Developments in Sedimentology, Volume 52: Cyclostratigraphy and the Milankovitch Theory

DEVELOPMENTS IN SEDIMENTOLOGY 52

Cyclostratigraphy and the Milankoviich Theory

FURTHER TITLES IN THIS SERIES VOLUMES 1-11, 13-15, 17,21-25A, 27, 28,31,32 and 39 are out of print 12 R.G.C. BATHURST CARBONATE SEDIMENTS AND THEIR DIAGENESIS 16 H.H. RlEKE 111 and G.V. CHlLlNGARlAN COMPACTION OF ARGILLACEOUS SEDIMENTS 18A G.V. CHlLlNGARlAN and K.H. WOLF, Editors COMPACTION OF COARSE-GRAINED SEDIMENTS, I 188 G.V. CHlLlNGARlAN and K.H. WOLF, Editors COMPACTION OF COARSE-GRAINED SEDIMENTS. II 19 W. SCHARZACHER SEDIMENTATION MODELS AND QUANTITATIVE STRATIGRAPHY 20 M.R. WALTER, Editor STROMATOLITES 258 G. LARSEN and G.V. CHILINGAR, Editors DIAGENESIS IN SEDIMENTS AND SEDIMENTARY ROCKS 26 T. SUDO and S. SHIMODA, Editors CLAYS AND CLAY MINERALS OF JAPAN 29 P.TURNER CONTINENTAL RED BEDS 30 J.R.L. ALLEN SEDIMENTARY STRUCTURES 33 G.N. BATURIN PHOSPHORITES ON THE SEA FLOOR 34 J.J. FRlPlAT, Editor ADVANCED TECHNIQUES FOR CLAY MINERAL ANALYSIS 35 H. VAN OLPHEN and F.VENIALE, Editors INTERNATIONAL CLAY CONFERENCE 1981 36 A. IIJIMA, J.R. HEIN and R. SIEVER, Editors SILICEOUS DEPOSITS IN THE PACIFIC REGION 37 A. SlNGERand E. GALAN, Editors PALYGORSKITE-SEPIOLITE: OCCURRENCES, GENESIS AND USES 38 M.E. BROOKFIELD and T.S. AHLBRANDT, Editors EOLIAN SEDIMENTS AND PROCESSES 40 8. VELDE CLAY MINERALS-A PHYSICO-CHEMICALEXPLANATION OF THEIR OCCURENCE 41 G.V. CHlLlNGARlAN and K.H. WOLF, Editors DIAGENESIS, I 42 L.J. DOYLE and H.H. ROBERTS, Editors CARBONATE-CLASTIC TRANSITIONS 43 G.V. CHlLlNGARlAN and K.H. WOLF, Editors DIAGENESIS, I I 44 C.E. WEAVER CLAYS, MUDS, AND SHALES 45 G k . ODlN, Editor GREEN MARINE CLAYS 46 C.H. MOORE CARBONATE DIAGENESIS AND POROSITY 47 K.H. WOLFand G.V. CHILINGARIAN,Editors DIAGENESIS, 111 48 J. W. MORSE and F.F. MACKENZIE GEOCHEMISTRY OF SEDIMENTARY CARBONATES 49 K. BRODZlKOWSK1andA.J. VAN LOON GLACIGENIC SEDIMENTS 50 J.L. MELVIN EVAPORITES, PETROLEUM AND MINERAL RESOURCES 51 K.H. WOLFand G.V. CHILINGARIAN,Editors DIAGENESIS, IV

DEVELOPMENTS IN SEDIMENTOLOGY 52

Cyclostratigraphy and the Milankovitch Theory W. SCHWARZACHER Department of Geology, The Queen's University of Belfast, Belfast BT7 1NN, United Kingdom

ELSEVIER Amsterdam - London - New York -Tokyo

1993

ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 P.O. Box 21 1,1000 AE Amsterdam, The Netherlands

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Data

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V

PREFACE

So stehen wir vor rnancher ungelosten Frage, und man mochte die Jugend beneiden urn die Entdeckungen welche ihr vorbehalten sind.

Eduard Suess

Cyclostratigraphy is a term that probably was first used publicly at a meeting organised by A.G. Fischer and I. Premoli-Silva in Perugia (Italy) in 1988. The name well expresses the intention of this branch of geology, which is to use cycles of any description for the construction and improvement of the stratigraphic framework. Practically all cycles which represent identifiable time periods, are related somehow to planetary movement and therefore they are “astronomical cycles” and this simply means that they can be explained ultimately by the laws of astronomy. This book will deal in particular with the so-called Milankovitch cycles, a term which will be used to describe any cycle which is related to changes in the orbital elements of our planet. In his famous theory of the ice ages, Milankovitch linked orbital variation to the amount of solar radiation that reaches the earth and so determines the climate. The last thirty years have seen the rediscovery of the Milankovitch theory. New evidence that has been derived largely from the study of Pleistocene deep sea sediments, together with renewed calculations of the earth’s orbital elements, have removed many of the doubts which have been associated with astronomical control theories. Already a cyclostratigraphy of the Pleistocene is firmly established. The present status of prePleistocene stratigraphy could be compared with the state of biostratigraphy when William Smith started to collect his first fossils. Whether it will ever be possible to construct a consistent stratigraphic framework that is based on sedimentary cycles, will have to be decided in the future. As will be explained however, cycles can be regarded as approximate time units and they are therefore extremely useful in the study of many of the problems connected with stratigraphy and sedimentology. The time-measuring qualities of Milankovitch cycles are indeed the most important aspect of this text and the time units of ka (thousand years, kilo anni) and Ma (million years) will be used to describe the cycles. The book is exclusively concerned with pre-Pleistocene stratigraphy. However, as the Milankovitch theory and most of the modern methods have been developed

vi

PREFACE

for the Pleistocene, the Pleistocene literature is highly relevant to the subject. This literature is already vast and involves many different areas such as astronomy, climatology, and various mathematical methods. As a geologist, it is clear that while writing this book I had to migrate into subjects which are not my own and to rely heavily on the published literature. It is hoped that not too many errors have occurred during such incursions. Although I have made a fairly extensive survey of the Milankovitch cycle literature, only a selection could be used in the second part of the book which deals with the description of actual examples.

vii

ACKNOWLEDGEMENTS

I am particularly indebted to A. Berger who read a draft of my chapter on Milankovitch theory and suggested many improvements. I am grateful to M. Ripepe who supplied me with data from the Piobbico bore hole, to I. Penn who provided me with geophysical data from British Geological Survey wells which penetrated the Kimmeridge, and to W. ten Kate and A. Sprenger who provided me with several diagrams from their thesis. I thank Miss M. Pringle from the School of Geosciences Queen’s University Belfast for drafting most of my diagrams. If this book is readable it is entirely due to my wife who corrected my English in the most tactful way.

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CONTENTS

........................................................... Definition of cycles ......................................................................... The quantitative description of cycles ........................................................ The complexity of cycles .................................................................... The regularity of cycles ..................................................................... The origin of cycles ......................................................................... Conclusions ................................................................................

4 5 7 7 9 10

OSCILLATING SYSTEMS ...................................................

11

Linear systems.............................................................................. Non-linear systems ......................................................................... Stochastically driven oscillators .............................................................. Examples of oscillating systems .............................................................. Climatic oscillators: the Ghil oscillator ....................................................... The Saltzman oscillator ..................................................................... Geological oscillators ....................................................................... Concluding remarks ........................................................................

11 14 17 19 21 22 24 27

THE MILANKOVITCH THEORY ............................................

29

The planetary system ....................................................................... The precession ............................................................................. The frequency stability of the orbital elements ............................................... The insolation .............................................................................. Milankovitch cycles and climate ............................................................. The Pleistocene climate .....................................................................

30 34 36 40 43 44

Chapter 1.

Chapter 2.

Chapter.3.

Chapter 4.

INTRODUCTION

METHODS OF ANALYSIS ...................................................

Stratigraphical sections as stochastic processes ............................................... The spectral analysis ........................................................................ The estimation of spectra. .................................................................. Cospectral analysis ......................................................................... Power spectra in stratigraphical analysis...................................................... Walsh spectra ............................................................................... The Walsh spectrum in stratigraphy .......................................................... The role of spectral analysis in cyclostratigraphy.............................................. The filtering of sections ..................................................................... The effects of non-stationarity ...............................................................

1

49 51 52 55 58 59 63 65 65 67 69

CONTENTS

X

Complex demodulation ..................................................................... Conclusions ................................................................................

.

Chapter 5

THE RELATION BETWEEN TIME AND SEDIMENT ACCUMULATION

70 72

...

The random walk model ............................. The rate of sedimentation ................................................................... ............ The completeness of the record ................................... ............. Cycles with precise time periods ................................... Cycles with changing lithology ............................................................... The effect of random variations on a periodic signal ..................................... Bioturbation ............................................................................... Methods of finding a mapping function z . t ...................................................

.

73 74 78 79 81 82 85 88 90

STRATIFICATION AND STRATIFICATION CYCLES ........................

93

Marl-limestone sedimentation .............................................................. Stratification patterns ........... ................................................... Stratification cycles ......................................................................... ..... The numerical description of bedded sequences .................. Stratigraphic trends ......................................................................... The recognition of stratification cycle boundaries .............................................

94 97 99 101 104 105

EXAMPLES FROM THE CARBONIFEROUS ................................

107

The Lower Carboniferous ................................................................... The Carboniferous limestones of north-west Ireland ...................... The Benbulbin shale and Glencar limestone. ................................................. The Dartry limestone ....................................................................... The cyclostratigraphic interpretation of the SIigo sequence .................................... The cyclicity of Yorkshire and North Wales ................................................... The Pennsylvanian (Upper Carboniferous) cycles .............................................

107 108

Chapter 6

Chapter 7.

.

Chapter 8

TRIASSIC: CARBONATE PLATFORMS .....................................

The Northern Calcareous Alps .............................................................. The cyclicity of the Dachstein limestone ..................................................... Quantitative studies ......................................................................... The ?tans-Danubian Central range .......................................................... The Dolomites. ............................................... ........................ Absolute time estimates ..................................................................... The similarity of cycles .......................................................... Subsidence ......................................................... .. Sea level fluctuations ....................................................................... The geometry of cycle formation ............................................................ The platform-basin interaction .............................................................. Concluding remarks ........................................................................

109 116 117 119 122 125 125 128 130 131 135 136 137 139 139 142 145 146

xi

CONTENTS

.

Chapter 9

SOME JURASSIC EXAMPLE

..........................

149

The lower Jurassic in Britain ................ ........................ .... The Kimmeridge clay .. ................................................................. The German Upper Jurassic ................................................................ Examples from %thyan regions ..............................................................

149 152 154 157

Chapter 10.

159

EXAMPLES FROM THE CRETACEOUS ....................................

The Umbria and Marche region of Italy ...................................................... The Majolica ............................................................................... The Scisti a Fucoidi ......................................................................... The Scaglia bianca .......................................................................... The western %this and the Atlantic ......................................................... The epicontinental seas ........... ............................................ .. The Cretaceous-Tertiary boundary .......................................................... Concluding remarks ........................................................................

159 159 162 166 172 175 177 180

NON-CARBONATE CYCLES ................................................

183

The Permian evaporites ..................................................................... Lacustrine environments: the Lokatong and Passaic Formations. ............. The Green River Formation ................................................................. Shallow marine environments ............................................................... Fluviatile environments ..................................................................... Prodelta turbidites ................................. .....................................

183 186 188 190 193 194

Chapter 11.

Chaprer 12.

CYCLOSTRATIGRAPHY AND MILANKOVITCH CYCLES ..................

Practical cyclostratigraphy ................................................................... The recognition of Milankovitch cycles ...................................................... The causes of sedimentary cycles ............................................................ REFERENCES

............................................................................

SUBJECT INDEX

.........................................................................

197 197 198 204 209 221

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1

Chapter 1

INTRODUCTION

The idea that geological history developed in a series of distinct stages has a long tradition. Cuvier and Lamark believed in a sequence of evolutionary stages which affected both the biosphere and the environment. Suess (1892) found evidence of eustatic sea level fluctuations which caused transgressions and regressions and had an important influence on the development of our planet. Stille (1924) thought that tectonic activity followed a world wide pattern which operated over definite periods of time. Similarly, Umbgrove (1947) in his “Pulse of the earth” apparently found evidence of a clear natural division of the stratigraphic column. More recently, Vail (Vail, 1988; Vail et al., 1991) and his associates have provided a detailed scheme of sea level changes which has become known as sequence stratigraphy. Vail subdivides the cycles into six different orders, according to the lengths of time which they represent. The six orders represent intervals ranging from 50 Ma to 10 ka. It is thought that at least the low frequency cycles are tectonic in origin but there is unlikely to be a single mechanism which can account for all the cycles. It is not assumed that sequence cycles represent repeated equal time intervals. Cyclostratigraphy in contrast, is concerned specifically with cycles which represent equal time intervals and in particular with time intervals which can be associated with definite time periods. This means that in practice, cyclostratigraphy is largely concerned with cycles which are astronomically controlled. In the high frequency range, these are the daily and annual cycles and in the low frequencies, these are the Milankovitch cycles, which range from approximately 10 ka to 2 Ma. Some of Vail’s cycle orders overlap the Milankovitch cycles. However, his order terminology will not be used in the following text because unless very strict definitions are employed, it can lead to confusion. Cycle orders can only be established when several cycles of different lengths are present simultaneously. In such a case, the first order cycle is always the longest cycle and the higher orders can only be determined when the lower orders are present. In other words, the concept of cycle orders is completely relative. The alternative approach is to define cycle orders by time limits and this has been proposed by Vail. Therefore the identification of a cycle order is based on finding the absolute time duration of a cycle. In the Vail scheme, the Milankovitch cycles are covered by the third to sixth order and the important Milankovitch periods of 100 to 20 ka, cover three orders. To find out the order of a cycle, one would have to determine its duration with an

2

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

accuracy of at least 10 ka. In the rare cases where this is feasible, it should also be possible to identify the cycle as belonging to the Milankovitch group. In this approach, Milankovitch cycles are kept separate from sequence stratigraphy or any other stratigraphic system because they form a well defined group of nearly time periodic cycles which are directly linked to planetary movement. The effects of planetary movement on the environment are well within our experience: day or night, tidal movements and the seasons are all determined by the movement of the earth in the solar system. Such environmental changes can be recorded in sediments and the study of astronomical cycles is therefore an important part of the study of sedimentary cycles in general. In the last century astronomers made a detailed study of the planetary system and came to the conclusion that the earth’s orbit and the position of its rotational axis undergo slow cyclic variations, with periods ranging from thousands to hundreds of thousands of years. These slow cycles have come to be known as Milankovitch cycles. It was probably the astronomer John Herschel (1830) who first suggested that the long range changes in the earth’s orbit ought to be observed in the geological record. Lye11 discussed the possibility of astronomically caused climate changes in some detail in his “Principles” (1867). The idea of orbital control was taken up seriously by Croll (1875) who suggested that the ice ages are caused by the variations in the distance between the earth and the sun. Croll was also the first to attempt to fit actual stratigraphic observations to a system of astronomical cycles. The first connection between sedimentary cycles and orbital control was made by Gilbert (1895) who observed the very regular alternations of limestone and shale in the Niobrara chalk at the base of the Rocky Mountains in Colorado. Gilbert argued that there is no “purely terrestrial” phenomenon which could cause such regular changes and he therefore suggested that astronomical cycles must be responsible for the climatic changes which determined the limestone shale sedimentation. Gilbert’s argument is still valid but his calculation of the duration of the Cretaceous period, which was based on the sedimentation rates which he deduced from his cycles, was somewhat uncritical. Important is the work of Bradley (1929) who was probably the first geologist to describe a 21,000 year cycle based on measured sections. Bradley gave reasonable petrographic evidence for the existence of true varves in the Eocene Green River formation in the Central Western United States and he used these varves to calculate some sedimentation rates. He observed a regular and approximately 3 m thick alternation between oil shale and marl stones. According to his calculations, this cycle had an average duration of 21,630 years and he suggested that it was due to the cycle of the precession of the equinoxes. However it was Milankovitch who worked on the first proper analysis of the astronomical control theory from 1913 onwards. In his monumental work (Milankovitch, 1941), he published new calculations of the orbital variations and

INTRODUCTION

3

showed quantitatively how these determine the amount of solar radiation which is the ultimate driving force of our climate. Milankovitch estimated climatic changes in terms of shifting geographical latitudes and showed that the ice ages, as far as their durations were known, seemed to fit his calculations. The theory was not very popular at the time and there are several reasons for this. Possibly the strongest arguments against climatic cyclicity came in fact from climatologists. When Milankovitch calculated the amounts by which the solar radiation changes through orbital variations, he found them to be very small and indeed most climatologists did not believe them to be large enough to cause such drastic climatic changes, as were represented by the glacial and interglacial periods. Geologists also objected because the time scale of the Milankovitch glacial cycles did not fit the Pleistocene stratigraphy very well, as it was known at that time. Indeed, the almost general acceptance of the Milankovitch theory came only after a much more detailed stratigraphy became available from a study of deep sea sediments. Two new methods in particular have helped to re-establish the Milankovitch theory, the most important one being the study of oxygen and carbon isotopes (Emiliani, 1955; Emiliani and Geiss 1958) which provided a quantitative record of palaeotemperatures and land locked ice masses. The finding of magnetic reversals in the cores of deep sea sediments was also of considerable importance. This enabled an accurate determination of sedimentation rates to be made and therefore a precise timing of the isotope records. Time series analysis of such data showed that they contained exactly the frequencies which were predicted by the Milankovitch theory (Hays et al., 1976). By comparing the ratios of the observed frequencies with the theoretical ratios, it was possible to gain support for the theory, completely independently of any absolute time determinations. Establishing an astronomical control theory for pre-Pleistocene sediments is much more difficult. Not only are the climatic variations in most geological periods less pronounced .than in the Pleistocene but also, absolute timing becomes more difficult and much less accurate in the earlier geological periods. In the absence of precise timing, regularity of sedimentary cycles becomes a very important argument in favour of astronomical control. The regularity can either refer to the thicknesses of the beds, so called rhythmites (Sander 1930, 1936), or it may consist of repeated groups which contain a constant number of beds (Schwarzacher, 1947). Such groups of beds can be regarded as being a mixture of cycles with different frequencies and the number of beds in a group again provides a ratio which can be used to provide support for the idea of astronomical control independently from absolute time determinations (Schwarzacher, 1954,1964, 1975). With the rediscovery of the Milankovitch theory in the Pleistocene, more authors have considered the possibility of pre-Pleistocene sediments having been formed under the influence of Milankovitch cycles. It is usually assumed in such theories that it is the climate which determines the cyclicity. It is still uncertain whether other effects exist, such as for example a cyclicity in the earth’s magnetic field.

4

CYCLOSTRATIGRAPHYAND THE MILANKOVITCH THEORY

DEFINITION OF CYCLES

A sedimentary cycle is a group of different lithologies or textures which is

repeated regularly in a sequence. This definition which is based on observable criteria, must be regarded as a preliminary definition; it is unfortunately not quite sufficient for our purpose in cyclostratigraphy. Repetitive events can be either produced by episodic processes or by processes governed by oscillating systems. It is the latter which will be regarded as being responsible for true sedimentary cycles and the existence of an oscillating system is regarded as being essential for the formation of stratigraphically useful cycles. Oscillating systems are dynamic systems of physical quantities which are capable of development, and this development leads of necessity to repetition. Whether sedimentary cycles are associated with oscillating systems will always involve geological arguments and purely descriptive criteria are not sufficient for recognising cycles as being cycles in the above sense. To link the definition of sedimentary cycles to the existence of an oscillating system is a logical step and defining stratigraphic cycles in this way is in agreement with the terminology of Einsele et al. (1992), who differentiate between cyclic and event stratigraphy. The difference between cyclic and episodic events however, will not be regarded as relative (Seilacher, 1992). On the contrary, it is absolute and an oscillating system either exists or it does not. If an oscillating system exists, then any cycle generated by it will have time-measuring qualities. Repetitive episodic events clearly lack this quality. Whether one can identify an oscillating mechanism is of course another matter and it is likely that under some circumstances, it will be impossible to tell whether events are cyclic or episodic. When trying to classify the origin of sedimentary cycles, Brinkmann (1925) differentiated between autonomous cycles (stratification) and externally induced cycles. Beerbower (1964) made a similar classification into two types which he called autocycles and allocycles. Autonomous cycles are equivalent to allocycles and they are generated by systems which can oscillate. For example, Beerbower considers fluviatile sedimentation on flood plains, where the fluctuation of a meandering river can produce cycles. Allocycles on the other hand, are driven by an outside force which is oscillating. They are therefore some kind of extension to an autocyclic system. Wical examples for the so-called allocycles are sedimentary cycles which have been produced by tidal forces or climatic fluctuations. In physics, the definitions are more precise. One differentiates between conservative and non-conservative oscillating systems. The conservative system is an ideal system which is without friction and therefore oscillates continuously. Although such systems do not really exist, some processes like planetary motion or the movement of a pendulum over short intervals, can be treated in this way. The non-continuous systems are called dissipative and self oscillating. The dissipative system is an oscillation which eventually disappears. A typical example for this is

INTRODUCTION

5

the damped pendulum. In contrast, self oscillating systems are systems which can create periodic processes at the expense of a non-periodic source of energy. Typical examples for such systems are steam engines or mechanical and electric clocks. The differentiation between dissipative and self oscillating is one of convenience and if one were to apply this nomenclature to oscillators that actually occur in nature, one might come to the conclusion that all systems must be classified as self oscillating, but that they are often driven by very discontinuous energy sources. With these definitions from physics in mind, it is possible to clarify the meaning of the terms autocycle and allocycle. The allocycle is an induced cycle and therefore each allocycle needs an autocyclic system to drive it. Whether the allocyclic system exists or not is irrelevant for the autocycle. Considering that most autocycles will have allocyclic systems attached to them, it becomes obvious that the differentiation between the two can become quite meaningless. It is possible for example, that two autocyclic (self oscillating) systems can interfere with each other and that resonances may develop. In this case, both are the inducing and reacting systems, or in established geological nomenclature they are both auto and allocyclic at the same time! In some of the more recent literature, the term auto cycle is used almost as if it meant that it is of non-Milankovitch origin and at the same time, it is implied that autocyclic processes provide an easy explanation for sedimentary cycles. The latter is not true and as will be discussed in detail later, it is extremely difficult to find terrestrial oscillating systems which can be made responsible for cyclic sedimentation within the Milankovitch frequency range.

THE QUANTITATIVE DESCRIPTION OF CYCLES

When quantitative methods are used for describing cycles, there are three attributes of cycles which are of outstanding importance. These are the dimensions of the cycle, meaning its thickness as well as its regional distribution, the complexity and the regularity. The previously given definition of cycles clearly includes such sedimentary features as lamination and stratification, because they are repetitive. Like fossil assemblages, cycles vary in their stratigraphic importance and some may only be used for local correlations while others may be traced world wide. Whether individual beds or even laminae are of stratigraphic value, depends on the type of sediment and the detail of stratigraphic resolution which is attempted. For example, many laminae in shales, are useless for stratigraphy because they cannot be traced. On the other hand, the varves which are found in evaporites for example, can be extremely important. In order to indicate the various scales of cycles, some geologists have used terms like micro, meso and macro cycles, a nomenclature which is fairly harmless as long as no genetical implication is associated with such terms. It is possibly better

6

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

practice to make a direct reference to the scale by using such terms as millimetre, centimetre or metre cycles. Depending on the rate of sedimentation, vertical thickness is more or less related to the time interval represented by the cycles. The regional distribution of cycles, which is equivalent to their lateral dimensions, is very much determined by the sedimentary environment, and regional studies are therefore very important for understanding the mechanism of cycle formation. Such studies depend on the traceability of cycles and this in turn often depends on their scale. Larger cycles and groups of beds are more easily traced. The regional change of cycles can often provide valuable information, such as the direction of sediment transport and sediment sources, which may have played an important role in their formation. However, a number of studies have shown that it is possible to trace beds and even individual laminae over very large areas. It is very often wrong to assume automatically that stratification is a locally restricted feature. Sometimes, tracing lithologies on a regional scale may also be used to decide between the primary and diagenetic origin of structures in the sediment. The regional extent or the area over which cycles are effective, will be called the domain of the cycle generating system. This domain should be related to the physical dimensions of the oscillating systems if possible. This is usually only the case if the mechanism which generates the cycles is known in some detail. For example, periodic processes such as oscillations of water levels in a lake (seiches) are caused by a system which is well defined within given geographical boundaries. Similar effects of tidal activity can be related to the earth, moon, sun system and within these boundaries, they can be regarded as autocyclic but they can only be observed where tides are active, that is, in the sea. It is clear from both of the examples that the physical size and boundaries of the system do not as a rule, coincide with the region or domain in which the effect of the system can be observed. A knowledge of the domain is the first step towards understanding the mechanism of the oscillating systems which generate the cycles. The latter are of great importance when interpreting cycle producing mechanisms. Cycles which can be observed world wide must be due to large scale variations of either exogenic or endogenic forces. A cycle which occurs only locally can possibly be explained by some local process of sedimentation. If the domain is outside the region of the cycle producing system, one can speak of induced cycles. The recognition of domains is clearly difficult, not only because the environmental conditions may not be suitable for the recording of cycles, although the effects of the oscillating system may be present, but also because different environments will react quite differently to the same system. Further complications can arise when an observed cycle and the domain associated with it, is the combination of either two oscillating processes, or when it is a combination of an oscillation process with either an aperiodic or a random process. Any such mixed processes, which almost certainly are associated with any cyclic

INTRODUCTION

7

sedimentation, make the differentiation between autocyclic and allocyclic not only very difficult but also, as said previously, meaningless.

THE COMPLEXITY OF CYCLES

Any cycle must contain at least two different recognisable lithologies or textures, otherwise there would be no variation. Generally speaking, a cycle consisting of only two alternating lithologies is considered to be less complex than for example, a coal measure cycle which may contain ten or more different lithologies. However, many changes in the sediment are continuous and it then becomes quite arbitrary as to how a cycle is divided and the number of lithologies which make up a sequence is therefore not important in explaining cyclicity. On the other hand, complex cycles are more easily recognised and this may help considerably when correlating cycles. In the older literature, great emphasis was put on the symmetry of cycles, that is, whether lithologies are arranged as ABCABC... or ABCBAE3C... but the genetic significance of such different types is probably small and mostly reflects the way in which the sediment reacts to the cyclic stimulus of the environment. A complexity which is different from lithological complexity, is found when a sequence contains several approximately phase coherent cycles of different order. In such cases, some very complex cyclic patterns may arise and these can help in identifying the cycle mechanism. The best examples for this type are Milankovitch cycles. Such cycles are a mixture of different frequencies and they can be identified if the ratio between the frequencies can be evaluated. This mixing of cycles often finds its expression in the grouping of apparently identical beds. The groups are called stratification cycles (Schwarzacher, 1987) or in a more loose terminology they can be referred to as a “bundle” (Schwarzacher, 1952).

THE REGULARITY OF CYCLES

There are three categories of regularity: a regularity in the sequence of rock types, a regularity with respect to the thickness of the cycle and finally a regularity in the time interval which is represented by the cycle. One can therefore speak of a sequence cycle, a thickness cycle or a time cycle. Much of the background work in cyclostratigraphy consists in establishing the degree of this regularity. Mathematics gives a very precise definition of periodicity in a sequence. If a variable x ( t ) observed at different times t , shows the functional property: ~ ( t ) = ~ ( t + n p ) n = l , 2 , 3 ...

then this sequence is periodic and the constant p is called the period.

(1-1)

8

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

Sine functions are typical examples of periodic functions. Such functions can also be used as building blocks for more complex periodic data by adding them. Such complex data however, are periodic in the strict sense (eq. 1-l), only if the ratios of all the possible pairs of frequencies form rational numbers. Any combination of sine or cosine waves in which this is not the case, leads to almost periodic data which is often referred to as quasi-periodic. Quasi-periodic sequences usually arise when two or more unrelated periodic processes are mixed. As will be seen later, the Milankovitch cycles are the result of planetary movement around the sun and since each of the major planets has its own characteristic period, the mixture of these leads to quasi-periodicity. The definition given by eq. 1-1, can be applied to stratigraphic sequences by replacing time with the stratigraphic position, if this is preferred. By testing whether the condition of eq. 1-1, is fulfilled, any sequence can be examined for space or time periodicity. It is clear however, that geological cycles will not follow a precise mathematical definition and one has to allow for the fact that cycles are only more or less identical, fixed intervals and statistical methods to measure this “more or less”, will permit the measurement of cyclicity in quantitative terms. Of the three types of regularity, which are sequence, thickness and time, cyclostratigraphy is most concerned with cycles that mark equal time spaces. Indeed, our interest in cycles which are of regular thickness or in the order of lithologies, is largely due to the possibility that such cycles also represent regular cycles in time. Time of course cannot be measured in the same way as stratigraphic thickness but it can be estimated by making assumptions about sedimentation rates. Under normal steady conditions of sedimentation, it is always more likely that cycles of equal thickness represent equal time intervals. This is because any other mechanism would imply that fluctuations in sedimentation rates combined accidentally in such a way that cycles of equal thickness resulted. On the other hand, time regular cycles are very likely to be recorded as cycles of somewhat unequal thickness because fluctuations in sedimentation rates are always present. This rule which connects the thickness of cycles with time, was first formulated by Sander (1930) and it is still of fundamental importance. The exceptions to the rule are cycles which carry no time information, that is, layers which have been deposited so fast that they can be regarded as being instantaneous and which for some reason have formed in approximately equal thicknesses. Such layers could result for example, from repeated turbidity currents which had approximately the same volume, or from sedimentation which was caused by fluctuations of the transporting power of a stream with a certain capacity. Sander’s rule tells us that we cannot restrict our attention to cycles which are regular in thickness. It further tells us that if we do find some regularity in thickness, the associated time regularity must have been higher. Again this is because it is unlikely that a higher regularity is the result of random fluctuations.

INTRODUCTION

9

THE ORIGIN OF CYCLES

Introducing time as the variable of primary interest in cyclostratigraphy, leads from descriptive analysis to the more speculative considerations regarding the origin of cycles. We will concentrate on primary sedimentary cycles, where the sediment is regarded as being a record of conditions in the environment. The sedimentary cycle therefore, simply reflects the cyclic fluctuations of environmental conditions. It has been claimed that cycles can also originate from post-depositional diagenetic processes but such explanations have never been proved and they are highly unlikely. It would be very difficult to visualise a diagenetic process, for example, the unmixing of carbonate clay sediments, which follows the original sedimentary bedding without having any connection with it. If however, primary differences in sediment composition or texture existed, then diagenesis can accentuate, alter and sometimes eliminate primary cyclicity. The post-depositional changes therefore have to be carefully examined and taken into consideration. Traditionally, two explanations of the origin of cycles have been favoured. These are either changes in water depth, which are largely caused by tectonic movements, or changes in climate. The effects of such general causes may overlap and an analysis of the sedimentary record may not permit a clear differentiation to be made between the two. For example, it is not immediately obvious whether an increase in current activity represents the shallowing of an environment or an increase in storm activity. Similarly, an increased terrestrial influx can be caused either by the uplift of some source area, or by an increased run off due to increased precipitation. Since the sedimentary environment is an extremely complex system in which most of the variables interact and depend upon each other, any interpretation of the sediment which only partly reflects this complexity, cannot provide a full reconstruction of the environment. It is well known that some types of sediments can record specific changes in the environment better than others. In general, clastic sediments are good indicators of changes in current velocities, turbulence and the general energy of the environment, whereas carbonates often record changes in biological activity and chemistry in general. The geographical setting of the depositional area is also important. Small scale changes in water depth are more dramatically recorded near the margins of a basin. Changes in planktonic assemblages are probably better preserved in pelagic sediments. Changes in sedimentation can usually be explained by immediate changes in the environmental conditions but why such changes occur, is more difficultto decide. It is very important that the evidence provided by the sediment is not over-interpreted in order to fit or apparently fit, one preconceived model.

10

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

CONCLUSIONS

This introductory discussion attempted to show the difficulties and problems involved in using sedimentary cycles as a stratigraphic tool. To be of practical use, cycles must either be periodic or at least, nearly periodic in time. They must also be recognisable over large distances. Both properties are very difficult to establish, particularly if not much is known about the mechanism generating the cycles. Indeed, one can often only come close to identifying this mechanism by establishing the time and space regularities of the cycles. A number of important questions need to be asked. What kind of oscillating systems are available for generating the sedimentary cycles? How can we recognise the effect of such systems? In particular, how can we recognise Milankovitch cycles, which are our prime interest? Closely related to the last problem is the question of how geological time can be measured and how reliable the sediments are, as time recorders.

11 Chapter 2

OSCILLATING SYSTEMS

Cycles of any kind are generated by a mechanism which is called the oscillating system. Sedimentary cycles largely reflect the environmental conditions and the environment therefore is part of the oscillating system which is responsible for the cycles. The oscillating system is a dynamic system which may be visualised as a system of variables and parameters which describe the conditions of the sedimentary environment at any time during the sediment formation. The most important feature of the system is that its variables should interact with each other in such a way that repeated oscillations are generated. A sedimentary series that has been produced in this way, is basically different from a sequence which consists of a series of isolated events and the recognition of sedimentary cycles depends on recognising some oscillating system. In classical physics, dynamic systems are described by variables such as positions, movements and masses which are connected by differential equations. Systems are very often greatly simplified, sometimes to such an extent that they become unreal. For example, the pendulum without friction, which will be discussed shortly, is an abstraction which does not actually exist but which helps us to understand the real process. The use of simple systems which have been borrowed from physics as models for sedimentary cycles should be understood in a similar way. We will attempt to replace very complex processes by simple models in order to obtain a better understanding of cycle formation. To be successful in this, requires considerable abstract thinking and simplification. Nevertheless, both systems, whether they are model or real world, have the same components such as masses, distances and movements in common.

LINEAR SYSTEMS

One of the most common assumptions that are made when analysing a physical process is that one is dealing with a linear system. This means essentially that the rules of operating the system are not changed whilst the system is operating and in particular, that all the parameters of the system remain constant. The number of variables which are necessary to describe a system, depend on its complexity. To become an oscillating system, it is necessary for at least one

CYCLOSTMTIGRAPHY AND THE MILANKOVITCH THEORY

12

Fig. 2-1. A simple oscillating system.

of the variables to change and repeatedly return to the same value. For example, in a one-dimensional system, a single co-ordinate and the rate of change of this co-ordinate constitutes a minimum description. In order to be oscillating, one of the co-ordinates has to be repeatedly visited. We will consider the simple oscillator which is illustrated in Fig. 2-1 as a first example. A body is suspended by two springs. The body can only move in the x direction and it is not affected by gravity or friction. We can describe the system by saying that the change of position, which means the acceleration of the body, only depends on its position. If one adds mass to the body, one can say that the force acting on the point is proportional to its displacement: mx = -kx

(2-1)

In this equation, m represents the mass and k is a constant which describes the stiffness of the spring and x is the position of the point. By setting k / m = wt one can write the equation as:

x = - 0 o2x

(2-2)

with the general solution: x = A coswo t

+ B sinwo t

By assuming that at t = 0, x = xo and i = i o one finds the solution: x = K cos(w0

+ (p);

i = y = -Kwo sin(w0 t

+

(p)

(2-3)

where K = d m and tan (p = - B / A = io/w xo. An important way of representing an oscillating process, is through its phase plot. In the present example, this is the graph of the position against the speed of the point in the x , y plane or phase plane. This function is called the phase path which in parametric form, is given by eq. 2-3. The co-ordinate form of this is: XL

yL

K2

K2wi

-+-=1

(2-4)

This is clearly the equation for an ellipse (see Fig. 2-2) and because the parameter

OSCILLATING SYSTEMS

13

Fig. 2-2. Phase portrait of a frictionless oscillator. The x co-ordinate gives the position and the y co-ordinate gives the speed.

K depends on the initial conditions, there is a whole family of identical ellipses which would fill the phase plane completely. The important feature of the phase paths in this example, is that they are closed curves. This means that each value is revisited again and again and that the process therefore is not only oscillating but is also periodic and so fulfils the conditions of eq. 1-1. The system which has been discussed, is typical for systems which are called conservative systems. These are systems which need no external energy to keep them operating because they do not lose any energy. Systems of this type are an idealisation and they do not exist in the real world. However, they provide very good models, either for processes which are only observed over short time intervals or for systems which due to their size and inertia, change so slowly that the loss of energy can be neglected. In most real systems, energy losses due to friction are inevitable. For example, the oscillating spring can be made more realistic by introducing a frictional force. It is reasonable that such a force should be proportional to the velocity of the moving mass and the resulting oscillation is therefore: X+2hi++w(:x=O

(2-5)

The factor 2h = b/m is equivalent to the coefficient of friction divided by the mass and w i = k/m as before. The equation has the general solution: x = A ell'

+ B eA2'

(2-6)

where h l , A2 are the roots of the quadratic equation: 2h + h2 + w i = 0. If h2 < 00 then one is dealing with a damped oscillating system and setting w = one finds the specific solution:

,/-

x = K e-h cos(wt + p)

i= y = -Kwe-h' [ h cos(wt

+ q ) sin(wt + q ) ]

(2-7)

14

CYCLOSTRATIGRAPHYAND THE MILANKOVITCH THEORY

Fig. 2-3. Phase portrait of an oscillator with friction. Co-ordinates as in Fig. 2-2.

+

where K = d m and tan rp = -(xo hxo)/wo xo If h2 =- w i then the system becomes aperiodic and no longer oscillates. The solution of eq. 2-6 under this condition is: x = A e-""

+ B e-92"

and

y = q l A e-"" - q 2 e-92"

(2-8)

JG.

where q1,2 = h T The systems represented by eqs. 2-7 and 2-8 are clearly no longer conservative and such systems are called dissipative. The difference between such systems and the previous one is clearly seen in their phase portraits (Fig. 2-3). The phase path of eq. 2-7 shows that the maximum displacement of the oscillating point as well as its speed, decreases exponentially. The spiral shaped orbit will eventually end in the origin of the x , y co-ordinate system and movement will therefore stop. The number of revolutions, which means the number of oscillations, will depend on the amount of damping that in its turn is determined by h, which is the coefficient of friction. The oscillations produced by such systems are clearly not periodic in the sense of eq. 1-1and the time interval T = 2n/w is sometimes called the conditional period. If one takes successive maxima of x , one finds the ratio between them to be ehTor e2hn'oo and the quantity hT is called the logarithmic decrement. In the case of the aperiodic system (eq. 2-8), no oscillations are generated and the phase path approaches the origin directly.

NON-LINEAR SYSTEMS

If one proceeds from simple models to more realistic situations, one automatically enters the field of non-linear systems. In the example of the harmonic oscillator (eq. 2-1) we assumed an elastic force which is linearly dependent on displacement, as Hook's law tells us. It is well known that this is only valid within very small limits.

OSCILLATING SYSTEMS

15

Once the spring which acts as an elastic force is overstretched, it will behave quite differently. Similarly, it was assumed in eq. 2-5 that the friction of the oscillator is exactly proportional to the speed and this again is only true as an approximation. The equations are most likely to be non-linear in real systems, whereby it is particularly common that certain thresholds must be reached before some reaction occurs. It is far beyond the scope of this book to treat non-linear problems in detail but since a number of “cycle models” have been specifically based on non-linear systems, some basic concepts will be outlined. We introduced the phase portrait earlier when considering simple systems like the mechanical system: mx = f ( x )

(2-9)

In this case, the phase portrait is given by plotting speed against displacement. If we assume unit mass, the phase path can be written in parametric form as: dx dY dY = f(x)y and therefore: dt = y; dt = f ( x ) dx The last equation is easily integrated and one obtains:

(2-10)

Y2 + V ( x ) = h (2-11) 2 In this equation h is a constant. The quantity y2/2 = mx2/2 is in fact the kinetic energy of the system and V ( x ) = f ( x ) dx is the work done on the system. We will assume that V ( x ) = z for a certain oscillating system, is known and the phase path can be drawn in the plotted in the z , x plane. Since y = y , x plane (see Fig. 2-4). Now h is the maximum available energy and if z becomes

Jm

b

A‘

X

Fig. 2-4. Energy levels and phase portrait of a non-linear system. The z axis gives the energy, x and y gives the position and speed. The maximum energy of the system is given by h .

16

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

greater than h there is no more energy available and therefore no movement can take place. Therefore the work function together with the constant energy determine the phase path and several possibilities can arise. Closed paths clearly represent periodic movements. If V ( x ) is totally above h but touches it at a point, this corresponds to a minimum of potential energy and therefore represents a stable equilibrium. On the other hand, if the tangential h represents an energy maximum, the equilibrium is unstable. If, as shown in Fig. 2-4, the phase path consists of two or more closed curves which are just touching, one speaks of a separatrix. It is easily seen that a slight decrease in the available energy would transform the unstable system into a stable one consisting of a number of periodic movements. The equilibrium positions in conservative systems depend on there being constant available energy. In more general non-linear systems they can be controlled by some critical parameter. If one plots the position of equilibria against a controlling parameter, one obtains the so called bifurcation diagram. Bifurcations are points at which branching occurs and this implies that two stable equilibria coexist side by side or in other words, that the problem has multiple solutions. A very important concept in the study of non-linear oscillations is the attractor. This can either be a singular point in the phase diagram or it may be a closed path which is called the limit cycle and which is shown in Fig. 2-5. An example for the point attractor is the origin of the co-ordinates in the phase portrait of the damped harmonic oscillator (Fig. 2-3). Attractors of this type lead to dissipative systems. In contrast, limit cycle attractors lead to asymptotically stable periodic movement. A system which contains such a limit cycle attractor is called a self-oscillating system. The oscillations in such systems are not dependent on the initial conditions and they are entirely determined by the properties of the system itself. Irrespective of whether an oscillating particle is outside or inside the limit cycle, it has the tendency to approach it and in this way, ensures the stability of the oscillation. Such

Fig. 2-5. Phase diagram of a self-oscillating system with limit cycle.

17

OSCILLATING SYSTEMS

systems therefore can produce periodic movement from a non-periodic source of energy. The two-dimensional attractor can be generalised to attractors with higher dimensions. If the phase space in such systems is sectioned in different directions, one finds different periodicities. This means that such attractors can be used to model the quasi-periodic behaviour which results from combining apparently unrelated periodic processes. Finally, attractors may have fractal dimensions and such attracting objects will lead to a chaotic behaviour of the system.

STOCHASTICALLYDRIVEN O S C I L L ~ Q R S

With the exception of idealised conservative systems, any oscillator needs a source of energy to drive it. If the driving force is a random variable, then the system takes on the behaviour of a random or stochastic process. For example, eq. 2-5 describing a harmonic oscillator, could be written as:

+

. iA

i i

+ A ~ =x ~ ( t )

(2-12)

where ~ ( t is) a random number which changes with every time increment. The equation describes the motion of a randomly disturbed pendulum. Yule (1927) visualised this pendulum as being kept in a room with many little boys armed with peashooters shooting at the pendulum. Stochastic processes are of fundamental importance in time series analysis, where one is interested in variables which change with time and which are usually given in the form of discrete values: x(t);

t = 0, 1 , 2 , . . . n

(2-13)

One can therefore write eq. 2-13 as a difference equation and assuming that the time intervals are evenly spaced one finds: Xt

= -a1

XI-1 - a 2 Xt-2

+

&I;

a1 < a 2 < a1

(2-14)

This is the equation of a second order autoregressive process. It is autoregressive because the state of variable x at time t , can be obtained by regression from the previous states. However, there is also a random term which affects the outcome and this can only be treated statistically. Autoregressive processes can have any order and the process of order k is usually abbreviated as AR(k). A process which can also provide oscillating models is the moving average or MA process. In this process, the value x ( t ) depends on the weighted sum of random variables in the past; the process can be written as: (2-15)

18

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

By mixing AR and MA processes, one arrives at the ARMA process which can be used to construct a very wide range of models that can either oscillate by themselves or they may be driven by some oscillating process. The main statistics which are used in time series analysis are: the mean

l R

a2 = var ( x > = -

E(x

the autocorrelation

-/A)~ n pr = covar ( x , x,-,)/var ( x )

the power spectrum

fo =

the variance

/ 2n 1

(2-16)

Po0

-m

eiorprd t

Mean and variance are familiar from elementary statistics. The autocorrelation function and its Fourier transform the power spectrum, are specific for time series analysis. The autocorrelation is particularly useful for describing the relationship between values which are t measuring points apart. It can be shown (cf. Jenkins and Watt, 1968) that the autocorrelation function for the second order process is: (2-17)

where: tan+ = -tanfl, 1 - a2

cost9 = --

2fi

andaz > a: 4

The correlation function of this process is therefore very similar to the solution of eq. 2-5. It is again a damped harmonic with a conditional period which is determined by the regression coefficients and these also determine the exponential decay. The resulting time series of such a system, however, is not only unpredictable but also unrepeatable. The latter means that if the process were to be started for a second time with exactly the same initial values, the outcome would be different. Stochastic processes in this respect differ from chaotic oscillations which may result from a fractal attractor. Chaotic sequences are unpredictable but repeatable if the initial conditions are precisely duplicated. Autocorrelations of second and higher order are very common in stratigraphical time series and they cannot be dismissed simply as random variations. They represent true oscillating systems which suffer, however, from a continuous change in amplitude and phase and therefore produce quite irregular results. One must keep in mind, however, that such systems could be driven by periodic processes, in which case, resonance cycles could result.

