that the use of frequency modulation permits us to considerably attenuate the scattering level from the near zone even for application of the first harmonics in radar operation. The increasing of harmonic number leads to further attenuation of the scattering level. The clutter power from the far zone is 2

PfarFM ≈ ␥

P t ␭ 2G 0 ␪ 0 (4␲ )

3

⭈

␲c 2 ␹ 4␻ m

(7.39)

The signal to clutter ratio is q FM ≈

冉

8 ␴t ␭m ␻ r 2 ⭈ ⭈ I m sin m ␥ (2␲ )2m 2␩ r 4␪ 2 1 c 0

冊

(7.40)

Here ␭ m is the wavelength of modulation. Supposing that the target is at the range corresponding the maximum of the Bessel function, one can simplify (7.40) to q FM ≈

␴t ␭m ␴t 2.8 2.8 ⭈ = ⭈ ␥ (2␲ )2m 2␩ r 4␪ 2 ␥ ␲ m 2␩ r 3␪ 2 0 0

(7.41)

If the antenna aperture is constant with change in wavelength (7.41) can be transformed to the form q FM ≈

a 2␴ t ␭ m ⭈ (2␲ )2␥ m 2␭ 2␩ r 4 11.2

(7.42)

It is seen from (7.42) that for a constant aperture, the SCR increases with the decreased wavelength. As an illustration, the dependences q (r ) are presented in Figure 7.13 for the following radar parameters: ␭ m = 3 ⭈ 103 m, ␴ t = 1 m2, ␪ 0 = 30 mrad. The values of ␩ have been used for rain of the same intensity as in Figure 7.12 for the continuous signal. The result of comparing Figure 7.12 with Figure 7.13 shows that sinusoidal frequency modulation permits us to improve the SCR by 40–50 dB in comparison with the continuous signal. This results from the rain clutter rejection in the near zone. Thus, the analysis of land (sea) and rain clutter rejection in the MMW band for several signal types permits us to make the following conclusions. The use of periodic uncoded pulsed sequences with unambiguous target range is preferable. Otherwise, a considerable decrease of SCR takes place due to backscattering contributions from the ambiguities closer than the target.

311

References

Figure 7.13

The continuous signal with frequency modulation SCR dependence on range (for ␴ t = 1 m2, ␭ m = 3 ⭈ 103 m, m = 1, ␸ 0 = ␪ 0 = 30 mrad) for rain intensities of 4 mm/hr at wavelengths of (1) 3 cm, (2) 8 mm, and (3) 4 mm, and of 1 mm/hr at wavelengths of (4) 3 cm and (5) 8 mm.

The use of complex signals with pulse modulation permits us to increase the clutter rejection if the processed pulse width is less than the uncoded pulse width. Otherwise, the clutter accumulation from the large area or volume illuminated by the transmitted pulse leads to decrease of the SCR and increases the clutter power from the side lobes of the ambiguity function. For CW signals, the clutter power from the near zone increases and frequency modulated CW signals do not provide clutter rejection to levels obtained for the pulsed sequence.

References [1] [2] [3] [4] [5]

Skolnik, M., Radar Handbook, New York: McGraw-Hill, 1990. Barton, D. K., Modern Radar System Analysis, Norwood, MA: Artech House, 1988. Nathanson, F. E., J. P. Reilly, and M. N. Cohen, Radar Design Principles, 2nd ed., New York: McGraw-Hill, 1991. Kulemin, G. P., and V. B. Razskazovsky, Scattering of Millimeter Waves by Earth’s Surface for Small Grazing Angles, Kiev: Haukova Dumka, 1987 (in Russian). Kulemin, G. P., ‘‘Influence of the Propagation Effects on Millimeter Wave Radar Operation,’’ SPIE Conf. Radar Sensor Technology IV, Vol. 3704, Orlando, FL, April 1999, pp. 170–178.

