The Hilbert-Huang Transform in Engineering

Author: Norden E. Huang | Nii O. Attoh-Okine

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DK342X Half Title pg 5/18/05 9:51 PM Page 1

The Hilbert-Huang Transform in Engineering

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The Hilbert-Huang Transform in Engineering Edited by

Norden Huang Nii O. Attoh-Okine

Boca Raton London New York Singapore

A CRC title, part of the Taylor & Francis imprint, a member of the Taylor & Francis Group, the academic division of T&F Informa plc.

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Published in 2005 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2005 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-10: 0-8493-3422-5 (Hardcover) International Standard Book Number-13: 978-0-8493-3422-1 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe.

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Preface Data analysis serves two purposes: to determine the parameters needed to construct a model and to confirm that the model constructed actually represents the phenomenon. Unfortunately, the data — whether from physical measurements or numerical modeling — most likely will have one or more of the following problems: • • •

The total data are too limited. The data are nonstationary. The data represent nonlinear processes.

These problems dictate how the data can be analyzed and interpreted. This book describes the formulation and application of the Hilbert-Huang transform (HHT) in various areas of engineering, including structural, seismic, and ocean engineering. The primary objective of this book is to present the theory of the Hilbert-Huang Transform (HHT) and its application to engineering. The presentation of the book is such that it can be used as both a reference and a teaching text. The authors of the individual chapters provide a strong theoretical background and some new developments before addressing their specific application. This approach demonstrates the versatility of the HHT. The book comprises 13 chapters, covering more than 300 pages. These chapters were written by 30 invited experts from 6 different countries. The book begins with an introduction and some recent developments in HHT. Chapter 2 uses HHT to interpret nonlinear wave systems and provides a comprehensive analysis on the assessment of rogue waves. Chapter 3 discusses HHT applications in oceanography and ocean-atmosphere remote sensing data and presents some examples of these applications. Chapter 4 presents a comparison of the energy flux computation for shooting waves of HHT and wavelet analysis techniques. In Chapter 5, HHT is applied to nearshore waves, and the results are compared to field data. In Chapter 6, the author uses HHT to characterize the underwater electromagnetic environment and to identify transient manmade electromagnetic disturbances, where the HHT was able to act as a filter effectively discriminating different dipole components. Chapter 7 presents a comparative analysis of HHT and wavelet transforms applied to turbulent open channel flow data. In Chapter 8, nonlinear soil amplification is quantified by using HHT. Chapter 9 extends the application of HHT to nonstationary random processes. Chapter 10 presents a comparative analysis of HHT wavelet and Fourier transforms in some structural health-monitoring applications. In Chapter 11, HHT is applied to molecular dynamics simulations. Chapter 12 presents a straightforward application of HHT to decomposition of wave jumps. Chapter 13, the concluding chapter, presents

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perspectives on the theory and practices of HHT. It attempts to review HHT applications in biomedical engineering, chemistry and chemical engineering, financial engineering, meteorological and atmospheric studies, ocean engineering, seismic studies, structural analysis, health monitoring, and system identification. It also indicates some directions for future research. One important feature of the book is the inclusion of a variety of modern topics. The examples presented are real-life engineering problems, as well as problems that can be useful for benchmarking new techniques. The studies reported in this book clearly indicate an increasing interest in HHT and analysis for diversified applications. These studies are expected to stimulate the interest of other researchers around the world who are facing new challenges in new theoretical studies and innovative applications. Norden Huang NASA Nii O. Attoh-Okine University of Delaware

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Acknowledgments The editors are grateful to the contributing authors. We also wish to express our thanks to Tao Woolfe, B. J. Clark, and Michael Masiello of Taylor & Francis for providing useful feedback and guiding the editors throughout the complex editorial phase. Norden E. Huang would like to thank Professors Theodore T. Y. Wu of the California Institute of Technology, and Owen M. Phillips of the Johns Hopkins University for their guidance and encouragement throughout the years, without which the Hilbert-Huang Transform would not be what it is today. Nii O. Attoh-Okine, the co-editor of the book, wishes to express his gratitude to his parents, Madam Charkor Quaynor and Richard Attoh-Okine, for their support and encouragement through the years.

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Contributors Nii O. Attoh-Okine Civil Engineering Department University of Delaware Newark, Delaware Brad Battista Information Systems Laboratories, Inc. San Diego, California Rodney R. Buntzen Information Systems Laboratories, Inc. San Diego, California Marcus Dätig Civil Engineering Department Bergische University Wuppertal Wuppertal, Germany Brian Dzwonkowski Graduate College of Marine Studies University of Delaware Newark, Delaware

Ping Gu Department of Civil Engineering University of Illinois at UrbanaChampaign Urbana, Illinois Norden E. Huang Goddard Institute for Data Analysis NASA Goddard Space Flight Center Greenbelt, Maryland Paul A. Hwang Oceanography Division Naval Research Laboratory Stennis Space Center, Mississippi Lide Jiang Graduate College of Marine Studies University of Delaware Newark, Delaware

Colin M. Edge GlaxoSmithKline Pharmaceuticals Harlow, United Kingdom

Young-Heon Jo Graduate College of Marine Studies University of Delaware Newark, Delaware

Jonathan W. Essex School of Chemistry University of Southampton Southampton, United Kingdom

James M. Kaihatu Oceanography Division Naval Research Laboratory Stennis Space Center, Mississippi

Michael Gabbay Information Systems Laboratories, Inc. San Diego, California

Michael L. Larsen Information Systems Laboratories, Inc. San Diego, California

Robert J. Gledhill School of Chemistry University of Southampton Southampton, United Kingdom

Stephen C. Phillips School of Chemistry University of Southampton Southampton, United Kingdom

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Panayotis Prinos Department of Civil Engineering Aristotle University of Thessaloniki Thessaloniki, Greece

David W. Wang Oceanography Division Naval Research Laboratory Stennis Space Center, Mississippi

Ser-Tong Quek Department of Civil Engineering National University of Singapore Singapore

Quan Wang Department of Civil Engineering National University of Singapore Singapore

Jeffrey Ridgway Information Systems Laboratories, Inc. San Diego, California

Wei Wang Physical Oceanography Laboratory Ocean University of China Shandong, P. R. China

Torsten Schlurmann Civil Engineering Department Bergische University Wuppertal Wuppertal, Germany Martin T. Swain School of Chemistry University of Southampton Southampton, United Kingdom Puat-Siong Tua Department of Civil Engineering National University of Singapore Singapore Albena Dimitrova Veltcheva Port and Airport Research Institute Yokosuka, Japan Cye H. Waldman Information Systems Laboratories, Inc. San Diego, California

Yi-Kwei Wen Department of Civil Engineering University of Illinois at UrbanaChampaign Urbana, Illinois Adrian P. Wiley School of Chemistry University of Southampton Southampton, United Kingdom Xiao-Hai Yan Graduate College of Marine Studies University of Delaware Newark, Delaware Athanasios Zeris Department of Civil Engineering Aristotle University of Thessaloniki Thessaloniki, Greece

Ray Ruichong Zhang Division of Engineering Colorado School of Mines Golden, Colorado

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Contents Chapter 1

Introduction to Hilbert-Huang Transform and Some Recent Developments...........................................................................1 Norden E. Huang

Chapter 2

Carrier and Riding Wave Structure of Rogue Waves........................25 Torsten Schlurmann and Marcus Dätig

Chapter 3

Applications of Hilbert-Huang Transform to Ocean-Atmosphere Remote Sensing Research..................................................................59 Xiao-Hai Yan, Young-Heon Jo, Brian Dzwonkowski, and Lide Jiang

Chapter 4

A Comparison of the Energy Flux Computation of Shoaling Waves Using Hilbert and Wavelet Spectral Analysis Techniques ....83 Paul A. Hwang, David W. Wang, and James M. Kaihatu

Chapter 5

An Application of HHT Method to Nearshore Sea Waves...............97 Albena Dimitrova Veltcheva

Chapter 6

Transient Signal Detection Using the Empirical Mode Decomposition .................................................................................121 Michael L. Larsen, Jeffrey Ridgway, Cye H. Waldman, Michael Gabbay, Rodney R. Buntzen, and Brad Battista

Chapter 7

Coherent Structures Analysis in Turbulent Open Channel Flow Using Hilbert-Huang and Wavelets Transforms..............................141 Athanasios Zeris and Panayotis Prinos

Chapter 8

An HHT-Based Approach to Quantify Nonlinear Soil Amplification and Damping............................................................. 159 Ray Ruichong Zhang

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Chapter 9

Simulation of Nonstationary Random Processes Using Instantaneous Frequency and Amplitude from Hilbert-Huang Transform .........................................................................................191 Ping Gu and Y. Kwei Wen

Chapter 10 Comparison of Hilbert-Huang, Wavelet, and Fourier Transforms for Selected Applications .............................................213 Ser-Tong Quek, Puat-Siong Tua, and Quan Wang Chapter 11 The Analysis of Molecular Dynamics Simulations by the Hilbert-Huang Transform .....................................................245 Adrian P. Wiley, Robert J. Gledhill, Stephen C. Phillips, Martin T. Swain, Colin M. Edge, and Jonathan W. Essex Chapter 12 Decomposition of Wave Groups with EMD Method......................267 Wei Wang Chapter 13 Perspectives on the Theory and Practices of the Hilbert-Huang Transform .........................................................................................281 Nii O. Attoh-Okine

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1

Introduction to Hilbert-Huang Transform and Some Recent Developments Norden E. Huang

CONTENTS 1.1 1.2 1.3

Introduction ......................................................................................................1 The Hilbert-Huang Transform .........................................................................9 The Recent Developments .............................................................................16 1.3.1 The Normalized Hilbert Transform ...................................................16 1.3.2 The Confidence Limit ........................................................................19 1.3.3 The Statistical Significance of IMFs .................................................21 1.4 Conclusion......................................................................................................22 References................................................................................................................22

1.1 INTRODUCTION Hilbert-Huang transform (HHT) is the designated name for the result of empirical mode decomposition (EMD) and the Hilbert spectral analysis (HSA) methods, which were both introduced recently by Huang et al. (1996, 1998, 1999, and 2003), specifically for analyzing data from nonlinear and nonstationary processes. Data analysis is an indispensable step in understanding the physical processes, but traditionally the data analysis methods were dominated by Fourier-based analysis. The problems of such an approach were discussed in detail by Huang et al. (1998). As data analysis is important for both theoretical and experimental studies (for data is the only real link between theory and reality), we desperately need new methods in order to gain a deeper insight into the underlying processes that actually generate the data. The method we really need should not be limited to linear and stationary processes, and it should yield physically meaningful results. 1

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The development of the HHT is motivated precisely by such needs: first, because the natural physical processes are mostly nonlinear and nonstationary, there are very limited options in data analysis methods that can correctly handle data from such processes. The available methods are either for linear but nonstationary processes (such as the wavelet analysis, Wagner-Ville, and various short-time Fourier spectrograms as summarized by Priestley [1988], Cohen [1995], Daubechies [1992], and Flandrin [1999]) or for nonlinear but stationary and statistically deterministic processes (such as the various phase plane representations and time-delayed imbedded methods as summarized by Tong [1990], Diks [1997], and Kantz and Schreiber [1997]). To examine data from real-world nonlinear, nonstationary, and stochastic processes, we urgently need new approaches. Second, the nonlinear processes need special treatment. Other than periodicity, we want to learn the detailed dynamics in the processes from the data. One of the typical characteristics of nonlinear processes, proposed by Huang et al. (1998), is the intra-wave frequency modulation, which indicates that the instantaneous frequency changes within one oscillation cycle. Let us examine a very simple nonlinear system given by the nondissipative Duffing equation as ∂2 x + x + ε x 3 = γ cos ωt , ∂t 2

(1.1)

where ε is a parameter not necessarily small and γ is the amplitude of a periodic forcing function with a frequency ω. In Equation 1.1, if the parameter ε were zero, the system is linear, and the solution would be easy. For a small, however, the system is nonlinear, but it could be solved easily with perturbation methods. If ε is not small compared to unity, then the system is highly nonlinear. No known general analytic method is available for this condition; we have to resort to numerical solutions, where all kinds of complications such as bifurcations and chaos can result. Even with these complications, let us examine the qualitative nature of the solution for Equation 1.1 by rewriting it in a slightly different way, as ∂2 x + x (1 + ε x 2 ) = γ cos ωt , ∂t 2

(1.2)

where the symbols are defined as in Equation 1.1. Then the quantity within the parentheses can be regarded as a variable spring constant or a variable pendulum length. With this view, we can see that the frequency should be changing from location to location and from time to time, even within one oscillation cycle. As Huang et al. (1998) pointed out, this intra-frequency frequency variation is the hallmark of nonlinear systems. In the past, there has been no clear way to depict this intra-wave frequency variation by using Fourier-based analysis methods, except to resort to the harmonics. Even by the classical Hamiltonian approach, in which the frequency is defined as the rate of change of the Hamiltonian with respect to

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3

the action, we still cannot gain any more insight, for the definition of the action is an integration of generalized momentum along the generalized coordinates; therefore, there is no instantaneous value. Thus, the best we could do for any nonlinear distorted waveform in the earlier approaches was to refer to harmonic distortions. Harmonic distortion is, in fact, a rather poor alternative, for it is the result obtained by imposing a linear structure on a nonlinear system. Consequently, the results may make perfect mathematical sense, but at the same time they have absolutely no physical meaning. The physically meaningful way to describe the system should be in terms of the instantaneous frequency. The easiest way to compute the instantaneous frequency is by the Hilbert transform, through which we can find the complex conjugate, y(t), of any real valued function x(t) of Lp class, 1 y(t ) = P π

∞

x (τ)

∫ t − τ dτ ,

(1.3)

−∞

in which the P indicates the principal value of the singular integral. With the Hilbert transform, we have z (t ) = x (t ) + j y(t ) = a (t ) e j θ(t ) ,

(1.4)

where

(

a(t ) = x 2 + y 2

)

1/ 2

; θ(t ) = tan −1

y . x

(1.5)

Here a is the instantaneous amplitude and θ is the phase function; thus the instantaneous frequency is simply ω= −

dθ . dt

(1.6)

As the instantaneous frequency is defined through a derivative, it is very local and can be used to describe the detailed variation of frequency, including the intra-wave frequency variation. As simple as this principle is, the implementation is not at all trivial. To represent the function in terms of a meaningful amplitude and phase, however, requires that the function satisfy certain conditions. Let us take the length-of-day (LOD) data, given in Figure 1.1, as an example to explain this requirement. The daily data covers from 1962 to 2002, for roughly 40 years. After performing the Hilbert transform, the polar representation of the data is given in Figure 1.2, which is a random collection of intertwined looping curves. The corresponding phase function is given in Figure 1.3, where random finite jumps intersperse within

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The Hilbert-Huang Transform in Engineering

FIGURE 1.1 (See color insert following page 20). Data of Length-of-Day measure the deviation from the mean 24-hour-day.

FIGURE 1.2 (See color insert following page 20). Analytic function in complex phase plane formed by the real data and it Hilbert Transform. It shows not apparent order. After the EMD, the annual cycle is extract and plotted also.

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FIGURE 1.3 The phase function of the analytic function based on the Length-of-Day data.

the data span. If we follow through the definition of the instantaneous frequency as given in Equation 1.6 literally, we would have a totally nonsensical result, as given in Figure 1.4, where the instantaneous frequency is equally likely to be positive or negative. Unfortunately, this is exactly the procedure recommended by Hahn (1996). To show how this should not be the case, let us consider the three curves given by the following three expressions x1 = sin ωt ; x 2 = 0.5 + sin ωt ;

(1.7)

x3 = 1.5 + sin ωt ; shown in Figure 1.5. All three curves are perfect sine functions, but with the mean displaced: for x1, its mean is exactly zero; for x2, its mean is moved up by half of its amplitude; for x3, its mean is moved up by 1.5 times its amplitude. As a result, the curve represented by x3 is totally above the zero reference axis. If we perform the Hilbert transform to all three functions given in Equation 1.7, we would get three different circles in the phase plane, with the centers of two of the circles displaced by the amount of the added constants, as shown in Figure 1.6. Consequently, the phase functions from the three circles will be different, as shown in Figure 1.7: for x1, the phase function is a straight line; for x2, the phase function is a wavy line, but the general trend still agrees with the straight line; for x3, the phase function is also wavy, but the variation is always within ± π.

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The Hilbert-Huang Transform in Engineering

FIGURE 1.4 Instantaneous frequency obtained form derivative of the phase function without decomposition first. The values are equally like to be positive as negative.

FIGURE 1.5 (See color insert following page 20). Model data to illustrate the fallacy of the instantaneous frequency without decomposition.

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FIGURE 1.6 (See color insert following page 20). The analytic function in complex phase plane of the data given in Figure 1.5. Phase Angles for Test Data 20 sin x 0.5 + sin x 1.5 + sin x

Phase Angle: radians

15

10

5

0

−5

0

100

200

300

400 500 Time: (100) second

600

700

800

FIGURE 1.7 (See color insert following page 20). Phase function of the model function given in Figure 1.5.

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The Hilbert-Huang Transform in Engineering

FIGURE 1.8 (See color insert following page 20). Instantaneous frequency computed from the cosine model functions consist of the identical cosine function with different displacements.

Based on these phase functions, the instantaneous frequency values are very different for the three expressions, as shown in Figure 1.8: for x1, the instantaneous frequency is a constant value, which is exactly what we expected; for x2, the instantaneous frequency is a variable curve with all positive values; for x3, the instantaneous frequency is a highly variable curve fluctuating from positive to negative values. Even if we are prepared to accept negative frequency, the result of three different values for the same sine wave with only a displaced mean is very unsettling: some of the results are, of course, nonsensical. The only meaningful result is from the sine curve with a zero mean. What went wrong was the fact that two of the curves do not have a zero mean, or the envelopes of the curves are not symmetric with respect to the zero axis. Thus, before performing the Hilbert transform, we have to preprocess the data. In the past, any preprocessing usually consisted of band-pass filtering. For some of the data from linear and stationary processes, this band-pass filtering method will give the correct results. For data from nonlinear and nonstationary processes, however, the band-pass filter will alter the characteristics of the filtered curve. The problem with the filtering approach is that all the frequency domain filters are Fourier-based, which means they are established under linear and stationary assumptions. When the data are from nonlinear and nonstationary processes, such Fourier-based analysis will surely generate spurious harmonics, which are mathematically necessary but physically meaningless, as discussed by Huang et al. (1998, 1999). Considering these points leads us to this conclusion: The correct preprocessing for data from nonlinear and nonstationary processes will have to be adaptive and

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FIGURE 1.9 Test data to illustrate the procedures of Empirical Mode Decomposition also known as sifting.

implemented in the time domain. The only method presently known to achieve this is based on the Hilbert-Huang transform proposed by Huang et al. (1996, 1998, 1999, and 2003). This method is the subject of the next section.

1.2 THE HILBERT-HUANG TRANSFORM The Hilbert-Huang transform is the result of the empirical mode decomposition and the Hilbert spectral analysis. As the EMD method is more fundamental, and it is a necessary step to reduce any given data into a collection of intrinsic mode functions (IMF) to which the Hilbert analysis can be applied (see, for example, Huang et al., 1996, 1998, and 1999), we will discuss it first. An IMF represents a simple oscillatory mode as a counterpart to the simple harmonic function, but it is much more general: by definition, an IMF is any function with the same number of extrema and zero crossings, with its envelopes, as defined by all the local maxima and minima, being symmetric with respect to zero. Obtaining the EMD consists of the following steps: For any data as given in Figure 1.9, we first identify all the local extrema and then connect all the local maxima by a cubic spline line as the upper envelope. We repeat the procedure for the local minima to produce the lower envelope. The upper and lower envelopes should cover all the data between them. Their mean is designated as m1, as shown in Figure 1.10, and the difference between the data and m1 is the first proto-IMF (PIMF) component, h1: x (t ) − m1 = h1 .

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(1.8)

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FIGURE 1.10 (See color insert following page 20). The cubic spline upper and the lower envelopes and their mean, m1.

This result is shown in Figure 1.11. The procedure of extracting an IMF is called sifting. By construction, this PIMF, h1, should satisfy the definition of an IMF, but the change of its reference frame from rectangular coordinate to a curvilinear one can cause anomalies, as shown in Figure 1.11, where multi-extrema between successive zero-crossings still existed. To eliminate such anomalies, the sifting process has to be repeated as many times as necessary to eliminate all the riding waves. In the subsequent sifting process steps, h1 is treated as the data. Then h1 − m11 = h11 ,

(1.9)

where m11 is the mean of the upper and lower envelopes of h1. This process can be repeated up to k times; then, h1k is given by h1( k −1) − m1k = h1k .

(1.10)

Each time the procedure is repeated, the mean moves closer to zero, as shown in Figures 1.12a, b, and c. Theoretically, this step can go on for many iterations, but each time, as the effects of the iterations make the mean approach zero, they also make amplitude variations of the individual waves more even. Yet the variation of the amplitude should represent the physical meaning of the processes. Thus this iteration procedure, though serving the useful purpose of making the mean to be zero, also drains the physical meaning out of the resulting components if carried too far. Theoretically, if one insists on achieving a strictly zero mean, one would

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Data and h1 10

h1 data

8 6

Amplitude

4 2 0 −2 −4 −6 −8 −10 200

250

300

350

400 Time: second

450

500

550

600

FIGURE 1.11 (See color insert following page 20). Comparison between data and h1 , as given by Equation (1.9). Note most, but not all, riding waves are eliminated in h1.

Envelopes and the Mean: h1 10

Data: h1 Envelope Envelope Mean: m2

8 6

Amplitude

4 2 0 −2 −4 −6 −8 −10 200

250

300

350

400 Time: second

450

500

550

600

FIGURE 1.12 (A) (See color insert following page 20). Repeat the sifting using h1 as data.

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The Hilbert-Huang Transform in Engineering

Envelopes and the Mean: h2 10

Data: h2 Envelope Envelope Mean: m3

8 6

Amplitude

4 2 0 −2 −4 −6 −8 −10 200

250

300

350

400 Time: second

450

500

550

600

FIGURE 1.12 (B) (See color insert following page 20). Repeat the sifting using h2 as data.

FIGURE 1.12 (C) (See color insert following page 20). After 12 iterations, the first Intrinsic Mode Function is found.

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probably have to make the resulting components purely frequency-modulated functions, with the amplitude becoming constant. Then the resulting component would not retain any physically meaningful information. Thus to attain the delicate balance of achieving a reasonably small mean and also retaining enough physical meaning in the resulting component, we have proposed two stoppage criteria. The “stoppage criterion” actually determines the number of sifting steps to produce an IMF; it is thus of critical importance in a successful implementation of the EMD method. The first stoppage criterion is similar to the Cauchy convergence test, where we first define a sum of the difference, SD, as T

∑h

k −1

SD =

(t ) − hk (t )

2

t =0

;

T

∑h

2 k −1

(1.11)

(t )

t =0

then the sifting will stop when SD is smaller than a preassigned value. This definition is a slight modification from the original one proposed by Huang et al. (1998), where the SD was defined simply as T

SD =

∑ t =0

hk −1 (t ) − hk (t ) hk2−1 (t )

2

.

(1.12)

The shortcoming of this old definition as given in Equation 1.12 is that the value of SD can be dominated by local small values of hk–1, while the definition given in Equation 1.11 sums up all the contributions over the whole duration of the data. Even with this modification, there is still a problem with this seemingly mathematically sound approach: in this definition, the important criterion that the number of extrema has to equal the number of zero-crossings has not been checked. To overcome this practical difficulty, Huang et al. (1999, 2003) proposed an alternative in a second stoppage criterion. The second stoppage criterion is based on a number called the S-number, which is defined as the number of consecutive siftings when the numbers of zero-crossings and extrema are equal or at most differing by one; it requires that number shall remain unchanged. Through exhaustive testing, Huang et al. (2003) used this S-number method of defining a stoppage criterion to establish a confidence limit for the EMD, to be discussed later. When the resulting function satisfies either of the criteria given above, this component is designated as the first IMF, c1, as shown in Figure 1.12c. We can then separate c1 from the rest of the data by X (t ) − c1 = r1 .

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(1.13)

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Data and residue: r1 10

Data Residue: r1

8 6

Amplitude

4 2 0 −2 −4 −6 −8 −10 200

250

300

350

400 Time: second

450

500

550

600

FIGURE 1.13 (See color insert following page 20). Comparison between data and the residue, r1, after the first IMF, c1, is removed. Notice the residue behaves like a moving mean to the data that bisect all the waves.

This resulting residue is shown in Figure 1.13. Since the residue, r1, still contains information with longer periods, it is treated as the new data and subjected to the same process as described above. This procedure can be repeated to all the subsequent rj’s, and the result is r1 − c2 = r2 , .

...

(1.14)

rn−1 − cn = rn By summing up Equation 1.13 and Equation 1.14, we finally obtain n

X (t ) =

∑c

j

+ rn .

(1.15)

j =1

Thus, we achieve a decomposition of the data into n IMF modes, and a residue, rn, which can be either a constant, a monotonic mean trend, or a curve having only one extremum. Recent studies by Flandrin et al. (2004) and Wu and Huang (2004) established that the EMD is a dyadic filter, and it is equivalent to an adaptive wavelet. Since it is adaptive, we avoid the shortcomings of using an a priori–defined wavelet basis, and we also avoid the spurious harmonics that would have resulted. The

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components of the EMD are usually physically meaningful, for the characteristic scales are defined by the physical data. The sifting process is, in fact, a Reynolds-type decomposition: separating variations from the mean, except that the mean is a local instantaneous mean, so that the different modes are almost orthogonal to each other, except for the nonlinearity in the data. Having obtained the intrinsic mode function components, we can apply the Hilbert transform to each IMF component and compute the instantaneous frequency as the derivative of the phase function. After performing the Hilbert transform to each IMF component, we can express the original data as the real part, RP, in the following form: n

X(t) = RP

∑ a (t) e j

i ∫ ω j (t) dt

.

(1.16)

j=1

Equation 1.16 gives both amplitude and frequency of each component as a function of time. The same data, if expanded in a Fourier representation, would have a constant amplitude and frequency for each component. The contrast between EMD and Fourier decomposition is clear: the IMF represents a generalized Fourier expansion with a time-varying function for amplitude and frequency. This frequency–time distribution of the amplitude is designated as the Hilbert amplitude spectrum, H(, t), or simply the Hilbert spectrum. With the Hilbert spectrum defined, we can also define the marginal spectrum, h(), as T

h( ω ) =

∫ H(ω ,t)dt.

(1.17)

0

The marginal spectrum offers a measure of total amplitude (or energy) contribution from each frequency value. It represents the cumulated amplitude over the entire data span in a probabilistic sense. The combination of the EMD and the HSA is known as the Hilbert-Huang transform for short. Empirically, all tests indicate that HHT is a superior tool for time–frequency analysis of nonlinear and nonstationary data. It has an adaptive basis, and the frequency is defined through the Hilbert transform. Consequently, there is no need for the spurious harmonics to represent nonlinear waveform deformations as in any of the a priori basis methods, and there is no uncertainty principle limitation on time or frequency resolution resulting from the convolution pairs possessing a priori bases. Table 1.1 compares Fourier, wavelet, and HHT analyses. From this table, we can see that the HHT approach is indeed a powerful method for the analysis of data from nonlinear and nonstationary processes: it has an adaptive basis; the frequency is derived by differentiation rather than convolution — therefore, it is not limited by the uncertainty principle; it is applicable to nonlinear and nonstationary data; and it presents the results in time–frequency–energy space for feature extraction. This basic development of the HHT method has been followed

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TABLE 1.1 Comparisons between Fourier, Wavelet, and Hilbert-Huang Transform in Data Analysis Fourier

Wavelet

Hilbert

Presentation Nonlinear Nonstationary Feature Extraction

A priori Convolution: global, uncertainty Energy–frequency No No No

Adaptive Differentiation: local, certainty Energy–time–frequency Yes Yes Yes

Theoretical Base

Theory complete

A priori Convolution: regional, uncertainty Energy–time–frequency No Yes Discrete: no Continuous: yes Theory complete

Basis Frequency

Empirical

by recent developments that have either added insight to the results or enhanced their statistical significance. Some of the recent developments are summarized in the following section.

1.3 THE RECENT DEVELOPMENTS We will discuss in some detail recent developments in the areas of the normalized Hilbert transform, the confidence limit, and the statistical significance of IMFs.

1.3.1 THE NORMALIZED HILBERT TRANSFORM It is well known that, although the Hilbert transform exists for any function of Lp class, the phase function of the transformed function will not always yield physically meaningful instantaneous frequencies, as discussed above. In addition to the requirement of being an IMF, which is only a necessary condition, additional limitations have been summarized succinctly in two theorems. First, the Bedrosian theorem (1963) states that the Hilbert transform for the product of two functions, f(t) and h(t), can be written as H [ f (t ) h(t )] = f (t ) H [ h(t )] ,

(1.18)

only if the Fourier spectra for f(t) and h(t) are totally disjoint in frequency space, and if the frequency content of the spectrum for h(t) is higher than that of f(t). This limitation is critical, for we need to have H [ a(t ) cos θ(t ) ] = a(t ) H [cos θ(t )] ;

(1.19)

otherwise, we cannot use Equation 1.5 to define the phase function. According to the Bedrosian theorem, Equation 1.19 is true only if the amplitude is varying so

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slowly that the frequency spectra of the envelope and the carrier waves are disjoint. This has made the application of the Hilbert transform even to IMFs problematic. To satisfy this requirement, Huang and Long (2003) have proposed the normalization of the IMFs in the following steps: starting from an IMF, we first find all the maxima of the IMFs, defining the envelope by spline through all the maxima and designating the envelope as E(t). Now, we normalize the IMF by dividing the IMF by E(t). Thus, we have the normalized function with amplitude always equal to unity. Even with this normalization, we have not resolved all the limitations on the Hilbert transform. The new restriction is given by the Nuttall theorem (1966). This theorem states that the Hilbert transform of cosine is not necessarily the sine with the same phase function for a cosine with an arbitrary phase function. Nuttall gave an error bound, ∆E, defined as the difference between y(t), the Hilbert transform of the data, and Q(t), the quadrature (with phase shift of exactly 90°) of the function: 0

T

∆E =

∫ y(t) − Q(t)

t =0

2

dt =

∫ S (ω) dω , q

(1.20)

−∞

where Sq is Fourier spectrum of the quadrature function. The proof of this theorem is rigorous, but the result is hardly useful, for it gives a constant error bound over the whole data range. For a nonstationary time series, such a constant bound will not reveal the location of the error on the time axis. With the normalized IMF, Huang and Long (2003) have proposed a variable error bound based on a simple argument, which goes as follows: let us compute the difference between the squared amplitude of the normalized IMF and unity. If the Hilbert transform is exactly the quadrature, then the squared amplitude of the normalized IMF should be unity; therefore, the difference between it and unity should be zero. If the squared amplitude is not exactly unity, then the Hilbert transform cannot be exactly the quadrature. Consequently, the error can be measured simply by the difference between the squared normalized IMF and unity, which is a function of time. Huang and Long (2003) and Huang et al. (2005) have conducted detailed comparisons and found the result quite satisfactory. Even with the error indicator, we can only know that the Hilbert transform is not exactly the quadrature; we still do not have the correct answer. This prompts the suggestion of a drastic alternative, eschewing the Hilbert transform totally. To this end, Huang et al. (2005) suggest that the phase function can be found by computing the arc-cosine of the normalized function. A checking of the results so obtained has also proved to be satisfactory. The only problem is that the imperfect normalization will give some values greater than unity. Under that condition, the arc-cosine will break down. An example of the normalized and regular Hilbert transforms is given in Figure 1.14, from the data given in Figure 1.12c. There are three different instantaneous frequency values: the instantaneous frequency from regular Hilbert transform, the normalized Hilbert transform, and the generalized zero-crossing, which can serve as the standard in the mean. It is easy to see that the normalized instantaneous

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Comparison of Instantaneous frequency from Diﬀerent Methods

0.2

0.15

Data IF-H IF-NH IF-Z

Amplitude

0.1

0.05

0

–0.05 0

50

100

150

200 Time : second

250

300

350

400

FIGURE 1.14 (See color insert following page 20). Comparison of the instantaneous frequency values derived from different methods. Note that the Instantaneous from IMF is still not correct when the amplitude fluctuates too much. The normalized Hilbert Transform, however, gives a much better instantaneous frequency when compared with the values derived from the generalized-zero-crossing method.

frequency is very close to the zero-crossing values, while the regular Hilbert transform result gives large undulations that will never result in the mean as given by the zero-crossing method. The high undulation results from the large changes of amplitude and some nonlinear distortions of the waveform, both of which will cause the envelope to fluctuate as shown in Figure 1.15. In the normalization scheme, the smooth spline helps to eliminate many of the undulations in the resulting instantaneous frequency. One can also see that the problem of the regular Hilbert transform occurs always at the location where either the amplitudes change drastically or the amplitude is very low, as predicted by the Nuttall theorem. The normalized Hilbert transform alleviates the problems substantially. Finally, the error index is given in Figure 1.16; here we can see that the error is also small in general, unless the waveform is locally distorted. Even over the large error location, the index values are smaller than 10%, except for the end region, where the end effect of the Hilbert transform causes additional problems. Thus the normalized Hilbert transform has helped to overcome many of the difficulties of the regular Hilbert transform, and it should be used all the time.

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Comparison of Amplitude from Diﬀerent Methods 10 Data A-H A-NH A-Z

9 8

Amplitude : cm

7 6 5 4 3 2 1 0 0

50

100

150

200 Time : second

250

300

350

400

FIGURE 1.15 (See color insert following page 20). The instantaneous amplitude (or the envelope) of the test data. Note the improvement in adopting the spline envelope over the simple analytic function.

1.3.2 THE CONFIDENCE LIMIT The confidence limit for the Fourier spectral analysis is routinely computed. The computation, however, is based on the practice of cutting the data into N sections and computing spectra from each section. The confidence limit is defined as the statistical spread of the N different spectra. This practice is based on the ergodic theory, where the temporal average is treated as the ensemble average. The ergodic condition is satisfied only if the processes are stationary; otherwise, averaging them will not make sense. Huang et al. (2003) have proposed a different approach, using the fact that there are infinitely many ways to decompose one given function into different components. Even using EMD, we can still obtain many different sets of IMFs by changing the stoppage criteria. For example, Huang et al. (2003) explored the stoppage criterion by changing the S-number. Using the length-of-day (LOD) data, they varied the S-number from 1 to 20 and found the mean and the standard deviation for the Hilbert spectrum given in Figure 1.15. The confidence limit so derived does not depend on the ergodic theory. By using the same data length, there is also no downgrading of the spectral resolution in frequency space through subdividing of the data into sections.

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Error Index for Normalized Hilbert Transform

0.1 0.08 0.06 0.04

Amplitude

0.02 0 –0.02 –0.04 –0.06 –0.08 –0.1 0

50

100

150

200 Time : second

250

300

350

400

FIGURE 1.16 (See color insert following page 20). The Error Index of the normalized Hilbert transform; it has large value whenever the wave form deviated from a smooth sinusoidal form. But their values are, in general, small except near the ends.

Additionally, Huang et al. (2003) invoked the intermittence criterion and forced the number of IMFs to be the same for different S-numbers. As a result, they were able to find the mean for specific IMFs. Figure 1.17 shows the IMF representing variations of the annual cycle of the length of day. The peak and valley of the envelope represent the El Niño events. Of particular interest are the periods of high standard deviations, from 1965 to 1970 and from 1990 to 1995. These periods turn out to be the anomaly periods of the El Niño phenomena, when the sea surface temperature readings in the equatorial region were consistently high based on observations, indicating a prolonged heating of the ocean, rather than the changes from warm to cool during the El Niño to La Niña changes. Finally, from the confidence limit study, an unexpected result was the determination of the optimal S-number. Huang et al. (2003) computed the difference between the individual cases and the overall mean and found that there is always a range where the differences reach a local minimum. Based on their limited experience from different data sets, they concluded that an S-number in the range of 4 to 8 performed well. Logic also dictates that the S-number should not be too high (which would drain all the physical meaning out of the IMF) nor too low (which would leave some riding waves remaining in the resulting IMFs).

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Error Index for Normalized Hilbert Transform

0.1

MeanAnnual Envelope M+std M–std std

0.08 0.06 0.04

Amplitude

0.02 0 –0.02 –0.04 –0.06 –0.08 –0.1 1965

1970

1975

1980 1985 Time : second

1990

1995

2000

FIGURE 1.17 (See color insert following page 20). The mean envelope of the annual cycle IMF component from LOD data. The peaks of the envelope are all aligned with El Nio events, when the additional angular momentum imparted to the atmosphere from the over heated Equatorial ocean water. The large scatter of the envelope periods in 1065-70 and 199095 represent periods of El Nio anomalies.

1.3.3 THE STATISTICAL SIGNIFICANCE

OF

IMFS

The EMD is a method of separating data into different components by their scales. There is always the question: on what is the statistical significance of the IMFs based? In data that contains noise, how can we separate the noise from information with confidence? This question was addressed by both Flandrin et al. (2004) and Wu and Huang (2004) through the study of signals consisting of noise only. Flandrin et al. (2004) studied the fractal Gaussian noises and found that the EMD is a dyadic filter. They also found that when one plotted the mean period and root-mean-square (RMS) values of the IMFs derived from the fractal Gaussian noise on log–log scale, the results formed a straight line. The slope of the straight line for white noise is –1; however, the values change regularly with the different Hurst indices. Based on these results, Flandrin et al. (2004) suggested that the EMD results could be used to discriminate what kind of noise one was encountering.

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Instead of fractal Gaussian noise, Wu and Huang (2004) studied the Gaussian white noise only. They also found the relationship between the mean period and RMS values of the IMFs. Additionally, they have also studied the statistical properties of the scattering of the data and found the bounds of the data distribution analytically. From the scattering, they deduced a 95% bound for the white noise. Therefore, they concluded that when a data set is analyzed with EMD, if the mean period-RMS values exist within the noise bounds, the components most likely represent noise. On the other hand, if the mean period-RMS values exceed the noise bounds, then those IMFs must represent statistically significant information.

1.4 CONCLUSION HHT is a relatively new method in data analysis. Its power is in the totally adaptive approach that it takes, which results in the adaptive basis, the IMFs, from which the instantaneous frequency can be defined. This offers a totally new and valuable view of nonstationary and nonlinear data analysis methods. With the recent developments on the normalized Hilbert transform, the confidence limit, and the statistical significance test for the IMFs, the HHT has become a more robust tool for data analysis, and it is now ready for a wide variety of applications. The development of HHT, however, is not over yet. We still need a more rigorous mathematical foundation for the general adaptive methods for data analysis, and the end effects must be improved as well.

REFERENCES Bedrosian, E. (1963). On the quadrature approximation to the Hilbert transform of modulated signals. Proc. IEEE, 51, 868–869. Cohen, L. (1995). Time-Frequency Analysis. Prentice Hall, Englewood Cliffs, NJ. Diks, C. (1997). Nonlinear Time Series Analysis. World Scientific Press, Singapore. Daubechies, I. (1992). Ten Lectures on Wavelets. SIAM, Philadelphia. Flandrin, P. (1999). Time-Frequency/Time-Scale Analysis. Academic Press, San Diego, CA. Flandrin, P., Rilling, G., and Gonçalves, P. (2004). Empirical mode decomposition as a filterbank. IEEE Signal Proc. Lett. 11 (2): 112–114. Hahn, S. L. (1996). Hilbert Transforms in Signal Processing. Artech House, Boston. Huang, N. E., and Long, S. R. (2003). A generalized zero-crossing for local frequency determination. U.S. Patent pending. Huang N. E., Long, S. R., and Shen, Z. (1996). Frequency downshift in nonlinear water wave evolution. Advances in Appl. Mech. 32, 59–117. Huang, N. E., Shen, Z., Long, S. R. (1999). A new view of nonlinear water waves — the Hilbert spectrum. Ann. Rev. Fluid Mech. 31, 417–457. Huang, N. E., Shen, Z., Long, S. R., Wu, M. C., Shih, S. H., Zheng, Q., Tung, C. C., and Liu, H. H. (1998). The empirical mode decomposition method and the Hilbert spectrum for non-stationary time series analysis. Proc. Roy. Soc. London, A454, 903–995. Huang, N. E., Wu, Z., Long, S. R., Arnold, K. C., Blank, K., Liu, T. W. (2005). On instantaneous frequency. Proc. Roy. Soc. London (submitted).

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Huang, N. E., Wu, M. L., Long, S. R., Shen, S. S. P., Qu, W. D., Gloersen, P., and Fan, K. L. (2003). A confidence limit for the empirical mode decomposition and the Hilbert spectral analysis. Proc. Roy. Soc. London, A459, 2317–2345. Kantz, H., and Schreiber, T. (1997). Nonlinear Time Series Analysis. Cambridge University Press, Cambridge. Nuttall, A. H. (1966). On the quadrature approximation to the Hilbert transform of modulated signals. Proc. IEEE, 54, 1458–1459. Priestley, M. B. (1988). Nonlinear and nonstationary time series analysis. Academic Press, London. Tong, H. (1990). Nonlinear Time Series Analysis. Oxford University Press, Oxford. Wu, Z., and Huang, N. E. (2004). A study of the characteristics of white noise using the empirical mode decomposition method. Proc. Roy. Soc. London, A460, 1597–1611.

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2

Carrier and Riding Wave Structure of Rogue Waves Torsten Schlurmann and Marcus Dätig

CONTENTS 2.1 2.2

Introduction ....................................................................................................26 Rogue Wave Measurements...........................................................................27 2.2.1 Preliminaries.......................................................................................27 2.2.2 Fourier-Based Spectral Analysis of Water Waves .............................28 2.2.3 Power Spectral Densities ...................................................................28 2.2.4 Unwrapped Phase Spectra .................................................................33 2.2.5 Time-Dependent Spectral Analysis....................................................36 2.2.6 Disintegrated Components .................................................................38 2.2.7 Estimated Power Spectral Densities of Disintegrated Modes...........41 2.3 Relations Between Mean Period and Energy Density ..................................43 2.3.1 Motivation ..........................................................................................43 2.3.2 Characteristic Quantities ....................................................................43 2.3.3 Averaged Periods of IMF...................................................................45 2.3.4 Interaction Between Energy Density and Averaged Periods ............47 2.3.5 Accordance with Theoretical Treatments ..........................................48 2.4 Summary and Discussion...............................................................................54 Acknowledgments....................................................................................................55 References................................................................................................................55

25

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ABSTRACT A new interpretation of nonlinear wind–wave systems is proposed by means of the empirical mode decomposition and Hilbert spectral analysis on rogue waves. Based on data series recorded in the central North Sea, it is shown that the inherent structure of real sea states is assembled of a finite number of carrier and riding wave components. These predominant constituents are tentatively determined through ordinary Fourier-based spectra but inevitably proven by disintegrating the recorded data series into characteristic oscillations with the empirical mode decomposition. Spectral components are not considered a random collection of free waves, each obeying the usual dispersion relation, but are effectively nondispersive bound-wave components of an ensemble of single dominant carrier waves. Relations and interactions between energy density and averaged periods of the intrinsic modes are determined and correspond to previously done investigations on white noise and length-of-day data (Huang et al., 2003).

Keywords Time–frequency analysis techniques, Hilbert spectral analysis, empirical mode decomposition, irregular water waves, perturbation expansion approach, rogue waves, wave breakdown and modulation, wave self-focusing

2.1 INTRODUCTION A significant number of reported damages to ships and offshore structures suggest the existence of rogue waves (or freak waves), which are characterized by single exceptional large wave heights [1]. Rogue waves are defined as transient waves, existing only in one specific location in one particular instant in time. The assumption that these waves occur is preliminary, based on superposition of an infinite number of linear (free) wave components that coincide in phase during storm events (see e.g.: [2], [3], or [4]). Other attempts to explain them focus on wave–current interaction processes as a possible mechanism to generate or to enlarge the possibility of those abnormal events [5]. Extensive work has been carried out to study these phenomena, either to prove their existence by attempting to record rogue waves or to examine primary driving mechanics and perform numerical simulations (see e.g.: [3], [4], [25], [7], and [8]). Other researchers reproduce transient waves under laboratory conditions and derive useful kinematic insights in order to establish advanced design criteria for offshore structures and deep-sea vessels (see e.g.: [1], [9], [10], or [11]). However, simple superposition of free wave components appears not to be an adequate tool to simulate rogue waves, since linear wave theories fail to describe crest and trough heights; they possess no Gaussian or even near-Gaussian statistics [12]. Taking the mathematical approach of the perturbation expansion theory to higher order degrees and, yet, allocating wave–wave interaction between nonlinear components often makes little physical sense, although numerical models [10] and analytical approaches [13] match data measured from laboratory experiments exceptionally well. Whether those approaches are appropriate to fit recorded extreme wave events from real sea states is still uncertain. Recent research results propose different, more complex underlying hydrodynamic mechanisms [14, 15] that are based on phenomena defined as wave breakdown

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and focusing and self-organization. However, it is not in the scope of the present paper to deepen those nonlinear theories; rather, it is our purpose to identify the embedded structure of rogue waves to determine physical insights about their occurrence in accordance with those newly developed theoretical explanations. Therefore, the present paper relates to the newly developed Hilbert-Huang transformation (HHT) pioneered by Huang et al. [16, 17]. The numerical procedure of the key element, the so-called empirical mode decomposition (EMD), is made clear in principle, although we direct the interested reader to study the elementary publications by Huang et al. In this context, we first examine the generation of validated Fourier-based spectra and phase representations, since these are understood to be most familiar to the reader associated or working in ocean engineering or other related fields. The performance and limitations of those time-invariant techniques provide an introduction to the second section. Here, we apply the EMD and emphasize an appropriate calculation of the embedded modes from the original signals. Next, we quantify a certain essential parameter within the intrinsic mode function (IMF) and attempt to discover internal relationships between associated energy densities and averaged periods. We then compare our data with proposed classifications on this decomposition technique performed for synthetically generated noisy data by Flandrin et al. [18] and Wu and Huang [19]. We demonstrate that our data accord, tentatively, with the above-mentioned recently established, more complex generation mechanisms of rogue waves. Finally, we summarize the main outcomes of the present investigation and offer conclusions for future research.

2.2 ROGUE WAVE MEASUREMENTS 2.2.1 PRELIMINARIES The present paper investigates rogue waves recorded in the central North Sea during January 1995, particularly one extreme event when a gigantic wave with an approximate wave height Hmax = 25.6 m appeared during a severe storm with a significant wave height Hs = 11.9 m (20-min interval). Time series of the water surface elevations were measured by a downward-oriented laser device (fsampling = 2.1333 Hz) installed at a transparent jacket platform (Draupner) operated by STATOIL, Norway, in water depths of 70 m. Figure 2.1 shows this event and clearly demonstrates the unusual asymmetry of this transient wave, as the single crest height exceeds Hcrest = 18.5 m with adjacent troughs of only less than one third that height (Htrough = 7 m). Haver and Andersen [12] point out that calculations using second order theory reveal the probability of obtaining this particular event to be 1:1000. From a general statistical point of view, this abnormal transient wave is indeed categorized as a rogue wave, since its relations Hmax/Hs 2 and Hmax 10 correspond to basic conventions established through the List of Sea State Parameters [20] and Kjeldsen [21]. Figure 2.2 shows another 20-min recorded signal, recorded immediately after the data series that contains the abnormal event shown in Figure 1.1. Obviously, no second rogue wave is registered in this signal, although the severe storm continued in the central North Sea with an almost constant significant wave height Hs = 12.0 m (20-min interval).

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20

15

η(t) (m)

10

5

0

−5

−10

0

200

400

600 time (s)

800

1000

1200

FIGURE 2.1 Signal 1: New Year’s Eve rogue wave 01/01/1995, Draupner platform, STATOIL, Norway.

2.2.2 FOURIER- BASED SPECTRAL ANALYSIS

OF

WATER WAVES

Various idealized spectra are used in ocean engineering to study how wind interacts with the upper surface of the ocean. The most fundamental is that proposed by Pierson and Moskowitz [22], who assume that as the wind blows steadily for a long time over a large area, the waves would come into equilibrium with the wind. This concept of a fully developed sea is used to interpret all forthcoming spectral representation of the measured sea states in the central North Sea.

2.2.3 POWER SPECTRAL DENSITIES Figure 2.3 through Figure 2.10 show power spectral densities and corresponding phase spectra for both time series. These are qualitatively evaluated in regard to a general Pierson-Moskowitz spectrum (PM spectrum). The Nyquist frequency for each Fourier-based time-invariant spectrum is fNyquist = fsampling /2 = 1.0667 Hz. The raw power spectral density of the “New Year’s Eve” rogue wave is shown in a double logarithmic plot in Figure 2.3, with constant frequency resolution df = 1/(2Ttotal) = 4.1667 × 10-4 Hz. The spectral energy distribution of this wave is renowned for illustrating a fully developed real sea state according to a typical PM spectrum: a characteristic steep positive energy gradient rising from low infra-gravity waves to

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20

15

η(t) (m)

10

5

0

−5

−10

0

200

400

600 time (s)

800

1000

1200

FIGURE 2.2 Signal 2: Sea state recorded immediately after the New Year’s Eve rogue wave 01/01/1995, Draupner platform, STATOIL, Norway.

peak components and a relatively gentle negative energy slope adjacent to the peak waves on the “tail” of the spectrum that approximately follows a f–5 law [23]. According to the fundamental perturbation expansion approach, which is understood to simulate nonlinear water waves accurately, the latter part of the spectrum is usually recognized to contain the high order nonlinear harmonics of the full wave train. This particular spectrum of the New Year’s Eve rogue wave exhibits peak components with frequencies between 6 × 10–2 Hz and 9 × 10–2 Hz, but highly misleading spectral fluctuations make an overall interpretation of a biased spectrum impossible. It turned out to be most effective to proceed with some spectral smoothing techniques. Figure 2.4 displays the estimated power spectral density using Welch’s method. Here, the raw input signals with N = 2,560 data points is subdivided into four, five, and ten sections of equal length (nwin = 640, 512, and 256, respectively), to give an idea of the most appropriate spectral smoothing method. With an overlap between the individual windows of 32 data points (5%, 6.25%, and 12.5% of window length), each subdivision is carried out by means of Welch’s averaged modified periodogram method of spectral estimation. The Nyquist frequency remains constant. In respect to the raw data series, spectral frequency resolution becomes coarser with more data segmentation; df = 1.6667 × 10–3 Hz, 2.0833 × 10–3 Hz, and 4.1667 × 10–3 Hz for Welch spectra with equal-length sections nwin = 640, 512, and 256, respectively.

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103

102

PSD (m2/Hz)

101

100

10−1

10−2

10−3 10−2

10−1 Frequency (Hz)

100

FIGURE 2.3 Raw power spectral density of signal 1.

Evidently, energy distribution is represented in an efficient way, as Figure 2.4 shows. The overall shape of the spectrum is smoothly resolved for all window lengths and exhibits peak component frequencies around approximately fpeak = 7 × 10–2 Hz, similar to the raw input spectrum. In fact, two of those estimated Welch spectra [pwelch(640,32) and pwelch(512,32)] reveal a double peakedness of this sea state. The predominant mode is located at a frequency fpeak1,1st = 5.7 × 10–2 Hz, and the second, less energetic, yet governing carrier wave component is at fpeak2,1st = 8.3 × 10–2 Hz. The third averaged spectrum [pwelch(256,32)] is incapable of resolving both components and instead shows a joint peak range of components in the range 5 × 10–2 Hz to 9 × 10–2 Hz. It is noteworthy at this stage that Olagnon and van Iseghem [24] study measured extreme waves from the North Sea and categorize these recorded abnormal events in terms of spectral bandwidth, multiple peakedness, and other characteristics. They find that that the percentage of double peaks detected in the spectra for severe sea states with extreme waves is smaller than the percentage in single-peaked sea states, but they also find that there is no double-peaked sea state at all for the 10% “most extreme” waves in their selection. Thus, the present spectral examination seems to disagree with their observations, but we must ask, how is double-peakedness defined in this context? Olagnon and van Iseghem [24] dispute their statistical outcome that

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103 pwelch(640, 32) pwelch(512, 32) pwelch(256, 32) 102

PSD (m2/Hz)

101

100

10−1

10−2

10−3 10−2

10−1 Frequency (Hz)

100

FIGURE 2.4 Estimated power spectral density of signal 1 made using Welch’s method.

only multidirectional waves can generate double-peaked spectra. But, in the case of the New Year’s Eve rogue wave, a clear unidirectional wave field emerged from the severe storm, as Trulsen [25] significantly points out, since the measured response of the platform was found to be solely unidirectional. It is therefore concluded that the waves were essentially long-crested. So, our result shown in Figure 2.4, a particular double-peakedness with very closely resolved carrier wave components, can be validated. We will show that both dominating carrier wave groups fpeak1,1st and fpeak1,2nd are resolved as well by the EMD. Moreover, higher order nonlinear superharmonics — either bounded or dispersive (free) wave components of higher frequency — are evident at fpeak1,2nd = 1.01 × 10–1 Hz and fpeak2,2nd = 1.03 × 10–1 Hz, as shown in Figure 2.4. These specific riding wave groups show significant smaller energy fraction than the dominant wave modes at peak frequencies in the spectrum and are clearly evident from all three Welch spectra. Energy portions of infra-gravity components and other higher order superharmonics are of minor importance. Power spectral densities for the second data series of the sea state are shown in Figure 2.5 and Figure 2.6. Obviously, we see no significant differences to the raw spectrum in Figure 2.3, as high-frequency fluctuations disrupt the overall spectral energy content and make an appropriate interpretation difficult. Smoothed estimated

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103

102

PSD (m2/Hz)

101

100

10−1

10−2

10

−3

10−2

10−1 Frequency (Hz)

100

FIGURE 2.5 Raw power spectral density of signal 2.

spectra obtained with Welch’s methods also identify the double-peaked dominant components, but this time they are less meaningful. Only the first subdivided Welch frequency representation [pwelch(640,32)] helps to reveal the preceding carrier components, at fpeak1,1st = 5.7 × 10–2 Hz and (now slightly lower) at fpeak2,1st = 7.5 × 10–2 Hz. Nevertheless, the rest of the spectrum from this data series is mostly identical to the one examined before — even the riding wave components are evident at fpeak1,2nd = 1.01 × 10–1 Hz and fpeak2,2nd = 1.03 × 10–1 Hz. From these simple spectral estimates obtained with Welch’s method, the prevailing components, which are supposed to be accountable for the occurrence of this particular abnormal rogue wave, are evaluated, leading to a physical model of two (or three) main carrier wave components and two riding wave components. Their phase orientation and time-dependent dispersive propagation are supposed to be in charge for any transient phenomena occurring at sea. At this stage, we can conclude that it is not possible to understand the occurrence of rogue waves through an examination of power spectral densities alone. Transient data are completely misinterpreted, since essential time-dependent information is cancelled out through ordinary Fourier-based techniques. The occurrence of rogue waves on the open sea is therefore simply disregarded in Fourier-based power spectra density representations. Nonetheless, to support this mathematical approach on the whole, we now look more closely at the corresponding phase spectra.

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103 pwelch(640, 32) pwelch(512, 32) pwelch(256, 32) 102

PSD (m2/Hz)

101

100

10−1

10−2

10−3 10−2

10−1 Frequency (Hz)

10

0

FIGURE 2.6 Estimated power spectral density of signal 2 made using Welch’s method.

2.2.4 UNWRAPPED PHASE SPECTRA Figure 2.7 presents unwrapped phase spectra for the raw signal of the data series containing the abnormal wave event. The radian phase angles are corrected by adding multiples of ±2 when absolute jumps between consecutive array elements are greater than radians. Informational content is to some extent undefined since phase deviations between components within the whole raw spectrum are large, and significant phase fluctuation occurs. Figure 2.8 uses Welch’s method to evaluate the phase spectrum from the averaged unwrapped phases corresponding to the power spectral density in Figure 2.4. Initial phases for both previously derived dominating carrier wave groups, at fpeak1,1st = 5.7 × 10–2 Hz and fpeak2,1st = 8.3 × 10–2 Hz, and both subsequent riding waves, at fpeak1,2nd = 1.01 × 10–1 Hz and fpeak2,2nd = 1.03 × 10–1 Hz, are significantly dissimilar from all three Welch spectra. However, it seems reasonable to study the unwrapped phases between the prevailing components. It is most important to learn about how these wave groups are structured among themselves and about their individual dependence on each other. Table 2.1 summarizes derived results for the unwrapped phases from signal 1. These results indicate that, from a mathematical point of view, all predominant wave groups certainly do not coincide in phase, and that the New Year’s Eve rogue wave is not due to that physical constellation — although we do observe that the

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100 0

Unwrapped phase (radians)

−100 −200 −300 −400 −500 −600 −700 −800 10−2

10−1 Frequency (Hz)

100

FIGURE 2.7 Raw unwrapped phase (radians) spectra of signal 1.

phase lag between the components tends to be constant and, therefore, is mainly independent of the window length of the Welch spectrum. Otherwise, however, this analysis technique is categorically unqualified to examine supposed elementary generating mechanisms, since it is mainly based on averaging the unwrapped phases from the individual windows. Therefore, implemented variations are pronounced excessively and yield to misleading results. To validate this conclusion, we similarly analyzed the second signal. Calculated results for the raw unwrapped phase spectra of the second data series, shown in Figure 2.9, are almost identical to those of the first data series. Note that these unfiltered phase spectra are rather unsatisfactory for physical interpretation. Again, Welch’s estimated spectral phase representations are applied, and results are shown in Figure 2.10. Previously proposed effects are shown to be valid for this data series as well, since all three Welch representations produce significant dissimilar phase spectra. This indirectly points to the fact that both underlying peaks and both higher order superharmonic wave groups are partially out of phase. Table 2.2 summarizes the derived results for the unwrapped phases from signal 2. Consequently, we conclude that no anomalous effects concerning driving mechanisms of tentative rogue waves are apparent in the data series. Fourier-based analysis seems to be effective as long as the mathematical model attempts to formulate the wave groups as stationary and composed solely of linear components

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3π

pwelch(640, 32) pwelch(512, 32) pwelch(256, 32)

Unwrapped phase (radians)

2π

π

0

−π

−2π 10−2

10−1 Frequency (Hz)

100

FIGURE 2.8 Estimated unwrapped phase (radians) spectra of signal 1 made using Welch’s method.

TABLE 2.1 Unwrapped Phases of Significant Components from Signal 1

pwelch(640,32) pwelch(512,32) pwelch(256,32)

fpeak1 ,1st

fpeak2 ,1st

fpeak1 ,2nd

fpeak2 ,2nd

0.8 0.4 0.9

0.5 0 0.6

1.1 0.7 0.9

1.3 0.2 0.6

that are superimposed in a random sea state. Based on perturbation expansion approaches, Fourier-based analysis seems to match results from Figure 2.8, as peak and corresponding superharmonic components coincide in phase. We know from theoretical models based on Stokes wave theories that these higher order superharmonics are bounded to the fundamental wave train. The phase mismatch between fundamental and superharmonic components evident in Figure 2.10 disproves this assumption and hints at the fact that both dominating components in the spectral

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100 0

Unwrapped phase (radians)

−100 −200 −300 −400 −500 −600 −700 −800 10−2

10−1 Frequency (Hz)

100

FIGURE 2.9 Raw unwrapped phase (radians) spectra of signal 2.

representation are individual dispersive wave trains — in essence, a carrier and a riding wave component that both propagate and modulate autonomously. That the corollaries we have derived are really appropriate for a tentative physical model of phenomena like the New Year’s Eve rogue wave in the central North Sea is rather doubtful, since each of the spectra we have studied here is exclusively based on time-invariant Fourier techniques. How can we prove the underlying physics or generating mechanisms of a transient phenomenon with time-ignoring mathematical spectral instruments? Moreover, as we applied Welch’s spectral estimates to derive those results, one might raise the question, what is the average confidence limit embedded in this time-invariant method? Are those phase spectra validated in representing the physical model for generating abnormal waves of extreme wave heights? In the next section we try to find answers to these queries as we apply a newly developed time-dependent spectral analysis method, the empirical mode decompositon and the Hilbert transform, to either strengthen or refute these theoretical attempts.

2.2.5 TIME- DEPENDENT SPECTRAL ANALYSIS Time–frequency dependent analysis methods are novel approaches in the field of applied mathematics, having rapidly emerged as common subjects of research and application in scientific and engineering investigations in the last two decades [26].

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3π pwelch(640, 32) pwelch(512, 32) pwelch(256, 32)

Unwrapped phase (radians)

2π

π

0

−π

−2π 10−2

10−1 Frequency (Hz)

100

FIGURE 2.10 Estimated unwrapped phase (radians) spectra of signal 2 made using Welch’s method.

TABLE 2.2 Unwrapped Phases of Significant Components from Signal 2

pwelch(640,32) pwelch(512,32) Pwelch(256,32)

fpeak1 ,1st

fpeak2 ,1st

fpeak1 ,2nd

fpeak2 ,2nd

0.2 0 0.6

0.5 0.1 0.7

0.1 0.1 0.2

0 0.1 0.4

Consequently, these techniques are also applied to ordinary wave data and are approved in the coastal and ocean engineering disciplines. Liu [27, 28, 29] introduces the wavelet transform technique in its entirety and analyzes coastal wave data sets from the Great Lakes. Chien et al. [30] analyze nearshore observed extreme wave events off the east coast of Taiwan. Schlurmann [31, 32] previously studied time–frequency representations of rogue waves in the Sea of Japan and focused on a direct comparison between wavelet and Hilbert spectra.

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However, within the present study, another time-dependent spectral analysis tool, the empirical mode decomposition (EMD), is applied to both data series recorded in the central North Sea. As mentioned, this technique has been developed by Huang et al. [16, 17, 33] and recently extended by Flandrin and his group [18, 34]. The EMD is capable of disintegrating transient, nonlinear data into its own embedded characteristic oscillations on a non–a priori basis. This feature constitutes the essential difference from other time–frequency analysis techniques, such as wavelet transformations. Huang et al. provide a general overview of EMD and its recent applications [16, 17, 33], and full details are given by Dätig and Schlurmann [36].