OSCILLATING SYSTEMS

19

EXAMPLES OF OSCILLATING SYSTEMS

The simplest dynamic systems are based on the first order differential equation of the type:

dx (2-18) dt = f(x) If f ( x ) is a monotonic function, then the phase space of the system becomes one-dimensional and reduces to a “phase line” and periodic movement, which relies on a closed path, is therefore impossible. Periodic motion in this case can only be generated if f ( x ) is multivalued. A good example for this is the behaviour of an oven which is thermostatically controlled (Andranov et al., p. 235). The example can be adopted for a large number of geological problems in which certain thresholds have to be reached. In the case of the oven, we have the heat balance given by: (2-19) In this, 6 is the temperature of the oven, C is its thermal capacity and W the power supplied to the oven by a heater, k is the thermal loss by radiation. The oven has a temperature sensitive regulator of the on off type. Switching on occurs at temperature 61( W = WO)and switching off occurs at temperature ( W = 0) whereby 61 < 6 < 62.The last condition is a characteristic feature of the switch mechanism and it makes W = W ( 6 ) a two valued function. The power temperature relation and the corresponding phase line are illustrated in Fig. 2-6. The latter consists of two half lines, one (I) with 6 > 61 corresponding to the switching on and the second one (11) with 6 < 4 corresponding to the switching off. The passage from the first half line to the second half line occurs at point c for 6 = 61 and the reverse at point a, for 6 = 62.The phase path a b c d is closed and periodic switching should take place. The solution of eq. 2-19 for the half line I is: B A e-klct and for half line 11: (2-20)

Fig. 2-6. Phase diagram of a thermostatically controlled oven. 61 and 62 are the critical switch on and switch off temperatures.

20

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

9 4

Fig. 2-7. The temperature 19 of a thermostatically controlled oven as function of time 1.

The system has a single equilibrium I9 = Wo/kwhich is the temperature at which power and heat loss compensate each other. If Wo/k> 01 the oven will heat to the first switch off time and from then on it will operate in a self-oscillating mode. The time temperature record will be periodic and will consist of alternating exponential curves which represent the heating and the cooling of the oven (Fig. 2-7). A further analysis of the problem shows that the oscillating period of the process is: (2-21)

where a = Wo/k. For a second slightly more complicated example (Andranov et al., 1966, p. 158), we consider a damped oscillator (eq. 2-5) which has the following property: i+ 2 h i +wox =

0

forx < 0

wi

forx > o

(2-22)

The model corresponds to an oscillator in which power is only applied in one direction. Such a situation can be realised for example, in an electrical circuit where an electronic device becomes active only if a certain positive (or negative) potential is reached. It will be assumed that x and i are continuous, particularly when f passes through zero. The phase plan will consist of half turns (Fig. 2-8). The lower half, y < 0 will be centred around the equilibrium at the origin of the co-ordinates (0, 0). The upper half is centred around an equilibrium with the co-ordinates 0, 1. Like in the linear case there is an exponential change in the sha e of the ellipses which is determined by the logarithmic decrement d = 2nh - h2 and one can therefore determine the so-called sequence function which gives the consecutive points where a phase path intersects the x axis. One finds:

J

21

OSCILLATING SYSTEMS

Fig. 2-8. Phase diagram of the non-linear oscillator described in eqs. 2-21 and 2-22.

x = 1 + e-d/2 = x1 e-'

(2-23)

If x1 < X then the distance XI x2 decreases and x = XI = x 2 , consecutive points fall on the same position and this of course signifies a closed path. This path is the limit cycle and it indicates a periodic oscillation which is asymptotically stable. The stability is assured by the fact that the limit cycle can be reached also from a point which starts outside the circle.

CLIMATIC OSCILLATORS: THE GHIL OSCILLATOR

The climate is the result of many interacting processes which are driven essentially by solar energy. A number of mathematical models have been constructed to investigate and at least partly explain the behaviour of this complex system. A particular problem which has occupied many climatologists is to explain the recent ice ages and this is a problem which of course has also been the main focus of the Milankovitch theory. As will be discussed in the next chapter, such theoretical climate models can be regulated by varying the solar radiation. However, some are capable of producing oscillations on their own and they are therefore genuine self-oscillatingsystems. The first model to be discussed is a deterministic model which investigates the coupling between land ice masses and the average planet surface temperature (Ghil and Taventzis, 1982). The model consists of two differential equations for temperature T and the latitudinal extent of the continental ice cover L: dT

CT = = Q{l- [aland(L)]l dt

(1 - Y ) %,an

dL C L = - = ( L ~ , , / L ) % { [ l + E(T)]LT - L ) dt

(TI

- k(T - TO)

(2-24)

22

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

The temperature variation is proportional to the heat capacity of the atmospherehydrosphere complex and it is determined by the insolation Q. Strong nonlinearities are introduced through the albedo (II which for the landmasses is largely a function of the ice cover ( L ) and for the sea, a function of ( T ) . The last term in the first equation represents the emitted long wave radiation which is determined by two constants k and To. The second equation determines the change in ice cover which is proportional to C L .This coefficient is equivalent to the temperature coefficient CT and determines the dynamic relaxation time of continental ice sheets. L* is a measure of ice coverage which takes the geometry and dynamic behaviour of ice sheets into account. LkaX refers to a continent when it is completely covered by ice. The coefficient E is the ratio between the accumulation and the ablation and it is represented in the model by a piece wise linear function of temperature. LT is the latitude of the snow line arising from the assumed geometry of the ice sheets. The second equation is also highly non-linear. By making reasonable assumptions about the parameters of the model and by setting the right hand side of the two equations to zero, one finds the isoclines of the two vector fields which when combined, contain three critical points of interest. The phase portrait for this situation can be constructed by numerical integration. It appears that two of the critical points are saddles of a separatrix and the central critical point is a centre leading to a limit cycle. The ratio p = C T / C Lcan be used as a bifurcation parameter and Fig. 2-9 shows the phase portrait for three increasing values of p. In the first example there is a single point attractor at U and this develops into a limit cycle which is indicated by a dashed circle. The limit cycle attractor indicates that periodic oscillations will develop. With the parameters chosen by Ghil, the period of the cycles turned out to be of the order of 10 ka.

THE SALTZMAN OSCILLATOR

Saltzman investigated a number of mathematical climate models (Saltzman, 1985), of which the simplest system that can generate oscillations is a two variable model. This model connects the ice mass with the ocean surface mean temperature 19. Denoting the deviations from the norm with the symbols i j 8, one can write:

(2-25)

The q’s and the 9 ’ s are positive parameters. The oscillations in this model arise from interplay of temperature and sea-ice cover which determines the heat loss and the exchange of oceanic and atmospheric COz. The latter in turn has a destabilising

OSCILLATING SYSTEMS

23

Fig. 2-9. Examples of the Ghil oscillator with an increasing bifurcation parameter (after Ghil and Tavantzis, 1983).

effect on the mean temperature of the ocean. The period of this feedback system is determined by a phase lag between the mean temperature and the extent of ice cover. Saltzman showed that with an appropriate choice of parameters, the system passes into a self-oscillating limit cycle with time periods of around 1 ka. It is of particlllar interest to investigate what happens if such a system is stochastically disturbed. A system which passes from a stable steady state into a self-oscillating limit cycle can be transformed to the normal form: dz = (B dt

+ i O 0 ) z - CZI ZI 2

(2-26)

The procedure is outlined by Nicolis (1984) in her analysis of the problem. In this normal form, z is a combination of real and imaginary contributions from the original variables, q ;/?gives the distance from the bifurcation point and C = p + iv is a combination of all the remaining variables. The new variable z can be written in polar co-ordinates z = eip and this can be separated into an amplitude and phase part: (2-27)

24

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

Fig. 2-10. Self oscillating process which has been disturbed at point A.

The analysis shows a very interesting difference in behaviour when the radial and the phase components of this oscillation are disturbed. If the phase path is displaced for example at point A (Fig. 2-10) then (because of its stability) the oscillation will re-establish itself and the amplitude will return to the full radius rs, for example at point Ao. The phase, however, will have changed in the time taken from the disturbance to the return and it is extremely unlikely that the return to r will happen exactly 237 times after the disturbance. The effect of this is an oscillating record which may contain possibly quite unpredictable jumps in phase. It will be seen later that this is indeed a very characteristic feature of stratigraphic records containing hiatuses or jumps in sedimentation rates.

GEOLOGICAL OSCILLATORS

Climatic fluctuations are important factors in the formation of sedimentary cycles but sedimentation involves many more aspects and processes which combine to determine a sedimentary sequence. To name but a few important variables which interact, there is the accommodation space and the sediment supply, accumulation rates and conditions of preservation, water-chemistry and bioproduction. All of these are interrelated and can form feedback loops and self-oscillating systems. Although a large number of variables exists which can lead to the development of oscillating systems, the geological literature contains very few, if any, examples which have been properly or which have been convincingly established. The only exceptions are high frequency processes such as ripple formation and similar hydraulics related sedimentation processes. For example, in a review paper on cyclic sedimentation (Wells, 1960) only one of the proposed theories: the Robertson Plant control theory (see later) represents a proper oscillating system in which the variables interact in such a way that oscillations should result. On the other

OSCILLATING SYSTEMS

25

hand, Beerbower (1964) who is responsible for the definition of autocycles in geology, lists a number of so-called autocyclic mechanisms all of which would, using modern nomenclature, give rise to event stratification. For example, sedimentation processes connected with meandering rivers cannot be regarded as part of an oscillating process in the sense that each stage in the development leads by necessity to a repetition of the process. The various channel shifts and overbank flows on fluviatile flood planes are largely governed by random events and they are episodic. Algeo and Wilkinson (1988) who believe that many sedimentary cycles are “autocyclic in nature” quote examples which have nothing to do with oscillating systems. One autocyclic process which is given as an example of a mechanism which could provide oscillations in the Milankovitch frequency range, is the stick slip fault theory (Cisne, 1986). This unlikely theory has never been well established. Genuine autocyclic systems can be found in some of the schemes which have been proposed to explain the sedimentation in the coal measure cycles (cf. Duff et al., 1967) which are common in the Middle and Upper Carboniferous. A tectonic theory which was developed by Bott (1964), is a good example of an oscillating system. The basic principle of this model is the interplay between the uplift of a source area and the sinking of a basin due to sedimentary loading. The rise of a denuded source area is believed to produce a mantle flow which helps the depression of the basin. An important feature of the model is that the faults which surround the basin restrict the subsidence to a well-defined area, which must be larger than the area of uplift. There must be a time lag between subsidence and uplift for the model to oscillate and this can be provided by the slow movement of the mantle material. The model represents a closed feedback system which at least in theory, is capable of oscillating. Without going into a discussion of the geological feasibility of this model, two important features can be seen immediately. The system is clearly strongly damped, not only because of “frictional” forces but also because the different specific gravities of the transported and the displaced material must cause large energy losses. The system is therefore not capable of generating sustained oscillations. The second consideration concerns the scale of the system. If it is assumed that the time-lag which determines the frequency of the oscillations is caused by the relatively low speed of the mantle flow, one can make rough estimates of the size or domain of the system. For instance if one assumes a flow rate of 1 cm per year, which is slow compared with known rates of ocean floor spreading, then frequencies of 1/100 ka would be generated by basins which are only a kilometre wide. This means that the mechanism could only account for cyclicities on a very small scale. It is possible to increase the length of the cycles by modifying the theory by introducing some critical thresholds. For example, one could assume that sediment loading has to reach a certain limit before mantle flow takes place. Any such assumption, like the switching mechanism in the thermostat example, would make the process highly

26

CYCLOSTRATIGRAPHYAND THE MILANKOVITCH THEORY

non-linear but without any data on such a process, the model cannot be further developed. Another self-regulating process which has been proposed to explain coal measure type cycles has been put forward by Robertson (1952). Robertson envisaged that plant growth in the coal swamps had an important effect on sedimentation. Plants slow down currents and therefore enable sediment to be deposited and so the swamp spreads. Eventually the land is raised above the water level and no more sediment can be deposited. The system has not been evaluated in quantitative terms but the model could work because there is a link which acts in both directions between the sediment supply and subsidence. If such a mutual link between the variables is not present, the system cannot oscillate. A good example for an incomplete system which cannot operate is the Ginsburg (1971) model which has often been quoted as an “explanation” of cyclic deposits on carbonate platforms. The model assumes the constant subsidence of a wide shelf area of abundant carbonate production. Tidal currents shift such sediments towards the shore where they build up into a prograding bank. This tidal sediment will gradually cover the shelf and so reduce the carbonate production, which is supposed to come to a stop. The system then rests until it has again sunk deep enough to produce more sediments. The fallacy of this argument lies in the assumption that the carbonate production actually stops. Although there is a clear connection between advancing sediments and diminishing carbonate production, there is no connection between sediment and subsidence. It is only subsidence which could renew the carbonate production and under these circumstances, the system comes to an equilibrium between advancing tidal sediment and carbonate production. The Ginsburg model can be made to oscillate if a maximum and minimum water depth is introduced, which determines a critical range in which sedimentation can take place. By introducing such thresholds, the model becomes very similar to the discussed model of a thermostatically controlled oven which also needs two critical values at which switching occurs. Unfortunately, in this case the model no longer explains the process and simply consists of a rule which states that sedimentation should be turned on and off at regular intervals. If such a rule is incorporated into a computer program, not surprisingly, the computer will draw regular cycles. In palaeoecological and related sediment production problems, the “predator prey” relationship may become an important factor in generating sedimentary cycles. If two species live together in the same area whereby one of them provides the food for the second, oscillations can develop. As the food supply for the preying species diminishes, it itself will decline in number and this gives the prey time to recover. The classical example for this relationship is the Canadian lynx which has been extensively trapped by man and the production of lynx pelts in the last fifty years shows a roughly seven year fluctuation. The lynx time series is also a good example of an oscillating system which is best treated as a stochastic process and

OSCILLATING SYSTEMS

27

the data can be very reasonably modelled by a second order auto regressive process. The need for a stochastic driving force indicates that it is not stable enough to generate sustained oscillations and that the development of such a system is only possible in a statistical sense.

CONCLUDING REMARKS

If one applies the theory of oscillations to geological problems, one finds that there are two types of oscillating systems which differ considerably in their behaviour. The first is the planet system which is controlled by well known physical laws and which will be discussed in the next chapter. Its outstanding characteristic is its stability which is greater than in any other system that could have geological effects. It is also, thanks to its size, capable of generating cycles with periods which are very much longer than any cycles we know of in the terrestrial environment. The stability which of course is not absolute, permits us to treat the astronomical complex as a conservative system for considerable time spans. The second type of system which could have generated sedimentary cycles are self-oscillating systems. An attractive feature of these systems is their ability to return to a periodic movement, even after they have been disturbed. This means in practice that similar-looking cycles would be generated, as long as the system and its parameters have not changed. The only models based on self-oscillating systems which have been analysed to some extent, are climatic models. These are global models with a world wide domain and this restricts the length of the cycles which they can generate. If one considers models which involve the atmosphere and the oceans, an upper limit for possible time delays is given by the mixing time of the oceans. This is estimated to be between 1 and 2 ka. After this time, any disturbance in the ocean temperature or composition will have evened out. The atmospheric mixing time is of course much shorter. To obtain longer periods, climatologists have had to use additional variables and in both of the discussed models these are the inertia of the climatic system introduced through snow and ice. In other climatic models, use has been made of the slowness of the isostatic adjustments which determines the geometry and distribution of large glaciated areas. Climatologists claim that with the involvement of ice it is possible to construct models which generate cycles with periods of 100 ka, but such models involve major glaciations and they apply only to the Pleistocene or times in which similar extensive glaciations occurred. Tectonic models are probably capable of generating much longer cycles but because of the strong damping, one would not expect widely distributed and sustained cycles to be caused by any known tectonic process. On the other hand, the complexity of sedimentary processes suggests that many relatively short self-

28

CYCLOSTRATIGRAPHY A N D THE MILANKOVITCH THEORY

oscillating cycles should be expected. Such cycles would show abrupt changes in phase after they have been disturbed and they may often degenerate into stochastically driven cycles of such complexity that they become chaotic.

29 Chapter 3

THE MILANKOVITCH THEORY

We are very familiar with the fact that our climate undergoes strong seasonal variations. The reason for this is the well known movement of our planet around the sun in an orbit called the ecliptic. Since the earth’s axis of rotation is inclined at an angle of approximately 23 degrees, the northern hemisphere receives more sun during the northern hemisphere summer and the southern hemisphere has more sun during the northern hemisphere winter. The two days in the sidereal year during which both hemispheres receive equal amounts of sunlight are called the equinoxes and the position of the spring or vernal equinox is shown in Fig. 3-1 by the symbol y . The positions of the summer and winter solstices,which are the longest and shortest days in the year, are indicated by S S and WS. Since the sun is in a focal point of an elliptical orbit, there is a position where the earth is closest to the sun, which is called the perihelion, P and a point of greatest distance from the sun, called the aphelion A. At the present time, the perihelion is close to the winter solstice. If all the quantities which determine the seasons were to remain fixed, the climate averaged over a whole year would remain constant for all time. In fact, the orbits of all the planets including the earth, undergo variations. The elliptical shape of the orbit itself changes and the inclination of the earth’s axis carries out a complex movement along the cone shaped surface which is shown diagrammatically in

W

P

A

S

Fig. 3-1. The earth sun system. A and P are the aphelion and perihelion. WS and SS are the winter and summer solistices. y is the position of the vernal equinox. The longitude (j, of the perihelion is measured from the vernal equinox and the obliquity E is the angle between the terrestrial north pole N and the celestial pole.

30

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

Fig. 3-1. This precessional movement produces a migration of the vernal equinoxes which is measured by its longitude w. In addition to this, there is a continuous change of the angle E . All of these changes cause a slow change in the climate. The theory which explains such long range climatic changes has become known as the Milankovitch theory. The basic idea of this theory is however much older and it has also been considerably developed since the work of Milankovitch. Nevertheless it is fair to associate his name with this theory as he was the first to study in detail the three basic aspects of the theory of astronomical climate control. The three sections which Milankovitch investigated were: the problem of celestial mechanics (which is concerned with the calculation of the orbital elements of our planet), the question of how much solar energy is received by the surface of the earth and finally, what the climatic consequences of this energy supply are.

THE PLANETARY SYSTEM

Although the Greeks already realised that the earth rotates around the sun, it was Copernicus who first developed the idea of the planetary system. The theory of planetary movement goes back to the work of Johannes Kepler (1571-1630) who discovered his famous three laws of planetary motion which were derived largely from observations but also from geometrical arguments. Kepler’s first law states that all planets circle the sun in elliptical orbits, whereby the sun is situated in one of the focal points (Fig. 3-2). The ellipses are defined by the half axes a, b or alternatively, by the major half axis, a and the eccentricity, e. The eccentricity is defined as: e=

d n a

or:

Fig. 3-2. The quantities used in describing the orbit of a planet.

(3-1)

THE MILANKOVITCH THEORY

31

e = J m where p is called the parameter of the ellipse. The parameter is useful for presenting the orbit of a planet in heliocentric co-ordinates, which in this form is written as:

r =p/(l+ecosq)

(3-2)

In this equation, r is the distance of the sun from the planet and q is the angle of the ray connecting the sun and the planets with some reference direction. Kepler’s second law refers to the speed with which the planet moves around the sun. Kepler formulated this law by stating that the area swept by the line connecting the sun to the planet in equal time units, is constant. This is stated in heliocentric co-ordinates:

r 2 @ = constant

(3-3)

Kepler’s third law relates the time a planet takes to complete its orbit (the so-called orbital period T ) , to the major half axis a, of the ellipse. He found:

a 3 /T~ = constant

(3-4)

Kepler’s laws are fully explained by Newton’s theory of gravitational attraction. The sun generates a gravity field which will interact with any planet moving in this field. The movement of such a planet in a heliocentric co-ordinate system is given by the three differential equations: x

+ p y / r 3 = 0; I + p z / r 3 = o (3-5) In these equations, P = (1 + m)k2 where k2 is the Gaussian constant of gravity

+ p / r 3 = 0;

y

and m is the mass’of the planet which is expressed as a fraction of the sun’s mass, that is itself assumed to be unity. If we use polar co-ordinates in the orbital plane, we can write eq. 3-5 as: i . - i b = ,-P r and

r219=h;

h=

(3-6) 2na2(1 - e2) T

(3-7)

In fact the last equation (eq. 3-7) represents Kepler’s second law, with T representing the orbital period as before. Integration of eq. 3-7 gives r=

k2 p[1+ e cos(I9 - w ) ]

(3-8)

32

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

/ 0

Fig. 3-3. The planetary orbit QNP in relation to a reference system, X,Y,Z . P is the perihelion, N is the node and i is the inclination of the orbit. The longitude of the node is measured from the reference axis x and the longitude of the perihelion is measured from the node in the plane of the orbit.

e and w are constants of integration. With 0 c e c 1 this is the equation of an ellipse which proves Kepler's first law. Finally it can be shown that:

Equation 3-9 is Kepler's third law with a slight correction. Newton showed that the ratio of the half axis over the orbital period is not constant for all the planets, as it contains both the mass of the sun and the mass of the planet. Since the latter is very small compared with the mass of the sun, the discrepancy is quite small. Before considering the behaviour of two or more planets, a new co-ordinate system for comparing different orbits will be useful (Fig. 3-3). We take the orbital plane of the earth and the ecliptic of the year 1950.0 as a reference. The x y plane is taken in the orbital plane, with the x axis pointing towards the vernal equinox of the reference date. The z axis is at right angles to the ecliptic. Any orbit will intersect the reference plane at two points. The ascending node N ,is fixed by its longitude Q, which is measured from the x axis and the inclination i, of the orbit is measured at this intersection. A total of six orbital parameters are necessary to describe the orbit completely. These are: the major half axis a, the eccentricity e, the inclination i, the longitude of the ascending node 52, the longitude of the perihelion (measured from the ascending node) w and E which is the mean longitude at the epoch. The epoch is the time when measurements began. The consideration of additional planets very much complicates the problem. In fact, only the case of a single planet orbiting the sun, permits a rigorous solution to be made. When three or more planets are considered, the orbits interfere with each other and eq. 3-6 becomes: 3 aR

x + p x / r -;

ax

aR y + p y / r 3 = -; aY

aR z+pz/r3 = -

az

(3-10)

33

THE MILANKOVITCH THEORY

In these equations, R is called the disturbing function and it is made up from the masses of the remaining planets and the distances between them. Solutions can only be found by stepwise approximations. Instead of calculating the position of the planets directly, it is more convenient to calculate the orbital elements and their changes in time. This is the method originally adopted by Lagrange (1871) who gave six differential equations, each containing a disturbing function for the six orbital elements. As a first step, orbits are calculated for each planet, treating them as a two body problem. The interference represented by the perturbation can then be considered, using the positions of the planets, which have been individually calculated. The approximate integration is achieved by developing the Lagrange equations into trigonometric expansions. A considerable simplification is achieved by considering only the long term changes of the orbital elements, which means that the short periodic terms in the disturbing function can be ignored. The solutions of the long term variations can be obtained from the following equations: e sin x =

C Mj sin(gjt +

Bj>

(3-11) sini cos Q =

C

~i

cos(sit + ~

i >

This very compact way of obtaining the orbital parameters is due to Bretagnon (1974) who also carried out the considerable computational work needed to obtain the amplitudes M and N , the mean rates gj, si and the phase constants Pj and S i . A full tabulation of these constants is found in Berger (1978a, b). The values for the first five terms of the expansion are given in Table 3-1. TABLE 3-1 Values for the expansion of eq. 3-11 (after Berger, 1978) Amplitude

Second/year

Mj

Ej

0.0816 0.0162 0.0130 0.0098 -0.0034

4.207 7.346 17.587 17.221 16.847

Degree 28.620 193.788 308.307 320.199 275.371

Amplitude Ni

Second/year Si

Degree 8;

0.0276 0.0200 0.0120 0.0076 0.0051

0 -18.83 -5.61 -17.82 -6.77

106.15 248.51 11.98 277.44 305.03

34

b

CYCLOSTRATICRAPHY AND THE MILANKOVITCH THEORY

Oll

012

0.4 015 0.6 0.7 Million years before present

0.3

0.8

0.Q

1.0

Fig. 3-4. Eccentricity, obliquity and precessional indices for the last one million years.

Berger also calculated direct expansions of the eccentricity e, and the precessional index esinw (see later) which is of palaeoclimatic importance. A graph of these two quantities for the last 1 Ma is given in Fig. 3-4. The amplitudes and periods of such expansions can be used directly to plot a power spectrum. There is a strong maximum at 412.8 ka and other frequency maxima occur in a group of between 123 ka and 95 ka (see Fig. 3-5). There is a third group of maxima which is at approximately 2 Ma. Such a spectrum should also be found in any stratigraphic record where the cyclicity is caused exclusively by the eccentricity. Even if completely unbiased records of the eccentricity signal cannot be expected in a sedimentary record, a similar grouping of frequency maxima in sedimentary cycle spectra, is a strong indication that eccentricity may be responsible. Fischer et al. (1989) introduces a rather unfortunate confusion into the description of eccentricity cycles by proposing the indexed abbreviation El to Es for the periods of 100, 400, 1290,2030 and 3400 ka. The order of indices does not correspond to the amplitudes and since Fischer (1990) changed the system in which the strongest period at 413 ka is supposed to be called E3, the system of indexing is best avoided.

THE PRECESSION

The precession is the reaction of the earth’s rotational momentum to disturbances from the sun and the moon. The effect of the other planets is small and can be neglected. The precessional movement of the axis follows a roughly cone-shaped envelope (see Fig. 3-1) and it causes a slow backward movement of the vernal node. The perihelion moves in the opposite direction and both effects combine to increase the longitude of the perihelion. In palaeoclimatic reconstructions, one needs to know in particular, the longitude of the moving perihelion and the inclination of the

35

THE MILANKOVITCH THEORY 112.8 ka

0.010

94.9 ka 0.008

128.2 ka

99 a!

0.006

3 c .-

F

4

2035.4 ka

0.004

0.002

1 10

20

30

50

40

60

70

io

90

1

1

110

cycledl OMa

Fig. 3-5. Spectrum of the eccentricity.

earth's axis E , called the obliquity:

+ C Ai COS(f i t + esinw = C Pj sin(ait + E

= E*

Sj)

(3-12)

Sj)

The first five constants, according to Berger are given in Table 3-2. The general precession E* has the value of 23'320556. TABLE 3-2 Constants for eq. 3-12 (after Berger, 1978) Amplitude

Second/year

Ai

h

-2462.22 -857.32 -629.32 -414.28 -311.76

3 1.609 32.620 24.172 31.983 44.828

Degree

250.903 280.833 128.306 292.725 15.375

Amplitude

Second/year

Degree

pi

ai

ti

0.0186 0.0163 0.0130 0.0099 -0.0034

54.646 57.785 68.297 67.660 67.286

32.01 192.18 311.69 323.59 282.16

36

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY i 1 .O ka

2400

9.7 ka

2000

s

.-C

1600

. x e .-m 3

0

3.6 ka 1200

800

400

200

2

cyclesll0Ma

Fig. 3-6. Spectrum of the obliquity.

Berger (1978) has calculated the trigonometric expansions for both the obliquity and the precessional parameter esinw and the amplitude spectra of the two are given in Fig. 3-6 and Fig. 3-7. The frequencies which appear in the expression 3-11 and 3-12 can be traced back to seventeen fundamental periods (Berger, 1988) which are the sixteen values: e sin R and sin i sin SZ for the eight planets and the general precession. It is found that the frequencies of the eccentricity of the earth’s orbit are linear combinations of the e, R ,and R, i system.

THE FREQUENCY STABILITY OF THE ORBITAL ELEMENTS

It is of the utmost importance for geologists to obtain some idea about the stability and accuracy of the astronomical solutions or to put it more directly: how far back into the past can we use the astronomical parameters? The question really consists of three parts: how accurate are the astronomical solutions, are there any

THE MILANKOVITCH THEORY

0.01I

37

!3.7ka

0.01t

22.4 ka

0.014

18.9 ka 0.012 a,

2 0.010 -6.

E

a

0.008

0.006

0.004

0.002

L

0

500

cycles/l OMa

Fig. 3-7. Spectrum of the precession.

slow changes in the system which determines the earth’s movement and how stable is the planetary system as a whole? As has been shown, most of the functions which are used in astroclimatology are mixtures of sine or cosine waves, the frequencies of which have no rational relationship. The resulting functions therefore are quasi-periodic and do not satisfy eq. 1-1. This absence of pure periodic behaviour in a quasi-periodic history or time series does not mean that such a series is unpredictable or cannot be reconstructed. However, the predictability does depend on the accuracy of the initial data and the stability of the solutions used in the complex problem of orbit analysis. The stability of different solutions can be tested by actually generating time series which can then be examined and compared in the frequency and time domain (Berger, 1984). Comparing the best data available in 1989, Berger finds that after 3 Ma, the solutions for the eccentricity differ by one third of a cycle, the precession differs by a full cycle and the obliquity is out by one fifth of a cycle. In Berger’s analysis, the time interval of 3 Ma was divided into four equal parts. The resulting

38

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

subseries were compared with each other through their power spectra. It was found that the three investigated parameters, eccentricity, precession and obliquity showed a gradual development in the frequency content of their spectra. The eccentricity shows a change in power of the 400 ka and 100 ka periods. Sometimes the first is more important and sometimes the second is more important. The same phenomenon of alternating importance is found for the 23 ka and 19 ka period of the precession. In the obliquity record, it is always the 41 period which is strongest but its power also varies with time and at one stage it practically disappears. The accuracy of the astronomical calculations clearly depends on the initial conditions, which must have been derived from observations. Berger (1984) comes to the conclusion that a better knowledge of the masses of the planets will have no significant influence. The planetary eccentricities and inclinations have much less effect on the solutions than the masses and they are therefore adequately known. He concludes that for the last 3 to 4 Ma, the astronomical frequencies are sufficiently stable to provide not only a basis for palaeoclimatic studies but they can also be used for absolute dating, if the climatic data are known. The situation is different for the older pre-Pleistocene cyclicities,when significant changes in the system could have occurred over long time intervals. That is particularly the case for any cycle containing the general precession, which means the obliquity and the precessional index. The precession, as has been pointed out, depends on the rotational speed of the earth, the momentum of inertia around the polar axis and the principal inertia axis in the equatorial plane. The distance between the earth and the moon is very important also. Each of these variables is likely to have changed in geological time. The rotational speed must have decreased throughout geological time due to tidal friction. The lengths of the days therefore have increased and this is suggested by the growth rings of Devonian corals which indicate shorter days in the past. The distance between the earth and the moon is gradually increasing and this in turn will change the oblateness of the earth, changing the shape and momentum. Tectonic changes and the movement of plates will have an effect on the inertia axes. With a shorter earth-moon distance, the precession must have been larger and the precession and obliquity cycles must have been shorter. Berger et al. (1989) estimated and calculated the lengths of the precession related periods using a model which took account of decreasing day length, the continuously increasing earth moon distance and the changes in inertia over the last 400 Ma. The results of this are shown in Table 3-3 and Fig. 3-8. The theory of planetary orbits is based on Newtonian mechanics, which assumes absolute stability for the finally achieved orbits. The considerable changes in the precession related cycles raise the more general question of how stable the planetary system can be. The answer to this problem, which has occupied astronomers since Lagrange is not known. From the geological evidence we know, obviously, that the planetary system has existed for as long as we have had geological records. For the

THE MILANKOVITCH THEORY

39

TABLE 3-3 Estimated length of paleoclimatic periods (after Berger et al., 1989) ~~

Present Upper Cretaceous Lower Permian Upper Carboniferous Middle Devonian Lower Silurian

Ma

Precession (years)

Obliqity (years)

0 12 270 298 380 440

19000 18641 17545 17272 16562 16014

41600 39328 34227 32954 29649 27097

23000 22474 20868 20468 19428 18625

54000 51100 42250 40403 34309 29884

last 500 Ma, our records have been sufficiently detailed to make it very unlikely that major changes occurred in the form of some catastrophic event. The meteorite impacts which for example, are assumed for the Cretaceous-Tertiary boundary did not as far as we can tell, produce any evidence of sudden changes in the seasons or the length of day. We have no reason or evidence for assuming that, apart from minor modifications, the planetary system was different in Precambrian times

: : i l l : : Precession

l5

400

300

Ma

200 before present

100

Fig. 3-8. Change of obliquity and precession in the geological past (Berger et al., 1989).

0

15

40

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

from the present. This does not mean that changes have not occurred nor does it mean that such changes, if they took place, were predictable. Buys and Ghil(l982) point out that the solar system is of incredible complexity and certainly cannot be adequately described by linear systems. Many solutions as has been seen, are quasiperiodic but between these, there can be aperiodic solutions which may indicate chaotic or random behaviour. One can only hope that geologists can eventually contribute towards an understanding of such problems.

THE INSOLATION

According to the Milankovitch theory, the secular variation of the orbital elements will change the amount of solar energy which reaches the earth at various times and this will cause a change of climate. The amount of radiation which a planet receives from the sun is called the insolation and its intensity is inversely proportional to the square of the distance of the planet from the sun. The amount of energy which falls on to one square centimetre of the earth’s surface which is at right angles to the sun’s rays is 1.95 cal/cm2 minute, or 1353 W/m2 expressed in watts. This value is called the solar constant Jo and it is believed to have remained unchanged for a long time. The definition of the constant assumes a mean distance of the earth from the sun and a completely transparent atmosphere. We will use the following symbols:

w

= ~0 = a = r = p =

6 =

A = S = HO = JO =

energy (Watt/m2) geographical latitude the major half axis of the ecliptic the distance of the earth from the sun the distance of the earth from the sun measured in units of a: p = r / a = (1 - e2/[1 + e cos(w - A)] the longitude of the perihelion. Its numerical value is calculated from the annual general precession and the longitude of the perihelion plus 180 degrees. the true longitude of the earth counted anticlockwise from the vernal equinox. the declination of the sun the hour angle of the sun. the solar constant at a distance a.

Making use of the inverse square law, one can calculate the mean energy which each part of the earth receives over a full day:

W

Jo

= -p 7t

-2

(Hosinf?+cosrpcos6sinHo)

(3-13)

41

THE MILANKOVITCH THEORY

For the polar latitudes where there are no sunsets, the equation simplifies to: JO -2

W = --p

n

sinpsinS

(3-14)

Using eq. 3-13, one can calculate the seasonal variation of the energy and one finds for the equinoxes: Spring Autumn

equinox:

W

Jo n

= -(1

+ 2e cos 6cos p)

-

(3-15)

The energy flow during solstices on the other hand, is far more complicated. For latitudes with sunrises and sunsets one obtains: (3-16) For latitudes without sunsets:

W$;F = fJo( 1T e sin 6sin p sin E )

(3-17)

It is relatively easy to calculate the insolation for specific astronomical seasons. For example, the mid-month insolation can be calculated by increasing the longitude of the earth (A) for each month by 30 degrees. Starting from the 21st March = 0, April = 60, May = 90. . . one can obtain an insolation for any time and geographical location. Such distributions have been calculated by Berger (1978) and an example for the present is shown in Fig. 3-9a. The times given in such astronomical calculations are not true calendar dates. If similar calculations are to be carried out for the past, it must be taken into account that the climatic seasons have shifted in relation to the astronomical seasons. To avoid the difficulty of shifting seasons, Milankovitch (1941) introduced the so-called caloric half year. He divided the year into two seasons of equal length, one of which receives the maximum radiation and the other the minimum. This method simplifies procedures but hides some climatologically important features. Berger (1978) provided a method of calculating precise monthly or even daily radiation values, which give more information. This can be illustrated by comparing the power spectra of the caloric seasons with those of the mid month insolations for various geographical latitudes. Berger and Pestiaux (1984) calculated insolations for the last 800 ka and they also compiled the power spectra for these values. The calculations for the caloric seasons show that in the high latitudes, the obliquity variation with its cycle of 41 ka is more important. In the latitudes from 60 to 0 degrees, the precessional periods become more important. It can also be seen that the caloric half year approach has a smoothing effect on the seasonal variation and that the mid month approach gives considerably more detail. For example, the variation of the insolation which is based on the caloric half year and calculated over the last 1 Ma, is less than 3%, whereas Berger found

42

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

a 80 60 40

g 20 .-2 0

-9

-20 -40 -60

-80

.

,

t

""1

-60

0

-80 J ' F

M

A ' M ' J J Month

A

S

0

N

D

Fig. 3-9. (a) The present day insolation calculated from the astronomical parameters, (b) on top of the atmosphere, incident radiation on the earth's surface and (c) absorbed radiation. After Picot and Berger (1988).

maximal variations of 12% when considering mid-month insolation values over the same time interval. Berger (1979) also showed that following the annual variation of the insolation for a given latitude through time, can reveal important differences in the insolation history. He called the graphs which show such developments the "insolation signatures" and he could link them to known stratigraphical events.

THE MILANKOVITCH THEORY

43

MILANKOVITCH CYCLES AND CLIMATE

Solar radiation is the most important source of energy and it is practically the only source of energy for driving the climatic processes. The astronomical climate theory permits us to calculate the radiation, but this is not enough for the development of a theory which is sufficiently detailed to reconstruct the ancient climate accurately. The reason for this is the complexity of the climate system and our limited knowledge of it. The insolation values which were calculated by Milankovitch and later scientists (Vernekar, 1972; Berger, 1978) are ’’ top of the atmosphere ” values, meaning that they represent the intensity of the solar radiation before it passes through the atmosphere. When radiation penetrates the atmosphere, it is partly absorbed and partly reflected by the earth’s surface and by internal layers like clouds in the atmosphere, for example. The proportion of radiation which is lost by reflection back into space is called the albedo. The short wave radiation of sunlight is absorbed by the surface of the earth and this in turn emits longer wavelength radiation, which heats the atmosphere. Milankovitch estimated the temperature of the surface by treating it as a black body, following Kirchhof’s law which gives the relationship between absorption and emission as a function of temperature and wavelength. The temperature values were transformed into climatic values which were expressed in terms of a nominal geographical latitude. Milankovitch did not attempt to treat climate as a dynamic system. Modern calculations investigating the distribution of effective radiation on the earth’s surface are due to Tricot and Berger (1988) who used the delta-Eddington approximation of radiation transfer in a three layer model. The radiation reaching the earth’s surface was plotted as a function of months and geographical latitude. It has already been shown that according to the astronomical theory, the top of the atmosphere radiation maximum is at the north pole during the northern hemisphere Summer and at the south pole during the northern hemisphere Winter (Fig. 3-9a). The earth’s surface at the polar regions receives less radiation. This is because the attenuation caused by the atmosphere is stronger in the polar regions and also because the angle of incidence of the sunlight is lower and therefore the passage through the atmosphere is much longer. The maximum radiation on the surface therefore, is in the equatorial regions but there are submaxima in the polar areas (see Fig. 3-9b). Because the albedo due to cloud and ice distribution is different in different latitudes, the amount of the absorbed radiation is different from the incident radiation (see Fig. 3-9c). Using the same atmosphere model, one can calculate the absorption values for the past, from the insolation distribution before it penetrated the atmosphere. Since all climatic processes are of a dynamic nature, it is particularly interesting to investigate energy gradients. The differences in the absorbed radiation at 30 degrees and at 70 degrees latitude reveal an interesting effect (Tricot and Berger, 1988).

44

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

As one would expect, the long term variation of radiation in the polar latitudes of the northern hemisphere, has a strong obliquity component and this results in a gradient which has a strong 41 ka periodicity, when calculated for the top of the atmosphere. By the time the atmosphere is penetrated, the polar component will have been strongly attenuated and therefore the gradient on the surface will have a well-developed periodicity in the 23 ka wavelength range. As Tricot and Berger point out, the example clearly shows how careful one will have to be when comparing astronomical frequencies with frequency spectra from geological records.

THE PLEISTOCENE CLIMATE

According to Milankovitch, it is cool summers and relatively mild winters which lead to the development of glacial periods. Climatologistsat this time agreed that an essential factor in glacier formation is a summer temperature that is cool enough for the snow and ice which has accumulated during the winter, to persist (Koppen and Wegener, 1924). A relatively mild winter would have an increased circulation and therefore a higher precipitation, which would lead to accumulations of more snow and ice. Milankovitch, in collaboration with Koppen (Milankovitch, 1941), fitted his calculated radiation curve to the Pleistocene stratigraphy and he recognised the four major stages of glaciation which were known at the time. A comparison (see Fig. 3-10) of the younger part of the Milankovitch curve with modern isotope data, taken from the time adjusted stacked core of Martinson et al. (1987), shows that the agreement is reasonable but that there are also discrepancies. Without doubt, the major difficulty which Milankovitch had, was the incomplete knowledge of Pleistocene stratigraphy.

20

I

I

00

I

I

I

I

1

I

100 140 180 Thousand years before present

I

220

I

I

200

Fig. 3-10. The original Milankovitch curve (solid line) compared with modern isotope data (dotted line).

THE MILANKOVITCH THEORY

45

Our present detailed knowledge comes from the study of deep sea sediments. The most important development in studying these sediments was the use of isotope variations as a stratigraphic tool. Emiliani (1955) showed that the isotope record he had available agreed reasonably well with the Milankovitch radiation curve (Emiliani and Geiss, 1957). It was found that the oxygen and carbon isotopes which have been incorporated into the calcareous skeletons of foraminifera, can be used to reconstruct the isotope composition of the oceans in the past. Of particular importance is the ratio between "0 and l60as well as the ratio between the carbon isotopes 13C and "C. Such ratios are expressed in 6 l 8 0 and 613C values which measure the isotope concentrations in permil linked to a reference standard. The S l 8 0 values were believed originally to reflect largely water temperatures, but later work showed that the index is also a very good indicator of continental ice. The definition of the 6 1 8 0values is taken in such a way that during the cold periods, the values are positive and the values decrease in the warmer periods when they can become negative. A second important development was the recognition of magnetic reversal zones in cores which contained foraminifera. This enabled the relatively accurate timing of isotope events and the accumulating observations made it possible to develop a very accurate isotope stratigraphy (Shackelton and Opdyke, 1976). In this stratigraphy, warm or relatively ice free stages are given even numbers and cold stages are given uneven numbers. The actual length of such stages is quite variable. There are at least 63 well developed stages to the top of the Olduvai (Shackelton et al., 1990) which is just below the Pleistocene-Pliocene boundary (Harland et al., 1990). The reliability of isotope data as stratigraphic markers is possibly best judged by the fact that one can without much difficulty, correlate cores from the Atlantic with cores from the Pacific with an accuracy completely unknown in any other field of stratigraphy. Power spectral analysis of time series which were based on isotope variation, showed periodicities in the wavelengths of 100 ka, 40 ka and 21 ka, precisely as predicted by the Milankovitch theory( Hays et al., 1976). It was particularly pleasing to find that the isotope record showed good evidence of two frequency peaks corresponding to the 19 and 23 ka peaks which are part of the precessional index spectrum (see Fig. 3-7). The frequency analysis of the isotope data must be regarded as being a most convincing proof that orbital variations determined the Pleistocene climate and that this basic concept of the Milankovitch theory must be accepted. As a first attempt at relating isotope climatic indicators to the astronomical parameters, straight forward regression methods have been used. For example, a relatively simple empirical formula can be used to predict the climatic curves from known astronomical data. Such a palaeoclimatic indicator, called ACLINl is given by Kukla et al. (1981):

46 a!=

CYCLOSTRATIG RAPHY AND THE MI LANKOVITCH THEORY

(3-18)

The indicator a! is given for time t and w1 and el are the longitude of the perihelion and the eccentricity at that time. The value € 2 is the obliquity and w;! the-longitude of the perihelion at time t 2 . Time tl is separated from t 2 by the time difference of a 90 degree change in perihelion longitude corresponding to a time lag of between 5 and 6 ka. The time lag or phase shift between the isotope climate curves and the astronomical insolation is very important if attempts are to be made to tune the stratigraphical data to the astronomical insolation curve. Various authors have found the time lag to be between 5 ka to 10 ka, but there are indications that this lag can be variable and frequency dependent (Imbrie et al., 1984). Although the frequencies found in the deep sea records coincide reasonably well with the theoretical frequencies of the Milankovitch theory, the distribution of their power needs a more elaborate explanation. The spectra calculated by Hays et al. (1976) show that by far the highest power (which means the highest variance as will be explained later) is contributed by the low frequencies that correspond to the 100 ka eccentricity. The theory of insolation however, suggests that the variation caused by the eccentricity should be extremely small. The maximum effect of this variation occurs in the perihelion and aphelion positions, where it may be responsible for insolation changes of 14-17% during eccentricity maxima (e = 0.075) (Berger, 1989). Since the position in which the eccentricity has its maximum effectiveness migrates through the seasons, the climatic influence of the eccentricity should be much smaller than is suggested by the isotope record. A number of explanations have been suggested to explain this discrepancy. Hays et al. (1976) in their original paper, suggest that a strong non-linear response to the eccentricity signal could be responsible for the high power of the 100 ka period. Such a non-linearity could be introduced by the interaction of the relatively simple radiation cyclicity with the highly complex climatic system. The previous chapter discussed the possibility of climatic oscillators which could either be self-oscillating or driven by the quasi-periodic radiation input. The interference of the two systems can lead to beat frequencies which may fall into the 100 ka band. The climate of the Pleistocene was without doubt most strongly influenced by the development of large ice accumulations. Such glacier formation depends on the position of the so-called snow line which marks the boundary of an area in which snow does not melt during summer. The extent of ice fields generated by glaciers therefore, not only depends on insolation, but also on the distribution of land and sea and on the morphology of the ice fields. The latter is not only determined by topography but also by the rheological properties of ice and by the isostatic reaction of the crust which it is loading. In particular, the last two factors introduce long delay times and therefore may be very important in the explanation of low frequency

THE MILANKOVITCH THEORY

47

climatic variations. The relation between insolation and ice growth is certainly highly non-linear. It is estimated for example, that the melting of ice sheets is five to ten times quicker than their growth (Weertman, 1976). The deterioration of ice sheets is considerably accelerated when a continental glacier complex reaches sea level. Once the margin of the sea is reached, progress terminates and the formation of proglacial lakes and marine incursions speeds up the retreat (Pollard, 1984). Climatic models using the physical processes of ice sheet formation, frequently have built in feed back mechanisms which lead to autocyclicity. The two climatic models which have been discussed in the previous chapter turned out to be true self-oscillating systems. However, as it was seen from the analysis by Nicolis, the model generates oscillations which can be quite stable in amplitude but not in phase. Oscillations of this type soon lose their predictability and therefore could not explain the Pleistocene climatic cycles by themselves. Predictability can be restored by adding a small periodic force to the process and eq. 2-26 can be written as: (3-19)

In this, we is the external forcing frequency which is supposed to have a very small amplitude E . The complex valued coefficients describes the coupling of the forcing with the normal form variable. Nicolis (1985) finds that the system can become stable when it goes into resonance with the forcing function. This can only occur in the immediate vicinity of the bifurcation point i.e. when = 0 and when we = k o e , where k is an integer. This means that the self-oscillatingfrequency must be a harmonic of the forcing frequency. In the case of the Saltzmann oscillator, resonance could occur directly with the eccentricity component of the oscillation, in which case it would be strongly amplified. In the case of the Ghil oscillator, the self-oscillating frequency could be a harmonic of the 100 ka cycle. When the system is in resonance, the phase will lock on to the forcing cycle and the system will again become predictable. The results of this analysis are important for all complex systems which have a self-oscillating character and which therefore are liable to random disturbances. Such systems cannot produce phase consistent records unless they are forced by a periodic process. To explain the high power of the low frequency 100 ka cycle in terms of climatic oscillations, the formation of ice sheets seems to be necessary. This assertion seems to be supported by the disappearance of the large 100 ka cycles with an amplitude of 1.5%0 6 l 8 0 PDB prior to the onset of extensive glaciations in the middle of the Pliocene. But temperature variations with periods around 100 ka are still recorded by isotope data in the Middle Miocene (Shackelton, 1979) and the occurrence of repeated cooling events cannot be excluded for earlier periods. The relationship between eustatic sea level fluctuations and major glaciations is well known. A change of 1.5%0 in 6l80 which is about average in the Pleistocene,

48

CYCLOSTFUTIGRAPHY AND THE MILANKOVITCH THEORY

corresponds to a sea level change of 130 m if the present day configuration of sea and land is assumed. Shackelton (1986) gives the analytical accuracy of oxygen isotope determinations as O.l%o. This corresponds to a sea level change of approximately 10 m. A sea level fluctuation of this magnitude could be caused by the freezing or thawing of 3.6 x lo6 km3 of ice, which is approximately 15% of the present day Antarctic ice. Therefore it seems difficult to exclude the effect of glaciations on the climate, even for geological ages for which we have no records of major glaciations. It is not known for certain whether other processes than glaciation are involved in the formation of low frequency climatic cycles.