312

Clutter Rejection in MMW Radar [6] [7] [8] [9] [10] [11]

[12] [13]

[14]

[15]

Woodward, P. M., Probability and Information Theory with Applications to Radar, Oxford: Pergamon Press, 1953; Dedham, MA: Artech House, 1980. Cook, C. E., and M. Bernfeld, Radar Signals: An Introduction to Theory and Application, New York: Academic Press, 1967; Norwood, MA: Artech House, 1993. Berkowitz, R. S., (ed.), Modern Radar: Analysis, Evaluation, and System Design, New York: John Wiley, 1965. Marcum, J. I., ‘‘A Statistical Theory of Target Detection by Pulsed Radar,’’ IRE Trans., Vol. IT-6, No. 2, 1960. Bakut, P. A., et al., Problems of Radar Statistical Theory, Moscow, Russia: Soviet Radio, 1963 (in Russian). Kulemin, G. P., ‘‘The Clutter Rejection in Short-Range Radar with Wideband Pulsed Signals,’’ PIERS Workshop on Advances in Radar Methods, Baveno, Italy, July 20–22, 1998, p. 73. Kulemin, G. P., ‘‘Clutter Rejection in Short-Range Radar with Uncoded and Wideband Signals,’’ J. Electromag. Waves and Applications, Vol. 14, No. 2, 2000, pp. 245–260. Kulemin, G. P., ‘‘Land and Rain Clutter Rejection in Millimeter Band Radar with Continuous and Pulsed Signals,’’ SPIE Conf. Radar Sensor Technology VII, Orlando, FL, April 2002, Vol. 4744, 2002 (to be published). Kulemin, G. P., and V. I. Lutsenko, Detection and MTI Features by Near-Range Radar with Use of Some Signal Types, Preprint No. 136, Inst. Radiophys. Electr., Kharkov, Ukraine, 1979, p. 28 (in Russian). Skolnik, M., Introduction to Radar Systems, New York: McGraw-Hill, 1962, p. 100.

About the Author Gennadiy P. Kulemin received a diploma degree in radiotechnic engineering from Kharkov Polytechnic Institute in 1960, a Ph.D. in 1970, and a doctor of science degree in electronic systems in 1987. Dr. Kulemin has been an assistant professor and senior tutor on the Electronic Systems Faculty of the Kharkov Aviation Institute. Since 1966, he has worked at the Institute of Radiophysics and Electronics of the Ukrainian National Academy of Science. Presently, he is a principal scientific researcher within the Millimeter Radar Department and a professor at Kharkov Military University. His main areas of interest are backscattering from targets, land, and sea; radar remote sensing of the Earth in microwave bands; and millimeter band radar systems research. He has been investigating the experimental and theoretical aspects of these problems. Dr. Kulemin is the author of Scattering of Millimeter Radiowaves by the Earth’s Surface at Small Grazing Angles, edited in Russian, and has written more than 200 publications on scattering problems in microwaves and radar efficiency. He is a member of the Academy of Science of Applied Radioelectronics, a member of Commission F of URSI and the Ukrainian National Committee of URSI, and a member of the IEEE.

313

Index A Absolute reflectivity, 283–84 Acoustic waves, 52 Aircraft freedom degree number for, 60 RCS, structure contributions, 12 remote sensing, 157–59 Ambiguity function SCRs and, 298 thumbtack, 301 Angel backscattering, 251 amplitude variance, 253 power spectra, 254 at S-band, 253 Angel-echoes, 250, 251 Angel RCS cumulative distribution, 252 Antenna aperture, 310 Asphalt dielectric constant, 89, 146, 147 spectral density, 147 surface roughness estimates, 146 surface statistical characteristics, 147 Atmosphere, turbulent, 254–56 Attenuation, 233–36 angular dependencies, 40 atmospheric, 233–36 coefficient, 234 data, 235 determination, 233 dust storms, 235–36 factor, 36

factor, dependence, 37 factor, experimental, 40 fog, 235 frequencies, 41 gas, 233–34 maximal, 39 maximal total, 40 microwave, 234 microwave total, 37 multipath, 288–90 rain, 235 temporal dependence, 41 Autocorrelation function, 102, 105 envelope, 105–6 exponential, 144, 145, 148 Fourier transform of, 102, 107 Gaussian surface, 144, 145, 148 of plowed field, 153 of roughness, 153 of scatterer velocities, 106 sea roughness, 181 spatial surface, 186 Azimuthal dependence, 218 B Backscattering angel, 251, 253, 254 of birds/insects, 13 cloud, 242 cloud, spectral parameters, 249 from forest, 130, 132