2.2.6 DISINTEGRATED COMPONENTS We processed recorded water surface elevations during the severe storm in the central North Sea to find governing individual wave components constituting the overall sea state. The present study makes use of the extrema-sifting algorithm and is based on the second characteristic stopping criteria for the numerical algorithm, which is obtained when the total number of zero-crossings is equal to or differs at most by one from the sum of maxima and minima derived from the upper and lower envelope of the data series, respectively. No curvature sifting or intermittency checking has been performed. By means of this technique the first data series and the signal recorded immediately after it are separated into specific components. The first signal is disintegrated into ten intrinsic mode functions (IMFs), and the subsequent data set is subdivided into a total of nine IMFs. Each computed disintegrated mode leads to some residual signal, which represents a kind of trend of the signal itself. The numbers of sifting operations to invoke the stopping criterion were nbits = [107,19,11,14,9,16,14,12,14] for the wave record containing the transient water wave and nbits = [80,24,13,13,9,8,5,4] for the second data series. The total number of siftings within the EMD, as well as the full amount of IMFs, correspond well to earlier studies on other data series (e.g., [32], [35], and [33]). The overall orthogonal indexes are IE1 = 0.1124 for the first signal and IE2 = 0.0736 for the second. The magnitudes of the latter parameter correspond closely to the results obtained by Huang et al. [33] on daily length-of-day data and demonstrate the overall numerical performance of the present analysis. Derived IMF for both data series are given in Figure 2.11 and Figure 2.12. It is obvious that c1 to c4 carry major parts of the embedded energy in each signal. Time-dependent amplitudes and time scales constitute the essential information generating the sea state. Simple inspection by eye proves the nonexistence of mode mixing, making further intermittency testing unnecessary. A closer look at the disintegrated IMF reveals that both sea states are composed of two (or three) carrier waves and one superimposed riding wave. The latter component represents the first IMF and mainly embeds the high frequency constituents of each signal. We know from the literature that the EMD decomposes any data sets beginning with the “detail” of the signal. Clearly, this technique cannot derive this particular rogue wave event as an individual component, as it is entirely enclosed in the high frequency part c1 of the measured water surface elevation. The abnormal wave is distinctively embedded in

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c1

η(t) (m)

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39

20 0 −20 10 0

c2

−10 10 0

c3

−10 5 0

c4

−5 2 0

c5

−2 1 0

c6

−1 0.5 0

c7

−0.5 0.5 0

c8

−0.5 0.5 0

c9

−0.5 0.1 0

rn

−0.1 0.5 0 −0.5

0

200

400

600 time (s)

800

1000

1200

FIGURE 2.11 The IMF components from extrema-based sifting without intermittency testing of signal 1.

the first IMF at around t = 280 sec, with an approximate wave height of 12 m. Compared to this unusually large wave height in c1, the rest of this particular mode is characterized by relatively low amplitudes. It is well accepted from preceding investigations that it is the individual time scale of an embedded mode that makes the EMD work successfully in disintegrating meaningful components from any given

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c1

η(t) (m)

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20 0 −20 5 0

c2

−5 20 0

c3

−20 5 0

c4

−5 5 0

c5

−5 1 0

c6

−1 1 0

c7

−1 0.5 0

c8

−0.5 0.05 0

rn

−0.05 0.4 0.2 0

0

200

400

600 time (s)

800

1000

1200

FIGURE 2.12 The IMF components from extrema-based sifting without intermittency testing of signal 2.

signal, and not its time-dependent amplitude. We can conclude that the existence of the New Year’s Eve rogue wave is a sudden and unexpected phenomenon within the data series, being part of the high frequency component of the sea state. Spectral analysis of sifted IMF will resolve further particulars. Regardless of this observation, the higher order modes c2 to c4 emerge as almost of constant period, but allowing time-dependent amplitude variations. This feature builds the main advantage of the

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103 x1(t) IMF1(t) IMF2(t)

102

IMF3(t) IMF4(t) IMF5(t)

PSD (m2/Hz)

101

IMF6(t) IMF7(t) IMF8(t)

100

10−1

10−2

10−3 10−2

10−1 Frequency (Hz)

100

FIGURE 2.13 Estimated power spectral density made using Welch’s method [pwelch (512,32)] from raw data (signal 1) and IMF components of signal 1.

EMD in regard to some Fourier-based analysis techniques applied earlier. Moreover, investigations carried out by Dätig and Schlurmann [36] on laboratory generated bichromatic waves processed with the EMD are expected to show that the sampling frequency of the present wave records (fsampling = 2.1667 Hz) is by far too low to permit distinguishing between closely frequency related components. This assumption is not applied and therefore not proven within the present investigation. Dätig and Schlurmann suggest employing certain artificial upsampling techniques to obtain further information on transient phenomena.

2.2.7 ESTIMATED POWER SPECTRAL DENSITIES

OF

DISINTEGRATED MODES

Spectral representations of the IMF from the first signal are shown in Figure 2.13. Here, we again apply Welch’s estimated power spectral density method with window lengths of 512 data points and an overlap of 6.25% (32 data points) to show the spectra in double logarithmic plots. Note that the additional window lengths [pwelch(640,32), pwelch(256,32)] used previously are neglected this time for the sake of easy interpretation. Nonetheless, the main results with other window lengths have been validated. In addition, the frequency distributions of the corresponding original data series are shown to achieve better comparison of the subsequent spectral representations of all disintegrated IMF.

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As assumed previously, it is now proven that c1 in Figure 2.13 contains all high frequency contents of the signal. From a cut-off frequency of about 1.1 Hz its distribution is identical to that of the original data series. This first IMF is extremely broad banded with no significant peak component. It ranges from 10–2 Hz up to the Nyquist frequency fNyquist = 1.0667 Hz with only slightly varying energy distribution, but it fails to embody any of the carrier or riding wave components mentioned earlier. The second IMF implements a substantially smaller frequency bandwidth, clearly constituting the higher order superharmonic components at around fpeak1,2nd = 1.01 × 10–1 Hz and fpeak2,2nd = 1.03 × 10–1 Hz and almost the full energy content of one of the dominant carrier waves at fpeak2,1st = 8.3 × 10–2 Hz. The subsequent mode, c3, contains the other dominating carrier wave group at fpeak1,1st = 5.7 × 10–2 Hz. Nonetheless, both components c2 and c3 show a certain degree of frequency overlapping, but they do individually characterize a predominant wave group inside the spectrum. At this stage, it is remarkable that the EMD automatically sifts those components that were previously identified from ordinary Fourier-based transformations. Higher order modes disintegrated from the EMD are situated in lower frequency regions, but these play only marginal role in the overall energy spectra. Although those disintegrated modes principally match observation from Fourier analysis, suggesting that the EMD is a successful method for analyzing transient water waves, it is cumbersome that this particular sifting procedure artificially creates energy modes in the infra-gravity wave range of the spectra. These are not evident in spectral representation of the original data set. Modes c1 to c8 do all show higher energy magnitudes in the lower frequency region than the water surface elevation from the measured sea state initially shows. Dätig and Schlurmann [36] have reported this effect in bichromatic wave groups and concluded that it is an artifact of the numerical sifting algorithm of the EMD. We do not clarify this mismatch within the present paper and suggest further investigations in future research projects. Figure 2.14 illustrates the spectral representation of characteristic oscillations of the second data set. Again, Welch’s estimated power spectral density method is applied, with identical specifications to the spectra shown in Figure 2.13. The bold solid line indicates the spectral distribution of the original data series. Again, c1 is broad banded and mainly shows wave groups with relatively high frequencies. It is identical to the initial signal from an approximate cut-off frequency of 1.1 10–2 Hz. In contrast to the first signal, the second IMF accords with the assumptions on carrier and riding waves stated earlier. Note that only mode c2 implements peak carrier wave groups as well as the assumed riding wave components. The EMD fails in distinguishing those characteristic oscillations and finally yields just nine modes compared to ten IMFs from the first signal. The third IMF, c3, shows a significant overlap with c2, but it cannot exclusively occupy the carrier wave group around frequencies of fpeak1,1st = 5.7 × 10–2 Hz. Higher order modes disintegrated from the EMD are also located in the lower frequency regions. Their influence on the overall appearance of the spectral decomposition is marginal and can be neglected. As already observed, cumbersome artificially created energy modes in the infra-gravity wave range of the spectra are also verified for the second signal and are due to numerical artifacts of the sifting procedure within the EMD.

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43

x1(t) IMF1(t) IMF2(t)

102

IMF3(t) IMF4(t) IMF5(t)

PSD (m2/Hz)

101

IMF6(t) IMF7(t)

100

10−1

10−2

10−3 10−2

10−1 Frequency (Hz)

100

FIGURE 2.14 Estimated power spectral density made using Welch’s method [pwelch (512,32)] from raw data (signal 2) and IMF components of signal 2.

2.3 RELATIONS BETWEEN MEAN PERIOD AND ENERGY DENSITY 2.3.1 MOTIVATION Flandrin et al. [18] show that the EMD essentially acts as a dyadic filter bank in stochastic situations involving broadband noise. Wu and Huang [19] point out that similar results have been obtained independently and provide evidence that in the case of white noise the product of the mean period and energy density of the intrinsic modes is constant. We examined both signals from the central North Sea to determine whether these idealistic self-determinations from synthetic data series are verified to any degree in natural signals.

2.3.2 CHARACTERISTIC QUANTITIES The first quantity that was measured was the total number of extrema and zero-crossings identified in the sifting process of the EMD. Next, the averaged period and corresponding standard deviation of each empirical mode was calculated. Flandrin

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TABLE 2.3 Data for Signal 1. (a) Number of Extrema, (b) Number of Zero-Crossings, (c) Mean Periods of IMFs in sec, (d) Corresponding Standard Deviation (STD) in sec mode k (a) (b) (c) (d)

1

2

3

751 751 3.2 2.3

255 256 9.4 4.2

142 142 16.9 5.4

4

5

6

85 84 28.4 8.4

47 47 51.0 11.5

27 27 89.3 22.6

7

8

9

10

16 15 155.7 30.6

9 8 260.2 65.1

6 5 416.5 52.1

1 2 180.9 NA

TABLE 2.4 Data for Signal 2. (a) Number of Extrema, (b) Number of Zero-Crossings, (c) Mean Periods of IMFs in sec, (d) Corresponding Standard Deviation (STD) in sec mode k (a) (b) (c) (d)

1

2

3

4

5

6

7

8

9

801 801 3.0 2.3

256 255 9.4 4.2

136 136 17.7 5.4

80 80 29.9 8.4

41 40 58.9 11.5

21 20 121.3 22.6

6 6 366.0 30.6

3 4 677.2 65.1

1 NaN NaN NaN

et al. [18] make clear that the average period of these zero-crossings is a rough indication of the mean embedded frequency of each IMF. The way this number varies from mode to mode is further indication of the hierarchical structure of the decomposed natural data series from the central North Sea. The standard deviation also suggests the bandwidth of the disintegrated mode. Table 2.3 summarizes results for the first signal, containing the abnormal wave. The same measurements and specifications were made for the second data set; these are listed in Table 2.4. Clearly, the natural data sets do not behave like white noise or any other artificially created data. Flandrin et al. [18] and Wu and Huang [19] suggest an interesting pattern in stochastic data: that the average period of any IMF component almost exactly doubles that of the previous one. They conclude that this behavior establishes the EMD as performing as a dyadic filter bank. This mathematical feature is not drawn from the present data sets. However, as Table 2.2 and Table 2.4 show, the first five modes of each signal correlate quite well. At least, the number of extrema, mean periods, and corresponding standard deviations are strongly related. This pattern is especially interesting given that we are analyzing two independent data series.

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10 9 8

Log2 (period)

7 6 5 4 3 2 1 0

0

1

2

3

4

5 6 Mode k

7

8

9

10

11

FIGURE 2.15 Circle: Log2 mean periods versus mode number k of signal 1. Solid line: Log2 linear fit. Dashed line: confidence limits.

2.3.3 AVERAGED PERIODS

OF

IMF

To help us better understand these parameters and how they emerge from their inherent hierarchical structure, Figure 2.15 and Figure 2.16 present the (log2) logarithm of the averaged periods (zero-crossings), which are plotted versus the mode number of the IMF. These quantities are illustrated by black, hollow circles. For simplicity, plus and minus one standard deviation for the 68% upper and lower confidence levels (dashed lines) have been adopted for all modes for signal 1 (nine modes) and signal 2 (eight modes). The tenth mode for the first and the ninth mode for the second signal are disregarded in this context, as they only represent the global trend of the individual decomposition. In detail, Figure 2.15 shows a kind of motivating characteristic. This log2(Tk) versus mode number k plot reveals linear behavior between modes c2 and c9, represented by the solid black line. It follows that: Tk = 2( ak + b )

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10 9 8

Log2 (period)

7 6 5 4 3 2 1 0

0

1

2

3

4

5 6 Mode k

7

8

9

10

11

FIGURE 2.16 Circle: Log2 mean periods versus mode number k of signal 2. Solid line: Log2 linear fit. Dashed line: confidence limits.

where k is the mode number and a and b are real valued coefficients. A least squares method determines a = 0.7877 and b = 1.7039 to fit the average period log2(Tk). Modes c1 and the residue c10 heavily deviate from the linear fitted function. As shown earlier, spectral distribution of the first IMF is very broad-banded and represents a kind of half-band high pass filter. It contains more or less exclusively the high frequency fractions of the original signal, so that the mean period of c1 is far less than that of the linear fitted function from c2 to c9. The dotted black line, which is constructed from simple extrapolation of the latter linear fitted function, makes this clear. The mathematical behavior of the other signal is shown in Figure 2.16. Here, only modes c2 to c5 appear to obey this strict (log2) linearity of the mean periods. Modes c1 and IMFk for 6 k 9 show strong divergence from this log2 linearity. Coefficients for the second data series are determined to be a = 0.8709 and b = 1.4917 to fit the averaged periods log2(Tk) with 2 k 5. This behavior tends to underestimate higher modes (6 k 9), whereas c1 is categorically overestimated, as the dotted black line reveals — again determined through extrapolation of the perfectly fitting function from modes c2 to c5. Flandrin et al. extend their method to determine whether those dyadic filter modes can be classified for a possible self-similarity behaviour of their data. This feature is (so far) unnecessary in application to

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the present data, since transient water waves from the central North Sea evidently disobey the dyadic filter bank characteristic.

2.3.4 INTERACTION BETWEEN ENERGY DENSITY

AND

AVERAGED PERIODS

Wu and Huang [19] provide evidence that for synthetically generated white noise the product of the mean period and energy density of the intrinsic modes is constant. As the Fourier spectral shape for each IMF has already been described, finding the total energy density Ek embedded in each characteristic mode is not difficult. Moreover, Tk is the averaged period of each IMFk, which is identical to that derived from counting the zero-crossings (see Table 2.3 and Table 2.4). However, Wu and Huang obtain a simple equation that relates the energy density and the averaged period: Tk ⋅ Ek = p (= const )

(2.2)

log10 (Tk ) + log10 ( Ek ) = q (= const )

(2.3)

or

Since it is impossible to normalize the present decomposed data series, it is not presumed that the p on the right side of Equation 2.2 is unity; therefore, q in Equation 2.3 is expected to differ from zero. Figure 2.17 presents this mismatch for both signals. Obviously, modes c1 and c4, up to the residuals c10 and c8, respectively, tend to follow this assumption. The product TkEk is nearly constant. Mean horizontal expectation lines can be located at approximately TkEk = 3 for the first signal and at TkEk = 4 for the second signal. Clearly, however, modes c2 and c3 deviate significantly (by a factor of about 15) from both approximated constants derived for the other modes. Why are these IMFs so different? Modes c2 and c3 unmistakably embed the largest energy fractions, since they both represent the major part of the disintegrated signals. Additionally, they involve relatively long mean periods, so that the product TkEk differs from the others. This is just another confirmation of their predominant status within the decomposed data sets. Figure 2.18 attempts to clarify the other assumptions established in Equation 2.3. The energy density log10(Ek) is plotted versus the mean period log10(Tk) of each mode from both data series. In addition, fitted log10-lines illustrate a close correspondence between all disintegrated modes. From a least squares fit, the solid black line characterizes expectation line data from the first signal; a dashed line pertains to the second signal. Both undeniably show similar parallel slopes and follow the mathematical expression: Ek = 10 cTk + d

(2.4)

where c = –2.0264 and d = 2.5001 are derived for the modes from the first data series and c = –2.0118 and d = 2.6944 are calculated for the IMF of the second data

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100 90 80 70

Tk* Ek

60 50 40 30 20 10 0

0

1

2

3

4

5

6

7

8

9

10

11

Mode k

FIGURE 2.17 Circle: product of energy density and mean period periods versus mode number k of signal 1. Square: product of energy density and mean period versus mode number k of signal 2.

series. Only modes c1 and the residuals c10 show significant divergence from being fitted. Clearly, Equation 2.4 offers an excellent correlation to these fitted points. Thus, we can prove for natural data series from the central North Sea that the energy density of the IMF and its averaged period follows a hyperbolic function. This outcome is in agreement with results obtained by Wu and Huang [19].

2.3.5 ACCORDANCE

WITH

THEORETICAL TREATMENTS

We have mainly considered the classification of results derived from the empirical mode decomposition in the context of overall performance, energy content, and mean period for accurate comparison with other data series, especially white noise. It is therefore necessary to return to the basic objectives and attempt to discover how the EMD performs on transient wave data from the central North Sea. We have mentioned that simple superposition of free wave components does not appear to be an adequate tool to simulate rogue waves. Linear wave theories fail to describe crest and trough heights, as they possess no Gaussian or even near-Gaussian statistics [12]. Also, higher order techniques — allocating wave–wave interaction between nonlinear components — often make little physical sense, so we might question whether those approaches are appropriate for matching recorded extreme

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1 0.5 0 −0.5

Log10 (Ek)

−1 −1.5 −2 −2.5 −3 −3.5 −4

0

0.5

1

1.5

2 Log10 (Tk)

2.5

3

3.5

4

FIGURE 2.18 Circle: relation of energy density and mean period periods of signal 1. Square: relation of energy density and mean period of signal 2. Solid line: Log2 linear fit, signal 1. Dashed line: Log2 linear fit, signal 2.

wave events from real sea states. As Onorato et al. [14], Pelinovsky et al. [15], and Trulsen and Dysthe [6] point out, the utility of those models has not yet been determined. They propose different, more complex underlying hydrodynamic mechanisms for the generation of rogue waves, based on phenomena named wave breakdown and focusing and self-organization. These phenomena are described by an envelope function on the basis of the nonlinear Schrödinger equation and higher order corrections (Dysthe equations), yielding an equation hierarchy that attempts to explain an abnormal wave within real sea states through a simple model. It is found numerically that rogue wave events are mainly related to the initial steepness and to the correlation length of the complex envelope of the initial time series. The effects of wave breakdown and focusing and self-organization arise through the concept of Benjamin-Feir [37] instability mechanisms, which cause local exponential growth in the amplitude of a wave train. Still unknown is how rogue waves can generate these instabilities since realistic oceanic conditions are not characterized by a single monochromatic wave perturbed by two small-amplitude parasitic waves in the adjacent side-bands, but instead by a complex spectrum whose perturbation of the carrier wave cannot be viewed as being small. Therefore, Onorato et al. [14] perform numerical simulations based on the envelope equations by using

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an inverse scattering technique, which finally calculates the maximum wave amplitude as a function of the wave steepness. Interestingly, Pelinovski et al. [15] indicate that for shallow water, the Korteweg-de Fries equation is an adequate tool to demonstrate wave focusing. They show that large-amplitude abnormal short-lived impulses can be generated from weak, so-called invisible deterministic components in the background of the random wind wave field. An initial disturbance representing a single crest, for example, evolves into a set of solitons (solitary waves) and an oscillating dispersive tail located in space according to the magnitudes of the individual celerity of each embedded component. They also make use of the nonlinear Schrödinger equation for describing the complex amplitude of the wave envelope in a deep water environment, and they make clear in this context that the well-known Benjamin-Feir effect of modulation instability can explain the appearance of abnormal waves in a periodic weakly modulated wave field with heights exceeding two to three times the unperturbed wave height. It is not in the scope of the present paper to deepen those nonlinear theories, but it is of main interest to identify the embedded structure of rogue waves to discover new physical insights about their occurrence that accord with these newly developed theoretical explanations. Nonetheless, numerical investigations suggest that the physical model of an infinite number of free waves that coincide is not sufficient to describe the occurrence of water waves of extreme wave height. This is in close correspondence with the work carried out by Trulsen and Dysthe [6], who simulate the measured time series of the New Year’s Eve rogue wave forward and backward in space to find the time histories at neighboring locations. Based on the modified nonlinear Schrödinger equations with exact linear dispersion, he concludes that the abnormal wave did not appear suddenly from nowhere; instead, he suggests that a short and tall wave group approached the platform about one minute before the rogue wave hit the structure. Both groups appear to split up and run away from a longer and smaller trailing group that becomes rather diffuse as it approaches the platform. The rogue wave that hits the structure arrives in the middle of the short leading group. After the impact, the whole wave group broadens, becomes less steep, and slows down slightly. A large trough, a so-called hole in the sea, is observed slightly upstream of the platform. All numerical observations strongly correlate with a long-established model for nonlinear wind waves created by Lake and Yuen [38]. They propose to characterize a wind–wave system by an ensemble of single nonlinear wave trains, which are considered to be a coherent bound-wave system. The associated energy is transported only at the group velocity of the dominant wave and is governed by nonlinear self-interactions of the type found in amplitude-modulated wave trains. Essentially, all of the energy in the resulting nonlinear wind–wave system is contained in the bound-wave components of a single dominant wave. The individual components do not propagate as free waves and do not obey the usual dispersion relation, but they are in spectral representations merely Fourier components needed to describe the complicated shapes of the individual waves as well as of their modulating envelope. Lake and Yuen propose that to a leading order the individual components still travel at the single wave speed C0 = ω0 /k0, where ω0 is the carrier frequency and k0

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is its corresponding wave number. The phase speed of the carrier wave component and the energy (or envelope) is equal to the carrier wave group celerity Cg0 = ω0 /2k0. Lake and Yuen further add that although the spectrum of these modulated wave groups appears more or less complicated, covers a relatively broad frequency range, and contains many components that are not harmonics of the carrier wave, it is in fact questionable whether the existence of a coherent carrier wave form is proven solely from visual inspection of the spectrum. Nevertheless, the wave system that produced the spectrum in their analysis has a coherent carrier wave, bound-wave components, and a single energy propagation speed, although — and this is important — it does not propagate without change of form. It is believed that this model is essentially valid for the present case in the central North Sea and that the EMD is categorically capable of deriving these self-modulating carrier wave components according to the Lake and Yuen model. Although we have established that this particular sea state is tentatively composed of at least two carrier waves and two additional riding wave components, we have not determined whether the latter ones are bounded or dispersive within the spectral representation. To attempt to prove our assumption, we simply add both intrinsic modes c2 and c3 and clarify how closely this reproduces the full water surface elevation. Figure 2.19 and Figure 2.20 show this comparison for both time series. The differences

x1(t) (m)

20 10 0

c2(t) + c3(t) (m)

−10

0

200

400

600

800

1000

1200

0

200

400

600

800

1000

1200

0

200

400

600 Time (s)

800

1000

1200

20 10 0 −10

∆x(t) (m)

20 10 0 −10

FIGURE 2.19 Top: Signal 1. Middle: Sum of c2 and c3 of signal 1. Bottom: Difference between signal 1 and sum of c2 and c3 of signal 1.

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x2(t) (m)

20 10 0

c2(t) + c3(t) (m)

−10

0

200

400

600

800

1000

1200

0

200

400

600

800

1000

1200

0

200

400

600 Time (s)

800

1000

1200

20 10 0 −10

∆x(t) (m)

20 10 0 −10

FIGURE 2.20 Top: Signal 2. Middle: Sum of c2 and c3 of signal 2. Bottom: Difference between signal 2 and sum of c2 and c3 of signal 2.

x(t) between the original signals x1(t) and x2(t), respectively, and the sum of modes c2 and c3 are shown individually in the bottom subfigures of Figure 2.19 and Figure 2.20. Obviously, only marginal divergence is noticed for both time series, mainly indicating that some higher frequency waves are missing. Most obvious is the omitted peak of the New Year’s Eve rogue wave, which is hidden in the high frequency component c1 disintegrated from the first signal. Fourier-based transformation, discussed earlier, suggests two peak carrier wave components located in the spectrum at fpeak1,1st = 5.7 × 10–2 Hz and fpeak2,1st = 8.3 × 10–2 Hz. These frequencies recalculated into periods are defined as Tpeak1,1st = 17.5 sec and Tpeak1,2nd = 12.0 sec. According to Table 2.3 and Table 2.4, the EMD proposes the mean period of components c2 and c3 to be about T2 = 9.4 sec and T3 = 16.9 sec to 17.7 sec, respectively. Mode c2 also embeds both previously identified superharmonic riding waves at approximately fpeak1,2nd = 1.01 × 10–1 Hz and fpeak2,2nd = 1.03 × 10–1 Hz, so that the averaged period is significantly smaller for that mode. Nevertheless, it is most interesting that both carrier waves and fractions of the riding wave components previously detected from Fourier-based spectra are validated in principle from the EMD. Moreover, if we now apply the Hilbert transformation to those derived components from the EMD, we can easily confirm Lake and Yuen’s proposed model, because time-frequency representation using the Hilbert spectral technique allows modulating amplitudes within time and frequency domains.

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0.20

c1

Instantaneous frequency (Hz)

0.16

0.12

c2

0.08

c3 0.04 c4 0

0

200

400

600

800

1000

1200

Time (s)

FIGURE 2.21 Hilbert spectrum of signal 1. Only modes c1 to c4 are plotted.

Figure 2.21 and Figure 2.22 present results from the Hilbert spectral analysis. Each figure renders the time–frequency dependent representations of modes c1 to c4. Evidently, amplitudes of the carrier wave components vary in time; a detailed measure in the form of colormaps is omitted here. Alternating colors exclusively indicates this tendency. Large amplitudes are designated by black colors, and smaller time-variant wave heights are shown by gray-shaded colors. Note that each Hilbert spectrum is created by use of an additional filter algorithm to smooth these distinct representations to a coarser time–frequency grid (400 × 400). Intrinsic modes c2 to c4 shown in Figure 2.21 are characterized as slowly varying time-dependent functions. While the instantaneous frequency of c4 is nearly constant, located at f = 0.04 Hz, both dominating carrier wave components c3 and c2 show slightly larger time-variant spectral deviations. The latter mode alternates around an average frequency of about f = 0.08 Hz, with the largest amplitude in the spectrum. The third IMF characterizes the other dominant carrier wave and oscillates around f = 0.06 Hz. As pointed out before, mode c1 contains the high frequency part of the signal and is therefore more or less absent in the Hilbert spectrum since the upper frequency limitation of this plot is set to f = 0.20 Hz. It seems obvious that the model proposed by Yuen and Lake could be validated generally with Hilbert spectral analysis. Results from the second data series are shown in Figure 2.22. Both dominating modes c2 and c3 show evidence of larger frequency variations in time domain.

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0.20

c1

Instantaneous frequency (Hz)

0.16

0.12

c2

0.08 c3 0.04 c4

0

0

200

400

600

800

1000

1200

Time (s)

FIGURE 2.22 Hilbert spectrum of signal 2. Only modes c1 to c4 are plotted.

Again, mode c4 is almost of constant frequency, located at f = 0.04 Hz, but the second and third IMFs do exhibit significant frequency fluctuations and hardly represent an oscillation around an average frequency. The first IMF is almost absent in the Hilbert spectrum.

2.4 SUMMARY AND DISCUSSION Though the existence of rogue waves cannot be elucidated from the spectra directly, the carrier and riding wave structure of real sea states is straightforward and strictly follows Yuen and Lake’s proposed model. So far it is not clear why those designated carrier wave components produce single extreme waves with exceptionally large wave heights. No crucial generating mechanism of self-focusing transient waves is deduced from the Hilbert spectra. We do observe the general trend that whenever those carrier components show a nearly time-invariant behavior, specifically showing approximately constant instantaneous frequencies — similarly to a monochromatic wave — rogue waves are more likely to occur. If the concept of Benjamin-Feir instability mechanism is proven to explain the occurrence of extreme wave events, as some researchers assume, this latter hypothesis based on the Hilbert spectral representation can be taken to support the theory established by Pelinovski et al. [15]. They ask how rogue waves can be generated

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from these underlying Benjamin-Feir instabilities yet be demonstrated only from laboratory data series. Realistic oceanic conditions are not characterized by a single monochromatic wave perturbed by two small-amplitude parasitic waves in the adjacent side-bands, but instead by a complex spectrum whose perturbation of the carrier wave cannot be viewed as being small. Thus, the pseudo-monochromatic behavior of carrier and riding wave components derived from the EMD and Hilbert spectral analysis may strengthen assumptions about principal driving mechanisms of rogue waves within random sea states.

ACKNOWLEDGMENTS The authors want to express their deep appreciation to Sverre Haver, STATOIL, Norway, for kindly providing the whole New Year’s Eve data set from the Draupner platform in the central North Sea. This paper was carried out within the framework of a research project SCHL503/5-1 funded by the Deutsche Forschungsgemeinschaft (DFG). The authors gratefully acknowledge this financial support.

REFERENCES 1. Clauss, G. 2002. Dramas of the sea: Episodic waves and their impact on offshore structures. Applied Ocean Engineering 24, 3:147–161. 2. Dean, R. 1990. Freak waves: A possible explanation. In: Water Wave Kinematic, A. Torum & O. T. Gudmestad, Eds. Dordrecht: Kluwer, pp. 743–756. 3. Sand, S. E., Ottesen Hansen, N. E., Klinting, P., Gudmestad, O. T., & Sterndorf, M. J. 1989. Freak wave kinematics. In: Water Wave Kinematic, A. Torum & O. T. Gudmestad, Eds. Dordrecht: Kluwer, pp. 535–549. 4. Rozario, J. B., Tromans, P. S., Taylor, P. H., & Efthymiou, M. 1993. Comparison of loads predicted using “Newwave” and other wave models with measurements on the term structure. Wave Kin. Env. Forces 29, 143–159. 5. Lavrenov, I. V. 1998. The wave energy concentration at the Agulhas current off South Africa. Natural Hazards 17, ASCE, 117–127. 6. Trulsen, K., & Dysthe, K. 1997. Freak waves: A three-dimensional wave simulation. Proc. 21st Symp. Naval Hydrodynamics, pp. 550–558. 7. Yasuda, T., & Mori, N. 1997. Occurrence properties of giant freak waves in sea area around Japan. J. Water, Port, Coast. Ocean Eng. 123, ASCE, 4:209–213. 8. Mori, N., Yasuda, T., & Nakayama, S. I. Statistical properties of freak waves observed in the Sea of Japan. 10th Int. Offshore Polar Eng. Conf. (ISOPE), Vol. 3, pp. 109–115. 9. Schlurmann, T., Lengricht, J., & Graw, K.-U. 2000. Spatial evolution of laboratory generated freak wave in deep water depth. 10th Int. Offshore Polar Eng. Conf. (ISOPE), Vol. 3, pp. 54–59. 10. Claus, G., & Steinhagen, U. 1999. Numerical simulation of nonlinear transient waves and its validation by laboratory data. 9th Int. Offshore Polar Eng. Conf. (ISOPE), Vol. 3, pp. 368–375. 11. Kway, J., Loh, Y., & Soon, C. E. 1998. Laboratory study of deep-water breaking waves. Ocean Eng. 25, Elsevier Sc., pp. 657–676.

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The Hilbert-Huang Transform in Engineering 12. Haver, S., & Andersen, O. J. 2000. Freak waves: Rare realizations of a typical population or typical realizations of a rare population? 10th Int. Offshore Polar Eng. Conf. (ISOPE), Vol. 3, pp. 123–130. 13. Schlurmann, T. 1999. Nonlinear interaction processes of multichromatic wave groups and the spatial propagation of freak waves. Ph.D. thesis, University of Wuppertal (in German). 14. Onorato, M., Osborne, A. R., Serio, M., & Damiani, T. 2001. Occurrence of freak waves from envelope equations in random ocean wave simulation. In: Rogue Waves 2000, Édition Ifremer, ISBN: 2-84433-063-0, pp. 181–191. 15. Pelinovsky, E., Kharif, C., Talipova, T., & Slunyaev, A. 2001. Nonlinear wave focussing as a mechanism of the freak wave generation in the ocean. in: Rogue Waves 2000, Édition Ifremer, ISBN: 2-84433-063-0, pp. 193–204. 16. Huang, N. E., Shen, Z., Long, S., Wu, M. C., Shih, H. H., Zheng, Q., Yen, N.-C., Tung, C. C., & Liu, H. H. 1998. The empirical mode decomposition and Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. R. Soc. London A 454, 903–995. 17. Huang, N. E., Shen, Z., & Long, S. 1999. A new view of nonlinear water waves: The Hilbert spectrum. Annu. Rev. Fluid Mech. 31, 417–457. 18. Flandrin, P., Rilling, G., & Gonçalves, P. 2003. Empirical mode decomposition as a filter bank. IEEE Signal Processing Letters Vol. 11, No. 2, pp. 112–114. 19. Wu, Z., & Huang, N. E. 2002. A study of the characteristics of white noise using the empirical mode decomposition method. Submitted to Proc. R. Soc. London A. 20. List of Sea State Parameters. 1986. International Association for Hydraulic Research (IAHR). Suppl. to Bulletin No. 52, General Secretary of PIANC, Belgium. 21. Kjeldsen, S. P. 1997. Examples of heavy weather damages caused by giant waves. Techno Marine 10, 24–28. 22. Pierson, W. J., & Moskowitz, L. 1964. A proposed spectral form for fully developed wind seas based on the similarity theory of S. A. Kitaigordskii. J. Geophysical Res. 69, 5181–5190. 23. Phillips, O. M. 1977. The Dynamics of the Upper Ocean (2nd ed.). London: Cambridge University Press. 24. Olagnon, M., & van Iseghem, S. 2000. Some cases of observed rogue waves and an attempt to characterize their occurrence. In: Rogue Waves 2000, Édition Ifremer, ISBN: 2-84433-063-0, pp. 105–116. 25. Trulsen, K. 2000. Simulating the spatial evolution of a measured time series of a freak wave. In: Rogue Waves 2000, Édition Ifremer, ISBN: 2-84433-063-0, pp. 265–273. 26. Meyer, Y. 1993. Book review: “An introduction to wavelets” by C.K. Chui and “Ten lectures on wavelets” by I. Daubechies. Bull. Am. Mathematical Soc. 28, 2:350–360. 27. Liu, P. L. 2000. Is the wind wave frequency spectrum outdated. Ocean Eng. 5, 577–588. 28. Liu, P. L. 2000. Wave grouping characteristics in nearshore Great Lakes. Ocean Eng. 27, 11:1221–1230. 29. Liu, P. L. 2000. Wavelet transform and new perspective on coastal and oceanic engineering data analysis. In: Advances in Coastal and Ocean Engineering Book Series, Vol. 6, Phillip L. F. Liu, Ed. World Scientific, pp. 57–101, Singapore. 30. Chien, H., Chuang, L., & Kao, C. C. 1999. A study on mechanisms of nearshore rabid waves, Proc. 2nd German-Chinese Joint Sem. Recent Dev. Coastal Eng., pp. 469–483.

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31. Schlurmann, T. 2001. The empirical mode decomposition and the Hilbert spectra to analyse embedded charactersitic oscillations of extreme waves. In: Rogue Waves 2000, Édition Ifremer, ISBN: 2-84433-063-0, pp. 157–165. 32. Schlurmann, T. 2002. Spectral analysis of nonlinear water waves based on the Hilbert-Huang transformation. J. Offshore Mech. Artic Eng. 124, ASME, 22–27. 33. Huang, N. E., Wu, M.-L. C., Long, S. R., Shen, S. S. P., Qu, W., Gloersen, P., & Fan, K. L. 2003. A confidence limit for the empirical mode decomposition and Hilbert spectral analysis. Proc. R. Soc. London A, FirstCite e-publishing, Vol. 459, pp. 2317– 2345. 34. Rilling, G., Flandrin, P., & Gonçalves, P. 2003. On empirical mode decomposition and its algorithms. Proc. IEEE-EURASIP, Workshop on Nonlinear Signal and Image Processing, Grado, Italy, 8–11 June 2003 (in press). 35. Veltcheva, A. D. 2002. Wave and group transformation by a Hilbert spectrum. Coastal Eng. J. 44, 4:283–300. 36. Dätig, M., & Schlurmann, T. 2003. Performance and limitations of the Hilbert-Huang transformation with short application to water waves. Submitted to Ocean Engineering. 37. Benjamin, T. B., & Feir, J. E. 1967. The disintegration of wave trains on deep water, Part I: Theory. J. Fluid Mech. 27, 417–430. 38. Lake, B. M., & Yuen, H. C. 1978. A new model for nonlinear wind waves. Part I: Physical model and experimental evidence. J. Fluid Mech. 88, 33–62.

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Applications of Hilbert-Huang Transform to Ocean-Atmosphere Remote Sensing Research Xiao-Hai Yan, Young-Heon Jo, Brian Dzwonkowski, and Lide Jiang

CONTENTS 3.1 3.2

Introduction ....................................................................................................60 Analyses of TOPEX/Poseidon Sea Level Anomaly Interannual Variation Using HHT and EOF .....................................................................62 3.3 Application of HHT to Ocean Color Remote Sensing of the Delaware Bay ......................................................................................64 3.4 Mediterranean Outflow and Meddies (O & M)from Satellite Multisensor Remote Sensing .........................................................................71 3.5 Conclusion......................................................................................................78 Acknowledgments....................................................................................................79 References................................................................................................................79 ABSTRACT The Hilbert-Huang transform (HHT) is a newly developed method for analyzing nonlinear and nonstationary processes. Its application in oceanography and oceanatmosphere remote sensing research is still in its infancy. In this chapter, we briefly introduce the application of this method in oceanatmosphere remote sensing data analyses and present a few examples of such applications. 59

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3.1 INTRODUCTION Spectral analysis is a very useful tool to analyze a time series signal. However, this method does not fully describe a data set that changes with time. The spectrum gives us the frequencies that exist over the entire duration of the data set. On the other hand, time–frequency analysis allows us to determine the frequencies at a particular time. Hence, the fundamental idea of time–frequency analysis is to understand and describe phenomena where the frequency content of a signal is changing in time. Scientists traditionally use short Fourier transform by sliding the window along the time axis to get a time–frequency distribution. Since it relies on the traditional Fourier spectral analysis, one has to assume the data to be piecewise stationary. Currently, the most famous time–frequency analysis method is wavelet transform. The most common method used is Morlet wavelet, defined as Gaussian enveloped sine and cosine wave groups with 5.5 waves (1). The problem with Morlet wavelet is the leakage generated by the limited length of the basic wavelet function, which makes the quantitative definition of the energy–frequency–time distribution difficult. Once the basic wavelet is selected, one has to apply it to analysis of all the data (2). Recently Huang et al. (3) introduced a new and potentially more robust method for time–frequency analysis. This method, the empirical mode decomposition–Hilbert-Huang transform (EMD-HHT), is applicable to both nonstationary and nonlinear signals. In real ocean and ocean atmosphere coupling, most processes are nonlinear and nonstationary. One example is that at the onset of El Nino: nonlinear Kelvin waves carry warm water from the western Pacific to the east (4). This process is exhibited as a nonlinear pattern in altimeter data. For this reason, we use the EMD-HHT technique in our El Nino study and in many of our other studies. Since the EMD-HHT is relatively new to the ocean remote sensing community, a brief summary of the technique based on Huang et al. (3) is given in this section. Basically, the EMD-HHT method requires two steps in analyzing the data. The first step is to decompose time series data into a number of intrinsic mode functions (IMFs). These functions must satisfy the following two conditions: (a) within the entire data set, the total number of extrema (as a function of time) and the total number of zero-crossings must either be equal or differ at most by one, and (b) at any point, the mean value of the envelope defined by the local minima (as a function of time) and the envelope defined by the local maxima (as a function of time) must be zero. The second step is to apply the Hilbert transform to the decomposed IMFs and construct the energy–frequency–time distribution, designated as the Hilbert spectrum. The presentation of the final results of the time–frequency analysis is similar to the wavelet transform method, which is a spectrogram (time–frequency–energy plot). For clarity, a spectral analysis of the corresponding signal is shown next to the spectrogram when we present our results in the next sections. The decomposition of the time series data (i.e. H(t)) into IMFs uses separately defined envelopes of local maxima and minima. Once the extrema are identified, all the local maxima are connected by a cubic spline to form the upper envelope. The procedure is repeated for the local minima to produce the lower envelope. Their mean is designated as m1(t), and the difference between the time series data and m1(t) is the first component, h1(t). One can repeat this procedure k times, until hk(t)

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is an IMF. Then, hk(t) = c1(t) is the first IMF component of the data. c1(t) should contain the finest scale or the shortest period component of the signal. Then, c1(t) is separated from the data, and the process is repeated until either the component cn(t) or the residue rn(t) becomes so small that it is less than a predetermined value, or when the residue rn(t) becomes a monotonic function from which no IMF can be extracted. After all IMFs have been determined, one can check the original data with the sum of the IMF components n

()

H t =

∑ c (t ) + r (t ) . 1

n

(3.1)

i =1

Thus, a decomposition of the data into n empirical modes and a residue rn(t) is achieved. The rn(t) can be either the mean trend or a constant. After IMFs of the data have been generated, the next step is to apply the Hilbert transform to each IMF time series. For an arbitrary time series X(t), one can define its Hilbert transform, Y(t), as:

()

1 Y t = π

( ) dτ .

∞

X τ

∫

t−τ

−∞

(3.2)

With this definition, X(t) and Y(t) form a complex conjugate pair, so one has an analytic signal Z(t) as

()

()

()

Z t = X t + iY t ,

(3.3)

where its amplitude function a(t) is

()

()

()

1

a t =  X 2 t + Y 2 t  2 ,

(3.4)

and its phase function (t) is

() ()

Y t  θ t = arctan  .  X t 

()

(3.5)

The instantaneous frequency is defined as

ω=

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( ).

dθ t dt

(3.6)

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Once the instantaneous frequency (as function of time) of each IMF has been generated, the final result of the time–frequency analysis is similar to the wavelet transform method, time–frequency–energy plot, or spectrogram. The energy density similar to Fourier transform can be generated by summing the spectrogram for the whole time series along a constant frequency for each frequency. From the definition of the phase function and instantaneous frequency, we can clearly see that for a simple function such as a = sin(t), the Hilbert spectrum is simply cos(t), and the phase function is a monotonic straight line. Hence, the spectrogram is simply a horizontal line for a whole time along fixed frequency, and its spectrum is simply a single peak at that particular frequency. In the next sections, we describe a few examples of applications of the HHT in ocean-atmosphere remote sensing data processing and research. These examples include an analysis of TOPEX/Poseidon sea level anomaly interannual variation using HHT and empirical orthogonal function (EOF), application of HHT to ocean color remote sensing of the Delaware Bay, and Mediterranean outflow and Meddies determined from satellite multisensor remote sensing.

3.2 ANALYSES OF TOPEX/POSEIDON SEA LEVEL ANOMALY INTERANNUAL VARIATION USING HHT AND EOF The measurement of sea surface height provides a unique opportunity to make significant contributions to many of the science objectives of oceanatmosphere remote sensing scientists working in the area of climate variability, oceanatmosphere interactions, and upper ocean dynamics. Satellite altimeter data can be employed to gain vital new understanding of the nature of the ocean’s circulation and ocean atmosphere coupling, both in terms of mean state and variability and interrelationships between this circulation, its variability, and global climate change. We used TOPEX/Poseidon (T/P) monthly sea level anomaly (SLA) data from January 1993 to August 2002, obtained from the University of Texas’ Center for Space Research (UT/CSR). The deviation of the sea surface was removed from a mean surface and was preceded by instrument corrections (ionosphere, wet and dry troposphere, and electromagnetic bias) and geophysical corrections (tides and inverted barometer). The availability of accurate altimeter data, such as that from T/P, with the root-mean-square (RMS) error as small as 2 cm (5), allows one to examine the annual and interannual sea surface variability due to El Niño Southern Oscillation (ENSO) and global climate change. The original spatial range was 65°S to 65°N, 180°E to 180°W; however, we limited the latitude range to 60°S to 60°N. The resolution of the data set was 1° by 1°. To isolate the signal of the interannual variation of SLA, the annual fluctuation was removed by subtracting the overall monthly mean for a given month from the individual monthly data for that respective month. According to calculation, the interannual energy accounted for about 60% of the total energy. Furthermore, to analyze the resulting time series, EOF method was applied. The first three EOFs, EOF-1, EOF-2, and EOF-3, accounted for 35.4%, 12.6%, and 9.8% of the total interannual energy, respectively.

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Interannual EOF-1 (35.4%)

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FIGURE 3.1 The spatial (upper) and temporal (lower) interannual EOF-1 (left) and EMD of the temporal mode (right) of the T/P-SLA.

Due to its important role in interannual variation, EOF1 was further analyzed using the HHT. Empirical mode decomposition (EMD) was applied, and EOF1 was decomposed into three IMFs plus a residue (Figure 3.1). The temporal mode of EOF1 shows a peak in late ‘97, which correspond to the strong 1997–1998 El Niño. To examine the effect of ENSO events on EOF1, the Southern Oscillation Index (SOI) was also used as a reference by decomposing it using EMD. The IMF4 of the SOI appeared to be a smoothed curve of the SOI, and Salisbury and Wimbush (6) used this to predict future ENSO events. Here a comparison was made between the EOF-1 and the SOI by calculating the correlation coefficient between the IMF-3 of the EOF-1 and IMF-4 of the SOI. The correlation coefficient was found to be as high as 0.85 (Figure 3.2). Considering this, the EOF-1 can be considered to behave under the impact of the ENSO, and we can further infer that the ENSO contributes to about one third of the interannual SLA variation. Moreover, we can reexamine the spatial mode of EOF1 (Figure 3.1) to see the global impact of the ENSO events on human activities. Besides the coastal areas of America, Australia, and Indonesia, which are most strongly and directly affected, those of east Africa and Japan also are undergoing remarkable ENSO variation. The

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Correlation = 0.85 0.1

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FIGURE 3.2 Correlation between negative IMF-4 of SOI (red) and IMF-3 of EOF-1 (blue) is 0.85. It suggests that the primary interannual EOF is impacted by ENSO.

impact even spread as far as Antarctica. On the other hand, coastal areas of the Atlantic Ocean are less affected by the ENSO events. In addition, the HHT spectrum of the EOF1 was investigated to extract the frequency information (Figure 3.3). The frequency distribution of the EOF1 is between 0 and 4 cycle/yr. The 2.5 to 4 cycle/yr range corresponds to the spectrum of the IMF1, which behaves in a relatively random pattern. The 0.5 to 2 cycle/yr range corresponds to the spectrum of IMF2, which has more energy within the frequency range 0.5 to 0.8 cycle/yr. The 0.2 to 0.4 cycle/yr range has much higher energy, which is shown as a dark red curve at the bottom part of the spectrum. This curve corresponds to the frequency feature of IMF3, i.e., a period of 2.5 to 5 yr, which can be considered as the typical frequency of ENSO events. The lowest frequency part of the 0 to 0.1 cycle/yr range corresponds to the frequency of the residue, which is the longterm trend. It has a period of tens of years or longer and, therefore, requires much longer time series to analyze.

3.3 APPLICATION OF HHT TO OCEAN COLOR REMOTE SENSING OF THE DELAWARE BAY The use of satellite-based ocean color data has become an integral part of oceanographic studies, aiding in the exploration of a number of important topics. The ability to collect ocean color data at high temporal and spatial resolutions, which only satellite sensors can provide, has given the oceanographic community a rich data

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HHT Spectrum of EOF-1

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FIGURE 3.3 The HHT spectrum of EOF-1. The dark red curve indicates high energy at frequencies between 0.2 and 0.4 cycle/yr, which corresponds to the Hilbert spectrum of IMF-3 of interannual EOF-1. It indicates that ENSO events have a typical period of 2.5 to 5 yr.

source. However, use of this data source is dependent on the premise that there are discernable relationships between reflectance exiting the water column and the constituents in it. A primary constituent of study has been chlorophyll-a on account of its connection to phytoplankton and thus to primary production and biomass (7). As coastal management strategies begin to focus on large-scale ecosystem based programs, the relationship between chlorophyll-a and primary production and biomass has the potential to provide a cost-effective alternative to traditional ship-based or point source sampling. Monitoring and assessing the health of ecosystem-size regions will be much more feasible with satellite-based parameters due to the frequent repeat periods and large spatial areas covered by satellite sensors. Thus, the quantitative assessment of water constituents from satellite-based ocean color data holds great promise for implementing dynamic management strategies. However, to interpret ocean color data appropriately, a full understanding of the periodicity of chlorophyll concentrations in a given area is essential. The purpose of this study is to examine the seasonal and interannual chlorophyll-a cycles from satellite data of the coastal region at the mouth of the Delaware Bay. Although quantifying chlorophyll concentration from satellite ocean color data has been successful in the open ocean, coastal areas (case II water) can still present problems. Water constituents associated with coastal regions, such as excessive chlorophyll-a concentrations, suspended sediments, and colored dissolved organic

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material, create an optically complex area in which traditional empirical models typically perform poorly (8). However, several studies have used an innovative neural network (NN) methodology successfully to develop ocean color algorithms in Case II water that outperform empirical methods (9, 10, 11, 12). NN based algorithms are advantageous on account of their ability to model complex, nonlinear geophysical transfer functions like those required to relate chlorophyll-a concentrations to the reflectance measured by a satellite sensor (13). The NN requires training data consisting of input data paired with the desired output, from which it “learns” the geophysical relationship. The ability of a NN to generalize (i.e. accurately predict the geophysical parameter) is assessed through a validation set of pair data that the NN has never encountered. Given the success of NNs in coastal water algorithm development, the chlorophyll-a data used in this study was produced from a NN trained and validated using the Sea-Viewing Wide Field-of-View Sensor (SeaWiFS) remotely sensed reflectance data paired with in situ chlorophyll data in the Delaware Bay and its adjacent coastal water. These paired data points were collected during a number of different days and seasons. The neural network showed significant improvement over the current SeaWiFS processing algorithm (Ocean Color 4 [OC4]) and was used to reprocess the satellite data used in this study (14). Over the course of the SeaWiFS project, daily high-resolution picture transmission (HRPT) images of the Delaware Bay and the adjacent coastal zone were subjectively viewed to identify all the cloud-free images of the Delaware Bay and the adjacent coastal zone from September 1997 to July 2003. This period covers almost six years of data, from which 468 images were collected, with an average of 6.6 images per month. The daily images were obtained at the L1A processing level and converted to a remapped L2 remote sensing reflectance (Rrs) product. The remapping used a standard Mercator projection with a fixed grid of 890 × 890 pixels. Each image ranges from 34°N to 42°N latitude and from 68°W to 78°W longitude, which covers approximately 9.58 × 105 km2. The Rrs product was then used as input for the optimal NN model mentioned earlier to produce daily chlorophyll-a maps of the Delaware Bay and the adjacent coastal zone. Since NNs are extremely good at interpolation but poor at extrapolation, only a limited subset (the 121 × 121 pixel Delaware Bay and adjacent coastal zone) of the entire SeaWiFS image was processed using the NN. This ensured that the model would only be applied to the region where it was specifically trained with in situ data. The SeaWiFS imagery was binned by month and geometrically averaged to produce an image of the mean monthly values of chlorophyll-a concentrations for each month. This organizational process created spatially coherent patterns of chlorophyll-a values at evenly spaced time scales that help reduce the effects of sensor and algorithm errors and minimize the influence of high frequency variability on seasonal distributions of chlorophyll-a concentrations (15). Furthermore, due to the fact that chlorophyll pigments tend to be lognormally distributed, the geometric means were used in the various statistical procedures applied to chlorophyll-a values in this study (15). A station at the mouth of the Delaware Bay (38.916°N, –75.100°W) was used as the geographic point from which a monthly times series of chlorophyll-a concentrations was produced. This monthly time series is shown in Figure 3.4.

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Chlorophyll-a (mg/m3)

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Hilbert-Huang transform (HHT) analysis provides an innovative way to study the time–frequency characteristics of time series, due to its ability to handle nonlinear and nonstationary data and to account for changes in frequency over time. This novel technique has been applied to the time series of chlorophyll-a concentrations at the mouth of the Delaware Bay, in order to better understand the forcing mechanisms associated with seasonal and interannual variations. The initial chlorophyll-a time series was decomposed into IMFs, to which the Hilbert transform can be applied. The resulting Hilbert spectrum and its associated marginal spectrum are shown in Figure 3.5. The marginal spectrum is conceptually similar to a spectral density graph produced by Fourier analysis; however, it is produced by summing the instantaneous energy associated with a constant frequency over the entire time period. Thus, the marginal spectrum provides a measurement of the frequencies that contain the largest amounts of energy. As expected, the dominant energy peak occurs at about 0.09, which is the approximate annual forcing frequency and corresponds to period of 11.1 months. This result is supported by the fact that several annual physical forcing factors, such as sea surface temperature, solar insolation, and stream flow, have been shown to strongly influence chlorophyll-a concentrations in the Delaware Bay and its adjacent coastal zone (8, 16). The fact that the chlorophyll-a periodicity, typically

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presumed to be annual, is not precisely 12 months is not unexpected, since the biological processes are not simply controlled by physical processes. Examining the Hilbert spectrum along the 0.09 frequency shows that there is generally high energy along this signal, with the exception of two time periods. From January 1999 to October 1999, the annual signal completely degenerates, and in the summer of 2001 there is a weakening of this signal. Without the ability of the HHT to ascertain instantaneous frequency, these breakdowns in the annual periodicity would be difficult to determine. In addition to the strong annual signal, there is a significant energy peak in the lower frequencies. The marginal spectrum indicates an energy peak at the 0.02 frequency, which corresponds to a 4.2-yr period. However, looking at the Hilbert spectrum shows that the energy at the 0.02 frequency is not constant; instead, it appears to be roughly oscillating around the 0.02 frequency. This energy begins to increase in frequency from mid-1999 to late 2001, when the signal briefly weakens and starts to decrease in frequency. This interannual signal is primarily captured in IMF 4 of the empirical mode decomposition (Figure 3.6). The relatively long period of the mode suggests that a slowly varying basin-scale process could be a forcing candidate. One such basin-scale process is the North Atlantic Oscillation (NAO). NAO is a measure of atmospheric conditions focused around the North Atlantic. The NAO can have a positive or negative phase, as determined by the NAO index, which is the surface sea-level pressure difference between the Subtropical (Azores) High and

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Applications of Hilbert-Huang Transform

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FIGURE 3.6 Interannual variation (IMF 4) for both the NAO index (blue line) and the chlorophyll-a (green line) time series.

the Subpolar Low. Both positive and negative phases affect basin-wide variations in the North Atlantic jet stream and storm tracks, as well as in large-scale modulations of the typical patterns of zonal and meridional heat and moisture transport. These alterations will consequently affect temperature and precipitation patterns and can extend from eastern North America to western and central Europe (17). Furthermore, recent studies have shown links between the NAO and hydrographic properties along the northeast Atlantic shelf slope and the Gulf of Maine, and zooplankton and chlorophyll-a concentrations within the Gulf of Maine (18). Thus, the interannual frequency of the NAO was examined. Monthly mean data of the NAO index from September 1997 to July 2003 were obtained from the NOAA Climate Prediction Center. The NAO index data are shown in Figure 3.4. This time series was also decomposed into its component IMFs in order to isolate the NAO interannual mode. After decomposing the NAO index time series, the fourth IMF contained an interannual signal that exhibited features very similar to the fourth chlorophyll-a IMF (Figure 3.6). To quantify this relationship, cross-correlation analysis was performed on the IMF 4 of both the chlorophyll-a concentrations and the NAO index. The results are shown in Figure 3.7. This study focused only on positive lag time (chlorophyll-a time series following the NAO time series), because it would not be sensible to assume that negative lags (chlorophyll-a time series leading the NAO time series) were meaningful. The correlation analysis reveals that the NAO and chlorophyll-a

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The Hilbert-Huang Transform in Engineering Maximum correlation is r = −0.67 at tiag = 10 (95% signiﬁcance level: rsig = 0.25) 1 Correlation coeﬃcient (r) 95% signiﬁcance level

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FIGURE 3.7 Correlation coefficient (blue line) and the related 95 % confidence interval (red dashed line). The maximum correlation coefficient with a positive lag (i.e. chlorophyll-a interannual variation following NAO index interannual variation) is –0.67, corresponding to a lag of 10 months.

concentrations have their highest correlation (correlation coefficient of –0.67) when chlorophyll-a concentration lags (follows) the NAO by 10 months. The result is significant at the 95% confidence level, as shown in Figure 3.7. Although a high correlation coefficient does not guarantee a relationship between the two times series, the fact that previous studies (mentioned earlier) have shown that the NAO can affect the interannual time scales of biological and physical processes in oceanic regions as distant as the Gulf of Maine suggests that this correlation is important. This study examined the seasonal and interannual variability in satellite derived chlorophyll-a data from the mouth of the Delaware Bay for a 6-yr period. By applying the HHT to the chlorophyll-a time series, a more complete understanding of its time–frequency characteristics was obtained. The resulting Hilbert spectrum displayed a strong annual signal, with an unexpected weakening at two brief intervals. In addition, the energy at interannual frequencies was compared to the NAO. The fourth IMF of the NAO index and the fourth IMF of the chlorophyll-a concentrations displayed strong similarities and had a cross-correlation coefficient of –0.67

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that was significant at the 95% confidence level. This relatively high correlation coefficient occurred at a lag of 10 months, which suggests that the possible interannual link between chlorophyll-a concentrations at the mouth of the Delaware Bay and the NAO index may result from chlorophyll-a concentrations responding to variations in the NAO index. Finally, the observations of both the weakening of the seasonal signal and the suggested teleconnection of the interannual variation between chlorophyll-a concentrations and the NAO are topics worthy of future investigation that would have been difficult to identify with traditional time series analysis techniques.

3.4 MEDITERRANEAN OUTFLOW AND MEDDIES (O & M) FROM SATELLITE MULTISENSOR REMOTE SENSING One of the most interesting and prominent features of the North Atlantic Ocean is the salt tongues originating from an exchange flow between the Mediterranean Sea and the Atlantic through the Strait of Gibraltar. The Mediterranean outflow through the strait is heavier than Atlantic water due to its higher salt content. Evaporation in the Mediterranean Sea raises water salinity to approximately 38.4 ppt, compared to 36.4 ppt in the eastern North Atlantic. After leaving the strait under the incoming lighter North Atlantic water, the Mediterranean outflow sinks and turns to the right, due to the Coriolis force, following the continental slope of Spain and Portugal. Eventually, the Mediterranean water leaves the coast and spreads out into the middle North Atlantic, forming a tongue of salty water and generating clockwise eddies in the process. These Mediterranean eddies, or Meddies, are rapidly rotating double convex lenses that contain a warm, highly saline core of Mediterranean water. They are typically 100 km in diameter, extend over 800 m vertically, and are located at a depth of 1000 m. Generally speaking, most of the remotely sensed oceanographic observations are confined to either the sea surface or the top part of the upper mixed layer. This limitation is not always true. Yan et al. (19, 20, 21) and Yan and Okubo (22) developed methods to infer the upper ocean mixed layer depths from multisensor satellite data. Stammer et al. (23) applied Geosat Exact Repeat Mission altimeter data to investigate the possible surface signal of the Meddies. In this section, we report a new method to study the O&M by satellite integration analyses of altimeter, scatterometer, SST, and XBT data with help of HHT. We computed the sea level difference, namely ∆η′ = η′Total − ηUL ′ = η′Total − ( η′T + η′S + ηW ′ ). η′Total is the deviation of the sea surface topography of the whole vertical water column, which can be measured by TOPEX/Poseidon (T/P) altimetry. ηUL ′ is the sea surface variation in the upper layer (400 m) due to thermal expansion (η′T ), calculated from XBT. Salt expansion ( η′S ) and wind stress (ηW ′ ) are computed from satellite scatterometer data. The HHT was applied to help interpret ∆η′. To estimate the sea level variation due to thermal expansion in the upper layer, sea surface height anomaly was calculated using monthly mean XBT (η′T ) data from the Joint Environmental Data Analysis (JEDA) Center by the integration of temperature down to the 400 m depth from January 1993 to December 1999, i.e.,

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0

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1 ∆ρ ρ0 ∆T

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The spatial resolution of XBT data is 73° × 61° longitude/latitude, which is interpolated to 1° × 1° longitude/latitude data for this study. It is possible that errors occurred due to interpolation of the time rates of change of the integrated upper ocean heat storage anomaly (5 W/m2, on average); this is discussed by White and Tai (25). We also checked interpolation error using 1° × 1° longitude/latitude optimum interpolation sea surface temperature (OISST). We compared interpolated sea surface temperature (SST) from XBT data and OISST by calculating correlation coefficients. Correlation coefficients between OISST and SST from XBT were over 95% with RMS 0.2°C, except for two small areas, and correlation coefficient 80% with RMS 1.2°C. We conclude that interpolation error of XBTs is less than 1°C at the sea surface. Because of scarcity of temporal and spatial salinity data, we estimated the effect of η′S in the upper layer indirectly. Two different correlation and RMS calculations were made: the first correlation is between η′T and OISST, and the second correlation is between η′Total and OISST. Good temporal and spatial agreement between SST and η′Total or η′T suggests that a robust regression between fields may have some physical significance with respect to thermal expansion, but a low correlation with high RMS may have some other variability in the mixed layer or below the mixed layer due to the salinity. It appears that the difference is due to a change in ocean salinity, which is reflected in the T/P sea level measurements but not in the η′T measurements. The first correlation coefficients were all over 90% with smaller RMS than the second ones, but the second ones were also over 60% to 80%, with larger RMS than the first ones. This result is caused by a warm and salty core below the mixed layer. Using Levitus 94 (26), the vertical structure of the salinity was examined to see the salinity distribution. The whole upper layer in our study domain has a uniform salinity distribution above 1000 m. We conclude that the anomaly of η′S is spatially uniform with only small variability. Wind effect on the η′Total signal was considered using scatterometer data from European Remote Sensing Satellite-1/2 (ERS-1/2) from January 1993 to December 1999. First we calculated the correlations between wind stress curl and the η′Total for the temporal variation. Negative correlation would be expected; rising sea level is associated with negative wind stress curl. Correlation coefficients showed a relation of about –30% in our study area, with southern areas having larger RMS than

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northern areas. Second, we considered the magnitude of sea level variation due to wind stress based on a linear barotropic vorticity equation, i.e., d η − g′ ≈ ∇x τ, dt f0ρg where η is the sea level height, ρ is the water density, g′ is the reduced gravity, f0 is the Coriolis parameter, and τ is the wind stress. We also estimated ηW ′ by using NASA’s scatterometer: NSCAT (August 1996 to June 1997) and Quikscat (June 1999 to present). The mean sea level variation due to wind stress was around 1 cm. The result of ∆η′ = η′T P − ηUL ′ is apparently some variation of the vertical water column, which is a response according to the different incoming or outgoing water mass under the upper layer. Sea level variation due to salinity (temperature) change using mass conservation with 800 m thick is around 60 cm (32 cm), with 1-ppt differences from background fluid (2°C), derived from the salt expansion coefficient 7.5 × 10–4/°C and thermal expansion coefficient 2.0 × 10–4/°C, respectively. The bottom pressure (deep current) variation is negligible at this region (26, 27). However, since the Meddy was a weakly stratified result of extensive salt fingering (28), there were regions of high stratification above and below the lens, where the background isopycnal surface becomes increasingly broader as it moves above the Meddy toward the sea surface. Because of this isopycnal compensation, the O&M are not revealed in the η′Total signal. This is why we cannot detect a Meddy with the altimeter observation alone. We computed the absolute differences of the sea surface height anomaly of ηUL ′ and η′Total to examine the trajectories of the O&M from January 1993 to December 1999. Figure 3.8 is an example of such trajectories from July 1998 to June 1999. One can see a strong signature (|∆η′|) toward west and south from July to November. Generally, southward Meddies are formed near 36°N by separation of the frictional boundary layer at sharp corners (29) and in the Canary Basin (30, 31, 32). The southward travelling Meddies can be explained by the strongest low frequency zonal motions driven by baroclinic instability (33) and the influence of the neighboring mesoscale features (cyclonic vortices or Azores Current meanders) in the regions (34). The northward O&M over 36° to 40°N are also shown from August to October. Furthermore, a strong signature over 40°N in February is considered to be the influence of the returning Gulf Stream. From March to June the weak O&M signatures are found because of the weak salinity deviation in 1000 m depth examined from climatological data. To investigate the reasons for the change in the direction of propagation of Meddies, the stream function (ψ) was computed by using the T/P altimeter. This representation of the stream function permits observations of the interactions between the sea surface gradient and the Meddies’ propagation. The computation of the sea surface height anomaly η′ in terms of the usual dynamic variables is straightforward, if the flow is assumed to be quasigeostrophic: The stream function (ψ) is defined as

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FIGURE 3.9 Comparisons of Meddy trajectories with stream functions (Equation 3.7) for January, April, July, and October 1994.