49

Chapter 4

METHODS OF ANALYSIS

This chapter will deal with some aspects of quantitative stratigraphical analysis, particularly with methods for the recognition of sedimentary cycles. The first requirement for any analysis, is data. Since we are attempting to make the analysis, quantitative, such data should be measurable quantities which are used to describe the stratigraphical sequences. Ideally, we would like to have some variable x , which can be determined for any part of the section and expressed as a function of the stratigraphical position. This position will be either a distance z above or below a certain datum, or a time t, if that is known. In any case, we will try to reduce the stratigraphical description to functions x(z), z = 0, 1,2, , . . or x ( t ) , t = 0, 1 , 2 . .. Continuous variables are particularly useful for this analysis. Good examples are: chemical or isotope compositions, counts of fossils, grain size or, other physical properties such as resistivity and so on. The choice depends not only on which parameter gives the most meaningful description of the sediment, but also on how well it describes the environment in which the sediment originated. In the study of Milankovitch type sedimentary cycles, one may also be interested either in the reconstruction of astronomical parameters, or in the reconstruction of climatic conditions. Both are not directly observable and they have to be deduced from the sedimentary parameters and these are therefore sometimes called proxy variables. If it is impossible to obtain continuous variables, one may have to use descriptions which classify only a limited number of states. For example, an alternating sequence of sandstone and shales present just two states. In extreme cases, the description is restricted to recording the recurrence of a single event. A good and very important example is the record of bedding planes in a stratified sequence. This type of sequence is called a series of events and special methods are available for analysing such data. Irrespective of whether true time or stratigraphical thickness is used as the stratigraphical variable x, the series xo, XI, . . .x,, is called a time series, which is a term that can be applied to any consecutively ordered sequence of measurements. For most analytical procedures, it is necessary to have the observations at equal intervals in time or space. If such data are not available, it is possible to create a new series with equal intervals by interpolation from the available data..In the analysis of single event series, one can either look at the intervals between successive events, or one can count the number of events occurring in a given interval.

50

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

a

0

'D

rq

0

1

4 5 6 7 8 metres below the surface

3

2

9

10

11

12

b 20

-

fn 3

2 0

s

U

0

b

i

2

4

5

4

5

6

8

9

10m

C

3

6

7

8

9

10m

Fig. 4-1. Three types of stratigraphic data. a. The isotope record of core 28-238. b. Abundance of planktonic foraminifera in metres -10 to -20 of the Piobbico (Italy) core (after Premoli-Silva et al., 1989). c. Bed thicknesses in Scaglia bianca in the Contessa highway section (near Gubbio, Italy).

Fig. 4-1 illustrates three typical stratigraphical time series. Fig. 4-la is the already familiar 6'*O plot for a Pleistocene unconsolidated deep sea sediment. The original measurements (Shackelton and Opdyke, 1973) have been adjusted to give measurements at 5 cm. intervals, when measured from the sediment surface downwards. Fig. 4-lb is a series of planktonic foraminifera counts from the Aptian Albian of Piobbico in Italy (Premoli-Silva et al., 1989). The counts have been made at 1 cm intervals. Fig. 4-lc represents bed thicknesses from the Scaglia bianca (Cenomanian) in the same area (Schwarzacher, 1993a). In the first two examples, the horizontal scale is the stratigraphical position, which could be translated approximately into time. In the last example of bed

51

METHODS OF ANALYSIS

20

10

30

Number of beds

Fig. 4-2. Cumulative thicknesses of 30 beds from the Cenomanian in the Contessa highway section.

thicknesses, the horizontal scale gives the successive bed numbers, which can be translated into time, only if the beds actually represent equal time intervals. It is possible to recover the stratigraphical positions of the individual beds, by plotting the bed thicknesses as a cumulative curve (Fig. 4-2). This graph plots the consecutive bed numbers along the x axis against the added thicknesses, which of course represent the stratigraphical positions.

STRATIGRAPHICAL SECTIONS AS STOCHASTIC PROCESSES

The cumulative plot of bed thicknesses (Fig. 4-2), is an approximately straight line, but there are deviations which are quite unpredictable. These indicate that either the beds represent random time intervals, or that the sedimentation rates varied at random, or that both have undergone random disturbances. Indeed, it seems almost inevitable that any sedimentation process will involve random variations and this suggests that stratigraphical time series are best represented by stochastic processes. The stochastic process is a mathematical model which can produce a time series. Since random variables are involved, each time series which the process generates will differ in detail. A single time series is called a realisation of the process and all possible time series are called the ensemble, which if it were known, would provide all the statistical properties of the series. It is conventional to write the stochastic process X ( t ) with a capital letter and the sequence which it generates, with lower case letters: X ( t ) = X O , XI,. . .x,,.

52

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

Stratigraphical data can be regarded as being the realisation of a stochastic process, but it is clearly impossible to repeat the procedure, once the sequence has been deposited. In other words, geological time series are part of an ensemble which must be for ever unknown. Nevertheless, it is possible to obtain statistical information from a single realisation, if the process has ergodic properties. The latter means that one can substitute successive observations in time, for observations which are made at the same time on different realisations of the ensemble. An ergodic process which has been observed long enough, therefore provides the same information as observing a shorter process, which has been repeated many times. Ergodicity of the data has to be assumed, but one of the essential conditions for it, which is the stationarity of data, can and must be tested. Stationarity implies that the statistical properties of a time series remain unchanged in time. In particular, the statistics of eq. 2-16 which define the meaqvariance, autocorrelation and the spectrum of a time series, must remain unchanged in time. Stationarity therefore, means that it does not matter which part of a section is used to calculate the statistics. If the mean for example, is calculated in a certain part of a section, it should be identical with the mean that is calculated anywhere else in the section. Tests for stationarity can therefore be obtained by comparing statistics like the mean, variance, or autocorrelation which have been calculated from different parts of the section. Stationarity is often missing in stratigraphical data because various trendlike developments are normal during the long range history of sedimentary basins (Schwarzacher, 1975) and such trends must be removed before a further analysis is attempted.

THE SPECTRAL ANALYSIS

The classical method of searching for cyclicity in time series uses spectral analysis. This method looks at time series in terms of their frequency composition. The method is not one which comes naturally to geological thinking, since stratigraphy describes a sequence as a series of strata which follow each other in time. A sedimentary cycle for example, is a group of rocks which is repeated after a certain thickness has been deposited. In other words, geologists are used to thinking in terms of the wavelength, rather than the frequency. The two are easily converted and wavelength is l/frequency. The standard unit of frequency in physics is the cycle/s or the Hertz. Also, it is often expressed as angular frequency, which is measured in radians per unit time. The length of a period is then 2x01. In stratigraphical spectra, one finds various units being used, such as cycledcm or if time estimates are available, cycles/ka or cycles/Ma. Sometimes, it is useful to give the frequencies in cycles/total length of section. This is particularly helpful when one needs to know directly, how many cycles of a certain frequency have been observed.

53

METHODS OF ANALYSIS

A number of text books are available as introductions to spectral analysis and for the geologist who would like to avoid too much theory, the paper by Sprenger and ten Kate (1993a) is very useful. The following gives a very short introduction to this important method and is taken partly from Priestley (1981), who has written a particularly useful text. We have seen in eq. 1-1, that the definition of a periodic function involves the precise repetition of each value, after the fixed time period, p . It has been shown by the French mathematician J.B.J. Fourier (1768-1830) that any periodic function of whatever shape, can be represented by a combination of sine and cosine waves. Such a representation is obtained by developing the time series x(t), which is assumed to be periodic and with period 2T, into a Fourier series of the form: x(t) =

7+

00

c ( u , cos2nnt

+ 6, sin2rrnt)

(4-1)

m-1

The coefficients a, and b, are called the Fourier coefficients and they are the amplitudes of the cosine and sine waves with frequencies n/2n; n = 0 , 1 , 2 , . . . The Fourier coefficient can be calculated from the original function by the Euler-Fourier formula: a, =

l T

n = 0,1, 2 . . .

- Cx,cosnnt/T; T l

(4-2)

According to Parseval's relation, one can write:

+ x ( u : + bi) and setting: CO=

2

and C , =

id-',

(4-3)

one can express the variance as:

00

Total variance of X ( t ) =

C:

(4-4)

fl=O

If one assumes that X ( t ) represents an electric current, then the sum of the squared amplitudes represents the total power, or the energy per time unit which the current develops and the value C2, is the power or variance contribution that is provided by the frequency n/2rr. A graph which shows the powers lC;1 for the frequencies 1/2n, 2/2rr, . . . n/2n, is called the variance or power spectrum. Since the power in the case of periodic functions, is calculated for discrete frequencies, the power spectrum itself is discrete and is often known as a line spectrum. The function which describes the line spectrum in terms of frequency o = 2nlr is called the periodogram I (w).

CYCLOSTRATIGRAPHYAND THE MILANKOVITCH THEORY

54

Non-periodic functions cannot be developed into Fourier series, as a rule. However, valid spectra can be obtained when the sequence is quasi-periodic and of finite length. One makes the assumption that the sequence is periodic, with a period equal to the finite length of the sequence. A quasi-periodic mixture of sine waves will have a discrete line spectrum. This discrete line spectrum only differs from the spectrum of a periodic series, in that the ratios between the frequencies can be irrational. As we have seen, the astronomical solutions can be developed into such quasi-periodic functions and the spectra which are given in Figs. 3-5 to 3-7, are typical line spectra. When the length of a sequence and its assumed period T , goes to infinity, the differences between the frequencies f - f = 1/2T will go towards zero and one will obtain an infinite number of points. In this case, the frequency becomes a continuous variable. Replacing the summation in eq. 4-3 by an integral and using complex notation we can now write for all t: W

[

x(t) =

J

p ( f ) e2xiffdf

(4-5)

-W

where:

[, W

p(f)=

x(t)

e-2niff dl

(4-6)

p ( f ) is called the Fourier transform of x ( t ) and eq. 4-5 is called the Fourier integral. We will assume that these integrals exist. Replacing 2 ~fr by angular frequencies w, we can also write: 1 x(t) = G ( w ) eio' dw

Srn Srn fi

6

and

-03

1

x ( t ) e-('"

G(w) = -

(4-7)

dt

-00

The total variance of the process is: W

[,IG2(w)Idw W

x ( t ) dt =

(4-8)

As before, the quantity IG(w)I measures the energy density which is contributed by

the frequency band w , w + dw and the total variance will be finite. The use of discrete Fourier transforms in the analysis of random processes, leads to considerable difficulties. Random processes are clearly not periodic and an essential condition for treating aperiodic processes with Fourier integrals is that the sequence x ( t ) , becomes zero when t goes to infinity. This is obviously a contradiction of the stationarity condition. Furthermore, we have seen that the spectral properties of a stochastic process must involve an ensemble and in order to average

METHODS OF ANALYSIS

55

spectra it is therefore necessary to obtain a true estimate of the spectrum. Such an estimate should be of the form: (4-9) E indicates the expectation (average operator).

THE ESTIMATION OF SPECTRA

In order to obtain an unbiased estimate of the spectrum, Blackman and Tukey (1959) calculated the spectral distribution from its autocorrelation function R ( t ) . This function is the convolution of the process with itself and therefore it is identical with the Fourier transform of the spectrum. To achieve the averaging effect, which is essential for examining a single realisation of a stochastic process, the estimates of the autocovariances were restricted to a number M . M is smaller than the length N , of the sequence. The Blackman-Tukey (B-T) estimate of the spectrum is then given by: .

M

(4-10)

This is called a truncated periodogram and M is called the truncation point. The same equation can be written in a more general form as:

cA,

. , M

1

i(w)=-

- -

2n - M

R, C O S W t

(4-1 1)

The special case of eq. 4-10 is given by: A ( t ) = 1 for It1 < M

(4-12)

A ( t ) = O for It1 2 M

The A's are in fact weights which are used to reduce the errors which arise by setting all the autocorrelations beyond the truncation point, to zero. Different weight functions, which are known as lag windows in spectral analysis, can be used to improve the properties of the spectrum estimates. For example in Fig. 4-3 the Tukey window was used to calculate the spectrum of the isotope data in core V28-238 Remembering that the periodogram I (w) is in fact the transform of R ( t ) , we can obtain the same result by multiplying the periodogram by the transform of the A's. The estimate of the spectrum can also be written as: n

h(w) =

J

I(6)W ( W- 6) d 6

(4-13)

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

56 95

5,

I

9

I 4-

\

I

3-

&

Cycleslcm

Fig. 4-3. Blackman-Tbkey spectrum and ARl model. Isotope data from core 28-238.

and: (4-14)

The function W ( O ) , is the discrete Fourier transform of h ( t ) and is known as the spectral window. With the introduction of the fast Fourier transform, spectrum calculations are often performed directly; the results of the two methods are equivalent. An alternative approach to the estimation of spectra is possible, if it is known approximately what kind of stochastic process is responsible for a certain set of data. The theoretical spectra of autoregressive processes AR or mixed moving average autoregressive processes ARMA (see Chapter 2), can be expressed in terms of their auto regressions. One then estimates the regressions from the autocorrelations of the process and substitutes them into the theoretical expression. This is known as autoregressive spectral estimation. A method relying on autoregression estimates, is the maximum entropy method for which a very neat algorithm is available (Press et al., 1986). Burg (1972) was able to show that one can obtain a spectrum by maximising its information entropy, under the constraint that it conforms to the transform of the estimated autocorrelations. If N observations are available, one uses a lower number of M autocorrelations. The method has the disadvantage that no precise statistical tests are available for testing the significance of the frequency maxima, but the method has a very much higher resolution than the B-T method (see Fig. 4-4). This resolution can be increased by taking M to be relatively high, but the choice is

METHODS OF ANALYSIS

57

3-

434 2a E

.-

L

3 1-

I@ 01

0

A

.018

,032

,048

,084

,080

.096

,112

,128

,144

Cycleslcm

Fig. 4-4. Maximum entropy spectrum. Isotope data from core 28-238.

critical because with higher values of M ,spurious peaks appear in the spectrum. Observational series are invariably finite and the discontinuities at both ends of them are responsible for a considerable amount of power leakage, which increases the bias and therefore lowers the accuracy of the spectral estimates. This can be reduced by tapering to a certain extent, a process which gradually fades out the data at both ends. A simple way of achieving this effect is to multiply the whole series by a bell-shaped cosine function. However, such a procedure leads to loss of statistical information and increases the variance of the estimates. A new technique of spectral analysis which is called the multitaper technique, was introduced by Thomson (1982). A short description of the method is given by Park et al. (1987). A number of different tapers are designed in such a way that successive applications recover the information that was lost in the previous tapering process. The discrete Fourier transforms of the transformed sequences, are then combined to give a single spectrum. To obtain a good taper ( N , W), for a frequency interval 2 W and for N data, one minimises the leakage from the outside to the frequency f . One maximises the energy A of the chosen band in relation to the white noise of the sequence:

(4-15)

The solution of this equation leads to an eigenvalue problem with eigen vectors dh)( N , W); k = 0, 1 , 2 . . . N - 1. The vectors are approximated by discrete prolate spheroidal sequences which are also known as Slepian sequences.

58

CYCLOSTRATIGRAPHYAND THE MILANKOVITCH THEORY

The estimates of power in the successive frequency bands are statistically independent and can be tested for their significance. By comparing individual estimates with the continuous spectrum, a test for line components in the spectrum can be made. This is particularly important in the search for periodic and quasi-periodic components in the spectrum. The analysis has been applied very successfully to Pleistocene isotope data, where thirteen harmonic components have been identified, which are probably related to eccentricity, obliquity and precession (Thomson, 1990). The high frequency resolution of the multitaper method was also used to prove the existence of two maxima in the 100 ka region, in spectra from Albian sediments (Park and Herbert, 1987). The advantages of the various methods of spectral analysis when they are applied to the study of Pleistocene palaeoclimatic time series, have been discussed in detail by Pestiaux and Berger (1982) and Berger et al. (1991). For the examination of isotope records in particular, it is found advisable to use a variety of methods. The Thomson technique is recognised as being superior because of its ability to resolve closely spaced frequency maxima and also because of the relatively reliable significance tests which are available for such maxima. However, the method requires considerably more computing resources. For pre-Pleistocene stratigraphical records, which frequently preserve much less detail, the B-T method is still one of the most useful approaches. COSPECTRAL ANALYSIS

The relationship between the autocorrelation function and the spectrum, indicates the possibility of a similar relationship existing for the cross-correlation function of two or more time series. The cross covariance and its standardised form, the cross-correlation, is estimated in the same way as the autocorrelation, by calculating the expectance of the cross product with the time lag. We have, for the cross-correlation of two series X I ( t ) and X 2 ( t ) : (4-16)

In this R I , J ( T )represents the autocorrelation of series 1; R l , z ( t ) the crosscorrelation of series 1 leading with series 2 and similarly for R2,1 and R2.2. The auto spectra are defined as the Fourier transform of R I , and ~ R2,2 and in a similar way, the cross-spectra for R1,2 and R2,1 for example, can be written: (4-17)

The cross-spectrum is a complex quantity and can be written as:

METHODS OF ANALYSIS h1,2(W)

= C1,2(0) - iq1,2(w)

59 (4-18)

The real part is called the cospectrum and the imaginary part is called the quadrature spectrum. If one takes the absolute value of the cospectrum, one obtains the cross-amplitude spectrum: (4-19)

and the phase spectrum: (4-20)

The standardised form of the amplitude spectrum is known as the coherence. Cross-spectral analysis has been applied to a number of geological problems (see Schwarzacher, 1975) and it has proved particularly useful in comparing Pleistocene climatic indicators with orbital variation (cf. Pisias and Moore, 1981). It is very likely that multi-spectral analysis will become extremely important as soon as more numerous and more detailed stratigraphical time series become available.

POWER SPECTRA IN STRATIGRAPHICAL ANALYSIS

Spectral analysis of stratigraphical data provides a quantitative description which can have many uses. Two problems in particular however, will be discussed in this section: how can this analysis clarify the basic definition of sedimentary cycles and how can spectral analysis help to identify Milankovitch cycles? Our original definition of cyclicity in Chapter 1, depended on the repetition of identical groups of lithologies which are repeated at regular intervals. We have already seen that the perfect regularity which would lead to periodic sequences is unlikely, but we have not yet tried to determine how regular the repetition must be, to qualify as cyclic. Surely, a reasonable condition must be that the repetition pattern should be regular enough not to have arisen from random fluctuations. Such a preferred repetition should lead to a significant frequency maximum in the power spectrum. Very regular repetitions will produce sharp maxima and the spectra will tend towards line spectra. If one were dealing with unadulterated astronomical data, then line spectra would result. However, the stratigraphical series is most likely to be conditioned by the climatological record which is already itself, a distortion of the astronomical data. During sedimentation, further stochastic processes are incorporated and the resulting spectra are therefore continuous. In many examples of Pleistocene and pre-Pleistocene series, one finds that the general shape of the spectra suggests a “background” which could have been produced by some autoregressive process (AR) of a relatively low order. For

60

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

example, the spectrum of an AR(1) process falls exponentially from a maximum at zero frequency towards the higher frequencies. This type of spectrum is very common in spectra from isotope data (see Fig. 4-3). A similar situation is often found in examples from communication engineering, where one has a “signal” which is superimposed on to autocorrelated (coloured) noise. In such cases, one speaks of mixed spectra (cf. Priestly, 1981). In communication problems, the signal is usually a harmonic process which would lead to a line spectrum. This makes it possible to separate the two, at least theoretically. In the stratigraphical records, sedimentary cycles that are themselves the signal, are stochastic processes which have continuous spectra and the difference between the “signal” and the background becomes arbitrary, to a certain extent. Nevertheless, it is possible to show that such spectra have not arisen from random fluctuations and approximate statistical tests are useful for judging the significance of stratigraphic spectra. The variance of the spectral estimate h(w), decreases in proportion to the ratio M / N , which is the number of autocorrelations used in estimating the power, divided by the length of the record. The sample distribution of the individual estimates of h(w) is asymptotically a chi square, with a nominal degree of freedom depending on the window being used (cf. Jenkins and Watts, 1968). Knowing this distribution, one can construct some approximate confidence limits for the individual frequencies. If the frequency is plotted against the logarithm of the power, an upper and lower confidence region for a known spectrum can be drawn. Of course, the spectra of the background noise are not known but one can assume that they can be represented by some not too complicated AR process. For example, an AR(1) and an AR(2) process was fitted to the isotope data of core V28-238 and the theoretical spectra of these were calculated. Figure 4-5 shows the spectrum of the data, together with the spectrum of the AR(1) process to which a 95% confidence limit has been added. If it is true that the series represents a mixed spectrum in which a signal is superimposed on to red noise, then any frequency peak above the confidence limit can be regarded as being significant. Figure 4-5 shows a second confidence limit, again with a significance of 95% but in this case, it is based on a background which is assumed to be an AR(2) process. The general trend of this noise component which is slightly convex, may not fit the data as well as the AR(1) process. However, the difference is small and in this case, there are only three maxima which are themselves barely significant. The experiment shows that the choice of the model on which the confidence tests are based, is highly critical and that it is impossible to develop criteria for deciding the significance of frequency peaks without making some assumptions. For judging the reliability of stratigraphical spectra, it is often more important to compare spectra from different parts of the section and also to compare spectra from different localities, rather than attempting statistical tests on data about which, very little is known. The spectrum of the isotope data, however, is an excellent example for answering the second question which we posed at the beginning of this section. How can

METHODS OF ANALYSIS

-0.2 0

,005

,010

61

.015

,020

,025

,030

,035

Cycleslcm

Fig. 4-5. Blackman-nkey spectrum of isotope data with 95% confidence limits. Dashed line shows the A R l model and dotted line showes the AR2 model.

spectral analysis help to decide whether one is dealing with Milankovitch cycles? The problem in this case is relatively simple, as the cores are extremely well dated and the Bruhnes-Matuyama reversal has been recorded at a depth of 12 m. Using this datum, one can arrive at an average sedimentation rate of 1.64 cm/ka for this core and therefore give absolute time values to the frequency maxima; these periods are indicated in ka on the spectra. As is well known from the classical paper by Hays et al. (1976), the maxima around 110 ka, 40 ka, and 20 ka are in good agreement with the eccentricity, obliquity and precession cycles to be expected from the Milankovitch theory. More accurate dating using data from many cores, as is now possible with Pleistocene stratigraphy (cf. Imbrie et al., 1984), leaves no room for doubt about the relevance of the astronomical control of cycles. Accurate dating is only available in exceptional cases and it is almost completely lacking in pre-Pleistocene sequences. The spectrum however, presents evidence of the complexity of the cycles which in itself, is a feature needing explanation. The astronomical cycles arise from the modulation and in part, from the addition of three sinoids, representing eccentricity ( E ) , obliquity (0) and precession ( P). Their approximate frequencies in cycles/ka are 1, 1/41, 1/21 and their ratios are O I E x 2 and P I E zx 5. As has been discussed earlier, these ratios have probably changed because of the general lengthening of all the precession related cycles. Because such changes are nevertheless slow, each geological time should have its own characteristic set of ratios and provided that the sedimentation .rates within a cycle are reasonably constant, the ratios should be found in the frequency maxima of the spectra.

62

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

There are however, a number of properties of the spectrum which must be taken into consideration before any interpretation of the spectrum can be made. The first concerns the resolution of a particular analysis. The highest frequency which can be detected in any set of data depends on the spacing At, of the sample points. For a “cycle” to be recognised, it takes at least two sample points and therefore the highest frequency, which is known as the Nyquist frequency is: f~ = 1/2At. The resolution of frequencies which are multiples of the Nyquist frequency, depends on the amount of data which is available and the methods which are used. This can be expressed numerically, by the bandwidth of the window which is used in calculating the spectrum. The bandwidth measures the sharpness of the frequency peaks that are produced by the various windows. In our illustrations, we have used the expression for the band width that was given by Jenkins and Watts (1968). Priestly (1981), uses a different definition and his width is half the width used by Jenkins for the n k e y window, which is used in all the spectra. All band width measures for a given length of record, increase inversely with a decrease of the parameter M . On the other hand, with an increase of band width, the variance of the estimates decreases, which means that it is possible to choose the ratio M / N either to obtain very reliable spectra with low variance but very poor resolution, or to obtain high resolution spectra which are not such good estimates because of high variance. The effect of different M / N ratios is illustrated by the two plots of the V28-238 spectrum with M values of 250 and 50 (see Fig. 4-6). The lower number of lags in the second example has reduced all the frequency maxima to a single one which is clearly the predominant period of the 100 ka cycle. One may be tempted to take the ratio M / N to be as large as possible, in order to obtain great detail. However, this clearly reduces the smoothing which is necessary 16.0-

1. _ _ _ 4 _Bandwidth ________

12.8-

-. 8,6

H

1L,’‘ I

I\

’

6.4-

3.2-

01 0

0.005

0.bl

0.015 0.b2 Cycleslcm

0.d25

0.’03

0.g5

Fig. 4-6. Blackman-Thkey spectrum calculated with different frequency resolutions (bandwidth).

METHODS OF ANALYSIS

63

to obtain a consistent estimate of the power and then the frequency maxima which appear when one reduces the bandwidth become unreliable. It is therefore advisable to be very careful when attempting to increase the resolution of the spectra. It is often possible to compare spectra from different parts of a section or from different localities. If the same maxima occur again and again, one can be more confident that they represent something real. Apart from random fluctuations which contribute to the confusion of spectra with multiple maxima, subsidiary peaks can also be produced by harmonics. It is important to remember that the basic principle of the Fourier spectral analysis is the representation of the time series in terms of sine waves. The records of sedimentary cycles however, are hardly ever pure sine waves. Each “cycle” incorporates a number of additional frequencies which account for its shape and asymmetry. Such cycles produce multiple peaks in the spectrum which occur at frequencies 204, 3w0, . . ., multiples of the basic frequency W O . Such maxima are therefore highly suspect, particularly if the basic frequency maximum is strong. WALSH SPECTRA

A large number of stratigraphical data come in the form of attributes which can be coded in discrete classes, or in the form of presence or absence data that can be coded as zero-one strings. For example, limestone and marl sequences are particularly common two state systems. If the presence of a state, for example limestone, is coded as +1 and the absence of this state is recorded as -1, then the resulting time sequence will be a square wave consisting of rectangular pulses. Several functions are available to describe such data and one particular set of functions known as Walsh functions has been used in stratigraphic data analysis, (see Beauchamp, 1984). Walsh functions are similar to the orthogonal trigonometric functions, which means that they can be used to synthesise more complex functions by combining different orders of functions. The Walsh functions can only take two values of either + 1 or -1, and they are defined over a limited time interval T , which is called the time base. A particular Walsh function is defined by two arguments, the order number n, and the time base. The function is written as: WAL(n, T ) . The order number n refers to the number of zero crossings. The first seven Walsh functions are shown in Fig. 4-7. It can be seen that all the functions have a symmetry with respect to the midpoint of the time base. This is a straightforward bilateral symmetry for functions which have an even number of zero crossings and the uneven function becomes symmetric, after changing all +1 values to -1 in one half of the time base. The even and uneven Walsh functions can be classified into two types: WAL(2k, t ) = CAL(k, t ) WAL(2k - 1, t ) = S A L ( k , t )

(4-21)

64

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

WAL(7.T)

6 SAL(4.T)

WAL(6.T)

&

WAL(5,T)

2

WALi4,T)

CAL(3.T)

, I

I

SALi3.T)

I CAL(2,T)

WAL(3'T)

SAL(2.T)

WAL(2.T)

CAL(i,T)

WAL(i.T)

2

SAL(l'T)

WAL(0,T)

CAL(0,T)

Fig. 4-7. Walsh functions.

The CAL and SAL functions have the same relationship as the cosine and sine functions. It is seen from Fig. 4-7 that Walsh functions are not periodic functions and that the zero crossings do not mark unit space. Strictly speaking therefore, one should not use the term frequency when referring to the repetition of zero crossings and the term sequency has been proposed instead. In practice however, sequency and frequency are often interchanged and the Walsh spectral analysis, which is strictly speaking a sequency analysis, can be directly compared to power spectral analysis which is based on trigonometric functions. The similarity between the two methods is considerable and a time series X ( t )can be represented by Walsh functions as: X ( t ) = a0

+ a1 WAL(1, t ) + a2 WAL(2, t ) . . *

(4-22)

whereby the coefficients are given by:

1

1

ak =

f ( t >W a ( k , t ) dt

(4-23)

It is therefore possible to define a transform pair: 00

@k

f(t) = k=O

WL(k,t) (4-24)

The Walsh transform @ ( K ) is a function of the ordering of the Walsh function which determines the sequency. The function is equivalent to the periodogram and after smoothing, yields the Walsh power spectrum. The Walsh transform, unlike the

METHODS OF ANALYSIS

65

Fourier transform, is not invariant to the phase of the input signal and it is possible to obtain different spectra depending at which point the analysis of a stationary series is started. In relatively long sequences, this does not seem to matter, but in short sequences with relatively low frequencies, this can have a very strong effect. A possible way of avoiding this is to average the spectra using sequences with all possible starting points. Since only the CAL function is used it means that N / 2 transform operations are needed to calculate an averaged Walsh spectrum.

THE WALSH SPECTRUM IN STRATIGRAPHY

The Walsh spectral analysis was first applied to limestone marl sequences in the hope of obtaining better defined frequency maxima in data which were coded as plus and minus one (Schwarzacher, 1985). Walsh spectral analysis was independently and extensively used by Weedon (1985) in the examination of systems involving several states which could be represented by stepped curves. Walsh transforms have also been used to analyse more complex cycles (Schwarzacher, 1988, 1991), but it seems that for such problems, Fourier transforms are much more suitable. The method does have the advantages of providing very well-defined spectra for records with simple square wave patterns and at the same time, it is extremely fast, in particular if the method is used without averaging the starting positions. However, this can only be done with very long series. The disadvantages of the Walsh method are that the harmonics which can arise from the more complex wave forms can be very difficult to interpret and the results from a system containing more than two states depend very much on the method of coding. A detailed discussion of this is given by van Echelpoel (1991). The Walsh method is very much to be recommended for two state systems but Fourier transforms appear to be generally more useful. It is probably best to try out both methods but if this is not possible, one might give the Fourier transform the preference. THE ROLE OF SPECTRAL ANALYSIS IN CYCLOSTRATIGRAPHY

Spectral analysis is possibly the best way of establishing cyclic behaviour in stratigraphic records. It was seen that the height and sharpness of the frequency maxima is an indication of how well the cyclicity is developed. Genuine periodic and quasi-periodic processes will generate line spectra, but if the signal is contaminated with a random background, then it is not possible to establish the periodicity of a signal, unless the correlation structure of the process is known. The same is true for tests based on confidence limits. These again need a reasonable model against which the maxima can be tested.

66

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

The second and possibly even more important role of power spectra is the analysis of complex cycles which consist of several oscillations. The relative position of the frequency maxima in this case should provide characteristic data which may lead to an identification of the cycles. The Milankovitch theory for example, was only generally accepted after 100 ka, 40 ka and 20 ka cycles had been recognised in the Pleistocene isotope records. The identification of complex cycles is possible even without absolute time values, if the ratios between the maxima conform to a definite pattern. It is very important in such a study, to recognise harmonics which have been generated by the method of analysis, for example, through the asymmetry of the original signal. Quasi-periodic processes are easier to identify because there is no danger of confusing genuine additional frequencies with the harmonics which are likely to occur with both Fourier and Walsh transform methods. A very real effect of the astronomical climate theory, is the amplitude modulation of the precession signal by the eccentricity variation. This modulation would not show up in the power spectrum if the response to this signal was perfectly linear and if the signal was a pure sine wave. The recognition of amplitude modulated records may be even more difficult when using Walsh techniques. It was shown by van Echelpoel (1991) that many “bundled” cycles which are essentially amplitude modulated, would not be recognised by the Walsh method which relies entirely on zero crossings. Spectral analysis is a statistical method and the accuracy with which a cycle length or the relative positions of the frequency maxima can be determined, depends very much on the quantity and quality of the data. The different methods which are used in estimating the power spectra, seem to have very little influence on the results, because most stratigraphic data are subject to fluctuations in sedimentation rates which far outweigh any advantage that could be obtained by choosing a particular method of analysis. For example, in a study of Pleistocene isotope records (Berger et al., 1991), four extremely well correlated cores were compared using the maximum entropy, Blackman-nkey and Thomson method of spectrum estimation and the differences between the results were approximately 3% of the wavelengths of the various periods. Isotope values in the Pleistocene are probably the very best palaeoclimatic indicators that we have and similar accuracies cannot be expected for pre Pleistocene records. The amount of data which is available is again an important factor in deciding just how accurately the length of the cycles can be determined. Clearly, if a cycle is only repeated a very few times in a particular set of data, one cannot put too much reliance on the results of the spectral analysis. This decrease in reliability with low frequencies is reflected in the inversely increasing wavelength which is obtained. This must be kept in mind when reading power spectra. For example, in the data shown in Figs. 4-3 to 4-6 some time values are given for the lengths of the prominent cycles which are based on an assumed constant rate of sedimentation

METHODS OF ANALYSIS

67

of 1.644 cm/ka. The frequencies in this analysis have been calculated in units of 2.5 x cycles/cm. Thus a maximum at 116 f 1 cycles/cm corresponds to a time value of 20.97 f 0.18 ka but a maximum of 22 f 1 cycles/cm corresponds to 110.592::;; ka. In theory, the accuracy of an analysis can be increased by increasing the length of the record. However, this also means increasing the effect of irregular sedimentation rates and so very often, the degree of accuracy in determining the cyclicity is not improved.

THE FILTERING OF SECTIONS

Some sedimentary cycles consist of well-defined rock sequences and if there is at least one horizon which can act as a marker, definite cycle boundaries can be determined. For example, in sections where a transgressive sequence regularly follows a period of non-deposition or erosion, one will have no difficulty in deciding the cycle boundaries and it is therefore possible to count the cycles and determine their thickness distribution. On the other hand, cycle boundaries are difficult to establish in sections where the lithological variation is represented by some continuous variable which in addition, is disturbed by random fluctuations. Consequently, it is very difficult to count the cycles in a given stratigraphic interval and this of course, is one of the basic problems in cyclostratigraphy. Spectral analysis can provide the best average number of cycles but it cannot give information about the individual cycles and cycle boundaries. In such cases, a possible approach is to filter the data, in the hope that restricting the frequency range of the record makes it easier to interpret. Filters are designed to change the frequency composition of a time series and they can be understood as black boxes which have an input signal x ( t ) and an output signal y ( t ) . If we write for the Fourier transform of y ( t ) , Y ( w ) and for the transform of x ( t ) , X ( w ) then we can write the filter procedure in frequency terms as:

in which H ( w ) is called the transfer or frequency response function. The multiplication of the two transforms is equivalent to the operation of convolution in the time domain and eq. 4-25 can therefore also be written as: (4-26)

The function h ( t ) is the inverse transform of H ( w ) , a function which is referred to as the impulse response function. If x ( t ) consisted of a single value, an impulse, then y ( t ) would be the same as h ( t ) .

68

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

The actual filtering can be done either in the frequency or in the time domain. The latter is more convenient for many applications and the filtering is achieved by multiplying the input and a specific number of output values with a set of weights. A first order filter for example, can be written as: (4-27)

In this case, the filtering consists of combining K past input values and a single past output; x ( t ) again represents the input and y ( t ) the output. Filters which pass either low or high frequencies are known as lowpass and highpass filters and filters which pass only a selected frequency range are called band filters. A good introduction to the theory and practice of filter application is given in Otnes and Enochson (1978). The book also contains computer programs to generate various filter weights and programs to implement the filtering. A particularly common use of filters is the removal of random variations or noise by the application of lowpass filters. Since such fluctuations are usually much faster than the signal, these filters can be highly effective. In the filtering of random processes however, it is often impossible to decide on what can be regarded as noise and what is actually part of the signal. It is well-known that uncontrolled filtering can in fact generate some cyclicity, called the Slutsky effect. Controlling filtering is however, always possible through the frequency response function and this clearly indicates which frequency bias has been given to the data and can be compensated for, if necessary. Filtering in the frequency domain, can be achieved by multiplying the Fourier transform with a chosen frequency response function which is then followed by back transformation. The process is particularly simple when using Walsh transforms. In this case, the transforms of the unwanted frequencies are simply set to zero and the adjusted transform is again inverted to form the time domain data. The method is called sequency based vector filtering (Beauchamp, 1975) and was first applied to stratigraphic data by Weedon (1989) in an analysis of Jurassic limestone shale cycles. If Walsh functions are used in very narrow band filters, they produce an artificial grouping of amplitudes which is due to the non-unit space markings of these functions. This effect can very easily give the appearance of a regular bundling in a sequence. However, this is not caused, as it would appear, by an amplitude modulation of the original signal. The effect can be avoided by using Walsh functions, either in sufficiently wide band filters or better still, as low pass filters where they can be very effective (Schwarzacher, 1992). The Walsh filters have an advantage over filters based on Fourier transforms because any finite sequence is determined by a finite number of Walsh transforms and this permits a very sharply defined filtering.

69

METHODS OF ANALYSIS THE EFFECTS OF NON-STATIONARITY

The methods of spectral analysis and to a certain extent some of the filtering methods, assume that the time series provided by stratigraphical sections are stationary. This is hardly ever the case and corrections for this have to be made. Slow changes in the amplitudes of the cycles can be corrected relatively easily by subtracting a trend,which could be any smooth function, for example a polynomial. Changes in the lengths of the cycles which would result from changes in sedimentation rates, are more difficult to treat. A very good method of investigating frequency changes in the section, is to compare the power spectra of the subsections. To do this, the total section is subdivided into a number of short sections which may overlap and spectra are calculated for these short intervals. The power spectra can then be either stacked vertically (Premoli-Silva et al., 1989) or arranged in a three dimensional presentation, in which the successive spectra are aligned along the x axis and the frequency or sequency is aligned along the y axis. The power is shown vertically along the z axis (Beauchamp, 1975; Berger, 1989; van Echelpoel, 1991). Another, very effective way of showing spectral variation is due to Melnyk and Smith (1989), who arranged successive spectra in the x y plane using the x axis as time and y as the frequency axis. The powers on this map are shown by contouring (see Fig. 4-8). By following the positions of the maxima in such graphs, one can trace increasing or decreasing frequencies and obtain from this, some information about the changes in the sedimentation rates. For such studies, it is particularly helpful if the spectra are not plotted against a linear frequency scale but if instead, the spectra are plotted

Starting value for each pass (crn)

Fig. 4-8. Contoured power spectra of the Cenomanian in the Gubbio district.,Such diagrams can be used to show non-stationarity of sections (after Melnyk and Smith, 1989).

70

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

against a linear wavelength scale, which permits thickness changes to be read off the graph directly. Diagrams of this type are also helpful in judging the reliability of the frequency maxima, since they show clearly which maxima are persistent throughout the section and which maxima are only developed in short intervals of the sequence.

COMPLEX DEMODULATION

A somewhat more formal approach to the problem of the non-stationarity of the cycles, is sometimes possible through the method of complex demodulation. This method is best described as a local spectral analysis which can be applied to a process with the form:

(4-28) in which a ( t , w ) is a function which slowly changes and which is responsible for the non-stationarity. Assume that there are two stationary processes, for example: eiWtdz(w)

and A ( t ) =

ll

(4-29)

eiw'd a

Let us further assume that we have a filter available which cuts out all but the zero frequencies. Then the product of the two functions, after filtering is: x(t)A(t)

-

filter -+

Z(t) =

L

(4-30)

dz(w)da(w)

If a filter is taken in such a way that da(w) = 1 for only and da(w # wo) = 0 we have: A ( t ) = eiyl'

(4-31)

and Z(t) = dZ(wo)

Z(t) is complex and will give the amplitude and phase of frequency w~ as a function of time. This result can also be applied to non-stationary sequences in which the frequencies undergo slow changes. Instead of demodulating at a precise frequency, one takes a frequency band and finds again that A ( t ) approaches eiWOt.Therefore if the original series x ( t ) , t = 1, 2 . . . n is multiplied by sin wot and cos mot and passed through a lowpass filter, one can obtain: Zl(t) = filter X(t)sinwlt

(4-32)

Z2 ( t ) = filter X ( t ) cos wgt The amplitude at any given time is found from 2JZ; by tan-' Z1 (t)/Z2(t).

+ 2:

and the phase is given

71

METHODS OF ANALYSIS

The demodulation method is described in Priestley (1981) and Bloomfield (1976). The latter provides a complete computer program which includes the creation of suitable low pass filters. Complex demodulation was used by Pisias and Moore (1981) and Pisias et al. (1990), to extract the precession and obliquity signal from an approximately 700 ka long isotope record. Pisias used time adjusted data for this work. A comparison of the astronomical and extracted cycles, shows some similarity in the amplitude modulation. However, there are also discrepancies, particularly in the precession isotope relationship. These are thought to be due to non-linear response effects. It is however not quite clear from the paper of Pisias et al. (1990), whether the astronomical sequence is the palaeoclimatic precession index, nor is it known for certain what climatic effect this variable really has. One of the main attractions of complex demodulation, at least in theory, is the possibility of using it as a technique to determine sedimentation rates, under the hypothesis that the cycle signal had a fixed period, which is approximately true for Milankovitch cycles. Experiments with Pleistocene isotope data have not been very encouraging. In Fig. 4-9 for example, the isotope record of core V28238 (Shackelton, 1973) is shown, together with the demodulated frequency at 0.21 cycles/cm. This frequency corresponds to the 21 ka precession cycle, when average sedimentation rates are taken. The amplitude distribution clearly shows a modulation of the precession effect by the 100 ka eccentricity. However, at isotope stage 11 (which due to the 400 ka cycle of the eccentricity shows very little modulation), a phase shift occurs which makes the interpretation of the following record very difficult. It was also found that, as is to be expected, the phase and amplitude results of the demodulation are very sensitive to the frequency at which demodulation was carried out. If the demodulation frequency bands were widened, the results became quite meaningless. Despite these difficulties, which will be even more pronounced in pre Pleistocene sediments, the method of complex demodulation should be remembered in the study of sedimentary cycles.

0

1

2

3

4

5

6

7 8 Depth (m)

9

1 0 1 1

1 2 1 3

1 4 1 5

Fig. 4-9. Demodulation of the isotope record of core 28-238 (solid line). The dotted line is the 1/21 ka frequency component.

72

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

CONCLUSIONS

The quantitative study of cycles by spectral analysis, autocorrelation or filtering techniques should be approached with a good understanding of the methods but at the same time, geological common sense should not be forgotten. Stratigraphical data are never like the mathematical models for which the analyses have been developed. If such data are analysed blindly, by relying on “significance” tests, one can miss cyclostratigraphically important sequences. In a similar way pooling a large number of heterogeneous observations (Algeo and Wilkinson, 1988) is also unlikely to provide evidence for cycles which can be interpreted as having been caused by a particular mechanism. On the other hand, selection invariably produces bias and by a process akin to filtering, cyclicity can be suggested where none exists. It is precisely at this point, where geological judgement must come in and the right balance must be maintained between data selection, the method of analysis, and the conclusions one draws from it.