315

316

Backscattering (continued) from grass, 129, 130, 132 of human body, 12 land, 89–132 microwave, electromagnetic field, 54 from natural turbulence, 54 precipitation, 239–42 rain, 239, 241 sea models, 189–93 snow, 114–18, 239 soil, modeling, 138–45 from sonic perturbations, 41–55 spectra, 213–21, 251 from swamp, 131 from SWF, 54–55 theory, 236–38 from turbulent atmosphere, 254–56 from vegetation, 118–20 Beaufort scale, 172 Biological objects, power spectra, 70 Birds density in flocking places, 17 distribution by altitude, 17 mass, RCS dependence on, 13, 15 power spectra of, 73 RCS of, 16 scattering pattern, 14 velocity of, 71 Bistatic RCS, 80 Boiling surf formation, 177 Bubble bursting, 177 C Canonical decompositions, 271 Chi-square distribution, 57, 63 Clouds, 230–32 backscattering, 242 classes, 230 cumulonimbus, 230 defined, 230 formation, 231 height, 230–31

Index

microphysical characteristics, 231, 232 nonrain, spectral shape, 250 normalized RCS of, 245 physical characteristics, 232 stratus, 230 undulating, 230 water drop size distribution, 231 See also Meteorological formations Clutter maps contour highlighting, 280–82 development, 276–84 image interlacing, 278 initial data, 276–77 inputting/processing data on, 279 raster image processing module, 278–80 software input/processing components, 277–78 for vegetation, 278 Clutter modeling, 259–84 land, 259–67 map development, 276–84 sea, 267–76 Clutter power, 304, 308 for CW signals, 311 from far zone, 310 ratio, 304 Clutter rejection, 287–311 for CW signal, 302 general notes, 297–98 land, 298–305 rain, 305–11 sea, 298–305 Concrete dielectric constants of, 89 normalized RCS for, 110 Cone cylinder targets RCS cumulative functions, 77 RCS distributions, 64 scattering pattern, 8 Continuous-wave (CW) radars, 28 clutter contributions, 303

317

Index

clutter power, 311 clutter rejection for, 302 parameters, 28 SCR on range dependence for, 303 unmodulated, 306 Contours automatic highlighting of, 280–82 color and brightness highlighting, 281 surface recovery from, 282 See also Clutter maps Copolarizations, 201 Correlation factors, 197 Correlation function, 263, 274 Correlation radius estimate, 145 Cross-correlation function, 196 Cross-polarizations components, 221 power spectra, 220 total scattered signal, 201 D Depolarization coefficients, 197–98 coefficients, mean, 198 degree, 197 for horizontal polarization, 200 of scattered signals, 123–26, 197–202 in spikes, 212 weak, 198 Dielectric constants of asphalt, 146, 147 of concrete/asphalt, 89, 146 of corn leaves, 96 dependence on volumetric moisture, 90 dry snow, 92, 93 for grain, 97 normalized RCS and, 111 soil, 91 in sound wave field, 47 water, 237 wet snow, 92

Diffuse scattering coefficient, 77–78 Dirac function, 75, 129 Doppler frequencies, 221 shift, 249 spectrum width determination, 71 Drizzle, 227 Dual-channel polarization, 157 Dust storms, 232–33 attenuation, 235–36 formation, 232 E Echo(s) angel, 250, 251 angular dependencies, 31 duration as function of filter low-band frequency, 32 instantaneous, 33 power spectra, 62–72 power spectrum analysis, 31 power spectrum at different moments of time, 33 power spectrum shape, 32 sea, fluctuations, 83 tails, 32 Effective front width, 46 Effective radar cross section, 46 Electromagnetic field reflection from sound wave package, 49 from SWF, 43 Electromagnetic field scattering, 49 Erosion state classification, 167 situ measurements of, 168 Euler’s constant, 7 Explosion(s) bands, RCS of, 30 experimental attenuation factors, 40 fluctuation intensity reduction, 26–27 gas-like products refractivity, 21 products, 22 radar reflections from, 28–34

318

Explosion(s) (continued) RCS estimation, 23 refractive index fluctuations, 26 spatial-temporal characteristics, 23–28 turbulence local characteristics, 24 volume, 34 volume, dimension measurements, 30 F Facet model, 190–91 small grazing angles and, 190–91 uses, 190 Fast Fourier transform (FFT), 153, 154 Foam bubble structure, 177 dependences, 176 distribution parameters, 176 formations, 176 See also Sea Fog, 232 attenuation, 235 sea, 232 Forests backscattering, 130, 132 experimental values of model parameters for, 98 as homogeneous scatterers, 94 normalized RCS, 119, 122 normalized RCS vs. frequency, 123 RCS, 119 See also Vegetation Fourier-series expansion, 266 Fourier transforms autocorrelation function, 102, 107 fast (FFT), 153, 154 Fresnel coefficients, 100 Friction velocity, 188 G Gaps central frequency, 220