ψ= g

f

η′Total ,

(3.7)

0

where ψ is the surface quasigeostrophic stream function (35). This is illustrated in Figure 3.9. Due to the seasonal migration of the tropical water toward the north and polar water toward the south, we can see that the seasonal signals divide into a northern part and a southern part at 37°N. We compared the Meddies measured by the floats and the stream functions in all the months in the experimental periods and found that the Meddies preferred to travel toward the low streamlines and that they stayed within the northmost and southmost streamlines in April and October, respectively.

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3rd EMD mode using HHT at A(30W, 30N) with other places 5

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94 F(20W, 42N)

0 −5 93 5

94 G(30W, 42N)

0 −5 93 5

94 H(30W, 37N)

0 −5 93

94

FIGURE 3.10 A comparison of the third EMD modes using HHT are shown at given locations. The solid curve is for location A (refer to Figure 3.11 for the location) in all panels for comparison. The dotted curve is the individual EMD modes for those locations.

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Annual mean of |η′| with two experiments in 1994 5

45 AMUSE

4.5

SEMAPHORE G

F

4

E

3.5 40 3 2.5

D

H

2 35 1.5 1 0.5 30 A 30W

25W

B 20W

15W

C 10W

05W

0

FIGURE 3.11 Comparisons of annual mean |∆η′| signal from two float experiments in 1994. The gray scale and contours show the annual mean of the |∆η′| signal estimated from our method. All Meddies were discovered during the AMUSE experiment (stars) and SEMAPHORE experiment (circles) in 1994.

To find out the dominant signal of the sea surface interaction with the Meddies, the HHT was applied to their EMD modes. We chose eight places to investigate the dominant signal in our study area. Figure 3.10 showed that Place Group 1 (B, C, and D, shown in Figure 3.11) and Group 2 (H and G, shown in Figure 3.11) had a similar signal. However, Group 3 (E and F, shown in Figure 3.11) had slightly different signals from Groups 1 and 2. Consequently, there were seasonal fluctuations between location A and Group 3. To compute the dominant signal of the power for locations A and E, HHT was also employed; this is shown in Figure 3.12. In both, the dominant frequency was around f = 0.082 (1 year). However, location A had a lower frequency, f = 0.03 (33.3 months), which seems to indicate that wind stress with 33.3-month period produces sea surface forcing. The surface variations produce the baroclinic instability on the Meddies, which is related to southward translation. The southward translation of the Meddies due to the baroclinic effects were consistent with that discussed by Müler and Siedler (36) and by Käse and Zenk (37). If the current is vertically sheared in a stratified fluid, baroclinic instability can occur. Figure 3.12 also demonstrates that the comparison between field observations from two float experiments in 1994 and our computation was excellent.

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Frequency-Spectrum-Energy of the streamfuction at A

Energy

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FIGURE 3.12 Time–frequency–energy spectrum of the stream functions at locations A and E. Refer to Figure 3.11 for location A and E. One can see the high energy at the 33-month period at location A, but not at location E.

3.5 CONCLUSION Three examples of applications of the HHT method in ocean-atmosphere remote sensing research were illustrated in this chapter. These examples show that the HHT method is indeed a potentially very useful and powerful tool for ocean engineering and science studies.

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ACKNOWLEDGMENTS This research was supported partially by the National Aeronautics and Space Administration (NASA) through Grant NAG5-12745 and NGT5-40024, by the Office of Naval Research (ONR) through Grant N00014-03-1-0337, and by the National Oceanic and Atmospheric Administration (NOAA) through Grants NA17EC2449 and NA96RG0029.

REFERENCES 1. Chan, Y. T. (1995). Wavelet Basics. Boston, MA: Academic. 2. Huang, N., Long, S., Shen, Z. (1996). The mechanism for frequency downshift in nonlinear wave evolution. Adv. Appl. Mech. 32:59–111. 3. Huang, N., Shen, Z., Long, S., Wu, M., Shin, H., Zheng, Q., Yen, N.-C, Tung, C., Liu H. (1998). The empirical mode decomposition and Hilbert spectrum for nonlinear and nonstationary time seriesanalysis. Proc. Royal Soc. Mathematical, Phys. Eng. Science 454:903–995. 4. Yan, X.-H., Zheng, Q., Ho, C.-R., Tai, C.-K., Cheney, R. (1994). Development of the pattern recognition and the spatial integration filtering methods for analyzing satellite altimeter data. Remote Sens. Environ. 48:147–158. 5. Cheney, R., Miller, L., Agreen, R., Doyle, N., Lillibridge, J. (1994). TOPEX/POSEIDON: 2 cm solution. J. Geophys. Res. 99:24555–24563. 6. Salisbury, J. I., Wimbush, M. (2002). Using modern time series analysis techniques to predict ENSO events from the SOI times series. Nonlinear Proc. Geophys. 9:341–345. 7. Kiddon, J. A., Paul, J. F., Buffum, H. W., Strobel, C. S., Hale, S. S., Cobb, D., Brown, B. S. (2003). Ecological condition of U.S. mid-Atlantic estuaries. 1997–1998. Marine Pollution Bulletin 46(10):1224–1244. 8. Keiner, L. E., Yan, X-H. (1998). A neural network model for estimating sea surface chlorophyll and sediments from thematic mapper imagery. Remote Sens. Environ. 66(2):153–165. 9. Zhang, Y., Pulliainen, J., Koponen, S., Hallikainen, M. (2002). Application of an empirical neural network to surface water quality estimation in the Gulf of Finland using combined optical data and microwave data. Remote Sens. of Environ. 81:327–336. 10. Baruah, P. J., Oki., K., Nishimura, H. (2000). A neural network model for estimating surface chlorophyll and sediment content at Lake Kasumi Gaura of Japan. In: Conference Processing of GIS Development. Taipei, Taiwan, December. 11. Buckton, D., O’Mongain, E., Danaher, S. (1999). The use of neural networks for the estimation of oceanic constituents based on the MERIS instrument. Int. J. Remote Sensing 20(9):1841–1851. 12. Keiner, L. E., Yan, X.-H. (1997). Empirical orthogonal function analysis of sea surface temperature patterns in Delaware Bay. IEEE Trans. Geoscience Remote Sensing 25(5):1299–1306. 13. Zhang, T., Fell, F., Liu Z.-S., Preusker, R., Fischer, J., He, M.-X. (2003). Evalating the performance of artificial neural network techniques for pigment retrieval from ocean color in Case I Water. J. Geophys. Res. 108(C9):3286–3298.

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The Hilbert-Huang Transform in Engineering 14. Dzwonkowski, B. (2003). Development and application of a neural network based ocean color algorithm in coastal waters. Masters thesis, University of Delaware, Newark, DE. 15. Barnard, A. H., Stegmann, P. M., Yoder, J. A. (1997). Seasonal surface ocean variability in the South Atlantic Bight derived from CZCS and AVHRR imagery. Continental Shelf Res. 17:1181–1206. 16. Pennock, J. R. (1985). Chlorophyll distributions in the Delaware estuary: Regulation by light-limitation. Estuarine, Coastal and Shelf Science 21:711–725. 17. Hurrell, J. W. (1995). Decadel trends in the North Atlantic Oscillation regional temperatures and precipitation. www.cgd.ucar.edu/cas/papers/science1995/sci.htm. 18. Thomas, A. C., Townsend, D. W., Weatherbee, R. (2003). Satellite-measured phytoplankton variability in the Gulf of Marine. Continential Shelf Res. 23:971–989. 19. Yan, X.-H., Schubel, J. R., Pritchard, D. W. (1990). Oceanic upper mixed layer depth determination by the use of satellite data. Remote Sens. Environ. 32 (1):55–74. 20. Yan, X.-H., Okubo, A., Shubel, J. R., Pritchard, D. W. (1991). An analytical mixed layer remote sensing model. Deep Sea Res. 38:267–287. 21. Yan, X.-H., Niller, P. P, Stewart, R. H. (1991). Construction and accuracy analysis of images of the daily-mean mixed layer depth. Int. J. Remote Sensing 12(12):2573–2584. 22. Yan, X.-H., Okubo, A. (1992). Three-dimensional analytical model for the mixed layer depth. J. Geophy. Res., 97(C12):20201–20226. 23. Stammer D.,. Hinrichsen, H. H, Käse, R. H. (1991). Can Meddies be detected by satellite altimetry? J. Geophys. Res. 96:7005–7014. 24. Wilson, W., Bradley, D. (1996). Technical Report NOLTR. 66–103. 25. White, W. B., Tai, C.-K. (1995). Inferring interannual changes in global upper ocean heat storage from TOPEX altimetry J. Geophys. Res. 15:24943–24954. 26. Levitus, S., Boyer, T. P. (1994). World Ocean Atlas, Vol. 4, Temperature, NOAA Atlas NESDIS 4, U.S. Govt. Print. Off., Washington, D.C., 117. 27. Iorga, M. C., Lozier, M. S. (1999). Signature of the Mediterranean outflow from a North Atlantic climatology, 1. Salinity and density fields. J. Geophys. Res. 104:25985–26009. 28. Hebert, D. L. (1988). A Mediterranean salt lens. Ph.D. dissertation, Dalhousie University, Halifax, Nova Scotia. 29. D’Asaro, E. A. (1993). Generation of submesoscale vortices: A new mechanism. J. Geophys. Res. 93:6685–6693. 30. Armi, L., Stommel, H. (1983). Four views of a portion of the North Atlantic subtropical gyre. J. Phys. Oceanogr. 13:828–857. 31. Armi, L., Zenk, W. (1984). Large lenses of highly saline Mediterranean salt lens. J. Phys. Oceanogr. 14:1560–1576. 32. Richardson, P. L., Tychensky, A. (1998). Meddy trajectories in the Canary Basin measured during the SEMAPHORE experiment, 1993–1995. J. Geophys. Res. 103:25029–25045. 33. Spall, M. A., Richardson, P. L., Price, J. (1993). Advection and eddy mixing in the Mediterranean salt tongue. J. Mar. Res. 51:797–818. 34. Tychensky, A., Carton, X. (1998). Hydrological and dynamical characterization of Meddies in the Azores region: A paradigm for baroclinic vortex dynamics. J. Geophys. Res. 103:25061–25079. 35. Douglas, B. C., Cheney, R. E, Agreen, R. W. (1983). Eddy energy of the northwest Atlantic and Gulf of Mexico determined from GEOS 3 altimetry. J. Geophys. Res. 88:9595–9603.

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36. Müler, T. J., Siedler, G. (1992). Multi-year current time series in the eastern North Atlantic Ocean, J. Mar. Res. 50:63–98. 37. Käse, R. H., Zenk W. (1996). Structure of the Mediterranean water and Meddy characteristics in the northeastern Atlantic. In: Ocean Circulation and Climate: Observation and Modeling the Global Ocean, W. Krauss, Ed. Berlin: Gebrüder Bornträger, Berlin, pp. 365–395.

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4

A Comparison of the Energy Flux Computation of Shoaling Waves Using Hilbert and Wavelet Spectral Analysis Techniques Paul A. Hwang, David W. Wang, and James M. Kaihatu

CONTENTS 4.1 4.2 4.3

Introduction ....................................................................................................84 The Huang-Hilbert Spectral Analysis............................................................84 Shoaling Waves and Energy Flux Computation............................................85 4.3.1 Field Measurement.............................................................................85 4.3.2 Numerical Simulations.......................................................................88 4.4 Discussions.....................................................................................................91 4.4.1 Resolution and Nonlinearity ..............................................................91 4.4.2 Edge Effect.........................................................................................93 4.5 Summary ........................................................................................................94 Acknowledgments....................................................................................................94 References................................................................................................................94

83

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ABSTRACT The HHT analysis interprets wave nonlinearity in terms of frequency modulation instead of harmonic generation. The resulting spectrum energy concentrates in the neighborhood of the fundamental frequency in comparison with the FFT or wavelet spectrum. The energy ﬂux computation from the HHT spectrum is considerably larger than with the FFT or wavelet spectrum.

4.1 INTRODUCTION Fourier transform decomposes a nonlinear waveform into a fundamental frequency component and its harmonics. As a result, the energy of a nonlinear wave is spread into higher frequencies. This may produce difﬁculties in the interpretation of wave propagation and quantiﬁcation of wave dynamic properties such as the energy ﬂux of the wave ﬁeld. Recently, Norden Huang and his colleagues developed a new analysis technique, the Hilbert-Huang transformation (HHT). Through analytical examples, they demonstrated the excellent frequency and temporal resolution of HHT for analyzing nonstationary and nonlinear signals [1, 2]. With the HHT analysis, the physical interpretation of nonlinearity is frequency modulation, which is fundamentally different from harmonic generation. The HHT spectrum therefore retains its energy density near the fundamental frequency of the wave motion in comparison with the FFT or wavelet spectrum [3]. In this article, we brieﬂy describe the HHT technique and investigate its use in the calculation of the energy ﬂux of ocean waves. The resulting HHT spectrum is compared with the counterpart obtained by the wavelet method. The wavelet technique is based on Fourier spectral analysis but uses adjustable frequency-dependent window functions — the mother wavelets — to provide temporal/spatial resolution for nonstationary signals [4–7]. As expected, the Fourier-based analysis interprets wave nonlinearity in terms of harmonic generation; thus the spectral energy leaks to higher frequency components. The HHT interprets wave nonlinearity as frequency modulation, and the spectral energy remains near the base frequency. The computed energy ﬂux from the HHT method is much higher than that from the wavelet method.

4.2 THE HUANG-HILBERT SPECTRAL ANALYSIS Hilbert transformation was introduced to water wave analysis in the 1980s. Applications include the study of wave modulation leading to wave breaking [8], local properties of sea waves such as the group and pulse structure, ﬂuctuation of wave energy and energy ﬂux [9], and the measurement and quantiﬁcation of breaking events of wind-generated surface waves [10]. A key function of the Hilbert analysis is the derivation of local wave number in a spatial series or instantaneous frequency in a time series. To use the Hilbert transformation, proper preprocessing of the signals is critical. Large errors in the computed local frequency or wave number can occur when small waves are riding on longer waves or when sharp changes of the oscillation frequencies occur in the wave signal. Quantitative discussions on the riding

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wave problem [1] and on the rapid change of the oscillation frequencies [11] have been published and will not be repeated here. The key ingredient in the HHT is empirical mode decomposition (EMD), designed to reposition the riding waves at the mean water level. This is achieved through a sifting process that repeatedly subtracts the local mean from the original signals. The procedure decomposes the signal into many modes with different frequency characteristics, and thus also alleviates the problem of sharp frequency change in the original signal. Huang and his coworkers have provided extensive discussions on the EMD [1, 2]. From experience, even for very complicated random signals, a time series can usually be decomposed into a relatively small number of modes, M < log2 N. Each mode is free of riding waves and is suitable for Hilbert transformation to yield accurate local frequency of the mode. The spectrum of the original signal can be obtained from the sum of the Hilbert spectra of all modes. Extensive tests have been carried out, and the HHT technique proves to deliver very high frequency and temporal/spatial resolutions in analyzing nonstationary signals. It is also shown to be able to handle the task of analyzing nonlinear signals produced by exact solution with modulating oscillation frequencies, whereas Fourier-based techniques interpret the signals as superposition of harmonics [1–3].

4.3 SHOALING WAVES AND ENERGY FLUX COMPUTATION The Hilbert view of nonlinear waves is signiﬁcantly different from the conventional Fourier view. In particular, intra-wave modulation is a key signature of nonlinearity based on the Hilbert spectral analysis, while harmonic generation is typical of the Fourier spectrum. The difference in the interpretations of nonlinearity will lead to quantitative differences in the energy ﬂux computation.

4.3.1 FIELD MEASUREMENT An example of intra-wave modulation highlighted by the HHT analysis is illustrated in the analysis of shoaling swell shown in Figure 4.1. The three-dimensional (3D) surface wave topography was acquired by an airborne topographic mapper (ATM, an airborne scanning laser system) near Duck, N.C., three days after an extratropical storm passed through the area. The wind condition at the time of data acquisition was low, resulting in a relatively simple two-dimensional (2D) swell system propagating on mild slope bathymetry. More discussions of the environmental conditions, measurement techniques, and the bathymetry of the region are reported by Hwang et al. [12]. Figure 4.1 illustrates the spatial evolution of the airborne measured shoaling swell (Plot a), the Hilbert spectrum (Plot b), and the wavelet spectrum (Plot c) in the near coast region. For reference, the water depth based on the bathymetry database is shown in Plot d. The spectral energy is distributed narrowly about the dominant frequency. The distinctive intra-wave modulation in the neighborhood of the peak frequency of the wave spectrum can be clearly identiﬁed in the HHT spectrum (Figure 4.1b). In contrast, the wavelet spectrum distributes the energy into harmonics, as a result of interpreting nonlinearity as harmonic generation in the

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FIGURE 4.1 (a) A 3D surface wave topography of shoaling swell in the coastal region. (b) The HHT spectrum of the waves shown in (a). (c) The wavelet spectrum of the waves shown in (a). (d) The water depth from bathymetry database.

Fourier-based spectral processing (Figure 4.1c), as discussed earlier. It is noticeable that the second harmonic of the wavelet spectrum displays a spatial modulation characteristic similar to the intra-wave modulation of the HHT spectrum at the peak frequency. The energy ﬂux, F, of the wave ﬁeld can be computed from the wave spectrum, S, by

()

F x =

∫ S (ω; x )C (ω; x ) dω , g

(4.1)

where x is the spatial coordinate in the propagation direction, ω is angular frequency, and Cg is the wave group velocity. The computed results based on HHT and wavelet analyses are shown in Figure 4.2. The magnitude derived from the Hilbert spectrum is in general larger than that obtained by the wavelet spectrum. As explained in the last section, this is partly caused by the much higher spectral level in the lower frequency of the Hilbert spectrum compared to the wavelet spectrum. Because phase velocity and group velocity of gravity waves increase monotonically with wavelength, the increased low frequency spectral density translates to a higher level of the energy ﬂux in the wave ﬁeld. The source function, Q, of the wave system can be derived from the spatial rate of change of the energy ﬂux:

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(a)

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FIGURE 4.2 (a) Energy ﬂux computed from the shoaling swell shown in Figure 4.1. (b) The source function computed from the HHT spectrum. (b) The source function computed from the wavelet spectrum.

∂F =Q . ∂x

(4.2)

Figure 4.2b and c shows the derived source function based on the Hilbert and wavelet spectra, respectively. The magnitude of the Hilbert source term is considerably larger than the wavelet source term (notice the difference in scale of the two ordinates). The oscillatory nature of the source function highlights the complex nature of the wave conditions in the ﬁeld. The group structure can be introduced by many processes, such as wave nonlinearity, interaction among different wave components, and bathymetry-induced wave scattering. The group structure causes the source function to ﬂuctuate between positive and negative values, as can be visualized from Equation 4.1 and Equation 4.2. Further discussion of the group structure will be presented later. In appearance, the source functions obtained by the HHT and wavelet methods, as shown in Figure 4.2b and c, are considerably different. In reality, the apparent difference reﬂects the much higher temporal/spatial resolution of the HHT method. Figure 4.3 shows the results of the running average of HHT computation, with the wavelet result superimposed. With an averaging bin width of 60 elements, the running average of the HHT source function is almost identical to the wavelet source function. The spatial average (excluding the leading and trailing 90 m of the spatial coverage shown) of the source term is –2.49 × 10–4 for the wavelet processing, –5.64 × 10–4 for the HHT processing, and –4.78 × 10–4 for the running average of the HHT

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<EF(HHT)>=−6.373e−004, <EF(HHT)> =−5.522e−004, <EF(wavelet)>=−3.083e−004, offset=30 60

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processing. From this limited investigation, we conclude that the energy ﬂux computation may be off by a factor of two using FFT based processing techniques. As mentioned earlier, harmonic generation has created major difﬁculties for the analysis of wave dynamics due to the dispersive nature of water waves. For example, the phase and group velocities of each free wave component are frequency dependent, yet the harmonics (of nonlinear waves) are not dispersive. Therefore, the energy ﬂux (the product of group velocity and spectral density) of the harmonics associated with the nonlinear waves cannot be distinguished from the energy ﬂux of the free waves at the same frequency, and FFT based processing will always underestimate the energy ﬂux of the wave ﬁeld. The underestimation will reﬂect on the magnitude of parameters such as the dissipation rate or growth rate of a wave ﬁeld.

4.3.2 NUMERICAL SIMULATIONS To investigate further the inﬂuence of processing methods and wave nonlinearity on the quantitative results of energy ﬂux computation, numerical simulations were carried out using the refraction–diffraction (REFDEF) model [13]. The case simulated is a train of monochromatic waves (10 sec wave period and 0.25 m initial wave amplitude) propagating from offshore along a plane sloping beach. In the ﬁrst scenario, the nonlinear terms are turned off, and the waveform remains sinusoidal along the full computational domain (Figure 4.4Aa). In the second scenario, the nonlinear terms are turned on, and wave asymmetry becomes obvious in the nearshore region (Figure 4.4Ba). The spatial evolution of the wave spectra for the two

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(a)

(b)

FIGURE 4.4 Numerical simulations of shoaling waves. Left panels: linear simulation. Right panels: nonlinear simulation. (a) Waveform and water depth, (b) HHT spectrum, (c) wavelet spectrum, and (d) characteristic wave numbers kp and k2.

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linear

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FIGURE 4.5 Same as Figure 4.4 but for the energy ﬂux computation. (a) Energy ﬂux, (b) gradient of energy ﬂux, and (c) same as (b) but phase averaged. A: linear simulation. B: nonlinear simulation.

scenarios by the HHT processing method is shown in Figure 4.4b. The spectral density is concentrated in a very narrow frequency band, as expected from a monochromatic wave train. The lack of frequency modulation in the absence of nonlinearity is clear from a comparison of Figure 4.4Ab with Figure 4.4Bb. The result based on the wavelet method is shown in Figure 4.4c. The spectral energy is more spread out because of the ﬁnite window size, and the effect of nonlinearity is harmonic generation. The characteristic wave number is typically deﬁned as the peak wave number, kp, or weighted wave number computed from the spectral moment, kn = ( ∫knS(k)dk/ ∫S(k)dk)1/n. With the HHT analysis, kp and kn are almost identical for both linear and nonlinear scenarios because the spectral energy is conﬁned in a narrow frequency range. With the wavelet analysis, kp and kn are also very similar in the linear scenario, but they differ considerably in the nonlinear scenario as a result of harmonic generation. The results of phase-average kp and k2 for the two scenarios are shown in Figure 4.4d. Intra-wave frequency modulation as reﬂected by the oscillation of the characteristic wave number is clearly seen in the HHT analysis of the nonlinear simulation; it is also somewhat noticeable but much weaker in the wavelet analysis. For the linear case, the computed energy ﬂux and the gradient of energy ﬂux derived from the two processing methods are very similar (Figure 4.5a,b,c). The HHT results show small intra-wave modulation due to ﬁnite amplitude of the waveform. For the nonlinear case, the intra-wave modulation is greatly ampliﬁed by the

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nonlinearity of the wave system. The differentiation process in calculating the gradient of energy ﬂux sometimes produces large spikes in the computation. The results become very sensitive to small modiﬁcations in the data processing procedure. Figure 4.5 shows an example of the processed results.

4.4 DISCUSSIONS 4.4.1 RESOLUTION

AND

NONLINEARITY

Fourier-based spectral analysis methods have been widely used for studying random waves. One major weakness of the Fourier-based spectral analysis methods is the assumption of linear superposition of wave components. As a result, the energy of a nonlinear wave is spread into many harmonics, which are phase-coupled via the nonlinear dynamics inherent in ocean waves. In addition to the nonlinearity issue, strictly speaking Fourier spectral analysis should be used for periodic and stationary processes only. Wave propagation in the ocean is certainly neither stationary nor periodic. Using the HHT analysis, the physical interpretation of nonlinearity is frequency modulation, which is fundamentally different from the commonly accepted concept associating nonlinearity with harmonic generation. Huang et al. [1, 2] argued that harmonic generation results from the perturbation method used in solving the nonlinear equation governing the physical processes, thus the harmonics are produced by the mathematical tools used for the solution rather than being a true physical phenomenon. Through analytical examples, they demonstrated the excellent frequency and temporal resolutions of HHT for analyzing nonstationary and nonlinear signals. Here we give a few computational examples to illustrate the points discussed earlier. Figure 4.6 presents a comparison of HHT and wavelet analysis of three different cases [3]. Case 1 is an example of an ideal time (t) or space (x) series of sinusoidal oscillations of constant amplitude; the frequency (f) or wave number (k) of the ﬁrst half of the signal is twice that of the second half (Figure 4.6a). The spectra computed by the HHT and wavelet techniques are displayed in Figure 6b and c, respectively. The HHT spectrum yields very precise frequency resolution as well as high temporal resolution in identifying the sudden change of signal frequency at about the half point of the time series. In comparison, the wavelet spectrum has only a mediocre temporal resolution of the frequency change. There is also a serious leakage problem, and the spectral energy of the simple oscillations spreads over a broad frequency range (the contour interval is 3 dB in the spectral plots). Case 2 is a single cycle sinusoidal oscillation occurring at the middle of the otherwise quiescent signal stream (Figure 4.6d). The precise temporal resolution of the HHT method is clearly demonstrated by the sharp rise and fall of the HHT spectrum coincident with the transient signal, as illustrated in Figure 4.6e. In comparison, the wavelet spectrum is much more smeared, both in the frequency and temporal resolutions (Figure 4.6f). Case 3 is a sinusoidal function, y, with its oscillating frequencies subject to periodic modulation (Figure 4.6g)

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1000

0

−1 500

1500

0.3

(d)

y

0

−1 500

1

(a)

y

y

1

1000

1500

(b) 0.2

0

−1 500

550

3dB contours

600 (h)

k|f

k|f

k|f

0.2 0.1

0.1

0.1 1000

0 500

1500

0.3

1000

1500

(c) 0.2

0 500

550

(f) 0.3

600 (i)

k|f

k|f

0.2

k|f

0.2 0.1

0.1 0 500

(g)

(e) 0.3

0.2

0 500

1

0.1 1000 x|t

1500

0 500

1000 x|t

1500

0 500

550 x|t

600

FIGURE 4.6 Examples comparing HHT (middle row) and wavelet (bottom row) analysis of nonlinear and nonstationary signals. (a,b,c) Time series of the harmonic motion with the oscillation period doubled in the second half, and its HHT and wavelet spectra. (d,e,f) Transient sinusoidal function of one cycle. (g,h,i) Oscillatory motion with modulated frequencies.

()

(

)

y t = a cos ωt + ε sin ωt ,

(4.3)

where a is the amplitude, ω is the angular frequency, and ε is a small perturbation parameter. This is the exact solution for the nonlinear differential equation [1]

(

d2y + ω + εω cos ωt dt 2

)

2

(

y − 1− x2

)

0.5

εω 2 sin ωt = 0.

(4.4)

If the perturbation method is used to solve Equation 4.2, the solution to the ﬁrst order of ε is 1 y1 t = cos ωt − ε sin 2ωt = cos ωt − ε  1 − cos 2ωt 2

()

(

) . 

(4.5)

The HHT spectrum (Figure 4.6h) correctly reveals the nature of oscillatory frequencies of the exact solution (Equation 4.3). In contrast, the wavelet spectrum (Figure 4.6i) shows a dominant component at the base frequency and periodic oscillations of the second harmonic component.

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f(t)

1

f(t) f(t)

50

100

150 t

200

150 t

200

150 t

200

150 t

200

250

(a)

300 (b)

0 −1 0 1

50

100

50

100

250

300

exp(−0.01t)cosθ (expanded) Hilbert transform exp(−0.01t)sinθ (expanded)

0 −1 0 1

f(t)

cosθ (expanded) Hilbert transform sinθ (expanded)

0 −1 0 1

93

250

(c)

300 (d)

0 −1 0

50

100

250

300

FIGURE 4.7 An example illustrating the mirror-imaging approach to alleviate edge problems.

These three examples illustrate the excellent temporal (spatial) and frequency (wave number) resolution of the HHT method for processing nonlinear and nonstationary signals. The result from Case 3 is especially interesting, as it illustrates the unique ability of the HHT technique to reveal more accurately the nature of nonlinear processes with nonstationary oscillation frequencies. Solutions obtained through the perturbation method and analysis results obtained through Fourier-based techniques represent those processes as superpositions of harmonic motions.

4.4.2 EDGE EFFECT One problem frequently encountered in data processing involves the treatment of the two edges of the data segment. This problem is especially serious when the data segment is short, as in many transient processes. Figure 4.7a shows an example of one sinusoidal cycle (cos t) between 100 and 300 time marks, shown by the solid curve. The Hilbert transformation of the data segment (shown by x) deviates from its exact solution, sin t, at the two edges. Padding zeros to expand the data segment does not alleviate the problem (Figure 4.7a), but repeating the signal sometimes helps, as shown in Figure 4.7b. However, if the expanded data sequence contains discontinuities (Figure 4.7c), the edge problem persists. A technique to avoid discontinuities at the two edges when expanding the data segment is to mirror image the data, as illustrated in Figure 4.7d. Our experience indicates that the edge problem can be signiﬁcantly alleviated with about 30% expansion of the original data segment with the mirror-imaging method. More elaborate mirror-imaging techniques may include applying windowing (e.g., exponential decay) to the imaged segments (private communication, N. E. Huang, 2004). © 2005 by Taylor & Francis Group, LLC

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4.5 SUMMARY Analyzing nonlinear and nonstationary signals remains a very challenging task. Many methods developed to deal with nonstationarity are based on the concept of Fourier decomposition; therefore, all the shortcomings associated with Fourier transformation are also inherent in those methods. The recent introduction of empirical mode decomposition [1, 2] represents a fundamentally different approach for processing nonlinear and nonstationary signals. The associated spectral analysis (HHT) provides excellent spatial (temporal) and wave number (frequency) resolution for handling nonstationarity and nonlinearity [1–3]. The HHT spectrum also results in a considerably different interpretation of nonlinearity (frequency modulation, as compared to the traditional view of harmonic generation). Applying the technique to the problems of ocean waves, we found that the spectral function derived from HHT is markedly different from those obtained by the Fourier-based techniques. The difference in the resulting spectral functions is attributed to the interpretation of nonlinearity. The Fourier techniques decompose a nonlinear signal into sinusoidal harmonics; therefore, some of the spectral energy at the base frequency is distributed to the higher frequency components. The HHT interprets nonlinearity in terms of frequency modulation, and the spectral energy remains in the neighborhood of the base frequency. This results in a considerably higher spectral energy at lower frequencies and a sharper dropoff at higher frequencies in the HHT spectrum in comparison with the Fourier-based spectra [3]. The energy ﬂux computed by using the HHT spectrum is much higher than that obtained from the FFT or wavelet methods.

ACKNOWLEDGMENTS This work is supported by the Ofﬁce of Naval Research (Naval Research Laboratory Program Elements N61153 and N62435). This is NRL contribution BC/7330.

REFERENCES 1. Huang, N. E., Shen, Z., Long, S. R., Wu, M. C., Shih, H. H., Zheng, Q., Yuen, N. C., Tung, C. C., and Liu, H. H. (1998). The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis. Proc. R. Soc. Lond. 454A: 903–995. 2. Huang, N. E., Shen, Z., and Long, S. R. (1999). A new view of nonlinear water waves: The Hilbert spectrum. Annu. Rev. Fluid Mech. 31: 417–457. 3. Hwang, P. A., Huang, N. E., and Wang, D. W. (2003). A note on analyzing nonlinear and nonstationary ocean wave data. Appl. Ocean Res. 25: 187–193. 4. Shen, Z., and Mei, L. (1993). Equilibrium spectra of water waves forced by intermittent wind turbulence. J. Phys. Oceanogr. 23(9): 2019–2026. 5. Shen, Z., Wang, W., and Mei, L. (1994). Finestructure of wind waves analyzed with wavelet transform. J. Phys. Oceanogr. 24(5): 1085–1094. 6. Liu, P. C. (2000). Is the wind wave frequency spectrum outdated? Ocean Eng. 27: 577–588.

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7. Massel, S. R. (2001). Wavelet analysis for processing of ocean surface wave records. Ocean Eng. 28: 957–987. 8. Melville, W. K. (1983). Wave modulation and breakdown. J. Fluid Mech. 128: 489–506. 9. Bitner-Gregersen, E. M., and Gran S. (1983). Local properties of sea waves derived from a wave record. Appl. Ocean Res. 5: 210–214. 10. Hwang, P. A., Xu, D., and Wu, J. (1989). Breaking of wind-generated waves: measurements and characteristics. J. Fluid Mech. 202: 177–200. 11. Guillaume, D. W. (2002). A comparison of peak frequency-time plots produced with Hilbert and wavelet transforms. Rev. Scientific Inst. 73(1): 98–101. 12. Hwang, P. A., Walsh, E. J., Krabill, W. B., Swift, R. N., Manizade, S. S., Scott, J. F., and Earle, M. D. (1998). Airborne remote sensing applications to coastal wave research. J. Geophys. Res. 103(C9): 18791–18800. 13. Kaihatu, J. M., and Kirby, J. T. (1995). Nonlinear transformation of waves in ﬁnite water depth. Phys. Fluids 7: 1903–1914.

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5

An Application of HHT Method to Nearshore Sea Waves Albena Dimitrova Veltcheva

CONTENTS 5.1 5.2 5.3 5.4 5.5

Introduction ....................................................................................................98 Field Data.......................................................................................................99 Conventional Methods for Wave Data Analysis..........................................100 Hilbert-Huang Transform Method ...............................................................106 Application of the HHT Method .................................................................108 5.5.1 Offshore Waves During Different Sea Stages .................................112 5.5.2 Cross-Shore Transformation of Sea Waves .....................................114 5.6 Conclusions ..................................................................................................117 References..............................................................................................................118 ABSTRACT Real sea waves are nonlinear by nature, and this nonlinearity increases considerably in the shoreward direction. The presence of wave groups in the sea wave records shows that the wave process is also not stationary, as it is usually considered in conventional wave data analysis. In this work, the Hilbert-Huang transform (HHT) method for nonlinear and nonstationary time series analysis is applied to wave data from the nearshore area. The field data were collected at Hazaki Oceanographical Research Station (HORS), Port and Airport Research Institute, Japan, and covered different sea states. The importance of a proper choice of an analyzing technique for the correct understanding of the examined phenomenon is discussed. The ability of EMD, a key part of the HHT method, to extract into IMFs the different oscillation modes embedded in the sea surface elevation records is demonstrated. The variation of IMFs along the beach profile is examined in an attempt to investigate cross-shore transformation of sea waves. The dominant oscillations for each data record are determined, and their variations during different sea stages are investigated. 97

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5.1 INTRODUCTION Real sea waves are nonlinear and nonstationary by nature, and their profile is asymmetrical with respect to the zero level, with sharp crests and flat troughs, especially well pronounced in shallow water. The existence of small ripples riding longer waves, as well as an appearance of many small waves due to wave breaking, complicates the wave profile considerably. Real sea waves also appear in groups, which are time localized transient events. The wave data records always register a mixture of many wave systems with different periods and energy, generated by different sources. Through analysis of sea surface elevation record, we seek to determine characteristic time–frequency scales of the energy. The widely used methods of wave analysis are burdened with the traditional concept of sea waves: as a sine wave with definition domain from minus infinity to plus infinity. It is also assumed, according to linear wave theory, that within the framework of one wave the period is constant and the wave height is equal to twice the wave amplitude. In addition to the positive maxima and negative minima of the wave profile, positive minima and negative maxima are also often observed in the profile of real sea waves. The neglect of these actual features of sea waves leads to incompleteness and misinterpretation of the recorded sea state, and thus the analyzing method can distort the investigated phenomenon. The dominant approach in the existing methods of wave analysis is decomposition of the time series into component basis functions. Global sinusoidal components of fixed amplitude are the basis for data expansion in the commonly used Fourier spectral analysis methods. The parameters of water waves, determined by Fourier spectra, are integral characteristics of the sea state. Despite its being widespread and useful, the Fourier spectral technique has an important drawback: the Fourier spectrum contains no information about the timing of the analyzed process. Consequently, the local time variations of determined wave characteristics are impossible to track. To understand the sea wave process correctly, we need an adequate method for analysis. In the last decade the attention of scientists started to turn from globally estimated wave parameters to local properties of a wave system. The attention has been addressed to the time distribution of the wave energy, complimentary to the frequency distribution information provided by usual Fourier spectral analysis. Liu (1) suggested that the wave frequency spectrum approach is outdated, presenting the advantages of the wavelet spectrum in the investigation of wave growth. The differences between the Fourier and the wavelet spectrum are widely discussed by Massel (2) on the basis of the application of the wavelet transform for the processing of sea wave data. Huang et al. (3) also point out the necessity to break down the earlier paradigm of wave analysis, emphasizing that Fourier analysis is not a good method for studying nonlinear and nonstationary water waves. Several Fourier-based methods for frequency–time distribution of the energy are reviewed by Huang et al. (4): time window-width Fourier transform, evolutionary spectrum, principal component analysis, and wavelet analysis. These methods suffer from the demerits of Fourier spectral analysis — global definition of harmonic components and the use of linear superposition for decomposition of the data

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(assuming the stationarity and linearity of data). There is also a resolution problem inherent in all Fourier-based analysis due to the Heisenberg Uncertainty Principle. Wavelet analysis provides some improvement in the resolution by using adjustable windows, but the problems of the nonadaptive nature of these windows are still not solved completely. It has to be mentioned in addition that in these methods for frequency–time distribution of the energy, the basis for expansion of the data must be chosen in advance. An exception is the principal component analysis, in which the basis is derived empirically from the data. In this paper, we discuss the importance of the analyzing technique for the correct understanding of the examined phenomenon and pay special attention to the nonlinear and nonstationary behavior of real sea waves. We propose a new method for nonlinear and nonstationary time series analysis, the Hilbert-Huang transform (HHT) method, introduced by Huang et al. (4), as an alternative one for investigation of real sea waves. The empirical mode decomposition (EMD), a key part of the HHT method, extracts the energy associated with various intrinsic time scales into a set of intrinsic mode functions (IMFs). By virtue of their derivation, the IMFs have well-behaved Hilbert transform results, and the instantaneous frequencies can be calculated. The frequency–time distribution of energy is designated as a Hilbert spectrum. Any event in the data series can be localized in time as well as in frequency. We apply the HHT to the wave data from the nearshore area and demonstrate the ability of EMD to extract into IMFs the different oscillation modes embedded in the sea surface elevation records. The variation of IMFs along the beach profile is examined in an attempt to investigate cross-shore transformation of sea waves. The dominant oscillations for each data record are determined, and their variations during different sea stages are investigated.

5.2 FIELD DATA The wave data used in this work were collected at the Hazaki Oceanographical Research Station (HORS), Port and Airport Research Institute, Japan, during the period from 25 February to 1 March 1989. The HORS is located on the Pacific coast of Japan. The measurements were performed by six wave gauges mounted on a 427 m long pier and by three deepwater bottom wave gauges in an extension line of the pier, as shown schematically in Figure 5.1. The offshore distance and water depth of the wave gauges are listed in Table 5.1. The sea surface elevation was recorded every 6 h, and the records are 2 h long at a sampling data rate of 0.5 sec. Different sea stages were observed during the measurement period. The variations of offshore significant wave height (Hs) and mean wave period (Tmean) during the observation are presented in Figure 5.2. The sea conditions are classified into four different sea stages: calm, wave growth, wave decay, and post-storm stage, after Katoh et al. (5). The initial three terms of measurements were in a comparatively calm sea stage, when sea waves had a period of less than 6 sec and significant wave height was less than 2 m. Under the direct influence of a passing atmospheric depression with a strong north wind, growth of the sea waves was observed in the next three terms. The waves became greater than 3 m in significant wave height, but their periods remained short. In the wave decay stage, when the wind speed dropped, © 2005 by Taylor & Francis Group, LLC

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FIGURE 5.1 Location of wave gauges during the observations (HORS, PARI, Japan).

TABLE 5.1 Location of Wave Gauges Number of Wave Gauge Offshore distance (m) Water depth (m)

1

2

3

4

5

6

7

8

3000 24.00

2000 14.00

1000 9.00

380 5.65

320 5.05

215 2.33

145 1.81

80 0.30

the wave periods became longer than 9 sec. At the end of the observation, noted here as the post-storm stage, the wind changed its direction and the registered waves were small in amplitude, but with long periods. The wave gauges 7, 8, and 9 were located inside the surf zone during the calm sea stage; in other sea stages, the surf zone became wider and also included wave gauges 4, 5, and 6.

5.3 CONVENTIONAL METHODS FOR WAVE DATA ANALYSIS An example of sea surface elevation, recorded in the surf zone at 5.05 m water depth, is shown in Figure 5.3. The wave profile is irregular and considerably asymmetrical. There are small waves, resulting from wave breaking as well as ripples, riding the longer waves. The question is how to analyze properly such considerably nonlinear data time series, in view of the fact that the right choice of method is very important for the correct understanding of the investigated phenomenon. The conventional methods for wave data analysis propose estimation of wave characteristics by means of Fourier spectra or by analysis of time series of individual waves, determined by some criteria. The stochastic properties of sea waves, derived by different discretization methods, are presented and discussed in detail by Ochi (6). The identification of discrete waves from the record of sea surface elevation is nontrivial work, especially for the shallow water wave data. The three criteria most

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FIGURE 5.2 Offshore characteristics of sea waves during the observation period.

Sea surface elevatiom (m)

2

1

0

-1

-2 80

90

100

110 Time (s)

FIGURE 5.3 Example of a wave record from surf zone

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130

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TABLE 5.2 Discretization Criteria Criterion

Definition of Individual Waves

Zero-crossing criterion

A wave is determined by two consecutive (up–up or down–down) crossings of mean water level. A wave is determined by two consecutive maxima of surface elevation. A wave is determined as corresponding to a 2 advance of the phase angle in the complex plane.

Crest to crest criterion Orbital criterion

Mean Wave Frequency ω02 ω24 ω01

often used for determination of individual waves from the sea surface elevation records are briefly described in Table 5.2. The estimation of mean frequency depends on the discretization criterion and is determined as ω ij = 2π j −i m j mi , where mk is a kth spectral moment ∞

mk =

∫ f S( f )df , k

0

and S( f ) is a spectral density function. The application of the zero-crossing (ZC) method for determination of individual waves often meets difficulties in the treatment of small amplitude waves. They were considered as “non real” or false waves by Gimenez et al. (7), Kitano et al. (8), and Veltcheva and Nakamura (9), and were eliminated by some of the criteria proposed by Mizuguchi (10), Hamm and Peronnard (11), and Gimenez et al. (12). The main goal of removing the small waves, especially from the surf zone data, is to ensure the constancy of wave period from deep to shallow water. It must be stressed that the procedure of removing the small waves affects the estimation of wave characteristics, assuming a priori the linearity of sea waves. The orbital criterion, introduced by Gimenez et al. (12), uses the complex presentation of wave process. The analytical function ξ(t ) = X (t ) + jXˆ (t ) ,

(5.1)

corresponds to the vertical displacement of sea surface elevation X(t). Here Xˆ (t) is the Hilbert transform of the sea surface elevation,

( )

X t′ 1 Xˆ t = P dt ′ , π t − t′

()

∫

(5.2)

where P indicates the Cauchy principal values. The Hilbert transform is a convolution of the sea surface elevation X(t) with 1/t and thus emphasizes the local properties of X(t).

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The wave envelope, phase, and frequency are defined as

()

()

2 A(t ) = X t + Xˆ t

ϕ(t ) = arctg

Xˆ (t ) , X (t )

2

,

(5.3)

(5.4)

and ω (t ) =

dϕ , dt

(5.5)

respectively. The envelope A(t), defined by Equation 5.3, provides a direct measure of the wave amplitude and is widely used in the statistical analysis of the sea waves (13, 14, 15, 16, 17), while the phase information is usually not considered. Being an important physical quantity, the phase φ(t) associated with the sea wave process can also bear fundamental information about the dynamics of sea waves. But unfortunately, the phase is totally disregarded when conventional methods for wave analysis are employed to study the wave system. The phase φ(t) is a wrapped phase φ(t) ∈ [– π, π] or an unwrapped phase (t) ∈ [–∞, ∞], depending on its interval of definition. An example of the unwrapped and wrapped phase of sea surface elevation X(t) is shown in Figure 5.4. The unwrapped phase in Figure 5.4a increases smoothly with time, in contrast to the wrapped phase, which varies widely in the range [– π, π]. The frequency ω(t), defined by Equation 5.5 as the rate of phase change, is called the instantaneous frequency, and its value changes with time. This definition of frequency was adopted before by Huang et al. (18) and used by Cherneva and Veltcheva (19) for investigation of the local properties of sea waves and their group structure. Later, Huang et al. (4) emphasized the considerable controversy in this definition of instantaneous frequency, even with utilization of the Hilbert transform. The instantaneous frequency is, as mentioned by Cohen (20), “one of the most intuitive concepts, since we are surrounded by light of changing color, by sounds of varying pitch, and by many other phenomena whose periodicity changes. The exact mathematical description and understanding of the concept of changing frequency is far from obvious.…” Blindly applying the definition Equation 5.5 of instantaneous frequency to any analytic function may lead to one of the five paradoxes listed by Cohen (20). One of these paradoxes concerns the appearance of negative frequency: “Although the spectrum of the analytic signal is zero for negative frequencies, the instantaneous frequency may be negative.” The enlarged portion of the unwrapped phase shown in the upper left-hand corner of Figure 5.4a reveals the jumps up and down in the time variation of the unwrapped phase. The unwrapped phase function ϕ(t) is not always increasing function of time, and consequently the instantaneous frequency ω(t) will have negative values, which have no physical meaning.

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1000

Unwrapped phase [rad]

100

Unwrapped phase [rad]

800

600

90 80 70 60 50 80

100 120 Time [s]

140

400

200

0 0

200

0

200

400

600 Time [s]

800

1000

1200

Wrapped phase [rad]

4 2 0 -2 -4 400

600

800

1000

1200

Time [s]

FIGURE 5.4 The phase φ(t) of the analytical process. (a) Unwrapped phase; (b) wrapped phase.

The negative advances of the phase ϕ(t) produce loops or phase reversals in a polar diagram of the wave process (Figure 5.5), where the analytical function ξ(t) is presented as a radius-vector rotating in the complex plane. The magnitude of the vector is equal to the wave envelope A(t), estimated by Huang et al. (3), and the angle of rotation is equal to the unwrapped phase function ϕ(t). The arrows show the direction of rotation of the radius-vector. The rotation of the radius-vector with positive instantaneous frequency is called a proper rotation by Yalcinkaya and Lai (21) in their investigation of the phase dynamics of continuous chaotic flow.

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Sea surface elevation (m)

1.5

0 -1.5

0

1.5

-1 .5 Hilbert transform of sea surface elevation (m)

FIGURE 5.5 Presentation of sea surface elevation in complex plane.

For a clearer illustration, Figure 5.5 shows the polar diagram of the part of the wave record (thick, solid line) of Figure 5.3 between 85 sec and 112 sec. The trajectory of ξ(t) in the complex plane generally may have multiple centers of rotation. The majority of the individual zero-crossing waves in Figure 5.3 correspond to a completely closed trajectory of rotation around the origin of the coordinate system in Figure 5.5. The small ZC waves, around 90 sec and between 105 sec and 110 sec in Figure 5.3, are represented by loops in Figure 5.5. The centers of rotation of these loops differ from the origin of the coordinate system. The small riding waves in the wave profile also produce loops in the polar diagram, like the riding wave between 91 sec and 95 sec in Figure 5.3. These phase reversals, as seen in Equation 5.5, produce negative frequencies, which have no physical meaning. According Gimenez et al.’s (12) orbital criterion, any discrete waves that do not correspond to a 2π advance of the phase in the complex plane are considered as false waves and are simply removed. In this way, the wave process is again assumed to be a linear one, since only the completely closed orbit of water particles with 2π advance of phase angle is accepted as an individual wave. The small zero-crossing waves and ripples are indeed distinctive features of real sea waves from the surf zone, and thus it is incorrect to eliminate them by the orbital criterion for the purpose of achieving the positive frequency. Band-pass filtering,

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proposed by Melville (22), also does not resolve the problem with negative frequencies. Huang et al. (4) especially stressed the fact that the filtering operation itself contaminated the data with spurious harmonics. These unsuccessful attempts to resolve the problem of negative frequencies are probably attributable to an inappropriately chosen tool for investigation of nonlinear and nonstationary sea waves.

5.4 HILBERT-HUANG TRANSFORM METHOD Generally, the method of analysis affects the investigated phenomenon. The choice of an appropriate analyzing technique is very important for a correct understanding of the examined phenomenon, since the functions used in the analysis can be misinterpreted as characteristics of the investigated phenomenon (23). To reduce this risk, the analyzing function has to be chosen in accordance with the intrinsic structure of the field to be analyzed. The HHT method, introduced by Huang et al. (4), completely covers this requirement, since the expansion of the data is based on the local characteristics of the particular data. This method offers a unique and different approach for data processing. The HHT method was motivated “from the simple assumption that any data consists of different intrinsic mode oscillations” (3). The main idea of EMD, a key part of the HHT method, is to identify the time scale that will reveal the physical characteristics of the studied process, recorded in time series, and to extract these time scales into IMFs. A time lapse between successive extrema in the time series is defined as a time scale, and this is the essence of EMD. The EMD is a data sifting process to eliminate locally riding waves as well as to eliminate local asymmetry of the time series profile. Huang et al. (3, 4) present a procedure of sifting and several applications of the HHT method. On the basis of the results of EMD of the data, Veltcheva (24, 25) investigated the group structure of sea waves in the coastal zone. The time series X(t) is first decomposed by EMD into a finite number n IMFs (denoted Cj), which extract the energy associated with various intrinsic time scales, and residual Rn. The superposition of the IMF and residue reconstruct the data:

()

X t =

n

∑ C (t ) + R (t ) . j

n

(5.6)

j =1

By definition, the IMF has to satisfy two conditions. First, the number of extrema must be equal or differ at most by one from the number of zero-crossings; this is similar to the traditional narrow band requirements for the stationary Gaussian process. Practically, this condition corresponds to elimination of riding waves or small waves in the time series with multiple centers of rotation. The derived IMF corresponds to a proper rotation in the complex plane, as the center of rotation is the origin of the coordinate system. Second, the IMF has to have symmetric envelopes, defined by the local maxima and minima respectively. These upper and lower envelopes are determined by using cubic splines, as suggested by Huang et al. (4). This second condition ensures that the instantaneous frequency will not have fluctuations arising

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from an asymmetric wave profile. The important condition for the IMF is that only one maximum or minimum exists between successive zeros. Each IMF is determined by a sifting procedure, which is repeated several times in order to ensure that all the requirements of IMFs are satisfied. The set of IMFs obtained in this way is unique and specific to the particular time series, since it is based on and derived from the local characteristics of these data. IMFs could be considered as a more general case of the simple harmonic functions, but IMFs are claimed to have a physical meaning in additional to a mathematical one due to their specific derivation. In the second step, the Hilbert transform is applied to these IMFs:

()

1 Cˆ j t = P π

∞

∫

( )dt′

C j t′

(5.7)

t − t′

−∞

where P indicates the Cauchy principal value. The amplitude aj, the phase ϕj, and the instantaneous frequency ωj are calculated by a j (t ) = C 2j t + Cˆ 2j t ,

()

()

(5.8)

 Cˆ (t )  ϕ j (t ) = arctg   ,  C (t ) 

(5.9)

and

ω j (t ) =

( ).

dϕ j t

(5.10)

dt

By using Equation 5.8, Equation 5.9, and Equation 5.10, we can express the original data X(t) as a real part (Re) of the complex expansion

()

X t = Re

n

∑ a (t ) e ∫

()

i ω j t dt

j

,

(5.11)

j =1

which is considered a generalized form of the Fourier expansion. Here, both amplitude aj(t) and instantaneous frequency ωj(t) are functions of time t. Whereas the Fourier expansion is made in a global sense, the expansion by HHT is made in a local sense. It must be mentioned that, by definition, IMFs always have positive frequencies, because the oscillations in IMFs are symmetric with respect to the local mean. Thus, the HHT method solves the problem of negative frequencies, and there is no need © 2005 by Taylor & Francis Group, LLC

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for any additional elimination of small waves, which are indeed observed peculiarities of real sea waves. All the distinguishing features of sea surface elevation are taken into account, and the subjectivity in estimation of wave characteristics probably will be reduced. This is a direct result of the use of a more appropriate tool for examining nonlinear and nonstationary real sea waves. The frequency–time distribution of the amplitude or squared amplitude is designated as a Hilbert amplitude spectrum or Hilbert energy spectrum. In the present work, we use the Hilbert energy spectrum, determined as the frequency–time distribution of the squared amplitude. For simplicity, the Hilbert energy spectrum is denoted as the Hilbert spectrum H(ω,t). The frequency–time distribution of the energy allows us to determine which frequency exists at a particular time, whereas the Fourier frequency spectrum provides information as to which frequency exists generally in given data series. The time resolution of the Hilbert spectrum can be as precise as the sampling data rate. Since the Hilbert spectrum gives the best fit of a local sine or cosine function to the data, the frequency is uniformly defined by a local derivative of the phase function. The lowest extractable frequency is 1/T, where T is the duration of the record, and the highest frequency is 1/(l∆t), where l = 5 is the minimum number of points necessary to define frequency accurately, and ∆t is the sampling rate. Based on the Hilbert spectrum, the marginal Hilbert spectrum h(ω) can be defined as T

h(ω ) =

∫ H (ω, t)dt .

(5.12)

0

The marginal Hilbert spectrum represents the cumulated squared amplitude over the entire data span and offers a measure of total energy contribution from each frequency.

5.5 APPLICATION OF THE HHT METHOD For each wave record, a set of IMFs is obtained by EMD. Figure 5.6 presents detailed results of decomposition of sea surface elevation data, recorded at an offshore point with 24 m water depth at wave decay stage, with significant wave height of 3.18 m and mean period 9.96 sec. Figure 5.6a shows the data time series. The EMD method yields eight IMFs and residue, shown in Figure 5.6b through 5.6j. Since time scales are identified as intervals between the successive alternations of local maxima and minima, IMFs represent oscillation modes embedded in the data. The different IMFs correspond to the different physical time scales that characterize the various dynamical oscillations in the time series. A nearly constant (as to time scale) wave component dominates in each IMF, representing the carrier wave constituent at the specific time scale. The sifting procedure of EMD generates modes Cj(t), j = 1, …, n with degree of time variations in decreasing order. By construction, each mode Cj(t) generates a proper

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109

0.2

(f )

C5

X(t)

(a)

0

−3

0

C1

C6

0

0

C8

C3

−1.5

0

0

1000 2000 3000 4000 5000 6000 7000

0.1

(d)

0

1000 2000 3000 4000 5000 6000 7000

0

−0.1

1000 2000 3000 4000 5000 6000 7000

1.5

0

(h)

C7

C2

−3

(g)

0.1

(c)

0

1000 2000 3000 4000 5000 6000 7000

0

−0.1

1000 2000 3000 4000 5000 6000 7000

3

0

0.1

(b)

0

−3

−0.2

1000 2000 3000 4000 5000 6000 7000

3

0

0

−0.1

1000 2000 3000 4000 5000 6000 7000

(i)

0

1000 2000 3000 4000 5000 6000 7000

0.2

0.5

(j)

0

−0.5

R

C4

(e)

0

1000 2000 3000 4000 5000 6000 7000 Time (s)

0

−0.2

0

1000 2000 3000 4000 5000 6000 7000 Time (s)

FIGURE 5.6 EMD of wave record. (a) Data of sea surface elevation; (b) through (i) eight IMFs (C1 through C8); (j) residue R.

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The Hilbert-Huang Transform in Engineering

rotation in the complex plane of its own analytic function ψj(t) = aj(t) exp [iφj(t)]. The average rotation frequencies ω j (t ) = d φ j (t ) dt obey the order ω1 ≥ ω 2 … ≥ ω n because the sifting procedure picks the components with the fastest variations embedded in the original data first and those with the slowest variations last. The mean periods of the IMFs, shown in Figure 5.6, are correspondingly 3.4 sec, 8.4 sec, 16.6 sec, 35.3 sec, 89.2 sec, 235 sec, 572 sec, and 1540 sec. These oscillations not only have different time scales, but they also have a different range of energy. The limits of the vertical axes in Figure 5.6b through 5.6i are different for different IMFs. The most energetic in the decomposition is the second IMF, C2. The shortest oscillations in decomposition, presented by the first IMF, C1, have smaller amplitudes than those of C2. Physically, these high frequency oscillations, extracted in C1, may represent short period waves, such as wind waves in the growing stage, as well as the small ripples that ride longer waves. The third IMF, C3, takes second place in the energy hierarchy. The first three constituents in the decomposition, C1, C2, and C3, probably represent the contribution of the wind waves oscillations, while the lower frequency oscillations are characterized by the IMFs with higher index: C4, C5, and so forth. The complete presentation of the sea wave state recorded in this data can be considered as a composition of dominant wave oscillations, extracted in the second IMF and several (but finite number) components with smaller amplitudes. The contribution of different Intrinsic Mode Functions to the energy and frequency contents of wave data is investigated. The spectrum of wave record is compared with spectra of its Intrinsic Mode Functions in Figure 5.7 as estimated Fourier spectrum is shown in Figure 5.7a, while a marginal Hilbert spectrum h(ω) is presented in Figure 5.7b. Under a sampling rate of these data, the highest frequency that can be extracted by Empirical Mode Decomposition is 0.4 Hz. The highest frequency of Fourier spectrum is 1 Hz, but for simplicity and easy comparison with marginal spectrum, only until 0.35 Hz is shown in Figure 5.7a. The Fourier spectrum is estimated as an average of the raw spectra, calculated in the overlapped segments, by which the wave record is divided. The spectral estimations are also smoothed with a Hamming window. The Intrinsic Mode Functions represent different as period and energy oscillations, extracted by EMD from the wave data. The spectrum of the first IMF C1 (stars line) covers very well the tail of wave spectrum (thick solid line) in the both ñ Fourier and marginal spectrum. The peak of second IMF C2 (open circle line) coincides with the major peak of the spectrum of sea surface elevation, both Fourier and marginal spectrum. The marginal Hilbert spectra are projections of real frequency-time distribution of the energy, provided by Hilbert spectrum. There is basic difference in the meaning of frequency in Fourier and Hilbert spectrum. The presence of energy at a given frequency in Fourier spectrum means that a trigonometric component with this frequency and amplitude exists through the whole time span of the data. In the marginal Hilbert spectrum the existence of energy at a given frequency means that in the whole time span of the data, there is a higher probability for local appearance of such a wave.

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An Application of HHT Method to Nearshore Sea Waves

101

(a)

Fourier spectrum

100

10–1

111

Wave data C1 C2 C3 C4 C5 C6 C7 C8

10–2

10–3

10–4

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Frequency (Hz)

101

(b)

Marginal spectrum

100

10–1

Wave data C1 C2 C3 C4 C5 C6 C7 C8

10–2

10–3

10–4

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Frequency (Hz) FIGURE 5.7 Spectra of wave data and its IMFs. (a) Fourier spectra; (b) marginal Hilbert spectra.

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5.5.1 OFFSHORE WAVES DURING DIFFERENT SEA STAGES The peculiarities of the decomposition of offshore waves during different sea stages were examined. The energy and time characteristics of individual IMFs were estimated and compared. The zero-th moment m0Cj of the marginal Hilbert spectrum hj(ω) is proposed as a measure of the integrally determined energy of the jth IMF: C

m0 j =

∫ h (ω)dω . j

(5.13)

The mean value of the instantaneous frequency ω j , estimated by Equation 5.10, is used as a measure of the frequency of the jth IMF. Four cases, representative of each of the four different sea stages, were chosen. The characteristics of their decomposition are shown in Table 5.3. The energy of the IMFs with index higher than 4 is significantly lower than the energy of the first four IMFs. This tendency is confirmed by the analysis of all offshore wave records. From an energetic point of view, the first four IMFs contribute mainly to the energy contents of the wave data, measured at 24 m depth. For a complete description of the analyzed data, a complete set of all IMFs is necessary. In the wave growing stage, C2 has the highest energy, followed by C1, in agreement with the prevailing short period waves in the wave growing stage. The swell type of sea waves starts to dominate in the wave decay stage; consequently, C3 has energy comparable with or higher than the energy of C1. A similar tendency in the energy distribution of the IMFs is also observed in the post-storm stage, when the majority of the registered sea waves have a longer period. The energy hierarchy of the IMFs in decomposition of a wave record reflects clearly the specific peculiarities of the particular wave data. The different IMFs have different significance in view of the purpose of the specific study. If the signal of interest is captured in a single IMF, the investigation and eventual prediction of the examined phenomenon is facilitated, since the dynamical system is simpler. Salisbury and Wimbush (26) demonstrated this advantage of the EMD, analyzing only one IMF, which had the same mean frequency as a phenomenon of interest, instead of considering the original data series. An understanding of the characteristics of large sea waves is essential for many coastal engineering problems; therefore, special attention is paid here to the most energetic components in the decomposition. In this connection, the IMF most dominant in energy among the whole set of IMFs is determined and denoted Cm. In Figure 5.8, the energy of m0Cm is investigated as a function of the energy of the original wave data m0; the data from all observation terms and all water depths are used. The zero-th moments Equation 5.13 of the marginal Hilbert spectrum are again used as a measure of energy. The dominant IMF, Cm, contained about 55% of the energy of the wave data. The specific peculiarities of the wave process are well captured by EMD and reproduced by IMFs. Easy separation of the different time and energy associated with the oscillations in the data is well achieved by EMD.