73

Chapter 5

THE RELATION BETWEEN TIME AND SEDIMENT ACCUMULATION

The unique feature of astronomical cycles was emphasised in the previous discussion about oscillating systems and the Milankovitch theory in particular. Astronomical cycles are unique in the sense that they can almost be represented by periodic functions in the mathematical sense (see eq. 1-1) and they are caused by a nearly conservative oscillating system, in which damping over very long time periods is negligible. From this, it follows that perhaps the best proof for the orbital control of sedimentary cycles would be to show that they originated from time periodic, or at least quasi-time periodic processes. This involves many difficulties, not only because the astronomical cycles will be modified by their interaction with all the terrestrial processes, such as climate and sedimentation, but mainly because of the difficulty in reconstructing an accurate time history from the stratigraphic record. The stratigraphic record is measured in units of length and to transform the thickness of an accumulated sediment z into units of time t one needs to know z ( t ) which is a function that will be called the stratigraphic mapping function. The function contains invariably a certain stochastic component and the transformation from z into t will always contain some errors. It is well known that the accumulation of sediments is discontinuous in time. When the time steps are taken very close to one another, the accumulation can be very variable. When sedimentation is considered on a coarser time scale, the stratigraphic thickness z ( t ) becomes more continuous as a function of time. Ultimately, the discontinuity is a consequence of the sedimentation process, which is always an accumulation of particles. The time resolution of any stratigraphical record is determined by the depositional and post-depositional structures of the sediments. If a bed or lamina consists of homogeneous material, then it is often impossible to say which part of such a lithological unit is older, or which is younger and dating within a bed is therefore often impossible. Time resolution limits are also set by sediments which were deposited so quickly that they can be regarded as having been deposited instantaneously. For example, turbidites or any deposits which are separated by long intervals of non-deposition, are cases where the thickness interval up to the next depositional unit, carries no time information. The type of stratigraphical structure which is formed in this way, is known as an event stratification. Post-depositional lowering of the stratigraphical resolution can come about in many ways. A common process is bioturbation, which homogenises a certain

74

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

thickness interval of a sediment. However, there are many diagenetic and also post diagenetic processes which destroy or alter the primary sedimentary structures and in this way, contribute towards the loss of resolution. The smallest recognisable time-thickness related element, which can be regarded as a stratigraphic quantum, will be called the sedimentation step. In general, the distribution of the steps will have definite mean values and variances and similarly, they may represent a variable or a constant time interval. The latter is exceptional but can be realised, for example in tidal lamination or annual varves. THE RANDOM WALK MODEL

Based on these considerations, one can formulate some simple models to describe sediment accumulation. The follcwing mechanism is assumed. At discrete time intervals, a depositional step is laid down. This can either be positive and depositional, or negative and erosional. The decision between deposition and erosion is made at random at each stage and the resulting stochastic process is therefore called a random walk. In its simplest form, one can assume that only two moves in the random walk are possible; either a positive step of unit thickness occurring with probability p , or an erosional step occurring with probability q (see Fig. 5-1). To obtain accumulation, one assumes that deposition is more likely to occur than erosion and therefore that p > q and p q = 1. Under such simple conditions, the probability of finding the random walk after n iterations in position i, is given by the binomial distribution:

+

n! p ' ( l- p ) i - l i!(n - i)!

(5-1)

The end point of the random walk, z, represents the stratigraphical thickness in this simple model and the number of iterations, n, represents the time. The mean 1

/

z .

/-.

\

\

?

4

' r\ '' d

'

/

k .

OSL. \

. r '

1

,

1

, \

/

1

1

.

:,

v ,

I /

/ ? I

3

' /

,

. . .

.

1

/

time n

Fig. 5-1. A simple random walk. The interval LI leaves no stratigraphic record. It consists of k erosional steps and V steps to compensate for the erosion y.

THE RELATION BETWEEN TIME AND SEDIMENT ACCUMULATION

75

position of the end point is given by: P(Z> = np

(5-2a)

and the variance is given by: a2(z)= n p ( 1 - p )

(5-2b)

It is assumed in this model, that the sedimentation steps, or basic stratigraphical units, are of constant thickness. This condition can be relaxed and both depositional and erosional steps can have any probability distribution. If one writes 6 for the sedimentation increment, one can define the expected probability, that 6 is positive, by: /A+ =

E ( 6 + ) = E(616 > 0)

(5-3a)

and similarly, the mean erosional step can be written: p- = E(&) = E(6)S c 0 )

(5-3b)

The mean and variance of the sedimentation steps are: p = E ( p + - p - ) and u 2 = E ( 6 - p)2

As before, one can write for the position of the end point: P ( Z n ) = nw

(5-4a)

and u2(zn)= an

(5-4b)

The distribution of the end points will always tend towards a normal distribution, quite irrespective of the probability distribution of the individual steps. It is therefore possible to predict the end points, which means the accumulated thickness of the sediment, if the mean and variance of the individual steps are known. This result is important for the interpretation of sections where the cycles represent equal time intervals (as for example in varve analysis), and also for sections that represent approximately equal time intervals (as in sequences which contain positively identified Milankovitch cycles). In such cases, the cycle thickness, which has a known mean (1)and variance (a) can be used to predict the stratigraphical thickness z ( t ) , as a function of time. Time is determined obviously by counting the number of cycles in the section and the estimated stratigraphical thickness, after a deposition of n cycles, will be equal to p(zn) = np. The error associated with this predicted thickness is a2(zn) = nu2. As an example, we can use the thickness measurements of cycles from the Palaeocene of Gubbio (Italy), which have been plotted as a cumulative curve. In this case, the cycles have been interpreted as being 100 ka eccentricity cycles and the graph therefore represents a random walk in which stepwise sedimentation occurred

76

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

number of cycles

Fig. 5-2. Cumulative curve of possible 100 ka cycles. The numbering starts at the C/T boundary in the Bottacione gorge near Gubbio. Solid lines represent 95% confidence limits for predicting the total thickness, provided that the number of cycles is known.

at equal intervals. Since the mean and variance of the steps can be estimated, one can also predict the end point of the random walk and the error associated with this estimate. According to eq. 5-4b, this error increases continuously and the 95% confidence limit for the expected thickness is shown by solid lines in Fig. 5-2. The mean square deviation in time, which in our case is shown on the horizontal axis, is the error incurred when a time estimate is made from a given thickness measurement, without any reference to the number of cycles involved. In the particular sequence of which Fig. 5-2 is only a small sample, this error is approximately 25%. For example, using a mean thickness of 34 cm/cycle, one can state that 250 m of sediment represent 7.35 Ma f 1.8 Ma. The error is clearly quite high and this is due to the changes in the sedimentation rates in this particular section. In the example which has just been used, there is good evidence that the cycles represent nearly constant time intervals. However, the random walk model also shows that an age determination which is based on stratigraphical thickness, is always possible. This is also true when the steps are not exact time intervals. In eq. 5-4, time is given by n discrete time intervals. These can be of either equal or variable lengths. If one knows something about the distribution of this variability, one can again make some definite time estimates. Clearly, the less that is known about the variability of the time elements and sedimentation steps, the less accurate will be the thickness-time relationships. In the absence of any data, our knowledge is reduced to the well-known principle of superposition, which tells us that with increasing thickness, time also increases.

THE RELATION BETWEEN TIME AND SEDIMENT ACCUMULATION

77

The random walk model is the best possible way to predict the position of accumulating sediments from a given point. Time-thickness curves however, are often constructed from sequences in which two points are given, for example, a measured thickness for which two dates are known, which are either absolute or relative. In this situation, the regression line must go through the two points and the variance or standard deviation must be zero at the given dates. The problem has been treated by Ode11 (1975), who showed that in this type of correlation problem, one can make use of the so-called Brownian bridge, which defines a confidence region which is itself defined by the distance d, from the regression line (see Fig. 5-3). For a fixed time, points along a line at right angles to the regression line are approximately normally distributed and the 95% confidence region is found by: Prob.

[

d(t)

<<

,/m} = 0.95

(5-5)

One can obtain K from the variance u 2 ,of the point scatter and for 95% limit, K = (1.96)2u2. In eq. 5-5, T is the known time between the fixed points and if sufficient data are available, the confidence limits to the regression line can be drawn. The two points which have a known time value, have no error and the confidence limits join the regression line at this point. Odell’s formula can be interpreted as being the error which occurs after t time steps have accumulated and one can write:

a ( t )=

,/$ ( T - +)

to

Fig. 5-3.The confidence limit of a time-thickness relationship when two points are known.

(5 -6)

78

CYCLOSTRATIGRAPHYAND THE MILANKOVITCH THEORY

When T becomes very large, then (5-7)

u ( t ) = Ja’t

which is again the same as the error of a random walk without a fixed end. THE RATE OF SEDIMENTATION

The random walk model is an important aid to understanding the problems arising from making estimates of sedimentation rates. Such estimates are usually made by dating two points in a section and dividing the stratigraphical thickness by the time interval. That “rates” determined in this way depend strongly on the length of the interval which is used in such calculations, has been well documented (Sadler, 1981; Anders et al., 1987). The reason for this is that some random walks with intervals in which either no sedimentation, or negative sedimentation occurs, are never observed. Depending on how many non-depositional intervals are present in a section, the estimates of sedimentation will give different results (Tipper, 1983). It is possible to calculate the thickness of the sediment which accumulates, by adding only the positive sedimentation steps (Strauss, 1989). If one defines the initial expectation of positive steps only as:

E(z:,) = E(Z,IZ,

> 0)

(5 -8)

then according to the central limit theorem, this is normal and when standardised:

The distribution of Z, > 0 is equivalent to the distribution of Z, > a whereby a = - p m . Writing 4J and @ for the density and distribution function of the normal probability, one finds: E(z,,lz,

> a ) = $fJ@-fl

(5-10)

and using eq. 5-9: E(Z,IZ,

> 0) = n p

+U J G ~ J ~ / @ - ~

(5-11)

To obtain the rate of sedimentation R , the accumulated sediment which ignores sedimentation gaps, is divided by n (Strauss, 1989) and from eq. 5-11, one obtains:

R’(n)= p

IT +f i [exp(-np2/2a2)/~(pJG/~)]

(5-12)

The function of the “rate” depends on the time interval of the observations. The sedimentation rate decreases exponentially, and approaches a constant value

THE RELATION BETWEEN TIME AND SEDIMENT ACCUMULATION

79

which is a constant rate and valid over long time intervals. The theoretical “rate”time relation of Strauss agrees well with the empirical data collected by Sadler (1981). Sadler used very large but also unfortunately very mixed material, which showed that sedimentation rates did indeed seem to decrease with increasing time intervals. The data therefore show that the random walk model provides a very good representation of unsteady and discontinuous sedimentation. This is further supported by Sadler’s finding that the apparent rate-time relation is different for different depositional environments and therefore for different lithologies. As one would expect, in fluvial and terrigenous shelf deposits with many internal discontinuities, the rate-time relation is steeper than for example, in the carbonate platform environments. The tendency towards a rate (R’ in eq. 5-12), which is independent of the interval over which it is measured, is only well developed in deep sea sediments, where accumulation is probably more steady than in any other environment.

THE COMPLETENESS OF THE RECORD

The study of sedimentation through a random walk model, shows that the completeness of stratigraphic sections must be a function of their lengths. In fact, completeness C ( n ) can be defined as one minus the average number of time units which left no deposits.

C ( n )= 1- U / n

(5-13)

U is the number of units without deposition. The random walk is once again represented by positive and negative steps S+ and L , which are assumed to be exponentially distributed for simplicity. The steps have constant densities p1 and p2. The total number k , of erosional steps has a binomial distribution with the parameters n , p , q. The total erosion y , is the sum of k exponential variables with density p2 and is therefore gamma distributed with the parameters k , p2. A study of Fig. 5-1 will show that U , which is the number of units which leave no stratigraphic record, is U = k V ; k is the number of erosional steps and V is the time interval which is necessary to compensate for y , the total erosion. The distribution of V is Poisson, with mean y . V is also the number of positive steps needed for the compensation of y . It is:

+

E(UI k ) = k

+ p1k/p2

and

(5-14)

80

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

and the expected completeness is therefore: (5-15)

Clearly, when p = 1, then q = 0 and the completeness is C ( n ) = 1. When p / p 1 + q/pz then C ( n ) + 0. This means that when the average end point position of the random walk 0, the section will become fully incomplete. As one would expect, when n becomes large, then lim C ( n ) = p

n+m

(5-16)

- qp-/p+

One can show (Strauss, 1989) that eq. 5-11 is also valid when S+ and 8- have any distribution. The only condition is that p+ and p- remain constant. Strauss has also calculated the variance of the quantity C ( n )and finds: var [ C ( n ) ]=

1 4 n p ( l + P2/P1)

+ + (1 + 4)IP2/PIl

(5-17)

The standard error of C ( n ) therefore, is of the order of l / f i . As has already been pointed out, when the completeness of a section is being discussed, the length of the time is an important factor, especially with reference to the time units which are used to describe the section. Assume for example, that a stratigraphical record is detailed enough to record conditions every day for twelve hours. For the geologist, such a record would be considered to be incredibly detailed. However, if hours are taken as the time units, then the completeness is only 50%. The time unit in the case of the discrete random walk models, is fixed and stepwise sedimentation is assumed. Strauss showed in his analysis that it is possible to go from the discrete to the continuous case, by reducing the time intervals in the limit to zero and then the sedimentation history will be replaced by a Wiener process. The expected completeness is now the average of all the probabilities that are needed to obtain a positive deposit. The problem leads to a fairly complicated integral which can be solved numerically. One can use diagrams to determine the completeness for different time intervals. One needs to know however, the true average sedimentation rate and its standard deviation. To obtain the latter, it is necessary to date a large number of subsections to give an estimate for the completeness of the total section. The direct application of Anders et al. (1987) regression formula to a specific section (cf. Weedon, 1989) does not lead to any valid results. Completeness can however be estimated, if the accurate number of well identified cycles in a long dated interval is known. Alternatively, some estimates of completeness are obtained by comparing the number of cycles in several well correlated sections. The random walk model and its generalised form, the Wiener process, provide valuable fundamental insights into the problems of sedimentation. In practical applications of the model, one has to remember that accumulated thicknesses refer

THE RELATION BETWEEN TIME AND SEDIMENT ACCUMULATION

81

to the sediment as it was deposited and not to sediment that has been compacted or been modified diagenetically in any other way. Nevertheless, the theory shows that thickness distributions of accumulated sediments, which may or may not contain hiatuses, will tend towards the normal. Such distributions are fairly insensitive to the thickness distribution of the individual steps, provided that a large enough number of steps is involved and provided that the distribution of the gaps in the sequence is not determined by some systematic change in the sedimentation conditions.

CYCLES WITH PRECISE TIME PERIODS

The random walk model can be used to judge the accuracy with which beds or cycles represent a constant time interval. Let us first assume that a sedimentary section consisting of a uniform lithology, is marked at equal time intervals by distinct layers. Such markers are supposed to be instantaneous and without any time information. Using the model and assuming that each bed or cycle that is marked in this way consists of many sedimentation steps, then one can expect a normal distribution of the bed thicknesses. The best estimate of the sedimentation rate per cycle is the total thickness of the section divided by the number of beds. A normal bed or cycle thickness distribution is a necessary condition, if time within a cycle is to be measured by a linear thickness scale. Bed thickness distributions which are not normal and in particular, low order gamma distributions, are very common. They can often be explained by the composite nature of the beds, as for example with couplets consisting of two lithologies with different sedimentation rates (see Schwarzacher, 1975). Such composite cycles can be represented by random walks, but the time intervals representing the steps must be widened, possibly to such an extent that they comprise a complete cycle. In the random walk model, successive steps must have the same statistical properties and the widening of the steps can achieve this statistical uniformity. It is often necessary therefore, to have some method for comparing the different orders of the steps. For example, one can ask whether measurable laminae, beds and groups of beds can be treated by the same random walk using laminae as the basic step. This problem involves a comparison of the statistical properties of the steps. The comparison is made by using the relationship between the variance and the mean thickness of the accumulated sediment. According to eqs. 5-4, both the variance and the mean should increase linearly with the number of repeated steps. Therefore a plot of the mean versus the variance of the different orders of the steps should fall on a straight line, provided that the system is statistically homogeneous. In other words, the larger steps can be replaced by multiples of the smaller steps. An example of this test is shown in Fig. 5-4, where individual beds have been compared with the thicknesses of groups of beds which represent stratification cycles. In the

82

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

lo3W C

.-

L

>" lo2-

Carboniferous Sligo Cycles a Beds

do

lb

1

1000

Thickness (Cm)

Fig. 5-4. The comparison of cycles and beds with different thicknesses.

case of the Palaeocene limestone beds of the Gubbio (Italy) section, the two are equivalent. In the Carboniferous limestone beds of Sligo (N.W. Ircland), they are not.

CYCLES WITH CHANGING LITHOLOGY

In order to examine the process of recording a periodic signal somewhat more closely, let us assume that some environmental signal X ( t ) , can be represented by a sine wave. This signal will determine the type of sediment which is to be deposited and which will eventually determine the stratigraphic sequence. The simplest situation would be a direct transformation from the time process X (t) into a vertical sequence X (2): X ( t ) = A sinot

--f

x(z) = k A s i n o

(5-18)

It is well known and indeed, it is to be expected that a change in lithology also involves a change in the sedimentation rate. Assuming that the rate of sedimentation is strictly proportional to the environmental signal, then the transformation from time to stratigraphical thickness, generates curves which are known as cycloids (see Fig. 5-5). The function has the familiar geometrical interpretation of being the path of a point on a rolling wheel. The constant h is the diameter of the wheel and determines its progress. The constant A is the distance of the point from the centre of the wheel and it describes the amplitude or range of the lithological variation. The curves are best defined in parametric form:

THE RELATION BETWEEN TIME AND SEDIMENT ACCUMULATION

1 1

...........................................................

E 0 .c

'y

I2

0

83

1 1

+ A = ho

............................................................

KO+.

...........................................................

Time

Fig. 5-5. Cycloids.

X ( t ) = ho - A sinwt

(5-19)

Z ( t ) = hot - A C O S O ~ The second equation is, of course, the mapping function. To illustrate this model, we may assume that X ( t ) represents the carbonate production in a limestone-marl system. One can then imagine three geologically interesting situations. If hn = A, then the carbonate production will change periodically between A and zero. It can be seen that whenever the carbonate production stops, the sedimentation rate is zero (as shown in Fig. 5-5 centre). If A becomes greater than ho, then sedimentation with low carbonate production actually becomes negative and erosion occurs. In situations where ho c A, the effect of lowering the carbonate production simply involves a lowering of the sedimentation rates. Cycloids provide a very good first approximation for sedimentation in some limestone-marl sequences and they have been successfully used to model sedimentation in the Irish Carboniferous. It was possible to fit cycloids to limestone percentage curves by choosing the parameter ho in such a way that the area of one cycloid cycle agreed with the total limestone percentage of a complete sedimentary cycle. The parameter A was taken in such a way that the maximum of the cycloid became 100% (Schwarzacher, 1967). Schiffelbein and Dorman (1986) have examined the power spectra of cycloids and find, not surprisingly perhaps, that the Fourier spectra contain an increasing number of harmonics, the further the cycloid departs from the pure sine wave. They examine examples with a constant sedimentation rate, equivalent to parameter h in eq, 5-19 and a so-called distortion factor which, under the assumed conditions, is

84

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

equivalent to parameter A in eq. 5-19. By applying distortion in reverse, they try to find the least complex spectrum and in this way the original undistorted signal. The method does not seem to be very practical in a situation where the spectrum contains a number of frequencies which often are unknown. It is also unlikely that the conditions of eq. 5-17 are very realistic. The relationship between the lithology and the sedimentation rate is likely to be non-linear and the transformation from time into space is far more complex. Composite cycles which contain different lithologies, will almost certainly be composed of units which have accumulated at different rates. Kominz and Bond (1990) attempted to deal with this problem in situations where the cycles are known to be periodic and represent a constant time interval, T . If the cycle contains n lithological steps, each of which has a specific accumulation rate, then one can write:

T=Cyjz

(5-20)

i=l

whereby, the y ’ s are the reciprocals of the specific accumulation rates and the z’s are the corresponding thicknesses. In order to find n specific accumulation rates, one needs data from at least n cycles. Kominz and Bond claim that with more cycles, more accurate estimates can be obtained. However, the method makes several assumptions which may often prevent it from being of great practical use and sometimes it may yield some very misleading results. Although the method allows for the fact that the various facies within different cycles occupy different time intervals, it makes no allowance for the fact that the rate estimate depends on the length of the time interval from which it is estimated (see eq. 5-12). In cycle lengths of the order of 100 Ka and with facies which may vary from littoral to basinal, the rate factor can be quite important. The problem of completeness is particularly relevant if some cycles do not contain all the lithological types. The stratagem which was adopted by Kominz and Bond to select only complete cycles, is very questionable as it makes a selection which is possibly atypical. More important than any rate fluctuations within the cycles, are the possible rate fluctuations at the cycles’ boundaries. Indeed, many types of stratification and many types of cyclicity are caused by abrupt changes in sedimentation rates and this frequently includes sedimentation standing still, or even erosion. The situation then, is that sedimentation occurs for some fraction of the time period, but for possibly much longer periods, no sedimentation is recorded. As the time which is occupied by sedimentation is not normally known, no sedimentation rates can be estimated. In most situations, the recognition of such hiatuses and erosional gaps is only possible after a detailed sedimentological examination. This may be difficult if the sediments have been strongly altered by diagenesis. However, hiatuses should always be recognisable by proper sedimentological examination. The criteria for slow

THE RELATION BETWEEN TIME AND SEDIMENT ACCUMULATION

as

or no sedimentation are numerous: the truncation of structures, crusts, borings, the accumulation of fossils, increased bioturbation, to name but a few. Of course, it is often impossible to study long stratigraphic sections using intense sedimentological studies in every detail and other methods of finding discontinuities must be used. Kolmogorov showed that the distribution of preserved sediment lengths which have been interrupted by erosional gaps, has a typical truncated shape (Kolmogorov, 1951). Recognising such distributions could be an indication of incomplete sedimentation and may sometimes permit one to calculate the amount of eroded material (Mizutani and Hattori, 1972). Sedimentation breaks are a result of incompleteness and their importance is therefore a question of scale. If one measures time in fractions of a cycle length, the above examples may be very incomplete but if time is measured in units of cycles, the sequence is complete. In summarising the findings of this section, one comes to the following conclusions. A sediment which is marked at equal time intervals will lead to thickness intervals of approximately equal lengths, but because of the random nature of sediments, this will never lead to exactly equal thickness intervals. Homogeneous conditions within a cycle will lead to normal thickness distributions and the standard deviation of this is a measure of the inaccuracy of the sedimentary record. Non-normal distributions of cycle thicknesses are an indication that the subdivision of time within the cycles is uncertain. A hiatus within a cycle has the same effect but it becomes unimportant if stratigraphic time is measured in terms of the numbers of cycles. The removal of complete cycles can only be discovered either by sedimentological evidence, or by comparing cycle numbers in parallel sections.

THE EFFECT OF RANDOM VARIATIONS ON A PERIODIC SIGNAL

If one assumes that a sedimentary cycle represents a strictly periodic time process, then one can expect at least two types of disturbances to introduce irregularities. The first could be regarded as a superimposed error which can itself be interpreted as an observational error. This error is simply added to the sine wave of the signal. Thus (5-21) The error term E(Z)is either due to the effect of the geologist who observes the section, or it may also be due to mistakes that were made by the sedimentation process itself, which was supposed to record the environmental signal. No matter what particular parameter of a sediment is used to reconstruct an environmental variable, it will never represent that variable precisely, whether an isotope ratio is chosen to measure temperature, or a grain size distribution is chosen to measure environmental energy.

86

CYCLOSTRATIGRAPHYAND THE MILANKOVITCH THEORY

The process X ( z ) of eq. 5-21, is known as a harmonic process and has a line spectrum that is a single spike-like peak which is proportional to the variance of the disturbance (z). The autocorrelation function of such a record is a continuous cosine wave with an amplitude which is reduced from 1:

R(t)=

A2

(A2 + 2a2)

cos W t

(5-22)

a2is again the variance of E ( z ) . If one is dealing exclusively with superimposed errors like this, the frequency of the signal, as well as the variance of the disturbance, can be obtained directly from the cycles’ thicknesses. Unfortunately, this very simple situation is hardly ever realised in practice. The random walk model which was introduced earlier, makes it clear that the fluctuations of sedimentation are invariably incorporated into the stratigraphical record. The effect of such fluctuations on the periodic signal is shown graphically in Fig. 5-6. The distortions which are recorded on top of the observational errors, arise from irregularities in the time scale. The problem can be treated by making use of the theoretical results of problems which are found in communication engineering. A well known method of transmitting information by radio waves for example, is the frequency modulation or FM technique. The communication engineer actually differentiates between two related modulation processes of a signal with variable phase shift +(t>:

+

X ( t ) = Ao COS[WO t + (t)] + ( t ) is the modulation and the signal D ( t ) can either be

Fig. 5-6. The transformation of a signal into the stratigraphiciecord.

(5-23)

THE RELATION BETWEEN TIME AND SEDIMENT ACCUMULATION

@(t) = Dt or: W )=

87 (5-24)

1‘

(5-25)

D(t) dt

A case covered by eq. 5-24 is called phase modulation and case 5-25 is called frequency modulation. If the phase shift representing the time “error” is small compared with the frequency, one can calculate the power spectrum of the modulated wave. According to Middleton (1951) this is:

A2 h(w) =

2

1

00

exp[-D2C(t)]

0

[cos(wo

+ w ) t + cos(w0 - w)] dt

(5-26)

In this equation, D2 is the mean square deviation of the modulating function and C(t) is the covariance of this signal. For our purpose, it is interesting to see what happens to the carrier wave (which is a pure sign wave) when it becomes modulated by random noise. Middleton showed that Gaussian noise transforms the line spectrum into a normal probability curve which is centred on wo and which becomes flatter and flatter, with increasing variance of the noise D2. It follows from eq. 5-24 that the power of the sine wave at wo is reduced to:

4

h(wo) = - exp(-D2)

(5-27)

2n

The covariance of the random noise must disappear when t becomes large. The power spectra of a sine wave with noise modulated time scales are shown in Fig. 5-7. The frequency maximum becomes wider and eventually the spectrum will flatten completely, indicating white noise. Frequency modulation leads to very similar results. The power spectrum of frequency modulated waves differs from a phase modulated spectrum only by a factor of 1/w2. The relevance of frequency modulation becomes obvious when the transformation from time into stratigraphical thickness is regarded as being a random walk: ~ ( t=) (C X(Z)

+ E O ) + (C + + . . . (C + ~ 1 )

(

2

t

E!)

= ct

C i=O

Ei

and (5-28)

= ACOS w o t + w o C E j

It is clear that the summation sign in this equation is equivalent to the integral in eq. 5-25. The noise component in the model enters into the equation as a cumulative error and the process is no longer stationary. Indeed, the process is no longer ergodic and the covariance function of the noise changes with the increasing

88

CYCLOSTRATlGRAPHY AND THE MlLANKOVlTCH THEORY

WO

Fig. 5-7. Power spectra of sine waves recorded with an increasingly irregular timescale.

length of the record. This makes an analytical study practically impossible. However, for a limited sequence length, one can treat the random variable in eq. 5-26, as if it had a zero mean and a variance D. The power remaining in the modulated wave is now frequency dependent. A2

h(wo) = - exp(-D2w2) ( 5 -29) 2n The frequency dependence is illustrated in Fig. 5-8 which represents the result of simulated runs. In the experiment shown, a sign wave has been calculated using a time scale in which the time unit was replaced by a normally distributed number, with the mean equal to one and a standard deviation equal to one. The frequency of the wave was systematically increased and after each run, the power of the frequency maximum was plotted. As one would expect, the graph clearly shows how the power falls off exponentially with increasing frequency. Clearly, the very short waves disappear first in the random noise and consequently contribute less power or become eliminated.

BIOTURBATION

Bioturbation is a disturbance which is present in many sediments. Its effect in the stratigraphical record has been particularly well studied in deep sea sediments and it turns out that it is very similar to the effects of irregular sedimentation, as was discussed in the previous section.

THE RELATION BETWEEN TIME AND SEDIMENT ACCUMULATION

04

5

10

15

20

25

30

89

35

PacyclesJN

Fig. 5-8. Maximum power of sine waves with increasing frequencies recorded on a disturbed timescale.

Bioturbation is essentially a process of mixing the sediment and its effects can be interpreted as a low pass filter which is applied to the original record. A very simple but effective model for this process, was introduced by Berger and Heath (1968). It is assumed that at a certain depth below the surface, the sediment becomes mixed instantaneously. This is called the mixed layer which can be a few centimetres to several tens of centimetres thick. At the boundary between the mixed layer and the historical layer, an exchange of material takes place which follows the laws of diffusion, which causes any change in the new material (the mixed layer) to fade exponentially in the older material. Guinasso and Schint (1975) have elaborated the mixing model by making the mixing of the upper layer time dependent. The distribution of ash after a single volcanic event or the distribution of radioactive tracers, provide in fact an impulse response function for the bioturbation filter which is normally active all the time. It is therefore possible to test the mixing theory and in this way to calibrate the theoretical formulae. A variety of filters has been examined (Goreau, 1980), ranging from the box shaped low pass filter to filters with exponential fall off. The Fourier transform of the impulse response function (eq. 4-26) is the frequency response function and the power spectrum can be divided by the gain function, to find the distribution before filtering occurred. This operation corresponds to the deconvolution of the filtered sequence in the time domain and Dalfes et al. (1984) “reconstructed” a number of spectra of Sl80 data in Pleistocene deep sea cores. It was found that in each case, the higher frequencies around the 21 ka wavelength were attenuated, particularly if the ratio of the mixing depth over the accumulation

90

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

rate, is high. Unfortunately, it is difficult to assess the importance of bioturbation in quantitative terms because both the mixing depth and the diffusion constants are highly variable in different environments. It is also highly likely that the bioturbation itself will change with time (Schiffelbein, 1984). The bioturbation activity will follow such changes, particularly in cyclic sediments where ventilation and food supplies change. Dalfes et a1 rightly point out that the "red noise signal" which is very common in all the spectra of sediment recorded cycles, could largely be explained by bioturbation. Therefore it is no proof of a climatic history which followed such an autocorrelated trend. The fact that a similar spectrum can also originate from irregular sedimentation steps which become incorporated into the time history, as seen in the previous section, also shows however, that bioturbation is not the only mechanism which must be considered. Indeed, it has been known for some time that all stratigraphical sequences have either Markov properties, or higher order autocorrelated structures and that a large number of factors can be responsible for this. The inertia of an environment reacting to external stimulus processes, determines the amount and type of sediment as well as fluctuations in sedimentation rates, bioturbation, diagenesis, compaction and weathering. All of these are not independent random variables and therefore they contribute to the red noise. It is not possible at the present time, to single out one factor like bioturbation as an explanation and attempts to quantify such a hypothetical mechanism, by tuning it to observed data (Fischer and Ripepe, 1991) do not supply any real answers.

METHODS OF FINDING A MAPPING FUNCTION z, t

A major aim of any stratigraphical study is to find a time thickness relationship which can serve as a historical framework for any further study. The time scale which is used, can either be absolute, or relative and perhaps in arbitrary units. The graph of z against t , represents the stratigraphic mapping function and there are several methods which can be used to recover this function. Some of the methods have been developed by Pleistocene stratigraphers using oxygen isotope data, but any of these may also be useful in pre-Pleistocene stratigraphy. The simplest case is a system in which the age of the two positions is known. The two points, together with the two dates, define a straight line, the inclination of which is the sedimentation rate. Any further events in this section can be placed on this straight line and the age of each event is therefore found by linear interpolation. The accuracy of the method can be improved if more than one section is available and if the same events can be recognised in two or more sections. One then uses graphical correlation as introduced by Shaw (1964), to construct a composite section which will eliminate random fluctuations in the sedimentation rates, to some extent. Of course, it cannot remove the changes in sedimentation

THE RELATION BETWEEN TIME AND SEDIMENT ACCUMULATION

91

rates which are common to all the sections. Such composite standard thickness sections have been constructed for the Pleistocene by Prell et al. (1986). The main use of this type of standard is probably to discover abnormalities in sections such as, for example, artefacts which were introduced during coring. More detailed mapping functions can be obtained by tuning the section to either a known time standard like orbital variations for example, or to some other possibly unknown time units. In its simplest form, tuning is achieved by assuming that a sedimentary cycle does indeed represent a constant time interval. A stratigraphic variable can then be plotted on a scale which has been adjusted to give all the cycles equal thicknesses. Equally spaced values can be found by interpolation (cf. Schwarzacher, 1989). In the sections where a cyclicity may not be immediately obvious, the tuning can be done in stages. In the first stage, a few tie points between the section to be tuned and a reference section are chosen. The section is then stretched to bring the tie point into line. This often makes it easier to choose additional tie points and the tuning is done in this way, stage by stage. The method was applied very successfully by ten Kate and Sprenger (1992) to a variety of Cretaceous sections. A very elegant method of tuning has been developed by Martinson et al. (1982). The Martinson method involves finding a mapping function m ( z ) which relates a stratigraphical sequence of a variable X (z) such as isotope values, for example, to the values in a reference section (see Fig. 5-9). The aim of the mapping function is to make a particular record as similar to the reference section as possible and this is achieved by alternately squeezing and stretching the sequence. The mapping function can be approximated by a truncated Fourier series, containing up to thirty coefficients. The first coefficient in this series is given by the inclination of the straight line, which determines the amount of overall expansion or shortening of the tested series. This coefficient c a r b e based on a guess. All further coefficients are

t"

Reference section

Fig. 5-9. The Martinson et al. (1982) method of finding a mapping function.

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CYCLOSTRATIGRAPHYAND THE MILANKOVITCH THEORY

supplied by an iterative procedure which determines the best correlation between the two series, for each order of Fourier coefficient. Very complicated mapping functions can be constructed because of the orthogonality of the trigonometric approximation. The method is particularly useful for tuning a section to an astronomical time series, or in order to calculate insolation curves. In the case of the Pleistocene examples, an absolute time scale can be used and in this way, it has been possible to demonstrate fluctuating sedimentation rates on a very detailed scale (Martinson et al., 1987). If a large number of correlated and length adjusted sections are available, they can be averaged or “stacked” by superimposing them, after they have been reduced to zero mean and unit standard deviation (Imbrie et al., 1984).

93 Chapter 6

STRATIFICATION AND STRATIFICATION CYCLES

Stratification or bed formation, is one of the most characteristic features of any type of sediment and it is particularly useful in the study of sedimentary cycles. Many Milankovitch cycles have only been recognised because of the rhythmic changes in the pattern of stratification and a study of the vertical sequence of beds is therefore very important. Cycles consisting of groups of repeated beds have been called stratification cycles (Schwarzacher, 1987), or using a looser but more descriptive terminology, bundles (Schwarzacher, 1952). Unfortunately, beds are not easily defined. The definition must rely on the existence of bedding planes which form as the lower and upper limit of the bed. Bedding planes become visible because of some compositional variation in the sediment. However, it is generally accepted that the sediment within a bed can also vary; indeed the very common graded bed consists of continuously changing sediment. The bedding plane then, must represent a more intense or abrupt change in sedimentation, compared with the variation between the bedding planes. Consequently, the definition of beds depends on a relative variation in sedimentation and it is therefore possible that different facies react quite differently when beds are formed. Diagenesis may strongly change and accentuate bedding but there are very few geologists left who believe that bedding can develop spontaneously by diagenesis alone. Weathering of natural and artificial outcrops provides an added complication. Most sediments react differently to the various atmospheric agents and different bedding planes can develop depending on the duration and intensity of their exposure to them. Finally, it also depends on how observations are collected. Counting bedding planes from aerial photographs, or measuring a quarry face, or examining peels with a microscope, will yield a different number of bedding planes for the same stratigraphic interval. Despite these somewhat contradictory properties, beds are one of the most useful concepts in geology because beds constitute stratigraphic units which can be seen and easily recognised in the field. As was discussed in Chapter 4,the ideal stratigraphic time sequence should be based preferably on a continuous variable which has high environmental significance and which can be interpreted as being a continuous record of the stratigraphic history. The isotope data of Pleistocene sediments are a good example of such a variable. Unfortunately, good stratigraphic data like this are expensive, both to collect and process. For instance, a very simple method of obtaining quantitative data on a carbonate sequence, is the determination of insoluble residues. Suppose

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CYCLOSTRATIGRAPHYAND THE MILANKOVITCH THEORY

one is dealing with long sections representing let us say, 100 Ma, and one would like to resolve the 20 ka cycle for example. Then a reasonable interval to sample in the sequence would be 5 ka. This would necessitate two million analyses, which would be prohibitive for most geologists, both in expense and time. Beds on the other hand, very often appear to represent time ,intervals of between one to several tens of ka. If they have distinctive lithologies, one might hope that measuring the thicknesses of the individual beds could give some indication of the environmental conditions. Marl-limestone sequences are a particularly common example and will be used to study the bedding problem more closely in the following paragraphs.

MARL-LIMESTONE SEDIMENTATION

Sequences of alternating limestone and marl are known from the Cambrian to the present. A marl bed followed by a limestone bed is referred to as a marllimestone couplet and such a two bed combination forms a simple cycle. The relative proportions of limestone to marl can be extremely variable but in general, limestone predominates over marl and the latter is considerably more compacted. Some of the pioneer work on limestone-marl couplets was done by Seibold (1952) who examined the Jurassic limestones in Southern Germany. Seibold recognised one of the basic problems which has been discussed many times since then. This is the question as to which of the two components (clay or carbonate) has changed. Has the carbonate component remained constant and the clay influx fluctuated? Or has the clay fraction remained constant and the carbonate production fluctuated? The clay fraction in the last instance, must be terrestrial or volcanic and the carbonate is most likely to be related to bioproduction. Later, oceanographic work identified a third variable in the problem, the removal of carbonate by solution on the sea floor, which is called dissolution. Seibold identified yet another aspect which is important in understanding the problem of stratification. In the limestone-marl sequences which he examined, the carbonate percentage was a relatively continuous variable. However, at a critical calcium carbonate content of approximately 70%, the sediment changed its outcrop appearance from a soft back weathering marl, into a hard prominent bed which would be classified as limestone under field conditions. Not only does the precise carbonate content of what defines a limestone bed in the field vary with different degrees of weathering, but it also depends to a certain extent, on the superimposed and underlying lithologies. Seibold found that in the marl-rich parts of a sequence, a lower carbonate content was sufficient to be classified as limestone. Einsele and Ricken (1991) claim that the critical carbonate content for qualifying as a limestone bed, is between 65% and 85% in Europe and North America. It is clear therefore, that lithological logs which are based on field observations, have only a limited accuracy. This is less important in the analysis of single sections which are

STRATIFICATION AND STRATIFICATION CYCLES

95

measured under similar conditions in one locality, but it may become significant when comparing different sections from different localities. It is also important that Seibold’s observations suggest that the dependence of a lithological identification on its adjoining lithology, will introduce autocorrelation into these time series. The important problem of finding the mechanism which changes the relative proportions of the lime and clay fractions, cannot be solved by using the results of chemical analyses alone. Models for the various types of sedimentation (Arthur et al., 1984; Einsele and Ricken, 1991) have to be based on either fixed sedimentation rates or fixed time intervals. Under such assumptions the “models” simply state the obvious. Additional evidence must be used to solve the problem, such as direct observations on the dissolution of skeletal particles or the concentration of various geochemical indicators (Arthur and Dean, 1991). By far the most convincing evidence can be obtained from regional studies. For instance, when the marl beds systematically increase in the direction of palaeogeographically known land, then it is at the very least, likely that one is dealing with pulses of land derived material (Schwarzacher, 1968; Hattin, 1986). A specific example of such a situation will be discussed in Chapter 7. An added complication arises from the different behaviour of the marl and carbonate parts of the cycle during diagenesis. It is generally accepted that clayrich sediments undergo a much greater compaction than limestones. Indeed, some carbonate petrologists have claimed that limestones frequently show no compaction at all. Unfortunately, the estimation of compaction is often very difficult and requires the presence of distorted shapes which behave mechanically in a way that is identical to the rock surrounding them. For example, burrows filled with the same material as that surrounding them can be used but not burrows filled with a different material. For example, some burrows may have contained only water when deformation occurred. The often quoted shaliness of marls with their much better developed orientation of flat particles in the bedding plane, is not necessarily due to compaction and may in some cases, be the effect of different sedimentation conditions. That such changes occur, is sometimes indicated by the development of laminations towards the top of the limestone beds and by the trace fossils which are often quite different in the shales, compared to those in the limestones. However, one cannot generalise. There may be many different types of limestone-clay cycles and whenever possible, a detailed petrographic examination should be a part of any cyclostratigraphic study. Further diagenetic changes can follow compaction and it has been postulated frequently that there is an appreciable movement of carbonate material into the “limestone” part of the cycle. Such a process would clearly alter the thicknesses of the two lithologies and one has to consider this, when any attempt is being made to estimate the time intervals that are represented by the two layers. Such an estimate will always be difficult, if not impossible. Sometimes however, there are indications that the marl deposition was considerably slower perhaps, than the

96

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

limestone sedimentation. This can be indicated by laminations near the top of the limestone bed, which may become progressively thinner in an upwards direction. Another indicator is the higher fossil content per unit thickness, of a large number of the marl layers. Cycloid like plots of the limestone percentages against the stratigraphic position, would suggest that there was slower sedimentation at the place where the compressed part of an original sine wave falls on to the marl part as was discussed in the previous chapter. The relative amounts of limestone and marl in a sequence, are not necessarily constant and it may happen that the marl content diminishes and that limestone sedimentation takes over. The marl layers become thinner and thinner until they disappear and a bedded limestone results. The bedding planes which replace the marls have no thickness, but they are often developed with a variable degree of “intensity” which is best judged by their lateral persistence. Very strongly developed bedding planes which can be traced over large areas, have been called master bedding planes (Schwarzacher, 1958). Other bedding planes are less well developed and in a single outcrop, they may range from bedding planes which are laterally very persistent, to planes which may disappear within a few metres. Weakly developed bedding planes often disappear into parallel bedding stylolites which can also be laterally persistent over several metres. Bedding planes which have disappeared in one section, very often reappear in another section, in a place corresponding to their previously found position. This can be shown by tracing the persistent master bedding planes (Fig. 6-1). The formation of bedding planes in limestones can have different causes. When it is possible to trace the development of bedded limestones from the original marl-limestone couplets, it must be assumed that the bedding plane corresponds to either a short influx of clay, or to a slowing or perhaps even a cessation of the limestone deposition. Several petrographic criteria are available to suggest such a slowing down of the carbonate sedimentation. Amongst these are trace fossils, shell accumulations which often have an encrusting fauna and in extreme cases, emergence with soil or palaeokarst formation. Some geologists (Arthur, 1979; Bathurst, 1991; Ricken, 1986) have argued that bedded limestones without visible

M Master bedding plane st stylolites

Fig. 6-1.The variable lateral persistency of bedding planes.

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STRATIFICATION AND STRATIFICATION CYCLES

marl bands, are the product of diagenesis, the lime having moved from the marl beds into the limestones and leaving only traces of clay. This process which is difficult to prove or disprove, would lead to a considerable compaction of the marl beds. It would also indicate that the limestone beds may represent only a very short time interval and that most of the time interval is taken up by the now missing, marl beds. The effect of this hypothesis is therefore the same as the assumption of sedimentation pauses, which in many cases is the more reasonable model from the petrographic evidence. STRATIFICATION PAlTERNS

A number of patterns which occur in bedded sequences, can be best understood by assuming that the development of a bedding plane is the response to a signal with a variable intensity. This is indicated diagrammatically in Fig. 6-2. On the left of Fig. 6-2, it is assumed that a bedding plane is formed whenever the signal, for example clay content, reaches a maximum above a certain threshold level. Such a threshold may well be determined by the local weathering conditions, but it may also be influenced by a mechanism of sedimentation which as yet, is unspecified. The diagram on the right of Fig. 6-2, demonstrates how the same signal can give rise to quite different responses, depending on the thresholds. In this case it is assumed that the variance of the signal increases with its intensity. Such an increase of variability for instance, could be the result of unstable limestone production when the clay content becomes high. A similar effect could be obtained if the

Signal T3

Tz T I

Sienal

Response R3

Rz

Ri

Response

Tz TI

.. ... .

R2

R1

.. .... . .._.__- - - __.--

T,. T2, T3 are different thresholds

Fig. 6-2. The relationship between the signal and the formation of bedding planes. It is assumed that a signal must reach a certain threshold before a bedding plane is formed. On the left hand side, a modulated signal with three different threshold levels is shown and at the right a similar signal with noise is illustrated.

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CYCLOSTRATIGRAPHYAND THE MILANKOVITCH THEORY

thresholds which are responsible for the formation of bedding planes fluctuate. Both situations lead to an increased number of bedding planes near the maxima and if the fluctuations of the signal are in part due to noise, then the position of the bedding planes no longer represents the true maximum of the signal and additional beds are generated at and near the maximum. In the example shown in Fig. 6-2, the signal is chosen in such a way that all the maxima are equally spaced but the intensity of the signal is strongest every fifth maximum. If such a strong maximum generates a more persistent bedding plane, then a bundle of five beds is formed. It is easily seen that the thicknesses of the beds which can be formed by the different thresholds, correspond to either the short cycle, or to three times the short cycle, or to five times the short cycle. If the signal is not as symmetrical as shown in the diagram, then beds with twice the unit thicknesses can also be formed. It is also evident from the diagrams that in the presence of noise, beds can be formed which are not due to the signal. If the amplitude of the noise is smaller than that of the signal, one would expect to find evidence of noise near the maxima which is where the system is most sensitive. It has been seen in the previous chapter that individual beds which represent equal time intervals and which are the product of a large number of random steps of sedimentation, will tend to have a normal thickness distribution. The thickness distribution of a sequence in which the fusing of beds takes place is a gamma distribution, as long as the missing of bedding planes which causes the fusion, is strictly a random event (Schwarzacher, 1975). Such a distribution can vary from the extremely skewed negative exponential to the symmetrical normal distribution, depending on the statistical parameters. If the variance of the beds is not too high, polymodal distributions result. The highest mode will correspond to the basic bed thickness mean, with further modes at two or more times this value which represents the fusing of two or more beds. Figure 6-3 gives an example of a bed thickness

4 b

Fig. 6-3. Bed thickness distribution of beds in the Cenomanian Scaglia bianca (near Gubbio, Italy).