Index

duration distributions, 213 mean duration of, 212 probabilities, 211 spectral width, 220 Gas attenuation, 233–34 Gas-like products, 35 Gas wake fluctuations spectrum, 27 radar observation results, 32 radar reflections from, 28–34 Gaussian curve, 249 distribution, 106, 264 law, 246 random process, 210 surface, autocorrelation function, 144, 145 Green’s theorem, 99 Ground control points (GCPs), 162 H Hail, 229–30 fall zones, 230 formation, 229 normalized RCS of, 245 surface, 229 See also Meteorological formations Helicopters freedom degree number for, 60 power spectra of, 66 Hermittian polynomials, 274 Humans moving, power spectra, 71 power spectra, 70 swimming, power spectra, 72 I Image superimposing, 159–63 Inflatable boats, amplitude distributions, 62

319

Index

J Jet propulsion engines as intensive sound field, 53 sound radiation spectrum, 50 JONSWAP spectral model, 181 K Karhunen-Loeve transform, 266 Karman’s constant, 172 Kirchoff’s method, 4 gently sloping surfaces and, 99 normalized RCS estimations with, 100 Kirchoff’s model, 139 Kolmogorov-Obukhov theory, 255 L Land classification of, 89 clutter rejection, 298–305 forest, 94, 98 grassy terrain, 96 object power spectra, 66–67 objects, RCS of, 9 radar range influence, 292–97 roughness parameters, 90, 92 snow, 92, 93, 94 soil, dielectric constants, 91 surface types, 262 vegetation, 93 Land backscattering, 89–132 power spectra, 128 power spectrum model, 101–8 RCS models, 95–101 scattering surface, 103 simplified models of, 98 Land clutter modeling, 259–67 clutter stochastic process, 263 correlation function, 263 database development, 260 fluctuating component, 263 initial data, 259–62

input data, 261 peculiarities of, 262–67 requirements, 261 simulation algorithms, 265 surface model, 260 See also Clutter modeling Laplace function, 83 Laws-Parsons distribution, 237–38 Linear filtering, 266 Logarithmic Gaussian law, 82 Lognormal law, 246 Long-life reflections, 29 Lorentz-Lorentz relationship, 47 Low-altitude targets, 3 Low-RCS airborne vehicle detection, 2 M Magnetron oscillators, 39 Marine targets oscillations, 69 power index, 70 power spectra, 68 RCS distributions, 60 spectrum width, 70 Marshall-Palmer formula, 193 Marshall-Palmer raindrop size distribution, 234, 237 Meteorological formations, 227–56 clouds, 230–32 drizzle, 227 dust storms, 232–33 fog, 232 grouping, by size, 227 hail, 229–30 rain, 227–29, 241–44 sandstorm, 232 structure of, 227–33 thunderstorms, 230 Microwave attenuation, 234 causes, 35 total, 37

320

Index

Microwave scattering, sound perturbations from, 47–51 Millimeter wave (MMW) radar, ix advantages, 287 bands, 9 clutter rejection in, 287–311 efficiency estimation, 291 limitations, 287–88 measurements, 207 MTI systems of, 296 performance, 291 propagation effects on, 287–90 Mobile target indication (MTI), 296 Multichannel image processing, 159–66 filtering methods, 163–66 image superimposing, 159–63 Multichannel method application of, 150 capabilities analysis, 140 efficiency, 145–50 measurements, 141 Multichannel radar images after adaptive nonlinear vector filtering, 166 filtering methods, 163–66 Multipath attenuation, 288–90 estimation, 288 radar range influence, 290–92 for real terrain paths, 289 Multiple surface reflection, 78–84 Multiplicative noise, 158

cover coefficient and, 120 dependence, 108–9 dependence on incidence angle, 79 dependence on radar wavelength, 111 dependence on relative moisture, 112 dependence on wavelength, 100 dielectric constant and, 111 dual-polarization ratio of, 138 estimation of, 100 estimation with Kirchoff’s method, 100 forests, 119, 122 frequency dependence, 111, 121 frequency dependence for dry/wet snow, 118 Gaussian model vs., 128 of hail, 245 models, 120–23 for nadir radiation, 109, 112 pdf, 128, 206 of quasi-smooth surface, 108–9 of rain, 245 on range, 300 ratios for cross-polarized reception, 157 reflectivity and, 238 of rough surface, 99 for rough surfaces without vegetation, 109–14 sea, 193–97 of snow, 114 surface scattering, 116 of turbulent atmosphere, 256 volumetric scattering, 116 wind-dependent saturation, 203 See also RCS