© 2005 by Taylor & Francis Group, LLC

Growing Stage (Hs = 3.25 m, Tmean = 7.4 sec)

Decay Stage (Hs = 2.86 m, Tmean = 10sec)

Post-Storm Stage (Hs = 2.21 m, Tmean = 8.81 sec)

IMF

fp (Hz)

m0 (m2)

fp (Hz)

m0 (m2)

fp (Hz)

m0 (m2)

fp (Hz)

m0 (m2)

C1 C2 C3 C4 C5 C6 C7 C8

0.1920 0.1627 0.0987 0.0373 0.0187 0.0107 0.0053 0.0027

0.0797 0.0912 0.0309 0.0027 0.0007 0.0005 0.0002 0.0003

0.1120 0.1093 0.0907 0.0373 0.0160 0.0080 0.0053 0.0027

0.3257 0.5288 0.0865 0.0133 0.0044 0.0018 0.0013 0.0005

0.1053 0.0787 0.0693 0.0240 0.0107 0.0053 0.0027 0.0027

0.0693 0.5382 0.1580 0.0130 0.0030 0.0015 0.0007 0.0007

0.1947 0.0747 0.0800 0.0400 0.0187 0.0053 0.0027 0.0027

0.0473 0.2456 0.1232 0.0138 0.0050 0.0021 0.0004 0.0004

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Calm Stage (Hs = 1.53 m, Tmean = 5.2 sec)

An Application of HHT Method to Nearshore Sea Waves

© 2005 by Taylor & Francis Group, LLC

TABLE 5.3 Characteristics of Offshore Sea Waves and Their IMFs during Calm, Wave Growing, Decay, and Post-Storm Conditions

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1.4

C 2 −1 m0 m(m s )

1.2

1

0.8

2 −1

m0(m s )

Cm

↑ 0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

2 -1 m0 (m s )

FIGURE 5.8 Comparison of energy of the dominant IMF (m0Cm) and wave data (m0).

5.5.2 CROSS- SHORE TRANSFORMATION

OF

SEA WAVES

After the wave data, measured at different points of the nearshore, were disintegrated into IMFs, the cross-shore variations of the energy and frequency of the IMFs were traced in order to examine the transformation of sea waves. The energy of the IMFs, defined by the zero-th moment of the marginal Hilbert spectrum, is presented in Figure 5.9a, while the frequency of the IMFs, defined as a spectral peak frequency, is shown in Figure 5.9b. The horizontal axes in Figure 5.9 are an index of the first eight decomposition components Cj (j = 1, …, 8). The data from 24 m and 14 m water depth were collected in the region before breaking, while the other observation points, at 5.65 m water depth and less, were located inside the surf zone. The wave process in the surf zone is commonly considered to be a complicated one and associated with a broadband Fourier spectrum. Here, by virtue of the EMD method, these significantly nonlinear and complicated wave data series are practically decomposed into a small number of IMFs with a properly defined phase ϕ(t) respectively frequency ω(t). The energy of the first IMF, C1, characterizing the high frequency oscillations in the data, is significantly smaller than that of the second component for the case of offshore waves. However, in the shoreward region of the surf zone (2.33 m, 1.81 m, and 0.30 m water depths), the energy of C1 becomes comparable to the energy of C2, C3, and C4. The second IMF, C2, has the highest energy in the decomposition of the offshore waves. After breaking, the energy of the second IMF decreases

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115

0.8 24.0m 14.0m 9.00m 5.65m 5.05m 2.33m 1.81m 0.80m

0.7

Energy ( m 2s-1)

0.6 0.5 0.4

(a)

0.3 0.2 0.1 0 0

1

2

3

4 5 Ci, i =1,8

6

7

8

9

Peak frequency(Hz)

0.3 (b)

24.0m 14.0m 9.00m 5.65m 5.05m 2.33m 1.81m 0.80m

0.2

0.1

0 0

1

2

3

4 5 Ci, i =1,8

6

7

8

9

FIGURE 5.9 Decomposition of wave data measured in the cross-shore. (a) Energy of IMFs; (b) frequency of IMFs.

considerably. At the same time, the oscillations extracted in C2 become shorter. The energy and frequency of the lower frequency IMFs, C4, C5, C6, and C7, remain approximately constant in the process of cross-shore transformation. These peculiarities in the cross-shore variations of IMFs are reflected in the frequency–time distribution of the energy. A section of the Hilbert spectrum of sea waves in three different zones — offshore with water depth 24 m, in the seaward surf zone at 5.05 m depth, and in the shoreward surf zone at 1.81 m — is shown in

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(a) 24 m water depth

0.4

16

0.35

12

0.3 Fr 0.25 eq 0.2 ue nc 0.15 y( 0.1 H z) 0.05

10 8 1000

6

0.3 Frequency (Hz)

H (m2)

14 15 10 5 0 0.4 0.35

4

600 s)

400 m Ti

e(

0

0

2

1 1.5

1000

800

1000

800

1000

0.25 0.2 0.15

0.05 0.5

600 200

800

0.1

1

800 400

600

0.3

0.4 1000

400

0.35

Frequency (Hz)

2.5 4

0

200

0.4 3

3

0

Time (s)

(b) 5.05 m water depth

H (m2)

0.15

0.05

2

0 0

0.3 Fr eq ue 0.2 nc y( H 0.1 z)

0.2

0.1

800

200

0.25

0

s)

( me Ti

0

200

0

400

600

Time (s)

0 0 (c) 1.81 m water depth

0.4

1.5

0.35

H (m2)

1

1 0.5 0 0.4 0.3 Fr eq ue 0.2 nc y( Hz 0.1 )

1000

0.5

800 600

0 0

Tim

0.25 0.2 0.15 0.1 0.05

s) e(

400 200

Frequency (Hz)

0.3 1.5

0

0

0

200

400

600

Time (s)

FIGURE 5.10 A section of the Hilbert spectrum of sea waves, as a 3D plot in the left panels and as a 9 × 9 Gaussian filtered color coded map in the right panels. (a) Offshore zone; (b) seaward surf zone; (c) shoreward surf zone.

Figure 5.10. The Hilbert spectrum is presented as a three-dimensional plot in the left side panels. The color scale maps of the same sections of H(ω,t) in a smoothed form are shown in the right side panels. The Hilbert spectrum is smoothed with 9 × 9 weighted Gaussian filter. The smoothing operation degrades both the frequency and time resolution of H(ω,t). The color bars next to the diagrams denote the energy scale. The frequency–time distribution of the wave energy changes considerably due to wave transformation in a cross-shore direction. The Hilbert spectrum in Figure

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117

5.10a shows an energy concentration around 0.092 Hz, corresponding with the second component, C2, which has the highest energy in the decomposition in Figure 5.6. The magnitude of energy decreases, as revealed by changes in the energy scale of the color bars. A wide variation in the local frequency in the surf zone is evident in Figure 10b and c, in contrast with the offshore region. This broadening of the wave frequency range is partially due to wave breaking. The strong nonlinearity of the sea waves in shallow water, shown by the intra-wave frequency modulation or the variation of the local frequency within one wave, also contributes to a large variation in the local frequency in the Hilbert spectrum. On the basis of the results of EMD, real sea waves could be considered as a system of oscillations with different periods and amplitudes. It has to be stressed that the amplitudes and periods of these oscillations are not constant but instead are functions of time according to Equation 5.11. The wave system, as a whole, changes its content in cross-shore transformation. The process of wave breaking mainly affects the IMFs with the highest energy in the decomposition of offshore waves. Inside the surf zone, the frequency of each IMF tends to increase. These peculiarities in the spatial distribution of different IMFs agree with general principles of sea wave transformation in the cross-shore direction. The analysis of IMFs and the Hilbert spectrum provides information on the cross-shore transformation of sea waves. The EMD method, with its great ability to extract specific oscillation data embedded in the data, can facilitate the investigation of sea waves. Attention can be concentrated on a finite number of IMFs, each characterizing a dynamical system that is much simpler than the original one. The application of the HHT method to nearshore waves is a fertile area of research, as future studies will shed more light on the complicated dynamics of the coastal zone.

5.6 CONCLUSIONS We have discussed the importance of the analyzing technique for the correct understanding of the examined phenomenon and paid special attention to the nonlinear and nonstationary behavior of real sea waves. A review of widely used conventional methods for wave analysis shows that all of them assume a priori linearity and stationarity of sea waves; these assumptions consequently affect the estimation of wave characteristics. The Hilbert-Huang transform method for the analysis of nonlinear and nonstationary time series is proposed as an alternative for the investigation of the nonlinear and nonstationary nature of real sea waves. In this method, the oscillations embedded in the data are extracted by EMD into a set of IMFs without placing any subjective preliminary limitations on the nature of the investigated phenomenon. Easy separation of the different, as time and as energy associated with the oscillation in the data is well achieved by EMD. The dominant oscillations for each data record were determined and their variations during different sea stages were investigated. The energy hierarchies of the IMFs in decomposition of wave records observed during different sea stages reflect the specific peculiarities of the particular wave data. The process of wave breaking

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mainly affects the IMFs with the highest energy in the decomposition of offshore waves. Inside the surf zone, the frequency of each IMF tends to increase. The EMD method, with its great ability to extract oscillation modes embedded in the data, can facilitate the investigation of sea waves. Attention can be concentrated on a finite number of IMFs, each characterizing a dynamical system that is much simpler than the original one. The application of the HHT method to nearshore waves is a fertile area of research, as future studies will shed more light on the complicated dynamics of the coastal zone.

REFERENCES 1. Liu, P. C. (2000). Is the wind wave frequency spectrum outdated? Ocean Eng. 27: 577–588. 2. Massel, S. R. (2001). Wavelet analysis for processing of ocean surface wave records. Ocean Eng. 28: 957–987. 3. Huang, N. E., Shen, Z., and Long, S. R. (1999). A new view of nonlinear water waves: The Hilbert spectrum. Annu. Rev. Fluid Mech. 31: 417–457. 4. Huang, N. E., Shen, Z., Long, S..R., Wu, M. C., Shin, H. S., Zheng, Q., Yuen, Y., Tung, C. C., and Liu, H. H. (1998). The EMD and Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. R. Soc. London. Vol. 454: 903–995. 5. Katoh, K., Nakamura, S., and Ikeda, N. (1991). Estimation of infragravity waves in consideration of wave groups: An examination on basis of field observation at HORF. Rep. Port Harbour Res. Inst. Vol. 30, 1: 137–163. 6. Ochi, M. K. (1998). Ocean Waves: The Stochastic Approach. Cambridge Ocean Technology Series 6, 319 p. New York: Cambridge University Press. 7. Gimenez, M. H., Sanchez-Carratala, C. R., and Medina, J. R. (1998). Influence of false waves on wave records statistics. Proc. ICCE 1998, pp. 920–933. 8. Kitano, T., Mase, H., and Nakano, S. (1998): Statistical properties of random wave periods in shallow water: Analysis by utilizing false waves. Proc. Coast. Eng, JSCE, Vol. 45, 221–225. 9. Veltcheva, A., and Nakamura, S. (2000). Wave groups and low frequency waves in the coastal zone. Rep. Port Harbour Res. Inst. Vol. 39, 4: 75–94. 10. Mizuguchi, M. (1982). Individual wave analysis of irregular wave deformation in the nearshore zone. Proc. ICCE 82, pp. 485–504. 11. Hamm, L., and Peronnard, C. (1997). Wave parameters in the nearshore: A clarification. Coastal Eng. 32: 119–135. 12. Gimenez, M. H., Sanchez-Carratala, C. R., and Medina, J. R. (1994). Analysis of false waves in numerical sea simulations. Ocean Eng. Vol. 21, 8: 751–764. 13. Longuet-Higgins, M. S. (1958). On the intervals between successive zeros of a random function. Proc. R. Soc. Ser. A, 246: 99–118. 14. Tayfun, M. A. (1983). Frequency analysis of wave heights based on wave envelope. J. Geophys. Res. Vol. 88, C12: 7573–7587. 15. Bitner-Gregersen, E. M., and Gran, S. (1983). Local properties of sea waves derived from a wave record. Appl. Ocean Res. Vol. 5, 4: 210–214. 16. Hudspeth, R. T., and Medina, J. R. (1988). Wave group analysis by Hilbert transform. Coastal Eng. Vol. 1: 885–898. 17. Tayfun, M. A., and Lo, J. M. (1989). Envelope, phase, and narrow-band models of sea waves. J. Waterway Port Coastal Ocean Eng. ASCE, Vol. 115, 5: 594–613.

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18. Huang, N. E., Long, S. R., Tung, C. C., Donelan, M. A., Yuan, Y., and Lai, R. J. (1992). The local properties of ocean surface waves by the phase-time method. Geophysical Res. Lett. Vol. 19, 7: 685–688. 19. Cherneva, Z., and Veltcheva, A. (1993). Wave group analysis based on phase properties. Proc. 1st Int. Conf. Mediterranean Coastal Environ. (MEDCOAST ’93), Turkey, pp. 1213–1220. 20. Cohen, L. (1995). Time-Frequency Analysis. Prentice Hall Signal Processing Series, Alan V. Oppenheim, series editor, 299 p. 21. Yalcinkaya, T., and Lai, Ying-Cheng (1997). Phase characterization of chaos. Phys. Review Lett. Vol. 79, 20: 3885–3888. 22. Melville, K. (1983). Wave modulation and breakdown. J. Fluid Mech. 128: 489–506. 23. Farge, M. (1992). Wavelet transforms and their applications to turbulence. Annu. Rev. Fluid Mech. 24: 395–457. 24. Veltcheva, A. D. (2001). Wave groupiness in the nearshore by Hilbert spectrum. Proc. 4th Int. Symp. Ocean Wave Meas. Anal. (WAVES 2001), San Francisco, pp. 367–376. 25. Veltcheva, A. D. (2002). Wave and group transformation by a Hilbert spectrum. Coastal Eng. J. Vol. 44, 4: 283–300. 26. Salisbury, J. I., and Wimbush, M. (2002). Using modern time series analysis techniques to predict ENSO events from the SOI time series. Nonlinear Processes Geophysics, Vol. 9: 341–345.

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6

Transient Signal Detection Using the Empirical Mode Decomposition Michael L. Larsen, Jeffrey Ridgway, Cye H. Waldman, Michael Gabbay, Rodney R. Buntzen, and Brad Battista

CONTENTS 6.1 Introduction ..................................................................................................122 6.2 Empirical Mode Decomposition..................................................................125 6.3 EMD-Based Signal Processing....................................................................126 6.4 Experimental Validation Using Towed-Source Signals...............................135 6.5 Conclusion....................................................................................................137 Acknowledgment ...................................................................................................138 References..............................................................................................................138 ABSTRACT In this paper, we report on efforts to develop signal processing methods appropriate for the detection of manmade electromagnetic signals in the nonlinear and nonstationary underwater electromagnetic noise environment of the littoral. Using recent advances in time series analysis methods [Huang et al., 1998], we present new techniques for signal detection and compare their effectiveness with conventional signal processing methods, using experimental data from recent field experiments. These techniques are based on an empirical mode decomposition which is used to isolate signals to be detected from noise without a priori assumptions. The decomposition generates a physically motivated basis for the data.

121

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6.1 INTRODUCTION In this paper, we discuss application of nonlinear signal processing techniques to underwater electromagnetic data. Electromagnetic measurements provide a potentially useful complement to acoustic detection methods in littoral zones where high noise levels and reverberation in the littoral limit acoustic detection ranges. In shallow waters, for example, the ambient noise levels from wind-generated breaking waves are typically 10 to 100 times greater than those in deep water. Electromagnetic measurements are affected by noise as well. One source of electromagnetic fields in seawater is electrical currents induced by hydrodynamic flow in the Earth’s magnetic field. The ocean is a constantly moving conducting fluid [26], and ocean flows caused by gravity waves (wind waves and far-field swell), internal waves, turbulence, tides, and currents all result in electromagnetic field fluctuations. While the standard way of viewing the complexity of wave phenomena is to consider the ocean to be a random, Gaussian process, the importance of nonlinear interactions in water wave dynamics has been conclusively demonstrated [28]. Furthermore, turbulence is also often present at the bottom boundary layer due to nonlinear interactions between flows arising from tidal and wind-driven currents, wind waves, swell, and internal waves [4]. In the random sea assumption (used primarily in the deep ocean), the evolution of the sea surface is obtained as a superposition of waves of different frequencies and directions with random phases. The wave amplitudes are determined according to a measured power spectrum, the sea surface is modeled as a linear system, and energy transfer between wave modes is not accounted for. In littoral regimes, however, nonlinear wave interactions are significant and cannot be neglected. Higher-order statistics show that wave behavior in shallow water zones displays a distinctly non-Gaussian regime, where wave–wave interaction plays an important role and energy transfer between wave components occurs [18, 12]. A typical power spectrum of naturally occurring ocean-bottom magnetic and electric field data is shown in Figure 6.1. The most obvious peaks in both electric and magnetic spectra in Figure 6.1 are visible at 62 MHz, the dominant oceanic swell frequency. Swells have characteristic frequencies in the 50 to 100 MHz range, and surface waves driven by local winds are found at frequencies above the swell frequency, up to about 0.5 Hz. The effect of ocean swell on the magnetic field is significantly more pronounced than its effect on the electric field. An additional peak is also seen in the spectrum of the magnetic field data at twice the swell frequency (125 MHz). Frequency doubling effects due to the nonlinear interference of the swell with wave fields reflecting off local land masses have been previously observed [27, 24]. Spectral peaks above these frequencies are likely due to local turbulence, with the exception of the Schumann resonance discussed below. Another major component of underwater electromagnetic noise is actually due to sources of non-oceanic origin — geomagnetic, atmospheric, and ionospheric electromagnetic radiation. The Schumann resonance at 8 Hz is caused by worldwide lightning, which resonates in the earth-ionosphere cavity. Geomagnetic noise at lower frequencies originates in the ionosphere, is nonstationary, and increases in

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Electric ﬁeld spectrum, with and without ionospheric mitigation

Magnetic ﬁeld spectrum, with and without ionospheric mitigation

3

3

10

10

ionosphere unmitigated

102

101

0

Power (nT2/Hz)

Power (uV/m2/Hz)

ionosphere unmitigated ocean swell

102

ocean swell

101 10

instr. artifact

−1

10

10−2

123

ionosphere mitigated

10−3 Schmn res.

−4

10

−5

10

10−6 10−3

double freq.

10−1 10−2 10−3

10−1 Frequency (Hz)

100

101

instrument artifacts.

ionosphere mitigated

−4

10

10−5

(a) E-field 10−2

100

10−6 10−3

(b) B-field 10−2

10−1

100

101

Frequency (Hz)

FIGURE 6.1 Power spectra for ocean-bottom electromagnetic fields before and after noise mitigation.

power at lower frequencies, imparting the strong slope to the spectrum seen in Figure 6.1. These effects were observed in our experiments and in nearshore tests done by other researchers [16]. Geomagnetic noise is typically coherent over large distances and can be mitigated by using a remote reference station [17]. A transfer function can be calculated to account for the difference in the conductivity of the underlying geology and used to cancel out the effect of magnetospheric sources observed in the local sensor data. Figure 6.1 shows the spectrum before and after such noise reduction was performed. Characterizing the nonstationary and nonlinear character of ocean-bottom electromagnetic noise is important for detection of manmade electromagnetic (EM) anomalies. Ships and submarines generate electric fields in the surrounding water due to corrosion currents caused by the dissimilar metals used in their construction and also by onboard cathodic protection systems designed to mitigate such currents. The bronze propeller acts as one node of the vessel’s galvanic cell, while the sacrificial zincs on the hull act as nodes of opposite polarity. A current flows from the propeller through the drive shaft, bearings, and hull into the water, returning to the propeller. This current flow in the surrounding water can be well modeled as an electric dipole. The horizontal motion of the ship or submarine imparts characteristic ultra-low frequency signals to the field components, which change with the speed and range of the submarine. This signature is described as the “quasi-static” horizontal electric dipole (HED) signature. Impressed on the quasi-static dipole field is an AC modulation, which is caused by variations in the electrical resistance in the shaft as it rotates. A vessel passing a stationary electric field sensor generates a transient signal in the nonstationary background ambient electric field. Figure 6.2 shows an example of simulated horizontal electric field (E-field) components (Ex and Ey). For a stationary sensor, the measured dipole signal is a function of the vessel speed and its lateral distance from the sensor at the point of closest approach. The spectra of moving dipole signatures are generally confined to ultra-low frequencies (0.5 to 5 MHz) and thus fall into the underwater electromagnetic frequency regime

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Electric Field: Track8, Rec’r ‘k’ 1.5

x y

Electric Field, E (µV/m)

1

ELFE Detail

0.5

0

−0.5

−1

0

5

10

15

20 25 Time (mins)

30

35

40

45

FIGURE 6.2 Simulated time series of the two horizontal components of the electric field from a moving vessel. The dipole appears as a slow transient. The inset is an enlarged view of the cyclic modulation of the dipole.

dominated by ionospheric noise. When available, ionospheric noise removal can greatly aid detection of the moving dipole signal, as any reduction in spurious signal power translates directly into extended ranges [5, 1]. While remote reference mitigation techniques can remove much of this noise, specialized detection and extraction techniques are still necessary. A number of techniques for processing underwater electromagnetic data exist in the literature. A generalized likelihood ratio test was developed for HED detection using a 3-axis E-field sensor [6], but under simplifying Gaussian white noise assumptions. The use of higher-order statistics in detection algorithms has been investigated [3,20], and the wavelet transform has also proved useful for detection purposes [21,22]. In this paper, we investigate the use of signal processing techniques for underwater electromagnetic data which use the empirical mode decomposition (EMD) reported by Huang et al. [13]. The EMD was developed specifically to deal with the nonstationary and nonlinear characteristics of ocean wave data. In Section 6.2, we review the properties of the EMD. In Section 6.3, we describe a detection algorithm based on the conventional matched filter that employs the EMD for electromagnetic anomaly detection. Experimental results are presented in Section 6.4.

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6.2 EMPIRICAL MODE DECOMPOSITION The EMD facilitates calculation of physically meaningful instantaneous frequencies [13] by using the Hilbert transform. The Hilbert transform can be used to define the instantaneous amplitude and phase of an arbitrary time series x(t). The Hilbert transform of x(t) yields a time series y(t) given by y(t ) =

1 PV π

∫

∞

−∞

x (τ) d τ, t−τ

(6.1)

where PV denotes the Cauchy principal value. Hence, the Hilbert transform is the convolution of x(t) with 1/t, which consequently stresses the local nature of the signal. From the complex analytic signal formed via z(t) = x(t) + iy(t), an instantaneous amplitude and phase can be calculated. The instantaneous frequency corresponds to the time derivative of the phase of z(t). Direct application of the Hilbert transform to data may yield negative frequencies. Only positive instantaneous frequencies, however, correspond to physically meaningful oscillatory behavior. In order to avoid nonphysical instantaneous frequencies, the empirical mode decomposition is used to separate the data into well-behaved intrinsic mode functions (IMFs) from which the Hilbert transform will yield positive instantaneous frequencies. An IMF is a function satisfying two conditions: i) the number of extrema and the number of zero-crossings of the function must either be equal or differ at most by one, and ii) at any instant in time, the mean value of the envelopes defined by the function’s local maxima and minima must be zero. Each IMF has a characteristic frequency, although IMFs can overlap in frequency content. IMFs are found by using a recursive sifting procedure that generates the highest-frequency IMF first. This IMF is subtracted from the time series, and the process is iteratively applied to the difference until only a non-oscillatory residual, r, which represents the trend in the data, remains. Each IMF xn(t) has a variable instantaneous amplitude, an(t), and frequency, ωn(t), calculated via the Hilbert transform. The time–frequency distribution H(ωn,t) of the amplitude is known as the Hilbert spectrum. The original time series can be written as a sum of a finite number of IMFs:  N  x (t ) = Re  an (t )eiω n (t ) dt + r  .  n=1 

∑

(6.2)

Completeness of the EMD is assured in principle and depends only on the numerical accuracy of the sifting process. Orthogonality of the IMFs, while not guaranteed theoretically, is satisfied in practice. The Hilbert spectrum can be integrated over time to yield the Hilbert marginal spectrum, HMS (ω ) =

∫

T

0

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H (ω, t )dt,

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which represents the accumulated energy at each frequency over the entire data span and is related to the fraction of time that a given frequency can be observed in the system. The marginal spectrum is analogous to the Fourier power spectral density. The EMD leads to compact representation of nonlinear or nonstationary signals without higher-order harmonics or a proliferation of global modes needed to account for rapid transients. It is often observed that individual IMFs often correspond to identifiable physical processes in the data [13,14]. Recent work is yielding greater understanding of the behavior of the EMD and its potential for signal processing applications [8,7]. It appears to operate on fractional Gaussian noise as an almost dyadic filterbank, resembling a wavelet decomposition [10,9]. Furthermore, numerical experiments indicate that its impulse response is similar to a cubic spline wavelet [25].

6.3 EMD-BASED SIGNAL PROCESSING Electromagnetic noise in a littoral, oceanic environment is generally nonstationary and can also be nonlinear. The spectrum of such noise is colored, insofar as it rises with lower frequencies according to a power law. Hence, traditional detection methods based upon the assumption of the noise being white, stationary, and Gaussian may not be optimal. The empirical mode decomposition appears to distinguish signals originating from different physical processes by grouping them into a limited number of intrinsic mode functions [24]. This behavior suggests the use of empirical mode decomposition for increasing the effectiveness of detection of horizontal electric dipole in a littoral oceanic EM noise background. Such anomalies generated by moving sources are transient, with time scales lasting between 10 and 30 min, depending upon the ratio of the closest point of approach (CPA) to the velocity. The traditional method for detecting a deterministic signal in a noise background is the matched filter, which uses an estimate of the expected signal to filter the data. The matched filter is the optimal filter for detection of a deterministic signal in white Gaussian noise [15]. It has been used for HED detection by Blampain [2]. Other methods in the literature for detecting transient HEDs include a generalized likelihood ratio test [6] and a method based on the eigen-decomposition of the sample covariance matrix [23]. All three of these methods assume that the noise is spatially and temporally both white and Gaussian, an assumption that is not representative of the littoral oceanic environment. A method of transient magnetic signal detection in geomagnetic noise [22], based upon the application of wavelet packets, yielded mixed results. As a starting point for our analysis, we have used the matched filter method as our standard of comparison, with a priori knowledge of the target (i.e., bearing, CPA, velocity, and time of CPA). Electromagnetic signal anomalies generated by moving sea vessels are transient in nature. Results from simulated and actual data show that the transient quasi-static HED signature is often captured in one or two IMFs. This behavior is demonstrated with a simulated example without noise of a HED passing by at two different ranges, with a simulated double-frequency cyclic component whose propagation is modeled with the algorithm found in Orr [19]. The EMD of Ex generates five IMFs (see Figure 6.3). © 2005 by Taylor & Francis Group, LLC

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127

c0

Time Series and IMF Components 0.5 0 −0.5

c1

0.02 0 −0.02 c2

0.02 0

c3

0.5 0 −0.5

c4

−0.02

0.2 0 −0.2

c5

0.2 0 −0.2 0

100

200

300

400

500

Time

FIGURE 6.3 Simulated, combined dipole and cyclic signals at different ranges and their derived IMF components. The lower five graphs display the individual IMFs, showing the ability of the EMD procedure to separate signals with different characteristics.

Similar results are obtained for noisy signals. A typical track time series from the experiment, containing both the HED and cyclic signal generated by a towed source, is shown in the top graph of Figure 6.4. The cyclic signal (in the second) and the HED (in the seventh panel), highlighted in red, are easily identified in this close-pass track, which still contains noise. We investigated the performance of the matched filter for detecting a moving HED in the ocean using synthetic marine signals, which simulate a single dipole moving in a straight path past a single receiver. The expected signal is a function of the dipole amplitude and target range, bearing, and speed. In this study we elected to simplify the situation by assuming a priori knowledge of speed and range, in order to focus on the comparison between the matched filter and the empirical mode (EMD)-assisted matched filter (EMMF) to be described later. In practice, one would sweep through the set of realizations of speed, range, and HED dipole strength, and determine the parameters that maximize the matched filter output. The two components of the electric field were processed together in quadrature as a complex signal, with Ex as the real component and Ey as the imaginary component. A correlation filter was used with the magnitude of the output being the detection criterion, normalized to a maximum unity output. Through simulation it was determined that

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r

x11

x11

x7–10

x6

µ

x5

x4

x3

x1–2

x (t)

128

2 0 −2 0.2 0 −0.2 0.1 0 −0.1 0.2 0 −0.2 0.5 0 −0.5 0.5 0 −0.5 2 0 −2 0.5 0 −0.5 0.1 0 −0.1 0.1 0 −0.1

0

500

1000

1500

2000

2500

3000

Time (sec.)

FIGURE 6.4 EMD analysis of underwater electromagnetic data.

s(t)

Known target signature

x(t)

Target signal

EMD

N

i(t)

Constrained LSQ

xˆ(t)

Matched Filter

Detection γ(t) statistic

i=1

FIGURE 6.5 Block diagram of EMD-assisted matched filter, where the matched filter process is preceded by a least-squares fit of the IMFs (generated from the time series data).

the expected signal magnitude depends on the CPA range and target speed but not the target bearing. All bearings will have the same magnitude response, with differing phase responses. Thus the matched filter output depends only upon using the correct range and speed of the vessel. Building on the matched filter, an improved algorithm, the empirical mode matched filter, was developed to exploit the EMD’s ability to capture the signal in just a few of the IMFs. This method applies the matched filter to a signal generated using the IMFs as basis functions, rather than to the target signal x(t) itself. This improved detector performance. A block diagram of the processing steps of the algorithm is shown in Figure 6.5. Empirical mode decomposition was first applied

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129

to the time series to generate the IMFs αi(t), i = 1, …, N, where N is the number of IMFs. In the next step, the linear combination of the IMFs containing the target signal s(t) which best approximates the known target signature in the least-square sense was determined: N

min || s(t ) − ci ≥ 0

∑ c α || . i

i

i =1

The least-squares optimization was constrained so that the weights ci are positive. A matched filter based on the target signature s(t) was then applied to the signal xˆ (t), which is composed of the weighted linear combination of the IMFs, N

xˆ (t ) =

∑ cα , i

i

i =1

to generate a detection statistic λ(t). The underlying motivation for this technique is the tendency of the EMD to generate a physically motivated basis for the data. Individual IMFs can be associated with the dipole signal. Often the signal is captured in a single IMF. In such a case, it would be advantageous to apply a matched filter to this IMF, rather than to the whole time series, which may contain corrupting background signals. In general, it is not known a priori which IMF the dipole will be in; it may even be spread across several IMFs. Thus we employ the least-squares criterion above. The positive weight constraint stems from the fact that the IMFs contain no sign ambiguity — their sum is the complete time series. This constraint also reduces false alarms. A sample time series of this process from the experiment is shown in Figure 6.6. The process of weighting the IMFs results in an improved correlation with the target signal as compared to the full time series. We examined the performance of the EMMF using a statistical Monte Carlo analysis with real electromagnetic background data from the experiment and simulated target signals (which are shown above to match the actual towed source signals very well). This procedure validated that the EMMF robustly improves the probability of detection of the HED dipole signal in realistic oceanic/geomagnetic noise. A number of signal-conditioning and pre-processing steps were used as well. The simulation analysis was confirmed with detection results for real signals generated by the DC electric towed source. The first type of noise background tested for comparison between the linear matched filter (LMF) and the EMMF was white Gaussian noise generated by a random signal generator. The noise was band-limited by using a high-pass filter with a frequency cutoff of 0.5 MHz (2000-sec period), because this low-frequency cutoff does not change the moving dipole signals we wish to detect. A dipole signal was created with a source strength equal to that used in the experiment and numerically embedded in the white Gaussian noise background, which usually resulted in about

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1.5 raw signal LSQ wt signal model 1

0.5

µV/m

0

0.5

1

1.5

2

2.5

0

100

200

300

400

500

600

700

time (sec)

FIGURE 6.6 Comparison of matched filter and EMMF techniques.

3000 overlapping samples in the simulation. The window size was set to τ = 2.7 (τ = V/RCPA), which from experience yields an effective template for the matched filter. The desired Ex and Ey signals are shown in Figure 6.7. A τ parameter of 2.7 yields signals that are 9 min long for this geometry. Longer τ values include the tails of the signals, which contain little information, and shorter τ values may cut out valuable information about a signal’s shape that makes it different from the background. For this τ and constant source speed V, the signals lengthen in time proportional to their CPA offset RCPA. Receiver operating curves (ROC) curves were generated for the white Gaussian noise and signal by calculating distributions of the matched filter output (normalized to 0–1) for inputs of noise only and the noise plus signal. Once the distributions were created, ROC curves were extracted, utilizing a sliding threshold, to determine the probability of detection (PD) at different probabilities of false alarm (PFAs). The result, shown in Figure 6.8, confirms the expected theoretical behavior: that the standard LMF technique yields the best detection statistics for this type of noise. The LMF procedure outperforms the EMMF (also labeled as the “LSQ” in the figures that follow because the method involves a least-squares fit of the IMFs to the desired matched filter). As the source strength and signal-to-noise ratio decrease, both the LMF and EMMF ROC performances deteriorate, but the LMF maintains a slight advantage, as predicted by theory. Signal-to-noise ratio (SNR) is defined

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131

E-ﬁeld X/Y components, Tau = 2.7 1.5 1

E-ﬁeld (uV/m)

0.5 0 −0.5 −1 −1.5 −2

0

100

200

300 400 Time (seconds)

500

600

1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6 PD

PD

FIGURE 6.7 Ex (solid) and Ey (dotted) signals for a north–south track, where τ = 2.7, which yields a 9-min signal length at this CPA and speed.

0.5

0.5

0.4

0.4

0.3

0.3 0.2

0.2 0.1 0

0.1

LSQ LMF

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0

LSQ LMF

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

PFA

1

PFA

FIGURE 6.8 Comparative ROC curves for HED moving dipole detection in Gaussian white noise, for LMF method (lower curve) and EMMF/LSQ method (upper curve). The original source strength has been reduced to correspond to SNR values of –14 and –16.4. The LMF method outperforms the EMMF in Gaussian white noise, in accordance with signal-processing theory.

here as the peak excursion of the target signal versus the standard deviation of the background noise [11], normalized to a logarithmic dB number: SNR = 20 log10 ( P / σ ) , where P is the peak excursion of the target signal (of a specified component), and σ is the standard deviation of the background noise. This broadband definition is

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Source strength eﬀect on ROC curves, LMF vs. EMMF 1 SNR 10

0.9

SNR 7

0.8 0.7 PD (01)

SNR 5 0.6 0.5 0.4 0.3 0.2 EMMF/LSQ

0.1

LMF 0

0

0.2

0.4

0.6

0.8

1

PFA (01)

FIGURE 6.9 ROC curves for LMF process (light) and EMMF/LSQ process (dark), for several different SNRs. When the SNR is strong, the two processes achieve nearly identical detection results, but when the SNR is weaker, the EMMF process has a definite advantage.

calculated independently of frequency. The SNR for the ROC curves in Figure 6.8 is negative (printed on curves) and means that the signal amplitude is lower than the standard deviation of the noise. A SNR of –14, for example, corresponds to the peak signal value being one-fifth the noise standard deviation. The advantage of the LMF over the EMMF method is lost when an actual noise background is used. The background used in the following analysis consists of 52 h of experimentally measured underwater electromagnetic data, which is non-Gaussian with a colored spectrum and a nonstationary variance. In actual oceanic and geomagnetic noise, the EMMF method is superior. This is demonstrated in Figure 6.9, where we have displayed three representative curves together on one graph, utilizing SNRs between –5 and +14, much higher than the SNRs used in the white Gaussian noise process. A major characteristic of the real-world noise is its nonstationarity and high power at low frequencies, which can lead to anomalies that mimic the transient HED. The ROC performance curves in Figure 6.9 have been enhanced by two signal-conditioning procedures: high-pass filtering of the data to produce first-order stationarity (see above), and removal of a linear trend within each window being examined. The window length is determined by the τ value being sought. (For the

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133

PD versus source strength (dB) 1 0.9 0.8 0.7

Black: LSQ Red: LMF Blue: LSQ, 2x range Green: LMF, 2x range

PD (01)

0.6 0.5 0.4 0.3 0.2 0.1 0 0

10

20 30 Source Strength (dB Am)

40

50

FIGURE 6.10 Relative gain for HED detection using EMMF at set PFA (= 0.1) for two different ranges. PD is plotted as a function of source strength in dB. The EMMF method is in blue. For a given PD, the necessary source strength is lowered by 2 to 3 dB, for moderate PD.

simulations performed, we assume knowledge of such parameters; for detection in a field situation, one would sweep through a predetermined set of moving dipole parameters and choose the set that yields the highest matched-filter output.) Other signal-conditioning techniques were also explored and found to be less effective. For example, when we remove a mean value from the window rather than de-trending, the overall ROC curve degrades, but the EMMF displays a relative advantage over the linear matched filter. Figure 6.10 shows the effects of source strength and SNR on ROC performance curves. The source strength is represented in dB relative to 1 A-m. For low-tomedium source strengths (up to 35 dB), the EMMF method yields superior PDs at a given strength. Conversely, for a given PD, the necessary source strength to achieve that PD is reduced by 2 to 3 dB in the PD range of 0.4 to 0.8. This detection gain vanishes at high source strengths (high SNRs), where the two methods yield identical detection statistics. It must be emphasized that the analysis here has been done on “unreduced” data, which has not been differenced versus a remote reference and is a high-noise environment. Removing coherent geomagnetic noise by using a remote reference sensor yields a greater range.

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PD versus CPA; unreduced data EMMF LMF

1

PD (01)

0.8

0.6

0.4

0.2

Range FIGURE 6.11 Probability of detection versus CPA, for LMF (light) and EMMF (dark) methods. As the range is further increased and SNR decreases, the EMMF method excels, extending the detection range by 10 to 15% for a set PD, or increasing the PD by 10 to 25% for a given range.

We repeated this procedure for a greater range for unreduced data, which yields a nearly identical curve, but shifted to the right about 15 dB. Thus the EMMF method maintains its increase in effectiveness across range. The detection versus range behavior is summarized in Figure 6.11, where we plot PD versus CPA range for a set PFA = 0.1. Up to a certain range, the two methods again yield an identical high probability of detection, but as the range is further increased (and SNR decreases), the EMMF method (dark curve) is more effective, extending the detection range by 10 to 15% for a set PD, or increasing the PD by 0.1 to 0.2 for a given range. We see that the EMMF method flattens the falloff curve for low SNR regimes, thereby extending detection ranges. The inclusion of the EMD into the matched filter process adds detection gain in high-amplitude, colored, and nonstationary noise such as observed in the underwater electromagnetic environment. This effectiveness enhancement is seen generally in low to medium SNRs, whether caused by weaker source strength or a farther CPA. We also investigated the effectiveness of the EMMF method in a noise regime where the common-mode geomagnetic noise has been partially mitigated by using remote-reference techniques. This yields improved noise reduction so that an algorithm

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ROC curves, REDUCED data

Unreduced and reduced background noise 3

0.9

sigma = 0.23

2

0.8

1.5

0.7

1

0.6

PD (01)

E-ﬁeld amplitude (uV/m)

1

Unreduced reduced

2.5

0.5 0

0.5 0.4 0.3

−0.5

0.2

−1 −1.5

135

sigma = 0.090 0

5

10

15

20

0.1 25

30

35

40

0

EMMF/LSQ LMF

0

Time (hours)

0.2

0.4

0.6

0.8

1

PFA (01)

FIGURE 6.12 Effect of remote-reference noise reduction on matched filter methods. (A) Unreduced (top) and reduced (bottom) time series for background, X component, over 40 h. The geomagnetic noise removal not only lowers the variance, but also makes the resulting series more stationary. (B) ROC curves generated from reduced data, for EMMF (dark) and LMF (light) methods. The LMF method is more effective at low ranges/high SNR, whereas the EMMF method increasingly dominates at greater ranges.

comparison can be made in a background where oceanic and other processes dominate, rather than ionospheric sources. Figure 6.12A shows a plot of the unreduced and reduced time series. Visually, the reduced time series is more uniform across its entire length and has both lower variance and fewer outliers than the original time series, which is simply high-pass filtered. Histogram analysis at various positions in the time series also indicates that it has more a stationary variance, and spectral analysis shows it to have a flatter power spectrum at low frequencies and thus to be closer to white Gaussian noise. We repeated the Monte Carlo procedure for generating ROC curves as a function of range, with the results shown in Figure 6.12B. Detection ranges are noticeably extended. Also, the LMF method is actually slightly superior to the EMMF, at ranges where the SNR is high. However, as the range increases and the SNR decreases, the EMMF method becomes more effective than the linear matched filter, providing a flatter rolloff of PD versus range. Figure 6.13 shows PD performance versus range for the EMMF and LMF methods for reduced data. Here we have set the PFA to a constant (= 0.1), and calculated the PD as a function of range for both methods. The linear method has a slight advantage at lower ranges and higher SNR, whereas the HHT-enhanced method is more effective at long ranges and a small SNR. Overall, the EMMF method provides superior performance with flatter rolloff versus range.

6.4 EXPERIMENTAL VALIDATION USING TOWED-SOURCE SIGNALS We utilized a database of 17 tracks of a towed electromagnetic source collected over 6 sensors with various CPAs to test the relative effectiveness of the EMMF method

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PD versus CPA, REDUCED data 1 0.9 0.8 0.7

PD (01)

0.6 0.5 0.4 0.3 0.2 0.1 0 Range FIGURE 6.13 Summary graph of behavior of PD versus range for EMMF (dark) and LMF (light), after coherent geomagnetic noise removal. The LMF method is slightly improved at lower ranges, but overall the EMMF method provides a superior curve with flatter rolloff versus range.

versus the LMF method of detection for the moving HED signal on unreduced data. A total of 36 track/sensor combinations existed in this CPA range. For each individual track, assuming knowledge of the track bearing and CPA, we performed the algorithms discussed above and measured the correlation signal. Such signals generally produced a peak very near the CPA. To determine whether a detection occurred, we applied the matched filter to data measured at a remote sensor which contains only background noise, and calculated a threshold for a given false alarm level (a PFA level of 5% was used). The results are displayed in Figure 6.14A for the LMF method and Figure 6.14B for the EMMF method. As expected from the ROC analysis, the EMMF method is superior. EMMF correlations are uniformly higher with a much flatter falloff versus CPA than for the LMF method. Additionally, the threshold (actually a mean value of the thresholds over all 36 cases) for the EMMF method raises only slightly, which results in an overall increase in detectability using the EMMF. As displayed in Figure 6.15, in the 36 cases tested, the EMD-based method results in 24 detections, versus only 14 for the linear matched filter (points very near to the detection threshold were counted as detections), using a PFA of 5%. © 2005 by Taylor & Francis Group, LLC

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LMF-based correlation vs. threshold

EMMF-based correlation vs. threshold 1

(a)

0.9

0.9

0.8

0.8

Correlation (0–1)

Correlation (0–1)

1

137

0.7 threshold for PFA = 0.05 0.6 0.5 0.4

(b)

0.7 threshold for PFA = 0.05 0.6 0.5 0.4

0.3 A): 14/36 detections 0.2

0.3 A): 24/36 detections 0.2

0.1

0.1

FIGURE 6.14 Comparison of (A) the LMF method with (B) the EMMF method, utilizing signals created by the towed electric source. The EMMF has a flatter falloff with range and more detections (24/36 versus only 14 for the LMF).

Detections vs. range: LMF vs EMMF for data with coherent noise reduction

Number of detections, chances

Possible chances Detections EMD Detections LMF

Range

FIGURE 6.15 Comparison of number of detections of the LMF method (light) with the EMMF method (medium), utilizing signals created by the towed electric source.

6.5 CONCLUSION In summary, we have exploited the empirical mode decomposition to isolate physical signals into a discrete number of intrinsic mode functions in order to enhance the detection of transient HED signals in a background of geomagnetic and hydrodynamic noise. We have numerically compared this method to the standard matched filter by deriving ROC curves via a Monte Carlo simulation for experimental data. In white Gaussian noise, the LMF method yields superior ROC curves, in accordance

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with signal-processing theory. In a realistic, non-Gaussian, high-noise environment, the EMMF method outperforms its linear counterpart by enhancing detection ranges and minimum detectable source strengths. When it is possible to mitigate much of the geomagnetic noise by using remote reference methods, then the traditional LMF provides slightly superior detection at high SNRs, but the EMMF method yields a flatter PD versus range curve and significantly extends detection ranges for a given PD. The results for unreduced data were confirmed by applying the matched filter methods to experimental data, where the EMMF method increases detection ranges and flattens the detection curve with range. Further investigation of the optimal use of such methods is warranted.

ACKNOWLEDGMENT This work was funded by the Office of Naval Research and others. The authors are grateful for the experimental assistance of the Swedish Defense Ministry and the FOI, and for the assistance of Dave Rees of SPAWARSYSCEN San Diego.

REFERENCES 1. J. Berthier, F. Robach, B. Flament, and R. Blanpain. Geomagnetic noise reduction from sea surface, high sensitivity magnetic signals using horizontal and vertical gradients. In Proc. 1998 Oceans Conf., July 2001. 2. R. Blampain. Traitement en temps réel du signal issu d'une sonde magnetometrique. Doctoral thesis of engineering, INPG, Grenoble, 1979. 3. C. Chichereau, J.L. Lacoume, R. Blanpain, and G. Dassot. Short delay detection of a transient in additive Gaussian noise via higher order statistic test. In IEEE Digital Signal Proc. Workshop, pp. 299–302, 1996. 4. C.S. Cox, X. Zhang, S.C. Webb, and D.C. Jacobs. Electro-magnetic fluctuations induced by geomagnetic induction in turbulent flow of seawater in a tidal channel. In MARELEC, 3rd Int. Conf. Marine Electromagnetics, July 2001. 5. B. Deniel. Undersea magnetic noise reduction. In Proc. 1997 MTS/IEEE Conf., vol. 2, pp. 784–788, 1997. 6. R. Donati. Detection of the electric field due to corrosion phenomena. In Proc. 2nd Int. Conf. Marine Electromagnetics (MARELEC), Vol. 2, pp. 322–325, 1999. 7. P. Flandrin, P. Gonclaves, and G. Rilling. Detrending and denoising with empirical mode decompositions. In Proc. IEEE-EURASIP Workshop Nonlinear Signal Image Process. NSIP-04, to appear in 2004, available at http://www.inrialpes.fr/is2/people/pgoncalv/pub/emd-eusipco04.pdf. 8. P. Flandrin and P. Gonclaves. Sur la decomposition modale empirique. In Proc. GRETSI-03, 2003. 9. P. Flandrin, G. Rilling, and P. Gonclaves. Empirical mode decomposition as a filter bank. IEEE Sig. Proc. Lett., 11(2):112–114, 2004. 10. P. Flandrin. Empirical mode decompositions as data-driven wavelet-like expansions for stochastic processes. In Proc. Conf. Wavelets Statistics, 2003. 11. B.C. Fowler, H.W. Smith, and F.W. Bostick Jr. Magnetic anomaly detection utilizing component differencing techniques. Technical Report 154, Electrical Geophysics Research Lab., Univerisity of Austin Texas, 1973.

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12. K. Hasselmann. On the nonlinear energy transfer in a gravity wave spectrum. Jour. Fluid Mech., 12:481–500, 1962. 13. N.E. Huang, Z. Shen, S.R. Long, M.L. Wu, H.H. Shih, Q. Zheng, N. Yen, C.C. Tung, and H.H. Liu. The empirical mode decomposition and Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. R. Soc. London A, 454: 903–995, 1998. 14. N.E. Huang, Z. Shen, and S.R. Long. A new view of nonlinear water waves: the Hilbert spectrum. Annu. Rev. Fluid Mech., 31:417–57, 1999. 15. S.M. Kay. Fundamentals of Statistical Signal Processing, Volume 2: Detection Theory. Prentice Hall, Englewood Cliffs, N.J., 1998. 16. M.J. Morey, T. Bentson, and C. Osborn. TET E-field data analysis, summer 1991 experiment. Tech. Document 2482, NCCOSC RDT & E Divsion, 1993. 17. E.A. Nichols, J. Clarke, and H.F. Morrison. Signals and noise in measurements of low-frequency geomagnetic fields. J. Geophys. Res., 93:13743–13754, 1988. 18. M.K. Ochi. Ocean Waves: The Stochastic Approach. Cambridge Ocean Technology, Series 6. University Press, Cambridge, 1998. 19. A.M. Orr. Computational techniques for evaluating extremely low frequency electromagnetic fields produced by a horizontal electrical dipole in seawater. Dept. of Physics, King's College London, 2001. 20. A. Quinquis and D. Declerq. Magnetic noise subtraction with wavelet packets. In Proc. OCEANS '95. MTS/IEEE. Challenges of Our Changing Global Environment, Vol. 1, pp. 227–232, 1995. 21. A. Quinquis and C. Dugal. Using the wavelet transform for the detection of magnetic underwater transient signals. In Proc. OCEANS '94: Oceans Engineering for Today's Technology and Tomorrow's Preservation, Vol. 2, pp. 548–543, 1992. 22. A. Quinquis and S. Rossignol. Detection of underwater magnetic transient signals by higher order analysis. In Proc. Int. Symp. Signal Process. Appl., pp. 741–749, 1996. 23. J. Rantakokko. Localization of static electric and magnetic underwater sources. Technical Report FOA report No. FOA-R–99-01265-409–SE, Swedish Defense Research Establishment (F.O.A.), Stockholm, Sweden, 1999. 24. J. Ridgway, M.L. Larsen, C.H. Waldman, M. Gabbay, R.R. Buntzen, and C.D. Rees. Analysis of ocean electromagnetic data using a Hilbert spectrum approach. In Proc. 7th Exper. Chaos Conf., pp. 333–338, August, 2003. 25. G. Rilling, P. Flandrin, and P. Gonclaves. On empirical mode decomposition and its algorithms. In Proc. IEEE-EURASIP Workshop Nonlinear Signal Image Process. NSIP-03, 2003. 26. T.B. Sanford. Motionally induced electric and magnetic fields in the sea. J. Geophys. Res., 76:3476–3492, 1971. 27. S. Webb and C.S. Cox. Pressure and electric fluctuations on the deep seafloor: background noise for seismic detection. Geophys. Res. Lett., 11:967–970, 1984. 28. H.C. Yuen and B.M. Lake. Nonlinear dynamics of deep-water gravity waves. Adv. Appl. Mech., 22:67–229, 1982.

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7

Coherent Structures Analysis in Turbulent Open Channel Flow Using Hilbert-Huang and Wavelets Transforms Athanasios Zeris and Panayotis Prinos

CONTENTS 7.1 7.2

Introduction ..................................................................................................142 Results ..........................................................................................................143 7.2.1 Academic Signals.............................................................................143 7.2.2 Turbulent Signals .............................................................................143 7.2.3 Coherent Structures.........................................................................143 7.3 Conclusions ..................................................................................................150 References..............................................................................................................156 ABSTRACT In this study the Hilbert-Huang transform with empirical mode decomposition is applied for analyzing and quantifing turbulent velocity data from open channel flow. This novel technique, proposed by Huang et al. [1,2], is applied using experimental velocity signals in turbulent, fully developed open channel flow, with and without the imposition of bed suction. Signals were obtained with laser Doppler and hot film techniques [3,4]. Velocity data for the above flow were analyzed in previous studies, with traditional methods, for various suction rates for the description of the flow field and for the identification of the intense events related with coherent structures that contribute to the organized motion. Here, an attempt is made to exploit the novel decomposition method in the domain of joint frequency–time analysis of the turbulence data. 141

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7.1 INTRODUCTION Details for the instrumentation and experimental results from measurements of the application of suction from the bed in open channel have been presented in previous articles [3,4,5]. We mention for completeness of the present study that measurements were conducted with LDA (tracker) and hot film methods, the suction rate is based on discharge ratio (Qs/Qtot, Qs = suction discharge, Qtot = total discharge), the length of suction region L = 0.035 m, and the Re number based on upstream velocity Um and flow depth was Re = Umh/ = 59,000. The empirical mode decomposition (EMD) method proposed by Huang [1] and the Hilbert transform of the decomposed signal are summarized as follows: All the extrema of the signal x(t) are identified, and an interpolation for the maxima and the minima is attempted with cubic splines curves, following recommendations and proposals from previous work [1,2]. The local mean m(t) is computed, and the difference between the local mean m(t) and the initial signal is extracted. Until a stopping criterion [2] is achieved, an iteration process takes place. In this study the criterion that we adopt is that the number of extrema equals (or differs by one from) the number of zero-crossings. When this criterion has been achieved, the remainder is considered as an intrinsic mode function (IMF) and denoted c1; this IMF contains the shortest periods of the signal. The residual

()

()

x t − c1 = r t , is considered as a new signal and subjected to the iterative process repeatedly, until the last residue is smaller than a predetermined small value or is a monotonic function. Preliminarily, the problem of extrema interpolation is checked from the viewpoint of (a) it is time consuming especially when the signal belongs to the multifrequency ones as turbulent signals that needs a large number of iterations for a correct decomposition and (b) because of end effects propagation of the splines into the interior of the signal. We used different approaches as linear, linear with empirical modifications, and iterative process with preselected iteration numbers. With the criterion of the most perfect composition of the original signal we chose, for the present article, to extract IMFs with the originally proposed method [2] cubic spline with the addition of sine waveforms at the beginning and at the end of the analyzed data. With the aid of the Hilbert transform, Huang et al. [1,2] proposed the construction of the analytic signal for every IMF component. From them we get the time distributions of amplitude and phase

()

()

()

h t = c t + H c t 

( ( ))  ,

H c t φ t = tan −1   c t

()

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2

()



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143

where h(t) and φ(t) are the instantaneous amplitude and phase respectively. From this quantity, values for the instantaneous frequency are obtained as

()

f t =

()

dφ τ dt

With the extraction of IMFs as a time series, the tools for the HHT amplitude–frequency–time representation of a signal — the instantaneous frequency and the amplitude — are available, and, depending on the signal’s nature, their quality is better than or equal to the results of other well documented methods, such as short FFT, proper orthogonal decomposition, Wigner-Ville, and the wavelet decomposition method.

7.2 RESULTS 7.2.1 ACADEMIC SIGNALS Some academic signals are used for the estimation of the quality of decomposition applied in the present study. The traditional frequency shift sine signal (Figure 7.1a) is examined through frequency time variation of the first IMF with the point of frequency shift well localized. Figure 7.1c also gives the Hilbert spectrum and Figure 7.1d presents the FFD distribution. In Figure 7.2a the sawtooth signal is presented along with the distribution of the instantaneous frequency (Figure 7.1c) because this kind of variation is a candidate (and as the edge detection problem) as part of an average representative pattern indicating the presence of organised structures.

7.2.2 TURBULENT SIGNALS Figure 7.3 shows a typical velocity signal, and Figure 7.4 illustrates its analysis in eight IMF components. Figure 7.5 shows the distribution of these components in the frequency domain. Figure 7.5b presents the filtering approach of the Ci components divided by the root-mean-square (RMS) value of every component. Figure 7.6a and b gives the variation of frequency with time. For clarification only, Figure 7.6a shows the frequency–time distribution with a median filter of high order for a better visual observation of the different components. In addition, Figure 7.6c gives a representative energy–frequency–time distribution. Figure 7.7 presents the time variations in instantaneous energy IE(t) for a short time period for a velocity signal at y+ = 100 from hot film measurements.

7.2.3

COHERENT STRUCTURES

In turbulent wall flows, a great portion of the momentum transport is connected with the presence of organized structures in the form of burst structures in the region near the wall. According to well-documented models, streaks of low velocity start to oscillate and to lift up from the wall. The ejections from the decomposition of

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(a) 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0

1

2

3

4

5

3

4

5

t (sec)

(b) 40 35 30 f (Hz)

25 20 15 10 5 0 0

1

2 t (sec)

(c)

(d) 2

E (t)

EH (f )

1.5 1 0.5 0 1

10 f (Hz)

100

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

5

10

15

20

f (Hz)

FIGURE 7.1 (a) Frequency shift signal; (b) instantaneous frequency variation; (c) Hilbert spectrum; (d) FFT distribution.

these streaks, appropriately grouped, are characterized as burst structures. In the search for quantitative information, many techniques have been introduced during the past decades toward the identification of patterns and levels in the signal velocity connected with these organized structures. These techniques include the variable

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FIGURE 7.2 Sawtooth signal with (a) instantaneous frequency (b) variation with time.

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16 1

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Coherent Structures Analysis in Turbulent Open Channel Flow

18

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4 -1.5

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2 0

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0.4 0.2 0

u(m/s)

0

signal

5

10

15

20

25

30

35 t(sec) 40

FIGURE 7.3 Typical velocity signal.

interval time average (VITA) technique [6,7], quadrant splitting [8] of Re stresses, the temporal pattern average (TPAV) [10], linear stochastic estimation [9], and the proper orthogonal decomposition (POD) and wavelet decomposition–based techniques [11]. Traditional techniques, such as VITA (focusing on local variance that exceeds part of the whole signal variance) and quadrant splitting for velocity and Re stress, with empirical threshold criteria k = 1.0, predetermined time scale T* = 10, and H = 1 for quadrant splitting, respectively, have been applied in previous studies [3,4] to extract intense events in the flow field inside the porous suction region (x/L = 0.5) and beyond the suction region (x/L = 1.2) from hot film and LDA results. Figure 7.8 gives an example of results from the VITA technique. Here the effect of suction is evident at the exit of the suction region from the frequency appearance of the intense events. With the tool of signal decomposition, it is possible to overcome the shortcomings resulting from the single scale limitations of the VITA method. The VITA method does not permit the identification of events separated by a time smaller than that imposed by the VITA treatment. Also, a main disadvantage for VITA is the fact that it requires the selection of two rather subjective criteria (time scale and threshold). Wavelets decomposition allows intense events to be identified based on the value of the wavelet coefficients in the representation of the translation — scale plane using a nonsubjective threshold criterion. Analysis in the scale time plane of turbulent statistical measures gives scale dependent behavior for quantities such skewness and flatness factors. Strong non-Gaussian behavior of the smaller scales is an indication of the intermittency. For intermittency estimation at each scale, Farge [11] proposed the local intermittency index Im,n , the ratio of local energy to the mean energy at the respective scale. As this index takes an extreme value at a time instant, a strong percentage of energy and intermittency is found in the corresponding time and scale. With the above in mind, it is possible to use EMD to determine the ratio of local energy (amplitude squared) to mean energy for every IMF in the energy–frequency–time distribution. Figure 7.9 shows the above ratios with the 4th order Daub4 wavelet and for the HHT method. The next step in the identification of coherent events is to apply the appropriate thresholds. Universal proposed relations from wavelets such as Donoho and Johnstone [12] is considered that cannot contribute to these turbulent data because of the Gaussian consideration of the incoherent part. In this study, we applied different values for thresholding the ratio of instantaneous energy for every IMF mode. We propose the selection of a threshold value in the region where the number of the detected events are not independent from the

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IMF1

0.1 0.05 0 –0.05 –0.1

IMF2

147

IMF3 0.1 0.05 0 –0.05 –0.1 0.1 0.05 0 –0.05 –0.1

IMF4

0.1 0.05 0 –0.05 –0.1

IMF5

0.1 0.05 0 –0.05 –0.1

IMF6

0.1 0.05 0 –0.05 –0.1

IMF7

0.1 0.05 0 –0.05 –0.1

IMF8

0.1 0.05 0 –0.05 –0.1

IMF9

0.1 0.05 0 –0.05 –0.1

IMF10

m/s

0.1 0.05 0 0

Residual

5

10

FIGURE 7.4 (a) IMFs; (b) residual.

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40

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a)

E*103(m2/s)

1.0E+00

IMF1 IMF3 IMF5 IMF7 IMF9 ALL

1.1E-00

IMF2 IMF4 IMF6 IMF8 IMF10

1.2E-00

1.3E-00

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b) 1.E+02

10

(E/rms)*103(m2/s)

f(H) 100

IMF1 IMF3 IMF5 IMF7 IMF9 ALL

1.E+01 1.E+00

IMF2 IMF4 IMF6 IMF8 IMF10

1.E-01 1.E-02 1.E-03 1.E-04 1.E-05 1

10

f(H) 100

FIGURE 7.5 Frequency distribution of Ci.

threshold. Instead of using arbitrary predefined thresholds, we used relative thresholds (local value as percentage of the maximum, predefining only the existence of structure). Figure 7.10b presents instantaneous frequencies (that localize the corresponding amplitude squared) for a range of IMFs and for values 2 < IMF < 5, and Figure 7.10c shows Hilbert-Huang transforms of local intermittency thresholds based on these relative criteria. Having defined the points in time and IMF mode where strong events are identified, and having checked the decomposition process for the quality of reconstruction of the original time series and the degree of orthogonality, it is possible to

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f(Hz)

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10

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400

600

800

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N

FIGURE 7.6 ( a,b ) Frequency–time distributions of velocity signal; (c ) energy–frequency–time distribution.

select the corresponding points at every IMF that allow the coherent signal to be distinguished from the noncoherent signal in the reconstruction formula. As an example, Figure 7.11 presents the reconstruction of the organized and disorganized part of the signal applying a hard thresholding method for the Hilbert-Huang transform. With well-known strategies [14,15], it is possible to average the coherent pattern [6,7,13,14] and correct the phase jitter (through an iterative process with cross-correlation and a time shift between the ensemble average pattern and each individual pattern), in order to extract the averaged patterns for every turbulent quantity as streamwise, vertical velocities, and Re stresses. Figure 7.12 illustrates the shortcomings of the corresponding wavelets reconstruction in the case of the Daub4 wavelet, where the signature of the wavelet itself is observed in the reconstruction formula (N = 1024, m = 2, scale reconstruction). Figure 7.13 presents an IMF analysis (sum from coarser to finer scales Ri) of the intense events for two regions of the flow field where suction is applied in open channel flow. A stronger effect of suction is observed near the wall and at the exit of the suction region. Because of the indication of organized activity of the non-Gaussian behaviour of the turbulent data, we propose the analysis of the skewness and kirtosis coefficients,

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(a) 1

u (t)

0.8 0.6 0.4 0.2 0 0

2

4

6

8

10

6

8

10

(b) 0.5

IE (t)

0.15 0.1 0.05 0 0

2

4

FIGURE 7.7 (a) Velocity signal; (b) instantaneous energy distribution.

from the reconstruction formula (starting the composition of the initial signal from residual and the larger scales). Figure 7.14 a presents the skewness coefficient Sk for a short signal of N = 8192. The value of the whole signal is –0.23. Figure 7.14b shows a scale analysis of the same signal with Daub4 wavelet. Figure 7.15 shows, analysis with distributions for every IMF, of the turbulent quantities with the same length of signal for comparative reasons. Figure 7.15a presents the energy content of streamwise component Eiu and the vertical component Eiv, as percentage of the whole energy resulting from hot film measurements that lead to conclusions for the principal time scales. Figure 7.15b shows the ratio Eiu/Eiv. Figure 7.15c gives the percentage of energy for every IMF, from LDA measurements with and without suction application near the wall at the exit of the suction region.