STRATIFICATION AND STRATIFICATION CYCLES

99

distribution for beds measured in a Cenomanian section (Scaglia bianca Contessa highway section near Gubbio, Italy). The sequence, will be described in detail later, is well bedded and some beds show alternating marl and limestone while in other cases, the bedding plane only contains traces of clay. Any marl thickness has been added to the overlying limestone. The frequency curve gives the distribution of beds which were measured to an accuracy of one centimetre and the curve shows two maxima at 8 cm and 14 cm. The second maximum which is approximately twice the first maximum, is most likely to be the result of fusing two beds. One might suspect a third maximum at 24 cm, but the evidence for this is not convincing. For reasons which will be discussed later, the higher order maxima are not exact multiples of the 8 cm maximum. Nevertheless, the distribution definitely establishes that the mean thickness of the basic bed is 8 cm. STRATIFICATION CYCLES

If similar stratification patterns are repeated with some regularity, they can be called stratification cycles. The similarity of the groups can either be the similarity of a repeated lithological sequence, or of some repeated pattern in bed thickness, or simply a group consisting of a constant number of beds. In the limestone-marl facies, the best developed examples consist of a basal marl which is often quite thick and which is followed by several limestone beds that are separated by relatively thin marl bands. The transition into the more predominant limestone facies is often gradual, with the limestone beds increasing in thickness and the marl layers decreasing in thickness. The top boundaries of such cycles is often quite sharp and ends with the most massive limestones. Stratification cycles can change regionally from marl-rich facies into almost pure limestone successions, in a way that is similar to that seen in the simple marl-limestone sequence which was described earlier. Very often, one finds that the bedding plane corresponding to the original basal marl layer, is developed much more strongly and is laterally more persistent as a master bedding plane. The thickening upwards of the limestone beds is often preserved and so helps in identifying the cycle boundaries. However, the more central bedding planes of a cycle can be less intensely developed and this leads to a fusing of the central beds which will mask the trend of the beds thickening progressively in an upwards direction. Such irregularities are the reason for the mean thicknesses of beds being smaller than expected in the Scaglia bianca, which was discussed earlier. The fusion of beds is one of the difficulties which always arises when interpreting the distribution of bed thicknesses. The splitting of beds is a second factor which is just as disturbing. There are two types of bed splitting which can be of different significance. The first is simply the counterpart of bed fusion. This means that a bedding plane reappears in its expected position in another section. The second

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CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

type happens when additional bedding planes are formed by randomly distributed disturbances of the signal and by the variable reaction of the section to these variations. Experience has shown that the formation of additional bedding planes is most likely to occur near the expected position of the bedding planes. Very often, this is near master bedding planes. Once again, this counteracts the tendency of the cycles to thicken in an upwards direction and it can make the recognition of cycle boundaries very difficult indeed. A more detailed picture of stratification cycles can be obtained by considering the stratigraphical position of beds in relation to the cycle boundaries which can be traced over larger areas. Master bedding planes which form the boundaries of stratification cycles, are ideally suited to this. The position of a bedding plane in such a case, is expressed as a fraction of the bundle thickness which itself, is set to unity. This procedure largely eliminates the changes of bed thickness which are due to variations in sedimentation rates and according to our model, the frequency with which a bedding plane occurs is determined by the signal strength and the thresholds. As an example, the positions of bedding planes which were measured in very well correlated stratification cycles from the Carboniferous of North-West Ireland, are given in Fig. 6-4 (Schwarzacher, 1982). In this diagram, positions 0 and 1.0 represent the bundle boundaries which are formed by very persistent master bedding planes. The positions of subsidiary bedding planes within this bundle are variable. It is impossible to predict the positions of the bedding planes accurately from a knowledge of the cycle boundaries, but it is possible to give the probability of a bedding plane being encountered in any particular position and it is therefore possible to reconstruct an average cycle. This has been done in Fig. 6-4, where the positions of the most likely occurrences of bedding planes are shown. Positions which have a relatively low probability have been indicated by dashed lines in the diagram. The final characteristic of stratification cycles to be considered is the constant number of beds per cycle. In many marl-limestone successions and limestone-

I--

Reconstructed cycle 1 .o

0.5

____-

0.0 -0.0

2

4

8

8 10 12 14 18 18 Frequency

Fig. 6-4. The positions of bedding planes in stratification cycles of a Carboniferous limestone (Glencar limestone Co. Sligo, Ireland).

STRATIFICATION AND STRATIFICATION CYCLES

101

Number of beds in Triassic Lofer Cycles

Fig. 6-5.Frequency of bedding planes in Triassic limestone stratification cycles (Lofer near Salzburg, Austria).

dolomite sequences, one finds approximately five almost identical beds per cycle or bundle. Such an arrangement is much more indicative of Milankovitch cyclicity than any other form of regularity. The thickening of beds in an upwards direction, or a gradual change in the clay fraction could be explained by sedimentary processes like the gradual changes in a basin, or the increasing morphological maturity of a supply area. However, it is difficult to imagine that any such process must be repeated five times before anything else can happen. By far the easiest explanation is that such cycles are controlled by a time periodic process which would argue for astronomical control (Schwarzacher, 1947). Unfortunately, the vagaries of bed formation can make this criterion just as uncertain as any other indicator that relies exclusively on the evidence provided by bedding. Frequency plots of the number of beds in stratification cycles sometimes give useful information. Figure 6-5 gives the number of beds found in the Triassic limestone-dolomite cycles. The twofold maximum is very noticeable at five and three beds per cycle. The latter is caused by the two inner beds of the bundle becoming fused and reducing the normal five beds per bundle to three.

THE NUMERICAL DESCRIPTION OF BEDDED SEQUENCES

The aim of a numerical description is to replace the raw stratigraphic data by numerical values which are equally spaced and are geologically meaningful. The various methods for obtaining numerical data are illustrated in Fig. 6-6. Since the original data consist of only two states which are marl and limestone, they can be coded with plus or minus one (which transforms the section into a square wave). Alternatively, data can be obtained by measuring the thicknesses of the two lithologies and these can be expressed as a percentage of a predetermined interval.

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CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

cm

5 1 1

L-L-c--(

d

Bed thickness in cm 20

10

0

4

3

2

:~~ 0 1

2

1

om

Fig. 6-6. Various methods of representing a marl-limestone sequence.

In the example of Fig. 6-6b, an interval of 20 cm has been chosen for the section. The first method has the advantage that the interval which is to be coded can be taken as small as the accuracy of the original measurements allows and no loss of information occurs in this way. Square waves of this type are ideally suited to Walsh spectral analysis. The disadvantage of the second method is that the interval for which the proportion of either of the lithologies is calculated, has to be large enough to contain at least one marl and one limestone layer to become meaningful and this leads to some loss of information. The advantage is that the alternating layers are transformed into a continuous variable which can take any value between zero and one hundred. It is sometimes more easily interpreted as such and is suitable for spectral analysis that is based on trigonometric functions. Considerably less information is available when the marl beds have been reduced to bedding planes, or when the sequence contains only one lithology which is interrupted by bedding planes that make it into a series of events. Basically, there are again two ways of looking at such data. One can either record the thickness bed by bed, or one can record the density with which the bedding planes appear in some unit length. The resolution of the second method is limited by the unit interval which is used for counting the bedding planes. The bed by bed thickness data can be displayed in two ways. Either the already familiar cumulative curve can be used, where the horizontal axis gives the consecutive bed numbers and the vertical axis gives the stratigraphic position of the bed. Or bed thicknesses can be plotted against consecutive bed numbers (as shown in

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103

Fig. 6-6d). The cumulative curve has the advantage that the stratigraphic position of every bed is immediately available. Rapid variations of bed thicknesses, however, are not easily seen in the cumulative curve. Plotting by bed numbers, is better suited to showing rapid variations of bed thickness but the method makes it very difficult to locate individual beds in terms of their stratigraphic position. Their location can only be found by going back to the cumulative plot. If the beds are clearly defined in a particular section, then there is no difficulty. In many situations where the beds are not always clearly defined, different observers may often obtain different results. The whole record may be distorted because of the difficulty in recognising the beds and by the different interpretations that can be given by the different observers. The alternative is to plot the bed thicknesses along their stratigraphic position. Normally such a plot that gives the bed thicknesses at discrete intervals along the section, will involve a loss of information. This can be avoided if the points are taken at very close intervals. If a section, for instance, has been measured to the nearest centimetre one can choose the units along the x axis to be 1 cm and the section therefore can be recorded precisely. A record of this type will approach a square wave, each bed being represented by a rectangle where both its height and its width are proportional to the thickness of the bed. This is shown in Fig. 6-6b. Sections using this representation imitate to a certain extent, the appearance of a weathered outcrop, where the thicker and more prominent beds protrude more than the thin beds. The question as to which method to adopt, involves some difficult problems. The ultimate aim is, as ever, to obtain a time record of conditions during the period in which the sediment was formed. If there is any evidence that beds, or indeed more complete cycles represent time steps, then the choice is clear and the bed number can be used as a time unit. The thickness of a bed in this case is not determined by time but by the sediment supply within this time and it is therefore an important environmental indicator. If the bed thickness is determined primarily by the time span between two bedding plane events, then the rate of sedimentation must have been constant and therefore the bed thickness is not an environmental indicator, although the frequency with which bedding planes occur, still could be. The two possible extremes of constant time with variable sedimentation rates contrasted with variable time and constant sedimentation rates, are both situations which can be interpreted when they are recognised. It is, however, very possible that both factors are variable and in this case a clear interpretation is impossible. This is particularly the case when bed formation is determined by several threshold levels and when different orders of bedding result, that are difficult to recognise and are easily confused. As it is necessary to choose either bed numbers or stratigraphic position as a substitute time scale, one could very well choose whichever scale gives the best results for a particular purpose. However, if bed numbers are used, it is extremely important to make the stratigraphic position of all the beds easily available. This is best done by a cumulative curve or tabulation.

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CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

STRATIGRAPHIC TRENDS

A further understanding of the processes which determine the thicknesses of beds and cycles in general can be gained by a more detailed analysis of stratigraphical trends. The trend is a somewhat abstract concept. The term implies that it describes slow changes that are free from any apparent random fluctuation. However, this does not mean that random processes play no part at all in generating a stratigraphic trend. For instance, the random walk model of sedimentation assumes that random pulses of sediment arrive at possibly irregular times and that the path of the endpoints (which represents the accumulating sediment) develops a linear trend. The linearity of such a trend is due to the assumption that all the conditions have remained constant. If it is possible on geological grounds to identify the factors which determine the trend which might be constant sediment production or supply, or constant subsidence or water depth, then one can subtract the trend and examine the residue. Ikends are normally fitted to stratigraphic variables which are given as either a function of time or of stratigraphic position. If cycles are known to represent equal time intervals, then the cumulative plot of cycle thicknesses against cycle number represents the stratigraphic mapping function. Assuming that this function is approximated by a straight line, then deviations from it can be plotted and such graphs are known as Fischer plots. The rationale behind such constructions is that the straight line accumulation curve represents, the compensation of a mean subsidence. If overall conditions have remained constant, then the deviations represent changes in water depth. As with any trend removal, the correlation structure of the original data is changed in such operations. This is particularly the case, when trend is removed from cumulative data and strong serial correlation is introduced. The quality of the residual data very much depends on how well the trend function that has been removed, is represented by an actual physical process. The best results can be obtained from relatively long sections and by using least square methods for determining the trend. The method of simply connecting the beginning and end of a cumulative curve by a straight line which was used by Fischer (1964) should not be used because it does not allow any quantitative assessment to be made about how well the trends represents the data. There is no reason why one should not experiment with trend that are other than linear. Many geological processes, such as a constantly diminishing sediment supply or sedimentation in shallowing basins, lead to exponential trends and such trends often become obvious if a constant trend is subtracted (Schwarzacher, 1966,1975). The stratigraphic trend analysis can be interpreted on two levels. When constructing a cumulative curve, one substitutes a bed or a cycle as an estimate for a time unit. When such a substitution leads to a simple and geologically reasonable trend, one may take this in turn as proof that the choice of the unit was a good one and one may conclude that the bed or cycle does indeed represent

STRATIFICATION AND STRATIFICATION CYCLES

105

a time unit. Such arguments are best used when comparing different time scales (Schwarzacher, 1975). They could be for example, scales which are based on single beds, stratification cycles, or even groups of cycles. THE RECOGNITION OF STRATIFICATION CYCLE BOUNDARIES

There is no difficulty in mapping individual cycles in marl-limestone sequences, where the groups of beds forming a stratification cycle are clearly separated into distinct bundles by thick layers of marl. When such sequences degenerate into bedded limestone successions which have no marl interbeds, the recognition of cycles and in particular their boundaries, becomes very difficult. A numerically coded sequence of thickness values may still give clear results from a power spectral analysis, but this only gives the number of cycles which best fits a given interval. Spectral analysis is a statistical method and it does not give the beginning and end of each cycle, nor does it give any indication of where deviations from the ideal sine wave occur in the investigated interval. For relative dating and in particular stratigraphic correlation, one would like to know precisely how many cycles a particular sequence contains and one would be equally interested in any irregularities which could be used to recognise such cycles in different localities. Furthermore, comparing the number of cycles in identical stratigraphic intervals and in different localities, is the only way of testing the completeness of the section, on the scale of cycles. To obtain cycle boundaries in difficult situations, one can attempt different coding methods and combine the measured data with some of the less objective field observations. It is possible for example, to give measured thicknesses different weights according to whether bedding planes were well or poorly developed. Such thickness indices can be further improved by filtering techniques which deliberately accentuate their variation in the frequency band of the suspected cycles. It is also possible to combine the thicknesses with any semiquantitative observation which was made during field work, for example colour changes. Sometimes stratification cycles are still recognisable in field exposures, when the cyclicity is no longer particularly obvious in the numerical data. The reason for this is very often that measurements are taken along a single line across the section, whereas field observations in well exposed areas, include a very much wider area. The recognition of cycles in the field is often based on a distinct impression of the bundling of strata, which may not be obvious when one is only looking at the line of measurements. Such field classifications of cycles are clearly less objective, but should not be ignored in cyclostratigraphic work. One can use such data for comparing different localities and for comparing them with numerically derived data and if at all possible, with data from different observers. If such comparisons make sense and are consistent, then there is no reason to ignore them.

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107 Chapter 7

EXAMPLES FROM THE CARBONIFEROUS

The second part of this book will deal with specific examples of cyclic sedimentation and their cyclostratigraphic potential. The aim is not to give a comprehensive survey, but rather to choose examples which illustrate various aspects of cyclostratigraphic problems. Of prime importance will be the various arguments that can be used to demonstrate that the sedimentary cycles are Milankovitch cycles. This is particularly difficult for cycles from older formations, where absolute timing is much less accurate. It is also of interest to include cycles from different environments and especially from different environments of the same age. Too little is known about prePleistocene times for an accurate reconstruction of climatic conditions to be made, but it is known that most of the examples to be discussed, occurred during periods without major glaciations and therefore the mechanism of climatic variation must have been different from that of the Pleistocene. It is also hoped that the examples can be of value to stratigraphers and sedimentologists and that perhaps a more systematic review of known Milankovitch cyclicity in different formations will eventually lead to some valuable results.

THE LOWER CARBONIFEROUS

Although the Carboniferous has been for many years the classical period for the study of cyclic sedimentation, direct evidence for Milankovitch cyclicity is poor. The reason for this is clearly the lack of chronometric dates for the individual stages in which cycles have been investigated. This difficulty is common to all the older formations and unless several orders of Milankovitch cycles can be found, the identification of the cycles will always be uncertain. A special point of interest, however, in studying the Carboniferous examples, is the wide variety of different environments in which cycles have been observed. The examples from the Lower Carboniferous (Mississippian) in the northwest of Ireland represent a hemipelagic environment of more or less continuous sedimentation. In contrast to this, the contemporaneous Carboniferous limestone of northern England developed on shallow carbonate platforms, where sedimentation was frequently interrupted and the record is therefore incomplete. The well-known

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CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

Pennsylvanian cyclothems of North America and possibly some of the European coal measures, represent a very wide spectrum of facies and clearly indicate that sea level fluctuations must have played an important part in cycle formation.

THE CARBONIFEROUS LIMESTONES OF NORTH-WEST IRELAND

An extensive study of cyclic sedimentation was carried out in the Lower Carboniferous of north-west Ireland in the counties of Sligo and Leitrim (Schwarzacher, 1964). Examples from this area will be treated in considerable detail to demonstrate how petrographical analysis can be used to determine the cyclic changes, which may have been controlled ultimately by orbital variation. The Visean epoch of the West European Carboniferous has been divided into five stages from the Chadian to the Brigantian (George et al., 1976). Unfortunately, this system is not yet well enough established in the Irish exposures for precise dating. The local stratigraphic divisions which are based on lithologies, indicate a major transgression which is followed by a regression (Fig. 7-1). The basal sandstone that has been recognised as being partly deltaic and partly tidal, is followed by a marl which contains thin beds of limestone and which is called Benbulbin shale.

Harland et al 1989 Ma

332.9 Ma bl

Brigantian

15-

Ramsbottom 1977

-

N.Walas

Sligo

Settle

-7-- - Shgo T N George eta1 1976

Cyciostrat. L-p-p-

Asbian

10

-

Holkerian

I

D5a

j

-

\

middle

-

(GI Girvanella band

hndian

5-

D. septosa bands

-

Meenymore formatn Daltry Imst.

n Glencar imst.

:hadian

U

Benbulbin shale

0-

Mullaghmore sit 349.5 Ma bp

349.5 Ma bp

Fig. 7-1. The Lower Carboniferous stratigraphy of Irish and British examples.

EXAMPLES FROM THE CARBONIFEROUS

109

This is followed by approximately 100 m of Glencar limestone and 200 m of Dartry limestone. The sequence is overlain by the Meenymore Formation, which is a very shallow water deposit and supratidal in parts. The basal sandstone is thought to be late Arundian or early Holkerian and the Meenymore Formation is almost certainly Brigantian. The sediments between these two are thought to be largely Asbian in age and therefore the whole sequence represents a time interval of about 7 to 9 Ma. The palaeogeographical boundaries of the area were well-defined by structural north-eastern and south-western trends which determined the shape of the original basin. The Sligo syncline formed a small embayment which was probably closed to the north-east and open towards the south-west. The exclusively marine sequence from the Benbulbin shale to the Dartry limestone, shows from the bottom upwards a continuously decreasing marl content and an increasing carbonate content. The term marl is used loosely as a field term to describe a lithology which has approximately 30% insoluble residue. This value is an average and values as high as 50% have been found. A grouping of beds into stratification cycles is present throughout the sequence although it becomes very indistinct in some intervals.

THE BENBULBIN SHALE AND GLENCAR LIMESTONE

The best stratification cycles are found in the middle Glencar limestone. Particularly in the north-east part of the basin, limestone beds occur in well-defined groups of frequently five well-developed beds, which are separated from the next group by a thick marl layer (Fig. 7-2). The limestone beds are between 10 cm to 15 cm thick and are separated from each other by 2 cm to 5 cm thick marls. Only the basal marl is considerably thicker, usually between 40 cm to 60 cm thick. The marls are often well laminated and rich in fossils, predominantly fenestelid bryozoans. Elongated skeletal particles and crinoid stems often show a preferred orientation that is due to currents and some laminae have been truncated by erosion, which possibly indicates the formation of shallow ripples. The limestone in contrast, contains very little internal bedding and only some indistinct layers of slightly coarser fossil material. In the thicker limestone beds there are thin zones with skeletal particles that are aligned roughly parallel and which subdivide the bed without forming clearly defined bedding planes. Such divisions will be called sub-bedding planes. There is no preferred fossil orientation between such horizons and skeletal particles occur in all positions and give the impression of a sediment that has been thoroughly mixed. Much of this is probably due to bioturbation and burrows of the type Diplocraterion are often recognisable. Such burrows originate usually at bedding planes or sub-bedding planes and so they are often filled with a slightly more dolomitic sediment. The interpretation which has been given to the formation of the limestone beds is the following. Each bed consists of between one to several layers which

110

CYCLOSTRATIGRAPHYAND THE MILANKOVITCH THEORY

Fig. 7-2. The Middle Olencar limestone at lake Glencar, county Sligo, Ireland.

have been deposited very rapidly. The internal sub-bedding planes, which have an average spacing of about 5 cm, could be the remains of such layers. At possibly regular time intervals, the sedimentation conditions changed to reduced carbonate sedimentation and increased current activity. The supporting evidence for rapid sedimentation is found in selectively preserved areas, which contain groups of bryozoans and calcisponges which are covered with encrusting foraminifera in their original growth positions. The groups must have been covered rapidly to preserve them. A possible mechanism for such sedimentation is the occasional stirring up of carbonate muds, possibly by storms. If one examines the regional distribution of the marl and limestone thicknesses in a cycle, a very clear pattern emerges (Schwarzacher, 1968). For most of the cycles, the marl thicknesses increase towards the south-west but some cycles, which generally contain less marl, show a marl increase towards the south-east. The limestone thickness increases consistently towards the south-west which must be regarded as having been the more open part of the basin. It is likely but difficult to prove, that a certain part of the carbonate mud was brought into the Sligo bay from the more open part of the basin. The distribution of the marl thicknesses on the other hand, leaves no doubt that the clays are derived from the known land masses corresponding to the present day Donegal mountains in the north-west and the Precambrian Ox mountains in the south-east. Therefore the cyclicity of the Glencar

EXAMPLES FROM THE CARBONIFEROUS

111

limestone is to a large extent, a cyclicity in the supply of terrestrial material since it is impossible to explain the distribution pattern in any other way. It has been claimed that the stratification pattern of the Glencar limestone is of diagenetic origin (Walther, 1982) but such an interpretation bears no relation to any of the observable facts. Indeed the distribution of marl and limestones in this type of cycle is a fairly convincing demonstration that the sometimes proposed migration of carbonate from marls into limestones, could not have taken place. A decalcification of the 50 cm to 80 cm thick marls which is found at the base of the cycles, should clearly have released considerably more carbonate than the decalcification of the 2 cm to 5 cm thick marls that are between the limestone beds that are higher up in the cycle. However, both the amount of cementation and the amount of the insoluble residue of all the limestones is the same, irrespective of their relative positions to the basal marl. Furthermore, the carbonate content of most of the basal marls is only 40%, which is considerably lower than the carbonate content of the marls between the limestone beds, where it is usually around 60 to 70%. In most of the cycles, the marl layers decrease in thickness and the limestone beds increase in thickness in an upward direction. However, this apparent thickening of limestone beds towards the top of the cycle is partly caused by the fusing of two or more beds, a process which can be revealed by tracing individual beds laterally. The south-western exposures in the Sligo syncline have more beds per cycle than the area with higher marl sedimentation in the south-east. This is at first unexpected, as one normally assumes that a lower clay influx would encourage the fusion of beds. The most likely explanation of the phenomenon, however, is that the beds in the south-west exposures contain more sub-bedding planes as a result of rapid sedimentation steps. If the hypothesis that the steps represent storm deposits that are derived from the south-west is true, then it is possible that the layers became individual beds in this direction. The mechanism of bed formation raises an important question about the time that is involved in this process. It is very clear from the petrographic evidence, that a limestone bed in this sequence does not represent sediment which has steadily accumulated and which measures time by its continuous growth. On the other hand, with a few exceptions, one cannot classify the beds as event stratification, since they represent several events in most cases. This complex relationship between the thickness of beds and time, can be studied by computer simulation (Schwarzacher, 1975; 1976). lb obtain a first model, one makes the following assumptions. Sedimentation is discontinuous and sediment arrives in the form of discrete layers which for simplicity, we call storm layers. Such “storms” are randomly distributed in time but they occur with constant density. The amount of material that is deposited by each event is again random but its mean and variance can be estimated from actual measurements. It is further assumed that this process operates over a constant but unknown time interval, representing a bed. Based on these assumptions, distributions of bed thicknesses are calculated and a

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CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

comparison with the observed distributions permits an approximate estimation of the storm densities to be made. In the second stage of the model experiment, the assumption of a constant time interval for each bed is relaxed and distributions for variable time intervals are calculated. The results of such calculations showed that the bed formation of the Glencar limestone beds can be explained by a few randomly distributed sedimentation bursts which occurred over time intervals which were variable and had a coefficient of variation of 50%. If one assumes that the beds represent the 21 ka precession period, then the time error (standard deviation) that is associated with such a unit is 10.5 ka, which is a very large error indeed. The estimate of this variation depends on the assumption that the sedimentation events were randomly distributed. This is very unlikely and the large variation of 50% must be regarded as being an upper limit. A very important factor that contributes to the uncertainty of estimating the time which is represented by a single bed, is the difficulty of defining “beds”. The problem has been discussed in Chapter 6 , where it was pointed out that the formation of a bedding plane often depends on the marl percentage reaching a critical level and that sometimes, several subsidiary bedding planes may develop near this level. This is particularly the case with the Glencar limestone, where each bed contains a number of sub-bedding planes caused by the stepwise sedimentation. Lateral correlation has shown that the cycles can be traced throughout the basin but this is not the case for individual bedding planes (see Fig. 7-3). As was shown in the previous chapter, it is only possible to predict a likely position in which a bedding plane will develop. The diagram that is shown in the previous chapter (Fig. 6-4) is constructed from identical cycles in the middle Glencar limestone which

Fig. 7-3. Selected correlated sections of the middle Glencar limestone.

EXAMPLES FROM THE CARBONIFEROUS

113

were measured in different outcrops. The cycles are reduced to unit thickness and the positions of the bedding planes relative to the cycle boundaries, are given as frequencies. Because of the strong regional variation, the marl-limestone couplets are treated as single beds and this accounts for the well-defined maxima near the base of the cycle. The diagram suggests that the average cycle can be divided into five fairly evenly spaced beds. However, there are always additional bedding planes which increase the number of beds per cycle to between seven and nine. The number depends on the locality and is higher in the south-west part of the syncline, as mentioned earlier. In most of the Glencar limestone cycles, the bed thickness increases towards the top of the cycle and the top limestone is followed immediately by the basal marl of the next cycle. Occasionally, however, the top limestone can be split into several thinner layers and the pronounced asymmetry of the cycles is lost, particularly in the lower and upper parts of the sequence and the cycle boundaries are more difficult to determine. It is unfortunately a feature of many sections, not only in the Carboniferous, that cyclicity is well-developed in some parts of a sequence but very indistinct in other parts and may even disappear completely. This makes the counting of cycles over long stratigraphic intervals very uncertain and the misinterpretation of cycle boundaries is the most likely source of error. A relatively simple method of obtaining a quantitative description of the sequence, is to determine the limestone percentages in equal intervals of the section. In the original work on the Sligo successions (Schwarzacher, 1965), limestone percentages were calculated for every 20 cm of the measured sections. A more direct method can be used by coding the two lithologies, limestone and marl, with plus and minus one. Such a coding can be done if necessary at every centimetre and is therefore a very accurate representation of the sequence. In fact, the coded series with positive limestones and negative marl resembles the original outcrop with its protruding limestones and receding marls. If the coding is performed at very close intervals, then the series may be too detailed for cycles to be recognised and in this case, a filter can be used to remove the very short fluctuations. Walsh filter methods are particularly useful for marl - limestone coded data and they produce results which can be much more easily interpreted than the original series. A composite section through the Benbulbin shale and Glencar limestone (Fig. 7-4) gives an example of a filtered sequence. In this case, the cut-off frequency for the filter was taken at 250 cycles and the 2048 data points were obtained by coding the section at 8.5 cm intervals. The filtered section clearly shows some stretches with very well-developed cyclicity. In others, cycles are quite difficult to recognise. Well-developed cycles are seen particularly clearly in the Glencar limestone at 30 m to 50 m. From 80 m upwards a distinct grouping of about four cycles into a higher-order cycle seems to develop. In attempting to count the cycles, one comes across two difficulties. In some stretches of the section, such as 10 m to 20 m, the cyclicity is hardly developed at all and

114

CYCLOSTRATIGRAPHY AND THE MlLANKOVlTCH THEORY

1

DO

E

1i o

150

130

137 metres

Fig. 7-4. Filtered bed thickness data of the Benbulbin shale and Glencar limestone sequence (County Sligo, Ireland).

in other parts, some fluctuations which look like a cycle which is split into two or more and so counting becomes ambiguous. With different interpretations of the cycles and their boundaries, one will find between 60 to 90 cycles in this part of the section. Estimates of cycle numbers can be improved in two ways. A n approach which has been found very useful, is to interpret " cycle boundaries" in the field, rather than from collected records. Such interpretations are often more reliable if it is possible to see large exposures, where the persistency of certain beds and cycle boundaries can be judged much better than from a single record. Similar results can be obtained by interpreting two or more measured parallel sections simultaneously. A second and more objective approach is to determine the number of cycles in a given interval by spectral analysis. This method, as has been discussed, cannot find cycle boundaries but gives a statistical resuit in the form of a preferred frequency, which can be interpreted as being the average cycle that describes the section. The power spectrum of the combined Benbulbin shale and Glencar limestone section is given in Fig. 7-5. The section is 136 m long and has a strong peak at a wavelength of 209.2 cm. This indicates a total of 65 cycles for the complete interval. Significant peaks are also found at 4533.3 cm, 146.2 cm and 138.7 cm. A further peak at 824.2 cm is not significant when tested against the AR1 model (see Chapter 5 ) but it is nevertheless very clearly defined and can be found in a similar position in sub sections. The ratios of the low frequency maxima divided by the predominant frequency maximum are 21.6 and 3.9 and for the high frequencies, one obtains 0.66 and 0.53. These ratios are very important in the interpretation of the Glencar limestone cycles, since a ratio of 1 to 4 and 1 to 20 could indicate that one is dealing with the 100 ka, 400 ka and 2 Ma eccentricity cycles. Under this hypothesis, the higher frequency peaks would correspond to cycles of 70 ka to 53 ka and there is no direct explanation for such a cyclicity.

EXAMPLES FROM THE CARBONIFEROUS

115

01 E

.-

L

3

1

2

3

4 x

5 cycleslcm

6

7

8

Fig. 7-5. Power spectrum of the unfiltered Benbulbin shale and Glencar limestone section given in Fig. 7-4.

A study of Fig. 7-4 will show that the series cannot be regarded as being stationary and even without analysis, it can be seen that in the lower Glencar limestone cycles are developed more thickly than the cycles towards the middle and top of this sequence. To obtain further data for this change in Sedimentation rates, spectra1,analysis of short subsections were calculated and displayed as a contoured three-dimensional spectrum (see Fig. 7-6). The diagram is based on the spectra from a 20 m wide window which was moved in steps of 2 m along the section. The horizontal scale gives the wavelengths of the cycles and the vertical scale gives the stratigraphical position of the lower margin of the window. In reading this diagram, emphasis should be given to the occurrence of maxima since the general increase of power towards the low frequencies is largely due to the red noise which is part of every stratigraphic sequence. The diagram shows a very clear division into a lower (0-50 m), a middle (50-80 m) and an upper (80-120 m) Glencar limestone. There is a distinct shortening of the predominant cycle lengths in the middle Glencar limestone. This cycle seems to return to its original length in the higher part of the section. The noise and the shortness of the sections make it impossible to get any further information on the low frequency maxima. The resolution decreases with increasing length because of the non-stationary behaviour of the section. On the other hand, long sections are needed to resolve low frequency maxima. This is an unavoidable situation and it cannot be overcome by any more sophisticated method of analysis. Further progress could only be made by making definite assumptions about the sequence which, of necessity, are partly subjective. For example, tuning the section can be done by adjusting the cycle length to a constant thickness and in this way, compensating for changes in sedimentation

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CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

Fig. 7-6. Contoured spectrum of the Benbulbin shale and Glencar limestone section. Note that the horizontal scale gives wavelengths and not frequencies.

rates. In a similar way, complex demodulation needs a predetermined frequency which is to be demodulated (Schwarzacher, 1989). Also one has to make a priori assumptions about the analytical nature of the series, for instance the hypothesis that it is composed of sine waves.

THE DARTRY LIMESTONE

The marl layers in the Dartry limestone are reduced to thin films between the limestone beds and it is therefore impossible to measure marl percentages, as has been done for the lower part of the section. Since no other directly measurable variable was available, bed thicknesses were plotted at regular intervals. A useful series was obtained by recording the thicknesses at 20 cm intervals. The use of bed thickness for obtaining data on the cyclicity relies on there being systematic thickness changes for beds within a cycle. The Dartry limestone is similar to the Glencar limestone and often shows a thickening of the beds towards the top of a cycle. This should result in an asymmetric sawtooth like curve, when the

EXAMPLES FROM THE CARBONIFEROUS

117

thicknesses are plotted over several cycles. Unfortunately, the bed thicknesses again vary a great deal and the cycle boundaries are therefore not always well defined. In the original work on the Dartry limestone (Schwarzacher, 1964), bed thickness values were filtered by a running average of five and this gave twenty to twenty-one cycles with an average thickness of 300 cm. Power spectral analysis of the same data gives three frequency maxima at 1236 cm, 309 cm and 115 cm. All three frequency maxima are significant at 90% if tested against an AR1 model and the strongest maximum at 309 cm would give 22 cycles in the 69 m long section. This result is clearly in good agreement with the earlier findings. The ratios of the main maximum to the other maxima are 4.00 and 0.372; assuming that the main maximum represents the 100 ka eccentricity cycle, then the low frequency maximum fits the 400 ka maximum and the higher frequency would indicate a cycle of 37 ka. The latter happens to be precisely the value which Berger et al. (1989) calculated for the 54 ka obliquity cycle in the Lower Carboniferous. Despite this coincidence, one cannot attach too much importance to such figures because once again, the Dartry limestone sequence is not quite stationary and the cycle periods are average values for the complete section.

THE CYCLOSTRATIGRAPHICINTERPRETATION OF THE SLIGO SEQUENCE

Although the cyclicity of the Sligo sequence clearly falls into the general frequency band of Milankovitch cycles, it is nevertheless necessary to discuss critically, the astronomical origin of the cyclicity. Possibly the strongest argument for astronomical control is the persistence of the cycle pattern, despite the considerable change in lithology and facies in the sequence. The Benbulbin shale, which has only some 15 to 20% limestone, has the same repetitive pattern at a metre scale as the Dartry limestone which has more than 95% limestone beds. The repetition which is proved by significant spectra, indicates a persistent oscillating system which is quite independent of the local facies development and which therefore must be independent from locally changing conditions in the basin. This, together with the fact that the oscillations persist through several million years, excludes any local autocyclic origin. The actual timing of the Sligo cycles is less straightforward unfortunately. This is largely because of the uncertain stratigraphic dating of the sequence, together with the uncertainty of the Lower Carboniferous chronometric scale. The Harland et al. (1989) scale for the upper Dinantian is indicated in Fig. 7-1. Using this scale and the stratigraphic divisions suggested by George et al. (1976), the Glencar and Dartry limestones would fall into the Asbian and represent 3.4 Ma and the Benbulbin shale together with the basal sandstone, would be Holkerian in age and again represent 3.4 Ma. The cyclostratigraphic analysis of the Sligo sequence, excluding the basal sand-

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CYCLOSTRATIG RAP HY AND THE MI LAN KOV ITCH THEORY

stone, found between 80 to 100 well-developed cycles which, if it is assumed that one is dealing with the 100 ka eccentricity cycle, would represent 10 Ma. This is nearly twice the time suggested by the above scheme. In judging this discrepancy, one has to consider the uncertainty of the stratigraphic ages of the lower formations in particular. The classification of George et al. (1976) gives a Holkerian age for the basal Mullaghmore sandstone but Sevastopulo (1981) suggested an Arundian age which would come nearer to the cyclostratigraphicscheme that is based on the hypothesis of 100 ka cycles. The age of the top of the sequence is regarded as being more certain and is generally thought to be in the uppermost Asbian. It is succeeded by the shallow water facies of the Meenymore Formation which is assumed to be at the base of the Brigantian. If this age is accepted, then the Benbulbin shale would coincide with the base of the Arundian. It is also important to consider the uncertainty attached to the ages of the stages. These have been determined by Harland et al. (1989) by chron interpolation based on two tie-points: the base of the Chadian (349.5 Ma) and the top of the Brigantian (332.9 Ma). Chrons have been allocated in the following way: Chadian 2.0, Arundian 1.0, Holkerian 1.5, Asbian 1.5, and Brigantian 2.0. This is clearly arbitrary to some extent. Ramsbottom (1977) divided the Dinantian into eleven major cycles or mesothems. In this scheme, cycle D2a corresponds to the lower Chadian and cycle D6b to the top of the Brigantian. According to Harland’s scale (1989) the average length of such mesothems is 2.2 Ma. Figure 7-1 uses a chronometric scale, which is based on the assumption that the length of Ramsbottom’s mesothems is 2.0 Ma and it can be seen that this leads to a reasonable agreement with the Harland scale. If the Sligo data is plotted using this scale, a very clear picture emerges. The top of the Mullaghmore sandstone coincides with the top of the Arundian (D3). The Benbulbin shale and the lower Glencar limestone that consists of twenty 100 ka cycles are Holkerian. The middle Glencar limestone with again 20 cycles, is lower Asbian. The upper Glencar limestone is upper Asbian (D5b) and the Dartry limestone is lower Brigantian (D6b). The fact that one needs somewhat more time to accommodate the cycles, might suggest that one is dealing with shorter cycles, for example the obliquity induced cycle. However, even if one assumes that this latter cycle was not shorter in the past when compared with present day values, it is still too short and would lead to greater discrepancies with the chronometric scale. The hypothesis that the predominant cycle is caused by the 100 ka eccentricity variation is considerably strengthened by finding sub-maxima for both the 400 ka and 2 Ma periods which are known eccentricity cycles. The maximum found at 3.7 ka in the Dartry limestone could be an obliquity induced period but this is difficult to prove.

EXAMPLES FROM THE CARBONIFEROUS

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THE CYCLICITY O F YORKSHIRE AND NORTH WALES

The repeated facies changes in the Lower Carboniferous of Scotland, England and Wales are due to the widespread transgressions and regressions which make up Ramsbottom’s major cycles and which are closely related to the stages of Carboniferous stratigraphy (George et al., 1976). The mesothems or major cycles can be subdivided by a number of minor cycles or cyclothems which may be used, at least locally, as stratigraphic markers. Cyclothems like this, have been known for a long time in northern England and also in Scotland where alternating sand, shale, and limestone sequences are known as the Yoredale facies and where individual members of the cyclothems have been used as stratigraphic markers. However, minor cycles that are equivalent to the Yoredale cyclothems, were also developed as platform limestones which were deposited on the shelves of Yorkshire, Derbyshire and North Wales. Shallow shelf areas surrounding the deeper basin, which was situated in the Craven lowlands, were only flooded during the later stage of the Dinantian and in Asbian and Brigantian times. Systematic studies of the cycles were carried out in the Settle district of Yorkshire (Schwarzacher, 1958) and in North Wales (Somerville, 1979). The Asbian cycles of the Great Scar limestone in Yorkshire, consist of massive, light grey limestones which are 9-15 m in thickness and these are separated from each other by a 50-100 cm thick layer, which can be slightly darker in colour and weathers more easily. The cycle boundary is determined by a sharp bedding plane. In most of the surface exposures, the bedding plane appears simply as a gap between the massive limestones but Waltham (1971) could demonstrate that a number of these cycle boundaries represent palaeokarst surfaces with traces of shale and in some cases, plant remains. The predominant lithologies of the limestones are biosparites and biomikrites. Skeletal particles often show intense micritization and in this way, they may become indistinguishable from the matrix. Counts of organic material in thin sections showed that micritization increases towards the cycle boundaries. The limestones are seen to be vaguely laminated towards the top of some of the cycles, showing current bedding and occasionally, a preferred orientation of crinoid stems and other elongated fossils. Each cyclothem contains a number of bedding planes which are less persistent than the cycle boundaries. Nevertheless, these bedding planes appear in the same position in different localities, where individual cycles can sometimes be recognised by the pattern that is formed by these bedding planes. The cyclothems of North Wales are slightly more differentiated. A typical cycle consists of calcareous marls which are followed by even or wavy bedded, dark grey biosparites, and then thickly bedded pale grey limestone. The latter makes up the main part of the cycle. Somerville (1979) found very clear evidence that each cycle represents a marine transgression, followed by an actual emersion. The criteria for an emersion are paleocarst surfaces, K-bentonites (representing palaeosols) and

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laminated crusts that probably represent calcretes, which are typical of subaerial weathering. Similar paleocarst surfaces were also found in Derbyshire (Walkden, 1974). The Brigantian cyclothems of North Wales are generally more complex and the cycles that are between the emersion surfaces show evidence of internal cyclicity, which might suggest that such cycles can be subdivided further. In one example, a sequence of limestone, shale and coal is recorded and this approaches the type of cycle that is seen in the Yordale facies of northern England. There can be little doubt that the cycles of Yorkshire and North Wales represent time equivalent intervals, but a correlation between the two can only be tentative. In the Settle district, there is a distinct algal horizon, the Girvanella band, which is just above cycle nine and this is possibly the tenth or eleventh cycle from the base of the cyclic sequence. It is at the base of the second cycle in the Brigantian. On the other hand, if the cycles are counted upwards from the first well-developed cycle in both areas, then one finds fossiliferous bands containing the brachiopod Davidsonina septosa appearing at the same levels and there is also a certain agreement in the thicknesses of the cycles. For instance, cycle six is exceptionally thin in both areas. It is thought that each mesothem or major transgression brings new faunal elements and that this is the justification for a biostratigraphic classification. While this may be true, such a classification probably does not give a sufficiently clear resolution to make it possible to differentiate between the minor cycles. The wide distribution of the cyclothems, their unchanged persistence throughout Asbian as well as Brigantian times, together with the very convincing evidence for actual emersion, make eustatic sea level variations the most likely explanation for the cycle formation. Estimating the time interval that is represented by such cycles is more difficult because both the number of cycles and the time intervals are small. The development of cyclothems in the Settle district and North Wales is thought to have commenced during middle or late Asbian time. A similar development took place in Derbyshire, perhaps somewhat earlier. In Harland’s time scale (1989), the duration of the Asbian is given as 3.4 Ma and the subsequent Brigantian as 3.1 Ma. As already pointed out, this is based on “chrono interpolation” which must be arbitrary to some extent. There is also no real indication as to how much of the lower Asbian is missing in the cycle sequence. If it is assumed that eighteen cycles represent 5-6 Ma, then one obtains 277-333 ka per cycle, which is certainly not close to any Milankovitch frequency. The nearest period is the 400 ka eccentricity cycle but considering the uncertainty of the time scale, one might also consider the 100 ka period. In the latter case, one would have to assume a very large hiatus at the base of the limestones in Yorkshire which is about 73% of the Asbian, if one accepts the Harland time scale. If the 400 ka cycle is accepted, this is slightly longer than the Asbian and no space is left in the time scale for the supposed hiatus. Since it is impossible to estimate the time involved during the emersion of the cyclothems accurately, one can only estimate extreme

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minimum sedimentation rates of 10 cm/ka and 25 cm/ka, for the hypotheses of 100 ka and 400 ka cycles. Using the previously introduced chronometric scale, which is based on the assumption that Ramsbottom’s mesothems represent 2 Ma intervals and accepting the 400 ka cycle hypothesis, this would allow for a short hiatus at the base of the Asbian in Settle. It would not allow sufficient time to accommodate 115 m of Tynant limestone which, according to Somerville (1979), is thought to be of Lower Asbian age and which lies below the cyclic sequence which he described. Alternatively, if the 100 ka hypothesis is adopted, then the hiatus between the Holkerian and the Asbian would be excessively long. For this reason, the 400 ka cycle hypothesis is suggested to be the cause of the cyclothem formation and this solution is therefore shown in Fig. 7-1. The eight cyclothems which are recorded for the Brigantian, would allow sufficient space for the overlying sand passage beds in North Wales, as well as time for a short hiatus at the top of the Asbian. It should be noted that in all the areas where hiatuses have been stipulated, sedimentation is always conformable, apart from paleocarst surfaces, and nowhere is there evidence of the erosion of a complete cycle or part of a cycle. This may indicate that hiatuses may not have been as extensive as has sometimes been assumed. It is obvious that the tentative hypothesis of interpreting the Asbian and Brigantian cyclothems as 400 ka cycles, will need further proof and one particular approach could be to search for evidence of the 100 ka eccentricity cycle and perhaps even shorter cycles. Indications of shorter cycles within the cyclothems have been recorded in the Brigantian of North Wales and a number of Great Scar limestones show a distinct division into four approximately equal beds with a thickness of 1 to 2 m. If the 400 ka hypothesis is true, then this 100 ka signal is very much subdued when compared with the situation in Ireland. A fully complete stratigraphic record in the English examples can only be assumed at the 400 ka level, compared with the Irish examples which seem to be complete at the 100 ka level at least. At the same time, the English cyclothems, which look similar to the Irish stratification cycles, must serve as a warning not to judge simply by appearance. The regularity of the cyclothem thicknesses in the Yorkshire cycles for example, yielded an average thickness of 9.3 m with a coefficient of variation of 30% and this must be regarded as being a fairly large variation for an orbitally controlled cycle. As will be discussed in considerably more detail in the next chapter, the preservation of widespread transgression and regression cycles often requires a complimentary tectonic subsidence, otherwise the general sea level would have to rise indefinitely. Sinking movements are likely to be relatively continuous but regional differences will occur and even eustatically controlled cycles will therefore show regional deviations. The Lower Carboniferous can be interpreted in terms of Milankovitch cyclicity, as has been attempted in the previous section, and then one may reasonably

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ask whether a similar interpretation is possible for the remaining part of the system. No discussion of this problem in any detail is intended but some of the difficulties that are involved can be discussed in very general terms. It was seen in both the Irish as well as the English-Welsh example, that in order to construct a cyclostratigraphic scheme, it was necessary to alter the intervals in the current chronometric scale. In the English-Welsh examples, it was additionally necessary to ignore some biostratigraphic evidence, which suggests that the cyclic sequence should be restricted to the upper part of the Asbian. There are no real discrepancies with the biostratigraphic evidence in Ireland, largely one suspects, because the biostratigraphy is less firmly established there. The case for Milankovitch cyclicity in the Irish examples is considerably better than for the English-Welsh examples, simply because a much larger number of cycles is observable in the Irish ones and because there is reasonable evidence for at least three orders of frequency which approximately fit the 100 ka, 400 ka and 2 Ma cycles of eccentricity. The argument in favour of Milankovitch cyclicity in the shallow water deposits of England and North Wales relies partly on the knowledge of the probably orbitally controlled cyclicity in the deep Irish basins, at the same time and with the approximate timing which can be obtained by assuming that the Asbian and Brigantian stages have a length of 2 Ma. It is tempting to associate this 2 Ma interval, which is in fact quite close to the average duration of a large number of the Carboniferous stages, with the long eccentricity period. The twenty five stages of the Carboniferous chronostratic scale (Harland, 1989) have an average length of 2.92 Ma. The seven stages of the Namurian have an average length of 2.08 Ma. Although many of the stages are directly determined by a transgression-regression cycle, others contain more than one (Ramsbottom, 1977). It is clear that there are at least two good reasons why the number of cycles that is recorded or mapped in a given interval can be wrong. If a land surface cannot be reached by a transgression because of its vertical position, no sediment is deposited. Alternatively, the evidence of a transgressive cycle could have been removed subsequently by erosion. Of course, additional errors can also be introduced by misinterpretation.