N Nadir radiation RCS for, 109, 112 rms roughness height, 109 Nonlinear transformation, 272 Normalized RCS, 108–23 angular dependences, 112, 113, 114, 203 of clouds, 245 for concrete, 110

O Orbital speed, 215 Orbital velocity, 221 Oxygen absorption factor, 35

Index

P Peaks, 184–89 mean duration, 188 mean number of, 188 See also Sea Phillip’s generalized constant, 183 Phillips-Miles model, 178 Phillip’s spectrum, 181, 182 Pierson-Moscovitch spectrum, 179, 187–88 Plasma parameters, 36 Plunging, 174 asymmetry coefficients of, 175 defined, 174 See also Sea waves Point reflections, 250–54 angel-echoes, 250, 251 from angels, 252 origin, 250 Poisson’s law, 187 Polarization ratios, 142 analysis, 142 sea ice, 124 for vegetation, 126 Polarization(s) dual-channel, 157 HH, 149, 161, 162, 196, 207 horizontal, 199, 200 of scattered signals, 123–26 vertical, 198, 220 VV, 149, 161, 163, 196, 207 Power density, 78 Power series coefficients, 275 Power spectra of angel backscattering, 254 from atmospheric turbulence, 256 of backscattered signals, 218 backscattering, of land surfaces, 101 of biological objects, 70 cross-polarizations, 220

321

echo, 62–72 forest/grass backscattering, 130 Fourier-transform of autocorrelation function, 103 for GAZ-63 truck, 68 of helicopter, 66 in high-frequency region, 69 intensity, 219 for L-200 airplane, 65 of land backscattering, 128 land clutter, 129 of land objects, 66–67 of marine targets, 68 model, 101–8 of moving humans, 71 rain backscattering, 246 of scattered signals, 128–32 of seagull, 73 sea surface returns, 214 shape of backscattering signals, 250 swamp backscattering, 131 of swimming humans, 72 for tank, 67 width, 67, 69, 70 at X-band, 64–65 Precipitation backscattering, 239–42 clutter, 241 distributions, 229 intensity, 239 microwave scattering by, 236 volumetric normalized RCS of, 240 See also Drizzle; Rain; Snow Pressure jump illustrated, 43 power spectra of, 44 at SWF, 41 Probability density function (pdf), 1 amplitude, 56 chi-square, 57 experimental, 57 of normalized RCS, 128, 206

322

Probability functions amplitude, 207 of signal instantaneous values, 208 Propagation effects, 287–90 Pseudorandom number generator, 263 Pulsed radar, 28, 293 Pulse repetition frequencies (PRFs), 299 Q Quasi-smooth surfaces dependence on radar wavelength, 111 normalized RCS of, 108–9 R Radar cross section. See RCS Radar reflection from explosion and gas wake, 28–34 long-life, 29 mechanisms, 18–23 from shock wave front, 41–47 Radar tail, 32 Rain, 227–28 attenuation, 235 backscattering, 239, 241 backscattering power spectra, 246 clutter, 241, 242 clutter RCS, 243–44 clutter rejection, 305–11 distributions, 229 geometrical characteristics, 228 global models, 228 heavy, 228 intensity, 229 mean probability, 229 moderate, 228 normalized RCS of, 245 in precipitation zones, 227 probability, 228–29 radar range influence, 290–91 RCS, frequency dependence, 295 See also Meteorological formations Random access memory (RAM), 280

Index

Range land influence on, 292–97 multipath attenuation influence on, 290–92 normalized RCS on, 300 rain clutter influence on, 292–97 rain influence on, 290–92 SCR dependence on, 303 total RCS on, 294 Rayleigh distribution, 250 Rayleigh model defined, 56 experimental results and, 59 Rayleigh targets, 56 RCS aircraft, 12 of air targets, 10 bird mass dependence, 13, 15 of birds, 16 bistatic, 80 bounds, 16 clutter total, 295, 296, 297 crosswind, 29 decrease methods, 13 derivation, 4 electromagnetic wave polarization dependence, 6 estimation model, 14 evaluation, 5 experimental, 7 explosion, 20 explosion bands, 30 fluctuations, 1, 77 of insects, 16 of land objects, 9 of man, 13 of marine vessels, 7, 8 mean, of land targets, 11 mean values, 30 median value, 8 models, 3–7 in point of reception, 72–73