7.3 CONCLUSIONS Having established an approach using empirical mode decomposition to study open channel flow, we must mention the need for a systematic application in different flow situations. Nonstationary flows, vortex shedding over obstacle, and wake flows may comprise an ideal domain for application of this new technique. Further documentation of the sifting process is necessary, including the stopping criteria and extrema interpolation, especially for difficult environments such as turbulent flow where the appearance of all the frequency bands makes the overall process time consuming. Applications that demand real-time processing, such as active control, must be investigated under this new method with concepts such as the modulated

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1

151

Cq = 0 Cq = 0.056

0.9 0.8 0.7

f

0.6 0.5 0.4 0.3 0.2 0.1 0 0.001

0.01

0.1

1

y/h

FIGURE 7.8 Frequency of events with VITA analysis (y+ = 8, x/L = 1.2).

intra-wave frequency. Nonstationary flows with short time variations would be the field for comparison of HHT with other older techniques, such as those of entropy (MEM)–based methods. However, the final general impression remains the ability of this novel technique to exploit the Hilbert transform and the analytic signal for multifrequency signals to achieve better localization in the frequency time domain and avoid the shortcomings of wavelets coefficients, such as the strong smear effect.

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(a) 700 600

HHTLIM

500 400 300 200 100 0 (b) 100 90 80 70

LIM

60 50 40 30 20 10 0

FIGURE 7.9 Local intermittency measure with IMF and Daub4.

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t (sec) 10 8 6 4 2 0

0

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4

6

8

HHTLIM IMF5

t (sec) 10 8 6 4 2 0

0

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8 t (sec)

FIGURE 7.10 (a) Velocity signal; (b) instantaneous frequency variations; (c) HHTLIM with thresholds.

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(a) 0.2 0.1 0 −0.1 −0.2

(b) 0.2 0.15 0.1 0.05 0 −0.05 −0.1 −0.15 −0.2

FIGURE 7.11 (a) Disorganized part; (b) organized part.

0.4

u (m/s)

0.35 0.3 0.25 0.2

0

0.5

1

1.5 t (sec)

FIGURE 7.12 Reconstruction with Daub4 (N = 1024, m = 2).

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2

2.5

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1.6

155

Y+ = 8, x/L = 1.2

1.4 Y+ = 100, x/L = 0.5

Ns/N0 events

1.2 1 0.8 0.6 0.4 0.2 0 1

2

3 IMF

4

5

FIGURE 7.13 Ratio of intense events for suction and no-suction cases.

1

Sk

1

a)

Sk

b)

0.5

0.5 Ri

0 1

2

3

4

5

6

7

8

m

0 1

9 10 11

-0.5

-0.5

-1

-1

2

3

4

5

6

7

8

9 10 11 12

FIGURE 7.14 (a) Sk distributions with residuals; (b) Sk distribution with scales Daub4.

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30 0.2

Ei/Eiall

u v

0.16

20

0.12

15

0.08

10

0.04

IMF

0

1 2 3 4 5 6 7 8 9 10 11 12 13 14

0.1 Ei/Eall

Eiu/Eiv

25

5

IMF

0 1 2 3 4 5 6

7 8 9 10 11 12 13

no suction suction

0.08 0.06 0.04 0.02

IMF

0 1 2 3 4 5 6 7 8 9 10 11 12 13

FIGURE 7.15 (a) Percentage contribution of Eiu and Eiv; (b) ratio Eiu/Eiv; (c) effect of suction near the wall (y+ = 10).

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REFERENCES 1. Huang N., Shen Z., Long S. (1999). A new view of nonlinear water waves: The Hilbert Spectrum. Annu. Rev. Fluid Mech. 31: 417–457. 2. Huang N. E., Shen Z., Long S., Wu M., Shih H., Zheng Q. et al. (1998). The Empirical Mode Decomposition and the Hilbert Spectrum for non linear and non stationary time series analysis. Proc. R. Soc. London A 454, 903–995. 3. Zeris A. (2001). Turbulent flow in open channel with suction from the bed. Ph.D. thesis, Aristotle University, Thessaloniki, Greece, 2001. 4. Zeris A., Prinos P. (2001). Measurements of bed suction effects on an open channel flow with Laser–Doppler/Hot-Film Anemometry. EFHT Congr., pp. 1099–1103. 5. Zeris A., Prinos P. Open channel flow with suction from the bed. J. Hydraulic Eng., submitted 2004. 6. Blackwelder R., Haritonidis J. Scaling of the bursting frequency in turbulent boundary layers. J. Fluid Mech. Vol. 132, 87–103, 1983. 7. Bogard, D. G., Tiederman W. G. (1986). Burst detection with single point velocimetry measurements. J. Fluid Mech. Vol. 162, 389–413. 8. Lu, S. S., Wilmarth, W. W. (1973). Measurements of structure of Reynolds stress in a turbulent boundary layer. J. Fluid Mech. Vol. 60, 481–511. 9. Adrian R. J. On the role of conditional averages in turbulence theory. In Turbulence in Liquids, Zakin and Patterson, Eds., Princeton Science Press, 1977. 10. Wallace, J. M., Eckelmann, H., Brodkey, R. S. (1972). The wall region in turbulent shear flow. J. Fluid Mech. Vol. 54, 39–48. 11. Farge M. (1992). Wavelet transforms and their applications to turbulence. Annu. Rev. Fluid Mech. Vol. 24, 395–497. 12. Donoho D., Johnstone I. M. (1994). Ideal spatial adaptation by wavelet shrinkage. Biometrika, Vol. 81, 425–455. 13. Camussi R., Gui G. (1997). Orthonormal wavelet decomposition of turbulent flows: intermittency and coherent structure. J. Fluid Mech. Vol. 348, 177–199. 14. Raupach, M. (1981). Conditional statistics of Reynolds stress in rough wall and smooth wall turbulent flow. J. Fluid Mech. Vol. 108, 363–382. 15. Subramanian C. S., Rajagopalan S., Antonia R., Chambers A. (1982). Comparison of conditional sampling and averaging techniques in a turbulent boundary layer. J. Fluid Mech. Vol. 23, 335–362.

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8

An HHT-Based Approach to Quantify Nonlinear Soil Amplification and Damping Ray Ruichong Zhang

CONTENTS 8.1 8.2 8.3 8.4 8.5

Introduction ..................................................................................................160 Symptoms of Soil Nonlinearity ...................................................................161 Fourier-Based Approach for Characterizing Nonlinearity ..........................162 HHT-Based Approach for Characterizing Nonlinearity ..............................166 Applications to 2001 Nisqually Earthquake Data.......................................172 8.5.1 Detection of Nonlinear Soil Sites....................................................173 8.5.2 HHT-Based Factor of Site Amplification ........................................176 8.5.3 Influences of Window Length of Data ............................................178 8.5.4 Comparison of HHT- and Fourier-Based Factors for Site Amplification....................................................................................180 8.5.5 HHT-Based Factor for Site Damping ..............................................184 8.6 Concluding Remarks and Discussion ..........................................................185 Acknowledgments..................................................................................................187 References..............................................................................................................187 ABSTRACT This study proposes to use a method of nonlinear, nonstationary data processing and analysis, i.e., the Hilbert-Huang transform (HHT), to quantify influences of soil nonlinearity in earthquake recordings. The paper first summarizes symptoms of soil nonlinearity shown in earthquake ground motion recordings. It also reviews the Fourier-based approach to characterizing the nonlinearity in the recordings and demonstrates the deficiencies. It then offers the justifications of the HHT in addressing the nonlinearity issues. With the use of the 2001 Nisqually earthquake recordings and results of the Fourier-based approach as a reference, this study shows that the 159

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HHT-based approach is effective in characterizing soil nonlinearity and quantifying the influences of nonlinearity in seismic site amplification. Primary results are as follows: The first HHT-based component and the Hilbert amplitude spectra can identify abnormal high-frequency spikes in the recording at sites where strong soil nonlinearity occurs; this can help to detect the nonlinear sites at a glance. The HHT-based factor for site amplification is defined as the ratio of marginal Hilbert amplitude spectra, similar to the Fourier-based one that is the ratio of Fourier amplitude spectra. The HHT-based factor is effective in quantifying soil nonlinearity in terms of frequency downshift in the low-frequency range and amplitude downshift in the intermediate-frequency range. Hilbert and marginal damping spectra are identified in ways similar to Hilbert and marginal amplitude spectra. Consequently, the HHT-based factor for site damping is found as the difference of marginal Hilbert damping spectra, which can be extracted from the HHT-based factor for site amplification and used as an alternative index to measure the influences of soil nonlinearity in seismic ground responses.

8.1 INTRODUCTION Site amplification is the phenomenon in which the amplitude of seismic waves increases significantly when they pass through soil layers near the earth’s surface. It can be illustrated by considering the seismic energy flux along a tube of seismic rays, which is proportional to the impedance (density × wave speed) and squared shaking velocity. Since the energy should be constant in the absence of damping, any reduction in the impedance is compensated by an increase in the shaking velocity, thus yielding site amplification for seismic waves in soil layers. Site amplification is a key factor in mapping seismic hazard in urban areas (e.g., [1]) and designing geotechnical and structural engineering systems on soils (e.g., [2]). In general, site amplification is not linearly proportional to the intensity of input seismic motion at bedrock because of soil nonlinearity under large-amplitude earthquakes. The extent of soil nonlinearity can be characterized by the change of two dynamic features of soil layer, i.e., soil resonant frequency and damping, in the frequency-dependent site amplification. Consensus has been building that the site-amplification factors in the current codes overemphasize the extent of soil nonlinearity and thus potentially underestimate the level of site amplification. It has been demonstrated [3] that the recording-based amplification factors are larger than those in codes for a certain range of base acceleration intensity. In addition, some features of site-amplification factors used in codes and guidance for structural design contradict recent findings from the 1994 Northridge ground motion data set [4]. The aforementioned problem might exist partly because seismologists and engineers lack sufficient understanding of the underlying causes in nonlinear soil. For example, the influence of soil heterogeneity does not scale linearly, even when the soil is perfectly linear [5]. In other words, a linear elastic medium with random heterogeneity can change ground motion in a way similar to that caused by medium

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nonlinearity. Consequently, it is possible to interpret the motion influenced by random heterogeneous media as soil nonlinearity (e.g., in the form of damping [6]), which can distort the quantification of site amplification. The problem may also exist partly because there is lack of an effective approach to properly characterize nonlinear features of ground motion in recordings and then to quantify them. The objective of this study is to propose the use of a method for nonlinear, nonstationary data processing [7], referred to as the Hilbert-Huang transform (HHT) method, to characterize the soil nonlinearity from earthquake recordings. In particular, this paper first summarizes symptoms of soil nonlinearity shown in earthquake ground motion recordings from previous studies and literature. It will also review Fourier-based approaches to characterizing nonlinearity from earthquake motion and demonstrate their deficiencies. It then proposes to use the HHT method for characterizing soil nonlinearity in the motion. For illustration, the study analyzes a hypothetical recording to show the HHT-based characterization of nonlinearity. Finally, it examines the mainshock and aftershock of the 2001 Nisqually earthquake recordings to demonstrate the validity and effectiveness of the proposed approach for characterizing soil nonlinearity and the influences.

8.2 SYMPTOMS OF SOIL NONLINEARITY In general, the stress–strain relationship of a soil becomes nonlinear and hysteretic for a large-amplitude input excitation. Such nonlinearity and hysteresis correspond to a reduction of soil strength, increased soil damping, and deformed waveform of response in comparison with those of the linear case (e.g., [8–12]). With a reduction of soil strength such as shear modulus (G) for nonlinear soil, the shear-wave velocity (v = (G/ρ)½, where ρ is the soil density) and thus the fundamental resonant frequency (f = v/4h) of the soil layer with thickness h decrease. Therefore, seismic motion recordings over a nonlinear soil layer could show strong wave response at a lower resonant frequency than for the same layer with linear excitation. Accordingly, increased site amplification at the downshifted soil resonant frequency can be regarded as a signature of soil nonlinearity observable in ground motion records (e.g., [13, 14]). On the other hand, increased damping for nonlinear soil will decrease ground motion, thus moderating the site amplification. Since soil damping is typically frequency dependent, so is the change of damping for nonlinear soil. The increased damping of nonlinear soil is likely lower at higher frequencies [15]. The increased site amplification at the downshifted soil resonant frequency and the frequency-dependent increased damping imply that the change in site amplification due to soil nonlinearity should be strongly dependent upon frequency. Soil nonlinearity can also sometimes be inferred from abnormal high-frequency spikes and recorded waveforms that appear only in recordings that are close to the locations where strong nonlinearity occurred (e.g., [16]). The cusped waveforms and high-frequency spikes observed in the 2001 Nisqually earthquake recordings, for example, are symptomatic of a nonlinear response at some soil sites (e.g., [17]).

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8.3 FOURIER-BASED APPROACH FOR CHARACTERIZING NONLINEARITY In practice, the Fourier series expansion is frequently used for representing and analyzing recorded digital data of earthquake acceleration X(t), i.e., N

X (t ) = ℜ

∑

N

Aje

iΩ j t

=ℜ

j =1

∑[ A sin(Ω t) + iA cos(Ω t)] , j

j

j

j

(8.1)

j =1

where ℜ denotes the real part of the value to be calculated, i = (–1)∫ is an imaginary unit, amplitudes Aj are a function of time-independent frequency Ωj that is defined over the window in which the data is analyzed, and the Fourier amplitude spectrum is defined as N

F (Ω) =

∑A .

(8.2)

j

j =1

To apply this Fourier spectral analysis to estimate the influences of soil nonlinearity in the seismic wave responses at soil site or simply site amplification, two sets of recordings are typically needed [18], one at a soil site and the other at a referenced site such as bedrock or outcrop. For a frequency, the Fourier-based factor of site amplification (FF) for an earthquake event (either mainshock or aftershock) can then be found by

( )

FFs Ω =

Fs2,h1 + Fs2,h 2 Fr2,h1 + Fr2,h 2

,

(8.3)

where subscripts s and r denote respectively the soil and referenced sites, and subscripts h1 and h2 denote the two horizontal components. Note that Equation 8.3 is one of many representatives for site-amplification factor that can be the ratio of characteristics of seismic waves or spectral responses at a site versus referenced site. Since the wave paths and earth structures except the soil layer are almost the same for the soil and referenced sites, the factor for site amplification in Equation 8.3 eliminates approximately the influences of source from the earthquake event and thus provides essentially the dynamic characteristics of the soil. In addition, the recordings at the referenced site are generally believed to be the results of linear wave responses and the recordings at the soil site subject to the large-amplitude mainshock to be the results of nonlinear wave responses. Accordingly, comparing the factors from the mainshock and the aftershock could help us explore and quantify the influences of soil nonlinearity in site amplification. While the Fourier-based approach given here and similar methods are widely used, they have the following deficiencies in characterizing the nonstationarity of

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the earthquake motion that is caused by source, different types of propagating waves, and soil nonlinearity if the earthquake magnitude is large enough. A Fourier-based approach defines harmonic components globally and thus yields average characteristics over the entire duration of the data. However, some characteristics of data, such as the downshift of soil resonant frequency at a nonlinear site, may occur only over a short portion of a record. This is particularly true when the intensity of the seismic input to a soil layer is not strong, such that the soil becomes nonlinear over only a portion of the entire duration of motion and in only a certain frequency band. As a result, the averaging characteristic in Fourier spectral analysis makes it insensitive for identifying time-dependent frequency content. While a windowed (or short-time) Fourier-based approach can be used to improve the above analysis to a certain extent, it also reduces frequency resolution as the length of the window shortens. Thus, one is faced with a tradeoff. The shorter the window, the better the temporal localization of the Fourier amplitude spectrum, but the poorer the frequency resolution, which directly influences the measurement of downshift of soil resonance that typically arises in a low to intermediate frequency band. More important, a Fourier-based approach explains data in terms of a linear superposition of harmonic functions. Therefore, it is an appropriate, effective method for characterizing linear phenomena such as waves with time-independent frequency, rather than nonlinear phenomena with time-dependent frequency. An example of time-independent and time-dependent frequency waves is a hypothetical wave record y(t) = y1(t) + y2(t), where decaying waves y1(t) = cos[2πt + εsin(2πt)]e–0.2t have time-dependent frequency of 1 + εcos(2πt) Hz, with ε denoting a constant factor, and noise y2(t) = 0.05sin(30t) has time-independent frequency of 15 Hz. Note that the waves shown in Figure 8.1 with ε = 0.5 are physically related to one type of 1.5

1

Amplitude

0.5

0

−0.5 −1 −1.5

0

1

2

3

4

5 6 Time (sec)

7

8

9

10

FIGURE 8.1 A hypothetical wave recording, consisting of nonlinear waves and noise with frequencies 1 + 0.5cos(2πt) Hz and 15 Hz, respectively.

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FIGURE 8.2 Fourier and marginal Hilbert amplitude spectra of the recording in Figure 8.1.

water waves that result from a nonlinear dynamic process and are also representative of seismic responses at a nonlinear soil site (to be elaborated). The time-dependent frequency waves can be expanded into and thus interpreted by a series of time-independent frequency waves, as done by the Fourier spectral analysis in which y(t), or y1(t) in particular, can be interpreted as to contain Fourier components at all frequencies (see Equation 8.1, Figure 8.2, and Figure 8.3). Alternatively, the expansion of y1(t), i.e., y1 (t ) ≈ [−0.5ε + cos(2πt ) + 0.5ε cos(4 πt )]e −0.2t , for

ε << 1

(8.4)

suggests that the Fourier transform of y1(t) consists primarily of two harmonic functions centered respectively at 1 Hz and 2 Hz for ε << 1, and the widths of these harmonic functions are proportional to the exponential parameter 0.2, which is related to the damping factor. Note that Figure 8.1 and Figure 8.2 use ε = 0.5, which is not a small number in comparison to unity, and thus they have the third observable harmonic function at 3 Hz in Figure 8.2. Therefore, one can equally well describe y1(t) by saying that it consists of just two frequency components, each component having a time-varying amplitude that is proportional to e–0.2t. Indeed, if one were to examine the local behavior of y1(t) in the neighborhood of a given time instant, say

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FIGURE 8.3 Fourier components (fj, j = 1, 2, 3, 4, 5) of the recording in Figure 8.1 at selected frequencies (i.e., 10 Hz, 5 Hz, 2 Hz, 1 Hz, and 0.5 Hz).

t0, this is precisely what one would observe. The Fourier-based analysis or interpretation given here can also be seen in Priestley [19] and Zhang et al. [20], among others. Because the true frequency content of the waves y1(t) is bounded between 1 – ε and 1 + ε, much less than 2 Hz, analysis of the above example suggests that Fourier spectral analysis typically needs higher-frequency harmonics (at least 2 Hz for the example) to simulate the nonlinear waveform of the data. Stated differently, Fourier spectral analysis distorts the nonlinear data. Consequently, the Fourier-based approach in Equation 8.3 twists the influences of soil nonlinearity in site amplification. The above assertions are confirmed in Huang et al. [21] and Worden and Tomlinson [22], among others, with the aid of solutions to classic nonlinear systems such as the Duffing equation in general, and in Zhang et al. [23] with the nonlinear site amplification in particular. In theory, Fourier spectral analysis in general and Fourier-based approaches for site amplification in particular can be further used for evaluating damping factor. For example, the resonant amplification method or half-power method uses the amplitude change or width of the peaks at a certain frequency in the Fourier amplitude spectrum to find the damping factor of dynamic systems such as a soil layer (e.g., [24]). However, the distorted Fourier amplitude spectrum for nonlinear data

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will mislead the subsequent use for damping evaluation with nonlinear soil. For example, the damping factor evaluated at the first and second peaks in the Fourier amplitude spectrum in Figure 8.2 suggests that the damping is associated with frequency at 1 Hz and 2 Hz. In fact, the damping of the hypothetical record is dependent only on the true frequency content of the waves y1(t) bounded between 1 – ε and 1 + ε, or 0.5 Hz and 1.5 Hz with ε = 0.5 in Figure 8.1. Accordingly, a Fourier-based approach would misrepresent the influences of damping factor as it relates to soil nonlinearity.

8.4 HHT-BASED APPROACH FOR CHARACTERIZING NONLINEARITY A method for nonlinear, nonstationary data processing [7] can be used as an alternative to the Fourier-based approach for characterizing soil nonlinearity. This method, referred to as the Hilbert-Huang transform, consists of empirical mode decomposition (EMD) and Hilbert spectral analysis (HSA). Any complicated time domain record can be decomposed via EMD into a finite, often small, number of intrinsic mode functions (IMFs) that admit a well-behaved Hilbert transform. The IMF is defined by the following conditions: (1) over the entire time series, the number of extrema and the number of zero-crossings must be equal or differ at most by one, and (2) the mean value of the envelope defined by the local maxima and the envelope defined by the local minima is zero at any point. An IMF represents a simple oscillatory mode similar to a component in the Fourier-based harmonic function, but more general. The EMD explores temporal variation in the characteristic time scale of the data and thus is adaptive to nonlinear, nonstationary data processes. The HSA defines an instantaneous or time-dependent frequency of the data via Hilbert transformation of each IMF component. These two unique features endow the HHT with a possibly enhanced interpretive value, making it a useful alternative to Fourier components and amplitude spectra. The HHT representation of data X(t) is n

n

X (t ) = ℜ

∑ a (t)e j

iθ j ( t )

=ℜ

j =1

∑[C (t) + iY (t)] , j

j

(8.5)

j =1

where Cj(t) and Yj(t) are respectively the jth IMF component of X(t) and its Hilbert transform, Yj (t ) =

C j (t ′) 1 P dt ′ , t − t′ π

∫

where P denotes the Cauchy principal value, and the time-dependent amplitudes aj(t) and phases θj(t) are the polar-coordinate expression of Cartesian-coordinate expression of Cj(t) and Yj(t), from which the instantaneous frequency is defined as

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ω j (t ) =

d θ j (t ) . dt

167

(8.6)

It should be emphasized here that the instantaneous frequency has physical meaning only through its definition on each IMF component as in Equation 8.5 and Equation 8.6; by contrast, the instantaneous frequency defined through the Hilbert transformation of the original data is generally less directly related to frequency content [7]. Equation 8.5 and Equation 8.6 indicate that the amplitudes aj(t) are associated with ωj(t) at time t, or, in general, functions of ω and t. Similar to the Fourier amplitude spectrum, the Hilbert amplitude spectrum is defined as n

H (ω , t ) =

∑ a (t) , j

(8.7)

j =1

and its square gives the temporal evolution of the energy distribution. The marginal Hilbert amplitude spectrum, h(ω), defined as T

h(ω ) =

∫ H (ω, t)dt ,

(8.8)

0

provides a measure of the total amplitude or energy contribution from each frequency value, in which T denotes the time duration of the data. In comparison with Equation 8.2, the Hilbert amplitude spectrum H(ω,t) provides an extra dimension by including time t in motion frequency and is thus more general than the Fourier amplitude spectrum F(Ω). While the marginal amplitude spectrum h(ω) provides information similar to the Fourier amplitude spectrum, its frequency term is different. The Fourier-based frequency (Ω) is constant over the sinusoidal harmonics persisting through the data window, as seen in Equation 8.1, while the HHT-based frequency ω varies with time based on Equation 8.6. As the Fourier transformation window length reduces to zero, the Fourier-based frequency (Ω) approaches the HHT-based frequency (ω). The Fourier-based frequency is locally averaged and not truly instantaneous, for it depends on window length, which is controlled by the uncertainty principle and the sampling rate of data. Recordings that are stationary and linear can typically be decomposed or represented by a series of time-independent frequency waves through the Fourier-based approach in Equation 8.1. If the jth IMF component, i.e., Cj(t) in Equation 8.5, corresponds to a Fourier component with a sine function at a time-independent frequency, the Hilbert transform of the sine function, i.e., Yj(t) in Equation 8.5, can be found to equal the cosine function at the same frequency in opposite sign. Because the sign can be changed with adding a constant phase, the above analysis essentially leads to the consistence between Fourier- and HHT-based approaches in general,

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and Fourier and marginal Hilbert amplitude spectra in particular, in characterizing linear phenomena with time-independent frequency. For recordings that are nonstationary and nonlinear, such as large-magnitude earthquakes, many studies have showed [21] that the marginal Hilbert amplitude spectra can truthfully represent the nonlinear data in comparison with Fourier amplitude spectra. As a result, an HHT-based approach for characterizing nonlinear site amplification can be proposed that is similar to a Fourier-based approach. The HHT-based factor of site amplification (FH) is defined

( )

FH s ω =

hs2,h1 + hs2,h 2 hr2,h1 + hr2,h 2

.

(8.9)

While this HHT-based factor could provide an alternative insight in characterizing and quantifying the influences of nonlinear soil in site amplification, the role of damping in nonlinear site responses is not discerned from the general features of soil nonlinearity, which should be implicitly involved in the factor. To single out the damping factor from the HHT-based factor in Equation 8.9, which is associated with amplitude aj(t) in Equation 8.7 and Equation 8.8, the physical meanings of the jth IMF component that forms the amplitude aj(t) in Equation 8.5 is first examined below. Since all the IMF components are extracted from acceleration records that are the result of seismic waves generated by a seismic source and propagating in the earth, they should reflect the wave characteristics inherent to the rupture process and the earth medium properties. Indeed, with the aid of a finite-fault inversion method, Zhang et al. [25] examined signatures of the seismic source of the 1994 Northridge earthquake in the ground acceleration recordings. That study looks over only the second to fifth IMF components because they are much larger in amplitude than the remaining higher-order, low-frequency IMF components. The first IMF component was not investigated in that study because it contains information that is not simply or easily related to the seismic source (e.g., wave scattering in the heterogeneous media). That study shows that the second IMF component is predominantly wave motion generated near the hypocenter, with high-frequency content that might be related to a large stress drop associated with the initiation of the earthquake. As one progresses from the second to the fifth IMF component, there is a general migration of the source region away from the hypocenter with associated longer-period signals as the rupture propagates. In addition, that study shows that some IMF components (e.g., the fifth IMF) can exhibit motion features reflecting the influences of nonlinear site condition. Because of the significance of IMF components that directly relate amplitudes aj(t), Equation 8.5 can be rewritten as n

n

X (t ) = ℜ

∑ j =1

© 2005 by Taylor & Francis Group, LLC

a j (t )e

iθ j ( t )

=ℜ

∑ Λ (t)e j

j =1

− ϕ j ( t )+ iθ j ( t )

(8.10)

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169

where time-dependent amplitudes Λ j(t) can be interpreted as the source-related intensity, ϕj(t) are the exponential factors characterizing the time-dependent decay of the waves in the jth IMF component due to damping, and a j (t ) = Λ j (t )e

− ϕ j (t )

.

(8.11)

Similar to the description of the instantaneous frequency in Equation 8.6, the instantaneous damping factor can be defined as η j (t ) =

d ϕ j (t ) . dt

(8.12)

With the aid of Equation 8.11, the Hilbert damping spectrum can be found as n

D(ω , t ) =

∑

n

η j (t ) =

j =1

 a j (t )

(t )  Λ j

∑ − a (t) + Λ (t)  . j =1

j

(8.13)

j

The marginal Hilbert damping spectrum is T

d (ω ) =

∫

n

D(ω , t )dt =

0

T

∑∫ j =1 0

(t )   a j (t ) Λ + j  dt = d a (ω ) + d Λ (ω ) . −  a j (t ) Λ j (t ) 

(8.14)

Equation 8.14 indicates that the marginal Hilbert damping spectrum consists of two terms: one is from the time-dependent amplitudes aj(t) that are related to marginal and Hilbert amplitude spectra, and the other is from source-related intensity, i.e., time-dependent amplitudes Λ j(t). It is of interest to note that the definition of instantaneous damping factor in Equation 8.12 and subsequent spectra in Equation 8.13 and Equation 8.14 are different from those in Salvino [26] and Loh et al. [27]. For recordings of impulse-induced or ambient linear vibration responses, some IMF components can be extracted from the data that are related to certain vibration modes [28–30]. Consequently, Λj(t) are constant and ηj(t) are proportional to the damping ratio and damped frequency. The modal damping ratio can then be found. This is essentially the same as those in Salvino [26] and Loh et al. [27], if the latter can judicially relate the IMF components to the vibration/wave modes. For recordings to an earthquake, Λj are functions of time and unknown, which are dependent upon the seismic source. Their influences in the site amplification, however, can be removed if two recordings at soil and referenced sites are used. Similar to the HHT-based factor of site amplification, the difference of marginal Hilbert damping spectra at soil and referenced sites, or HHT-based factor of site damping, could approximately eliminate the influences of the source that is associated

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The Hilbert-Huang Transform in Engineering

c1

0.5 0

c2

−0.5 0.5 0 −0.5

c3

0.02 0 −0.02

c4

0.02 0

c5

−0.02

×10−3

10 5 0 0

1

2

3

4

5 Time (sec)

6

7

8

9

10

FIGURE 8.4 The five IMF components of the recording in Figure 8.1.

with Λj and thus provide essentially the characterization of the damping in the soil site. The HHT-based factor of site damping can be found as d ∆ (ω ) = [d s ,h1 (ω ) − dr ,h1 (ω )]2 + [d s ,h 2 (ω ) − dr ,h 2 (ω )]2 (8.15) ≈ [d

a s ,h1

(ω ) − d

a r ,h1

(ω )] + [d 2

a s ,h 2

(ω ) − d

a r ,h 2

(ω )]

2

where use has been made in the last approximation of the fact that the source-related damping terms at the soil and referenced sites are approximately equal, i.e., dsΛ (ω) drΛ (ω). Finally, comparing the HHT-based factors of site damping from the mainshock and the aftershock could help quantify the influences of nonlinear soil damping in site responses. To illustrate the HHT-based characterization of nonlinearity, the hypothetical record in Figure 8.1 is analyzed again. Figure 8.4 shows the five IMF components decomposed from the data by EMD. The first and second components (c1 and c2) capture the noise and primary waveform, while the other three (c3 to c5), with negligible amplitudes, represent the numerical error in the EMD process. Comparing Figure 8.3 and Figure 8.4 suggests that some IMF components can be not only more physically meaningful than the Fourier components, they can also be in principle used to explore the damping factor with the use of, e.g., the consecutive peak values

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Hilbert Spectrum (Hypothetical Water Wave)

Frequency (Hz)

20 18

0.9

16

0.8

14

0.7

12

0.6

10

0.5

8

0.4

6

0.3

4

0.2

2

0.1

0

0

1

2

3

4

5 6 Time (sec)

7

8

9

10

FIGURE 8.5 Hilbert amplitude spectra of the recording in Figure 8.1.

and corresponding time elapse in the second IMF component, if that component is related to a free-vibration response or forced response at a certain mode. The Hilbert amplitude spectrum in Figure 8.5 shows a clear picture of temporal–frequency energy distribution of the data, i.e., primary waves with frequency dependence modulated around 1 Hz and bounded by 0.5 Hz and 1.5 Hz, noise at 15 Hz, and the decaying energy of the primary waves with the color changing from the red/yellow at the beginning to the dark blue at the end of the record. In contrast, the Fourier amplitude spectrum in Figure 8.2 not only loses the information pertaining to temporal characteristics of the motion, but, more important, it also distorts the information of the record by introducing higher-order harmonics, notably at 2 Hz and 3 Hz. For comparison, the marginal amplitude spectrum of the recording is also plotted in Figure 8.2, showing truthfully the energy distribution of the motion in frequency. Figure 8.6 shows the marginal Hilbert damping spectrum, calculated without using any smooth function, which was not so done in the calculation of the above marginal Hilbert amplitude spectrum with the use of Hilbert-Huang Transformation Toolbox [31]. While oscillation of the curve in Figure 8.6 is originally due to the numerical calculation of a· j(t) and a· j(t)/aj(t) that subsequently influences the computation of the spectra in Equation 8.13 and Equation 8.14, the estimated mean damping factor at frequency 0.5 to 1 Hz is around the true value of 0.2. Compared with no

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Marginal Damping Spectrum

Wave Data–Marginal Damping Spectrum

100

10−1

10−2

100

101 Frequency (Hz)

FIGURE 8.6 Marginal Hilbert damping spectrum of the recording in Figure 8.1.

damping in the record at frequency larger than 1.5 Hz, the very small mean damping factor in Figure 8.6 (about 0.04, lower at a factor of five than the damping factor of waves) due to the aforementioned unavoidable, cumulated numerical error is still acceptable. It is believed that the oscillated curve in Figure 8.6 can be improved by replacing it with the instantaneous mean curve. The mean curve can be found by many methods, one of which is the use of the summation of all the IMF components, excluding the first IMF component, that are extracted from the data of the oscillated curve with one sifting process. The large damping factors at around 1.5 Hz are due to the numerical error caused by the transition of two damping factors from 0.2 to 0.04, although such abrupt damping change is unlikely in practice. The large damping factor below 0.5 Hz is caused by the high-order, low-frequency IMF components (primarily from the third to fifth IMFs). The error in overestimating damping factor at very low frequency can be theoretically minimized if high-order, low-frequency IMF components with very small amplitudes are judicially not used in the damping calculation.

8.5 APPLICATIONS TO 2001 NISQUALLY EARTHQUAKE DATA In this section, the HHT-based approach is used to analyze the recordings of the M6.8 mainshock and the ML3.4 aftershock of the 2001 Nisqually earthquake at SDS and LAP. The SDS is a station on artificial fill with nearby liquefaction, and the average shear-wave velocity in the top 30 m for the soft soil is estimated as Vs30 =

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Mainshock Time History Sds1

Acceleration (g)

0.3 0.2 0.1 0 −0.1 −0.2

0

5

10

15

20

25

30

35

40

30

35

40

1st Component Sds1

Acceleration (g)

0.1 0.05 0 −0.05 −0.1

0

5

10

15

20 Time (sec)

25

FIGURE 8.7 (a, top) NS-acceleration recording and (b, bottom) its first IMF of the Nisqually mainshock at SDS (soft soil).

148 m/s. The LAP is located over a stiff soil with Vs30 = 367 m/s. To examine the soil nonlinearity from recordings at the two stations, recordings at SEW are used as referenced ones, because Vs30 = 433 m/s at SEW is within the range of Vs30 values for typical rock sites in the western United States. Previous Fourier-based studies (e.g., [17]) suggest that SDS experienced strong soil nonlinearity during the mainshock while the LAP did not, with recordings at SEW as a reference.

8.5.1 DETECTION

OF

NONLINEAR SOIL SITES

Figure 8.7a shows the NS-acceleration recording of the 2001 Nisqually earthquake at SDS. The one-sided cusped waves and high-frequency spikes in the S-coda waves, notably between 20 sec and 25 sec, are similar to the waveform with sharper crests and rounded-off troughs in Figure 8.1, and have been concluded [17] to be symptomatic of nonlinear response at the soft soil site. Indeed, the high-frequency spikes are believed to appear only in recordings that are close to the locations where strong nonlinearity occurred (e.g., [16]). Therefore, singling out the spikes will help detect strongly nonlinear sites. Frankel et al. [17] used Fourier-based high-frequency band-pass filtering (10 to 20 Hz) to identify the

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Mainshock Time History Sds2

Acceleration (g)

0.4 0.2 0 −0.2 −0.4

0

5

10

15

20

25

30

35

40

30

35

40

1st Component Sds1 0.06 Acceleration (g)

0.04 0.02 0 −0.02 −0.04 −0.06

0

5

10

15

20 Time (sec)

25

FIGURE 8.8 (a, top) EW-acceleration recording and (b, bottom) its first IMF of the Nisqually mainshock at SDS (soft soil).

spikes. It should be noted that there exist other tools to identify the spikes. For example, Hou et al. [32] used a wavelet-based approach to characterize the spikes from nonlinear vibration recordings in the vicinity of a damaged-structure location subject to a severe earthquake. The disadvantages of these approaches, however, reside with the subjective selection of frequency band in a Fourier-based approach and subjective selection of another wavelet in a wavelet-based approach, among others. This study presents the effectiveness of the HHT-based approach in identifying the spikes. Figure 8.7b depicts the first IMF component of the NS-component of motion, clearly showing the two largest spikes between 20 sec and 25 sec. While the spikes in the recording of Figure 8.7a can be visualized without using any tools, the first IMF component in Figure 8.7b is shown simply for validation of the HHT-based approach in identifying the spikes. Indeed, the EW-component of the same recording in Figure 8.8a does not clearly show the spikes between 20 sec and 25 sec. The corresponding first IMF component in Figure 8.8b is, however, able to reveal them explicitly, suggesting that the first IMF component is effective at detecting the high-frequency spikes. This study also analyzed the horizontal recordings of the ML3.4 aftershock at the same location in Figure 8.9 and Figure 8.10, in which one does not observe the spikes in the S-coda waves in the corresponding first IMF

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Acceleration (g)

1

175

Aftershock Time History Sds1

×10

0.5 0 −0.5 −1

Acceleration (g)

2

0

5

10

×10−4

15

20

25

30

35

40

30

35

40

1st Component Sds1

1 0 −1 −2

0

5

10

15

20 Time (sec)

25

FIGURE 8.9 (a, top) NS-acceleration recording and (b, bottom) its first IMF of the Nisqually aftershock at SDS (soft soil).

components. These observations partially confirm that the nonlinearity-related spikes can be detected via the first IMF component. It should be pointed out here that the high-frequency spikes before the S-coda waves in the first IMF component (such as those before 20 sec in Figure 8.7b, Figure 8.8b, Figure 8.9b, and Figure 8.10b), which also show up in the Fourier-based high-frequency band-pass (10 to 20 Hz) approach, may not be simply, directly related to the soil nonlinearity. The aforementioned nonlinearity-related spikes can also be identified via Hilbert amplitude spectra. Specifically, the Hilbert amplitude spectra of the mainshock in Figure 8.11 and Figure 8.12 show two beams of high-frequency energy (5 Hz and above) between 20 sec and 25 sec, when the dominant energy of the motion is primarily below 5 Hz. In contrast, those of the aftershock in Figure 8.13 and Figure 8.14 do not show the distinguishing high-frequency energy beams in the same time period. This suggests that the first IMF component and Hilbert amplitude spectra can be used effectively to single out nonlinearity-related high-frequency spikes. Accordingly, the results will help us, at the first glance, to detect strong nonlinear soil sites.

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−3

Acceleration (g)

1

Aftershock Time History Sds2

×10

0.5 0 −0.5 −1

0

5

10

15

−4

8

×10

20

25

30

35

40

30

35

40

1st Component Sds2

Acceleration (g)

6 4 2 0 −2 −4

0

5

10

15

20 Time (sec)

25

FIGURE 8.10 (a, top) EW-acceleration recording and (b, bottom) its first IMF of the Nisqually aftershock at SDS (soft soil).

8.5.2 HHT-BASED FACTOR

OF

SITE AMPLIFICATION

Figure 8.15a shows the HHT-based factors of site amplification of the mainshock and aftershock at SDS. In calculating the factors in Equation 8.9, the correction for 1/R geometrical spreading in the recordings at SDS and SEW is not carried out since the hypocentral distances for the sites under investigation are similar. In addition, the marginal Hilbert amplitude spectra are not smoothed in the calculation, for the nonsmoothed spectra more clearly show the characteristics of the HHT-based approach. Examining Figure 8.15a shows the following: •

The profile of the HHT-based factor in the frequency band up to 2.5 Hz (referred to as the low-frequency range) is generally downshifted in frequency from the aftershock to the mainshock, with an average shift of approximately 0.7 Hz. This downshift is exemplified by the peaks of the aftershock at 0.95 Hz and 2.1 Hz downshifted to those of the mainshock at 0.25 Hz and 1.4 Hz, respectively. The degree of soil nonlinearity under the mainshock motion can be quantified by the frequency downshift.

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Nisqually Mainshock – 2-D Hilbert Spectrum - SDSns 0.18 101

0.16 0.14

Frequency (Hz)

0.12 0.1 100

0.08 0.06 0.04 0.02

10−1

0

5

10

15

20 Time (s)

25

30

35

40

FIGURE 8.11 Hilbert amplitude spectra of NS-acceleration recording of the 2001 Nisqually earthquake mainshock at SDS (soft soil).

•

•

The profile of the HHT-based factor in the frequency band 2.5 to 7 Hz (intermediate-frequency range) is generally reduced in amplitude from the aftershock to the mainshock, with an average difference of approximately a factor of 0.43. Note that the measure is carried out in the way that the averaged factor of 7 for the aftershock is reduced to 3 for the mainshock in frequency range 3 to 4 Hz. There is no evidence to support a difference in HHT-based factor from about 7 Hz up (high-frequency range) between the mainshock and aftershock.

To facilitate understanding of these observations, this study analyzes the HHT-based factor of the mainshock and aftershock at LAP in Figure 8.16a, which shows the following: •

In the low-frequency range (below 2.5 Hz), Figure 8.16a shows a downshift profile in both frequency and amplitude from the aftershock to mainshock that is similar to Figure 8.15a in the low to intermediate frequency range, but the former shows a smaller shift (about 0.1 Hz in

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Nisqually Mainshock – 2-D Hilbert Spectrum - SDSnew 0.11 0.1

101

0.09

Frequency (Hz)

0.08 0.07 0.06 100

0.05 0.04 0.03 0.02 0.01

10

−1

0

5

10

15

20 Time (s)

25

30

35

40

FIGURE 8.12 Hilbert amplitude spectra of EW-acceleration recording of the 2001 Nisqually earthquake mainshock at SDS (soft soil).

•

frequency and a factor of 0.2) than the latter (about 0.7 Hz and a factor of 0.43). In the intermediate to high frequency range, there is almost no difference in the two factors between the mainshock and the aftershock.

Comparison of the HHT-based factors at SDS and LAP suggests that SDS had severe soil nonlinearity during the mainshock and LAP had slight soil nonlinearity. Site LAP can also be regarded as having no soil nonlinearity under the mainshock if the aforementioned small downshift in both frequency and amplitude in the low-frequency range is the result of variation of data collection and sampling.

8.5.3 INFLUENCES

OF

WINDOW LENGTH

OF

DATA

The above results were obtained on the basis of 40-sec record lengths. It can be seen from Figure 8.7 through Figure 8.14 that the characteristics of the motion during the 40 sec change from one time interval to another. The noticeable features in different sections of the 40 sec (see Figure 8.7a and Figure 8.8a) are the dominant P-wave signals with high frequencies and low amplitude from 0 to 10 sec, followed by the dominant S-wave signals with intermediate frequencies and high amplitude from 10 to 20 sec, and finally the surface and coda waves with low frequencies and

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−4

×10

Nisqually Aftershock – 2-D Hilbert Spectrum - SDSns

6 101 5

Frequency (Hz)

4

3 100 2

1

10−1

0

5

10

15

20 Time (s)

25

30

35

40

FIGURE 8.13 Hilbert amplitude spectra of NS-acceleration recording of the 2001 Nisqually earthquake aftershock at SDS (soft soil).

intermediate amplitude from 20 to 40 sec. Accordingly, the HHT-based factor calculated over 40 sec is a characteristic averaged in frequency and amplitude, i.e., mixing low to high frequencies and amplitudes. It is also noticed that the soil nonlinearity is likely most prevalent in the strong S-wave motion. To characterize the soil nonlinearity precisely, one should select an appropriate window of motion for analysis in which the soil nonlinearity shows up most strongly. However, the selection of an appropriate window is subjective. Nevertheless, this study next examines the influence of window length on the HHT-based factor by selecting two windows, 0 to 10 sec and 10 to 20 sec. In the first window, 0 to 10 sec, where the P-wave signals are dominant with relatively small amplitudes and high frequencies, the soil is likely linear under both mainshock and aftershock motions, which should lead to the same factor in a broad spectrum of frequency for the mainshock and aftershock. This is verified in Figure 8.17a, except for a large difference in amplitude between 3 Hz and 6 Hz. The difference might be caused by unknown factors such as a lack of recorded frequency content in the referenced site or indeed by a change of physical properties in the soil. Further research is needed along this line. In the second window, 10 to 20 sec, where the S-wave signals are dominant with large amplitudes and intermediate frequencies, the soil is likely strongly nonlinear under

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−4

×10

Nisqually Aftershock – 2-D Hilbert Spectrum - SDSew

5.5 101

5 4.5

Frequency (Hz)

4 3.5 3 10

0

2.5 2 1.5 1 0.5

10−1

0

5

10

15

20 Time (s)

25

30

35

40

FIGURE 8.14 Hilbert amplitude spectra of EW-acceleration recording of the 2001 Nisqually earthquake aftershock at SDS (soft soil).

the mainshock. The downshift in both amplitude and frequency from the HHT-based factors between mainshock and aftershock is clearly seen in Figure 8.17b.

8.5.4 COMPARISON OF HHT- AND FOURIER- BASED FACTORS FOR SITE AMPLIFICATION To further illustrate the characteristics of the HHT approach, this study compares the HHT-based factors of site amplification at SDS shown in Figure 8.15a with the Fourier-based ones shown in Figure 8.15b (i.e., Figure 8.7 in Frankel et al. [17]). In the low-frequency range, Figure 8.15b shows a frequency-downshift profile from the aftershock to mainshock that is similar to Figure 8.15a, but the former shows a smaller shift (about 0.2 Hz) than the latter (about 0.7 Hz). Because of the averaging characteristic in Fourier spectral analysis, as indicated in Section 8.3, the frequency downshift measured from the HHT-based factors in Figure 8.15a may give a more truthful indication of the soil nonlinearity than that measured from Fourier-based factors in Figure 8.15b. In addition, the factor in the low-frequency range in Figure 8.15a is generally somewhat larger than the factors in Figure 8.15b. In the intermediate-frequency range, Figure 8.15b shows an amplitude-reduction profile from the aftershock to mainshock that is similar to Figure 8.15a, but the

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Nisqually Earthquake - Marginal Spectral Ratio (SDS/SEW)

1

Spectral Ratio

10

0

10

Aftershock Mainshock −1

10

10−1

100 Frequency (Hz)

1

10

Nisqually Earthquake - Fourier Spectral Ratio (SDS/SEW)

1

Spectral Ratio

10

100

Aftershock Mainshock

10−1 10−1

100 Frequency (Hz)

1

10

FIGURE 8.15 (A) HHT-based factor for site amplification at SDS (soft soil) for mainshock and aftershock of the 2001 Nisqually earthquake. (B) Fourier-based factor for site amplification at SDS (soft soil) for mainshock and aftershock of the 2001 Nisqually earthquake.

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Nisqually Earthquake - Marginal Spectral Ratio (LAP/SEW)

1

Spectral Ratio

10

0

10

Aftershock Mainshock

10−1 10−1

100 Frequency (Hz)

101

Nisqually Earthquake - Fourier Spectral Ratio (LAP/SEW)

Spectral Ratio

101

100

Aftershock Mainshock

10−1 10−1

100 Frequency (Hz)

101

FIGURE 8.16 (A) HHT-based factor for site amplification at LAP (stiff soil) for mainshock and aftershock of the 2001 Nisqually earthquake. (B) Fourier-ased factor for site amplification at LAP (stiff soil) for mainshock and aftershock of the 2001 Nisqually earthquake. © 2005 by Taylor & Francis Group, LLC

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Nisqually Earthquake - Marginal Spectral Ratio (SDS/SEW) ==> 0–10s

2

Spectral Ratio

10

1

10

10

0

Aftershock Mainshock

−1

0

10

10 Frequency (Hz)

101

Nisqually Earthquake - Marginal Spectral Ratio (SDS/SEW) ==> 10–20s

2

Spectral Ratio

10

1

10

10

0

Aftershock Mainshock −1

10

0

10 Frequency (Hz)

101

FIGURE 8.17 (A) HHT-based factor for site amplification with the use of recordings in window 0 to 10 sec at SDS (soft soil) for mainshock and aftershock of the 2001 Nisqually earthquake. (B) HHT-based factor for site amplification with the use of recordings in window 10 to 20 sec at SDS (soft soil) for mainshock and aftershock of the 2001 Nisqually earthquake. © 2005 by Taylor & Francis Group, LLC

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Nisqually Earthquake - Diﬀerence of Damping Factor (SDS-SEW) Aftershock Mainshock

Diﬀerence of Damping Factor

10−1

10−2

0

2

4

6

8

10 12 Frequency (Hz)

14

16

18

20

FIGURE 8.18 Difference of marginal Hilbert damping spectra at SDS and SEW for mainshock and aftershock of the 2001 Nisqually earthquake.

former shows a smaller reduction (about a factor of 0.2) than the latter (about a factor of 0.43). In the high-frequency range, Figure 8.15b shows a significant increase in the Fourier-based factor from the aftershock to mainshock, while Figure 8.15a does not. Following the discussion in Section 8.3 and numerical illustration in Figure 8.1 and Figure 8.2, the high-frequency content in the Fourier amplitude spectra may be influenced by higher-order harmonics used to represent a nonlinear waveform, which consequently increases the Fourier-based factor of the mainshock in the high-frequency range. This study also compares HHT- and Fourier-based factors at LAP in Figure 8.16a and b. Almost no fundamental difference in the factors is observed in terms of overall profile, amplitude value, frequency downshift, and amplitude reduction between the mainshock and aftershock, indicating that the two approaches are essentially consistent with each other in estimating linear or approximately linear site amplification.

8.5.5 HHT-BASED FACTOR

FOR

SITE DAMPING

Figure 8.18 shows the HHT-based factors for site damping at SDS calculated by Equation 8.15, an alternative perspective of soil nonlinearity from recordings. It

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reveals that the site damping during the mainshock is much larger than that in the aftershock at frequency 0.5 to 5 Hz, suggesting that strong soil nonlinearity occurred during the mainshock in this frequency band. The increased damping will decrease the amplified magnitude of seismic wave responses through the nonlinear soil and thus reduce the site amplification factor. This can be confirmed from Figure 8.15a, which shows that the HHT-based factor for site amplification is observably reduced for the mainshock from the aftershock in the similar frequency band of 0.5 to 7 Hz. Figure 8.18 also shows that the second largest increased site damping for the mainshock is in the frequency band 8 to 16 Hz. However, the HHT-based factors for site amplification in Figure 8.15a do not appear to change between the mainshock and aftershock. This can be explained as follows: On the one hand, the increased site damping for the mainshock over the frequencies 8 to 16 Hz will reduce the seismic wave responses in the same frequency band. Note that the soil nonlinearity typically occurs under large-amplitude seismic waves incident to the soil layer, and that the amplitude of the motion changes with time. This suggests that the soil nonlinearity is likely most prevalent in the strong S-wave motion or following surface or S-coda waves, but not in the P-wave motion during the mainshock. Figure 8.7a and Figure 8.8a reveal that except for the abnormal high-frequency spikes between 20 sec and 25 sec, the mainshock accelerations after the S-wave arrival at about 10 sec contain less high-frequency motion than does the aftershock in general, and in the high-frequency band of about 8 to 16 Hz in particular. This fact is consistent with the increased damping of the nonlinear soil during the mainshock in the high-frequency band of 8 to 16 Hz, which damped out the high-frequency wave responses in the recordings. On the other hand, the soil nonlinearity indeed introduces large-amplitude high-frequency spikes from 20 to 25 sec in the mainshock recordings in Figure 8.7a and Figure 8.8a. Together with the fact that the HHT-based factors in Figure 8.15a show the averaged characteristics of soil linearity and nonlinearity, these observations suggest that the overall high-frequency (around 8 to 16 Hz) content of the mainshock may be equivalent to that of the aftershock. This yields the same factors in Figure 8.15a in the frequency range 8 to 16 Hz. For further clarification, the HHT-based factor for site damping at LAP is examined. Figure 8.19 shows that the profiles of soil damping are essentially no different between the mainshock and aftershock, suggesting that site LAP is linear during the mainshock. This can be further verified from the HHT-based site-amplification factors shown in Figure 8.16a.

8.6 CONCLUDING REMARKS AND DISCUSSION This study investigates the role of the HHT method in effectively detecting soil nonlinearity and precisely quantifying soil nonlinearity in terms of site amplification and damping. In particular, it reveals that the first IMF component can help identify the high-frequency spikes related to soil nonlinearity, which can also be further confirmed by Hilbert amplitude spectra.

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Nisqually Earthquake - Diﬀerence of Damping Factor (LAP-SEW) Aftershock Mainshock

Diﬀerence of Damping Factor

10−1

10−2

0

2

4

6

8

10 12 Frequency (Hz)

14

16

18

20

FIGURE 8.19 Difference of marginal Hilbert damping spectra at LAP and SEW for mainshock and aftershock of the 2001 Nisqually earthquake.

The HHT-based factor of site amplification is defined as the ratio of marginal Hilbert amplitude spectra, similar to the Fourier-based factor that is the ratio of Fourier amplitude spectra. The HHT-based factor has the following relationships with Fourier-based one: • •

•

•

The HHT-based factor is essentially equivalent to the Fourier-based factor in quantifying linear site amplification. The HHT-based factor is more effective than the Fourier-based factor in quantifying soil nonlinearity in terms of frequency downshift in the low-frequency range and amplitude reduction in the intermediate-frequency range. In the low to intermediate frequency range, the HHT-based factor is generally larger than the Fourier-based factor, which is also dependent upon the window length in which the recordings are used. For nonlinear site amplification, the Fourier-based factor is increased with respect to the linear one at high frequencies; this phenomenon is not seen in the HHT-based factor. This difference may be due to the introduction of high frequencies in Fourier spectral analysis in the characterization of a nonlinear waveform.

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The HHT-based factor for site damping can be extracted from the HHT-based factor for site amplification and used as an alternative index to measure the influences of soil nonlinearity in seismic wave responses at sites. It should be pointed out that the results from this study rely mainly on the use of advanced signal processing techniques to explore and then quantify the signature of soil nonlinearity from recordings. They must, therefore, be validated by model-based simulation. Recently, significant advances have been made in borehole data collection and simulation techniques. These include data from 17 borehole arrays in Southern California [33] and from the Port Island vertical array for the 1995 Hyogoken Nanbu earthquake, geophysical data including the S-wave velocity profile in the top layer(s) at key strong motion station sites (Rosrine 2002 at http://geoinfo.usc.edu/rosrine, USGS open file reports), and simulated broadband ground motion for scenario earthquakes including nonlinear soil effects [34]. The above information should allow the further validation of the observations and results from this study.

ACKNOWLEDGMENTS The author would like to express sincere gratitude to Norden E. Huang at NASA; Stephen Hartzell, Authur Frankel, and Erdal Safak at USGS; Lance VanDemark from Colorado School of Mines; and Yuxian Hu from China Seismological Bureau and Jianwen Liang from Tianjin University of China for providing data, Fourier analysis and calculations, and more important, constructive suggestions. This work was supported by the National Science Foundation with Grant Nos. 0085272 and 0414363, and by the US-PRC Researcher Exchange Program administered by Multidisciplinary Center for Earthquake Engineering Research. The opinions, findings, and conclusions expressed herein are those of the author and do not necessarily reflect the views of the sponsors.

REFERENCES 1. Frankel, A., C. Mueller, T. Barnhard, D. Perkins, E. Leyendecker, N. Dickman, S. Hanson, and M. Hopper (2000), “USGS national seismic hazard maps,” Earthquake Spectra, v. 16, pp. 1–19. 2. Kramer, S.L. (1996) Geotechnical Earthquake Engineering, Prentice-Hall, Inc. Upper Saddle River, NJ. 3. Borcherdt, R.D. (2002) “Empirical evidence for site coefficients in building code provisions,” Earthquake Spectra, Vol. 18, pp. 189–217. 4. Hartzell, S. (1998) “Variability in nonlinear sediment response during the 1994 Northridge, California, earthquake,” Bull. Seism. Soc. Am. Vol. 88, 1426–1437. 5. O’Connell, D.R.H. (1999) “Influence of random-correlated crustal velocity fluctuations on the scaling and dispersion of near-source peak ground motions, Science, Vol. 283, pp. 2045–2050, March 26. 6. Yoshida, N. and S. Iai (1998) “Nonlinear site response and its evaluation and prediction,” The Effects of Surface Geology on Seismic Motion, (Irikura, Kudo, Okada and Sasatani, eds.), Balkema, Rotterdam, pp. 71–90.

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7. Huang, N.E., S. Zheng, S.R. Long, M.C. Wu, H,H. Shih, Q. Zheng, N-C. Yen, C.C. Tung, and M.H. Liu (1998). “The empirical mode decomposition and Hilbert spectrum for nonlinear and nonstationary time series analysis." Proc. R. Soc Lond., A 454 903–995. 8. Hardin, B.O. and V.P. Drnevich (1972a) “Shear modulus and damping in soils: measurement and parameter effects,” J. Soil. Mech. Foundations Div. ASCE, 98, 603–624. 9. Hardin, B.O. and V.P. Drnevich (1972b) “Shear modulus and damping in soils: design equation and curves measurement and parameter effects,” J. Soil. Mech. Foundations Div. ASCE, 98, 667–692. 10. Erdik, M. (1987) Site response analysis. In Strong Ground Motion Seismology (Erdik, M. and Toksoz, M., eds.), D. Reidel Publishing Company, Dordrecht, 479–543. 11. Vucetic, M. and R. Dobry (1991) “Effect of soil plasticity on cyclic response, J. Geotech. Eng., 117, 89–107. 12. Silva, W., S. Li, B. Darragh, and N. Gregor (1999). “Surface geology based motion amplification factors for the San Francisco Bay and Los Angeles areas,” A Pearl Report. 13. Field, E.H., P.A. Johnson, I.A. Beresnev, and Y. Zeng (1997) “Nonlinear ground-motion amplification by sediments during the 1994 Northridge earthquake, Nature, 390, 599–602. 14. Beresnev, I.A., E.H. Field, P.A. Johnson, and K.E.A. Van Den Abeele (1998) “Magnitude of nonlinear sediment response in Los Angeles basin during the 1994 Northridge, California, earthquake, Bull. Seis. Soc. Am., Vol. 88, pp. 1097–1084. 15. Joyner, W.B. (1999). “Equivalent-linear ground-response calculations with frequency-dependent damping,” Proceedings of the 31st Joint Meeting of the US-Japan Panel on Wind and Seismic Effects (UJNR), Tsukuba, Japan, May 11–14, 1999, pp. 258–264. 16. Bonilla, L.F., D. Lavallee, and R.J. Archuleta (1998). “Nonlinear site response: laboratory modeling as a constraint for modeling accelerograms,” The Effects of Surface Geology on Seismic Motion, (Irikura, Kudo, Okada and Sasatani, eds.), Balkema, Rotterdam. 17. Frankel, A.D., D.L. Carver, and R.A., Williams (2002) “Nonlinear and linear site response and basin effects in Seattle for the M6.8 Nisqually, Washington, Earthquake,” Bull. Seism. Soc. Am., v. 92, pp. 2090–2109. 18. Safak, E. (1997). “Models and methods to characterize site amplification from a pair of records,” Earthquake Spectra, Vol. 13, pp. 97–129. 19. Priestley, M.B. (1981). Spectral Analysis and Time Series, Vol. 2, Multivariate Series, Prediction and Control, Academic Press, London, 1981. 20. Zhang, R., S. Ma, E. Safak, and S. Hartzell (2003b) “Hilbert-Huang transform analysis of dynamic and earthquake motion recordings,” ASCE Journal of Engineering Mechanics, Vol. 129, No. 8, pp 861–875. 21. Huang, N.E., Z. Shen, and R.S. Long, (1999). “A new view of nonlinear water waves: Hilbert Spectrum.” Annu. Rev. Fluid Mech. 31,417–457. 22. Worden, K. and Tomlinson, G.R. (2001) Nonlinearity in Structural Dynamics: Detection, Identification and Modeling, Institute of Physics Publishing, Bristol and Philadelphia. 23. Zhang, R., L. VanDemark, J. Liang, and Y.X. Hu (2003) “Detecting and quantifying nonlinear soil amplification from earthquake recordings,” Advances in Stochastic Structural Dynamics (Zhu, Cai and Zhang, eds.), pp. 543–550.

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24. Clough, R.W. and J. Penzien (1993). Dynamics of Structures, Second Edition, McGraw-Hill, Inc., New York. 25. Zhang, R., S. Ma, and S. Hartzell (2003) “Signatures of the seismic source in EMD-based characterization of the 1994 Northridge, California, earthquake recordings,” Bulletin of the Seismological Society of America, Vol. 93, No. 1, pp 501–518. 26. Salvino, L.W. (2001) “Evaluation of structural response and damping using empirical mode analysis and HHT,” in CD ROM, 5th World Multiconference on Systemics, Cybernetics and Informatics, July 22–25, 2001, Orlando, Florida. 27. Loh, C.H., T.C. Wu and N.E. Huang (2001). Application of the empirical mode decomposition-Hilbert spectrum method to identify near-fault ground-motion characteristics and structural responses, Bull. Seism. Soc. Am. 91, 1339–1357. 28. Yang, J.N., Y. Lei, S. Pan and N. Huang (2002a). “System identification of linear structures based on Hilbert-Huang spectral analysis. Part I: Normal modes,” Earthquake Eng. Structural Dynamics, Vol. 32, pp. 1443–1467. 29. Yang, J.N., Y. Lei, S. Pan and N. Huang (2002b). “System identification of linear structures based on Hilbert-Huang spectral analysis. Part II: Complex modes,” Earthquake Eng. Structural Dynamics, Vol. 32, pp. 1533–1554. 30. Xu, Y.L., S.W. Chen, and R. Zhang (2003) “Modal identification of Di Wang building under typhoon York using HHT method,” The Structural Design of Tall and Special Buildings, Vol. 12, No. 1, pp. 21–47. 31. Hilbert-Huang Transformation Toolbox (2000). Professional Edition V1.0, Princeton Satellite Systems, Inc., Princeton, New Jersey. 32. Hou, Z., M. Noori and R. St. Amand (2000) “Wavelet-based approach for structural damage detection,” ASCE Journal of Engineering Mechanics, 126(7), pp. 677–683. 33. Archuleta, R.J. and J.H. Steidl (1998). “ESG studies in the United States: results from borehole arrays,” The Effects of Surface Geology on Seismic Motion, (Irikura, Kudo, Okada and Sasatani, eds.)., Balkema, Rotterdam, pp. 3–14. 34. Hartzell, S., A. Leeds, A. Frankel, R.A. Williams, J. Odum, W. Stephenson and W. Silva (2002). “Simulation of broadband ground motion including nonlinear soil effects for a magnitude 6.5 earthquake on the Seattle fault, Seattle, Washington,” Bull. Seism. Soc. Am. Vol. 92, No. 2, pp. 831–853.