THE PENNSYLVANIAN (UPPER CARBONIFEROUS) CYCLES

The Pennsylvanian (Upper Carboniferous) cyclicity of the central parts of North America have been regarded for many years as one of the prime examples of cyclic sedimentation. In the Kansas cyclothems for example (Moore, 1936), the sequence consists of a basal shale which sometimes contains coal fragments, followed by marine limestone and black shale, which may again be followed by a limestone. There is no doubt that each cyclothem represents a transgression and it is now thought that the black shale represents the maximum depth and the top limestone represents a regressive phase (Heckel, 1986). A problem which was discussed

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extensively in the 1940’s concerned the origin of such transgessions, which were almost exclusively attributed to tectonic movements that were more or less local (Weller, 1930). However, Wanless and Shephard proposed as early as 1936 that the cyclothems resulted from glacio-eustatic sea level fluctuations and that the cause of the transgressions and regressions was climatic and not tectonic. This idea was very much in advance of its time and was difficult to accept because it suggested the occurrence of at east fifty glacial and interglacial stages, when it was still believed that the Pleistocene was represented by only four glaciations. World-wide sea level fluctuations are now generally accepted and it is most likely that these are controlled by the Gondwana glaciation, which was a major glaciation. Whether the cyclicity can be attributed to Milankovitch cyclicity, is more difficult to prove and relies on two lines of evidence. The most direct way is to establish the length of the cycles by estimating their absolute time duration. The second argument relies on the frequency structure of the cycles. It was noted by R.C. Moore (1950) that the cyclothems of the Upper Pennsylvanian in Kansas show a regular grouping, which is repeated several times. Between major transgressions which are characterised by the appearance of black shales, three to four limestoneshale cycles are found that represent minor transgressions. Moore called such groups megacyclothems. Heckel (1986, 1990) attempted to construct a sea level curve by tracing the extent of the transgressions and regressions and he classified the marine cycles into three categories. The major cycles are cycles in which the transgression proceeded far enough on to the shelf to form conodont-rich shales at the northern limit of the outcrops in Iowa. Minor cycles typically lack conodont-rich shales and only cover the lower shelf. Intermediate cycles are transgressions which only reach the lower shelves of Iowa. Heckel’s major cycles correspond more or less to Moore’s megacyclothems. In Heckel’s sea level curve, eighteen major cycles were found in the interval from the top of the Cherokee to the base of the Wabaunse group and a total of 51 cycles was found altogether. Interpreting from Harland’s 1986 time scale, gives a time interval of 9.5 Ma. This gives a duration of 527 ka for the major cycles and 186 ka for the minor cycles. Using a somewhat arbitrary method of estimation, Heckel obtained values of between 235 ka and 393 ka for the major cycles and values of 44 ka to 118 ka for the minor cycles. Using yet another time scale, de V. Klein (1990) obtained values of 24 ka to 64 ka for the minor cycles and 129 ka to 216 ka for the major cycles. However, it is likely that these values are too short. These discrepancies clearly indicate the uncertainty which is involved in estimating the length of the Pennsylvanian cycles. Although Heckel’s results and the above estimate (which is based on Harland’s 1989 time scale) seem to indicate that the major cycles could be related to the 400 ka eccentricity cycle, estimates for the minor cycles do not confirm a frequency structure of one to four, which is what one would expect if both eccentricity cycles were present. In conclusion, one cannot produce any positive proof for Milankovitch cyclicity in the Pennsylvanian system of North America except that the ’cycles are of the

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order of orbital cycles and that no satisfactory alternative explanation has yet been given. It is perhaps of interest to note that both the Carboniferous limestones of England and the Pennsylvanian cyclothems seem to suggest periodicities which are around 400 ka, whereas the shorter eccentricity cycles are much less obvious.

125 Chapter 8

TRIASSIC: CARBONATE PLATFORMS

Carbonate platforms have been defined by Wilson (1975) as types of flat carbonate build ups which have been exposed to high energy environments. The platforms correspond to shallow shelves and are separated from a deeper basinal facies either by true organic reefs, or by early cemented slope deposits. In many cases, the margins of such build ups are also tectonically active and there is usually a strong contrast between the high sedimentation rates on the platforms and the lower rates in the basinal facies. The most profuse carbonate production occurs in very shallow water and indeed, many carbonate platforms contain widespread peritidal and subtidal environments. This automatically makes them very sensitive to sea level fluctuations and any cyclicity which is linked to such changes, ought to be particularly well recorded in such environments. Because sedimentation takes place near sea leve1,any recording of the cycles is highly dependent upon the subsidence of the sedimentation area. Thus the main problems of examining cyclic sedimentation on carbonate build ups are fourfold. First, it is necessary to establish the actual presence of cycles and in particular, whether they are Milankovitch related cycles. Secondly, there is the interpretation of the sedimentological evidence in terms of the palaeodepth and the environment. Thirdly,the relationship between sedimentation and subsidence must be evaluated. Fourthly, there is the elucidation of the sea level fluctuations, or any other environmental factors which may have been responsible for the cycles. Carbonate platforms are known throughout the stratigraphic column from the early pre-Cambrian (Grotzinger, 1986), Cambrian (Monninger, 1979), Devonian (Goodwin et al., 1986) and Cretaceous (Grotsch and Buser, 1991). Very extensive formations of carbonate build ups occurred during the Trias, along the margins of the Tethis ocean. There were particularly important developments during the Ladinian and Carnian and again during the Norian and Rhaetic. It was in the latter that carbonate cycles were studied in detail for the first time and these will be discussed first, although much better cyclicity was preserved in the lower Triassic build ups of the Dolomites. THE NORTHERN CALCAREOUS ALPS

During the Upper Trias, the north-west edge of the Thetis ocean was occupied by a shelf which was one hundred to two hundred kilometres wide and was separated

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from the open sea by a series of marginal reefs. The area behind the reefs was occupied by a lagoon in which the prevailing sediment was the Dachstein limestone and towards the ancient shore lines there was the Haupt Dolomite or Dolomia Principale. According to a palaeogeographical reconstruction (Haas, 1991), the deposition areas from north to south are: the Northern Calcareous Alps, the Trans-Danubian Central Range and the Dolomites. The regular stratification of the Dachstein limestone has attracted the attention of geologists for a long time. They realised that it was a shallow water deposit, from the fossil contents. Suess (1892), was the first to suggest that some of the bedding planes in the limestones were the actual emergence surfaces. He observed brecciated red limestone fragments that were embedded in reddish clay, which he likened to the terra rossa of karst areas. A. v. Winkler (1926) working on the wellbedded Upper lliassic limestones of Isonzo, attributes the stratification to periodic changes in the climate. The first detailed sedimentological description of the Dachstein limestone is found in the classical paper on carbonate petrology by Sander (1936). Sander was working in the Lofer area near Salzburg (Austria) and he recognised that the bed formation is due essentially to two facies. There is a massive grey limestone which forms beds that are 2 to 3 m in thickness and these are separated from each other by thinner “interbed” deposits which contain a wide variety of rock types. The most common development is a laminated limestone dolomite sequence, with laminae of between 1 mm and 2 mm in thickness. Dolomite and limestone alternate and the dolomite is either contemporaneous or very early diagenetic. Sander also recognised horizons in which the sediment had been reworked and he realised that the reworking had taken place before the sediment had hardened. In several places, filament-like structures were correctly interpreted by him as being of algal origin. Sander was not able to interpret the different facies correctly, largely because at that time, there was a profound ignorance of modern carbonate environments. He suggested that the laminated deposits indicated water that had been less turbulent but possibly deeper but at the same time, he did not exclude the possibility that emersion could have occurred. The first modern interpretation of the cycles in terms of environmental changes is due to Fischer (1964). He recognised that Sander’s laminated beds, with their typical bird’s eye structures and fenestral fabrics (which were both correctly described by Sander), must be interpreted as algal mats and from this it clearly followed that the laminated beds represented a time of low sea level. Fischer also noted the occurrence of reworked horizons and disconformities at the base of the algal beds and this led to his classification of three repeated facies which is shown in Fig. 8-1. Facies A is a disconformity which is followed by a horizon of reworking which contains mud pebble conglomerates and early diagenetic breccias. The limestones in this horizon are often reddish to greenish in colour and similar colouring is found in the areas nearer the reef, as well as in the reef itself. The red colour

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Fig. 8-1. Development of a typical Lofer cycle after Fischer (1964)A = emersion surface, B = supratidal algal mats. C = lagoonal sediments.

is less well-developed in the more northerly exposures and is completely absent in the Haupt dolomite facies. Associated with facies A, are solution cavities and cracks in the limestone C of the previous cycle and these are often partly filled with the red argillaceous limestone of facies A. Some of the solution cavities, which may be several centimetres in diameter, are aligned in a direction which is roughly parallel to the bedding plane, at levels of between one to several metres below the disconformity. Such levels could well represent the effect of the fresh water tables which formed during an emersion of facies A. Facies B is predominantly dolomitic, with Sander's laminated limestone-dolomite being interpreted by Fischer as algal mats. This interpretation is undoubtedly correct and the algal mats of facies B show all the features which are known from modern mats. Very often, one finds 50 cm to 100 cm thick algal laminates which have been crossed by vertical cracks. Fischer calls such cracks prism cracks and he attributes them to desiccation. Facies B also contains layers of homogeneous micritic limestone and occasionally, well pelleted limestones which may be faecal pellets from small gastropods. Generally speaking, the rocks of facies B are darker in colour than the remaining limestones and their fauna seems to have been quite restricted. Fischer thinks that facies B developed in an intertidal environment, by which he does not mean that the sediments were covered by daily tides but that they were only covered during extreme tides or perhaps during storms.

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Facies C contributes the massive limestone beds which make up 85% of the sequence, according to measurements made in Lofer (Schwarzacher and Haas, 1986). The light grey limestone beds are far from uniform throughout their thickness. They range from micrites to well-winnowed calcarenites and always contain a large amount of organic debris. Oncoids are fairly common and according to Fischer, they are likely to contain predominantly blue green algae. Amongst the macrofossils, the thick shelled clams of the type Megulodus and Conchodus are particularly noticeable and they are often in their original life position, forming layers which can be traced over many metres. Amongst other large fossils, one can find accumulations of large (20 cm high) gastropods which have obviously been moved by currents. The variety of organisms increases when approaching the outer margin of the platform, where a number of fossils appear, including corals, which are probably derived from the reef. Goldhammer et al. (1990) have remeasured a section which was also used by Fischer and they emphasise their finding that the Lofer cycles are in fact “regressive”. This could be expressed more clearly by saying that the regressive phase is more fully developed in these cycles. That any repeated sea level fluctuations must have a regressive and a transgressive stage, one assumes is clearly beyond dispute. Whether the sediments which record this change, do so in an identical manner for both the ascending and descending phases of the cycle is very questionable and it is difficult to decide whether a difference in the rates of transgression and regression took place. Indeed, in the Hungarian exposures of the Dachstein limestone, the disconformity is frequently preceded by an algal mat development which Haas and Dobosi (1982) call the B’ facies and which represents a regressive phase. The transgressive phase can also be developed. (The various modifications of the cyclic sediments will be discussed in more detail later.)

THE CYCLICITY OF THE DACHSTEiN LIMESTONE

The total thickness of the Dachstein limestone in the Lofer area is approximately 600 m. Since all the sediments in this sequence formed very close to sea level, this thickness not only indicates an average sedimentation rate, but also a crustal sinking rate which was responsible for the accommodation of the sediment. The cyclicity of the sediment can be considered either in a geometrical sense by describing the repetition of similar elements, or the cyclicity can be considered from the more interesting time aspect. The latter is related to the geometrical aspect through both the sedimentation rates and the crustal sinking rates. One study which was largely concerned with the geometry of the sequence, considered the Dachstein limestone as a homogeneous sediment which was subdivided by a series of bedding planes (Schwarzacher, 1954). It was recognised that the

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Fig. 8-2. Dachstein limestone. Hinterhorn, Loferer Steinberge, (Austria).

bedding planes coincided with Sander’s interbed facies, which as we now know, are the marine low stands of Facies B. Field observations established that there were two types of bedding planes. One of these (r) generated beds which were 2-3 m thick, and occurred in groups that were separated from each other by very persistent bedding planes (see Fig. 8-2). The latter were called R planes. The 3 m to 4 m thick r cycles were always more prominent near the master bedding planes of the R cycles and they were less prominent in the centre of the bundle. As was seen in the shale-limestone sequences, additional bedding planes sometimes developed and these can confuse the interpretation. The criterion which was used to differentiate between the boundaries of type R and the bedding planes of type r, was the traceability of the larger cycles and their prominent appearance in the exposed cliffs. Indeed, the R cycles were used to make a detailed map of the Lofer Steinberg area (Schwarzacher, 1948, 1952). The less prominent bedding planes of type r, are less persistent and the number of beds which make up an R cycle is therefore variable. It was realised that without any direct proof of time periodicity, it was necessary to show by some other means the regularities in the sequence and therefore the

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ratios between various bed thicknesses were investigated. This led to the discovery of the grouping of five beds of type r to form a megacycle of type R. Such groups were called bundles. The significance of such composite cycles is twofold. They are a strong indication that the time history of such cycles followed'a non-random process which must have been caused by some stable oscillating system because of its repetitive nature. Secondly, the fixed ratio between the two frequencies of 1:5 for example, gives additional evidence for the identification of the cycle. The argument that was used to show the non-random nature of a complex cycle was the following. The grouping into five beds can be understood as being a sequence of five events, four of which are of type r and which lead to the interbeds within the bundle. One is of type R and produces the stratification cycle boundary. The probability of type R occurring is 1/5. However, the probability of an ordered sequence: r r r r R, r r r r R, occurring, decreases rapidly with the number of repetitions. For example, the probability of two successive cycles of the type r r r r R, occurring by chance is 0.033 and the probability of three cycles of this type occurring is only 0.004 (Schwarzacher, 1947). The alternative to a chance occurrence is an ordered sequence and it is the intention here to see whether this can in fact be demonstrated. In some parts of the Lofer section, it can be seen very clearly that the R cycles are indeed composed of five r cycles and that they form bundles of beds which can be separated easily from the next bundle. In other parts of the sequence, the differentiation is less obvious and it is difficult to obtain a clear classification, particularly in the middle part of the Lofer succession. However, it has been estimated that the 600 m of Dachstein limestone contain 27 cycles of type R, with an approximate thickness of 20 m (Schwarzacher and Haas, 1986).

QUANTITATIVE STUDIES

The study which is mentioned above, relied heavily on field observations from a distance. Many of the exposures in Lofer are steep cliffs and they are not accessible. In areas where it is possible to walk along the ledges which mark the boundaries between the R cycles, facies A and B are often covered by scree. It is therefore impossible to obtain a continuous stratigraphic log and separate measurements had to be made of the thicknesses of the exposed R cycles and of the scree covered intervals that may contain some layers of facies C but which are probably not very thick. The average length of nineteen R cycle measurements was 13.6 m. Individually measured cycles of type r, gave an average thickness of 2.4 m. The ratio of R/r is therefore 5.6. An attempt can be made to use a more objective approach. A composite section of the Lofer Dachstein limestone can be coded into a two-state sequence consisting

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cycleslcm

Fig. 8-3. Walsh power spectrum of Dachstein limestone section coded as a two-state sequence of supratidal and intertidal sediments.

of the cliff forming facies C and the ledge forming facies A and B, with possible minor contributions from facies C. The Walsh power spectrum of this data is given in Fig. 8-3 and has a very characteristic shape, with a strong maximum at 17.13 m. This can be interpreted as representing the R cycle. There are further clear maxima at 2.6 m, 5.6 m and 7.9 m. The 2.6 m maximum is most likely to correspond to the basic r cycle and the low frequency maxima are respectively, 2.1, 3.0 and 6.6 times the wavelength of the basic cycle. The spectral analysis arrives at somewhat different results from the relatively few direct measurements which are available. The average thickness of the R cycle in particular, turns out to be higher in the spectral estimate. The number of beds per R cycle gives the higher number of 6.6, compared with 5.6 from the hand measured section. The discrepancy is caused by a different interpretation of the cycle boundaries. The field interpretation suggests that two, or possibly three, R cycles in the section are incomplete, or possibly condensed and this leads to a higher number of cycles and to a reduced average thickness. The spectral analysis, on the other hand, chooses objectively the best average period which fits the R cycles. The example illustrates that the “objective” methods are not always the best ones for any problem and that it is useful to look at both methods and to compare them.

THE TRANS-DANUBIAN CENTRAL RANGE

Considerably more detailed data are available from the Trans-Danubian Central range in Hungary. Here, a number of boreholes were made in the Bakony mountains in the Dachstein limestone and the Haupt dolomite and the cores were

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analysed by Haas (1991), Haas and Dobosi (1982), and Schwarzacher and Haas (1986). The general distribution of the facies is very similar to their distribution in the Northern Calcareous Alps. The Dachstein limestone is deposited on the outer shelf and the Haupt dolomite in the inner lagoon. Towards the upper Norian and the Rhaetic Dachstein limestone, transgressions occur over the Haupt dolomite and there is also a transition into the Kossen facies, in a way that is very similar to that seen in the Calcareous Alps. Since these studies by Haas were made on core material, a very detailed stratigraphic log was obtained. Haas differentiated between four facies. Facies A, B and C are like those of Fischer and facies B’ is a regressive algal facies. He also noted disconformities which were denoted d, so that the full cycle development would be d, A, B, C, B’. The full development of this cycle is relatively rare and Haas found that a number of sequence variations exist. Various types predominate in the different facies belts. Thus in the Haupt dolomite, the sequence d-BC predominates. Member B is usually thick and member A is often absent. In the transitional groups between the limestones and dolomites, cycles of d-BC are still common but types like d-A-C and d-B-C-B’ are also found. The lower Dachstein limestone contains the most complete cycles and although any combination is possible, the sequence d-A-C is relatively common, with facies B missing. According to Haas, this could be due to rapid transgressions in the upper Dachstein limestone. The types d-B-C and d-A-B-C are most common but both A and B are thinly developed. In his analysis, Haas uses the diastem as the defining boundary of a cycle, which leads him also to consider cycles which consist of a single member only. This traditional concept of a cycle boundary is acceptable when diastems are clearly defined features. However, it is less convincing in facies A, B and B’, where minor breaks in sedimentation are indeed part of the character of this facies. To avoid any confusion in the following analysis, a cycle will be taken as being any lithology between two successive tops of facies C. In other words, cycles which do not show evidence of marine transgressions are ignored. This will be called the basic cycle. Using this definition and the data from borehole PO-89 as supplied by Haas, the thickness frequency distribution of the basic cycles is given in Fig. 8-4. The distribution is quite symmetrical and nearly normal with a mean of 2.34 m and a standard deviation of 1.32 m. It is interesting that this distribution shows no evidence of the truncation which is very characteristic for facies B and facies A in particular (Fig. 8-5). The truncation could be evidence of erosion, according to the Kolmogorov model (Chapter 5 ) but there are other processes which lead to similar distributions (Schwarzacher, 1975). The grouping into stratification cycles is much less obvious in unweathered core material. In this particular section, a grouping is suggested by the occurrence of thicker layers of interbed facies. Based on this evidence, the total of a nearly 400 m thick sequence of Dachstein limestone and the transitional facies of borehole PO-89 appears to have approximately 25 stratification cycles, or bundles. Such cycles are

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Fig. 8-4. Thickness distribution of Dachstein cycles facies C. Data from bore hole PO-89, Bakony mountains (Hungary).

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15

20

Fig. 8-5. Thickness distribution of facies A. Bakony mountains (Hungary).

only occasionally composed of five well-defined beds and frequently, there are additional bedding planes. If one divides the rough average thickness of 16 m by the thickness of the basic cycles, meaning a group containing one facies, one finds an average of 6.83 beds per stratification cycle. A more objective method of examining the cyclicity is to use spectral analysis. Since the thickness of the facies A and B development seems to be the best indicator of bundling, the thicknesses of the combined A and B facies was sampled at 50 cm intervals and a BT spectrum was calculated of this record (Fig. 8-6).

134 x 102

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

\

12.27m

cycleslcrn

Fig. 8-6. Walsh power spectrum of data from bore hole PO-86. Data have been coded as a two state system; facies A + B and facies C.

The spectrum shows three significant peaks. The maximum of 2.52 m clearly corresponds to the basic cycle and a well-developed maximum at 12.27 m (which is 4.86 times the thickness of the basic cycle) indicates a grouping into stratification cycles. The third maximum of 4.65 m is 0.38 times the bundle maximum and if the 12 m maximum can be interpreted as being the 100 ka eccentricity maximum, then one might be tempted to interpret this peak as being the obliquity maximum. However,the maximum is also nearly twice (1.84 times) the wavelength of the basic cycle and therefore it could easily be the result of fusing two basic cycles. In contrast to the results from Lofer, the spectral analysis in this case finds 33 bundles, which is considerably more for the complete section than the visual estimate of 25 bundles. This may be partly because the bundles in the Hungarian

80 60 40 -

20 -

TRIASSIC: CARBONATE PLATFORMS

135

section are less well-defined and partly because stratification cycles are more difficult to recognise in a core which does not provide any opportunity for judging the lateral persistency of a horizon. If the basic cycles are used as units to plot a cumulative curve, then a linear trend can be fitted and the slope of this gives the slightly higher average value for the basic cycle of 2.84 m. Deviations from this trend are interesting (Fig. 8-7) and they show some fairly systematic fluctuations, with an average length of approximately 35 cycles. The superimposed and shorter fluctuations are the result of stratification cycles of type R. A spectral analysis of the trend deviation, indicates that the bundle consists of 6 basic cycles.

THE DOLOMITES

In late Permian and Lower Triassic times, the Palaeothetis transgressed towards the west and northwest and moved into the area of the present day Dolomites in northern Italy. By Ladinian times, a carbonate platform developed which formed a shelf with a very irregular and indented margin. In front of this shelf, relatively small and isolated carbonate build ups developed which were completely surrounded by basinal facies (Bosellini, 1991). One of these is the Latemar which has an almost atoll-like structure with a central lagoon and a surrounding reef. The “reef” is no longer regarded as being an exclusively organic structure but consists of slope deposits which have been cemented by early diagenetic processes. The lagoonal facies is represented by just over 400 m of well-bedded limestone which is known as the Latemar limestone. Hardie et al. (1987) and Goldhammer et al. (1990) describe and analyse the cyclicity of this area in great detail. According to them, the sequence represents a third order (1 to 10 Ma) cycle which is an unconformity bounded sequence. The use of the “order” terminology in their papers illustrates the confusion which can arise from it. Their third order would be a second order according to the Vail et al. (1991) classification. Their fifth order in Vail terms would be a sixth order. Only the fourth order is the same in both schemes! The Latemar sequence starts in the late Anisian with 250 m of lower platform facies, which is a subtidal lagoonal carbonate development.This predominantly marine sequence is interrupted at intervals of approximately 10 m by thin horizons that indicate subaerial exposure. No shorter cycles than the 10 m interruption can be seen. The lower platform facies is followed by the Latemar limestone proper. Hardy and Goldhammer divide it into three parts. The lowest 90 m is clearly cyclic and contains 73 cycles consisting of subtidal lagoonal limestones that are overlain by a 5-15 cm thick caliche crust which again indicates emersion. The average thickness of these cycles is 1.24 m and they are grouped into bundles which consist of five basic cycles that thin in an upwards direction.

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CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

1.0,

00

5

10

15

20

25

301

Lag number of cycles

Fig. 8-8.Correlogram of Latimar limestone cycles. After Goldhammer et al. (1990).

The lower cyclic facies is followed by 120 m of cyclic sediments which have been called the Tepee Facies because they contain frequent horizons with tepee formation. This series contains 295 cycles with an average thickness of 0.40 m, whereby the tepee horizons are interpreted as indicating prolonged subaerial exposure. Again, bundles containing five basic cycles are developed whereby tepee formation is particularly strongly developed at the bundle boundaries. The top part of the Latemar limestone is formed by 210 m of the upper cyclic facies, which is very similar in development to the lower cyclic facies.It consists of 230 cycles of marine lagoonal sediments which alternate with non-marine caliche horizons. The average thickness of the cycles is 0.96 m and stratification cycles with groups of five are very well-developed. This complex cyclicity can be well illustrated by correlograms of the thickness plots (Goldhammer et al., 1990) which are shown in Fig. 8-8.

ABSOLUTE TIME ESTIMATES

Absolute time estimates are very difficult to make in any part of the alpine and Trans-Danubian Tl[i.ias, mainly because the stratigraphic control is not wellestablished in any of these areas. According to Harland (1989), the Norian had a duration of 13.9 Ma followed by the Rhaetic of 1.5 Ma. This period of time is represented by approximately 2000 m of carbonates in Hungary and by approximately 1800 m in Lofer, of which 800 m are Dachstein limestone. Using these rough numbers, one can estimate that the 400 m of bore hole PO-89 represent approximately 3 Ma and that the 600 m of Dachstein limestone in the Lofer section represent 4.6 Ma. The estimated number of R cycles in PO-89 ranges from 25 to 33 and the estimate was 27 cycles for the Lofer section. This gives an estimate of 120 to 90 ka for the duration of the Hungarian R cycles and 170 ka for the duration of the Austrian ones. The estimate for the Lofer section in particular, is based on an

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137

average that contains a considerable thickness of Haupt dolomite which had a lower accumulation rate. This would increase artificially the values for the Dachstein limestone and therefore both estimates suggest a duration of the R cycles of about 100 ka. Consequently, r cycles have an estimated duration of 20 ka and this clearly makes the R and r cycles candidates for representing the 100 ka eccentricity and the 21 ka precession cycles. It is more difficult to obtain similar estimates for the Ladinian cycles of the Latemar limestone. Goldhammer et al. (1990) give a time scale in which the Ladinian commences at 240 Ma B.P. and ends at 228 Ma B.P. This time scale is considerably different from any other recognised time scale. For example, Harland (1989) has the Ladinian between 239.5 and 235 Ma B.P. with a duration of only 4.5 Ma. This time period is much too short to accommodate 598 cycles, if these really do represent 20 ka periods as the authors claim. It is possible that the stratigraphy or the absolute time scale needs revision. From the available data, one would estimate the length of the Latemar megacycle to be 37.6 ka.

THE SIMILARITY OF CYCLES

When comparing the cycles in the Northern Calcareous Alps, the TransDanubian central range and the Dolomites, one is first struck by the almost identical development of the facies and petrology in the three areas. In each case, the developments nearer the paleogeographical lend are peritidal dolomites that could have been Sabkha-like, according to Haas and which were areas that the sea invaded only occasionally. Large parts of the Dolomia Principale may have formed under similar circumstances. Dachstein limestone and subtidal Dolomia Principale developed in the deeper parts of the lagoon (Bosellini and Hardie, 1985). The Dolomia Principale contains the same fauna of Megulodus and large gastropods, as the Dachstein limestone. Both the dolomites and the limestones contain indications of periodic emersion, pisolitic breccias and tepee formations in the dolomite red soil like deposits and clear shrinkage cracks in the limestones. Prior to full transgression and sometimes also after emersion, large areas were covered by algal flats and there can be little doubt that the stratigraphic record indicates repeated changes in the relative sea level. The formation of the Latemar limestones occurred under somewhat different conditions. The lagoon in which they formed was not part of an enormous shelf but formed an isolated entity that was completely surrounded by a protective “reef”. It is possible that this special position was responsible for the very regular and probably uninterrupted formation of the cycles. The Ladinian examples are absolute proof that sedimentation repeatedly changed from marine to non-marine conditions, The tepee facies is an indication that the exposure to non-marine conditions must have been quite extensive.

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CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

TABLE 8-1

Thicknesses of R and r cyles in metres

R cycles Lofer PO-89 Dolomites

17.1-13.6 16.0-12.3 no data

r cycles (direct measurement)

r cycles

2.4 3.0 2.0

2.59 2.52

(from spectrum)

-

In all the examples, one finds a grouping of a basic cycle into a composite cycle containing an average of five shorter cycles. This is apparently extremely well-developed in the Ladinian examples but in all the other cases, one finds that the ratio between the most prominently developed order of cyclicity R/r, ranges from 6.6 to 4.8. The thicknesses of all the cycles in the Norian are of the same order and only the Ladinian cycles are consistently less (see n b l e 8-1). Goldhammer et al. (1990) are not in agreement with the above findings and they believe that “the worlds most famous carbonates” from Lofer are in fact chaotic and without any sign of Milankovitch cyclicity. They fail to mention that it is in Lofer that megacycles consisting of groups of five were first seen and described. They could not obtain evidence for cyclicity using the high resolution (Thomson, 1982) spectral techniques developed by Hinnov and Goldhammer. This is not surprising unless one believes in the naive concept that the cyclicity in the alpine Trias is due to sinusoidal sea level changes which are directly proportional to the thicknesses of the beds which they generate. The cyclicity of the Bias and indeed of most of the sediments, is often very unevenly developed. For example, at Lofer the best developed cycles with a clear grouping into sets of five are found at the base of the Dachstein Formation and again towards the top. In between is a thickness of 200 m, where the megacycles are reduced in thickness and quite irregular in their development. It would be impossible to demonstrate cyclicity in this region, without a comparison being made with the higher and lower parts of the sequence. Cyclicity is also less regular as the reef margin of the shelf is approached and the locality of Steinernes Meer, which Fischer (1964) measured and which Goldhammer et al. examined, is in this position. If cyclostratigraphy is to become a practical tool, it is vital that cyclicity should also be recognised in less obviously cyclic sections. This is sometimes made possible by comparing the lateral and vertical equivalent developments with sections where the cyclicity is clearly developed. In the examples of the alpine Trias which have been discussed, the evidence for relative sea level fluctuations that represent time intervals ranging from possibly 10 ka to 100 ka, is very clear. The approximate number of relative sea level changes was the same in all the areas and the simplest

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139

explanation for this must be that the level changes occurred simultaneously. If one can show clearly that such level changes follow a complex cyclic pattern in one area, it seems reasonable to accept that the mechanism producing the level fluctuations is the same in all areas. This does not exclude the possibility that some environments are strongly disturbed by other processes. It is perhaps not out of place to add a warning to searchers of stratigraphic rhythms not to allow simplistic computer generated models or unsuitable analytical techniques to override their geological common sense.

SUBSIDENCE

An essential part of the mechanism that is responsible for carbonate platform cycles, is the subsidence that creates the accommodation space for the accumulating sediments. In the Northern Calcareous alps, where there is an almost continuous record of the Norian shelf deposits, one finds sediment thicknesses of 2000 m and possibly more at the reef margins and some 1000 m of sediments in the Haupt dolomite. Considering that the sediments formed near sea level, this means that the crustal movement consisted of a tilt that was superimposed on to a continuous sinking movement. The shelf margin moved about twice as fast as the near shore dolomite areas. Some of this difference may, however, be due to differential compaction, since the reef possibly did not compact at all. The .subsidence values which are of the order of 10 to 15 cm/ka are high compared with normal oceanic subsidence, but they are close to other subsidence values for passive margins (van Hinte, 1978).

SEA LEVEL FLUCTUATIONS

There are possibly three basic reasons for the repetitive changes in water depth that were recorded by the cycles which have been discussed. (1) The basin is either stationary or sinking slowly and shallowing is being produced by filling the basin with sediment at a rate that is faster than the rate of subsidence. Repetition can only occur when the sediment supply is variable. (2) The sedimentation rate is constant but the subsidence can vary. (3) The sea level can undergo eustatic changes which could be superimposed on to the sedimentation subsidence process. The first process is perfectly feasible and mechanisms for this could be provided for instance, by climatically controlled sedimentation rates. Such a process for example, was suggested as an explanation for the variable supply of clastic material in the Pennsylvanian cyclothems of North America (Wanless, 1950). In the case of carbonate platforms, sedimentological arguments are against this process, particu-

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CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

larly where definite emersion has been established. That there is variable sediment production following the shallowing process, is beyond doubt. A number of authors (Goldhammer et al., 1990; Haas, 1991; Strasser, 1991) quote “autocyclic” processes and in particular, the “Ginsburg model” to explain variable self-regulating sediment production. The Ginsburg model, however, is completely unworkable and does not explain any cyclic sedimentation (Schwarzacher, 1993b). Among the few geologists who have analysed this model critically, is Grotzinger (1986) who writes: “The autocyclic model has a fundamental weakness in that it fails to explain the transition from prograded platform to submerged platform” (i.e. the cause of transgression)... “ the prograding wedge will reach a point where the limit of production is such that the tidal flats will remain more or less stationary...”. These important facts are ignored by the proponents of so-called autocyclic models and are replaced by the inconsequential statement that “at one point sediment accumulation is outpaced by continued subsidence’’ (Strasser, 1991) but unfortunately without telling us why. The second mechanism of intermittent subsidence, provides a possible but very unlikely explanation of the Triassic cyclothems. One process which has been proposed as a possible explanation of the Lofer cyclothems is “ stick slip faulting”. In this process, elastic strain energy (i.e. stress) is stored along the fault planes and then it is suddenly released (Cisne, 1986). The waiting times between slips would have to be considerably longer than the known earthquake frequencies (Schwarzacher, 1975) and Cisne claims that such frequencies have been estimated for normal faults in the western United States. However, the previously quoted subsidence rates are hardly indicative of a stable platform environment. It is also difficult to visualise a fault system that operated simultaneously or at least with the same frequency, over such wide areas as were occupied by the lliassic shelves. The difficulties in explaining complex stratification cycles by this theory would be vast. Some explanations of the cycles which have been discussed, have tried to combine unequal subsidence with sea level fluctuations. This is essentially Fischer’s (1964) explanation of the complex cyclothems in the Lofer area. Fischer assumed that the basic cycle is produced by a eustatic sea level fluctuation and the 15 to 20 m cycle (cycle R) is due to irregular subsidence. Fischer clearly missed the significance of the two cycles with a fixed frequencf ratio and he also ignored the constant phase relationship between the two. Consequently, he interpreted the basic cycle (which both Winkler (1926) and Sander (1936) attributed to the 21 ka cycle) as having been caused by the 40 ka obliquity variations. Lastly, one could explain the observed relative changes by assuming a complex eustatic movement of the sea level which is superimposed on to a steady and slowly changing crustal subsidence, if indeed the crustal subsidence changed at all. This is certainly the simplest explanation and the most widely accepted one, at least amongst those who regard the Milankovitch hypothesis critically.

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141

Eustatic sea level changes are generally believed to be due to two causes. There is either a change in the shape of the ocean basin, for which major tectonic processes must be postulated, or there is glacial control, which removes water from the oceans and stores it as ice on land. The second alternative is the more attractive one. Not only does it provide a direct link to climatic changes and therefore to possible astronomical control, but also it is difficult to see how repetitive changes of the ocean basin could occur, whereby each deformation would return again to its original shape, without any progressive development. The amplitude of a eustatic sea level movement can only be guessed at, but it is generally assumed that the fluctuations were around 10 to 20 m. With the present day ocean configuration, a lowering of the sea level by 10 m would involve a very minor glaciation that according to Shackelton (1986), would change the oxygen isotope composition of the sea water by 0.1 per ml. This is close to the analytical accuracy for Pleistocene samples. The I-iassic cycles themselves provide no accurate palaeodepth data for the maximum flooding stages of the shelf but a maximum of 20 m has been suggested. If one assumes that the shelf was some 200 km wide and that the maximum water depth was 20 m near the shelf margin, then one finds an average gradient of 1 in 10,000, which is a quite remarkably flat environment. If it is assumed for instance, that the tidal height on the shelf was only 20 cm, then the normal intertidal belt would be 2 km wide. It is also obvious that very small morphological irregularities must have had an enormous effect on the regional facies distribution. A low hummock for example, may make all the difference between subaerial, algal or marine development. An equally strong influence could have been effected by climatic variables such as the prevailing wind direction, which by raising the sea level perhaps by a few tens of centimetres could have flooded large areas. Practically nothing is known about the tidal activity in the area. It is possible that on the shelf, it was similar to present day conditions in the Mediterranean where tidal movement is relatively small. This would explain the quite remarkable absence of tidal channels. Although small-scale channelling which is only a few centimetres in depth is seen, large-scale channels are not found, even in areas where one can trace the “intertidal” horizons for several kilometres. Clearly, a large number of sedimentological problems have yet to be solved. Questions about the sediment production in the different environments of the platform are particularly important. Have most of the sediments been formed in situ? Has there been considerable lateral transport? For example, are the lagoonal calcareous sediments formed entirely in the lagoon, or has a large amount been washed in from the reef? Are the dolomites precipitated in situ, or have they been washed in from the lagoon and subsequently dolomitized? Is there any evidence of land derived material, or material distributed by wind?

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CYCLOSTRATIG RAPHY AND THE MI LAN KOV ITCH THEORY

THE GEOMETRY OF CYCLE FORMATION

The regular repetition of regressive and transgressive facies in the Triassic cycles indicates that sedimentation must have kept up with the overall subsidence and indeed, that the accumulation rates must have been determined exclusively by subsidence. If an equilibrium between subsidence and sedimentation is to develop, one has to assume that the potential sediment production was always more than sufficient to fill any structural depression and that during the marine as well as during the non-marine phase, a base level developed which prevented sediment accumulation beyond this critical level. This is the classical bypass concept of Barrel1 (1917) which assumed that any sediment above a certain level that was called the wave base, was usually dispersed by transporting it into the deeper psrt of the basin. Let it be assumed that the dispersive power in a bypass system decrcases with increasing water depth, w, and that the sediment production a ( w ) !s proportional to the water depth. In addition, if crustal sinking, R , is constant, then a base level given by R / a will always be reached. This will be irrespective of the absolute values of (1 or R (Schwarzacher, 1966). Instead of making the process dependent oh dispersal, one can also introduce various sedimentation processes which are known to be depth dependent. These could be for instance, lime production (depending on light penetration) or the production of plankton (depending on the depth of a water column). In order to examine the behaviour of a subsiding basin with a superimposed eustatic sea level fluctuation, the following simple model was examined (Schwarzacher and Schwarzacher, 1986). A basin subsides at a constant rate R, the sea level fluctuation is given by u = bsint and w is the water depth below the average sea level. Sedimentation is taken to be proportional to the water depth: Sedimentation = a ( u t w ) ,

(8-1)

where a is a factor determining the sediment production. Sedimentation is taken as positive when u + u >O and negative (erosion) when u + u
The equation has the solution: R a2b awb W ( t ) = - - -sin wt -cos wt a w2+a2 w2 + a2

+ C e-"'

At t = 0, let U I = wo which gives the integration constant as: R c=wo--+a

abw w2+a2'

(8-3)

TRIASSIC CARBONATE PLATFORMS

143

When the time in eq. 8-3 becomes large, the exponential term disappears and a mean equilibrium depth of R / a will be reached. This is exactly the same as in the case of a fixed sea level. The sea floor follows the sea level fluctuation but with a reduced amplitude given by: ab

A=nTP

(8-4)

The sine wave lags behind the sea level curve and is shifted by the phase w

tan4 = -a

(8-5)

It is interesting that the rate of the crustal sinking does not appear in the amplitude and phase of the sea floor movement relative to the average sea level. It should be noted, however, that the relations of eqs. 8-4 and 8-5 only hold for the assumed sinusoidal shape of the sea level changes and that any change like an asymmetrical rise of the level, will alter the relationship. The simple assumption of eq. 8-1 implies that with increasing depth, sedimentation increases without limit and an equilibrium between subsidence and sedimentation can therefore always be reached. In reality, sediment production will have some optimal depth but equilibria are still possible within given depth ranges and in principal, the model will behave in a very similar way over a wide range of sedimentation processes. For the interpretation of stratigraphic records, it is important to know which part of a cycle can be preserved. As can be seen from Fig. 8-9, for a fixed cycle frequency,

Fig. 8-9. Diagram to illustrate the preservation of sediments in an environment with fluctuating sea levels.

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CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

this depends entirely on the ratio R / a . The maximum thickness z(max) which can be preserved is: z(max) =

2nw

m

If R <2A, then only the depositional history of the cycle can be preserved. A is the amplitude of the sea floor movement (eq. 8-4). With larger sinking rates, some of the erosional history can also be preserved. Assuming A = 1, then we obtain the thickness record z(t) as function of time as: z(t) = R cos R - cost cos R

+ t sin R

t = 0, . . . 2n

(8-7)

The intervals ca9 be thinning in an upwards direction, but a maximum that is somewhere above the base is also possible in some cases. A number of authors have illustrated the geometrical problems arising from the combination of sinking basins and sea level fluctuations, using computer graphics (Dunn et al., 1986; Walkden and Walkden, 1990). With such programmes, one can experiment using a variety of superimposed curves to generate complex cycles. If one superimposes two curves, one with a frequency that is five times that of the second, then one cannot be surprised at generating “ bundles of five”. If such curves are rising asymmetrically, then one has a better chance of generating a complete bundle which may be preserved (cf. Schwarzacher and Schwarzacher, 1985). Any conclusions one draws from such geometrical models, suffer from not knowing either the shape of the sea level curve, or the shape of the subsidence curve, or the amount of sediment production, and the effect of the local topography. It has been possible in some cases, to relate definite facies changes to subsidence curves which have been obtained by plotting the deviation from the cumulative sedimentation curves (which are termed “Fischer plots” by some authors). If there is agreement between an observed facies and the type of facies one would expect from sea level high and low stands, then this can be taken as supporting a geometrical interpretation. In the Latemar sequence for example, Goldhammer et al. (1990) were able to show that if one assumes a constant average subsidence, then the cumulative sedimentation curve shows relatively slow changes over several millions of years. In their interpretation, this is associated with Vail type third order sea level changes. The different stages of the rising and falling third order cycles led to different facies developments, which supported the trend of the subsidence or the equivalent sea level curve. The relation between cyclostratigraphy and sequence stratigraphy which is demonstrated by the work of Goldhammer et al. is very important. This is not only because high sea levels and low stands can explain different facies developments, but mainly because cyclostratigraphy can provide time measurements which may help to obtain a better understanding of the processes which lead to the development of the sequences. It is clearly very desirable to repeat a study similar

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145

to the Latemar work that has been discussed, on sediments of identical age in quite different localities. Such parallel examinations could eventually eliminate all local variations and errors.