323

Index

probability functions, 73, 74–75 rain, 295 range dependence, 9, 10 of real targets, 7–17 relationships, 4 rough estimation of, 4 of small marine targets, 9, 11 surface scattering, 116 of SWF front, 46, 47 target, 3–17 total, 78 volumetric scattering, 116 wavelength dependence, 7 See also Normalized RCS RCS distributions, 75 anchored sphere, 63 cone cylinder targets, 64 jet airplanes, 59 piston engine, 59 quantiles, 77 Swerling models vs., 84 Reflected signal ratio, 50 Reflection coefficient as function of distance to pressure peak, 46 numerical derivation of, 50 for plane surface, 99 Reflection(s) mean RCS for, 80 point, 250–54 radar, 250–56 from semi-spherical thin layer zones, 252 specular, 251 surface, multiple, 78–84 Reflectivity absolute, 283–84 normalized RCS and, 238 Refractive focusing, 38 Refractive index, 256 Refractive index fluctuations for explosion products area, 27–28

intensity, 27 spatial-temporal spectra shape, 28 temporal, 24, 26 Refractivity coefficient, 44–45 of combustion products, 22 Regression analysis, 219 Rice’s formula, 187 Rough surfaces asphalt, 146 characteristics determination, 155 field values, 153 growth of, 110 measurement, 152 normalized RCS for, 109–14 scattering models, 139 spatial spectrum, 144 S Sandstorms, 232 Scattered signals depolarization of, 123–26, 197–202 fluctuations, 126 intensity, 137 mean power of, 236 power spectra analysis of, 246 power spectra of, 128–32 by spray, 200 stable component, 127 statistical characteristics of, 126–28, 242–50 temporal characteristics, 127 total power spectra in, 212 Scatterer velocities, 106 Scattering pattern of bird species, 14 calculation, 4 of cone-cylinder bodies, 8 of Convair-900 aircraft, 5 of man, 14 Sea clutter rejection, 298–305

324

Sea (continued) echo fluctuations, 83 foam, 176, 177 fog, 232 heavy, shadowing/peaks in, 184–89 ice, 124 normalized RCS, 193–97 spray, 177, 193 state, 220 Sea backscattering models, 189–93 radar spike characteristics, 209–13 spectra, 213–21 Sea clutter absence of reflectors, 268 characterization, 267 RCS model, 202–6 signal formation, 271 statistics, 206–9 total RCS, 184 Sea clutter modeling, 267–76 algorithm illustration, 269 algorithms, 268–76 algorithm stages, 270 initial stage, 270 peculiarities, 267–68 search for nonlinear transformation, 271 simulation process, 268 of stochastic process, 272 See also Clutter modeling Sea roughness autocorrelation function, 181 cause of, 179 characteristics, 171–84 determination, 171 features for small grazing angles, 171–89 Gaussian statistics, 182 power spectrum, 179 Sea surface backscattering from, 184 complexity, 171, 184

Index

shadowing of, 185 slope variance, 186 statistical description, 181 Sea waves asymmetry of, 174 breaking process, 175 height and slope angle, 173 plunging, 174, 175 secondary, 182 slope variance, 182 spilling, 174 Shadowing, 184–89 dependence of, 186 grazing angle vs., 187 mean function, 185 for small grazing angles, 191 zone, 185, 186 Shock wave front (SWF) backscattering from, 54–55 electromagnetic field reflection from, 43 expansion, 44 front, RCS of, 46, 47 parameter values, 43 pressure difference, 42 pressure jump at, 41 radar observation of, 43 reflections, 41–47 reflections, detection of, 51 turbulent atmosphere intersection, 43 width, 45 Shock wave ionized front (SWIF), 2 expansion, 20 high temperature, 19 propagation law, 18 radius, 18 spherical, 19 temporal dependence, 19 Shock wave propagation, 47, 51 Side-looking radar (SLAR), 137, 161 Signal amplitude, in point of reception, 72–73