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9

Simulation of Nonstationary Random Processes Using Instantaneous Frequency and Amplitude from Hilbert-Huang Transform Ping Gu and Y. Kwei Wen

CONTENTS 9.1 Introduction ..................................................................................................191 9.2 Hilbert-Huang Transform (HHT).................................................................194 9.3 IMF Recombination Method .......................................................................197 9.4 Improved Wen-Yeh Method.........................................................................200 9.5 Conclusions ..................................................................................................203 Acknowledgments..................................................................................................207 References..............................................................................................................207 Appendix: Relation between Hilbert Marginal Spectrum and Fourier Energy Spectrum ......................................................................................209

9.1 INTRODUCTION With the ever-increasing power of the computer and of the capacity of computational mechanics, simulation has become an indispensable tool in engineering. It is more so in earthquake engineering since earthquake ground motions are highly random and structural behaviors are nonlinear and inelastic, making random vibration solutions 191

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difficult to obtain. Generally speaking, structural responses are functions of the entire ground excitation time history. Current codes use a scalar intensity measure such as spectral acceleration or displacement for the earthquake demand on structures. These measures do not reflect satisfactorily the effects of near source or effects due to higher modes and other important structural response behaviors caused by ground motions. For performance evaluation of complex structural systems, simulation of earthquake ground motions and time history response analysis have played a more and more important role. Many models for simulation based on earthquake records have been proposed. For example, a stationary white noise model was first proposed [1], followed by filtered white noise models to reflect the site characteristics [2, 3, 4]. A high-pass filter was later added to remove the undesirable zero frequency content of the power spectral density function [5]. To account for the intensity variation with time, a uniformly (amplitude) modulated random process was first used [6, 7]. Nonstationary models with frequency content variation with time — in particular, the long-duration acceleration pulses observed in many near-fault earthquake records — have also been developed. Saragoni and Hart [8] proposed a modulated filtered Gaussian white noise model with different intensity and frequency content in three consecutive time intervals. The arbitrary selection of time intervals and the resultant abrupt change of frequency content in this model, however, are difficult to justify physically. The oscillatory process described by an evolutionary power spectral density suggested by Priestley [9], allowing both amplitude and frequency content variation with time, has been widely used in earthquake ground motion simulation. For example, Lin and Yong [10] formulated an evolutionary Kanai-Tajimi [2, 3] model as a filtered pulse train, where the filter represents the ground medium and the pulse train represents intermittent ruptures at the earthquake source. Deodatis and Shinozuka [11] simulated the earthquake ground motions using an autoregressive (AR) model based on selected parametric forms of the evolutionary power spectral density for given earthquake records. Li and Kareem [12, 13] further extended the method using the fast Fourier transform (FFT) and digital filtering technique to make it more computationally efficient. Der Kiureghian and Crempien [14] proposed an evolutionary earthquake model composed of individually modulated band-limited white noise processes and estimated the parameters of evolutionary power spectral density by matching the moments of each component process. Papadimitriou [15] produced a parsimonious nonstationary model by applying a second-order filter with slowly varying parameters to time-modulated white noise. Conte and Peng [16] proposed to use Thomson’s spectrum to estimate the evolutionary power spectra and proposed a model for simulation based on this. By extending the energy principle of wavelet transform developed by Iyama and Kuwamura [17], Spanos and Failla [18] and Spanos et al. [19] proposed methods using the wavelet spectrum to estimate the evolutionary power spectral density. The main challenge in evolutionary process-based methods lies in parameter estimation. In addition to estimating the power spectral density function, one also

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193

needs to know the modulation functions of frequency and time, which are not known a priori and are difficult to estimate. Motivated by an amplitude- and frequency-modulated process proposed by Grigoriu et al. [20], Yeh and Wen [21] proposed to model ground motion as I(t)ς(φ(t)), where I(t) is a deterministic intensity envelope that controls the amplitude of the ground motion, ς(φ(t)) is a stationary filtered white noise in time scale φ and depicts the spectral density form of the process, and φ(t) is a smooth, strictly increasing function that depicts the frequency modulation. The stationary white noise was filtered to have a Clough-Penzien spectrum [5] with zero mean and unit variance. By changing the time scale of the stationary filtered white noise, the spectral density of the stationary process becomes time varying: Sςς (t , ω ) =

 ω  1 . SCP  φ′(t )  φ′(t ) 

(9.1)

This method does not require the “slowly time-varying” assumption implied in the evolutionary spectrum method, and it gives a time-dependent power spectral density as a simple modification of the spectral density of the stationary process. However, the frequency modulation function is estimated by a zero-crossing rate procedure and is independent of the amplitude variation. It is therefore somewhat restrictive in nature and only suited for a certain special class of random processes. It has been applied to earthquake ground motion simulation with some success but lacks general theoretical basis for wider applications. Recently, Huang et al. [22] introduced a new method of spectral analysis of nonstationary time series based on empirical modal decomposition (EMD), Hilbert transform, and the concept of instantaneous frequency, commonly referred to as the method of Hilbert-Huang transform (HHT). Unlike the Fourier transform, the Hilbert transform emphasizes local behavior and does not decompose the signal into sine and cosine waves of fixed frequencies; thus it is suitable for nonstationary processes. HHT decomposes the signal into orthogonal functions called intrinsic mode functions (IMFs) directly based on data. The Hilbert transform of IMFs yields the instantaneous amplitude and instantaneous frequency. HHT can be used to overcome the difficulties of the Wen-Yeh simulation method since EMD provides a reasonable and physically meaningful decomposition of the nonstationary signal. The Hilbert transform yields instantaneous amplitude and frequency of each IMF, which can be used directly in the Wen-Yeh method. It bypasses the difficult tasks of estimating and constructing analytical functions for frequency and amplitude modulation. Also, since these two modulation functions are obtained from Hilbert transform of IMFs, the dependence between the two functions is preserved. In this study, an IMF recombination method for simulation of nonstationary processes is first proposed, followed by an improved Wen-Yeh method. After a brief introduction of HHT, we describe the IMF recombination method, followed by the improved Wen-Yeh Method. We then compare the two. These methods are demonstrated by numerical examples of simulation of earthquake ground motions.

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9.2 HILBERT-HUANG TRANSFORM (HHT) What follows is a brief description of the background of the HHT. Details can be found in Huang et al. [22]. First, note that a time series can be decomposed into a series of IMFs. An IMF is defined as a time series that satisfies two requirements: 1. Its number of extrema and number of zero-crossings are either equal or differ by one. 2. The mean value of the envelopes defined by the local maxima and minima is zero. The extraction of IMFs out of a time series requires a repeated “sifting” procedure called empirical mode decomposition. EMD is described briefly as follows: Connect all the local maxima by a cubic spline as the upper envelope and connect all the local minima by a cubic spline as the lower envelope. Then calculate the mean of the envelopes and subtract the mean from the time series. The result is the first component. Ideally it should have satisfied the requirements for an IMF, but in practice a gentle hump on a slope can be amplified to become a local extreme. So the procedure is repeated until an IMF is obtained. This is the first IMF. Subtract the first IMF from the original time series, then continue the procedure on the residue to obtain the second IMF. Repeat the process to obtain the third, fourth, fifth, and subsequent IMFs, until the residue is too small to be of any consequence or becomes a monotonic function from which no more IMF can be extracted. After the EMD, the time series X(t) can be expressed in terms of IMFs as follows: n

X (t ) =

∑ C (t) + r (t) j

(9.2)

n

j =1

in which Cj(t) is the jth IMF, and rn(t) is the residue that can be the mean trend or a constant. Taking Hilbert transform of Cj(t), D j (t ) =

C j (τ) 1 P dτ π t−τ

∫

(9.3)

in which P indicates Cauchy principal value. One can then form an analytical function Zj(t) = Cj(t) + iDj(t) = aj(t) e

iθ j ( t )

.

(9.4)

The instantaneous frequency of the IMF is given by ω j (t ) =

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d θ j (t ) . dt

(9.5)

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One can then express X(t) as follows: n

∑ a (t)e

X (t ) = Re{

j

iθ j ( t )

} + rn (t )

(9.6)

j =1

in which Re denotes the real part. Therefore, aj(t), j = 1, …, n, therefore control the amplitude variation; each has an instantaneous frequency at a given time. Collectively, the functions aj(t) and j(t) define the Hilbert amplitude spectrum. Similarly, aj2(t) and j(t) define the energy distribution with respect to time and frequency, called the Hilbert energy spectrum or simply Hilbert spectrum, H(,t). The marginal Hilbert energy spectrum is obtained by integration over the duration: T

h(ω ) =

∫ H (ω, t)dt .

(9.7)

0

Because of the local nature of the Hilbert transform and the concept of instantaneous frequency, the Hilbert marginal spectrum describes an energy contribution corresponding to a given frequency at a certain time but not the entire duration of the process, as would be the case for a Fourier spectrum. This is a fundamental difference between these two approaches. As an example, the method is applied to the well-known N-S component of the El Centro earthquake record of 1940 and the Newhall ground motion of the 1994 Northridge earthquake in SAC-2 ground motion records [23], representing far-source and near-source records respectively. The time histories of the ground accelerations (in cm/sec2) are shown at the top of Figure 9.1 (At the bottom of the figure, four simulated samples of the ground accelerations are shown based on the IMF recombination method described later in this paper.) The IMFs of the Newhall record are shown in Figure 9.2. The Hilbert spectra are calculated and shown in Figure 9.3. As can be seen, one distinct advantage of the Hilbert spectrum is that the energy distribution over time and frequency is always sharp. For example, a darker shade of color in the plot indicates higher energy, which can be clearly seen in the Newhall record from 4 to 10 seconds and below 20 rad/sec (3 cps), a frequently observed feature of near-source records; it can also be seen over a longer time period in the El Centro record. Figure 9.4 compares the marginal Hilbert energy spectrum with the Fourier energy spectrum obtained from the same record. There are some obvious differences due to the two different methods of spectral description. The scale of the Hilbert marginal spectra has been adjusted by a factor of π for better comparison. Details of comparison of these two spectra and the conversion factor are shown in the Appendix. Note that the Hilbert transform can be carried out by first performing the Fourier transform on the original signal, changing the phase of each positive frequency component by /2 and the phase of each negative frequency component by –/2, and finally performing an inverse Fourier transform to obtain the Hilbert transform of the sample [24]. One can use FFT to make the procedure more efficient.

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FIGURE 9.1 Recorded (top) and simulated El Centro and Newhall ground motions using IMF recombination method.

In spite of its numerical difficulties — such as in end swing caused by spline fitting, convergence in the sifting procedure; ambiguity of the local symmetry requirement of IMFs; and dealing with intermittent signals, negative frequency, and end distortion in Hilbert transform — HHT represents a new approach to the challenging problem of time–frequency analysis of nonstationary signals, especially for slowly varying signals, that has many advantages over existing methods. Instead of decomposing the signal into predetermined analytical functions, it obtains the orthogonal functions (IMFs) directly from the data; thus it is more adaptive, and the resulting IMFs usually carry physical meaning. For example, it has been shown that IMFs of earthquake recordings have physical meanings related to seismic source mechanism [25]. Unlike the Fourier transform, the Hilbert transform emphasizes local behavior, as the Hilbert transform is the convolution of the signal with a window function of 1/t. Because of these characteristics, HHT represents an attractive alternative to other methods currently available, such as short-time Fourier transform, wavelet transform, instantaneous spectrum, double-frequency spectrum, Karhunen-Loeve decomposition, and so forth. The emphasis of HHT study so far has been on data analysis and interpretation of a deterministic time series of a real physical phenomenon [22, 26–33]. Much less has been done to further development of the method as a simulation tool for non-

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FIGURE 9.2 IMFs of the 1994 Northridge Newhall record (SAC LA14).

stationary random processes. Such extension is essential for performance evaluation of engineering systems, since the demands need to be modeled as random processes. This paper proposes to use the new approach for simulation of nonstationary random processes and to apply the simulation methods to earthquake engineering.

9.3 IMF RECOMBINATION METHOD The Hilbert spectral representation of a signal,  x (t ) = Re  

n

∑ a (t)e j

j =1

iθ j ( t )

  + rn (t ) 

suggests that the underlying random process can be represented by introducing random elements as follows:  X (t ) = Re  

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n

∑ j =1

a j (t )e

i [ θ j ( t )+ φ j ]

  + rn (t ) , 

(9.8)

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Hilbert Energy Spectrum for EI Centro Record 140

Frequency (rad/s)

120 100 80 60 40 20

0

5

10

15 Time (s)

20

25

Hilbert Energy Spectrum for Newhall Record 140

Frequency (rad/s)

120 100 80 60 40 20

0

5

10

15 Time (s)

20

25

30

FIGURE 9.3 Hilbert spectra of 1940 El Centro and 1994 Northridge Newhall ground motion.

in which φj is an independent random phase angle uniformly distributed between 0 and 2π. X(t) is therefore a random process. The process has the following mean, covariance, and variance functions:  µ X (t ) Re  

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n

∑ j =1

a j (t )e

iθ j ( t )

 iφ  E (e j )  + rn (t ) = rn (t ) 

(9.9)

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106

199

Hilbert Marginal Spectrum and Fourier Energy Spectrum for Newhall Record

105 104 103 10

2

101 100 10−1 10−2 10

−3

Fourier Energy Spectrum Hilbert Marginal Spectrum (adjusted)

10−2

10−1

100

101

102

103

rad/s

10−1

Hilbert Marginal Spectrum and Fourier Energy Spectrum for EI Centro Record

10−2 10−3 10−4 10−5 10−6 10−7

Fourier Energy Spectrum Hilbert Marginal Spectrum (adjusted)

10−1

100

101

102

rad/s

FIGURE 9.4 Comparison of Hilbert marginal spectrum (solid line) with Fourier energy spectrum (dashed line).

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1 K XX (t1 , t2 ) = 2

{

n

∑ a (t )a (t ) cos[θ (t ) − θ (t )] j

1

j

2

j

1

j

2

(9.10)

j =1

()

()

σ 2x (t) = E  X t − uX t 

2

}

=

1 2

n

∑ a (t ) 2 j

(9.11)

j =1

X(t), as given in Equation 9.8, has a Hilbert energy spectrum characterized by aj2(t) with instantaneous frequency d(j(t))/dt, for j = 1, …, n. Due to the Central Limit Theorem, X(t) approaches a Gaussian process for large n. This Hilbert spectral representation model suggests a simple method for simulation of a nonstationary process as shown in Equation 9.8. One can generate the random phase angles on computer and use Equation 9.8 to simulate the process. The name reflects the fact that it is essentially a method of recombination of the IMFs. The Hilbert spectrum of each simulated sample can be shown to be the same as that of the recorded ground motion. The physical basis for this method is that the frequency and amplitude of each IMF represent those of a sub-wave caused by a given physical mechanism of the rupture surface of the earthquake, as shown by Zhang [25]. For comparison with the traditional Fourier-based source inversion solution obtained by Hartzell et al. [34] for the Northridge earthquake based on 70 records, Zhang studied the same 70 earthquake records using HHT. The spatial distributions of the slip amplitude over the finite fault, each of which corresponds respectively to the second to fifth IMFs of those records, were plotted. It was found that the large slip amplitude regions are located at almost the same regions as those in Hartzell’s study, indicating that wave motions in each IMF are generated by a fraction of the source. In addition, the rupture with large-slip region scatters out to the northwest from the hypocenter, with the IMF changing from the second to the fifth, indicating that the seismic waves are generated sequentially from the dominant short-period signals to main long-period signals as the rupture propagates. Therefore, the IMFs are closely related to the source mechanism in general and the source heterogeneity and rupture process in particular. For future earthquakes of similar characteristics, the rupture surface spatial features are random and unpredictable. The IMF recombination method represents an efficient way to capture these variabilities. Simulations of the N-S component of the El Centro and the Northridge Newhall records (SAC LA14) were carried out, and sample time histories are shown in Figure 9.1. The simulated samples represent a reasonably statistical image of the intensity and frequency variation with time of the records. The method has also been extended to vector processes.

9.4 IMPROVED WEN-YEH METHOD The IMF recombination method has certain shortcomings compared to the Wen-Yeh method. They are as follows:

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• •

•

201

In the context of the Wen-Yeh method, the underlying stationary process for each IMF has only one frequency component. There is only a small number of random elements. For example, Equation 9.8 has only n random variables, n being the number of IMFs, which is usually not very large (usually <15 for earthquake accelerograms). No smoothing of aj(t) and j(t) is done to separate noise from the underlying true physical characteristics.

Taking advantage of both methods, an improved Wen-Yeh method is proposed. In this method, the recorded ground motion is first decomposed into IMFs. Then the smoothing of aj(t) and j(t) of each IMF is performed to separate noise from the underlying true physical characteristics, from which the underlying stationary random process of the IMF is obtained and simulated by using the well-known spectral representation method [35]. The frequency and amplitude of the simulated sample of the stationary random process are then modulated in time. The frequency modulation is done by changing the time scale as in the Wen-Yeh method. The frequency modulation function and amplitude modulation function are obtained directly from the Hilbert transform. Finally, the sample function of the process is simulated by combining the simulated IMFs. The procedure is described in detail as follows: 1. Decompose the record into IMFs Cj(t), j = 1, …, n, by EMD, and determine the aj(t) and j(t) for each IMF by Hilbert transform. 2. Smooth aj(t) and j(t) and denote the smoothed functions respectively as ajs(t) and js(t). 3. Obtain the reduced process Cjr(t) from Cj(t) by removing the amplitude modulation ajs(t) from Cj(t) and changing the time scale by ωjs–1 (t). Changing time scale is done by making the signal a function of θ instead of t, θ being the integration of js(t). Then resample the signal using an evenly spaced. 4. Simulate the reduced process Cjr(t) as stationary process and obtain samples of the process Sjr(t). 5. Restore the time scale by using the function ωjs(t) in Sjr(t), and restore the amplitude modulation by using ajs(t). The result is the simulated jth IMF Sj(t). Restoring time scale is done by first resampling the underlying stationary process of θ and then expressing the signal as a function of time t. 6. Add all Sj(t), j = 1, …, n. The procedure is entirely numerical and based on sample data. No analytical forms are assumed for the frequency and amplitude modulation or for the spectrum of the underlying stationary random process. It is therefore conceptually simple and easy to implement. Figure 9.5 shows eight simulated samples using this method along with the record for the Newhall Northridge earthquake. Figure 9.6 shows the response spectra of the samples. As can be seen, the long-duration pulses of the near-fault ground motion are reproduced well. Both the amplitude and frequency changes with time

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Simulated samples of Newhall LA14 1000 0 −1000 1000 0 −1000 1000 0 −1000 1000 0 −1000 1000 0 −1000 1000 0 −1000 1000 0 −1000 1000 0 −1000 1000 0 −1000

Original record of Newhall LA 14

0

5

10

15

20

25

30

FIGURE 9.5 Simulation of Newhall Northridge 1994 by improved Wen-Yeh method.

are reflected well in the simulated samples. The response spectra generally fluctuate around that of the record. Unlike samples generated by methods based on analytical models of the spectra, the response spectra of the simulated samples retain the jagged look of those of the records. The artificial smoothness of the response spectra of the simulated ground motions obtained by existing power spectrum-based methods has been criticized by seismologists and practicing engineers. To further illustrate the procedure, the steps leading to the first sample in Figure 9.5, shown below by figures, are as follows: 1. Decompose the record into IMFs (Figure 9.2). 2. Smooth aj(t) and j(t). Figure 9.7 shows a2(t) and a2s(t), and Figure 9.8 shows 2(t) and 2s(t) as an example. Here, 24-point rectangular window smoothing with zero-padding is used. 3. Obtain the reduced process Cjr(t). The reduced process C2r(t) is shown in Figure 9.9 as an example. As can be seen, both the amplitude and frequency are much more uniform than the original time history of the second IMF. Figure 9.10 illustrates the effect of changing time scale: it removes the frequency modulation from the signal and makes it stationary. 4. Simulate the reduced process Cjr(t) as a stationary process. A sample of S2r(t) is shown in Figure 9.11. © 2005 by Taylor & Francis Group, LLC

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203

Response Spectra of Newhall LA 14 Simulations using Improved Wen-Yeh Method (5% damping) 3.5 Simulation 1 Simulation 2 Simulation 3 Simulation 4 Simulation 5 Simulation 6 Simulation 7 Simulation 8 Original record

3

Pseudo Acceleration (g)

2.5

2

1.5

1

0.5

0

0

1

2

3

4

5 Period (sec)

6

7

8

9

10

FIGURE 9.6 Response spectra of Newhall LA14 simulations by improved Wen-Yeh method.

5. Restore the time scale and the amplitude modulation to obtain the simulated jth IMF Sj(t). The restored sample in Step 4 of the second IMF is shown in Figure 9.12 and compared with the original second IMF. 6. Add all Sj(t), j =1 to n. The result is Sample 1 in Figure 9.5. Further investigation is needed, including refinement of the smoothing process, modeling of the possible dependence among the IMFs. This method is being used in a study of the effects of near-fault ground motions on buildings and structures and in simulation of suites of uniform hazard ground motions for performance evaluation.

9.5 CONCLUSIONS Two new methods of simulation of nonstationary random processes are presented based on the Hilbert-Huang transform of sample observations, namely the IMF recombination method and the improved Wen-Yeh method. Both take advantage of EMD and the instantaneous frequency and amplitude of the Hilbert transform, thus overcoming difficulties in the estimation of the frequency modulation and interdependence of © 2005 by Taylor & Francis Group, LLC

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Instantaneous amplitude function of the 2nd IMF 500 Amplitude function Smoothed amplitude function

450 400 350 300 250 200 150 100 50 0

0

5

10

15

20

25

30

FIGURE 9.7 Original (dashed line) and smoothed (solid line) instantaneous amplitude function of the second IMF. Instantaneous frequency function of the 2nd IMF 120 Frequency function Smoothed frequency function

100

80

60

40

20

0

0

5

10

15

20

25

30

FIGURE 9.8 Original (dashed line) and smoothed (solid line) instantaneous frequency function of the second IMF.

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Reduced 2nd IMF 1.5

1

0.5

0

−0.5

−1

−1.5

−2

0

50

100

150

200

250

300

350

FIGURE 9.9 Reduced (stationary) second IMF.

1.5 1 0.5 0 −0.5 −1 −1.5

3

4

5

6

7

8

9

10

1.5 1 0.5 0 −0.5 −1 −1.5

40

50

60

70

80

90

100

110

120

130

FIGURE 9.10 Effect of change in time scale to remove frequency modulation: original (top); frequency modulation removed (bottom).

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Simulated sample of reduced 2nd IMF 2

1.5

1

0.5

0

−0.5

−1

−1.5

−2

0

50

100

150

200

250

300

350

FIGURE 9.11 Simulated sample of reduced (stationary) second IMF.

Simulated (solid) and original (dotted) 2nd IMF 500 400 300 200 100 0 −100 −200 −300 −400 −500

0

5

10

15

20

25

30

FIGURE 9.12 Comparison of original (dotted line) and simulated (solid line) second IMF.

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frequency and amplitude modulation functions faced by most currently available methods. These new methods extract and preserve the true physical features of the process. They have advantages over most currently available methods since assumptions about any functional forms of the spectra or about piecewise stationarity of the process are unnecessary; difficulties associated with estimation of these analytical functions from data are therefore bypassed. These methods are largely numerical and therefore suited for computer-aided model-based simulations. Numerical examples on earthquake ground motion simulation were carried out. The sample ground motions and records are shown to have the same time-varying intensity and spectral characteristics as in the samples from the underlying nonstationary process. The response spectra of the simulated ground motions retain the jagged features of the spectra of the real records. The proposed HHT method has been shown to have great potential in engineering applications when dealing with nonstationary processes.

ACKNOWLEDGMENTS This research is supported by the University of Illinois Research Board and the NSF Earthquake Engineering Research Center Program under Award Number EEC-9701785. Information exchange with C. H. Loh of National Taiwan University is greatly appreciated.

REFERENCES 1. Bycroft, G. N. (1960). White noise representation of earthquakes. J. Eng. Mech. Div., ASCE, Vol. 86, EM2: 1–16. 2. Kanai, K. (1957). Semi-empirical formula for the seismic characteristics of the ground. Bull. Earthquake Res. Inst., Univ. Tokyo, Vol. 35, 309–325. 3. Tajimi, H. (1960). A statistical method of determining the maximum responses of a building structure during an earthquake. Proc. Second World Conf. Earthquake Eng., Tokyo and Kyoto, Japan, Vol. II. 4. Housner, G. W. and Jenning, P. C. (1964). Generation of Artificial Earthquakes, J. Eng. Mech. Div., ASCE, Vol. 90, EM1: 113–150. 5. Clough, R. W. and Penzien, J. (1975). Dynamics of Structures. McGraw-Hill Book Co., New York. 6. Shinozuka, M. and Sato, Y. (1967). Simulation of nonstationary random processes. J. Eng. Mech. Div., ASCE, Vol. 93, EM1: 11–40. 7. Amin, M. and Ang, A. H.-S. (1968). Nonstationary stochastic model for earthquake motions. J. Eng. Mech. Div., ASCE, Vol. 94, EM2: 559–583. 8. Saragoni, G. R. and Hart, G. C. (1974). Simulation of artificial earthquakes. Earthquake Eng. Structural Dynamics, Vol. 2, 249–267. 9. Priestley, M. B. (1967). Power spectral analysis of non-stationary random processes. J. Sound Vibration, Vol. 6, 1: 86–97. 10. Lin Y. K. and Yong, Y. (1987). Evolutionary Kanai-Tajimi earthquake models. J. Eng. Mech. Div., ASCE, Vol. 113, 8: 1119–1137. 11. Deodatis, G. and Shinozuka, M. (1988). Auto-regressive model for nonstationary stochastic processes. J. Eng. Mech., ASCE, Vol. 114, 4: 1995–2012.

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12. Li, Y. and Kareem, A. (1991). Simulation of multivariate nonstationary random processes by FFT. J. Eng. Mech., ASCE, Vol. 117, 5: 1037–1058. 13. Li, Y. and Kareem, A. (1997). Simulation of multivariate nonstationary random processes: Hybrid DFT and digital filtering approach. J. Eng. Mech., ASCE, Vol. 123, 12: 1302–1310. 14. Der Kiureghian, A. and Crempien, J. (1989). An evolutionary model for earthquake ground motion. Struct. Safety, 6: 235–246. 15. Papadimitriou, C. (1990). Stochastic characterization of strong ground motion and application to structural response, Rep. No. EERL 90-03, California Institute of Technology, Pasadena, Calif., U.S.A. 16. Conte, J. P. and Peng, B. F. (1997). Fully nonstationary analytical earthquake ground-motion model. J. Eng. Mech., Vol. 123, 1: 15–24. 17. Iyama, J. and Kuwamura, H. (1999). Application of wavelets to analysis and simulation of earthquake motions. Earthquake Eng. Structural Dynamics, 28: 255–272. 18. Spanos, P. D. and Failla, G. (2002). Multi-scale modelling via wavelets of evolutionary spectra in mechanics applications. Proc. Int. Symp. Multiscaling in Mechanics, Messini, Greece. 19. Spanos, P. D., Tratskas, P. and Failla, G. (2002). Wavelets applications in structural dynamics. In: Structural Dynamics, EURODYN2002, Grundmann & Schueller (eds.). Swets & Zeitinger, Lisse, ISBN 90 5809 510 X. 20. Grigoriu, M., Ruiz, S. E., and Rosenblueth, E. (1988). Nonstationary models of seismic ground acceleration. Earthquake Spectra, 4: 551–568. 21. Yeh, C. H. and Wen, Y. K. (1990). Modelling of nonstationary ground motion and analysis of inelastic structural response. Structural Safety, 8: 281–298. 22. Huang, N. E., et al. (1998). The empirical mode decomposition and the Hilbert spectrum for nonlinear, nonstationary time series analysis. Proc. R. Soc. Lond. A, 454: 903–995. 23. Somerville, P., Smith, N., Punyamurthula, S., and Sun, J. (1997). Development of ground motion time histories for Phase 2 of the FEMA/SAC steel project. Report No. SAC/BD-97/04, SAC Joint Venture, Sacramento, California, U.S.A. 24. Bendat, J. S. and Piersol, A. G. (2000). Random Data, Analysis and Measurement Procedures, 3rd ed. John Wiley & Sons, Inc., New York. 25. Zhang R. (2001). The role of Hilbert-Huang transform in earthquake engineering. Proc. World Multiconference Systemics, Cybernetics Informatics, Vol. XVII, Orlando, Florida, U.S.A. 26. Huang, et al. (2001). A new spectral representation of earthquake data: Hilbert spectral analysis of Station TCU129, Chi-Chi, Taiwan, 21 September 1999. Bull. Seismological Soc. Am., Vol. 91, 5: 1310–1338. 27. Huang, N. E., Shen, Z. and Long, S. R. (1999). New view of nonlinear water waves: The Hilbert spectrum. Annu. Rev. Fluid Mech., Vol. 31, 417–457. 28. Loh, C. H., Wu, T.C. and Huang, N. E. (2001). Application of the empirical mode decomposition-Hilbert spectrum method to identify near-fault ground-motion characteristics and structural responses. Bull. Seismological Soc. Am., Vol. 91, 5: 1339–1357. 29. Yang, J. N. and Lei, Y. (2000). System identification of linear structures using Hilbert transform and empirical mode decomposition. Proc. Int. Modal Anal. Conf., Vol. 1, 213–219. 30. Zhang, R. and Ma, S. (2000). HHT analysis of earthquake motion recordings and its implications to simulation of ground motion. In: Monto Carlo Simulation, G. I. Schueller and P.D. Spanos (eds.), 483–490. A. A. Balkema Publishers, Lisse, the Netherlands.

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31. Zhang R. R. (2001). The signature of structural damage: An HHT view. In: Structural Safety and Reliability, Corotis et al. (eds.), Swets & Zeitlinger, ISBN 90 5809 197 X. 32. Salvino, L. W. (2000). Empirical mode analysis of structural response and damping. Proc. SPIE Int. Soc. Optical Eng., Vol. 4062, 503–509. 33. Gravier, B. M. et al. (2001). An assessment of the application of the Hilbert spectrum to the fatigue analysis of marine risers. Proc. Int. Offshore Polar Eng. Conf., Vol. 2, 268–275. 34. Hartzell, S., Liu, P., and Mendoza, C. (1996). The 1994 Northridge, California earthquake: Investigation of rupture velocity, risetime, and high-frequency radiation. J. Geophys. Res., Vol. 101, 20091–20108. 35. Shinozuka, M. and Deodatis, G. (1991). Simulation of stochastic processes by spectral representation. App. Mech. Reviews, Vol. 44, 4: 191–203

APPENDIX: RELATION BETWEEN HILBERT MARGINAL SPECTRUM AND FOURIER ENERGY SPECTRUM Consider a time series y(t) which allows Fourier transform, defined as follows: y(ω ) =

∫

∞

−∞

y(t )e − iωt dt

(9A.1)

Its complex conjugate is then y∗ (ω ) =

∫

∞

−∞

y(t )eiωt dt

(9A.2)

Now if we define the Fourier energy spectrum by E f (ω ) = y(ω )

2

(9A.3)

then the area under the Fourier energy spectrum for all frequency is given by ∞

E=

∞ ∞ ∞

∫ y(ω) y (ω)dω = ∫ ∫ ∫ y(t ) y(t )e ∗

1

−∞

2

−∞ −∞ −∞

∞ ∞

=

∫ ∫ y(t ) y(t )2πδ(t − t )dt dt 1

−∞ −∞

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2

1

2

1

2

− iωt1 + iωt2

d ωdt1dt2

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The Hilbert-Huang Transform in Engineering

∞

= 2π

T

∫ y (t)dt = 2π ∫ y (t)dt 2

2

−∞

(9A.4)

0

if y(t) is defined for 0 < t < T, and zero otherwise. Therefore, the area under the Fourier energy spectrum defined by Equation 9A.3 is equal to 2π times the total energy of the time series, defined by the integral of the square of the time series. The area under the Hilbert energy spectrum can be illustrated by a simple case of a time series that is a summation of n sine functions defined for a given time period 0 to T, n

y(t ) =

∑ A sin(ω t) for 0 < t < T j

j

(9A.5)

j =1

where T = mπ/2ωj for all j, m an integer. When we apply EMD to y(t), we in effect recover the n sine functions as n IMFs. From Equation 9.1, Equation 9.2, and Equation 9.3 (in the text), one can see that if Cj(t) = Ajsinωjt, the Hilbert transform of each of the sine functions produces Dj(t) = –Ajcosωjt, and a2j (t) = A2j . The Hilbert energy spectrum is therefore equal to A2j at ωj for all time. For a finite period of time T, the discrete marginal spectrum is equal to A2j T at ωj. The total area under the Hilbert marginal spectrum is therefore n

∑ A T. 2 j

j =1

The total energy over the time period is given by

∫

T

0

1 y (t )dt = T 2 2

n

∑ Aj

2

(9A.6)

j =1

Hence, the total area under Hilbert marginal spectrum is n

∑

A2j T = 2

∫

T

y 2 (t )dt

(9A.7)

0

j =1

More generally, for an arbitrary time series, n

y(t ) =

∑ a (t) ⋅ cos[θ (t)] j

j =1

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j

(9A.8)

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211

if we neglect the residual term from Equation 9.5. Its total area under the Hilbert marginal spectrum is n

T

∑ ∫ a (t)dt. 2 j

j =1 0

Its total energy over time is

∫

T

n

n

i =1

j =1

∑∑ ∫

y 2 (t )dt =

0

T

ai (t )a j (t ) cos[θi (t )]cos[θ j (t )]dt

(9A.9)

0

Due to the orthogonality of IMFs (see Huang et al. [22]), the cross-terms (i ≠ j) in Equation 9A.9 can be neglected. Equation 9A.9 reduces to

∫

n

T

y 2 (t )dt =

0

∑∫ j =1

T

a 2j (t ) cos2 [θ j (t )]dt

(9A.10)

0

When the IMFs have relatively smooth and sinusoidal shapes, for integer number of quarter-waves, we have

∫

T

1 2

a 2j (t ) cos2 [θ j (t )]dt ≈

0

∫

T

a 2j (t )dt

(9A.11)

0

For high-frequency IMFs, the number of quarter-waves is usually large, so the portion that does not contain a complete quarter-wave is small and can be neglected. For low-frequency IMFs, this portion cannot be neglected, but low-frequency IMFs usually have small amplitudes and contribute insignificantly to the total energy. Therefore one arrives at

∫

T

y 2 (t )dt ≈

0

1 2

n

∑∫

T

a 2j (t )dt

(9A.12)

0

j =1

The total area under Hilbert marginal spectrum is n

∑∫ j =1

T

a 2j (t )dt ≈ 2

0

which is the same as in Equation 9A.7.

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∫

T

0

y 2 (t )dt

(9A.13)

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Comparing Equation 9A.13 with Equation 9A.4, we conclude that the total area under the Hilbert marginal spectrum needs to be multiplied by a factor, to be comparable with the area under the Fourier energy spectrum defined by Equation 9A.3.

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10

Comparison of Hilbert-Huang, Wavelet, and Fourier Transforms for Selected Applications Ser-Tong Quek, Puat-Siong Tua, and Quan Wang

CONTENTS 10.1 Introduction ..................................................................................................214 10.2 Hilbert-Huang Transform.............................................................................215 10.3 Applications of HHT, WT, and FT to Experimental Data..........................216 10.3.1 Locating Crack in Aluminum Beam Using HHT ...........................216 10.3.2 Detection of Edges in Aluminum Plate...........................................219 10.3.3 Determination of Modal Frequencies of Aluminum Beam ............222 10.4 Concluding Remarks....................................................................................236 Acknowledgments..................................................................................................239 References..............................................................................................................243 ABSTRACT This paper illustrates the suitability of three different signal-processing techniques, namely, the Hilbert-Huang transform (HHT), the wavelet transform (WT), and the fast Fourier transform (FFT), under different practical situations in relation to structural health monitoring. Three experimentally obtained signals are analyzed. First, in the time-of-flight analysis of flexural wave propagation in an aluminum beam, HHT is found to be a more direct method compared to WT, as no knowledge of the actuation frequency is required. However in the case of WT, if prior knowledge of the actuation frequency is utilized combined with a refined scale search, WT produces more accurate results. In another experiment involving the time-of-flight analysis of acoustic Lamb wave propagation in an aluminum plate, piezoelectric actuators and sensors (PZTs) are used to excite and receive direct and reflected waves at a specific frequency. HHT and

213

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WT give similar results in terms of precision and accuracy. Last, a modified Fourier transform (FT) is investigated based on the concept of HHT and applied to the signals collected from the free vibration of an aluminum beam under three different support conditions. The proposed method is able to reveal the presence of higher vibration modes than can be revealed by conventional FT of the original signal. The results produced are close to that of the more common WT. It is believed that supporters of FT will be more receptive to this modified HHT method.

Keywords Hilbert-Huang transform, wavelet transform, fast Fourier transform, structural health monitoring, flexural wave, Lamb wave, free vibration

10.1 INTRODUCTION Signal processing has gradually become an indispensable tool in the field of engineering for extraction of important information from raw signals. In civil engineering, many practical and robust nondestructive evaluation (NDE) techniques developed for assessment of structural performance involve two key components, namely: (1) data acquisition and (2) signal processing and interpretation. Although the use of vibration measurements is a simpler and less costly method with respect to instrumentation system in comparison with infrared thermography, ground-penetrating radar, acoustic emission monitoring, and eddy current detection, a key factor for identifying damage precisely lies in having an appropriate data analysis method. The most well known conventional signal processing technique is the Fourier transform (FT). Fourier spectral analysis provides a general method for examining the global energy–frequency distributions of a given signal. FT has high resolution in the frequency domain but loses the resolution in the time domain. Despite this, FT has been employed in health monitoring. For example, Crema et al. [1] used a fast Fourier transform (FFT) analyzer to perform modal analysis to detect damage in composite material structures. A broadband impulse input was applied, and the damage that was induced by loads well above the working load caused a small decrease in the eigen-frequencies for several modes. However, it can be problematic to locate damage by using FT-based modal analysis techniques. A few methods are available for processing signals of nonlinear and nonstationary systems, of which the short-time Fourier transform (STFT) method and wavelet transform (WT) method are widely adopted. Cook and Berthelot [2] used the time-dependent energy spectrum from STFT to distinguish backscattered signals from a single crack against those from a series of closely spaced cracks on steel surface. Hurlebaus et al. [3] used the time–frequency spectrum obtained by performing STFT on the Lamb waves generated by laser to obtain the group velocity–frequency domain. The notches are located from the autocorrelation plot of a series of group velocity spectra at different assumed propagation distances. Wavelet analysis is another adjustable window signal-processing technique to handle nonstationary signals. Although WT appears similar to STFT, the basic wave components are not limited to sinusoidal functions. Rather, classes of functions have been proposed, from the simplest Haar wavelet to the more complicated higher order Debauchies wavelet functions, to address various problems of time–frequency

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215

resolution. WT has been widely used in the field of damage detection, such as in modal-based detection (Staszewski [4]). An early work using WT to obtain a dispersive relation of group velocity of flexural waves in beam was presented by Kishimoto et al. [5]. Lee and Liew [6] studied the sensitivity of WT in space domain for identifying the damage location of a beam for small damage not easily detected via conventional eigenvalue analysis. Quek et al. [7], on the other hand, used wavelet coefficients over the time domain to locate cracks in beams via the time-of-flight analysis of the wave propagation along the beam. However, WT has its shortcomings. Experience shows that it can produce many spurious harmonics under different scales that makes the analysis difficult or sometimes meaningless. The most recently introduced signal-processing technique for analyzing nonlinear and nonstationary time series data is the Hilbert-Huang transform (HHT) [8]. Empirical mode decomposition (EMD) of the data is first performed to segregate the signals into narrow band components, and the Hilbert transform (HT) is then applied on each mode. Its ability to analyze nonlinear and nonstationary time series data [8–10] has attracted many applications. For example, Zhang et al. [11] adopted HHT to assess structural damage from vibration recordings. Shim et al. [12] exploited HHT’s ability to exhibit nonlinear and nonstationary behavior to locate damages on bridges by using frequency sweep and controlled excitations. Quek et al. [13] also employed HHT to analyze structural dynamic responses to detect and locate anomalies in beams and plates. In addition, HHT has been used to analyze experimental data collected by oscilloscope to investigate the dominance of the lowest antisymmetric (A0) and symmetric (S0) modes of Lamb wave vary across the frequency range adopted for excitation and good agreement with, which agrees well with theoretical response prediction in terms of their relative dominance [14]. Individual signal-processing techniques have their own strength and weaknesses. Suitability of a particular technique may, therefore, be problem dependent, especially with respect to structural health monitoring problems. This paper compares the three signal-processing techniques, namely, HHT, WT, and FFT, for signals obtained experimentally. The first example concerns the time-of-flight analysis of flexural wave propagation in an aluminum beam, where WT and HHT are compared. Next, we compare the analysis from the two techniques on a two-dimensional problem involving the propagation of acoustic Lamb waves in an aluminum plate. Last, we apply a modified FT based on the EMD concept of HHT to the free vibration signals of an aluminum beam under three different support conditions, we compare these results with results from using conventional direct FFT, HHT, and WT to obtain the modal frequencies.

10.2 HILBERT-HUANG TRANSFORM Hilbert transform was first developed to process nonstationary narrow-band signals [15]. To apply it to signals in general, Huang et al. [8] proposed using the empirical mode decomposition technique to decompose any given signal into a set of narrow-band signals. Each component (which may be nonstationary) of the set is termed an intrinsic mode function (IMF) of the original signal, on the IMF where HT can be carried out.

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The Hilbert-Huang Transform in Engineering

The distinct features of a narrow-band signal, and hence an IMF, are that (a) over its entire length, the number of extrema and the number of zero-crossings either are equal or differ at most by one; and (b) at any point, the mean value of the envelope of the signal defined by the local maxima and the envelope defined by the local minima is zero. This forms the basis on which the EMD technique was developed. For the EMD to be applicable, it is assumed that (a) the signal has at least one maximum and one minimum; (b) the characteristic time scale is defined by the time lapse between the extrema; and (c) if the data are totally devoid of extrema but contain only inflection points, then the signal can be differentiated one or more times to reveal the extrema. Hilbert spectral analysis can then be performed on each IMF. MATLAB 6.1 is adopted for the implementation of the HHT in this study.

10.3 APPLICATIONS OF HHT, WT, AND FT TO EXPERIMENTAL DATA An accurate signal-processing technique to obtain the flight times is crucial to locate damages in structures. Both WT and HHT are potentially applicable. The advantages of HHT for damage detection in beams using flight times from narrow-band nonstationary signals have been discussed by Quek et al. [13]. Waves in plates are even more complex due to the dispersive nature of Lamb waves; hence, a technique that provides good representation of localized events in both the frequency and energy is vital for determining the location of the damage precisely. Another practical problem of interest is the determination of higher frequencies from free vibration data. In this section we examine a proposal to combine FT with HHT, using an aluminum beam under three different boundary conditions as illustration.

10.3.1 LOCATING CRACK

IN

ALUMINUM BEAM USING HHT

Quek et al. [7] presented a wavelet analysis on experimental data to locate a crack in an aluminum beam based on simple wave propagation considerations. The beam considered has geometrical and material properties given in Table 10.1. Three sets

TABLE 10.1 Geometrical and Material Properties of Aluminum Beam/Plate

Dimensions (mm) Young’s modulus, E (GPa) Shear modulus, G (GPa) Mass density, ρ (kg/m3)

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Beam

Plate

650 × 32 × 6 73.1 — 2790

600 × 600 × 2 102 26 2700

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217

FIGURE 10.1 Schematic view of wave propagation for impact at 200 mm position.

of data with different boundary conditions are analyzed, namely, fixed-fixed, simply supported, and cantilever conditions. The basic concept is shown in Figure 10.1, in a schematic view of wave propagation due to an impact. The piezoelectric sensor captures the wave signal, which contains timings of the direct, damage-reflected, and boundary-reflected waves. By estimating these timings with a signal-processing technique, the damage location can be deduced. In view of the dispersive nature of the wave, only a wave at a particular frequency, with a corresponding wave propagation velocity, is used so that accurate results can be obtained. HHT is applied in this study on the same three sets of signals. One of the signals obtained (and shown in Figure 10.2a) was decomposed via EMD into its IMF components, shown in Figure 10.2b. It can be observed that the first IMF component has higher amplitude than the other IMFs. This IMF component is then subjected to HT, and the corresponding frequency–time and energy–time distributions plotted in Figure 10.2c and d, respectively. From the energy–time distribution, the timings of the energy peaks in the signal, which are due to the propagating wave passing the sensor, are obtained. The frequencies corresponding to the energy peaks are also noted so as to establish that they do not deviate significantly from each other. It should be pointed out that not all IMFs are used, as the first IMF is distinct from the others and contains only the frequency of interest, whereas other IMFs do not. The results for the estimated damage position are summarized in Table 10.2 and compared with those obtained with Gabor wavelet analysis with coarse (discrete WT, scale = 12) and fine (“continuous” WT or CWT, scale = 12.2) resolution. The wavelet results have been discussed in detail by Quek et al. [7], including the 3D wavelet plot, and will not be repeated here. It can be observed that the HHT method is able to identify the damage position with reasonable accuracy; the results are better than the discrete wavelet results but worse than the continuous wavelet results. However, using CWT for this application requires determining an appropriate scale to process the final results. From past experience, this requires a fair amount of skill, as the selection can be rather subjective and may lead to varied end results (see, for

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(a)

Frequency (Hz)

10 0 −10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

−3

c(2)

0

×10

1.5

2

2.5

3

3.5

4

4.5

5

0 0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.5

1

1.5

2

2.5

3

3.5

4

4.5 5 −3 ×10

5 0 −5

0

1

2

3

4

5

5

−3

×10

0 −5

(d)

1 Residue

Amplitude

1

0 −5

(c)

Time, t(s)

5 4 3 2 1 0

0 0.5

5

Time, t(s)

5000 4000 3000 2000 1000 0

(b)

c(3)

10 8 6 4 2 0 −2 −4 −6 −8 −10

c(1)

The Hilbert-Huang Transform in Engineering

c(4)

Input

218

0

1

2

3

4

5

0 −1 −2

−3

×10

Time, t(s)

Time, t(s)

FIGURE 10.2 (a) Dynamic response of a beam; (b) IMFs of response signal after EMD; (c) frequency; (d) energy spectra of first IMF component.

TABLE 10.2 Comparison of Results Based on HHT Method and Wavelet Analysis Wavelet Analysis HHT Analysis

End Condition Fixed ended Simply Supported Cantilever

Scale 12

Scale 12.2

Estimated Damage Position

Error (%)

Estimated Damage Position

Error (%)

Estimated Damage Position

*Error (%)

437.5 471.7 403.8

–2.78 4.83 –10.26

492.0 493.0 542.0

9.33 9.56 20.00

450 457 457

0.00 2.00 2.00

Note: Actual damage position is 450 mm from left of the beam.

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Comparison of Hilbert-Huang, Wavelet, and Fourier Transforms

219 C

D 600

500

Sensor

400

Actuator 50 mm

300 75 mm Line crack

200

100

A

0

y

0

X 100

200

300

400

500

600 B

FIGURE 10.3 Schematic view of PZT positions on plate.

example, Table 3 of Quek et al. [7]). This is due to the shortcomings of WT, which produces false harmonics depending on the scale adopted, although WT also has the ability to achieve high resolution in both frequency and time domain. The advantage of HHT over WT in this case is that the HHT procedure is more direct and can be performed to obtain the desired accuracy for the results without prior knowledge of the excitation frequency (which in this case of impact load is broadband). It should be noted that there are other applications where CWT may be more appropriate.

10.3.2 DETECTION

OF

EDGES

IN

ALUMINUM PLATE

In this example, we adopted the time-of-flight analysis of acoustic Lamb wave propagation in a 600 mm × 600 mm aluminum plate to detect the plate’s edges, In the experiment, a through crack, 1 mm wide and 30 mm long, inclined at 30º to one side of the plate (side AB in Figure 10.3), was cut with a milling machine. The location of the center of the crack is (270, 290) mm. Piezoelectric sensors and actuators (PZTs) were used to excite and receive direct and reflected waves at a specific frequency. A Lamb wave with a narrow-band (600 kHz) actuation pulse (Figure 10.4) was activated based on the following equation:

(

)

X (t ) = cos 2πω t e − a (t −t0 )

© 2005 by Taylor & Francis Group, LLC

2

(10.1)

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220

The Hilbert-Huang Transform in Engineering

20 15

Amplitude (V)

10 5 0 −5 −10 −15 −20

1.5

2

2.5 Time (s)

3

3.5

4

−5

×10

FIGURE 10.4 Actuation pulse using Equation 10.1 at frequency 600 kHz.

where ω is the actuation frequency (Hz), and a and t0 are constants. This was done with a function generator (Yokogawa FG300 15 MHz Synthesized FG) to feed a PZT actuator (8.0 mm × 8.0 mm × 0.5 mm). The responses were captured by a PZT sensor placed at a distance away and channeled to an oscilloscope (Yokogawa DL716 16-Channel) for monitoring and analysis. Figure 10.3 shows the positions of the actuator and sensor. By estimating the flight time due to the propagation of either a S0 or an A0 Lamb wave, the distance traveled by the wave from the actuator to the sensor can be computed as li = ∆ ti cg,

(10.2)

where ti is the time of flight referenced from the actuation time, and cg is the group velocity of either the S0 or A0 mode, determined either theoretically or experimentally. The response signal captured by the PZT sensor (Figure 10.5a) was processed by both HHT and Gabor WT. The resulting energy spectra are given in Figure 10.5d and Figure 10.6. In the HHT analysis, the peaks in the energy–time spectrum for the IMF component containing the highest energy (in this case the first IMF) give the wave arrival times of interest. Based on these energy peaks, the distances traveled by the wave from the actuator to the sensor can be computed; these are summarized in Table 10.3. It can also be observed from the frequency spectrum (see Figure 10.5b) that the frequency corresponding to the energy peaks is very close to the actuation frequency, which indicates the accurate representation of localization events by HHT. On the other hand, Gabor WT analysis on the same signal response was performed based on the prior knowledge that the actuation frequency is 600 kHz, which corresponds to a scale of 20.2. Similar energy peaks were obtained, which accounted for the wave pulses on the different paths received by the sensor. The results obtained by WT analysis are also given in Table 10.3. These indicate

© 2005 by Taylor & Francis Group, LLC

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0

2

1 ×10

c(1)

0.05 0 −0.05 0.02 0 −0.02

c(5)

Time, t(s)

−4

1 0 −1

c(2)

(b)

(a)

c(3)

1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1

c(4)

Amplitude

Comparison of Hilbert-Huang, Wavelet, and Fourier Transforms

5

(c)

c(7)

10 5

c(8)

0 1

2

×10

Time, t(s)

1

A0/S0 (incident)

0.8

(d) A0 (crack)

0.6

−4

c(9)

0

c(10)

−5

Amplitude

c(6)

×10

c(11)

Frequency, (Hz)

15

A0 (boundary, CD)

0.4 0

Res

0.2 0

1

2

Time, t(s)

×10

−4

221

0

1

2

0

1

2

0 0.02 0 −0.02 0 0.01 0 −0.01 0 0.01 0 −0.01 0 ×10−3 5 0 −5 0 0.01 0 −0.01 0 0.2 0 −0.2 0 0.02 0 −0.02 0 0.1 0 −0.1 0 0.1 0 −0.1 0

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1 Time, t(s)

2

×10

−4

FIGURE 10.5 (a) Response signal captured by sensor on aluminum plate; (b) IMFs of response signal after EMD; (c) frequency; (d) energy spectra of first IMF component.

1.2E–03

Amplitude

1.0E–03

A0/S0 (incident)

8.0E–04 A0 (crack)

6.0E–04

A0 (boundary, CD)

4.0E–04 2.0E–04 0.0E+00 0.0E+00

5.0E–05

1.0E–04 Time (s)

1.5E–04

FIGURE 10.6 Gabor wavelet analysis of response signal given in Figure 10.5a.

© 2005 by Taylor & Francis Group, LLC

2.0E–04

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222

The Hilbert-Huang Transform in Engineering

TABLE 10.3 Results of Wave Propagation in Aluminum Plate

Signal Analysis Technique HHT WT a b c

Theoretical Velocity (m/s)

Distance Traveled by Incident Wave (mm)a

Distance Traveled by Reflected Wave from Line Crack (mm)b

Distance Traveled by Reflected Wave from Boundary (CD; mm)c

A0

A0

Error (%)

A0

Error (%)

A0

Error (%)

3134

52.9 52.7

5.80 5.40

196.1 196.4

–1.95 –1.80

425.0 424.0

–1.62 –1.85

Actual distance traveled by incident wave is 50 mm. Actual distance traveled by crack-reflected wave is 200 mm. Actual distance traveled by boundary (CD)-reflected wave is 432 mm.

that HHT and WT give similar results in terms of precision and accuracy. HHT does not require prior knowledge of the activation frequency.

10.3.3 DETERMINATION

OF

MODAL FREQUENCIES

OF

ALUMINUM BEAM

In this part of the study, we consider an aluminum beam under three support conditions, shown in Figure 10.7, with material properties similar to the plate in Section 10.3.2 (see Table 10.1). The theoretical modal frequencies of the beam were first obtained (see Table 10.4) by using the following equation [16]:

fn =

(β nl )2 2πl 2

EI , ρA

n = 1, 2, 3, ...

(10.3)

where l, E, I, ρ, and A are the length, Young’s modulus, moment of inertia, density, and cross-sectional area of the beam, respectively; n corresponds to the vibration modes; and βn is solved from the following equations, respectively, for cantilever, simply supported, and fixed-fixed conditions. cos β n l cosh β n l = −1

(10.4a)

sinβ n l = 0

(10.4b)

cos β n l cosh β n l = 1

(10.4c)

For the experiment, the free vibration responses of the beam under the three different support conditions (Figure 10.7) were monitored and collected by an oscilloscope (Yokogawa DL716 16-Channel; see Figure 10.8). FFT was first performed directly on

© 2005 by Taylor & Francis Group, LLC

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Comparison of Hilbert-Huang, Wavelet, and Fourier Transforms

Piezoelectric x Sensor

223

Aluminum beam

6 mm

(a) 32 mm

y 0 mm

150 mm Piezoelectric x Sensor

(b)

920 mm Aluminum beam

6 mm

32 mm

y 0 mm

800 mm

150 mm Piezoelectric x Sensor

Aluminum beam

6 mm

(c) 32 mm

y 0 mm

150 mm

800 mm

FIGURE 10.7 Schematic setup of aluminum beam under different support conditions: (a) cantilever; (b) simply supported; (c) fixed-fixed for modal frequency analysis.

the response signal. For example, Figure 10.9a and b shows the linear and logarithmic plots of the Fourier spectrum, respectively, for the cantilevered beam. In the linear plot of the Fourier spectrum (Figure 10.9a), the first two modal frequencies of the beam can be easily identified (5.99 Hz and 34.93 Hz respectively). However, the third mode may be easily overlooked due to the low energy of the displayed peak. On the other hand, the logarithmic scale plot of the spectrum (Figure 10.9b) exhibits the first three modal frequencies clearly. This is also true for the fixed-fixed support condition (see Figure 10.10). However, for the simply supported condition, it is not easy to distinguish the modes from the logarithmic scale plot of the Fourier spectrum. It can be observed in Figure 10.11b that besides the three fundamental modal frequencies, there are also a number of energy peaks at other frequency ranges having amplitudes of order close to the modal frequencies. Hence for beams with closely spaced frequencies, the Fourier spectrum may not be appropriate. EMD was performed on the responses given in Figure 10.8. Consider the results for the cantilevered beam: three decomposed IMF components and one residual component are obtained, as depicted in Figure 10.12a. The individual IMFs are subjected to FT, and the resulting spectra are shown in Figure 10.12b through d. The residual component is not subjected to FT as it represents the mean or residual trend of the captured response signal. Figure 10.12b through d clearly identifies the

© 2005 by Taylor & Francis Group, LLC

FFT Support Conditions

Simply supported

Fixed-fixed

1st 2nd 3rd 1st 2nd 3rd 1st 2nd 3rd 4th

5.86 36.73 102.86 22.77 87.03 195.84 49.33 135.97 266.56 440.64

EMD + FT

HHT

WT

Freq (Hz)

Error (%)

Freq (Hz)

Error (%)

Freq (Hz)

Error (%)

Freq (Hz)

Error (%)

5.99 34.93 96.80 22.89 81.80 165.80 48.00 138.00 268.00 446.00

2.20 –4.91 –5.89 5.17 –6.01 –15.34 –2.69 1.49 0.54 1.22

5.76 35.23 98.00 22.95 82.85 167.50 48.00 138.00 — 460.00

-1.72 –4.09 –4.73 5.44 –4.81 –14.47 –2.69 1.49 — 4.39

5.77 34.63 96.62 22.80 77.88 159.18 46.96 135.34 263.19 —

1.52 5.72 6.06 4.75 –10.52 –18.72 4.80 0.47 1.26 —

5.64 35.86 101.56 23.20 81.25 162.50 49.85 135.42 270.83 451.39

3.73 2.39 1.26 6.59 –6.65 –17.02 1.06 –0.41 1.60 2.44

The Hilbert-Huang Transform in Engineering

Cantilever

Mode

Theoretical Frequency (Hz)

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224

© 2005 by Taylor & Francis Group, LLC

TABLE 10.4 Analysis of Modal Frequencies of Aluminum Beam under Different Support Conditions Obtained by FFT, EMD Followed by FT, HHT, and WT

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Comparison of Hilbert-Huang, Wavelet, and Fourier Transforms

25

225

(a)

20 15 Amplitude

10 5 0 −5 −10 −15 −20 −25

0

0.2

0.4

0.6

0.8

1

Time, t(s) 15

(b)

Amplitude

10 5 0 −5 −10 −15

0

0.2

0.4

0.6

0.8

1

Time, t(s) 25

(c)

20 15 Amplitude

10 5 0 −5 −10 −15 −20

0

0.1

0.2

0.3

0.4

0.5

Time, t(s)

FIGURE 10.8 Free vibration response of aluminum under (a) cantilever; (b) simply supported; (c) fixed-fixed conditions.

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The Hilbert-Huang Transform in Engineering

4

2.5

×10

1st Mode = 5.99 Hz

(a)

Amplitude

2

1.5

1

0.5

0

nd 2 Mode = 34.93 Hz

0

50

rd 3 Mode = 96.8 Hz

100 150 Frequency (Hz)

200

105 1st Mode = 5.99 Hz 10

4

(b)

nd 2 Mode = 34.93 Hz

103 Amplitude

250

3rd Mode = 96.8 Hz

102 101 0

10

10−1

0

50

100 150 Frequency (Hz)

200

250

FIGURE 10.9 FFT analysis of response of cantilever beam given in Figure 10.8a, plotted on (a) linear scale and (b) log-scale.

first three modal frequencies of the beam (5.76 Hz, 35.23 Hz, and 96.8 Hz respectively). Even for the third mode, the appearance of the model frequencies is very distinctive because the EMD process has effectively decomposed the response signal into its respective components, which are narrow band. Hence, the energy peak of the third mode is not obscured by the first and second modes, which contain relatively higher energies. Similar analyses are done for the same beam under the two other support conditions, and the results (see Figure 10.13 and Figure 10.14) obtained are comparable to those obtained by the conventional direct FT. However, for the case of the fixed-fixed condition, the third modal frequency could not be identified by the FT method (Figure 10.14). Instead, the first and second modal frequencies are

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Comparison of Hilbert-Huang, Wavelet, and Fourier Transforms

1200

227

(a)

st 1 Mode = 22.89 Hz

1000

Amplitude

800 600 400 200 0

nd

2 Mode = 81.8 Hz 0

50

100 150 Frequency (Hz)

Amplitude

10

200

nd

2 Mode = 81.8 Hz

250

(b)

rd

st

1 Mode = 22.89 Hz

2

rd

3 Mode = 165.8 Hz

3 Mode = 165.8 Hz

100

−2

10

10

−4

0

50

100 150 Frequency (Hz)

200

250

FIGURE 10.10 FFT analysis of response of fixed-fixed beam given in Figure 10.8c, plotted on (a) linear scale and (b) log-scale.

each contained in two IMF components (fourth and fifth IMFs and second and third IMFs respectively; see Figure 10.14). On closer observation, the third and fourth modal frequencies can be spotted in the Fourier spectrum of the second IMF component, but they are obscured by the stronger presence of the second modal frequency. This setback of the proposed method can be explained by the fact that IMFs may contain mix modes. This may occur when the frequency of interest in an IMF is only contained along a particular segment of the signal and not throughout the entire time domain. This is the so-called intermittency problem discussed by Huang et al [17]. The original HHT was also performed on the decomposed IMF components of the response signals for the beam under the different support conditions. However, as mentioned earlier, the frequency of interest in an IMF may be contained only

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The Hilbert-Huang Transform in Engineering

1400

2nd Mode = 138 Hz

st

1 Mode = 48 Hz

1200

(a)

Amplitude

1000 800 600 400 rd 3 Mode = 268 Hz

200

Amplitude

0

10

4

10

3

10

2

10

1

10

0

0

100

st

200 300 Frequency (Hz)

400

500

nd

1 Mode = 48 Hz

(b)

2 Mode = 138 Hz 3rd Mode = 268 Hz th

4 Mode = 446 Hz

−1

10

−2

10 10

−3

0

100

200 300 Frequency (Hz)

400

500

FIGURE 10.11 FFT analysis of response of simply supported beam given in Figure 10.8b, plotted on (a) linear scale and (b) log-scale.

along a particular segment of the signal; hence, a weighted average using the amplitude is performed on segments of the frequency spectrum from HT for each IMF to obtain the modal frequencies of the beam. To illustrate, spectra for the beam under the three support conditions will be presented in detail individually. First, consider the HT of the IMFs for the cantilever beam (Figure 10.15). From the Hilbert spectra of the three IMF components, it can be observed that the EMD has effectively decomposed the original response signal into its different components, which are well contained within a narrow band, as the fluctuation of the frequency along the time axis is rather small. For this case, weighted averaging was performed over the entire time domain for each of the IMF components. The modal frequencies obtained are 101.56 Hz, 35.86 Hz, and 5.64 Hz from the first, second, and third IMFs, respectively, which corresponds well with the theoretical values.

© 2005 by Taylor & Francis Group, LLC

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Comparison of Hilbert-Huang, Wavelet, and Fourier Transforms

229

c(1)

5 0 −5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 (a)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

c(2)

10 0 −10

c(3)

20 0 −20 Residue

10 5 0 −5

Time, t(s)

180

rd

100 80 60

200 150 100

20

50 0

50

100

150

200

250

×10

0

50

100

150

200

250

4

st

(d)

1 Mode = 5.76 Hz

3

0

Frequency (Hz)

Frequency (Hz)

Amplitude

250

40

3.5

(c)

300

120

0

nd

2 Mode = 35.23 Hz

350

Amplitude

Amplitude

140

400

(b)

3 Mode = 96.8 Hz

160

2.5 2 1.5 1 0.5 0

0

50

100

150

200

250

Frequency (Hz)

FIGURE 10.12 (a) IMF components of response of cantilever beam given in Figure 10.8a; FT analysis of (b) first; (c) second; (d) third IMF component.