THE PLATFORM-BASIN INTERACTION

Because platforms over-produce sediment, a great deal of it passes into the deeper basins. Basinal deposits associated with platforms therefore contain a good deal of the platform history (Schlager and James, 1978; Schlager, 1991; Reijmer et al., 1991). The relationship between platform and periplatform deposits is particularly important because it provides a mechanism which explains how relatively minute sea level fluctuations can be recorded in very deep water. For example, at site 632 in the Exuma sound (Bahamas) in a water depth of 1996 m, one finds a record of up to 24 glacial and interglacial stages of the Pleistocene. During the interglacial sea level high stands, the frequency of the turbidity currents increased and this shifted large amounts of platform produced sediments into the basin. During the glacial lowstands, very little sediment was produced on the platform and deep water sedimentation therefore predominated in the basin. During Norian times, a number of basinal facies developed adjacent to the carbonate platforms in the Calcareous Alps. The Potschen Kalk (Pedata Schichten) is the basinal equivalent of the Dachstein limestone in the Dachstein area and it was examined in detail by Reijmer et al. (1990, 1991). Careful faunal analysis showed that the sequence which contains a large number of calciturbidites, consists of a mixture of basinal and platform and reef derived biota. During high stands, the contributions from the platform and the shallow patch reefs predominate. During low stands, basinal and deep reef faunal elements predominate. The situation therefore, was very similar to that of the Pleistocene Bahama banks. Spectral analysis of point count data of a faunal index defining basinal and platform elements, show persistent peaks at 18-23 m, with a subdued peak at 8-9 m and at 3-4 m. Reimer estimates the sedimentation rate of the Potschen kalk as being 10-15 mm/ka and he identifies the three spectral peaks with the 100 ka eccentricity, the 40 ka obliquity and the 21 ka precession. A similar interaction of basin with platform facies is envisaged for the Rhaetic sediments of the Lombardy basin in North Italy (Masseti et al., 1988). The basinal sedimentation is predominantly argillaceous and marly and it alternates with layers of limestone which are believed to be derived from nearby carbonate platforms. About twenty limestone-marl couplets combine into an asymmetric stratification cycle of 6 m thickness, the limestone beds increasing in thickness towards the top. The chronostratigraphic control in the Rhaetic is poor but it is believed that the 6 m cycle represents the 100 ka eccentricity and that the marl-limestone

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CYCLOSTRATIG RAPHY AND THE MI LANKOVITCH THEORY

couplets represent shorter fluctuations of about 5 ka in length, which is above the Milankovitch frequencies

CONCLUDING REMARKS

The question finally arises: can the Milankovitch theory really explain the cyclicity of the Triassic platform sediments and what are the pros and cons of using the theory as a stratigraphic working hypothesis? Possibly the weakest point in support of the theory is the absence of well-defined chronostratigraphic control points. Indeed, no clear division between Norian and Rhaetic has been established in some of the areas that have been discussed.. There are also very few regional data giving the total thicknesses of Triassic sediments, together with data on the thicknesses of the cycles. In very general terms, one finds that the thicknesses of the R cycles increase with the increase in the overall thickness of the Triassic sediments. For example, north of the Salzburg area, the Haupt dolomite is approximately 1000 m thick and the corresponding R type stratification cycles are between 2 m and 5 m thick. In the Dachstein limestone area, where the total sediment thickness can exceed 2000 m, the R cycles are between 15 m and 20 m thick. However, one would like to have much more accurate data on both the cycles and stratigraphic thicknesses before one could be certain that each area actually contained the same number of cycles. On the other hand, it is very well-established for all the areas, that the cycles can be directly explained by fluctuations in the sea level and since the order of magnitude and the frequency of such changes is the same everywhere, it is reasonable to accept eustatic sea level changes as being the primary cause. Such sea level changes occurred simultaneously and the cycles therefore represent time horizons, within the accuracy of their thickness. The appearance of stratification cycles wirh constant subdivisions forming groups of five is highly significant. Such complex cycles are equivalent to sedimentary cycles with constant thicknesses which cannot be explained by random processes. It is therefore suggested that the time dictate that was responsible for the eustatic sea level movement was periodic or very nearly periodic. Milankovitch cyclicity is the best explanation for regular sea level fluctuations and the rough time estimates that are available, make it likely that the 10 m to 20 m cycles of type R represent the 100 ka eccentricity cycle and that the 1 m to 3 m r cycle represents the precessional cycle of roughly 20 ka. The fact that in many parts of the sections, the cyclicity is only faintly suggested and indistinct is to be expected in the platform environment. It is the clearly developed cycles which demand explanation and not the absence of cycles which can always be easily explained. Periplatform deposits in deeper water have recorded a complete section and the spectral analysis of the Potschen Kalk yielded three frequency maxima which have

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been interpreted as the 100 ka, 40 ka and 21 ka Milankovitch periods. No definite evidence for longer cycles such as the 0.4 Ma and 2.0 Ma eccentricity cycles has been found in the Triassic sections which have been examined so far. Some fairly constant systematic deviations from a constant accumulation curve in the Hungarian example could represent a period of 0.6 to 1.0 Ma, but it is not possible to give an explanation for this fluctuation at the present time.

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Chapter 9

SOME JURASSIC EXAMPLES

With a few exceptions, the cyclostratigraphy of the Jurassic is practically unexplored. It is known, however, that cyclic sedimentation can be found throughout the Jurassic and Milankovitch cyclicity is very likely to be responsible for many of the stratification patterns found.

THE LOWER JURASSIC IN BRITAIN

The Lower Jurassic sediments of Hettangian and Sinemurian age which are found along the southern coast of England and in Wales, are known as the Blue Lias. This formation consists of fine-grained limestones 5 cm to 30 cm thick which alternate with dark mark. The shales represent an anoxic environment which is typically free of bioturbation and is well-laminated, while the limestones in contrast, are often strongly bioturbated. Despite these clear indications of primary origin, it has been suggested that the limestone-shale alternation is at least partially secondary in nature and is due to diagenetic processes such as “rhythmic unmixing”. According to Hallam (1986) such strong diagenesis could invalidate the hypothesis of climatically induced Milankovitch cyclicity which has been applied to the sediments. However, no reasonable diagenetic mechanism has yet been proposed which could generate complex cyclicity. Indeed, one would rather expect diagenesis to reduce or eliminate it. The already discussed migration of carbonate from the marl into the limestone beds, has often been postulated but never proven and it seems certain that if it actually does occur, it will need some primary fluctuations in sedimentation to initiate it. To obtain an objective criterion for the existence of cycles, Weedon (1985) measured several sections in the south of England and in Wales and presented the lithological change in a coded form. He differentiated between four lithologies which were arranged in the sequence of massive limestone, grey marl, dark marl and laminated limestone, with the codes +1.5, +1, 0, -1, -1.5. This sequence was chosen because it more or less represents the redox state of the environment. The massive limestone has a very low organic carbon content and the laminated limestone is very bituminous and can have an organic carbon content of more than 5%. Experiments showed that the values assigned to the codes are not very critical and indeed, it may have been better if a simple code like +1,0, -1 were to have been chosen.

150

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

The coded series were analysed by Walsh spectral analysis and two of his sections at Watchet (Somerset) and Lyme Regis (Dorset) produced clear and persistent maxima. The analysis indicated two preferential periods at 819 cm and 512 cm in the Watchet section and two maxima at 85 cm and 51 cm in the Lyme Regis section. Weedon estimated the time represented by the two cycles by assuming that an ammonite zone has a duration of 1 Ma. Applying this principle to the Watchet section, he found a length of 207 ka and 129 ka for the two cycles of 819 cm and 512 cm and 93 ka and 56 ka for the cycles in the Lyme Regis section. The ratio between the two frequencies is the same for both localities and despite the difference in thicknesses, Weedon thought that they represent the same cycles. He tentatively interpreted the longer cycle as being caused by the obliquity and the shorter cycle as being the result of the 21 ka precession cycles. This interpretation is not completely convincing. 'Rvo points in particular make it difficult to accept without any additional evidence. Firstly, it is surprising that there is no trace of an eccentricity cycle, which in many other examples is much more prominent than the obliquity cycle. Secondly, the sedimentation rates which would have been necessary to produce an 8 m thick obliquity cycle (200 mm/ka) are rather high, even if the hemipelagic sediments were formed relatively close to the shore. Although Weedon's analysis demonstrates a regularity of cyclic sedimentation which is best explained by a quasi-periodic climate control, it has not convincingly identified which Milankovitch cycle is responsible for the sedimentary rhythms. The Blue Lias cycles have also been described in a much quoted paper by House (1985), who estimated the lengths of the limestone-shale couplets by assuming that the ammonite zones of this sequence represent a time interval of 1 Ma to 1.25 Ma. This calculation leads House to believe a likely duration of the Lias cycles to be about 40 ka and he therefore thought that they were controlled by the obliqity cycle. Since the only support for this assumption is the estimation of a relatively short time interval in the estimated length of the Jurassic (which is itself uncertain) the interpretation must be regarded as very tenuous. Further light was thrown on the nature of the Lias cycles by the work of Smith (1989) who correlated Weedon's sections of Watchet and Lyme Regis with data from the Burton Row bore hole, a locality which is approximately 30 km east of Watchet in Somerset. Smith could show, by making cross-plots between Burton Row and Watchet, that there is a cycle by cycle correlation between the two sections and that the cycle maxima fall on to a straight line in the correlation plot (see Fig. 9-1). The ammonite zones which have been established in both localities, do not form a straight line when plotted and deviate from the cycle correlation by one to two cycle lengths. This means that the correlation which is based on ammonite zones, is only approximately correct. It is also clear from a comparison of the ammonite zones with the cycle numbers, that the former are of very unequal length and this makes the dating method of Weedon and House very doubtful. A correlation plot of the Burton Row borehole with the Lyme Regis exposure,

151

SOME JURASSIC EXAMPLES

rotlforme subzone t

80e

Ammonite zone bases

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m

o

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B

3

t

t.

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01+ I 410 400

I

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390 380 370 360 350 340

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Burton Row borehole: depth (m)

Fig. 9-1. Correlation cross-plot of Burton Row borehole and an outcrop of Blue lias near Watchet (Wales). Crosses represent cycle by cycle correlations and filled in circles represent correlated biozones. After D.G. Smith (1989).

shows evidence of a hiatus of seven cycles in the bucklandi zone. This hiatus was not evident from the biostratigraphic evidence and it coincides approximately with the 202 Ma sequence boundary which was postulated by Haq et al. (1987). Estimates of the Hetangian vary between 9 Ma and 3 Ma (4.5 Ma in Harland, 1989) and Smith therefore estimates the length of the Lias cycles to be 130 ka to 390 ka. This estimate brings the cycles firmly into the range of the eccentricity cycles but it is still uncertain whether they are controlled by the 100 or 400 ka eccentricity cycle. A mechanism which could have been responsible for the Lias cycles is discussed by Weedon (1985). He suggests that the link with climate is most likely to be variable precipitation. His interpretation is essentially that of a dilution model. A variable influx of terrigenous clay dilutes the amount of cocolithic carbonate. With increasing runoff, fresh nutrients are introduced which favour the productivity of ostracods and non-cocolithic algae and this organic over-production leads to the anoxic conditions of the bottom waters. The development of stratified water, due to the increased influx of fresh water may contribute to the isolation of the bottom waters. A higher part of the Lias sequence of Dorset was examined by Weedon and Jenkins (1990) who measured a short 8.85 m long section in the Belemnite Marls of Pliensbachian age. Samples were taken at 3 cm intervals and calcium carbonate and organic carbon determinations were made. The analysis showed that the two variables are negatively correlated and that the alternation between carbonate-rich

152

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

and organic carbon-rich marls is essentially the same as in the Blue Lias. Spectral analysis gave two maxima, one at 37.5 cm and a second maximum at 300 cm. Renewed estimates of the duration of the bedding cycle, make it now more likely that the couplets represent a precessional period, since the timing in this paper suggests a duration of 8 ka to 34 ka. The 300 cm cycle appears to be a vague amplitude modulation of the calcium carbonate and total organic carbon curves which could have any wavelength between 191 cm and 705 cm because of the frequency resolution of the short series. The authors call this fluctuation a “bundle cycle” without giving any further definition of this term and they emphasise the irregularity of this “cycle”. However, it does not seem surprising that the average frequency of a “cycle” which is repeated only three times in the series is not very reliable.

THE KIMMERIDGE CLAY

The second Jurassic formation which is believed to be controlled by Milankovitch cycles is the Kimmeridge clay (House, 1985). Again, this sequence is best exposed on the south coast of England and shows alternations between mudstones, calcareous mudstones, bituminous mudstones and oil shales. Near the base of the Kimmeridge clay, the cycles have a siltstone base that is overlain by a dark grey mudstone and a pale calcareous mudstone which is often terminated by an erosional surface. In the higher parts of the sequence, the cycles consist of brownish grey bituminous mudstones or oil shales that are followed by dark grey mudstones which pass into lighter calcareous mudstone (Cox and Gallois, 1981). The thicknesses of such “ cycles” are extremely variable, ranging from decimetres to several metres and indicate a relatively rapid change in sedimentation conditions. Superimposed on to these alternations, is a much longer cycle which is caused by a relatively regular change in the more clayey and more calcareous mudstones. This cycle has a thickness of approximately 70 m in the exposures of the Dorset coast and it is repeated six times in the Kimmeridge Clay. Thin well-cemented dolomitic limestone bands occur, particularly in the middle and upper part of the section. These are the so called stone bands, which are between 15 m to 30 m apart. The lithological changes of the Kimmeridge can be recognised in the geophysical logs of wells (Whittaker et al., 1985) and in this way, they can be traced underground over an area of several hundreds of kilometres. The thickness of the Kimmeridge Clay (which on the south coast is in excess of 600 m) decreases in a northerly direction and increasingly younger levels are missing. Figure 9-2 shows the smoothed gamma-ray log of a well near Warlingham south of London, where the Kimmeridge Clay is reduced to a thickness of 218 m and where the lower two ammonite zones are missing. The record shows the six somewhat irregular cycles, together with the ammonite zones which experts can identify from the geophysical

SOME JURASSIC EXAMPLES

'1

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153 ISCIELI

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MU

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750

800 Depth in metres

850

900

Fig. 9-2. Smoothed gamma ray log of the Kimmeridge in the Warlingham borehole (South England) The following zones have been identified: MU = Mutabilis, EU = Eudoxus, AU = Autissiodorensis, EL = elegans, SC = Scitulus, WH = Wheatleyensis, HU = Hudlestoni, PE = Pectinatus, PA = Pallasoides. Data from the British Geological Survey.

30001n'"" 2400

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1800

800

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Fig. 9-3. Power spectrum of the Warlingham gamma ray log.

logs (Whittaker et al., 1985). The biostratigraphic zones are again very uneven in length compared with the cycles, which is similar to what was found in the Lower Jurassic. A power spectrum of the unsmoothed gamma record (Fig. 9-3) has four maxima at 200 m, 36.4 m, 17.4 m and 8.3 m. The latter has very little power but it is found in other well records in an equivalent position. This is clearly all the evidence that the log can provide for the presence of the short cycles with decimetre to metre wavelengths. The power contribution of the short cycles is probably low because of the great irregularity of these fluctuations and possibly also because the gamma log attenuates the high frequencies, due to a limited resolution. When interpreting the spectrum in terms of Milankovitch cycles, one has to remember that the well-developed maximum at 36 m represents a total of only 5.96 cycles, in

154

CYCLOSTFUTIGRAPHY AND THE MILANKOVITCH THEORY

this sequence of the Kimmeridge Clay. Stratigraphically, the time series is slightly incomplete because the lowest two fossil zones are missing. According to the Harland 1989 time scale, the Kimmeridge Clay is equivalent to the Kimmeridgian which is represented by 2.6 Ma. One sixth of the Kimmeridge time is 433 Ma. This makes it likely that the 36.4 m maximum is due to the 413 ka eccentricity cycle and that the whole Kimmeridge Clay in the Warlingham section represents 2.46 Ma. If the 36 m maximum was found to be associated with either the two Ma or 100 ka eccentricity cycle, then the total times would be 12.1 Ma or 0.59 Ma. Both these values are far less satisfactory because they are either too high or too low. The 200 m maximum could be the 2 Ma eccentricity maximum but resolution at this wavelength is not good enough to support the correct interpretation of the 36 m maximum. On the other hand, the 8.3 m maximum fits well enough to be taken as the 100 ka eccentricity cycle but here the power is so low that it could be accidental. The 17.4 m maximum is not attributable to a Milankovitch cyclicity if the proposed interpretation is accepted, but it is practically twice the 8.3 m cycle and as such, would actually support the existence of this cycle. It seems very likely that the short oil shale, mudstone, calcareous mudstone cycles are partly related to the 100 ka eccentricity cycle and perhaps partly to the precession, but without more detailed work either on outcrops or on high resolution well logs, nothing can be decided. The evidence for Milankovitch cyclicity is still tentative. Again, this is because of the uncertain timescales. House (1985) gave a different interpretation of the Kimmeridge cyclicity. He counted the minor cycles for each of the ammonite zones and then calculated the lengths of the minor cycles, under the assumption that a single ammonite zone represents one sixtieth of the total Jurassic which he assumed to be 60 to 75 Ma long. This calculation (which contains errors) gives him a likely duration for a minor cycle of 41 ka. Not all the minor cycles used in the calculations are actually found in the outcrops and they have been “interpolated ”, meaning that they have been added. Others which do appear in the published data of the section have been omitted. This makes it difficult to accept the calculations and results of House.

THE GERMAN UPPER JURASSIC

The Germanic Upper Jurassic, which is normally referred to as the Malm, is divided into three parts, Malm alpha, Malm beta, and Malm gamma. Malm alpha and beta correspond to the Middle and Upper Oxfordian and Malm gamma to the Kimmeridgian. During the Upper Jurassic, large parts of southern Germany were flooded and extensive limestone and limestone-marl sediments were deposited in Swabia and Frankonia. The sea was in direct contact with the Tethys and the environment was essentially pelagic. The basin contained shallow parts which led to the development of reefs. In the non-reef areas, bedding was very regular and

SOME JURASSIC EXAMPLES

155

individual limestone and marl beds can be traced over more than 100 km (v. Freyberg, 1966). The formation of bedding in the Malm limestones was investigated in detail by Seibold (1952). Based on a large number of calcium carbonate determinations, it was possible to show that the amount of insoluble residue for successive layers, whether they were marl or limestone, remained nearly constant for long time intervals. This argues for a constant flux of clay and since limestone and marl beds contain the same quantity of clay per layer, they represent equal time intervals. Therefore, the limestone-marl alternation is a consequence of the changing carbonate flux which is superimposed on to the constant rate of clay deposition. The argument which Seibold uses, is based on the extremely low probability that a constant clay content for each layer could develop accidentally from a variable clay flux. Indeed this reasoning is the same as the reasoning behind Sander’s rule of rhythmic sedimentation. Sander’s rule is concerned with the practically impossible probability that beds of equal thickness develop from randomly changing sedimentation rates, operating over randomly changing time intervals. Seibold’s record differentiates between marl and limestone layers, which in the lower part (Malm alpha) alternate regularly so that the bedding cycle consists of two layers: marl and limestone. In the Malm beta, the marl layers are often missing and the bedding cycle is reduced to limestone beds separated by bedding planes. Taking account of this change, one finds 96 bedding cycles both in the Malm alpha and in the Malm beta and 39 bedding cycles in the Malm gama which is incomplete. To obtain a time estimate for the cycles is very difficult. If one equates Malm alpha and beta with the Oxfordian, one can choose a wide range of times. Harland (1989) gives the duration of the Oxfordian as 2.4 Ma and Kent and Gradstein (1985) give it as 7 Ma. This gives the bedding couplets a duration of 12.5 ka to 36 ka. The development of stratification cycles is particularly noticeable in the Malm alpha and beta, where a grouping of beds into bundles of five is common (see Fig. 9-4). This can be shown by spectral analysis of Seibold’s data. Figure 9-5 is the spectrum of the thicknesses of the first 100 couplets of Seibold’s section. His layers 1 to 200 have been added in pairs to make 100 marllimestone couplets. The spectrum shows a strong maximum at 19 couplets out of a hundred, meaning that there is a cycle of 5.2 couplets to form a group. If it is assumed that the 5.2 couplet peak represents the 100 ka eccentricity cycle, then the couplet time duration was 19 ka and this would fit well with the precession cycle for the Upper Jurassic. Other maxima are recorded at 211 ka, 82 ka, 57 ka and 40 ka. The last value could be related to the obliqity cycle but considerably more work is needed to confirm the identity of any of the Milankovitch cycles in the Germanic upper Jurassic. Cyclostratigraphic work in this sequence would be especially rewarding, since individual cycles can be traced over long distances. v. Freyberg (1966) has correlated individual beds of the Lower and Upper Oxfordian throughout Frankonia and it would be particularly interesting to examine the

156

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

Fig. 9-4. Lower Malm limestone Hornle Bruch near 'hbingen (Germany).

SOME JURASSIC EXAMPLES

157 lOOka

8000 -

8000 -

158ka

1I1

Assumed duration of couplet = 19ka

55.8ka

48.7ka

A

CoupletsllOO

Fig. 9-5.Power spectrum of combined marl-limestone beds from Malm alpha. Data from Seibold (1952).

distribution of the Milankovitch cycles. A pilot survey using v. Freiberg’s data found some evidence for a 100 ka cycle that produces groups of beds with a thickness of 250 cm to 300 cm (Schwarzacher, 1987a). However more work is needed to prove definite Milankovitch cyclicity.

EXAMPLES FROM TETHYAN REGIONS

Jurassic cyclic sedimentation has been described from the alpine as well as the Betic systems which were marginal to the Tethys. One of the most widely distributed facies in the tethyan area is the “ ammonitico rosso” that is a red limestone which alternates with thin layers of dark red clay. Bedding is frequently very irregular, with the average thickness of the beds ranging between 5 cm and 10 cm. Slightly thicker shale layers occur at regular intervals and produce a grouping into packets with thicknesses of 40 cm to 60 cm. In a preliminary study of the Liassic Adnet limestone from Adnet near Salzburg (Austria), an average of seven beds per bundle was found (Schwarzacher, unpublished). However, there is insufficient data as yet to make an analysis of the cycles. Better data are available from a Lower Jurassic section in the Breggia Gorge (Canton Ticino, Switzerland) in the Southern Alps. The sequence consists of the Morbio Formation which is of Pliensbachian age, followed by Toarcian Ammonitico rosso. Weedon (1989) measured three sections and generated a time series by coding the three occurring lithologies which were shale, marl and limestone as -1, 0, +l. This series was then analysed by Walsh power spectra. The coding interval

158

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

in the two lower sections was 1 mm to 1 cm and the 11 m long ammonitico rosso section, was coded to give 1024 data points. The spectral analysis of all three sections gave a strong maximum corresponding to the bedding cycle. The values obtained for the lowest to the highest section were 25.6 cm, 13.0 cm, and around 16.0 cm. The stratigraphically highest section also had a well-developed maximum at 85 cm. The lowest section had minor peaks at 256 cm, and 51 cm. To interpret the frequency maxima, Weedon made time estimates which were based on the assumption that each Jurassic zone represents 1 Ma. If sedimentation rates are based on this assumption, then they would be minimum rates and therefore they provide maximum estimates for the various shorter periods. Such estimates make it likely that the bedding cycle corresponds to the precession and the 256 cm and 51 cm cycle in the lower Morbio Formation could correspond to the 100 ka eccentricity and to the obliquity cycles. The 81 cm peak in the Toarcian section, could be the 100 ka eccentricity cycle. This interpretation has to be regarded as tentative. To obtain further confirmation for his Milankovitch interpretation, Weedon makes some calculations which attempt to quantify the stratigraphic completeness of the sections. This is done by counting the number of cycles for each fossil zone which is assumed to have a duration of 1 Ma. Since he finds similar values to those which have been quoted by Anders et al. (1987) for pelagic sediments, he regards this as supporting the assumption that the cycles represent 21 ka periods. The incomplete knowledge of the Jurassic chronostratigraphy makes any such attempt still very taniiniic

159

Chapter 10

EXAMPLES FROM THE CRETACEOUS

The Cretaceous is a period in which the cyclicity of sediments is widely developed and also relatively well studied. Cyclic marl-limestone sequences were deposited in the epicontinental sea of North America and equivalent pelagic sediments are recorded from large parts of the Atlantic. This newly opened sea connected with the Thetis and here, marl-limestone sequences have been studied in the Basco Cantabrian basin of south-east Spain and in the Vocontian basin in the south of France. Shelf and basinal developments are also found in the Apennines of Italy, where parts of the middle and upper Cretaceous have been studied in some detail and these examples will be discussed first.

THE UMBRIA AND MARCHE REGION OF ITALY

In the central Apennines, sediments of Jurassic and Tertiary age are exposed in a series of anticlines. In the area around the town of Gubbio (Umbria and Marche), several long and relatively undisturbed sections through the Cretaceous and Lower Tertiary are available (Cresta et al., 1989). They became famous through the discovery of the iridium anomaly at the Cretaceous-Tertiary boundary and through the excellent bio and magneto-stratigraphy which has been established in the sequence. A diagrammatic column of the lithostratigraphic units in these sections is given in Fig. 10-1. Cyclic sedimentation has been studied in the Barremian to Cenomanian sequences and to a lesser degree in the Paleocene and Eocene. THE MAIOLICA

The Maiolica is a light coloured, massively bedded, micritic limestone which contains calpionellids, foraminifera and radiolaria. Its age is Tithonian to middle Cretaceous and it represents a facies that is widespread throughout the Thetian realm. The cyclicity of the sequence was studied by Schwarzacher and Fischer (1982) in the Bottacione gorge near Gubbio. A section of 35 m in length was measured just below the Scisti a Fucoidi. The section is all Barremian in age and the sequence shows very well-developed bundling into stratification cycles of approximately one

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EXAMPLES FROM THE CRETACEOUS

161

Fig. 10-2. vpical Maiolica Limestone in the lower part of the Bottacione gorge near Gubbio (Italy).

metre in thickness (see Fig. 10-2). The bundle boundaries are determined by 1 cm to 8 cm of very dark shale and frequently also by black chert, which replaces the position of the shale layers and in this way becomes the cycle boundary. The beds within the bundles have bedding planes which may contain thin layers of marl but frequently contain no visible clay. Many bedding planes are weakly developed, laterally impersistent and are often replaced by horizontal stylolites, particularly towards the centre of the stratification cycles. This leads to the very characteristic appearance of the Maiolica stratification cycles, which have a massive central part and relatively thin beds (15 cm to 20 cm thick) at the cycle boundaries. Therefore, using bed thicknesses to describe the sequence, leads to curves which have minima at the cycle boundaries and maxima approximately at the centre of the cycle. The spectral analysis of a plot based on bed thicknesses, gave maxima at 400 cm and 100 cm and these were interpreted as corresponding to the eccentricity periods of 400 ka and 100 ka (Schwarzacher and Fischer, 1982). The individual, non-fused, beds with thicknesses between 10 cm to 20 cm, were interpreted as being the result of the 21 ka precession cycle. This interpretation was partly based on the frequency ratios of the observed periodicities and partly on rough estimates of the sedimentation rates in the sequence. Much more compelling evidence for the correctness of this interpretation was given by Herbert (1992) who was able to calibrate the results of his spectral analyses against the reversals of the M series, which in the meantime

162

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

had been located in the Maiolica sequence of the area. Herbert measured the section of the Cerbara gorge (some 40 km north of Gubbio) and found a duration for the bundles ranging from 93 ka to 117 ka and an average duration for the beds of 23.5 ka. He also found good evidence for a 400 ka cycle. Herbert’s results confirm convincingly, the existence of Milankovitch cyclicity in the Maiolica. Upper Barremian marls (Marne de Puaz) from the Dolomites which have been deposited in a more marginal position than the Maiolica show a well-developed cyclicity corresponding to periods of 101 ka to 113 ka, 36 ka to 56 ka and 21 ka to 27 ka (Claps Masseti et al., 1991). The sedimentation rate of the marls is estimated to be 25 mmlka.

THE SCISTI A FUCOIDI

With the Scisti a Fucoidi which follow the Maiolica limestone, a very pronounced change in the environment took place. The sedimentation changed from almost pure limestones to multicoloured red and green marls, dark shales and only very thin limestone beds. The sedimentation rate, as will be shown, decreased to half the value of the Maiolica sedimentation rate and several conspicuous black shale horizons occur in the sequence. The thickest of these horizons occurs near the base of the Scisti a Fucoidi and is known as the Selli horizon. It has an organic carbon content of about 4% and a hydrogen index exceeding 600, indicating a marine origin. However, numerous black shales which occur repeatedly in the cycles, have a much lower organic carbon content and lower hydrogen indices and the carbon is at least partly land derived (Pratt and King, 1986). The first recognition of Milankovitch cyclicity in the Scisti a Fucoidi is due to de Boer (1982). He estimated that the average time represented by a limestone-marl couplet in the section which he measured, was 23.4 ka and he found a superimposed thickness variation that caused a distinct stratification cycle of approximately five beds. Consequently, he interpreted the grouping of beds as being a result of the 100 ka eccentricity cycle and the formation of beds as a result of the precession cycle. To obtain further data, a group of Italian and American scientists cut a core through the Scisti a Fucoidi at Piobbico a small town some 50 km north of Gubbio (Fischer and Herbert, 1986). The core has a length of over 80 m and it penetrated a corrected thickness of 76 m of the Aptian-Albian sediments. Cyclicity can be seen throughout the core, but detailed information is only available for the upper part, from -9.6 to -16.82 m of the core. This corresponds roughly to the Ticinella prueticinensis zone of the late Albian, a short segment at the base of the Albian and a short segment in the middle Aptian. The limestone-marl couplets in the I: prueticinensis interval show a very welldeveloped grouping into stratification cycles with an average thickness of 50 cm. The cycle boundaries are well-defined by 1 cm to 3 cm thick black shale layers

EXAMPLES FROM THE CRETACEOUS

163

and the individual marly limestone beds show a regular variation in thickness. The central bed of the bundle is usually the thickest. This thickness variation appears to be fairly symmetrical in the analysed part of the core, but in many surface exposures, a distinct asymmetry is frequently seen. The limestones in such cases show a thickening upwards, so that the top of the thickest limestone provides the best cycle boundary. Geochemical examination (Fischer and Herbert, 1986) of the core clearly reflects the 50 cm cycles of the bundles. This is particularly noticeable in the calcium carbonate determinations which were carried out at 2 cm. intervals. To obtain a rapid quantification of the redox state, black and white photographs of the core were scanned with an optical densitometer. The grey values obtained in this way, closely follow the carbonate curve. Light colours coincide with high carbonate values. It was further found that the silica aluminium ratio followed the trend of the carbonates very closely (Herbert et al., 1986). The geochemical variation shows at least three orders of cyclicity. The most obvious period is the 50 cm bundle thickness. Superimposed on to this, is a shorter fluctuation corresponding to the individual limestone beds and a longer 200 cm wave, which modulated the amplitudes of the 50 cm cycle (see Fig. 10-3). Using an average accumulation rate of

CaCO, 90

70

50%

Fig. 10-3. Scisti a Fucoidi. Carbonate content from the Piobbico core and outcrop near the drilling site (Italy). After Fischer and Herbert (1986).

164

CYCLOSTRATIGRAPHYAND THE MILANKOVITCH THEORY

5 mm/ka, one finds that the bedding corresponds to the precessional 21 ka period, the stratification cycle corresponds to the 100 ka eccentricity and its modulation, to the 400 ka eccentricity cycle. Park and Herbert (1987) made a detailed analysis of the time series provided by the carbonate and densitometer data. They calculated Thomson power spectra from the Piobbico series and from a calculated astronomical eccentricity series, for comparison. Testing both spectra for sinusoids (line spectra), they found only one significant maximum corresponding to the 100 ka eccentricity cycle in the Piobbico series and two maxima at 128 and 97 ka in the calculated astronomical eccentricity. The series was too short to resolve the finer details of the astronomical quasiperiodicities around the 100 ka region. To obtain a higher resolution of the Piobbico record, an attempt was made to correct for uneven sedimentation rates which would have distorted the time scale. This tuning procedure was based on establishing the frequency of the most prominent cycle at around 100 ka, in thirty short overlapping sections of the time series. It was found that the frequency changed systematically throughout the record and as one would expect, followed the 400 ka cycle which is so clearly visible in the data. Using the frequency change as a correction of the time scale, resulted in a tuned record with a spectrum which showed frequency peaks at 128 ka and 97 ka. The fact that this tuned record can resolve the 100 ka eccentricity into its two major components, must be taken as very convincing evidence that one is dealing with Milankovitch cyclicity. The tuned spectra also had significant maxima at 800 ka, at 39 ka and 24.9 ka, 23.6 ka and 17.6 ka. The 800 ka maximum, is twice the wavelength of the most important eccentricity maximum which itself is not present in the tuned spectrum. The 39 ka maximum is close to the modern 41 ka obliquity maximum but the analysis did not yield any periods which could be clearly attributed to the precessional frequencies. The 24.9 ka, 23.6 ka and 17.6 ka maxima are not thought to be directly related to the modern 23.7 ka, 22.4 ka and 17.6 ka periods of the precession. Park and Herbert point out that the observed periods should be shorter and not longer than the modern ones, that their significance is low and that there are several other maxima of similar significance which bear no relation to the Milankovitch cyclicities. The absence of the 400 ka cycle in the tuned record is probably best explained by the fact that this cycle modulates the 100 ka and because it determines to a large extent, the variable sedimentation rate. Tuning the sequence, should remove its effect to a large extent. It is well-known to the sedimentologist, but perhaps not always realised in the analysis of stratigraphic time series, that there is hardly any property of sediments or indeed any stratigraphic variable, which is not in some way correlated with the rate of sedimentation. To examine further such relationships through various models, would appear to be a promising but complex problem which could yield interesting cyclostratigraphic and sedimentological results. The absence of the precessional cycle in the analysed sequence could be the result of a non-linear time scale. It is generally assumed that the limestone in a

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marl-limestone couplet represents much faster sedimentation rates than the marl. Therefore, equidistant sampling would introduce a distortion of the time scale. It is also doubtful whether the chemical composition of the sequence has been adequately sampled in order to give a sufficiently accurate shape for the carbonate variation curves. Photographs of the core show that the sediment is well banded, indicating relatively abrupt changes in lithology. Such abrupt changes are very clearly shown by the detailed foraminifera counts at 1 mm intervals (Premoli-Silva et al., 1989), which give distinct step-like functions (see Fig 4.lb). The calcium carbonate variation which was sampled at 2 cm intervals, is spiky and does not bring out the essentially banded structure of the sequence. The densitometer measurements, which apparently were made at 0.2 mm intervals (Fischer and Herbert, 1986) must have been filtered or processed in some way to obtain the published values. Unfortunately, the authors did not give any details about such procedures. Park and Herbert (1987) emphasise that the relationship between the redox state of the environment and the photometrically determined grey values of the core is complex and without doubt non-linear. It is very likely that the original signal has been changed considerably from relatively simple sinusoids, into a record containing many harmonics. It is perhaps significant that the spectra of the foraminifera counts from the same stratigraphic interval, show well-developed frequency maxima corresponding to 44 ka and 29 ka and that these maxima are present in records which have not been tuned. The three most frequently discussed possible explanations of limestone-marl sedimentation are: variable carbonate production, variable influx of clay and dissolution of the already precipitated carbonate. There is good evidence that dissolution did not a play role in producing the clay-rich layers in the Scisti a Fucoidi because it is specifically in the dark shales where the preservation of the often thin-walled foraminifera, is best. It is further found that the thickness of a marl-limestone couplet is strongly correlated with the overall carbonate content of the couplet and therefore the carbonate production must have been the main variable producing the cycles (de Boer and Wonders, 1982). It can be shown by calculating the total amount of clay and carbonate over the thickness of a 100 ka bundle, firstly that both the biogenic and detrital components fluctuate in phase with the 400 ka (200 cm) cycle. Secondly, it can be shown that the fluctuations of the carbonate values considerably exceed the fluctuations of the clay. Herbert et al. (1986) have calculated flux rates for the biogenic components and found an average of 0.84 g/cm2/ka for the carbonate and 0.11 g/cm2/ka for the biogenic silica. Both values fluctuate strongly within a bundle and for example, carbonate fluxes may vary from 0.5 g/cm2/ka to 2.0 g/cm2/ka. However, the average Albian accumulation rates are constant and almost identical with the average rates for the modern oceans. It is impossible to decide whether the terrigenous component also changed within the time interval of a couplet, because the time represented by the two lithologies is not known. This is also true for the rates of organic carbon, which in most

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cyclic black shales does not exceed 1.5% and which, from its low hydrogen index, is probably of non-marine origin. The organic carbon content therefore is part of the detrital components and could very well be an eolian product (Prat and King, 1989). The quantitative examination of micro fossils in the Scisti a Fucoidi has shown that there is a very close correlation between carbonate percentages and the count of nannofossils and so-called micarbs (Erba, 1991). Micarbs are crystallites derived from coccoliths and foraminifera. This relationship has been established in three examined parts of the Piobbico core from the Middle Aptian to the late Albian. Palaeoecological indices based on various nannofossil species of the Upper Albian, suggest a regular fluctuation between periods with high fertility and intervals of lower fertility and warmer water. This work therefore links the formation of limemarl cycles directly to fluctuations of surface water fertility. The distribution of the foraminifera provides a less clear picture. The abundance of planktonic foraminifera is not directly correlated with the carbonate content. Pure limestones contain relatively low numbers of forams and this is also the case for the dark shales. The maximum abundance lies in the marls. It is believed that in a complete precessional cycle, conditions pass from being unfavourable to carbonate formation through to a state when marls developed and forams thrived and then to a state of maximum fertility, which is dominated by calcareous nannofossils (Premoli-Silva et al., 1989).It is possible that such a sequence of events could have been responsible for the previously mentioned frequency maximum at 14 ka which is approximately twice the precessional frequency.

THE SCAGLIA BIANCA

Towards the top of the Scisti a Fucoidi, sedimentation changes from multicolored marls, black shales and limestones into a bedded limestone succession. In the lower part of the series, the limestone beds are still separated from each other by marl layers. However, with increasingly higher stratigraphic positions, the thicknesses of the marl layers decrease and individual beds become less well-defined. Bedding planes are often occupied by only thin films of clay and some bedding planes are no longer laterally persistent. Near the top of the Scaglia bianca, a very conspicuous black shale horizon is developed, that is called the Bonarelli horizon and which marks an anoxic event that can be traced world wide. The horizon is close to the Cenomanian-Turonian boundary which is approximately 3 m above it (Cresta et al., 1989). Stratification cycles have been described in the Scaglia bianca by de Boer (1982, 1983) de Boer and Wonders (1982) and Schwarzacher and Fischer (1982). The cycles in these studies were recognised from a regular variation of bed thickness which in the field, can be seen as individual bundles (see Fig. 10-4). Since such cycles contain approximately five beds, they were interpreted as being 100 ka eccentricity

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Fig. 10-4. Cenomanian Scaglia bianca from the Contessa highway section near Gubbio (Italy).

cycles and the beds were thought to be the result of precessional variation. This interpretation agreed with the very rough estimates of sedimentation rates that were available at the time. A quite substantial confirmation of this interpretation, comes from the similarity of the Scaglia bianca cycles to the cycles of the Scisti a Fucoidi. In fact, there is a gradual transition from the more marl-rich Scisti into the limestones of the Scaglia. This change involves a considerable increase in carbonate production and therefore also an increase in sedimentation rates. As the Milankovitch nature of the cycles in the Scisti is very well-established, one has some basis for assuming the same for the Scaglia bianca. If Milankovitch cycles are to be useful for detailed stratigraphic work, one has to go beyond the demonstration of their cyclic nature. It will be necessary to recognise the individual cycles and to establish the accurate number of cycles in a given stratigraphic interval. Furthermore, it should be possible to correlate between different localities by correlating cycle by cycle. The use of cycles in detailed stratigraphy, was tested by measuring two sections: the Contessa section, which is along the highway from Gubbio to Cantiano and the Petrano section, which is near the summit of Monte Petrano, along a small road leading to the village of Palecano (Schwarzacher, 1993a). The two sections are approximately 15 km apart. Both sections contain the uppermost Albian and go as far as the Bonarelli horizon. The base of the sections was taken at the first limestone of the Piobbico bore hole

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cm

Fig. 10-5.Bed thickness index (see text) of the Cenomanian Scaglia bianca. The dashed vertical lines are cycle boundariesd which have been identified in the field.

at -70 cm (Tornaghi et al., 1989). At this level, the change from Scisti lithology into Scaglia lithology takes place. The Albian-Cenomanian boundary is approximately 750 cm above this zero datum. The lower part of the Scaglia bianca shows some very well-developed bundling. Groups of between four to seven beds are separated by up to 8 cm thick marl layers. Apart from the distinct marl layers, there is also a systematic change in the thickness of the beds. Assuming that the marl layer is taken as the base of the cycle, then bed thicknesses increase in an upwards direction. The maximum thickness is reached shortly before the top of the cycle. The stratification cycles therefore produce an asymmetrical bed thickness plot and this is clearly seen in the lower third (0 to 20 m) of Fig. 10-5. The figure was constructed by plotting the thicknesses of the beds at 1 cm intervals and the plot is therefore based on 6522 points. To differentiate between limestone and marl as identified in the field, limestone thicknesses have been plotted in a positive direction and marl thicknesses as negative values. The definition of a bed, as repeatedly discussed, must be based on some agreed criteria and here, not only well-developed beds but also beds formed by intermittent bedding planes, have been used to derive a bed thickness. It can be seen from the bed thickness plot (Fig. 10-5), that marl layers which are

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thicker than 1 cm, are almost absent in the upper two thirds of the section. This makes bundling much less obvious and since the bedding is also less well-defined, the cycle boundaries can no longer be established definitely in the section from 30 m to 50 m. The loss of distinct cyclicity in the central part of the sequence is found in both localities and it was also noted by de Boer (1983), who measured a section close to the Monte Petrano locality. The change in cycle development therefore, cannot be accidental, but must be a genuine change in the signal. Such a change could be due either to changes in the basin behaviour, for example tectonic trends, or to long-range sea level changes, or it could have been a time interval during which the eccentricity variation was relatively low. It is most likely to be a combination of all the factors. The loss of cyclicity also means that the central part of the two sections in the two localities are difficult to correlate. Stratigraphic correlation on a cycle by cycle basis can be established in the lower 30 m of the section and in the upper part, from approximately 50 m upwards. In the higher parts, correlation is further aided by the occurrence of black chert bands which can be followed over large areas (Cresta et al., 1989; Boudin, 1992). To obtain estimates for the number of cycles in the central part of the sections, two methods can be used. Digital filters can be used to simplify the bed thickness curve and this can help to interpret the sections in terms of cycles. Alternatively, one can rely on cycle boundaries which have been determined during field work. Such a procedure is of course biased by the observer, but it is often better than trying to interpret some measured data which may not be representative. The field approach is particularly indicated when a section is well exposed and when stratification cycles can be traced over some distance and this allows a more precise determination of the cycle boundaries. Cycle boundaries which have been recorded in the field, are shown in Fig. 10-5 by dashed vertical lines. The reliability of the various cyclostratigraphic interpretations, can be judged by comparing the results using different methods and from different localities. Comparing the field observations for the complete sections, one finds 63 cycles from the base of the section to the Bonarelli horizon in the Contessa section and 66 cycles in the Petrano section. The number of average cycles can be obtained by spectral analysis. Since the sequence is not stationary, the sections were divided into three 20 m long pieces and spectra were calculated for each of them. This resulted in 61 cycles for the Contessa highway section and 64 cycles for the Monte Petrano section. An alternative method which is perhaps more objective than the two previous methods, is to estimate the sedimentation rate in the lower third of the sequence, where the cycles are well-developed. This rate can then be applied to the upper two thirds of the section, after it has been corrected for the increase in sedimentation rate, which is reflected by a general increase in bed thickness as well as an increase in the thicknesses of the bundles. Using this method, one obtains a value of 6.57 Ma for the interval of the section.

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Since the bundle forming cycle is being interpreted as the 100 ka eccentricity cycle, one can express the difference between the various estimates in time. One finds a maximum discrepancy of 0.5 Ma or 7% for the 6.1 Ma to 6.6 Ma long sections. In the measured section, approximately 8 bundles belong to the topmost Albian and 57 stratification cycles are between this level and the Bonarelli horizon. The Cenomanian-nronian boundary is approximately 3 cycles above the Bonarelli. Unfortunately, there is as yet, no way of estimating the time taken up by this anoxic interval, but there is no reason for assuming automatically that it was a particularly long interval. If the duration of the Bonarelli is ignored, then one obtains an estimate of 6 Ma for the duration of the Cenomanian. Even taking the uncertainties into account, this value is slightly lower than the value of 6.6 Ma given by Harland (1989). Estimating the time interval would be considerably helped if longer cycles like the 400 ka and the 2 Ma eccentricity cycle could be recognized. However, the evidence for a 2 Ma cycle in the data was disappointing when the thicknesses of the 100 ka stratification cycles were plotted against consecutive cycle numbers. This graph shows three reasonably equally spaced double peaks with maxima at 7.14 ka, 25.30 ka and 40.51 ka, and 100 ka eccentricity cycles. The maxima are roughly 1 Ma apart and since they are in pairs, the spacing could be 2 Ma. However, the last pair in particular is too long and considering that there are only three repetitions, the evidence for a 2 Ma modulation of the 100 ka cycle is not good. Slightly more convincing evidence for an effect of the 2 Ma eccentricity cycle, is obtained by averaging the thicknesses of beds at intervals of 50 cm. If this record is plotted against time and assuming a constant sedimentation rate (10 mm/ka), one obtains a curve which is shown in Fig. 10-6 (the upper curve in the Cenomanian interval). The lower curve represents the eccentricity which was calculated with the aid of Berger’s (1978) algorithm for the years 100 Ma B.P. to 60 Ma B.P. In this

Ma b.p.

Fig. 10-6. Lower curve: Filtered eccentricity values showing the 2 Ma cycle. Upper curve: bed thicknesses of the Cenomanian averaged over 50 cm intervals.