Index

Signal-to-clutter ratios (SCRs), 298, 310 dependence on range, 303 at Doppler bandpass filter output, 302 losses, as function of relative pulse width, 302 losses, dependence, 301 pulse duration and, 299 signal ambiguity function and, 298 for uncoded pulsed sequence, 305 for wideband signals, 300 Small grazing angles facet model and, 190–91 scattering elements near wave crests, 216 shadowing for, 191 two-scale model for, 192 Small perturbation model, 139 Snow air-snow boundary, 116, 117 backscattering, 114–18, 239 cover, 117 dry, 92, 93, 94, 117 normalized RCS of, 114, 117 wet, 92, 117 See also Precipitation Soil backscattering modeling, 138–45 clay content, 152 cultivation methods, 152 dielectric constants, 91 drying, 151–52 normalized RCS dependencies, 115 parameters, estimation of, 137–50 RCS variations due to moisture, 138 remote moisture determination, 142 spatial correlation radius, 155 Soil erosion aircraft remote sensing, 157–59 determination from ratio images, 166–68 experimental determination, 150–59 from multichannel remote sensing data, 157

325

radar measurement results, 155–57 RCS vs. incidence angle, 156 set and technique of measurement, 150–51 state, situ measurements of, 168 state classification, 167 statistical/agrophysical characteristics, 151–55 Soil moisture, 113 content determination, 140 content sensitivity, 141 measurements at reference points, 151 weighted, 151 Sonic perturbations, radar backscattering of, 41–55 Sound absorption coefficient, 51 Sound oscillations, 52 Sound perturbations atmospheric pressure variation during, 48 electromagnetic field reflection, 49 first phenomenon, 49–50 microwave scattering from, 47–51 Sound wave energy loss, 51 intensity, 51 propagation, 47, 51 Spatial grids, 254 Spectral width histograms, 248 intensity, 247 proportional to wavelength, 247 Specular reflection, 251 Specular scattering coefficient, 288–89 Spheroid, geometry, 201 Spikes central frequency, 220 characteristics, 209–13 depolarization in, 212 duration distributions, 212, 213 maximal, 211 mean number of, 210 probability, 210–11

326

Spikes (continued) probable duration, 211 spectral width, 220 statistics, 210 See also Sea backscattering Spilling, 174 Spray, 177 contribution, 204 effect, 193 signal scattering by, 200 volume normalized RCS of, 193 See also Sea Square-law detection, 131 Statistical characteristics, 55–72 diffuse scattering surface influence on, 72–78 echo power spectra, 62–72 estimation of, 142 models, 55–58 real, 58–62 of scattered signals, 126–28, 242–50 surface influence on, 72–84 Stratification, 250 Superimposed images, 159–63 interpolation of, 161 multiplicative noise, 161 Super-refraction, 203 Surface recovery, 282 Swerling models, 57–58, 73 illustrated, 58 models 1 and 2, 74, 75 models 3 and 4, 74 RCS distributions vs., 84 use of, 57 Synthetic aperture radar (SAR), 137 T Tanks, power spectra, 67 Target RCS, 3–17 models, 3–7 real, 7–17 See also RCS

Index

Targets acceleration, 63 cone cylinder, 8, 64 nonfluctuating, 81 Rayleigh, 56 statistical characteristics of, 55–72 velocity, 63 Target statistical models, 55–58 analysis, 55 experimental pdf and, 57 Swerling, 57–58 Target to surface, 84 Thumbtack ambiguity function, 301 Trucks, power spectra, 68 Turbulence, 52 Turbulent atmosphere, 254–56 backscattering from, 254–56 normalized RCS of, 256 power spectra, 256 Two-scale model, 191–92 defined, 191 for small grazing angles, 192 U Upwind-to-crosswind ratio, 196 V Vector filtering methods, 165 Vector sigma filters, 165–66 advantage, 166 noise-suppressing efficiency, 166 Vegetation, 93 backscattering from, 118–20 clutter maps for, 278 normalized RCS seasonal dependence, 119 penetration depth for, 115 polarization ratios, 126 RCS angular/frequency dependences and, 118 rough surfaces without, 109–14

327

Index

Volumetric water content of corn leaves, 96 for grain, 97 W Wedge, geometry, 199 Weibull distribution, 128 Wiener-Khinchin theorem, 263 Wind velocity, 69 critical, 173 dependence of single foam formation, 176

effective incidence angle and, 80 lower height dependence on, 251 mean, 172 in sea roughness, 171 sea wave height dependence on, 173 spike mean number dependence on range for, 189 Z Zacharov-Philonenko-Toba spectrum, 183