Next, for the simply supported beam, it can be observed in the HT spectra shown in Figure 10.16 that the frequency is stable only over a certain time range for each IMF. For example, for the first IMF, the frequency has a small fluctuation between

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The Hilbert-Huang Transform in Engineering

c(1)

5 0 −5

c(2)

5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 (a)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 −5

c(3)

10 0 −10 Residue

1 0 −1 −2

Time, t(s)

45

rd

3.5

30 25 20 15

2 1.5 1 0.5

50

100

150

200

250

900

rd

700

0

50

100

150

200

250

(d)

1 Mode = 22.95 Hz

800

0

Frequency (Hz)

Frequency (Hz)

Amplitude

3

5 0

(c)

2.5

10 0

nd

2 Mode = 82.85 Hz

4

Amplitude

Amplitude

35

4.5

(b)

3 Mode = 167.5 Hz

40

600 500 400 300 200 100 0

0

50

100

150

200

250

Frequency (Hz)

FIGURE 10.13 (a) IMF components of response of simply supported beam given in Figure 10.8b; FT analysis of (b) first; (c) second; (d) third IMF component.

0.1 sec and 0.3 sec; hence, a weighted average on the frequency is done over this time range. The frequency obtained is 159.18 Hz, which is closer to the third vibration mode of the beam. Similar analysis is done for the second IMF (between

© 2005 by Taylor & Francis Group, LLC

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Comparison of Hilbert-Huang, Wavelet, and Fourier Transforms

231

c(1)

10 0

c(2)

−10 0 5

c(3) c(4)

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5 (a)

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 −10 0 5 0 −5 0 2

c(5)

0.1

0 −5 0 10

Residue

0.05

0 −2 0 0.5 0 −0.5

0

Time, t(s)

0.7

90

(b)

Amplitude

Amplitude

70

0.5 0.4 0.3 0.2

60

rd

3 Mode (268 Hz)

50 40

th

30

4 Mode (460 Hz)

10 0

100

200

300

400

500

0

600

0

100

600

nd

300

400

500

st

600

400 300 200

(e)

1 Mode = 48 Hz

300

Amplitude

Amplitude

350

(d)

2 Mode = 138 Hz

500

200

Frequency (Hz)

Frequency (Hz)

250 200 150 100

100 0

(c)

20

0.1 0

nd

2 Mode = 138 Hz

80

0.6

50 0

100

200

300

400

Frequency (Hz)

500

600

0

0

100

200

300

400

500

600

Frequency (Hz)

FIGURE 10.14 (a) IMF components of response of fixed-fixed beam given in Figure 10.8c; and FT analysis of (b) first; (c) second; (d) third; (e) fourth IMF; (f) fifth IMF component.

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232

The Hilbert-Huang Transform in Engineering

45

st 1 Mode = 48 Hz

40

(f )

35 Amplitude

30 25 20 15 10 5 0

0

100

200

300 400 Frequency (Hz)

500

600

FIGURE 10.14 (continued)

0.12 sec and 0.28 sec), and the result (77.88 Hz) gives the second vibration mode. For the third IMF, a rather stable fluctuation of the frequency is observed over the entire time domain; hence, a weighted average is taken over the entire data range. This obviously gave the result corresponding to the first fundamental frequency (22.80 Hz). The Hilbert spectra for the IMF components of the fixed-fixed condition are shown in Figure 10.17. Considering the first IMF, a large fluctuation of the frequency over the entire time domain is observed. This can be explained by the fact that the filtered component effectively contains the noise in the original response signal; hence, meaningful modal parameters of the beam can be extracted from it. The second IMF reveals a stable frequency region with high amplitude over the time domain between 0.05 sec and 0.12 sec. Taking the weighted average over this time range gives the second modal frequency (135.34 Hz). For the third IMF, two stable frequency fluctuation regions with high amplitude can be identified, namely, from 0 to 0.16 sec and from 0.16 to 0.46 sec. By performing a weighted average on the two regions, two fundamental frequencies are obtained (138 Hz and 263.19 Hz). A similar stable region is observed for the fourth IMF, from 0.04 to 0.15 sec; this corresponds to an averaged frequency of 46.96 Hz, which gives the first modal frequency of the beam. In the fifth IMF, the two stable regions can be pinpointed, from 0 to 0.15 sec and from 0.24 to 0.48 sec. These two regions give a weighted average frequency of 42 Hz and 43.3 Hz respectively. Last, wavelet analysis using the Morlet function was also performed on the three signals of the beam under the three support conditions. WT was first done on the original response signals for the beam over a large scale range. By observing the dominant energy contents in this overall spectrum, different scales were zoomed in to obtain the dominant frequency contents. For example, WT was first performed for the vibration response of the cantilevered beam given in Figure 10.8a over a wide scale range of 1 to 256. This spectrum plot (see Figure 10.18a) shows three regions with a consistent existence of a corresponding frequency throughout the

© 2005 by Taylor & Francis Group, LLC

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Frequency, (Hz)

Frequency, (Hz)

Comparison of Hilbert-Huang, Wavelet, and Fourier Transforms

500 450 400 350 300 250 200 150 100 50 0

100 90 80 70 60 50 40 30 20 10 0

233

2.0

1.0 (a)

0 0

0.2

0.4

0.6 Time, t(s)

0.8

1

8.0

4.0 (b)

0 0

0.2

0.4

0.6 Time, t(s)

0.8

1

60

20

Frequency, (Hz)

50 40 10

30

(c)

20 10 0

0 0

0.2

0.4

0.6 Time, t(s)

0.8

1

FIGURE 10.15 HT spectrum for (a) first, (b) second, and (c) third IMF component for response of cantilever beam as given in Figure 10.12a.

entire time domain, namely, in scales 105 to 183, 14 to 27, and 4 to 12. By focusing toward these regions, a more refined dominant frequency can be deduced, namely at scales of 144, 22.66, and 8, which corresponds to frequencies of 5.64 Hz, 35.86

© 2005 by Taylor & Francis Group, LLC

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Frequency, (Hz)

234

The Hilbert-Huang Transform in Engineering

500 450 400 350 300 250 200 150 100 50 0

4.0

2.0 (a)

0 0

0.2

0.4

0.6

0.8

1

Time, t(s) 300

3.0

Frequency, (Hz)

250 200 1.5 (b)

150 100 50 0

0 0

0.2

0.4

0.6

0.8

1

Time, t(s) 350

8

Frequency, (Hz)

300 250 200

4

(c)

150 100 50 0

0 0

0.2

0.4

0.6

0.8

1

Time, t(s)

FIGURE 10.16 HT spectrum for (a) first, (b) second, and (c) third IMF component for response of simply supported beam as given in Figure 10.13a.

Hz, and 101.56 Hz respectively. These results give the three fundamental frequencies of the beam. The wavelet analyses for the other two support conditions are presented in Figure 10.19 and Figure 10.20.

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FIGURE 10.17 HT spectrum for (a) first, (b) second, (c) third, (d) fourth, and (e) fifth IMF component for response of fixed-fixed beam as given in Figure 10.14a.

The results obtained by the different signal analysis methods are compared in Table 10.4. It can be observed that all the methods produce results of approximately the same order of error. The methods are equally feasible in determining the modal

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FIGURE 10.17 (continued)

frequencies of the beam. However, these methods do contain imperfections. For example, the direct FT method may not be very effective for cases where the modal frequencies have energy of same order as that of other frequency ranges. Similarly, for the modified FT with EMD process, the identification of certain modal frequencies may not be possible when the IMF components contain mixed modes. For HHT, the definition of the stable frequency region over the time domain, although subjective, is least problematic. For WT, identification of the precise scale from a range of scales for the dominant frequency can be ambiguous.

10.4 CONCLUDING REMARKS A comparison of three signal-processing techniques is presented. In the first study of flexural wave propagation in an aluminum beam, HHT is shown to be a more direct method compared to WT when no knowledge of the actuation frequency is available. In the second experiment, involving acoustic Lamb wave propagation in an aluminum plate, HHT and WT give similar results for the analysis of the propagating Lamb waves actuated and received by the PZTs. Good WT results hinge on the fact that the actuation

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FIGURE 10.18 Morlet WT analysis for response of cantilever beam given in Figure 10.8a at scales (a) 1–256, (b) 144, (c) 22.66, and (d) 8 with sampling rate 1000 samples/sec.

frequency is narrowband and known a priori. For the last case, the combination of the EMD process from HHT with FT on the decomposed IMF components has proven to be capable of revealing modal frequencies of the aluminum beam having relatively lower energy contents. These modal frequencies may be overlooked by the traditional direct FFT, unless a logarithmic plot is done for the energy spectrum. However, the problem of mixed modes within individual IMFs needs to be addressed. The original © 2005 by Taylor & Francis Group, LLC

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22.66

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FIGURE 10.18 (continued)

HHT with knowledge of the problem does provide good results when it is used with an averaging concept. These examples show that HHT is a suitable tool for processing nonstationary wave propagation signals encountered in damage detection studies. It provides a good representation of localized events in both the frequency and energy of any transient signal collected. Indeed, it is a simple signal-processing tool to use and provides reasonably good results. © 2005 by Taylor & Francis Group, LLC

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FIGURE 10.19 Morlet WT analysis for response of simply supported beam given in Figure 10.8b at scales (a) 1–256, (b) 33.5, (c) 10, and (d) 5 with sampling rate 1000 samples/sec.

ACKNOWLEDGMENTS The authors wish to thank the National University of Singapore for providing the support to perform this study.

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FIGURE 10.19 (continued)

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1958 1855 1752 1649 1546 1443 1340 1237 1134 1031 928 825 722 619 516 413 310 207 104 1

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FIGURE 10.20 Morlet WT analysis for response of fixed-fixed beam given in Figure 10.8c at scales (a) 1–2042, (b) 163, (c) 60, (d) 28, and (e) 18 with sampling rate at 10000 samples/sec.

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FIGURE 10.20 (continued)

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FIGURE 10.20 (continued)

REFERENCES 1. Crema L B, Peroni I and Castellani A (1985). Modal Tests on Composite Material Structures Application in Damage Detection. Proc. Int. Modal Anal. Conf. Exhibit Vol. 2, pp. 708–713. 2. Cook D A and Berthelot Y H (2001). Detection of Small Surface-Breaking Fatigue Cracks in Steel Using Scattering of Rayleigh Waves. NDT & E Int. 34:483–492. 3. Hurlebaus S, Niethammer, Jacobs L J and Valle C (2001). Automated Methodology to Locate Notches with Lamb Waves. Acous. Soc. Am. ARLO 2(4):97–102. 4. Staszewski W J (1998) Structural and Mechanical Damage Detection Using Wavelets. Shock Vibr. Digest 30(6):457–472. 5. Kishimoto K, Inoue H, Hamada M and Shibuya T (1995). Time Frequency Analysis of Dispersive Waves by Means of Wavelet Transform. ASME J. Appl. Mech. 62: 841–846. 6. Lee Y Y and Liew K M (2001). Detection of Damage Locations in a Beam Using the Wavelet Analysis. Int. J. Struct. Stab. Dyn. 1(3):455–465. 7. Quek S T, Wang Q, Zhang L and Ong K H (2001). Practical Issues in Detection of Damage in Beams Using Wavelets. Smart Mater. Struct. 10(5):1009–1017. 8. Huang H, Norden E, Zheng S, Long S R, Wu M C, Shih H H, Zheng Q, Yen N C, Tung C C and Liu H H (1998). The Empirical Mode Decomposition and Hilbert Spectrum for NonLinear and Nonstationary Time Series Analysis. Proc. R. Soc. Lond. A 454:903–995. 9. Huang H, Norden E, Zheng S and Long S R (1999). A New View of Nonlinear Water Waves: The Hilbert Spectrum. Annu. Rev. Fluid Mech. 31:417–457.

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10. Huang H, Norden E, Chern C C, Huang K, Salvino L W, Long S R and Fan K L (2001). A New Spectral Representation of Earthquake Data: Hilbert Spectral Analysis of Station TCU129, Chi-Chi, Taiwan, 21 September 1999. Bull. Seism. Soc. Am. 91(5):1310–1338. 11. Zhang R R, King R, Olson L and Xu Y (2001). A HHT View of Structural Damage from Vibration Recordings Proc. ICOSSAR ’01 (San Diego) – CD. 12. Shim S H, Jang S A, Lee J J and Yun C B (2002). Damage Detection Method for Bridge Structures Using Hilbert-Huang Transform Technique. 2nd Int. Conf. Struct. Stab. Dynam., Dec. 16–18, 2002, Singapore. 13. Quek S T, Tua P S and Wang Q (2003). Detecting Anomaly in Beams and Plate Based on Hilbert-Huang Transform of Real Signals. Smart Mater. Struct. 12(3): 447–460. 14. Tua P S, Jin J, Quek S T and Wang Q (2002). Analysis of Lamb Modes Dominance in Plates via Huang Hilbert Transform for Health Monitoring. Proc. 15 KKCNN Symp. Civil Eng., Ed. S T Quek and D W S Ho, Dec. 19–20, 2002, Singapore. 15. Hahn S L (1996). Hilbert Transform in Signal Processing. London: Artech House Boston. 16. Graff K F (1975). Wave Motion in Elastic Solids. Oxford: Clarendon Press. 17. Huang H, Norden E, Zheng S and Long S R (1999). A New View of Nonlinear Water Waves: The Hilbert spectrum. Annu. Rev. Fluid Mech. 31:417–457.

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11

The Analysis of Molecular Dynamics Simulations by the Hilbert-Huang Transform Adrian P. Wiley, Robert J. Gledhill, Stephen C. Phillips, Martin T. Swain, Colin M. Edge, and Jonathan W. Essex

CONTENTS 11.1 Introduction ..................................................................................................246 11.2 The Hilbert-Huang Transform .....................................................................249 11.3 Molecular Dynamics ....................................................................................251 11.4 Reversible Digitally Filtered Molecular Dynamics.....................................254 11.5 Limitations ...................................................................................................262 11.6 Conclusions ..................................................................................................262 References..............................................................................................................263 ABSTRACT Proteins are an integral component of all living systems, and obtaining a detailed understanding of their structure and dynamics is of considerable importance. Molecular dynamics computer simulations are an important tool in this process. However, the length of simulation required to probe the full range of motions available to proteins

245

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is prohibitively expensive. As a solution to this problem, we have developed a technique, Reversible Digitally Filtered Molecular Dynamics (RDFMD), that can enhance or suppress specific frequencies of motion. We have demonstrated that RDFMD is able to amplify the low frequency vibrations in the system that are responsible for largescale changes in structure. In this paper we describe the use of empirical mode decomposition (EMD) and the Hilbert-Huang transform (HHT) to probe the effect of RDFMD on the internal vibrations of molecular systems, including a simple protein, in a molecular dynamics (MD) simulation. EMD allows the decomposition of the MD trajectory into intrinsic mode functions (IMFs) that are shown to be of physical relevance to the input signal. In particular, spontaneous changes in molecular shape are shown to be accompanied by increased energy in low frequency IMFs. Moreover, we show that the HHT enables a superior analysis of the MD trajectories than Fourier-based techniques and provides essential information that may be used to determine the optimum parameters for the RDFMD method.

11.1 INTRODUCTION Proteins are polymer chains made up from 20 possible monomer units (amino acids), each sharing a common backbone structure with differing functional side chains (1). YPGDV, a simple protein used in later discussion, consists of five amino acids: tyrosine, proline, glycine, aspartic acid, and valine (Figure 11.1). It is convenient to consider the amino acid chains (the primary structure) in terms of relatively rigid and stable secondary structure units. These are classified as either α-helices, where a section of coiled amino acids forms a rod-like helix, or β-sheets, where amino acid strands form layered structures. These units of secondary structure are connected by flexible loop regions, and secondary structure units can themselves associate to form compact globular units called domains that move in a correlated fashion. Understanding the functions and mechanisms of biological molecules requires detailed knowledge of their structure and internal motions. The arrangement, or conformation, of a protein is of considerable importance to its biological function (2, 3). For example, Creutzfeldt-Jakob Disease (CJD) is a fatal neurodegenerative

FIGURE 11.1 Pentapentide YPGDV (Tyr-Pro-Gly-Asp-Val) with backbone dihedral angles φi and ψi labeled.

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disorder that is generally believed to be caused by conformational change in the prion protein (4). The prion (proteinaceous infectious particle) theory proposes that the prion protein (a normal constituent of mammalian cells) undergoes a conformational change to an active form that induces the same change in further prion proteins. Protein aggregation then occurs, which is associated with cell death. Large-scale conformational motion occurs predominantly by rotation about the single bonds in the protein, the so-called dihedral angles, since a small change in these angles can have a significant effect on the structure of the protein. The important dihedral angles in the main-chain of YPGDV are marked in Figure 11.1. Anharmonic, low frequency motions in dihedral angles are responsible for conformational changes (5, 6). There are several experimental methods for determining three-dimensional structures of molecules, of which X-ray crystallography (7) is undoubtedly the most common. The molecular arrangements seen in X-ray crystallography are, however, “snapshots” of a protein that is not in its natural environment. Often, proteins can be crystallized in a variety of arrangements that differ significantly (8, 9), and it is believed that crystal-packing forces can have a significant influence on the conformer observed (10). Analytical techniques, such as essential dynamics (11) or normal mode analysis (12), can be applied to experimentally obtained structures in order to give limited information about the flexibility of the protein (13). Another prominent experimental technique for studying protein structures is nuclear magnetic resonance (NMR) (14). Here data is gathered about the protein structure when it is in solution, so that the crystal-packing artifacts associated with X-ray crystallography are avoided. Unfortunately the method is time consuming owing to the need for significant isotopic replacements to allow for the spectra to be assigned. Circular dichroism (CD) is a form of light absorption spectroscopy that is very sensitive to the secondary structure of proteins (15). The relative amounts of disordered, helical, and sheet secondary structure can be determined, although problems can be encountered if side chains also contribute to the spectrum. Experimental methods are generally unable to follow atomic coordinates with sufficient time resolution to describe the nature of many protein motions. For example, the T4 lysozyme (T4L) enzyme consists of two main domains linked by a hinge region, and genetic mutants have been crystallized with a range of hinge angles (8). Unfortunately, the natural form of the enzyme can only be crystallized in a closed state in which there is no space for known substrates to bind. Although NMR data suggest that the closed state is abundant in solution (16), there is significant evidence that an opening event must occur. The structure of this open conformer is thought to be similar to that crystallized from a mutated form of the protein (17). Experimental methods are unable to observe this opening/closing event. To increase our understanding of the mechanisms of conformational change, a technique is therefore required that can follow atomistic detail on the time scales over which the relevant motions occur. Molecular dynamics (MD) is a computational modeling tool that follows the trajectories of atoms through time. Improved software and the decreasing cost of

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computer power now allow routine simulations of large protein systems of around 10 ns. Over this time scale, conformational events such as the opening of the two large flaps in HIV-1 protease can be observed (18). A classical description of atoms is generally used in MD; the energy of the system is assumed to be independent of the location of electrons (BornOppenheimer approximation), so atoms are not polarizable and are described by a point location only. MD effectively solves Newton’s Second Law,F = ma, for each atom, using a finite difference integrator to overcome the many body problem. One such integrator, the Verlet algorithm (19), is shown in Equation 11.1. The time step, δt, must be sufficiently small to keep the system energy stable, yet as large as possible for efficient time evolution of the system (a 2 fs time step is commonly employed). Forces are obtained from the gradient of a potential energy function. This function has separate terms for bond stretching, angle deformations, dihedral motion, and non-bonded interactions. The parameters used for the potential energy function are optimized with data from vibrational spectra, experimental geometry information, and ab initio quantum mechanical calculations (20).

(

)

() (

)

()

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(11.1)

Simulations can be run with constant energy, although various thermostats (21, 22, 23, 24) and barostats (22, 23) can be applied to reproduce more biologically relevant constant temperature and constant pressure ensembles. Frequency analysis of molecular dynamics trajectories yields spectra that can be compared to infra-red experimental data. The majority of intramolecular motions are wavelike in nature and exist either as localized oscillations in single bonds or angles, or as collective motions (or modes) characterized by coupled vibrations. For example, the amide I mode in proteins that originates from the carbonyl stretching vibration also includes a contribution from the backbone carbon–nitrogen bond (25). Postsimulation digital filtering has been applied to molecular dynamics simulations to remove high frequency motions that are irrelevant to conformational changes. One simple approach Fourier transforms the atomic trajectories, applies a frequencydomain filter, and inverse Fourier transforms the result (26). Alternatively the filtering can be done in the time-domain by using non-recursive digital filters (27). These methods have proved useful for the visualization of low frequency motions that are important for conformational change. Although MD has been used to study protein conformational change, the time scales over which these motions occur are generally longer than those accessible by conventional simulation routes. Methods to overcome this problem are therefore required. Reversible Digitally Filtered Molecular Dynamics (RDFMD) is a method that applies digital filters to the velocities in an evolving simulation to promote low frequency motions. RDFMD has been successfully applied to a range of systems (28, 29). The analysis of an RDFMD simulation cannot be achieved with sufficient time resolution by using traditional Fourier-based methods, especially given the discovery of nonstationary frequency targets (30). The Hilbert-Huang transform provides the necessary analysis tools, as presented here.

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11.2 THE HILBERT-HUANG TRANSFORM The Hilbert-Huang transform (HHT) is a combination of empirical mode decomposition (EMD) and the Hilbert transform (HT). The method has been applied across a broad range of fields, including biology (31, 32), geophysics (33), and solar physics (34). Numerous applications to simple systems and comparisons to Fourier-based techniques have also been performed (30, 31). The Hilbert transform takes a time-domain signal and transforms it into another time-domain signal, unlike the Fourier transform, which moves from the time to the frequency domain. The HT of a real-valued function, x(t), over the range –∞ < t < +∞ is another realvalued function, h(t), which is the convolution of x(t) with 1/πt (Equation 11.2). In this equation, P signifies the Cauchy principal value. The rapid decay of 1/(t–u) biases the transform heavily to points close to t, and thus the HT acts on time-localized data. ∞

h(t ) = Ρ

∫ π (t − u) du x (u)

(11.2)

−∞

In practice, the Hilbert transform can be computed by taking the Fourier transform of the data, setting the negative frequency components to zero, doubling the positive frequency components, and back-transforming (35). This method relies on the infinite replication of the data set required by the Fourier transform, yielding unreliable regions at the start and end of the transformed data due to the discontinuities introduced. These regions are excluded from our analysis. It can be shown that the signal magnitude is unchanged by HT, but the phase is adjusted by π/2 (35). The original signal, x(t), and its Hilbert transform, h(t), may be considered part of a complex signal, z(t):

() ()

()

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(11.3)

This can also be written in terms of amplitude, A, and phase, φ:

()

()

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(11.4)

where

()

()

()

(11.5)

h t  φ t = tan −1  . x t 

(11.6)

A t = x 2 t + h2 t and

()

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The rate of change of the phase angle with time is the frequency of motion occurring. The instantaneous frequency, f(t), is thus defined as:

()

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()

1 dφ t . 2π dt

(11.7)

The minimum frequency that can be reliably sampled by this method is 1/T (31), where T is the time length of the data set. For an instantaneous frequency to be physically meaningful, the signal must carry no riding waves and must be locally symmetrical about its mean point, as defined by the envelope of the local maxima and minima. Real-world phenomena that are described by signals matching these criteria are not common. EMD is a recently developed signal-processing method. It decomposes a signal, which may be nonstationary, into a set of intrinsic mode functions (IMFs) suitable for the Hilbert transform (31). An IMF is defined as a wave in which the number of extrema and the number of zero-crossings differ by a maximum of one and where, at any point, the mean of the envelope defined by the local maxima and minima is zero. To generate an IMF, the local mean is repeatedly determined and subtracted from the data set until the number of extrema and zerocrossings in the residual data differ by at most one (this process is termed “sifting”). The local mean is found by taking the mean of a curve through all of the maxima and a curve through all of the minima. These curves are defined, in this work, by cubic spline functions. The algorithm proceeds by subtracting each generated IMF from the remaining signal until either the recovered IMF or the residual data is small, or the residual has become a trend component with only a single maximum and minimum. The final residual of the data is similar in concept to the DC component of a Fourier transform, except that it contains the overall trend of the data. Once a signal has been decomposed into a set of IMFs, each IMF is separately Hilbert transformed, giving the amplitude and instantaneous frequency at discrete points in time. If the IMF is a harmonic oscillator of variable amplitude and frequency, then its signal energy at time t can be written in terms of the signal amplitude, A, and frequency, v:

( ) ( ) A (t ) .

E t ∝v t

2

2

(11.8)

In the HHT spectra presented in this work, the energy in each IMF is calculated and presented using Equation 11.8. This paper presents the application of the HHT method to the analysis of MD and RDFMD simulation trajectories, with a view to obtaining a better understanding of system dynamics, validating the assumptions underlying the RDFMD methods, and deriving some of the parameters associated with the RDFMD approach.

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FIGURE 11.2 The bond length (C=O) and angle (CA-C-N) used in the frequency analysis.

11.3 MOLECULAR DYNAMICS The YPGDV protein is a common target for conformational studies by molecular dynamics methods (36) and NMR (37), and it has been used as an RDFMD test case (29). Results included here have been derived from previously published constant energy (30) and RDFMD (29) simulations. The versatility of computer simulation allows the selection of specific internal coordinates for analysis. For example, it is trivial to extract the progression of an angle or bond during a constant energy YPGDV simulation and to locate the frequencies of motion with a windowed Fourier transform. Figure 11.2 illustrates a bond-stretching and angle-bending motion in a section of YPGDV, and Figure 11.3 shows the Fourier transform of the simulation trajectory for these degrees of freedom. These harmonic motions are stationary throughout the simulation, and the analysis extracts the expected results: the amide I vibration between 1600 cm–1 and 1800 cm–1, the angular motion at around 500 cm–1, and collective backbone motions between 800 cm–1 and 1200 cm–1. The angular motion has a large component at the bondstretching frequency of 1600 to 1800 cm–1, showing that the angles are coupled to the bond stretches. However, the bond motion has no low frequency component. The time integral of the HHT spectrum, the marginal spectrum (data not shown), is broadly similar to the Fourier spectrum. However, the HHT analysis can be continued to greater depth. Figure 11.4 shows the HHT for the angle-bending CA-C-N vibration between glycine and aspartate, and Figure 11.5 presents the spectrum for C=O bondstretching vibration. The Hilbert spectra show energy flowing into and out from the high frequency vibrations at the same times, demonstrating the coupled nature of these vibrations. Figure 11.6 plots the highest frequency IMFs of the glycine–aspartic acid backbone angle (CACN) and of the glycine carbonyl length (C=O); these are clearly in phase. Subtracting the first IMF from the original signal and taking the Fourier transform of the result shows that this IMF is responsible for all the vibrational motion above approximately 1250 cm–1. The signals of most relevance to conformational change are the low frequency dihedral motions. These are generally nonstationary, and Fourier analysis provides a poor description of conformational events (30). By looking at the energy in low frequency regions as a function of time, Hilbert techniques allow recognition of

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FIGURE 11.3 Fourier transform of backbone degrees of freedom using a Hanning window and a 50-point running average.

events that occur simultaneously across several dihedrals. Figure 11.7 plots the eight main-chain dihedral angles of YPGDV that are most important for conformational change (see Figure 11.1) as a function of time in a region where a spontaneous conformational transition occurred. The energy in the 0 to 10 cm–1 region obtained by integrating the HHT spectrum is also shown, and the three highest energy peaks occur from 15 to 22 ps, corresponding to a rearrangement of dihedrals ψ2, φ3, and ψ3. Several dihedrals are often seen moving at once, as this reduces the need for significant solvent reorganization. It is necessary to show that the EMD method produces physically relevant IMFs. Results for the φ3 data of the spontaneous transition are shown in Figure 11.8. The low frequency IMFs follow physical motions from the signal, and there is an obvious separation between the high frequency (IMFs 1 to 5) and low frequency motions (IMFs 6 to 10). The application of EMD as a signal-smoothing technique is also clear. It is a requirement of the HHT method that frequencies of motion be separable. Without this, artifacts may enter the data (31). This is inherent in the method because, in each sifting iteration, the extracted IMF tries to follow the highest frequency motion in the remaining signal. Meaningful separation is not always guaranteed for dihedral data due to the flexible nature of the simulation model, and so EMD results must always be interpreted with caution. However, we have shown the physical relevance of IMFs and, as Figure 11.8 shows, we find that a single IMF describes

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Time/ps

FIGURE 11.4 HHT of bond angle trajectory CA-C-N.

conformational events. Sometimes the frequency separation of IMFs is not complete, and the physical relevance of a single IMF is not clear. A blind analysis of a single IMF is therefore not sensible, but by averaging results over several signals or by integrating over long time periods, a more robust analysis may be obtained. In this section the ability of the HHT to yield extra information over and above that available by a conventional Fourier analysis has been demonstrated. The coupled nature of vibrational motion in molecular systems has been clearly shown, as has the ability of EMD to smooth signals of high frequency data that are not required and can indeed obscure overall trends. The analysis of conformational change events has revealed that these motions are associated with low frequency vibrations, vindicating the assumptions underpinning the RDFMD method. The physical relevance of the IMFs obtained from these signals has been shown, although the separability of scales required for HHT does not always apply. Consequently, the EMD results must always be interpreted with caution.

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2000

Frequency/wavenumbers

1800

1600

1400 0

10

20

30

Time/ps

FIGURE 11.5 HHT of bond length trajectory C=O.

11.4 REVERSIBLE DIGITALLY FILTERED MOLECULAR DYNAMICS The basis of the RDFMD method is the application of digital filters to the atomic velocities in an ongoing molecular dynamics simulation to promote low frequency motion and hence increase the rate of conformational change (29). The digital filters used in RDFMD are linear phase, non-recursive filters designed in MATLAB (38). A time series of input velocity data is required, and a mathematical function is applied to the data to produce modified output velocities. For example, to apply a filter to a single atom in the simulation, the atom’s velocity is stored for every time step until a buffer of 2m + 1 velocities, v, is obtained. The filter is a list of 2m + 1 real coefficients, ci, which is applied to the velocity buffer to produce a single output velocity, v ′, as shown in Equation 11.9. The application of a filter to a signal (the velocity buffer in this case) may affect the frequencies in the signal in different ways — it may enhance the amplitudes of some frequency components and reduce the amplitudes of others. The filter coefficients can be chosen to control the effect of

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Signal Amplitude

1st IMF of Gly/Asp CA-C-N 1st IMF of Gly C = 0

0

0.1

0.2

0.3

Time/ps

FIGURE 11.6 The first IMFs of the coupled angle, Gly / Asp CA-C-N, and bond motion, Gly C=0.

Angle / degrees

300

200

Energy in 0 -10 cm-1 region

100

0

0

10

0

10

20

30

20

30

Time / ps

FIGURE 11.7 Top: The eight dihedral angle trajectories around a spontaneous conformational transition. Bottom: The energy in the 0 to 10 cm–1 region extracted using HHT.

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FIGURE 11.8 YPGDV ϕ3 data during a trajectory containing a spontaneous conformational transition. Top: Signal (solid line) and residual (dashed line). Upper middle: Sum of high frequency IMFs (1 to 5). Lower middle: Low frequency IMFs (6 to 10). Bottom: Sum of low frequency IMFs (6 to 10).

the filter on the input signal. For our purposes, the desired frequency response is an amplification of low frequencies (typically those lower than 100 cm–1) with no change in the amplitude of higher frequencies. The coefficients of the digital filter are chosen such that the actual frequency response of the filter matches the desired response as closely as possible (see Figure 11.9). The longer the list of coefficients, the closer the actual frequency response is to the desired frequency response. However, short filters are desirable as they require fewer data points and hence less simulation. We have found filters of 1001 coefficients (i.e., m = 500) to be sufficient. m

v′t =

∑c v

i t −i

.

(11.9)

i =− m

The RDFMD method generally applies a digital filter to a set of atoms in an MD simulation that are of particular interest to the simulator (e.g., just the solute and not the solvent, or just a region of a protein). A normal MD simulation is performed to fill a buffer of length 2m + 1 (corresponding to the number of coefficients in the filter) for each atom of interest, and the filter is applied to each of the atoms to enhance or suppress motion in the chosen frequency ranges. A digital filter affects the phase of each frequency component in the signal as well as the frequency, but, by using a property of linear phase filters, the phase can also be controlled. The output of a linear phase filter is in phase with the midpoint of the filter input buffer. It is for this reason that the system state (coordinates, accelerations, and forces) is

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1.5 Actual frequency response Desired frequency response

Frequency Response

1

0.5

0

−0.5

0

25

50 Frequency/wavenumbers

75

100

FIGURE 11.9 The desired and actual frequency responses for a 1001 coefficient filter.

saved as the midpoint and then restored after the application of the filter — ensuring that the new velocities are in phase with the atomic positions. After applying a filter, we allow the system to evolve for a number of steps d. Another filter may then be applied in order to build up gradually the kinetic energy in the low frequency modes. We have been using a value of d = 20 time steps (29), and as a safeguard against unrealistic, high energy events, the filter series is terminated early if the kinetic energy of the system exceeds that equivalent to a temperature of 2000 K. Since new velocities are applied at the midpoint of a buffer, if d is less than m, molecular dynamics must be run backward in time to fill the buffer, as shown in Figure 11.10. This is possible with the velocity Verlet MD integrator (39). This flexible method allows many filters to be applied, with subsequent filter applications positioned anywhere in time relative to the first. It is, however, possible that the system is not approaching a conformational transition, and amplifying low frequency motions will have little effect. It is for this reason that the number of filter applications in a series is limited to ten, after which the simulation proceeds as normal from the end of the final filter buffer. Once filter applications have been terminated (either by reaching the maximum number of filters allowed or by heating to the temperature cut-off), 20 ps of MD are performed, giving a new starting structure from which the filtering process can be repeated. Prior to the RDFMD simulation, a desired frequency target must be found. A short MD simulation of YPGDV has been used for this purpose. Since conformational motions are complex with non-stationary frequencies, Fourier methods yield

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FIGURE 11.10 RDFMD sequence showing the delay parameter, d, and the filter length, 2m + 1.

little information. Figure 11.11 presents a spontaneous conformational change and shows the energy present in defined frequency ranges. These data are derived by integrating the associated HHT spectrum over frequency. It is clear that the spontaneous conformational change is associated not only with vibrations in the 0 to 10 cm–1 region but also with vibrations up to 50 cm–1. Furthermore, there are other locations in the trajectory where energy is present in the region 25 to 50 cm–1. These data suggest that although the conformational change itself is associated with very low (< 10 cm–1) frequency motion, to enhance conformational change, the digital filter should be designed to amplify vibrations up to 50 cm–1. For this paper, a set of RDFMD simulations of YPGDV in water solvent has been performed. The digital filter was applied to all atoms of YPGDV and not to the solvent. A starting conformation was chosen, and ten sets of random initial velocities were generated. From each set, seven filter application series were run, using digital filters to amplify the 0 to 25 cm–1 range by two, three, four, five, six, seven, and eight times. The filters used 1001 coefficients, and each filter series consisted of up to ten filter applications using a delay, d, of 20 steps and a temperature cut-off of 2000 K. Once a filter has been applied to a buffer, the next buffer should contain an amplification of any low frequency motions present in the system. The HHT method can help demonstrate this, as Figure 11.12 shows. The top of the figure shows a representative example of a dihedral angle trajectory progressing through two filter buffers. Only the data corresponding to the middle section of the buffers are shown; the filter is applied at step 500. The bottom graph of the figure is the sum of the EMD trend data and the IMFs with components in the low frequency (0 to 50 cm–1)

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150 Signal 0–10 0–25 0–50

100

Dihedral Angle/degrees

Signal Energy/Arbitrary Units

200

50

0

10

20

30

0

Time/ps

Dihedral angle/degrees

FIGURE 11.11 YPGDV ψ2 data during a trajectory showing a spontaneous conformational transition. The signal and the energy in low frequency bands (cm–1), derived from the HHT spectrum, are shown.

1st buﬀer 2nd buﬀer

170

160

Displacement/degrees

150

170

160

150 450

500 Timesteps through buﬀer

550

FIGURE 11.12 YPGDV ψ2 data. Top: Signal before and after the application of a digital filter. Bottom: Sum of the residual and the low frequency IMFs showing the trend in the data.

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Dihedral angle change (RMSD)

30 Filter 1 Filter 2 Filter 3 Filter 4 Filter 5 20

10

0 Energy

FIGURE 11.13 Change in YPGDV backbone dihedral angles as low frequency kinetic energy is successively amplified.

region. The dihedral motion is clearly amplified either side of the central point in the direction of the low frequency trend. The amount of additional motion in the backbone dihedral angles caused by the application of a filter to the system should correlate with the amount of energy in the low frequency region at the point of filter application. To test this hypothesis, we define a measure of the additional dihedral motion going from one buffer to the next as the average of the root-mean-square deviations (RMSDs) of the eight dihedral angles in the central region of the buffer (the midpoint ± 100 steps). The amount of energy in the low frequency region is calculated by taking the instantaneous frequencies and amplitudes of the dihedral IMFs at the midpoint of the buffer and calculating the energy at each frequency, as described previously. To estimate the amount of energy in the filter amplification region, these energies are multiplied by the filter’s frequency response, scaled from unity at 0 cm–1 to zero at 100 cm–1. Finally, the energies are summed to provide our measure of the kinetic energy available for amplification. Figure 11.13 plots the dihedral angle RMSDs against the energy amplified for five successive applications of a ×5 amplifying filter. The energy and the dihedral motion are clearly correlated, though the data is quite spread. We also see a steady increase in the energy as the digital filter is applied to successive filter buffers. Fewer points are displayed for the later buffers due to simulations terminating as they exceed the temperature cut-off. A trend line has been fitted to the entire data set by a least-squares procedure, plotted to the extremes of the data set only.

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FIGURE 11.14 Change in YPGDV backbone dihedral angles as low frequency kinetic energy is successively amplified using different digital filters.

This analysis can be extended to investigate the effect of different amplification filters. High amplification filters would intuitively yield greater induced conformational change, although the temperature cutoff will be reached more quickly. In Figure 11.14, the analysis reported in Figure 11.13 has been extended to include different amounts of amplification by the digital filters. Only the trend lines obtained by least-squares fitting for all the filters are reported. As expected, as the filter’s amplification factor increases, so does the induced conformational change. It was found that the data for amplification in excess of ×6 were poorly correlated, presumably as a result of rapid simulation termination due to the temperature cut-off being exceeded, and consequently these data are not presented. The controlled increase of energy in low frequency motions is likely to be of greater importance to larger protein systems, for which lower temperature cutoffs may be required. The energy in the low frequency region reached a maximum at an amplification factor of ×3. However, factors of ×4 to ×6 all reach significant levels of induced conformational change, and higher factors do this in fewer buffers and thus in less computer time. It is believed that the tradeoff between slowly building up energy in low frequency motions and efficiently inducing a conformational response will have to be determined for each system to which RDFMD is applied. In this section, HHT has been applied to examine the hypothesis underpinning the RDFMD technique, namely that the amplification of low frequency vibrations in a molecular dynamics simulation of a protein increases the amplitude of dihedral angle vibrations and hence conformational change. Supporting this hypothesis, a correlation has been observed between the energy available for amplification by the

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digital filter and the resulting RMSD in dihedral motion. As expected, using larger amplifying filters increases the energy in the critical degrees of freedom more rapidly, although there will inevitably be a compromise between the need for efficiency by amplifying as rapidly as possible and the requirement that the conformational transitions be physically sensible without significantly disrupting the protein structure, for which gentle amplification will be required. Finally, more detailed analysis of spontaneous conformational transitions has confirmed that, although such transitions are indeed associated with very low frequency motions (< 10 cm–1), they are also associated with higher frequencies (25 to 50 cm–1). Analysis of the remainder of the simulations trajectory has identified other occasions where significant energy resides in this frequency region, but without associated conformational change. We suspect that these conditions are associated with possible conformational change events. Amplification of frequencies of up to 50 cm–1 may therefore be more efficient in terms of driving conformational change.

11.5 LIMITATIONS The limitations of the HHT method as applied to molecular dynamics simulations are essentially threefold. First, analysis of short (1001 time step) RDFMD filter buffers results in a low frequency limit in the Hilbert transform of 16.7 cm–1. Given that this limit lies in the frequency range over which amplification is taking place, it restricts our ability to analyze the simulation behavior over the content of the filter buffer. Second, conformational transitions often leave large residuals, giving dihedral angle plots similar to step functions. EMD produces physically unrealistic IMFs in an attempt to recreate the unusual signal given by the data less the residual, as shown in Figure 11.15. Although the data could be windowed to reduce the size of the affected regions, low frequency resolution would be lost. This limitation is due to the nature of the data and suggests that the HHT must be used with care, with constant checks of the physical relevance of the IMFs. Third, the HHT requirement of separable scales is not necessarily met by signals from molecular dynamics simulations. Great care must therefore be taken when performing analyses by, for example, careful examination of the IMFs to confirm their physical relevance, repeated analysis over many different simulations, averaging over a number of simulation signals, and integration of the HHT spectra over frequency and time.

11.6 CONCLUSIONS In this paper the application of the HHT method to the analysis of molecular dynamics computer simulations has been reported. The non-stationary data associated with these simulations are unsuitable for analysis with Fourier methods. However, it has been shown that the HHT method does yield useful insight into the dynamics. In particular, the assumptions underpinning the RDFMD method of enhancing conformational change, namely that low frequency vibrations are associated with conformational change and that non-recursive digital filters may be used to amplify these vibrations, have been confirmed. HHT analysis has also suggested

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FIGURE 11.15 YPGDV ψ2 data. Top: Signal and residual (dotted line). Middle: Signal after the residual is removed (equivalent to the sum of the IMFs). Bottom: IMFs generated by EMD.

that frequencies greater than those earlier identified with spontaneous conformational change may in fact be more appropriate targets for RDFMD. However, the presence of large residuals and an incomplete separation of scales indicate that HHT cannot be applied to these simulations as a “black box” and that care must be taken.

REFERENCES 1. Creighton, T. E. (1993). Proteins structures and molecular properties. 2nd ed. New York: W. H. Freeman and Company, pp. 192–193. 2. Gerstein, M., Lesk, A. M., Chothia, C. (1994). Structural mechanisms for domain movements in proteins. Biochemistry. 33(22):6739–6749. 3. Källblade, P., Dean, P. M. (2003). Efficient conformational sampling of local sidechain flexibility.J. Mol. Biol. 326:1651–1665. 4. Cardone, F., Liu, Q. G., Petraroli, R., Ladogana, A., D’Alessandro, M., Arpino, C., Di Bari, M., Macchi, G., Pocchiari, M. (1999). Prion protein glycotype analysis in familial and sporadic Creutzfeldt-Jakob disease patients. Brain Res. Bull. 49(6): 429–433. 5. Caliskan, G., Kisliuk, A., Tsai, A. M., Soles, C. L., Sokolov, A. P. (2003). Protein dynamics in viscous solvents. J. Chem. Phys. 118(9):4230–4236. 6. Rosca, F., Kumar, A. T. N., Ionascu, D., Ye, X., Demidov, A. A., Sjodin, T., Wharton, D., Barrick, D., Sligar, S. G., Yonetani, T., Champion, P. M. (2002). Investigations of anharmonic low-frequency oscillations in heme proteins. J. Phys. Chem A. 106(14): 3540–3552. 7. Rhodes, G. (1993). Crystallography Made Crystal Clear. 2nd ed. San Diego, Calif.: Academic Press.

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8. Zhang, X., Wozniak, J. A., Matthews, B. W. (1995). Protein flexibility and adaptability seen in 25 crystal forms of T4 lysozyme. J. Mol. Biol. 250:527–522. 9. Hayward, S. (1999). Structural principles governing domain motions in proteins. Proteins: Struct. Funct. Genet. 36:425–435. 10. Gogo, N. K., Skrynnikov, N. R., Dahlquist, F. W., Kay, L. E. (2001). What is the average conformation of bacteriophage T4 lysozyme in solution? A domain orientation study using dipolar couplings measured by solution NMR. J. Mol. Biol. 308: 745–764. 11. Amadei, A., Linssen, A. B. M., Berendsen, H. J. C. (1993). Essential dynamics of proteins. Proteins: Struct. Funct. Genet. 17(4):412–425. 12. Buchner, M., Ladanyi, B. M., Stratt, R. M. (1992). The short-time dynamics of molecular liquids. Instantaneous-normal-mode theory. J. Chem. Phys. 97(11):8522– 8535. 13. de Groot, B. L., Hayward, S., van Aalten, D. M. F., Amadei, A., Berendsen, H. J. C. (1998). Domain motions in bacteriophage T4 lysozyme: a comparison between molecular dynamics and crystallographic data. Proteins: Struct. Funct. Genet. 31: 116–127. 14. Sanders, J. K. M., Hunter, B. K. (1993). Modern NMR Spectroscopy. 2nd ed. Oxford: Oxford University Press. 15. Johnson, W. C. (1990). Protein secondary structure and circular dichroism — a practical guide. Proteins: Struct. Funct. Genet. 7(3):205–214. 16. Fischer, M. W. F., Majumdar, A., Dahlquist, F. W., Zuiderweg, E. R. P. (1995). 15N, 13C, and 1H NMR assignments and secondary structure for T4-lysozyme. J. Magn. Reson., Ser. B 108:143–154 17. Mulder, F. A. A., Mittermaier, A., Hon, B., Dahlquist, F. W., Kay, L. E. (2001). Studying excited states of proteins by NMR spectroscopy. Nat. Struct. Biol. 8(11): 932–935. 18. Scott, W. R. P., Schiffer, C. A. (2000). Curling of flap tips in HIV1 protease as a mechanism for substrate entry and tolerance of drug resistance. Structure 8: 1259–1265. 19. Verlet, L. (1967). Computer experiments on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules. Phys. Rev. 159:98–103. 20. Weiner, P.W., Kollman, P.A. (1981). AMBER: assisted model building with energy refinement. A general program for modelling molecules and their interactions. J. Comput. Chem., 4:287–303. 21. Woodcock, L. V. (1971). Isothermal molecular dynamics calculations for liquid salts. Chem. Phys. Lett. 10:257–261. 22. Berendsen, H. J. C., Postma, J. P. M., van Gunsteren, W. F., Di Nola, A., Haak, J. R. (1984). Molecular dynamics with coupling to an external bath. J. Chem. Phys. 81: 3684–3690. 23. Anderson, H. C. (1980). Molecular dynamics simulations at constant pressure and/or temperature. J. Chem. Phys. 72:2384–2393. 24. Hoover, W. G. (1985). Canonical dynamics: equilibrium phase-space distributions. Phys. Rev. A31:1695–1697. 25. Creighton, T. E. (1993). Conformational properties of polypeptide chains. In: Proteins structures and molecular properties. 2nd ed. New York: W. H. Freeman and Company, pp. 192–193. 26. Sessions, R. B., Dauber-Osguthorpe, P., Osguthorpe, D. J. (1989). Filtering moleculardynamics trajectories to reveal low-frequency collective motions — phospholipaseA2. J. Mol. Biol. 210(3):617–633.

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27. Levitt, M. (1991). Real-time interactive frequency filtering of moleculardynamics trajectories. J. Mol. Biol. 220(1):1–4. 28. Phillips, S. C., Essex, J. W., Edge, C. M. (2000). Digitally filtered molecular dynamics: the frequency specific control of molecular dynamics simulations. J. Chem. Phys. 112(6):2586–2597. 29. Phillips, S. C., Swain, M. T., Wiley, A. P., Essex, J. W., Edge, C. M. (2003). Reversible digitally filtered molecule dynamics. J. Phys. Chem. B. 107(9):2098–2110. 30. Phillips, S. C., Gledhill, R. J., Essex, J. W., Edge, C. M. (2003). Application of the Hilbert-Huang transform to the analysis of molecular dynamics simulations. J. Phys. Chem. A. 107(24):4869–4876. 31. Huang, N. E., Chen, Z., Long, S. R., Wu, M. L. C., Shih, H. H., Zheng, Q. N., Yen, N. C., Tung, C. C., Liu, H. H. (1971). The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis. Proc. R. Soc. London Ser. A Math. Phys. Eng. Sci. 454:903–995. 32. Echeverria, J. C., Crowe, J. A., Woolfson, M. S., HayesGill, B. R. (2001). Application of empirical mode decomposition to heart rate variability analysis. Med. Biol. Eng. Comput. 39(4):471–479. 33. Chen, K. Y., Yeh, H. C., Su, S. Y., Liu, C. H., Huang, N. E. (2001). Anatomy of plasma structures in an equatorial spread F event. Geophys. Res. Lett. 28(16): 3107–3110. 34. Komm, R. W., Hill, F., Howe, R. (2001). Empirical mode decomposition and Hilbert analysis applied to rotation residuals of the solar convection zone. Astrophys. J. 558(1):428–441. 35. Bendat, J. S. (1985). The Hilbert transform and application to correlation measurements. Denmark: Brüel & Kjær. 36. Wu, X. W., Wang, S. M. (2000). Folding studies of a linear pentamer peptide adopting a reverse turn conformation in aqueous solution through molecular dynamics simulations. J. Phys. Chem. B. 104(33):8023–8034. 37. Dyson, H. J., Rance, M., Houghten, R. A., Wright, P. E., Lerner, R. A. (1988). Folding of immunogenic peptide-fragments of proteins in water solution. 2. The nascent helix. J. Mol. Biol. 201(1):201–217. 38. MATLAB 5.3.0. Natick, MA: The MathWorks Inc. (1999). 39. Swope, W. C., Andersen, H. C., Berens, P. H., Wilson, K. R. (1982). A computersimulation method for the calculation of equilibrium-constants for the formation of physical clusters of molecules — application to small water clusters. J. Chem. Phys. 76(1):637–649.

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Decomposition of Wave Groups with EMD Method Wei Wang

CONTENTS 12.1 Introduction ..................................................................................................267 12.2 Decomposition of Intermittent Small-Scale Fluctuations from the Large Wave ...................................................................................269 12.3 Method for Decomposing Wave Groups .....................................................272 12.4 Validations ....................................................................................................274 12.5 Conclusions ..................................................................................................280 References..............................................................................................................280 ABSTRACT By adding a proper real temporary time series that can induce a series of additional extrema, intermittent small ﬂuctuations can be accurately decomposed from the large waves they ride on. We can thus effectively solve the mode mixing problem that arises when the amplitudes of the components of a time series are very different. Another kind of mode mixing arises when the frequencies of the wave components are close to each other, such as in wave groups. Here a temporary complex time series is introduced to shift down the frequencies of the components. This will greatly enlarge the ratio of the frequencies in a wave group, and the wave groups can then easily be decomposed with the empirical mode decomposition method.

Keywords wave groups, EMD method

12.1 INTRODUCTION Being among the most energetic events in the ocean wave ﬁeld, wave groups play an important role in ocean wave research. The signiﬁcant characteristic of a wave 267

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group is that the group velocity Cg is equal to the energy propagation speed of the wave ﬁeld; this is widely used in ocean wave models such as WAM. The formation of wave groups is assumed analytically to be the result of the superposition of several cosine waves with almost the same frequency. In practice, this kind of analysis often cannot be achieved since no method up to now has been available to decompose a wave group efﬁciently into several simple components. This problem has been the main obstacle in wave group studies. Feasible methods that might be used to decompose wave groups are Fourier transform, wavelet transform (Chan 1995), and empirical mode decomposition (EMD) analysis (Huang et al. 1998, 1999). For a continuous uniform wave group, in which the wave group arises from the superposition of continuous constant-amplitude cosine waves, Fourier transform can distinguish the frequencies of all the components accurately. For an intermittent wave group, in which the wave group arises from the superposition of intermittent cosine waves, Fourier transform can yield only a collection of frequencies that will not represent the actual components. The reason is that Fourier transform is a high-resolution frequency detector with no resolution in the time domain; as a result, the intermittency of wave groups will show as additional frequencies in the spectrum, and the actual frequencies of the wave components will be altered to new locations. Compared with Fourier transform, wavelet transform is more suitable for dealing with intermittent signals. It has higher resolution in the time domain and lower resolution in the frequency domain. Theoretically, intermittent wave groups can be decomposed by wavelet transform when the frequencies of the wave components are widely different, but this method would fail when the frequencies are only slightly different. In practice, the lower resolution of wavelet transform in the frequency domain will also induce a rich distribution of harmonics around the real components and will ultimately result in difﬁculties in the distinction of the actual wave group components. In consideration of the resolution in the time–frequency domain, the most acceptable method is EMD. Compared to the Fourier transform and the wavelet transform, the EMD method has higher resolution in both the time and frequency domains. As an example, Huang et al. (1998) decomposed a wave group that came from the linear sum of two cosine waves into eight components with up to 3000 sifting processes and an extremely stringent criterion. Although the ﬁrst two components contained most of the energy and were similar to the original components, the results are not acceptable for the following reasons: 1. The extra components would not have any physical meaning. 2. Using too many sifting cycles is not suitable for intermittent wave groups because “too many sifting cycles could reduce all components to a constant-amplitude signal with frequency modulation only and the components would lose all their physical signiﬁcance” (Huang et al. 1999). A signal can be decomposed by the EMD method only when the ratio of the frequencies is larger than approximately 1.5 and when the number of sifting cycles is no more than 20. This is a limitation of the EMD method and reﬂects a kind of mode mixing.

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2 C1

2

1

0 −2

0

0

500

1000

1500

2000

2500

3000

0

500

1000

1500

2000

2500

3000

2 C2

−1 −2

269

0

500

1000

1500

2000

2500

3000

0 −2

FIGURE 12.1 The original time series and the ﬁrst two IMF components decomposed with the EMD method. The solid line in left panel is the original signal, the dashed line in the left panel is the envelope, and the lines in the right panel are the IMF components.

Another kind of mode mixing is introduced by the intermittency of the small-scale ﬂuctuations riding on large waves. Huang et al. (1999) gave a method for solving this kind of mode mixing, but deﬁning which signal in the intrinsic mode function (IMF) components is the one we want may be difﬁcult, as shown in Figure 12.1. In the ﬁrst part of the present paper, we improve the limitation on the ratio of the frequencies of the wave components to 1.2 by introducing a high frequency temporary wave. However, decomposition of a wave group remains impossible with this method, since the ratio of the frequencies for a typical wave group is always smaller than 1.1. In the second part of this paper, instead of improving the frequency resolution of the EMD method, we introduce a temporary complex signal to shift the frequencies of the components down. This will greatly increase the ratio of the frequencies in the wave group, allowing it to be decomposed easily with the EMD method.

12.2 DECOMPOSITION OF INTERMITTENT SMALL-SCALE FLUCTUATIONS FROM THE LARGE WAVE In general, a ﬂuctuation with high frequency and small amplitude can only change the large-scale wave proﬁle slightly, but not alter the stationarity of the large scale wave (not introduce additional extreme value). In such a case, the EMD method will take no account of the existence of the small-scale ﬂuctuations. Moreover, if we carry out the Hilbert transform, the Hilbert spectrum will treat this small-scale wave as a nonlinear characteristic of the large-scale wave itself, which seriously distorts the original physical meaning. Only when the small wave is superimposed on the wave crest or trough of the large-scale wave can it affect the stationarity of the large-scale wave, and in this case the EMD method can decompose it. Apparently, if the envelope of this kind of signal is very close to the dominant wave proﬁle, the intermittent signal with high frequency and small amplitude can be effectively separated. Based on this idea and the characteristics of the EMD method, we introduced a temporary signal with high frequency and large amplitude (called the temp-signal hereafter). As a result, the IMF components with high frequency derived through the EMD method will contain the temp-signal along with

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the high frequency ﬂuctuations in the original signal, and it can easily be separated by subtracting the temp-signal from the IMFs. But an optionally selected temp-signal will not match the original signal. Hence, the selection of the temp-signal is very important. For the case of a large wave with single frequency, containing an intermittent signal with high frequency and small amplitude, we can write: a cos ω1t f (t ) =  1 a1 cos ω1t + a2 cos ω 2t

t ∈(0, t1 ) ∪ (t1 + 2π / ω 2 , T1 ) t ∈[t1 , t1 + 2π / ω 2 ]

(12.1)

t ∈(t1 , T1 )

(12.2)

where ω2 ω1, a1 > a2. The temp-signal is deﬁned as X tmp (t ) = a3 cos ω 3t The data series with the temp-signal is a1 cos ω1t + a3 cos ω 3t  X (t ) =  a1 cos ω1t + a2 cos ω 2t + a3 cos ω 3t

t ∈(0, t1 ) ∪ (t1 + 2π / ω 2 , T1 ) (12.3) t ∈[t1, t1 + 2π / ω 2 ]

The temp-signal should satisfy the following requirements: • • •

The temp-signal can introduce a series of additional extrema, so its frequency should not be near that of the dominant wave; that is, ω3 ω1. a3 should be large enough to introduce a series of extrema to conceal the discontinuity caused by the intermittency. The envelope of X(t) calculated by spline function S(t), must satisfy ε a1, where ε = S(t) – a1 cos ω1t.

It is easily found that in order to restrict ε to smaller than 5%, the suitable choice of the temp-signal satisﬁes 4T2/5 < T3 T1, in which T1, T2, and T3, are the periods of the signal of the dominant wave, the ﬂuctuations, and the temp-signal, respectively. In addition, the amplitude of the temp-signal should be large enough to form a series of extrema in the original signal. For the signal of Figure 12.1, the mathematical description is:   2π  t  , as t < 1264 or t > 1296  cos    640  X (t ) =  cos  2π t  − 0.02 sin  2π t  , as 1264 ≤ t ≤ 1296  32    640  

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1.5

X(t) + Xtmp (t)

1 0.5 0 −0.5 −1 −1.5

0

500

1000

1500 t

2000

2500

3000

FIGURE 12.2 The shape of the original signal with the superposition of the temp-signal.

0.03

(a)

0.02 0.01

0.03 0.01

0

0

−0.01

−0.01

−0.02

−0.02

−0.03 1200

(b)

0.02

1250

1300

1350

1400

−0.03 1200

1250

1300

1350

1400

FIGURE 12.3 Comparison of the original small-scale ﬂuctuation and the one decomposed with the EMD method. The solid line represents the decomposed signal, and the dashed line represents the original signal. In the left panel, the temp-signal is deﬁned by atmp = 0.2 and Ttmp = 60. In the right panel, the temp-signal is deﬁned by atmp = 50 and Ttmp = 120.

The signal includes a small-scale ﬂuctuation with frequency 50 times larger and amplitude 20 times smaller than those of the dominant wave. Based on the discussion above, the temp-signal  2π  X tmp (t ) = atmp cos  t  Ttmp  should be selected in the range 26 < Ttmp < 640. Here we chose atmp = 0.2 and Ttmp = 60. The composed signal is shown in Figure 12.2. After processing with the EMD method and Deng et al.’s (2001) method of dealing with the boundary condition, the intermittent small-scale ﬂuctuation is decomposed successfully, as shown in Figure 12.3a. As a comparison, different choices of atmp and Ttmp were used, and the result was shown in Figure 12.3b. The above example represents a case where the small-scale intermittent ﬂuctuation is located on the crest of the dominant wave. For a case where the small-scale

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1

(a)

A

0.5

0.03 0.01 0

0

−0.01

−0.5 −1

(b)

0.02

−0.02 0

500

1000

1500

2000

2500

3000

−0.03 1950

2000

2050

2100

2150

FIGURE 12.4 Same as Figure 12.3, but the intermittent small-scale ﬂuctuation is changed to the position marked by A in the left panel. The temp-signal is selected as atmp = 50 and Ttmp = 120.

2

(a)

1.5 1

1

(b)

0.5

0.5 0

0

−0.5 −1

0

500

1000

1500

2000

2500

3000

−0.5 1100

1200

1300

1400

1500

1600

FIGURE 12.5 Decomposition of signals with singularities. The temp-signal is selected as atmp = 50 and Ttmp = 120.

intermittent ﬂuctuation is located at an arbitrary location other than crest or trough, the stationarity of the dominant wave will not be changed. As a result, decomposing by the EMD method will give no result. Even under such a condition, the method we propose can work. Figure 12.4 shows the original signal and the decomposed results obtained with our method; the original signal is deﬁned by Equation 12.4, but the position of the intermittent small-scale ﬂuctuation is changed. Signals in which singularities exist, such as the one shown in Figure 12.5, can also be decomposed with this method. In addition to offering the advantages already mentioned, this method can be used to decompose wave groups when the ratio of the frequencies of the components is not less than 1.2 and the distortion of each component is quite small even after a large number of sifting cycles.

12.3 METHOD FOR DECOMPOSING WAVE GROUPS As mentioned above, the key point in wave group decomposition is the ratio of the frequencies of wave group components. Let us consider the wave group as y(t ) = cos(ω1t ) + cos(ω 2t + φ),

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(12.5)

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in which ω2 > ω1. The ratio of the two frequencies is α = ω2/ω1. Typically α is smaller than 1.1, but it can be greatly enlarged when we are subtracting a temporary frequency, ω0, in the numerator and denominator simultaneously. For a real signal, such a downshifting is hard to achieve. Theoretically, Equation 12.5 can be rewritten in a complex form as y(t ) = Re Y (t )  = Re  exp(iω1t ) + exp(i(ω 2t + φ))  ,

(12.6)

where Y(t) is the complex form of y(t), which can easily be constructed by using the Hilbert transform. For an arbitrary time series, y(t), we can always have its Hilbert transform, ˜y (t), as y (t ) =

1 P π

∞

y( τ ) d τ, −∞ t − τ

∫

(12.7)

where P indicates the Cauchy principal value. This transform exists for all functions of class Lp (Huang et al. 1998). Therefore, the complex function Y(t) can be formed as Y (t ) = y(t ) + iy (t ).

(12.8)

The downshifting can easily be achieved by multiplying Y(t) by a function, exp (–iω0t), to yield a new complex signal: Z (t ) = Zr (t ) + iZi (t ) = Y (t ) ⋅ exp(−iω 0t ) =  exp(iω1t ) + exp(i(ω 2t + φ))  ⋅ exp(−iω 0t )

(12.9)

= exp i(ω1 − ω 0 )t  + exp i((ω 2 − ω 0 )t + φ)  , The ratio of the frequencies for signal Z(t) is α = (ω2 – ω0)/(ω1 – ω0). For a well-selected ω0, α can be much larger than 1.5, the limitation in the EMD method. Zr(t) and Zi(t) can easily be decomposed, either with the EMD method or the method described in Section 12.2, and represented by their IMF components as: n

Zr (t ) =

∑C

rk

(t ) + rnr ,

k =1 n

Zi (t ) =

∑ C (t) + r . ik

k =1

© 2005 by Taylor & Francis Group, LLC

ni

(12.10)

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The combination of the corresponding IMF components of Zr(t) and Zi(t) will reconstruct Z (t) as a sum of complex components Ck(t): Z (t ) = Zr (t ) + iZi (t ) n

=

∑ (C (t) + iC (t)) + (r r

i

k =1

k

nr

+ irni )

(12.11)

n

=

∑ C (t) + R . k

n

k =1

According to Equation 12.9, Y(t) can be written as Y (t ) = Z (t ) ⋅ exp(iω 0t ) n

=

∑

Ck (t ) ⋅ exp(iω 0t ) + Rn ⋅ exp(iω 0t ).

(12.12)

k =1

Finally, the original signal y(t) may be represented as: y(t ) = Re Y (t )   n  = Re  Ck (t ) ⋅ exp(iω 0t ) + Rn ⋅ exp(iω 0t )   k =1 

∑

(12.13)

n

=

∑ c (t) + r , k

n

k =1

where ck(t) are IMF components of the original signal, and rn is the trend or mean value.

12.4 VALIDATIONS We offer two examples to verify the performance of the present method. The ﬁrst is a continuous uniform wave group with a phase shift in one of the components. For ω1 = 2π/30, ω2 = 2π/32, and φ = π/6, the wave proﬁle represented by Equation 12.5 is given in Figure 12.6. In this case α = 1.067. With ω0 = 2π/28, the wave proﬁle of Z(t), governed by Equation 12.9, is shown in Figure 12.7. Here, a and b represent the real and imaginary components of Z(t), respectively. Under such conditions, the ratio of the frequencies of the components is approximately 2. Thus, Zr(t) and Zi(t) can easily be decomposed into a series of IMF

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2 1.5 1

y (t)

0.5 0 −0.5 −1 −1.5 −2

0

200

400

600

800

1000

t

FIGURE 12.6 Wave proﬁle represented by Equation 12.9 with ω1 = 2π/30, ω2 = 2π/32, and φ = π/6.

Zr (t)

2 0 −2

Zi (t)

2 0 −2 0

200

400

600

800

1000

t

FIGURE 12.7 Wave proﬁle represented by Equation 12.10 with ω0 = 2π/28. (a) The real component of Z(t); (b) the imaginary component of Z(t).

components. Figure 12.8 shows the wave proﬁles of all the IMF components of the real and imaginary parts. Results calculated from Equation 12.11, Equation 12.12, and Equation 12.13 are shown in Figure 12.9 and Figure 12.10. The solid lines in Figure 12.9 and Figure 12.10 correspond to the ﬁrst and second modes in Equation 12.5, respectively. For comparison, the actual modes are also plotted as dotted lines in the same ﬁgures. The agreement is excellent.

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2

2

1

1 C1i

C1r

276

0 −1 −2

0 −1

0

200

400

600

800

−2

1000

0

200

400

2

2

1

1

0 −1 −2

600

800

1000

600

800

1000

t

C2i

C2r

t

0 −1

0

200

400

600

800

−2

1000

0

200

400

t

t

FIGURE 12.8 The IMF components of Zr(t) and Zi(t). (a,b) IMF components of Zr(t); (c,d) IMF components of Zi(t).

1.5 1 0.5 0 −0.5 −1 −1.5

0

200

400

600

800

1000

t

FIGURE 12.9 Comparison of the ﬁrst IMF component calculated from Equation 12.11, Equation 12.11, and Equation 12.13 (solid line) and the ﬁrst mode of Equation 12.5 (dotted line).

Figure 12.11 illustrates the differences between the IMF components and the actual modes. It is clear that the differences are negligible. The second example is an intermittent wave group. The intermittent wave group is formed as 0, as t < t1 or t > t2 y(t ) =  cos ω1t + cos ω 2t + φ , as t1 ≤ t ≤ t2

( )

© 2005 by Taylor & Francis Group, LLC

(

)

(12.14)

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1.5 1 0.5 0 −0.5 −1 −1.5

0

200

400

600

800

1000

t

FIGURE 12.10 Comparison of the second IMF component calculated from Equation 12.11, Equation 12.11, and Equation 12.13 (solid line) and the second mode of Equation 12.5 (dotted line).

1 0.5 0 −0.5 −1

0

200

400

0

200

400

600

800

1000

600

800

1000

1 0.5 0 −0.5 −1

t

FIGURE 12.11 Differences between IMF components and actual modes. The upper panel shows the differences between the ﬁrst IMF component and the ﬁrst mode in Equation 12.5. The lower panel shows the differences between the second IMF component and the second mode in Equation 12.5.

For ω1 = 2π/30, ω2 = 2π/32, φ = π/6, t1 = 1740, and t2 = 3150, the wave proﬁle represented by Equation 12.14 is shown in Figure 12.12. With ω0 = 2π/28, the wave proﬁle of Z(t), governed by Equation 12.10, is shown in Figure 12.13, where a and b represent the real and imaginary components of Z(t), respectively. In the ﬁrst example (shown in Figure 12.9, Figure 12.10, and Figure 12.11), only negligible errors exist at the ends because we have conﬁned end effects with a method of adding characteristic waves at the ends. In the second example, however,

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2 1.5 1

y (t)

0.5 0 −0.5 −1 −1.5 −2

0

500

1000

1500

2000

2500 t

3000

3500

4000

4500

5000

FIGURE 12.12 Wave proﬁle represented by Equation 12.14 with ω1 = 2π/30, ω2 = 2π/32, φ = π/6, t1 = 1740, and t2 = 3150.

2

(a)

Zr (t)

1 0 −1 −2

2

(b)

Zi (t)

1 0 −1 −2

0

500

1000

1500

2000

2500 t

3000

3500

4000

4500

5000

FIGURE 12.13 Wave proﬁle represented by Equation 12.10 with ω0 = 2π/28. (a) The real component of Z(t); (b) the imaginary component of Z(t).

end effects are hardly conﬁned because the ends of the wave groups are not at the ends of the data set. If the ends of the wave groups are left unattended, wide swings caused by spline ﬁtting will exist at the ends and will propagate to the interior and ulterior as the number of sifting iterations increases. Figure 12.14 and Figure 12.15 show the ﬁnal results.

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2

2

(a)

0

0

−1

−1

−2

−2

2

2

(b)

1

(d)

1 Ci2

Cr2

(c)

1 Ci1

Cr1

1

0 −1 −2

279

0 −1

0

1000

2000

3000

4000

−2

5000

0

1000

2000

3000

4000

5000

t

t

FIGURE 12.14 The IMF components of Zr(t) and Zi(t): (a,b) IMF components of Zr(t); (c,d) IMF components of Zi(t).

2 (a) 1 0 −1 −2

2 (b) 1 0 −1 −2

0

500

1000

1500

2000

2500 t

3000

3500

4000

4500

5000

FIGURE 12.15 Comparison of the IMF components calculated from Equation 12.11, Equation 12.12, and Equation 12.13 (solid line) and the two actual modes of Equation 12.14 (dotted line).

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12.5 CONCLUSIONS The present paper introduces two methods for decomposing a wave group into a series of components of relatively single scale. The ﬁrst method was used not only in solving the problem of decomposition of wave groups but also as an improved method of EMD for a variety of uses. The second method can decompose wave groups of small α, but the sensitivity to the selection of the temporary frequency is still an open question.

REFERENCES Chan, Y.T. (1995). Wavelet basics. Boston: Kluwer. Deng, Y.J., W. Wang, C.C. Qian, Z. Wang and D.J. Dai. (2001). Boundary-processing technique in EMD method and Hilbert transform. Chinese Science Bulletin, 46(11), 954–960. Huang, N.E., Z. Shen, S.R. Long, et al. (1998). The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. R. Soc. Lond. A, 454, 899–995. Huang, N.E., Z. Shen and S.R. Long. (1999). A new view of nonlinear water waves: the Hilbert spectrum. Annu. Rev. Fluid Mech., 31, 417–457.

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13

Perspectives on the Theory and Practices of the Hilbert-Huang Transform Nii O. Attoh-Okine

CONTENTS 13.1 Introduction ..................................................................................................282 13.2 Basic Ideas ...................................................................................................282 13.2.1 The Hilbert-Huang Transform .........................................................282 13.2.2 Starting Point....................................................................................288 13.3 Current Applications ....................................................................................289 13.3.1 Biomedical Applications ..................................................................289 13.3.2 Chemistry and Chemical Engineering.............................................289 13.3.3 Financial Applications......................................................................290 13.3.4 Meteorological and Atmospheric Applications ...............................290 13.3.5 Ocean Engineering...........................................................................291 13.3.6 Seismic Studies ................................................................................292 13.3.7 Structural Applications.....................................................................295 13.3.8 Health Monitoring............................................................................296 13.3.9 System Identiﬁcation........................................................................296 13.4 Some Limitations .........................................................................................297 13.5 Potential Future Research ............................................................................298 References..............................................................................................................299 Addendum: Perspectives on the Theory and Practices of the Hilbert-Huang Transform............................................................................302 13A.1 Analytical .........................................................................................302 13A.2 Hybrid Method.................................................................................302 13A.3 Bidimensional EMD ........................................................................303 13A.4 Some Observations...........................................................................304 References..............................................................................................................304 281

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ABSTRACT The theory and practice of the Hilbert-Huang transform (HHT) provide another way of analyzing nonlinear and nonstationary time series. The central idea of HHT is the empirical mode decomposition (EMD), which is then decomposed into a set of intrinsic mode functions (IMFs) and analyzed by using the Hilbert spectrum. This chapter is primarily a summary of current applications of HHT and suggests areas for future research based on my personal views.

13.1 INTRODUCTION The Hilbert-Huang transform (HHT) provides a new method of analyzing nonstationary and nonlinear time series data. It allows the exploration of intermittent and amplitude-varying motions. The classical signal analysis theory and practice have been dominated by the Fourier transform, which represents the signal system in the frequency domain. The Fourier works well for strictly periodic or stationary random functions of time. To address the issues of nonstationarity and nonperiodic functions, various methods have been introduced in the literature. These include wavelet analysis, the Wigner-Ville distribution, and the evolutionary spectrum. Unfortunately, these methods failed in one way or another [Huang et al., 1998]. HHT is emerging as a powerful signal-processing tool. During the past ﬁve years, a great deal has been learned about the computation, interpretation, and implementation of the HHT technique. This work has been scattered in a wide variety of ﬁelds, including signal processing, structural health monitoring, earthquake engineering, nondestructive testing, transportation engineering, ocean engineering, and ﬁnancial and biomedical applications. No one has drawn the new work together in a comprehensive manner. This chapter is primarily a summary of my own current views. My failure to mention a particular contribution should not be taken as an indication of disinterest or disagreement. After this general introduction, basic ideas of HHT are presented in a manner that should be accessible to readers with no previous familiarity with HHT. The paper then discusses various application areas, outlines some perceived “difﬁculties,” and ﬁnally highlights a future research direction of HHT.

13.2 BASIC IDEAS 13.2.1 THE HILBERT- HUANG TRANSFORM Huang et al. [1998] developed the Hilbert-Huang transform to decompose any time-dependent data series into individual characteristic oscillations by using the empirical mode decomposition (EMD). The decomposition is developed from the assumption that any data or signal consists of different intrinsic mode functions (IMFs), each representing embedded characteristics of the data/signal. The decomposition can be viewed as an expansion of the data in terms of the IMFs. The HHT method is a two-step data analyzing method that consists of EMD and Hilbert spectral analysis (HSA). The ﬁrst step is the EMD, by which a completed time

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history of the signal is decomposed into a ﬁnite and often small number of IMFs such that they may be Hilbert transformed. The EMD is based on local characteristics in the time scale of the data, making the approach direct, posterior, and adaptive. The IMFs admit well-behaved Hilbert transforms. An IMF is deﬁned by the following two criteria: 1. The number of extrema and zero-crossings must either be equal or differ by no more than one. 2. At any instant in time, the mean value of the envelope deﬁned by the local maxima and the envelope of the local minima is zero. With the above deﬁnition, any function can be decomposed as follows: • •

Identify all the local extrema, then connect all the local maxima by cubic spline as upper envelope. Repeat the procedure for the minima to produce the lower envelope. The upper and lower envelope should cover all the data. If the mean is designated as m1 and the difference between the data and m1 is the ﬁrst component h1, then x (t ) − m1 = h1

(13.1)

where h1 is an IMF. The mean m1 is given by the sum of local extrema connected by cubic spline: m1 =

L +U 2

where U is the local maximum and L is the local minimum. The IMF can have both amplitude and frequency modulations. In many cases, there are overshoots and undershoots after the ﬁrst round of processing, which is termed sifting. The sifting process serves two purposes [Huang et al., 2001]: eliminating riding signals/proﬁle and making the signals or proﬁle more symmetric. The sifting process has to be repeated many times. In the second sifting process, h1 is treated as the data and as the ﬁrst component. h1 is almost an IMF, except some error might be introduced by the spline curve-ﬁtting process. To treat h1as new set of data, a new mean is computed. Then, h1 − m11 = h11 .

(13.2)

After repeating the sifting process up to k times, h1k becomes an IMF. That is: h1( k −1) − m1k = h1k .

© 2005 by Taylor & Francis Group, LLC

(13.3)

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Let h1k = c1, the ﬁrst IMF from the data. c1 should contain the ﬁnest scale or the shortest period component of the data/proﬁle. Now c1 can be separated from the rest of the data by x (t ) − c1 = r1 .

(13.4)

Since r1 is the residue, it contains information of longer period component; it is now treated as the new data and subjected to the same sifting process. The procedure is repeated for all subsequent rjs, and the result is r1 − c2 = r2; ..., rn−1 − cn = rn;

(13.5)

c2 is now the second IMF of the data. The sifting process can be stopped by predetermined criteria: either when the component of cn or the residue rn becomes so small that it has a predetermined value of substantial consequence, or when the residue rn becomes a monotonic function from which no IMF can be extracted. Summing Equation 13.4 and Equation 13.5 yields the following equation: n

x (t ) =

∑ c +r , j

n

(13.6)

j =1

where cj is the jth IMF and n is the number of sifted IMFs. rn can be interpreted as a trend in the signal/proﬁle. The cj has zero mean. Due to the iterative procedure, none of the sifted IMFs derived is closed analytical form [Schlurmann, 2002]. The IMFs can be linear or nonlinear based on the characteristics of the data. The IMFs are almost orthogonal and form a complete basis. Their sum equals the original data [Salisbury and Wimbush, 2002]. The EMD then picks out the highest-frequency oscillation that remains in the signal. Flandrin et al. [2003] established how EMD can be used as a ﬁlterbank. They represented the signal as follows: K

x (t ) = mk (t ) +

∑ d (t) k

(13.7)

k =1

where mk(t) stands for a residual “trend,” and the modes {dk(t), k = 1, …, K} and dk(t) stand for details. The orthogonality of the IMF components can be checked as follows, if Equation 13.6 is expressed as n +1

x (t ) =

∑ c (t). j

j =1

© 2005 by Taylor & Francis Group, LLC

(13.8)

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That is, one has included the last residual or data trend as an additional element: n +1

x (t ) = 2

n +1

n +1

j =1

k =1

∑ c (t) + 2∑ ∑ c (t)c (t). 2 j

j

j =1

k

(13.9)

If the decomposition is orthogonal, then the cross terms given in the second part on the right hand side should be zero when they are integrated over time. The overall index of orthogonality OI can be deﬁned as follows:  c (t )c (t ) . IO = ∑ ∑ ∑  x (t ) T

t =o

  

n +1

n +1

j =1

k =1 2

j

k

(13.10)

To ensure that the IMF components retain enough physical sense of both amplitude and frequency modulations, we can limit the size of the standard deviation (SD), computed from two consecutive sifting results as T

SD =

∑ |h

k −1

t =0

(t ) − hk (t ) |2 hk2−1 (t )

A value of SD between 0.2 and 0.3 was used. In the next step, the Hilbert transform is applied to each of the IMFs, subsequently providing the Hilbert amplitude spectra a signiﬁcant instantaneous frequency. The Hilbert transform of each IMF is represented by

∑ a (t) exp(i ∫ ω(t)dt) n

x (t ) = Re

j

(13.11)

j =1

Equation 13.7 gives both the amplitude and frequency of each component as a function of time. The Fourier representation would be as follows: ∞

x (t ) = Re

∑ a exp j

iω j t

(13.12)

j =1

with both aj and j constants. The Fourier transform represents the global rather than any local properties of the data/signal. The major difference between the conventional Hilbert transform and HHT is the deﬁnition of instantaneous frequency. The instantaneous frequency has more physical meaning through its deﬁnition of each IMF component, while

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the classical Hilbert transform of the original data might possess unrealistic features [Huang et al., 1998]. This implies that the IMF represents a generalized Fourier expansion. The variable amplitude and instantaneous frequency enable the expansion to accommodate nonstationary data. Huang et al. [2001] demonstrated that the Fourier- and wavelet-based components and spectra may not have a clear physical meaning as in HHT. For example, the wavelet-based interpretation of a pavement proﬁle is meaningful relative to selected mother wavelets [Attoh-Okine, 1999]. Furthermore, Hilbert analysis [Long et al., 1995] is based on almost noncausal singular information. Therefore, at any given time, the data or signal has only one amplitude and one frequency, both of which can be determined locally. These represent the best information at that particular time. Physically, the deﬁnition of instantaneous frequency has true meaning for monocomponent signals, where there is one frequency, or at least a narrow range of frequencies, varying as a function of time. Since most data do not show these necessary characteristics, sometimes the Hilbert transform makes little physical sense in practical applications. In the present method, the EMD is used to decompose the signal into a series of monocomponent signals. Furthermore, to extract signiﬁcant information, the time–frequency–amplitude joint distribution [a(t), (t), t] can be developed. This joint distribution in 3D space can be replaced by [H(,t)], where x(t) = H(,t). This ﬁnal representation is referred to as the Hilbert spectrum. With the Hilbert spectrum deﬁned, we can deﬁne a marginal spectrum:

∫

h(ω ) =

t

H (ω, t )dt .

(13.13)

0

The marginal spectrum offers a measure of the total amplitude (or energy) contribution from each of the frequency values [Huang et al., 1999a]. The degree of stationarity is deﬁned as follows: D(ω ) =

1 T

∫

T

0

(1 −

H (ω, t ) )dt . h(ω ) / T

(13.14)

The closer the above equation is to zero, the more stationary the system is and the instantaneous energy: IE (t ) =

n

H (ω, t ) =

∫

∑ j =1

ω

H 2 (ω, t )d ω

_ H j (ω, t ) ≈

(13.15)

n

∑ A (t) j

(13.16)

j =1

≅ by deﬁnition. In Equation 13.16, H j is the jth component of the total Hilbert spectrum. The equation

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n

h(ω ) =

∑

(ω ) ≈ _ hj

j =1

n

∑∫ j =1

287

T

a j (t )dt

(13.17)

0

provides a measure of the total amplitude contribution for each frequency value [Zhang et al., 2003]. By using the Hilbert spectrum, ﬁrst and second order time-moments and the frequency-dependent order can be derived. The ﬁrst and second order of time-dependent moments are as follows: ∞

∫ ∫

mω ,1 (t ) =

−∞ ∞

ωH (ω, t )d ω (13.18)

−∞

∞

mω ,2

H (ω, t )d ω

∫ ω H (ω, t)dω − m (t ) = ∫ H (ω, t)dω 2

−∞ ∞

ω ,1

(t )

(13.19)

−∞

The ﬁrst and second frequency-dependent distributions are as follows:

∫ ∫

mt,1 (ω ) =

∞

−∞ ∞ −∞

∞

mt ,2

∫ (ω ) = ∫

−∞ ∞

tH (ω, t )dt (13.20) H (ω, t )dt

t 2 H (ω, t )dt

−∞

H (ω, t )dt

− mt ,1 (ω )

(13.21)

The marginal in the time and frequency domains can be calculated as follows:

mω ,0 (t )

∫

mt,0 (ω ) =

© 2005 by Taylor & Francis Group, LLC

∞

H (ω, t )d ω

∞

∫

∞

∞

H (ω, t )dt

(13.22)

(13.23)

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The information content of the marginal mt,0() is much reduced [Huang et al., 2001]. The time marginal corresponds to the instantaneous power |x(t)|2 of the signal, and the frequency marginal corresponds to the spectral energy density |X()|2. Huang et al. [2003a] developed a conﬁdence limit for HHT. The ensemble mean and standard deviation of IMF sample sets obtained with different stopping criteria are calculated and form a simple random set. The conﬁdence limit for EMD/HSA is then deﬁned as a range of standard deviations from the assembled mean. The authors attempted to establish a conﬁdence limit for the EMD/HSA approach as a statistical measure of the validity of the results. The method of establishing a conﬁdence limit depends on factors like stopping criteria, the maximum number of siftings, intermittence, sifting methods, and the nomenclature needed for the conﬁdence limit. The method does not invoke the ergodic assumption. Qu and Jarzynski [2001] did a comparative study of different time–frequency representations of Lamb waves. The authors noted that the time–frequency accuracy of the Hilbert spectrum is dependent on the accuracy of the EMD. That is, if the decomposition into IMFs does not capture the signal’s behavior, the resulting Hilbert spectrum will not give precise time–frequency results.

13.2.2 STARTING POINT The starting point of HHT is the Hilbert transform. The Hilbert transforms were originally developed to integrate equations [Duffy 2004]. Instead of expressing a function of time with its Fourier transform, which depended only on frequency, the Hilbert transform yields another temporal function: 1 fˆ = π

∫

∞

−∞

f (τ) dτ t−τ

(13.24)

Gabor (1946) developed the concept of analytic signal: Y (t ) = f (t ) + i ∧ f (t )

(13.25)

Y (t ) = a(t )eiθ(t )

(13.26)

or

which is a local time-varying wave with amplitude a(t) and phase (t). The Hilbert transform can be considered to be a ﬁlter that simply shifts the phase of its input by –/2 radians. Unfortunately, most signals are not band limited. One of Huang’s notable contributions was to devise a method — the sifting method — that transforms a wide class of signals into set of band-limited functions. The instantaneous frequency may possess negative values. Lai (1998) developed a proper method to implement positive frequencies. The Hilbert spectrum is similar to the Fourier spectrum in a physical sense, but the Fourier approach is more global. In the Hilbert

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spectrum, at any given time, the signal has only one amplitude and one frequency, both determined locally [Long et al., 1995].

13.3 CURRENT APPLICATIONS 13.3.1 BIOMEDICAL APPLICATIONS Huang et al. [1999b] analyzed the pulmonary arterial pressure on conscious and unrestrained rats. The authors analyzed the signal by using HHT and compared the Hilbert spectrum to the Fourier spectrum. The Hilbert spectrum clearly depicts the ﬂuctuations of the frequency with time, whereas the Fourier spectrum gives only the distribution of energy over frequencies without allowing the frequency of the oscillations to be variable in the whole time window. Huang et al. [1999c] also studied the signals obtained from pulmonary hypertension. Their analysis attempted to address the linearity and nonlinearity of the dependencies of blood pressure on other variables. Huang et al. [1999c] developed a generalized function connecting the IMFs and an analytic equation: M k (t ) = IMFk + IMFk −1 + … + IMFn M k (t ) = A + B(t − t0 )e

− t −t 0 T1

+ C (1 − e

− t −t 0 T2

); t0 ≤ t t1

(13.27)

(13.28)

where t0 is the instant in time when oxygen concentration drops suddenly, t1 is the instant when oxygen concentration increases suddenly, A is the mean value of Mk(t) before time t0, and B, C, T1, and T2 are constants.

13.3.2 CHEMISTRY

AND

CHEMICAL ENGINEERING

Phillips et al. [2003] performed a comparative analysis of HHT and wavelet methods. They investigated a conformational change in Brownian dynamics (BD) and molecular dynamics (MD) simulations. Most of the MD trajectories were generated by using a reversible digitally ﬁltered MD method. The authors used HHT to analyze the trajectories. The HHT analysis of BD trajectories was able to isolate characteristic frequencies of motion. The authors compared their results with the Morlet wavelet, and the results yielded similar information. The authors also tried to use Akima spline, but this did not enhance the algorithm. Wiley et al. [2004] used HHT to investigate the effect of reversible digitally ﬁltered molecular dynamics (RDFMD), which can enhance or suppress speciﬁc frequencies of motion. HHT was applied to conformational motion, which occurs predominantly by rotation about single bonds in a protein, called dihedral angles. A small change in these angles can have signiﬁcant effects on the structure of the protein. Unfortunately, there were limitations of the HHT method as applied to the MD simulations. These were:

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• • •

The analysis of short RDFMD ﬁlters buffers results in a low frequency limit in the HHT. EMD produces physically unreliable IMFs. In an attempt to recreate the usual signal given by the data, less the residual. The HHT requirement of separable scales is not necessarily met by signals from MD simulations.

Montesinos et al. [2002] applied HHT to signals obtained from BWR neuron stability. The authors used data from a nuclear power plant that were recorded during startup of the plant. They show a sharp peak produced by sudden activity insertion due to the change of the recirculation pump velocity from low to high speed. The authors attempted to recognize the IMF that is related to the stability of the BWR. They further used an autoregressive model to analyze the fourth IMF and showed that the IMFs are mutually orthogonal.

13.3.3 FINANCIAL APPLICATIONS Huang et al. [2003b] applied HHT to nonstationary ﬁnancial time series and used a weekly mortgage rate data. The latter authors illustrated the differences among Fourier, wavelet, and Hilbert spectral analysis. The authors relate the slope to the global level of the data. By using the HHT, the authors address the question of volatility, introducing a new measure of volatility, designated as “variability,” based on EMD applied to produce IMF components. The variability was deﬁned as the ratio of the absolute value of the component(s) to the signal at any time: V (t; T ) =

Sh (t ) S (t )

(13.29)

where T corresponds to the period at the Hilbert spectrum peak of the high-passed signal up to h terms, and h

Sh (t ) =

∑ c (t). j

(13.30)

j =1

The unit of this variability parameter is the fraction of the market value. This is a simple and direct measure of the market volatility.

13.3.4 METEOROLOGICAL

AND

ATMOSPHERIC APPLICATIONS

Salisbury and Wimbush [2002], using Southern Oscillation Index (SOI) data, applied the HHT technique to determine whether the SOI data are sufﬁciently noise free that useful predictions can be made and whether future El Niño southern oscillation (ENSO) events can be predicted from SOI data. The authors were able to identify an IMF that roughly corresponded to a 3.6-yr mean interval between ENSO warm

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events. The authors further developed a dynamical model using selected IMFs and SOI data that can make a predictive estimate. Pan et al. [2002] used HHT to analyze satellite scatterometer wind data over the northwestern Paciﬁc and compared the results to vector empirical orthogonal function (VEOF) results. Duffy [2004] used HHT to analyze three different data ﬁles: • • •

Sea level heights Incoming solar radiation Barographic observations

The HHT persistently detected periodic features such as tides, snowmelt, and heavy precipitation events in terms of sea level heights. Using the radiation data, Duffy [2004] observed anomalies during ENSO and its effect on the aerosol concentration in the troposphere. Some of the IMFs of the observation data could be paired with meteorological events, such as the passage of a cold front, a warm front, and a trough. Generally, the HHT is capable of discerning mesoscale and synoptic events. With a ﬁve-day time series of temperature, Lundquist [2003] demonstrated how the HHT allows simultaneous identiﬁcation of the stationary diurnal temperature cycle as well as intermittent and nonstationary cooling events such as frontal passages and density currents. One of the IMFs obtained by the authors appears to be associated with inertia gravity waves.

13.3.5 OCEAN ENGINEERING Schlurmann [2002] introduced the application of HHT to characterize nonlinear water waves from two different perspectives, using laboratory experiments. The author examined monochromatic and transient (freak) waves and compared the results with wavelet analysis. The HHT was able to identify superharmonic frequencies, which were not evident in the wavelet spectrum, in both cases. Schlurmann and Datig [2005] applied HHT to rogue waves (freak waves). The authors determined statistical characteristics of the IMFs and then deduced that the natural data sets do not behave like white noise. The statistical analysis demonstrated that the number of extrema, the mean periods, and the corresponding standard deviation of the IMFs are strongly related to each other. The authors developed an equation relating the average period and mode number. Various coefﬁcients were deﬁned for different ranges of modes. Veltcheva [2002] applied HHT to wave data from nearshore sea. The IMF was able to group the waves into calm and wave-decay stages at the offshore. Veltcheva [2004] also compared the HHT and Fourier transforms for nearshore sea waves. The author noted that the process of wave breaking mainly affects the IMF with the highest energy in the decomposition of offshore waves. Wang [2004] presented an improved EMD for the decomposition of wave groups. Hwang et al. [2004] compared the energy ﬂux computation of shoaling waves using HHT, FFT, and wavelet techniques. They found that the associated HHT spectral analysis provides superior spatial (temporal) and wavenumber (frequency) resolution for handling nonstationarity

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and nonlinearity. They also found that the energy ﬂux computed using the HHT spectrum is higher than that from wavelet and FFT techniques. Larsen et al. [2004] used HHT to characterize the underwater electromagnetic environment and identify transient manmade electromagnetic disturbances. The authors collected ﬁeld data based on in-water magnetic data. The HHT was able to act as a ﬁlter, effectively discriminating different dipole components. Therefore, the method helps in detecting moving electric dipoles in the ocean. The authors further developed an empirical mode–matched ﬁlter. The main idea is to apply a matched ﬁlter to a signal generated by using IMFs as basis function, rather than to the target signal. In the empirical mode–matched ﬁlter, EMD is initially applied to the signal (time series) to generate the IMFs, i(t), i = 1, …, N, where N is the number of IMFs. The second step involves determining the linear combination of the IMFs containing the target signal s(t) that is best approximated to the known target signature in the least-square sense: N

min s(t ) − ci ≥ 0

∑ c α . i

i

(13.31)

i =1

A matched ﬁlter based on the target signature s(t) is applied to the modeled signal, which is composed of weighted linear combination of the IMFs. The authors attempted to use receiver operator characteristics (ROC) curves, HHT, and the matched ﬁlter to address the probability of detection. Zeris and Prinos [2004] compared the results of using HHT and wavelet transform for turbulent open channel ﬂow data. Laser Doppler and hot ﬁlms techniques were used to obtain signals in turbulent open channel ﬂow, with and without the imposed bed suction. The authors were fairly successful in the application. Unfortunately, it is not clear how the authors did the comparative analysis.

13.3.6 SEISMIC STUDIES Huang et al. [2001] used HHT to develop a spectral representation of earthquake data. The HHT offers a better representation of the earthquake signals in the low frequency range. The authors were able to address the shortcomings of various methods typically used in earthquake spectral analysis. For example, the use of Fourier-based methods and Duhanle integral assumes stationarity in the data; in addition, the selection of an absolute maximum in the responses makes the HHT spectrum less dependent on the record length. Therefore, the design spectrum, deﬁned as the envelope of a collection of response spectra, will inherit the shortcomings of the response spectrum: the underrepresentation of the most critical low frequency range and its dangerous energy content. In the HHT approach, the frequency is instantaneous and is not determined by convolution. Loh et al. [2001] used EMD and HHT to analyze the free ﬁeld ground motion and to estimate the global structural characteristics of building and bridge infrastructures. The process was used to identify damage from the response data.

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Chen et al. [2002a] used HHT to determine the dispersion curves of seismic surface waves and compared their results to Fourier-based time–frequency analysis. The surface-wave dispersion measurement is essential for the study of the crust and upper mantle structures, earthquake source mechanisms, and inelastic properties of the earth. The authors applied EMD to the seismogram and obtained the IMFs from which the Hilbert spectrum is calculated. Then, for each frequency, the time of the maximum amplitude corresponds to the group arrival at which the group velocity can be obtained. The result of group velocity and phase velocity determined by using HHT is more accurate than by Fourier-based methods. Shen et al. [2003] applied HHT to ground motion and compared the HHT result with the Fourier spectrum. The HHT is capable of identifying the occurrence and properties of destruction near fault ground motions. The authors used the method as a demonstration tool in the estimation of seismic loading in bridge design. Magrin-Chagnolleau and Baraniuk [undated] applied HHT to decompose a seismic trace into different oscillatory modes and were able to extract instantaneous frequencies of these modes. Chun-xiang and Qi-feng [2003] made a comparative study of wavelet and HHT using earthquake record. The IMF directly decomposed from the original data, whereas wavelet components are decomposed according to mother wavelets, which means the results can be inﬂuenced by the selected mother wavelets. The Hilbert spectrum clearly presents the energy distribution with time and frequency. Zhang et al. [2003] examined the applicability of HHT to dynamic and earthquake loading motion recordings. The IMF was capable of capturing low and high frequency components. The authors commented how EMD can act like a ﬁlter by grouping the IMFs. The authors deﬁned EMD-based high frequency motions as the summation of the ﬁrst few IMF components, whereas the EMD-based low frequency motion is the summation of the remaining components. Unfortunately, the authors failed to give the exact number of IMFs needed to be summed. This choice, highlighted by the authors, is very subjective. Wen and Gu [2004; Gu and Wen, 2004] extended the HHT and applied their methods to earthquake ground motions. The authors express x(t) as follows: n

x (t ) = Re

∑ a (t) exp j

iθ j ( t )

+ rn (t )

(13.32)

j =1

and introduced j, an independent random phase angle uniformly distributed between 0 and 2: n

x (t ) = Re

∑ a (t) exp j

i [ θ j ( t )+ φ j ]

+ rn (t ) .

(13.33)

j =1

This makes x(t) a random process. The process then has the following mean, covariance, and variance functions:

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 µ x (t ) = Re  

 iθ iφ  a j (t )e j (t ) E (e j )  + rn (t ) = rn (t ) , 

n

∑

K XX (t1, t2 ) =

j =1

1 2

n

∑ a (t )a (t ) cos[θ (t ) − θ (t ) , 1

j

2

j

1

j

2

j

(13.35)

j =1

1 σ x (t ) = E [ x (t ) − µx (t )] = 2   

2

(13.34)

2  

n

∑ a (t). 2 j

(13.36)

j =1

The authors extended the approach to a vector process and developed the cross-correlation covariance as follows: n

x k (t ) = Re

∑a

jk

(t ) exp

i [ θ jk ( t )+ φ jk ]

+ rnk (t ) .

(13.37)

j =1

With the modiﬁed approach, the spectra of the simulated samples compare well with those of the record. Zhang [2005] used a modiﬁed HHT to estimate the damping factor of nonlinear soil and its role in seismic wave responses at soil sites, using data from earthquake recordings. Zhang presented the following equation: n

x (t ) = Re

∑

a j (t ) exp

iθ j ( t )

n

= x (t ) = Re

j =1

∑ Λ (t) exp

[ − ϕ j ( t )+ iθ j ( t )]

j

(13.38)

j =1

where j(t) can be interpreted as the source-related intensity, j(t) is exponential factor characterizing the time-dependent decay of waves in the jth IMF component due to damping, and n

a j (t ) = Re

∑ Λ (t) exp j

− ϕ j (t )

.

(13.39)

j =1

The authors deﬁned the instantaneous damping factor as: η j (t ) =

© 2005 by Taylor & Francis Group, LLC

d ϕ j (t ) . dt

(13.40)

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The Hilbert damping spectrum is deﬁned as follows: n

D(ω, t ) =

 .a j (t )

.Λ j (t ) 

∑ − a (t) + Λ (t)  dt j =1

j

(13.41)

j

whereas the marginal Hilbert damping spectrum is:

d (ω ) =

∫

T

n

D(ω, t )dt =

0

∑∫ j =1

T

0

 .a j (t ) .Λ j (t )  a Λ + −  dt = d (ω ) + d (ω ) .  a j (t ) Λ j (t ) 

(13.42)

Equation 13.4 consists of two components, the time-dependent amplitude, which is related to marginal and Hilbert amplitude spectra, and the source-related intensity time-dependent amplitude. The new approach can be used to evaluate site damping and to address the site ampliﬁcation factor for the inﬂuences of soil nonlinearity in seismic analysis. The authors suggested that their approach must be validated by model-based simulation.

13.3.7 STRUCTURAL APPLICATIONS Quek et al. [2003] illustrate the feasibility of the HHT as a signal process tool for locating an anomaly in the form of a crack, delamination, or stiffness loss in beams and plates based on physically acquired propagating wave signals. The authors considered four different cases: 1. Experimental data of a damaged aluminum beam to estimate the damage and boundary conditions. 2. Delamination in a sandwiched aluminium plate. 3. Healthy and damaged states of reinforced concrete (RC) slab based on experimental response time history. 4. Boundary and damage location of an aluminum square. It appears that the authors were successful in all cases. Quek et al. [2004] did a comparative analysis of HHT, wavelet transform, and FFT in damage detection. Although the HHT was found to be a more direct method compared to the wavelet transform, the authors provide similar results in terms of precision and accuracy. The authors advocated the combination of EMD with Fourier transform on the decomposed IMF components. They ascertained that this combination is capable of revealing modal frequencies with relatively low energy content. Using HHT, Li et al. [2003] analyzed the results of a pseudodynamic test of two rectangular reinforced concrete bridge columns. The authors conducted extensive laboratory investigation to model a near-fault ground acceleration of the Chi-Chi Earthquake. The HHT was used to analyze the signal output from various columns tested. The aim of these analyses was to understand the structural behaviors of the

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columns under near-fault ground acceleration by observing the characteristics of the Hilbert spectrum. The authors compared their results with Fourier transform results. The authors drew various conclusions, notably that HHT can be used to obtain the instantaneous natural frequencies of the bridge columns and to understand the relationship between frequency changes and bridge conditions. Furthermore, the HHT method outperforms the FFT method. Finally, the authors noted during the experimental investigation that the more wide spread the distribution of the Hilbert spectrum, the more severe the damage of the column.

13.3.8 HEALTH MONITORING Pines and Salvino [2002] applied HHT in structural health monitoring. The authors analyzed a civil building model with and without the presence of damage. The authors also used the same model to simulate a seismic loading. The authors investigated two case studies: one looked at ground ﬂoor damage and the other at top ﬂoor damage. To correctly identify damage structure, the authors used the concept of phase dereverberation and applied EMD to dereverberation time series. To address characteristic frequency change as a result of damage, the authors suggest that the reverberated/dereverberated time domain response and the phase angles of both dereverberated/reverberated signal can be used as indicators of damage. Yang et al. [2004] used HHT for damage detection, applying EMD to extract damage spikes due to sudden changes in structural stiffness. This enabled them to identify damage time and locations. They also used the method to identify natural frequencies and damping ratios of the structure before and after damage. The authors noted that the capability of detecting damage spikes in a signal is a statistical variable depending on the severity of damage and level of noise pollution. This implies that when the severity of damage is low or the level of noise pollution is high, the current method cannot detect the damage. HHT was also used to detect cracks at rivet holes in thin plates by using Lamb waves [Osegueda et al., 2003]. The authors conducted experimental studies on a plate with a hole and with a notch. Area scanning was used to obtain time series information on the wave propagation characteristics. The location of the notch was detected accurately, and the extent of ﬂaw was accurately predicted for all the notches, except for the 3.2 mm long notch. The extent of ﬂaw was overpredicted for the shortest notch. Yu et al. [2003] used HHT for fault diagnosis of roller bearings. The accelerative vibration signal of roller bearings was analyzed. A wavelet was used to decompose the signal to decrease the inﬂuence of lower frequency noise, after EMD was applied to the signals.

13.3.9 SYSTEM IDENTIFICATION Chen and Xu [2002] explored the possibility of using HHT to identify the modal damping ratios of a structure with closely spaced modal frequencies and compared their results to FFT. The damping ratios determined by using the combined HHT

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and random decrement technique (RDT) are much better than those resulting from the fast Fourier approach. In their studies, to identify instantaneous frequency and damping ratio, the measured response time history of the MDOF system is decomposed into IMFs. The authors checked the validity of model by applying it to a real civil engineering structure. The results show that the FFT-based method fails to identify modal damping ratios when two modal frequencies are too close. Xu et al. [2003] compared the modal frequencies and damping ratios in various time increments and different winds for one of the tallest composite buildings in the world. The results were then compared to FFT. Once again, the HHT method outperformed the FFT. The FFT-based results signiﬁcantly overestimate the ﬁrst lateral and longitudinal modal damping ratios of the building. Yang et al. [2002] developed a damage identiﬁcation–based HHT. The authors applied the technique to the ASCE structural health monitoring benchmark structure. The approach was quite successful in identifying damage sections. Yang et al. [2003a] used HHT to identify multi-degree-of-freedom linear systems by using measured free vibration time histories. The authors proposed band ﬁlter and EMD cases with polluted signals and high modal frequencies. Failure to introduce the band-pass ﬁltering approach will result in a large number of siftings. The Fourier spectrum of the acceleration signal helps to identify the range of natural frequencies. The acceleration signal is processed through band-pass ﬁlters. The time history obtained from each band-pass proceeds through EMD. A linear least-square ﬁt procedure is proposed to identify natural frequencies, damping ratios, mode shapes, mass matrix, damping matrix, and stiffness matrix. Yang et al. [2003b] extended their work [Yang et al., 2003a] to identify general linear structures with complex modes by using free vibration response data polluted by noise. Simulated results show that the approach is capable of identifying the complex mode shapes as well as the mass, stiffness, and damping matrices. This can help with the complete dynamic characteristics of general linear structure.

13.4 SOME LIMITATIONS Quek et al. [2003] noted the following: • •

The approximate envelopes obtained using the local maxima and minima do not always encompass all the data. The use of discrete signals instead of continuous introduces numerical uncertainty in the true extrema.

The spline at the beginning and the end of the envelope can result in large variation, which is very critical in a low frequency signal. Deng et al. [2001] discussed the use of a cubic spline in calculating the lower and upper envelopes. The authors mentioned the data extension method as key in addressing this issue. They used a two-step extension method based on a neural network approach. The original signal was extended forward and backward, and a neural network was used to learn the envelopes.

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Chen and Feng [undated] proposed a technique to improve the HHT procedure. The authors noted that the EMD has its limitation in distinguishing different components in narrow-band signals. The narrow band may contain either (a) components that have adjacent frequencies or (b) components where the frequencies may not be adjacent, but one of the components has a dominant energy intensity much higher than the other components. The improved technique is based on beating-phenomenon waves. Olhede and Walden [2004] compared HHT to a maximal-overlap discrete wavelet transform (MODWT) and wavelet packet transforms (MODWPT). The authors replaced EMD projections by a wavelet-based one and developed a Hilbert spectrum via wavelet packets. The authors claimed the MODWPT and MODWT to be superior to the HHT using a particular class of Fejer-Korovkin wavelet ﬁlters. It appears that their assertion may be true only for nonlinear data. Attoh-Okine [2004] did a comparative analysis of HHT, MODWPT, and MODWT on pavement proﬁle data and deduced that the HHT outperforms the MODWPT and MODWT in the case of nonlinear and nonstationary data. Datig and Schlurmann [2004] did the most comprehensive studies on the performance and limitations of HHT with particular applications to irregular waves. The authors did extensive investigation into the spline interpolation. The authors discussed using additional points, both forward and backward, to determine better envelopes. They also did a parametric study on the proposed improvement and showed signiﬁcant improvement in the overall EMD computations. The authors noted that HHT is capable of differentiating between time-variant components from any given data. Their study also showed that HHT was able to distinguish between riding and carrier waves.

13.5 POTENTIAL FUTURE RESEARCH The HHT is becoming one of the most useful signal-processing techniques. The idea of cubic spline needs a critical look, although it has been addressed by a few authors. Another area that needs further research is proper interpretation of all of the IMFs. This depends on a variety of factors, especially the proper understanding of the system. Researchers have to address what criteria should be used to eliminate IMFs within a particular signal. Is the statistical analysis and distribution of the IMFs the answer? What are proper approaches for using the HHT technique as a ﬁlter method? Comparative analyses of HHT, FFT, and wavelet techniques have been well addressed, but more research is needed to investigate the MODWPT and MODWT, and which of these approaches is more applicable than the other. The use of computational intelligence has not been addressed in the overall HHT applications. For example, how can neural networks be used effectively in constructing the upper and lower envelopes? The use of HHT in image analysis and two-dimensional problems also needs to be explored.

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REFERENCES Attoh-Okine, N. O. (1999). Application of Wavelets in Pavement Proﬁle Evaluation and Assessment. Proc. Estonian Acad. Science, 5, 53–63. Attoh-Okine, N. O. (2004). Comparative Analysis of Hilbert-Huang Transform and Maximal Overlap Discrete Wavelet Packet Transform. Working paper, University of Delaware Engineering Department. Chen, C. H., Li, C. P., and Teng, T. L. (2002a). Surface-Wave Dispersion Measurements Using Hilbert-Huang Transform. TAO 13, 2:171–184. Chen, J., and Xu, Y. L. (2002). Identiﬁcation of Modal Damping Ratios of Structures with Closely Spaced Modal Frequencies. Struct. Eng. Mech. 14, 4:417–434. Chen, Y., and Feng, M. Q. (Undated). A Technique to Improve the Empirical Mode of Decomposition in the Hilbert-Huang Transform. p. 24. Chun-xiang, S., and Qi-feng, L. (2003). Hilbert-Huang Transform and Wavelet Analysis of Time History Signal. Acta Seismological Sinica 166, 4:422–429. Datig, M., and Schlurmann, T. (2004). Performance and Limitation of the Hilbert-Huang Transformation with an Application to Irregular Water Waves. Ocean Eng. 31, 14–15: 1783–1834. Deng, Y., et al. (2001). Boundary-Processing Technique in EMD Method and Hilbert Transform. Chinese Science Bull. 46, 11:954–961. Duffy, D. (2004). The Application of Hilbert-Huang Transform to Meteorological Datasets. J. Atmospheric Oceanic Technol. 21, 599–611. Flandrin, P., Rilling, G., and Goncalves, P. (2003). Empirical Mode of Decomposition as a Filter Bank. IEEE Signal Process. Lett. Gabor, D. (1946). Theory of Communication. Proc. IEE 93, 429–457. Gu, P., and Wen, Y. K. (2005). Simulation of Nonstationary Random Processes Using Instantaneous Frequency and Amplitude from Hilbert-Huang Transform. In this Volume. Huang, N. E., et al. (1999a). A New View of Nonlinear Water Waves: The Hilbert Spectrum. Annu. Rev. Fluid Mech. 31, 417–57. Huang N. E., et al. (2003a). A Conﬁdence Limit for the Empirical Mode Decomposition and Hilbert Spectrum Analysis. Proc. R. Soc. London A, 459, 2317–2344. Huang, N. E., Chern, C. C., Huang, K., Salvino, L. W., Long, S. R., and Fan, K. L. (2001). A New Spectral Representation of Earthquake Data: Hilbert Spectral Analysis of Station TCU129, Chi-Chi Taiwan, 21 September 1999. Bull. Seismological Soc. Am. 91, 5:1310–1338. Huang, N. E., Shen, Z., Long, S. R., Wu, M. C., Shih, E. H., Zheng, Q., Tung, C. C., and Liu, H. H. (1998). The Empirical Mode Decomposition Method and the Hilbert Spectrum for Non-Stationary Time Series Analysis. Proc. R. Soc. London A 454, 903–995. Huang, W., Shen, Z., Huang N. E., and Fung, Y. C. (1999b). Engineering Analysis of Biological Variables: An Example of Blood Pressure Over 1 Day. Proc. National Acad. Science 95, 4816–4821. Huang, W., Shen, Z., Huang N. E. and Fung, Y. C. (1999c). Nonlinear Indicial Response of Complex Nonstationary Oscillations as Pulmonary Hypertension Responding to Step Hypoxia. Proc. National Acad. Science 96, 1834–1839. Huang, N. E., Wu, M.-L., Qu, W., Long, S. R., Shen, S. S. P., and Zhang, J. E. (2003b). Application of Hilbert-Huang Transform to Non-Stationary Financial Time Series Analysis. Appl. Stochastic Models Bus. Industry 19, 245–268. Hwang, P. A., Wang, D. W., and Kaihatu, J. M. (2005). A Comparison of the Energy Flux Computation of Shoaling Waves Using Hilbert and Wavelet Spectral Anlysis Techniques. In this Volume.

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Lai, Y. (1998). Analytic Signals and the Transition to Chaos Deterministic Flows. Phys. Rev. E 58, 6:R6911–6914. Larsen, M. L., Ridgway, J., Waldman, C. H., Gabbay, M., Buntzen, R. R., and Battista, B. (2005). Nonlinear Signal Processing of Underwater Electromagnetic Data. In this Volume. Li, Y. F., Chang, S. Y., Tzeng, W. C., and Huang, K. (2003). The Pseudo Dynamic Test of RC Bridge Columns Analyzed Through the Hilbert-Huang Transform. Chin. J. Mech. A 19, 3: 373–387. Loh, C.-H., Wu, T. C., and Huang, N. E. (2001). Application of the EMD-Hilbert Spectrum to Identify Near-Fault Ground Motion Characteristics and Structural Responses. Bull. Seismological Soc. Am. 191, 5: 1339–1353. Long, S. R., Huang, N. E., Tung, C. C., Wu, M. L., Lin, R. Q., et al. (1995). The Hilbert Techniques: An Alternative Approach for Non-Steady Time Series Analysis. IEEE Geoscience Remote Sensing Soc. Lett. 3, 3–11. Lundquist, J. K. (2003). Intermittent and Elliptical Inertia Oscillation in Atmospheric Boundary Layer. J. Am. Meteorological Soc. 60, 2261–2273. Magrin-Chagnolleau, I., and Baraniuk, R. G. (undated). Empirical Mode Decomposition Based Time Frequency Attribute. pp 4. Montesinos, M. E., Munoz-Cobo, J. L., and Perez, C. (2002). Hilbert-Huang Analysis of BWR Neutron Detector Signals: Application to DR Calculation and to Corrupted Signal Analysis. Ann. Nuclear Energy 30, 715–727. Neithammer, M., et al. (2001). Time–Frequency Representations of Lamb Waves. J. Acoustical Soc. Am. 109, 5: Pt. 1, 1841–1847. Olhede, S., and Walden, A. T. (2004). The Hilbert Spectrum via Wavelet Projections. Proc. R. Society A 460, 2044: 955–975. Osegueda, R., Kreinovich, V., Nazarian, S., and Roldan, E. (2003). Detection of Cracks at Rivet Holes in Thin Plates Using Lamb-Wave Scanning. Proc. SPIE 5047, 55–66. Pan, J., Yan, X. H., Zheng, Q., Liu, W. T., and Klemas, V. V. (2002). Interpretation of Scatterometer Ocean Surface Wind Vector EOFs over the Northwestern Paciﬁc. Remote Sensing Environ. 84, 53–68. Phillips, S. C., Gledhill, R. J., and Essex, J. W. (2003). Applications of the Hilbert-Huang Transform to the Analysis of Molecular Dynamics Simulations. J. Phys. Chem. A 107, 4869–4876. Pines, D., and Salvino, L. (2002). Health Monitoring of One-Dimensional Structures Using Empirical Mode Decomposition and the Hilbert-Huang Transform. Proc. SPIE 4701, 127–143. Qu, J., and Jarzynski, J. (2001). Time Frequency Representation of Lamb Waves. J. Acoust. Soc. Am., 109(5) Part 1, 1841–1847. Quek, S. T., Tua, P. S., and Wang, Q. (2003). Detecting Anomalies in Beams and Plate Based on the Hilbert- Huang Transform of Real Signals. Smart Mater. Structures 12, 447–460. Quek, S. T., Tua, P. S., and Wang, Q. (2004). Comparison of Hilbert-Huang, Wavelet, and Fourier Transforms for Selected Applications. In this Volume. Salisbury, J. I., and Wimbush, M. (2002). Using Modern Time Series Analysis Techniques to Predict ENSO Events from SOI Time Series. Nonlinear Processes in Geophysics 9, 341–345. Schlurmann, T. (2002). Spectral Analysis of Nonlinear Water Waves Based on the Hilbert-Huang Transformation. Trans. ASME 124, 22–27. Schlurmann, T., and Datig, M. (2005). Carrier and Riding Wave Structure of Rogue Waves. In this Volume.

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Shen, J. J., Yen, W. P., and Fallon, J. O. (2003). Interpretation and Application of Hilbert-Huang Transformation for Seismic Performance Analyses. Advanced Mitigation Technologies 657–666. Veltcheva, A. (2005). An Application of HHT Method to the Nearshore Sea Waves. In this Volume. Veltcheva, A. D. (2002). Wave and Group Transformation by a Hilbert Spectrum. Coastal Eng. J. 44, 4: 283–300. Wang, W. (2005). Decomposition of Wave Groups with EMD Method. In this Volume. Wen, Y. K., and Gu, P. (2004). Description and Simulation of Nonstationary Processes Based in Hilbert Spectra. J. of Eng. Mech. 130, 8: 942–951. Wiley, A. P., Gledhill, R. J., Phillips, S. C., Swain, M. T., Edge, C. M., and Essex, J. W. (2005). The Analysis of Molecular Dynamics Simulations by the Hilbert-Huang Transform. In this Volume. Xiang, S. C., and Feng, L. Q. (2003). Hilbert-Huang Transform and Wavelet Analysis of Time History Signal. Acta Seismological Sinica 16, 4: 422–429. Xu, Y. L., Chen, S. W., and Zhang, R. C. (2003). Modal Identiﬁcation of Di Wang Building Under Typhoon York Using the Hilbert-Huang Transform Method. The Structural Design of Tall and Special Buildings 12, 21–47. Yang, J. N., Lei, Y., Pan, S., and Huang, N. (2003a). System Identiﬁcation of Linear Structures Based on Hilbert-Huang Spectral Analysis. Part 1: Normal Modes. Earthquake Eng. Structural Dynamics 32: 1443–1467. Yang, J. N., Lei, Y., Pan, S., and Huang, N. (2003b). System Identiﬁcation of Linear Structures Based on Hilbert-Huang Spectral Analysis. Part 1: Complex Mode. Earthquake Eng. Structural Dynamics 32: 1533–1554. Yang, J. N., Lei, Y., Lin, S., and Huang, N. (2004). Hilbert-Huang Based Approach for Structural Damage Detection. J. Eng. Mech. 130, 1: 85–95. Yang, J. N., Lin, S., and Pan, S. (2002). Damage Identiﬁcation of Structures Using Hilbert-Huang Spectral Analysis. 15th ASCE Eng. Mech. Conf., New York. Yu, D., Cheng, J., and Yang, Y. (2003). Application of EMD Method and Hilbert Spectrum to the Fault Diagnosis of Roller Bearings. Mechanical Syst. Signal Process. 19, 2: 259–270. Zeris, A., and Prinos, P. (2005). Coherent Structures Analysis in Turbulent Open-Channel Flow Using Huang-Hilbert and Wavelets Transforms. In this Volume. Zhang., R. R. (2004). An HHT Based Approach to Quantify Nonlinear Soil Ampliﬁcation. In this Volume. Zhang, R. R., Demark, L. V., Liang, J., and Hu, Y. (2004). On Estimating Site Damping with Soil Non-Linearity from Earthquake Recordings. Int. J. Non-Linear Mech. 39, 1501–1517. Zhang, R. R., Ma, S., Safak, E., and Hartzell, S. (2003). Hilbert-Huang Transform Analysis of Dynamic and Earthquake Motion Recordings. J. Eng. Mech. 129, 8: 861–875.

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ADDENDUM: PERSPECTIVES ON THE THEORY AND PRACTICES OF THE HILBERT-HUANG TRANSFORM Recently there have been innovative applications and major improvement to EMDHHT. These improvements have spread to general analytical approaches, hybrid applications, and bidimensional HHT.

13A.1 ANALYTICAL Coughlin and Tung [2004] used EMD to extract the solar cycle signal from stratospheric data. The authors highlighted some difﬁculties in using EMD — especially the inﬂuence of the end point in the sifting process. They addressed a method of reducing the inﬂuence of the end effects on the internal solution. Coughlin and Tung [2004] extended both the beginning and end of the spline by introducing a wave equation of the form:  2πt  Wave Extension = A sin  + phase + local mean   p 

(13A.1)

The typical amplitude, A, and period p, are determined by the nearest local extrema.

()

()

( )

( )

Abeginning = max 1 − min 1

Aend = max N − min N

()

()

( )

( )

(13A.2)

pbeginning = 2 time max 1 − time min 1

pend = 2 time max N − time min N

where max(1) and min(1) are the ﬁrst two local extrema in the time series and max(N) and min(N) are the last two local extrema. This approach reduces the large swings in the spline calculation. This approach is quite different from the one proposed by Dätig and Schlurmman [2004].

13A.2 HYBRID METHOD Iyengar and Kanth [2004] used HHT to decompose India monsoon data. The authors argued that the ﬁrst IMF is nonlinear and the remaining IMFs are linear. This is a little questionable, since it is not clear how the authors determined these properties. The authors used a neural network to forecast the nonlinear part of the data. Lee et al. [2003] explored the vibration mechanism of a bridge at both global and local behavior. They proposed a combination of EMD, HHT, and a nonlinear parametric model as an identiﬁcation methodology. The approach was successful in identifying abnormal signals indicating the structural condition of the bridge decks based on the time–frequency spectrum.

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Loutridis [2004] used EMD to decompose vibration signals from gear systems. The author developed an empirical law that relates the energy of the intrinsic modes to the crack magnitude of the gear. Using this technique, the degree of crack can be predicted based on energy obtained from the IMFs.

13A.3 BIDIMENSIONAL EMD Nunes and coworkers developed a method for analyzing two-dimensional (2D) time series which was used for textural analysis [2003a]. The bidimensional sifting process they presented in an earlier paper [Nunes et al., 2003] is as follows: • • • • •

Identify the extrema of the image I by morphological reconstruction based on geodesic operators. Generate the 2D “envelope” by connecting the maxima points with a radial basis function (RBF). Determine the local mean m_{i}, by averaging the two envelopes. Subtract I – m_{i} = h_{i} Repeat the process.

The authors used a radial basis function in the form:

()

()

s x = pm x +

∑ λ Φ( x − x ) N

i

i

(13A.3)

i =1

radial basis function (RBF); pm is mth degree polynomial in d variables; λ are RBF coefﬁcients; xi are the RBF centers. The stopping criteria for Nunes et al. [2003] approach is based on similar to 1D case used, using the standard deviation. Linderhed [2002] developed a 2D EMD and presented a coding scheme for image compression purposes. In the coding scheme, only the maximum and minimum values of the IMFs are used. Linderhed [2004] also developed the concept of empiquency, short for empirical mode frequency as a frequency measure. Empiquency is deﬁned as one half the reciprocal distance between two consecutive extrema points. Linderhed [2004] developed a sifting process for 2D time series. The symmetry criterion on the IMF envelope is relaxed, and the stop criterion is based on the condition that the IMF envelope is close to zero. Hence, there is no need to check for symmetry. The new sifting process is as follows: 1. Find the positions and amplitudes of all local maxima and minima in the input signal in_{lk}(m,n), where m and n denote spatial dimensions of the image. 2. Create upper and lower envelopes by spline interpolation, denoted by e_{max}(m,n) and e_{min}(m,n).

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3. For each position, calculate the mean of the upper and lower envelopes:

(

(

)

(

emax m, n + emin m, n

)

emlk m, n =

)

2

(13A.4)

4. Subtract the envelope mean signal from the input signal:

(

)

(

)

(

hlk m, n = inlk m, n − emlk m, n

)

(13A.5)

The process is terminated if the envelope mean signal is close to zero:

(

)

emlk m, n ≺ ε

(13A.6)

for all (m,n). If it is not, the process repeated until:

(

)

(

hl( k +1) m, n = hlk m, n

)

(13A.7)

When the termination criterion is achieved, k = K, and the IMF is deﬁned as:

(

)

(

)

)

(

cl m, n = hlK m, n

(13A.8)

5. The residue is deﬁned as r_{l}(m,n):

(

)

(

rl m, n = inl1 m, n = cl m, n

)

(13A.9)

6. The next IMF is found as follows:

(

) (

in(l +1)1 m, n = rl m, n

)

(13A.10)

7. Just as the 1D series the signal can be represented as:

(

)

(

)

x m, n = rL m, n +

L

∑ c ( m, n ) j

(13A.11)

i =1

Liu and Peng [2005] proposed a boundary-processing technique for bidimensional EMD. The authors proposed two techniques, but unfortunately the two algorithms developed by these authors are similar to algorithms proposed by Nunes et al. [2003] and Linderhead [2004].

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13A.4 SOME OBSERVATIONS The use of bidimensional EMD-HHT is beginning to emerge. Few results indicate that the 2D HHT outperforms traditional wavelet and Fourier analysis. As with the 1D HHT, there are difﬁculties regarding the type of spline and the effect of the end points on the overall sifting process. The use of different RBFs, such as thin plate splines, cubic spline, and multiquadrics, will be worth exploring in the bidimensional HHT.

REFERENCES Coughlin, K. T. and Tung, K. K. (2004). 11-Year Solar Cycle in the Stratosphere Extracted by the Empirical Mode Decomposition Method. Adv. Space Res. 34, 323–329. Dätig, M. and Schlurmann, T. (2004). Performance and Limitations of the Hilbert-Huang Transformation with an Application to Irregular Water Waves. Ocean Eng. 31, 14/15: 1783–1834. Iyengar, R. N. and Kanth R. S. T. G (2004). Intrinsic Mode Functions and a Strategy for Forecasting Indian Monsoon Rainfall. Meteorol. Atmospheric Phys. (online publication). Lee, Z. K., Wu, T. H. and Loh, C. H. (2003). System Identiﬁcation on Seismic Behavior of an Isolated Bridge. Earthquake Eng. Structural Dynamics 32, 1797–1812. Linderhed, A. (2003). 2-D Empirical Mode Decomposition: In Spirit of Image Compression. SPIE Proc. 4738, 1–8. Linderhed, A. (2003). Image Compression Based on Empirical Mode Decomposition. Proc. SSBA Symp. Image Anal. 110–113. Linderhed, A. (2004). Adaptive Image Compression with Wavelet Packets and Empirical Mode Decomposition. Dissertation No. 909 submitted to Linkoping Studies in Science and Technology. Liu, Z. and Peng, S. (2005). Boundary Processing of Bidimensional EMD Using Texture Synthesis. IEEE Signal Process. Lett. 12, 1: 33–36. Loutridis, S. J. (2004). Damage Detection in Gear Systems Using Empirical Mode Decomposition. Eng. Structures 26, 1833–1841. Nunes, J. C., Bouaoune, Y., Delechelle, E., Niang, O. and Bunel P. (2003). Image Analysis by Bidimensional Empirical Mode Decomposition. Image Vision Comput. 21, 1019–1026. Nunes, J. C., Niang, O., Bouaoune, Y., Delechelle, E., and Bunel P. (2003a). Bidimensional Empirical Mode Decomposition Modiﬁed for Texture Analysis. SCIA 2003, Lecture Notes on Computer Science 2749, 171–177.

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