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curve, higher frequencies have been removed by applying a moving average 300 times which left the 2 Ma period as the strongest cycle. If the astronomical solutions are carried as far back as 100 Ma, cumulative errors will occur. However, ten Kate and Sprenger (1992), as discussed later in this chapter, have shown that stratigraphic data can be fitted to the eccentricity curve as far back as 65 Ma B.P. Sensitivity experiments which were carried out by the authors (A. Sprenger, personal communication, 1992), showed that phase and amplitude changes were relatively small and that dating around the CretaceousTertiary boundary can be accurate to the nearest 10 ka. Similar experiments at 100 Ma B.P., again showed a considerable stability of the calculated eccentricities and it is therefore suggested that the eccentricity curve can be used as a time scale, at least for the Upper Cretaceous. The stage boundaries which are shown in Fig. 10-6 are according to the Harland (1989) time scale. The comparison of the calculated eccentricity and the sedimentation rates, suggests that the Albian-Cenomanian boundary which Harland gives as -97 Ma, is in agreement with cyclostratigraphic dating. This comparison assumes that high sedimentation rates are associated with high eccentricities. Sedimentation rates found by de Boer (1983) show similar fluctuations when moving averages of bed thicknesses were calculated and plotted as a function of successive bed numbers. He found four maxima which were approximately 100 beds apart. If it is assumed that beds are an expression of the precession cycle and represent approximately 20 ka, then the observed fluctuations are of the order of 2 Ma. However, de Boer’s long cycles are very uneven and this is similar to the discussed variation in the 100 ka cycle. The interpretation of such cycles must be regarded as tentative. Melnyk and Smith (1989) have compared de Boer’s curve of bed thicknesses with the logs of two oil wells in the Adriatic, some 100 km east of the Gubbio area. A similar number of maxima to de Boer’s sequence is found in the filtered spontaneous potential logs of these wells. Although the correlation of the three records is very good, thanks to the easily recognisable Bonarelli horizon, it is not clear from the paper how the filtering was performed and how de Boer’s sequence was transformed to be comparable to the well logs. Melnyk and Smith and de Boer estimate the duration of the Cenomanian to be 7 Ma. There are a number of possible ways to make these time estimates more accurate. An obvious way is to use a proxy variable, which is better than bed thickness, for describing the sedimentary history. The bed thickness is difficult to define and therefore different orders of beds can be confused. This means that the variable loses its geological significance as a measure of sedimentation rates. Furthermore, using bed thicknesses as proxy variables, also means that periodicities of the order of beds and smaller than beds, cannot be resolved. If beds really are determined by the precessional cycle, then this is above the Nyquist frequency of the data. To give a better definition to the cycles, some measured quantity such as chemical composition, should be determined to tune the section. A tuned section would allow

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one to determine the changes in sedimentation rates more closely and this could provide better evidence for longer cycles such as the 400 and 2000 ka eccentricity cycle.

THE WESTERN TETHIS AND THE ATLANTIC

Cyclic sedimentation in the Cretaceous has been reported from many localities and a considerable amount of work has been done on the Lower Cretaceous in the south of France and in the south-east of Spain. Pelagic sediments consisting of marls and limestones, have been deposited in the Vocontian Basin which covers an area of approximately 12,000 km2. Stratification patterns show a variety of styles, ranging from bedded limestones to very regular marl-limestone, or marl, marly limestone, limestone cycles (Cotillion, 1991). Some of the marl-limestone cycles can be traced over the whole basin (Cotillon et al., 1980) and from the basin to the marginal hemipelagic limestones and shales (Fery and Monier, 1987). Cotillon and Rio (1984) measured the Valangian and Hauterivian sections near Angles and Vergons, approximately 25 km south-east of Digne. They plotted the thicknesses of the major cycles. A major cycle in their nomenclature, is any marl to marl couplet or multiplet which is thicker than 10 cm. The thickness plots or cyclograms, show a very well-developed cyclicity of groups consisting of four to six bed cycles. The duration of these groups was estimated to be 65 ka to 100 ka and an orbital control of their origin has been suggested. Cotillon and Rio compared the Vocontian sediments in the south of France with the Atlantic sediments of the same age in the Blake-Bahama basin at Deep Sea Drilling site 534 and with two cores from the south-eastern Gulf of Mexico (sites 535 and 540). The lithologies in these localities are strikingly similar to the Vocontian pelagic sediments. There is a regular change from limestones with more than 80% calcium carbonate to marls with less than 60% calcium carbonate but the structures of the dark shales in the Atlantic sites are different and frequently finely laminated. This may be due largely to differences in bioturbation in the Bahama sites. Minor bioturbation is only seen in the limestones and the shales are not disturbed. In the Vocontian basin, bioturbation has removed any lamination in the shales. The difference is most likely to be due to the different palaeodepths of the localities. The depth of the Blake-Bahama basin has been estimated to be 3000 m and that for the Vocontian basin to be between 500 m and 1500 m. Using bed thickness plots of the Atlantic and Vocontian sites, a tentative correlation between the localities was attempted. Although this does seem not to be possible on a cycle by cycle basis, it does at least show that the cyclicity in both realms is very similar. Cotillon (1984, 1987) derived relative sedimentation rates for the Beriasian to Aptian interval. He showed that such rates changed in a similar way in both the

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Atlantic and Vocontian basins. This information would support the correlation. The derivation of the rates that Cotillion uses, is somewhat unconventional. They are derived from the frequency of minor fluctuations of calcium carbonate content of the sediments. The relationship between the sedimentation rates and the frequencies of minor fluctuations, is an empirical one which is difficult to explain. It is interesting that the average period of the sedimentation rate fluctuations, is near 2 Ma. Both the Blake-Bahama Formation and part of the sequence near Angers in the Vocontian basin, were studied by Zehui Huang (1991). In this study, data from DSDP holes 391c, 534a and 603 have been examined. Using core photographs, visual colour estimates were made and the lithologies and structures were coded in such a way that they could be expressed in a time series, with values between plus and minus one. These series were step functions and were analysed using Walsh power spectra. The spectra showed frequency maxima at approximately 20 cm to 30 cm, 50 cm to 70 cm and occasionally at 200 cm. Zehui Huang interpreted the maxima around 60 cm as being due to the obliquity. Maxima which could have corresponded to the 100 ka eccentricity cycle are rare, but this could be due mainly to the shortness of the series. The average core length in the bore hole is 9.5 m and the eccentricity maximum is expected to be at 200 cm or more and therefore it would not have been repeated sufficiently to provide a good estimate. After calibrating the spectra in terms of Milankovitch cycles, the observed number of cycles was used to calculate accumulation rates that range from 85 mm/ka to 281 mm/ka. Zehui Huang’s calculations suggest that there was a steady increase in sedimentation from the Beriasian to the Hauterivian. The rates therefore show no similarity to the sedimentation rates found by Cotillion. Zehui Huang also measured two sections in the Angers Vergons area, a 31 m long section in the Valangian and a 14.4 m long section in the Hauterivian. The sections were coded as a two state system of shale and limestone and analysed by Walsh power spectra. The analysis gave maxima around 60 cm (corresponding to the bedding cycle) and maxima between 120 cm and 190 cm, as well as a very minor maximum at 300 cm. The latter was interpreted as being the 100 ka eccentricity maximum and the main maxima at between 120 cm and 190 cm was interpreted as being the obliquity maximum, corresponding to approximately 41 k. Once again the sections are too short to allow the 100 ka eccentricity maximum to be properly developed, if the sedimentation rates that Zehui Huang deduces are correct. Considering the unusually high power that his analysis attributes to the obliquity maximum in both the Atlantic and Vocontian basin and considering that the frequency maxima in his analyses showed a wide range, one would like to see the analyses extended to longer and perhaps properly tuned sections. The cyclicity of the lowest Cretaceous in the Vocontian basin and in southeastern Spain was examined by ten Kate and Sprenger (1988) and Sprenger and ten Kate (1993b). The Spanish section is 77.5 m long and is late Beriasian in age. It comes from the Miravete Formation near Caravaca (Province Murcia) and it

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comprises the two Calpionellid zones D1 and D2. The marl-limestone sequence is well-bedded, each bedding cycle consisting either of marl-limestone, or marl-marly limestone-limestone alternations. To obtain an unbiased record of the stratigraphic variation, the sequence was sampled at intervals of 7 cm in the limestones and at 5 cm intervals in the shales. Cubic splines were used to obtain a series of 1239 equally spaced data points. Power spectral analysis of this series, gave maxima at 3000 cm, 666 cm and maxima around 133 cm to 92 cm. Since the sedimentation rates for the 77 m long section were not known, it was assumed that the total section could represent anything between 0.5 Ma and 5 Ma. When calculating the lengths of the various cycles represented by the frequency maxima in the spectrum, it was found that a match with Milarikovitch cycles is only obtained when the length of the sequence is taken to be between 1.1 Ma and 1.5 Ma. In this case, the 3000 cm cycle matches the 400 ka eccentricity cycle, the 666 cm cycle matches the 100 ka eccentricity cycle and the shorter cycles are a match for the precession. Sprenger and ten Kate (1992) subdivided the total section into 10 overlapping segments and calculated individual spectra for each subsection. This showed that only the central part of the sequence developed a well-defined 100 ka eccentricity maximum. This is probably because of changes in sedimentation rates in the lower and higher parts of the section. The changes in sedimentation rates appear to be systematic and proceed from segment to segment, so that it is likely that the 400 ka eccentricity cycle is responsible for this change. Such a modulation by the 400 ka cycle would be similar to the situation observed in the Piobbico bore hole which was discussed earlier in this chapter. The power spectrum of the central part of the section from 11 m to 67 m, has very well-defined frequency peaks. If the most prominent maximum is fixed at 100 ka, then there are two clear precessional maxima at 17 ka and 20 ka, as well as two (albeit not very strongly developed) maxima at 52.9 ka and 40.9 ka which agree well with the obliquity periods. A second section which contains almost the complete D1 zone of the Beriasian, was measured in the Vocontian basin near La Faurie (south-east France). This section is about 950 km north west of the Spanish section and 740 samples were taken and analysed for calcium carbonate. Since it was established that the Spanish section near Caravaca showed only minor changes in the sedimentation rates, it was taken as a standard and the French section was tuned to it. The tuning was achieved by visually choosing tie lines between the sections and by linearly shrinking or expanding the segments defined in this way, to equal thicknesses and by using the Caravaca section as a standard. This procedure increased the quality of the spectrum of the French section and maxima corresponding to the 100 ka eccentricity and the 20 ka and 17 ka precession cycles were found. Peaks with higher frequencies were interpreted as being due to relatively rapid changes in the sedimentation rates. The tuning also brought out a maximum at 42.9 ka that corresponds to the obliquity and this maximum is very much better developed than in the Spanish

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section. The higher power of the obliquity maximum is in accordance with the higher palaeolatitude of the French section, compared with the Spanish locality.

THE EPICONTINENTAL SEAS

During the Middle and Upper Cretaceous, large parts of the West European continent and the central interior of the North American continent, were flooded by relatively shallow seas which deposited semi-pelagic and pelagic sediments. These sediments will not be discussed in detail but they are important for cyclostratigraphic studies, largely because individual cycles can be traced over very long distances. Facies changes or the absence of facies changes, in such regional studies can give an added insight into the formation of cycles. In the United States, two Cretaceous horizons show particularly well-developed cyclicity. These are the Bridge Creek Member of the Greenhorn Formation and the Fort Hayes Member of the Niobrara Formation (Hattin, 1986). The Bridge Creek Member can be traced over several thousand miles but the main facies change occurs in a west to east direction. The west provided the source of clay that continuously decreases towards the east, where massive chalk is deposited. The cycles that consist of marl-limestone alternations on a decimetre scale, decrease in thickness in an easterly direction. Based on this regional distribution, Hattin (1971) interpreted the cycle as being the result of variable clay influx. The cycle could have been controlled by climatic variations (Baron et al., 1985), when arid periods alternated with more humid periods. During the arid periods, the terrestrial runoff was low and the water column was well circulated and supported a benthic fauna that caused extensive bioturbation in the limestones. During the pluvial periods, the sediment laden runoff formed a widespread layer of low salinity water near the surface, bottom conditions became anoxic and bioturbation in the shales was minimal. Fisher (1980) believes that the best estimate for the duration of a marlshale cycle is 40 ka, which would make it obliquity controlled. According to Eicher and Diner (1991), the hypothesis of a brackish water layer on the surface is not tenable because of the highly diverse foraminifera assemblage in the marl stones. For this reason, the authors favour a productivity model, rather than the dilution model which was outlined above. The younger Niobrara Formation and in particular the Fort Hays member, was deposited under very similar conditions to the Bridge Creek member (Hattin 1986). The limestone-marl couplets according to Fischer (1991), represent a time interval of 28 ka. The beds appear in sets of five and these are interpreted as corresponding to the 100 ka eccentricity cycle. A proper time series analysis of the American Cretaceous sequences appears to be lacking. In particular, the question of whether the obliquity or the precessional cycle was predominantly developed, has not yet been resolved.

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There are also very few quantitative data about the cyclicity in the European relatively shallow water sediments. A detailed quantitative study is available from two cores taken from the Albian Gault clay in the south of England (Erba et al., 1992). Spectral analysis of the abundance of Biscutum constans and Watznaueria burnesae, which are two calcareous nannofossils together with calcium carbonate determinations, all gave frequency peaks at around 100 ka and relatively strong peaks corresponding to 50 ka to 40 ka. The timing is based on an average sedimentation rate of 5 mm/ka and therefore it cannot be very accurate but the Milankovitch nature of the cycles is well established by the spectra. As expected, it was found that the abundance of B. constuns which is an indicator of surface water fertility, is negatively correlated with W barnesae, which is linked to lower fertility. However both abundances are out of phase with the calcium carbonate content. Thus the latter cannot be linked to surface fertility. The Cenomanian chalk marls of the Anglo-Paris basin show very distinct cyclicity and marl and limestone cycles of decimetre scale can be traced over a very wide area (Gale, 1989). An attempt to interpret the chalk marls in terms of Milankovitch cycles was made by Hart (1987), who determined the average thicknesses of the limestone-marl couplets in different intervals of the Cenomanian. He obtained an estimate of the number of cycles by dividing the measured thicknesses of each interval, by the mean couplet thickness. In this way, he estimated a total of 101 couplets for the Cenomanian. He assumed that the bedding is controlled by the obliquity cycle which he took as representing 41.5 ka and which gave him a total time of 4.91 Ma for the section he investigated. Allowing for missing cycles at the base of the sequence, he estimated the time interval of the Cenomanian to be 5.9 Ma. Gale (1989) found that the limestone-marl couplets are arranged in groups of five and range in thickness from 1 m to 5 m. Such stratification cycles have been interpreted as representing the 100 ka eccentricity cycle. Gale compiled a section of the complete Cenomanian, using French and English exposures and found a total of 44 stratification cycles. This would make the duration of the Cenomanian 4.4 Ma, which is considerably less than the estimate obtained in the Gubbio area, which was 6.5 Ma. Gale believes that the discrepancy could be due to a different biostratigraphic dating of the base of the Cenomanian, or possibly to a considerable gap at the Albian-Cenomanian boundary. The existence of numerous small gaps seems a less likely explanation, otherwise it would not be possible to correlate the cycles. It must be remembered that the timing is done by counting 100 ka intervals and if gaps occur within such cycles, they are not relevant as long as the cycle itself is counted. The discrepancies between Hart and Gale are considerable. Hart estimates that the Cenomanian contains 101 plus possibly 42 bedding cycles, and Gale for the same interval, estimates 220 bedding cycles. However, Gale recognises that in some cases, the beds can fuse and that the assumed 100 ka cycle may then consist of only a single

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bed. Despite the difference in the estimates, both authors arrive at approximately the same duration for the Cenomanian. This clearly arises from the two different assumptions that were made about the length of the bedding cycle which Hart believes to be controlled by the obliquity and which is approximately twice as long as the precession favoured by Gale. Cottle (1989), working on the abundance of foraminifera in the Turonian chalk, also attributes considerable importance to the obliquity. It is generally believed that the obliquity influence becomes stronger with higher latitudes, but it must be noted that there is as yet, no well documented example of sedimentation in which the obliquity cycle has more power than the eccentricity cycle. Regular stratification is also found in the Upper Cretaceous chalk and bedding planes and flint layers are often formed by a change in bottom current activity that is either responsible for omission surfaces with evidence of winnowed sediment, or actual erosion surfaces that are often developed as hardgrounds (Kennedy and Garison, 1975). Such cycles have been called scour cycles (R.O.C.C. Group, 1986). The average interval represented by bedding planes is estimated to be 40 ka to 50 ka and stratification is believed to be controlled by the obliquity cycle (Hart, 1987).

THE CRETACEOUS-TERTIARY BOUNDARY

The cyclicity near the Cretaceous-Tertiary (C/T) boundary has been investigated by ten Kate and Sprenger (1992) in three Spanish localities and in the Gubbio area by Schwarzacher (1987). The C/T boundary is well established in the Bottacione gorge near Gubbio, Italy (Alvarez et al., 1977). It occurs in the Scaglia rossa where, apart from an approximately 40 cm thick group of white limestone beds followed by 2 cm of clay, there is no lithological change. The following Paleocene and Eocene Scaglia rossa consists of thin limestone beds interrupted by thin layers of dark red mark. The average bed thickness near the boundary is 6 cm and the beds show a definite bundling into groups which are 20 cm to 50 cm thick. The boundaries of such stratification cycles are often formed by slightly thicker marl layers, which are more easily weathered and are in this way responsible for the bundling effect. The positions of the cycle boundaries were determined visually during field work and in some cases, this can be prone to bias. However, in the cumulative presentation shown in Fig. 10-7, interpretation errors only affect the horizontal scale and not the vertical. Exact positions can be found in Schwarzacher (1987). The straight line in Fig. 10-7 represents the time-accumulation relationship, as calculated from the magnetic chrons 29N to 26N and which gave a sedimentation rate of 3.81 mm/ka. Assuming that the recorded cycles each represent 100 ka precisely, one obtains an accumulation rate of 3.34 mm/ka and this means that the interpretation of the cycles as 100 ka eccentricity cycles is very convincing. Figure 10-8 shows the deviations of

178

3474 64.5

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

I

64

63.5

63

62.5

62

Ma b.p.

61.5

61

60.5

,

60

I

59.5

Fig. 10-7. Cumulative positions of cycle boundaries above the C/T boundary plotted as 100 ka increments. The straight line represents constant sedimentation at a rate of 3.81 mm/ka, which is derived from magnetic chrons 29N and 26N. Bottacione gorge near Gubbio (Italy).

Fig. 10-8. Deviations of cumulative cycle thicknesses from a constant sedimentation rate (dotted line) compared with eccentricity values (solid line with a nominal scale).

the cumulative plot from a fitted straight line and it indicates systematic changes in the sedimentation rates that have a period of approximately 2 Ma. If one compares the deviations with the calculated eccentricity curve for the interval of 64.5 Ma B.P. to 59.5 Ma B.P., one finds that the two are in phase and that high sedimentation rates coincide with high eccentricity values. This is similar to what was found for the Scisti a Fucoidi. The Bottacione section near Gubbio is unfortunately somewhat disturbed by tectonics and minor slumping. The cycle description is entirely based on field descriptions. Much better data come from the Spanish localities, investigated by ten Kate and Sprenger (1992). The northernmost locality is in the Basque country near Zumaya in the Bay of Biscay. The Maastrichtian here, consists of reddish purple calcsiltites and marls

EXAMPLES FROM THE CRETACEOUS

179

with a variable carbonate content. The sequence also contains a large number of turbidites, ranging in thickness from 0.1 cm to 12 cm. A 6 cm thick clay layer separates the Danian from the Maastrichtian, which is developed as well-defined limestone-marl couplets. This part of the section also contains turbidites. The two localities, Agost and Relleu, are within 30 km of each other in the province of Alicante in the south-east of Spain and approximately 600 km from the northern locality. The lithologies in these localities consist mainly of olive grey to cream coloured marls that are interrupted by calcsiltite layers in the Maastrichtian and by massive, strongly indurated calcsiltites, in the Danian. Because the sedimentation rates in the Maastrichtian are about twice as high as in the Tertiary, samples in the lower part were taken at 4 cm intervals and at 2 cm intervals in the upper part. At the Zumaya locality, samples in the Maastrichtian were taken at 10 cm intervals and in the Danian at intervals which averaged 4 cm. For each sample, the calcium carbonate content was determined and for the Zumaya samples, the magnetic susceptibilitywas also determined. The first correlation between the three sections was based on the position of the C/T boundary, which was known accurately in all the sections, as well as on the known positions of the four magnetic reversals. These few tie lines provided some sedimentation rates which showed a wide variation from about 6 mm/ka to 35 mm/ka. The sections were roughly adjusted to a constant sedimentation rate and were compared with the calculated eccentricity curve for the interval of 69 Ma B.P. to 63 Ma B.P. this is shown in Fig. 10-9. In this comparison, it is already possible to find some visual matches between the sections. By repeatedly adjusting for sedimentation changes, it is possible to achieve a very high match indeed between the eccentricity and the three sections (see Fig. 10-10). If a power spectrum of the adjusted sequence is calculated, very clear Milankovitch periodicities appear. The 400 ka and 100 ka eccentricity maxima are well-developed and at least two precessional maxima can be seen. Maxima in the obliquity bands are present but weakly developed (see Fig. 10-11). When transforming the stratigraphical sequence into a time series, turbidites which are regarded as having been formed instantaneously, have been eliminated. The frequency of the occurrence of turbidites itself showed a distribution which was, at least in parts, influenced by Milankovitch cycles. Spectra of the turbidity frequencies show the same maxima as the carbonate spectra but in addition, they show some relatively strong power in the obliquity band. Since it was possible to fit the sedimentation history to the calculated eccentricity, it was also possible to give an accurate age to the C/T boundary. Under the assumption that the quasi-periodicities of the eccentricity are stable, the CIT boundary can be dated as 64.56 Ma. This is the first time that an important prePleistocene event has been dated exclusively with cyclostratigraphic methods and it is also probably the most accurate datum in pre-Pleistocene stratigraphy.

180

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

AGOST

RELLEU

CaCO3 (XI

CaCO3 (XI

IMOOTHEt

IMOOTHEt

uu

3

I

I

Fig. 10-9. The analyses of three sections across the Cretaceous-Paleogene boundary, compared with the eccentricity. From ten Kate and Sprenger (1992).

CONCLUDING REMARKS

The study of Cretaceous cycles has contributed considerably to the understanding of cycle formation, as far as sedimentary (and by implication) palaeoclimatic conditions are concerned. Such problems will be discussed in a later chapter but with regard to the cyclostratigraphic working methods, the following has emerged. Experience from the Cretaceous examples, has shown that it is probably most profitable and often considerably more convenient, to work on core material and

181

EXAMPLES FROM THE CRETACEOUS

ZUMAYA CoC03

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bore hole data rather than on surface exposures. However, the latter cannot be neglected, not only because nobody can afford an indefinite number of bore holes, but also because cyclostratigraphic studies must go together with regional sedimentary studies (see, for example, Hattin, 1986). It is obvious that meaningful

182

CYCLOSTMTlGFUPHY AND THE MILANKOVITCH THEORY el

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02

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0.05 50.0

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16.7

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Frequency (cyclei/A) C y c l s l e n g t h i n ka

Fig. 10-11. Power spectrum of the tuned carbonate content from the Zumaya section (Spain).

quantitative data, such as faunal and chemical analyses are better than substitutes such as field identifications of rock types or bed thicknesses. However, such methods will again have to be used if nothing better is available. Particularly important is the development of automated or at least rapid data acquisition methods, such as optical scanning (Fischer and Herbert, 1986) and magnetic susceptibility measurements (ten Kate and Sprenger, 1992), which are particularly promising. The second topic concerns the possible achievements of cyclostratigraphic analyses. The examples so far have demonstrated that the study of cyclicity is particularly important for calculating sedimentation rates and for recording changes in sedimentation rates. The observation that shorter cycles, for example the 100 ka eccentricity, are modulated through changes in sedimentation rates by longer cycles such as the 400 ka and 2 Ma periods, has implications in stratigraphic correlation. It is clearly easier to correlate long cycles than shorter cycles and by using the hierarchial properties of Milankovitch cycles, it should be possible to make very accurate correlations. Attempts to time complete stratigraphic stages like the Cenomanian for example, will undoubtedly be achieved eventually but it may be necessary to obtain data from composite sections to eliminate local disturbances and to limit global effects such as non-cyclic sea level changes. Finally, the analysis of cyclicity near the C/T boundary has shown that it is possible to achieve the ultimate aim of any stratigraphic method, which is to obtain a reliable absolute date for stratigraphic events. The possibility that sedimentary proxy data can be directly compared with the eccentricity, will make it possible t o produce at least the skeleton of a time scale for the Upper Cretaceous.

183

Chapter 11

NON-CARBONATE CYCLES

A study of the Milankovitch cycle literature shows clearly that the majority of the examples which have been investigated, involve the sedimentation of carbonates. This does seem to indicate that carbonates are particularly sensitive to the type of changes which result from variable insolation. This sensitivity may be largely the result of the organisms which determine carbonate sedimentation to a considerable extent and which are sensitive to climatic changes for biological reasons. Other factors favour carbonates as a recording medium for Milankovitch-type cycles. In particular, the sedimentation rates of many carbonate environments are such that cycles of 10 ka to 100 ka duration are of decimetre to metre scale and are easily seen. There are, however, other environments in which Milankovitch cycles have been recorded. These are sometimes at least as sensitive or even more sensitive than carbonate sediments. A study of such systems is important because they provide additional and sometimes more direct evidence of climatic variations.

THE PERMIAN EVAPORITES

The dependence of evaporite formation on climatic controls has been known for many years. Anderson (1982, 1984) was able to demonstrate that the effect of orbitally induced climate changes could be seen in the sedimentation of the Permian evaporites in the Castile Formation of New Mexico and Texas (U.S.A.). This formation has been deposited in the Delaware basin, which was a basin that was almost completely surrounded by reefs and which in this way, became isolated from the open sea. The only communication with the sea was through a narrow gap containing a shallow sill, which must have prevented any deep water circulation. The Castile Formation is underlain by siltstones that change into laminated mudstones with layers which are rich in organic material. The evaporite sequence starts with a basal limestone and is approximately 600 m in thickness, with clear laminations throughout. It contains four anhydrite units measuring from 30 m to 200 m in thickness and these are separated from each other by units containing halite. The basal limestone is well laminated, consisting of an evenly grained mosaic of calcite crystals that are separated by thin films of organic material. The anhydrite units contain laminae which commence with a thin layer of dark limestone and

184

CYCLOSTRATI GRAPHY AND THE M ILANKOVITCH THEORY

rnrn

7

04 0

50

I

100

1

150

200 ka

Fig. 11-1. Smoothed plot of absolute thickness of laminae in the Permian Castile formation ( U S A . ) . After Anderson (1984).

are followed by the anhydrite, which is an interlocking aggregate of subhedral to euhedral crystals. The thicknesses of the laminae are very even, with an average of 1.9 mm. It has been possible to correlate individual laminae over a distance of 113 km and it is likely that most of the laminae can be correlated throughout the basin (Anderson, 1982). The high regularity of the laminae and their lateral persistence, makes it almost inevitable that one is dealing with genuine varves. Since the lamination can be recognised in all the lithologies, it has been possible to construct a varve chronology for the whole Castile Formation. With this data, one can construct a time series of the annual carbonate and sulphate production in this evaporite basin. Figure 11-1 shows the sulphate production for a time interval of 120 ka. The thicknesses of the anhydrite lagers are represented by taking running averages over 8 ka. The series shows a fairly regular cyclic variation. If one takes the spacing of the maxima in this curve, one finds an average wavelength of 19.2 ka. The theoretical precessional cycles for the late Permian are 21.5 ka and 17.8 ka (Berger et al., 1989) and the observed period therefore is in good agreement with these values. Furthermore, the sulphate curve shows a broad trend which could have easily been produced by a 100 ka cycle. It seems therefore very likely, that the evaporite formation was partly controlled by the precessional and eccentricity cycles. Spectral analysis showed no evidence of an obliquity cycle. Apart from the long range variation seen in Fig. 11-1, there is a very noticeable shorter cycle with an average thickness of 4 m. It is caused by a gradual increase in the anhydrite part of the varves and a decrease in the carbonate. This may lead to a stage when anhydrite is followed by halite and halite varves are developed, which can be up to 10 cm in thickness. This cycle clearly indicates a salinity change in the

NON-CARBONATE CYCLES

185

Castile basin which occurred periodically. Not all the salinity cycles are complete and the halite part may be missing. The time interval represented by this cycle, is between 2 ka and 3 ka with an average of 2.7 ka. The amplitude of this cycle increases noticeably with stratigraphic position. Anderson interprets this as being an effect of the progressive shallowing of the basin, indicating that the hydro-climatic variable which is responsible for this cycle, became increasingly effective. Additional insight into the mechanics of cycle formation, is gained by looking at the cospectra of sulphate and carbonate production. Anderson finds that there is a strong positive coherence between these two variables at fluctuations which are about 600 years. Fluctuations with a wavelength of 2 ka to 3 ka, show a distinct negative coherence which again becomes positive for fluctuations above 10 ka. A possible explanation for this, is that the short and very long variations are directly caused by temperature changes. A rising temperature has the effect of lowering the solubility of carbonates and at the same time increasing the evaporation, so that both the carbonate and sulphate production increase together. The 2 ka to 3 ka cycle on the other hand, could be due to the freshening of the Delaware basin waters by an influx of lower salinity water from the open sea. It is known from the relationship of the basin sediments to the surrounding reefs, that the water depth in the basin must have been quite deep and this implies that the evaporated water must have been replaced occasionally. The introduction of fresher water reduces the solubility of calcium carbonate and sulphate differentially and carbonate production can continue, while the sulphate production decreases or stops. The result is a negative correlation between the two components (Dean and Anderson, 1978). This effect could be heightened by the carbonate that has been produced due to organic activity and which would have increased with the influx of fresh water. The evidence of the carbonate sulphate coherence seems to suggest that cycles which are of the order of precession cycles, are directly related to temperature changes and this could be a direct function of insolation. On the other hand, fluctuations of the order of the eccentricity cycle show a similar coherence. The temperature changes due to eccentricity variations, however, would have been quite small and this seems to indicate that the dependence of sulphate and carbonate production on astronomical variations, is probably quite complex and cannot be interpreted simply in terms of temperature changes. The salinity cycle has an average wavelength of 2.7 ka and is not easily explained by the normal Milankovitch frequencies. Anderson (1982) draws attention to the similarity of this cycle to the periods found in neoglacial cycles of the Holocene. For example, Pestiaux et al. (1988) found evidence of increased climatic variability in frequencies of 1/26 ka to 1/37 ka, in the spectra of isotope data from late Pleistocene deep sea sediments. Short cycles like these, have been explained as corresponding to combination tones of the two precession cycles with a 19 ka and 23 ka period, together with the 41 ka signal of the obliquity. Very similar combinations can be generated with the slightly shorter 21.5 ka and 17.8 ka precessional cycles of the late Permian. However, the

186

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

orders that are needed to generate short cycles like these are high (between five and nine) and one would expect to find some evidence of the lower orders also. Clearly the problem should be investigated further, using high resolution spectral analysis. It must also be kept in mind that the salinity cycles are short enough to have been generated by a purely terrestrial system of oscillations in the oceans and atmosphere.

LACUSTRINE ENVIRONMENTS: THE LOKATONG AND PASSAIC FORMATIONS

Towards the end of the Trias and prior to the opening of the Atlantic, a number of rift valleys developed along the eastern margin of North America. Some of these basins were occupied by lakes and were filled with lacustrine sediments and igneous rocks. The largest of these basins is the Newark basin in the states of New Jersey and Pennsylvania (U.S.A.). The Newark basin runs for approximately 400 km in a south-westerly to north-easterly direction and is about 30 km wide. There is a general dip towards the north-west. The oldest Middle Carnian Stockton is surfacing along the south-eastern margin of the basin and is at a depth of over 6000 m, at the north-west margin. The Stockton Formation is followed by the Upper Carnian Lockatorig Formation and the Norian Passaic Formation. Van Houten (1964) first recognised the cyclic nature of the Lockatong Formation and differentiated between two types of cycles which he named detrital and chemical cycles. The more common one is the detrital cycle which consists of three phases or divisions (Olsen, 1984): a basal massive or platy grey mudstone which sometimes contains current bedding, a fine dark laminated shale and a grey massive mudstone (Fig. 11-2). The base is interpreted as being the transgressive stage. The laminated part represents the high stand of the lake and the top of the cycle records a renewed regression. The lake frequently dried up completely, as indicated by shrinkage cracks and intraformational breccias. The detrital cycles are usually about 5 m in thickness. The chemical cycles are slightly thinner and contain analcym and dolomite, which can form pseudomorphs after gypsum, glauberite and halite. Van Houten assumed that the chemical cycles resulted from the drying of lakes, which had not reached an outlet and therefore accumulated more salt during their lifetime than lakes with detrital cycles which had been through flowing, at least for some time. Van Houten noticed further that chemical cycles do not occur at random but that they form clusters of five to six, in which one to four of them may be chemical cycles. Such bundles have a thickness of around 25 m and they can be further grouped into bundles of four, resulting in a 100 m cycle. Van Houten’s work was most successfully continued by Olsen (1984, 1986), who was able to trace and correlate individual cycles over large distances. He found that the Newark basin represented one large lake that covered at least an area of 7000 km2. The deep water facies of millimetre thick laminae, which have been interpreted as vanes,

187

NON-CARBONATE CYCLES

(%)

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40

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=

Microlaminated black claystone Laminated grey claystone Thin bedded massive grey siltstone

0

Fig. 11-2. A representative cycle of the Late Triassic Lockatong Formation ( U S A . ) . After P.E. Olsen (1984).

exists over the whole area of the lake and deposition must have occurred at a level below any possible disturbance by waves. Based on the wide fetch of the lake surface and assuming normal wind Speeds, Olsen estimates that the depth of the lake during its high water stages must have been in excess of 100 m. The lateral tracing of cycles also showed that the entire lake became very shallow or dried out completely during regressional stages. The water level fluctuations within one cycle have therefore been very considerable. Olsen designed a ranking scale that was based on the sediment fabric, total carbon content and preservation of fossils. These are all properties which can be related to a relative scale of the water depth during sediment formation. This scale can be used to give a detailed description and it can be used to create a continuous time series which can be analysed. Power spectra have been calculated for a number of the sections and Fig. 11-3 gives the spectrum of the middle Lockatong Formation. The spectrum has a maximum at 96 m, two very prominent peaks at 32 m and 25 m and two well-developed peaks at 6.1 m and 5.5 m. In order to assess these spatial data in terms of time, van Houten and Olsen used two methods: the estimation of sedimentation rates from varve counts and a comparison of the sequence with radiometric dates. The first method gave an average rate of 314 mm/ka and the second method gave between 200 mm/ka and 314 mm/ka, depending on which time scale was being used. Using a 240 mm/ka rate, gave the following values for the peaks in Fig. 11-3: 400 ka, 133 ka, 100 ka, 25.5 ka and 22.8 ka. This must be regarded as being a very good

188 21.8

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY 133ka

1 01

15.6

C

400

100ka

L

3

9.4

3.2 0.060

0.i3Q 0.208 Cycles/metre

0.277

0.322

Fig. 11-3. Power spectrum of the Middle Lockatong Delaware river section (U.S.A.). After P.E. Olsen (1986)

match for the eccentricity and precession periods. Both the frequency ratios and the absolute values, strongly support the interpretation that one is dealing with true Milankovitch cycles. The spectrum also has a statistically insignificant maximum at 41.9 ka, which could be an effect of obliquity variations. The locality must have been very close to the palaeoequator, where the effect of the obliquity variations would have been low. It is tempting to assume that the record of the water level fluctuations in the Newark lake, is directly related to insolation, in the sense that high insolation means high temperatures and high evaporation and therefore low water levels. However, although this connection must certainly exist in principle, the functional connection between water level and insolation may be quite complicated. Evaporation and precipitation are part of a complex climatic system which is not a linear function of insolation. As in the case of the Permian evaporites, one has to assume that the strong eccentricity signal indicates that the seasonality of the climate has been an important factor. Perhaps it was the seasonality that largely determined the precipitation.

THE GREEN RIVER FORMATION

The Green River Formation of Wyoming and Colorado (U.S.A.) is not only one of the most studied lake deposits, but it has also considerable historical interest for Cyclostratigraphy. As mentioned in the first chapter, Green River cycles of oil-shale and mudstone were interpreted by Bradley (1926) as corresponding to Milankovitch cycles. He based this interpretation on sedimentation rates, which he obtained from laminated oil shales and which he interpreted as being lacustrine varves. Bradley originally thought that the Green River sediments were deposited in a large perennial lake which he named lake Gosiute. With the additional evidence

NON-CARBONATE CYCLES

189

that came in particular from boreholes, these original interpretations had to be somewhat modified. The Green River Formation consists of three members. The lowest Tipton shale is some 50 m thick and contains mainly oil-shales. It is followed by over 100 m of alternating shales and mudstones which are called the Wilkins peak member, and these are followed by about 70 m of Laney shale. Modern interpretations tend to explain the Green River Formation as being composed of sediments that formed in a large Playa Lake basin (Eugster and Surdam, 1973). During the Tipton shale time, a very shallow lake may have existed but this was surrounded by large Playa-like mudflats. The Wilkins Peak member was the most arid period, when lacustrine conditions were restricted to very shallow ephemeral lakes which could give rise to oil shale formation and frequently, layers of trona (sodium carbonate) and occasionally halite were formed. In the Laney shale period, the lake probably changed from an alkaline lake into a fresh water lake, a transition which probably indicates the change from a closed to an open basin regime (Surdam and Stanley, 1979). According to Bradley’s original interpretation, the varves in the oil-shales were formed by a precipitation of carbonate during a seasonal warming of the lake and this was followed by annual algal blooms that were responsible for the organic content of the shales. This interpretation is clearly impossible in a Playa lake environment. Eugster and Hardie (1975), when describing the oil shales in the Wilkins Peak member, assume that the organic material was derived from bottomdwelling blue green algae and fungi. During occasional floods, these were covered by carbonate and quartz grains that were brought in from the surrounding Playas. Such floods could have been seasonal, but were not necessarily so. However, there are short cycles in the laminations of the oil-shales which confirm their varve nature. Bradley noticed a regular change in the varve thicknesses, with an average length of slightly under twelve years. This was interpreted as being an effect of the 11 year sunspot cycle and this is confirmed by the results of power spectral analysis (Crowley and Duchon, 1986; Ripepe et al., 1991). Power spectral analyses show frequency maxima clustering around 1/11 years and 1/5 years. The eleven year cyclicity is very likely to have been caused by the sun spot cyclicity and is therefore indirect proof of the varve nature of the laminae. The shorter cycle, which is approximately one half of the solar cycle, has been interpreted as being a harmonic by Crowley et al. However, Ripepe et a1 believe it to be real and describe it as an El Niiioj-type oscillation which is a somewhat controversial climatic cycle occurring on the north-westem coast of Peru at the present time. Although the varve nature of the oil-shale laminae seems to be confirmed, it is doubtful whether one can use sedimentation rates derived from the oil-shales, to calculate the total duration of the Green River Formation. Considering how frequently the lacustrine sedimentation was interrupted, the sedimentation rate can only be given as a very rough estimate. The most prominent larger scale cycles are seen in the Wilkins Peak member, with cycles consisting of flat pebble conglomerates, oil-shales and mudstones that

190

CYCLOSTRATIGRAPHY AND THE MILANKOVITCH THEORY

indicate an expanding and shrinking lake. This basic cycle is often modified by the appearance of lime sandstones and also trona. The average thickness of the cycles is between 246 cm (from data given by Bradley, 1926) and 235 cm (deduced from gamma-ray logs, see Fischer and Roberts, 1991). Both Bradley and Fischer et al. interpret this cycle as corresponding to the precessional period and if one uses an average sedimentation rate of 120 mm/ka, one obtains time values of 19.6 ka to 20.5 ka and this appears to confirm the precessional cycle interpretation. Fischer et al. believe that they can see some regular grouping in the maxima of the gamma and sonic velocity logs of a well in the Tipton and Wilkins Peak members. They interpret this grouping as corresponding to eccentricity modulation of the precession. Unfortunately, this interpretation has not yet been supported by spectral analysis and it is quite clear that still more work can be done in this classical locality of cyclostratigraphy.

SHALLOW MARINE ENVIRONMENTS

An interesting example of a cyclic non-carbonate deposit is the Boom clay formation of Belgium (Fig. 11-4). This Lower Oligocene (Rupelian) was deposited in an embayment at the southern end of the Rupelian North sea. It is exposed in

Fig. 11-4. Boom clay. Swenden pit near Terhagen aproximately 30 km north of Brussels.

NON-CARBONATE CYCLES

191

an area running from Tongeren (75 km east of Brussels) in a west-northwesterly direction, to the present day North sea. It reaches a maximum thickness of about 60 m. The sedimentology of this formation has been investigated in detail by Vandenberghe (1987) and the cyclicity in particular by van Echelpoel (1991, 1993) and van Echelpoel and Weedon (1990). The most characteristic feature of the Boom clay is a very regular colour banding on a decimetre scale that was produced by a variation in the silt content of the stratigraphic sequence. If the percentage of particles greater than 32 p m is plotted against the stratigraphic position one finds a very symmetrical curve which is almost sine shaped. It reaches a maximum of approximately 50% somewhere in the centre of the silt layers, and a minimum of about 10% in the centre of the clay layers. The transition from clay to silt and back, is a gradual one and there has been no recognisable break in the sedimentation. There is, however, the possibility that minor layering in the sediment has been eliminated by bioturbation. Fossils in the clay are not common and they show no signs of extensive transport. Some lamellibranches (nuculena) have been found in their life position, which indicates a low energy environment. The individual layers are very persistent laterally and can be traced over distances of at least 100 km. The overall thickness of the formation and in particular, the thicknesses of the individual clay layers, increase in a northerly direction and this is away from the palaeoshore line. Taking the sedimentological evidence into account, Vandenberghe and van Echelpoel concluded that the direct cause of the cyclic sedimentation must have been changing turbulence conditions near the sea floor. There does not seem to be any indication of variable sediment supply from different sources for example. Rather, it is visualised that the overall supply was constant but that different bottom turbulence removed periodically greater or lesser amounts of the finer sediments. This model would imply that minor stages, such as individual storms which are not recorded in the sequence, have been eliminated very probably by bioturbation. To examine the cyclicity of the Boom clay, two types of time series were used by van Echelpoel. The first came from a 21.46 m long section in which 151 samples were analysed for grain-size and the percentage of grains larger than 32 pm was chosen as the most representative parameter. Interpolation was used to obtain equally spaced sample points at 16 cm intervals and power spectra were calculated using the Blackman Tukey method. In addition to this, time series were created by coding silt clay layers as + 1 and -1 respectively and such series were analysed using Walsh power spectra. The spectra were calculated by sampling the square wave that was generated by the +1, -1, coding method at a constant interval of 4 cm. This limit of sampling density was chosen to make allowances for the inaccuracies in determining the boundaries between the silt and clay, which are diffuse. The results of the two methods are very comparable. The Fourier spectrum gave two maxima at a wavelength of 100 cm and 46 cm and the averaged Walsh spectrum of a 25.6 m long section gave maxima between 137 cm and 102 cm and two maxima at

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47 cm and 43 cm. Using radiometric dates for the Oligocene biozones P19 and P20, gave average sedimentation rates of 8.1 mm/ka to 8.4 mm/ka, depending on which time scale was used. This would give the 100 cm cycle a length of 120 ka, which coincides well with the duration of the 100 ka eccentricity cycle. The 47 cm cycle has a corresponding length of 55.7 ka and this has been interpreted as being a result of the obliquity cycle. The alternative to this interpretation is that the 100 cm cycle should be interpreted as the 41 ka obliquity. This would involve a sedimentation rate of 2.4 mm/ka which, compared with the calculated sedimentation rate of 8.3 mm/ka, would imply a stratigraphic loss of 71%. This is not very likely, considering the environment in which layers of a few decimetres in thickness can be traced over distances of more than 100 km. The interpretation given by van Echelpoel (1992) and van Echelpoel and Weedon (1990), does, however, pose the question as to why there is no evidence of the precessional cycle, which should have an approximate wavelength of 20 cm. As pointed out by van Echelpoel and Weedon, this is beyond the Nyquist frequency of the grain-size data but it is not beyond the Nyquist frequency of 1/8 cm, which was used in the calculation of the Walsh spectra. It seems unlikely that bioturbation removed all the evidence of a 20 cm cycle and this problem seems to need still further investigation. Sedimentation changes that have been produced by changes in the turbulence of bottom waters, can be linked to climatic changes by several processes. Possibly the most direct connection, is a variation in storm frequencies or intensities. Such a model, however, makes it difficult to understand how the effect of individual storms became integrated to generate the relatively smooth cycle that is recorded in the sediments and demonstrated by the steady grain-size changes. Therefore, a more likely explanation would be a change in the depth of the wave base which could be partly due to changes in the storm intensities and partly due to some glacio-eustatic sea level fluctuations. Such fluctuations may have been similar to the Pleistocene situation determined by the 100 ka eccentricity cycle. If this cycle predominated, it could explain the absence of the precession signal. Van Echelpoel made use of the Boom clay cycles as potential time markers and calculated trends of sediment supply and trends in the distribution of coarser and finer material. Two types of trend were recognised. The first was a change that was not necessarily periodic, with an approximate wavelength of 50 m. This was a slow change which could be best explained by a change in the sediment supply, possibly caused by tectonic movement in the southern source area. A faster trend with a period of 20 m is possibly due to eustatic sea level fluctuations which would change the depth of the wave base and in this way, would act directly on the turbulence near the sea floor.

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FLUVIATILE ENVIRONMENTS

Fluviatile sediments, one may think, are not the most promising medium to record Milankovitch-type cycles. Nevertheless, it has been claimed that the depositional pattern of upper Devonian fluviatile sediments in East Greenland can be explained by orbitally controlled processes (Olsen, H., 1990). The study of river deposits is particularly interesting, since the size and discharge of rivers is predominantly a function of the available water and therefore precipitation. In this way they are directly dependent on climate. Olsen describes a 215 m long section in the Upper Devonian of the Kap Graah group in Gunar Anderson land (East Greenland), which is a sequence with a total thickness of 1100 m. The series contains a large number of channel sandstone bodies that are separated by flood basin siltstones, sheetsplay and crevasse sandstones and minor aeolian sandstones. The sandstone bodies are of single and multi-storey type, each storey representing the remains of a pointbar. The base of each storey is formed by an erosion surface which represents the lateral thalweg migration of a shifting meander loop. The internal structures of the sandbodies make it clear that they have been generated by single channel rivers and not by a braided river system. Olsen reconstructed the width and depth of the channels, to obtain an estimate of the cross section. The mean velocity of the channel-forming flow can be obtained from the average grain-size. The cross section and flow, permit the calculation of the channel forming discharge. A plot of the calculated discharges against the stratigraphic position of the units, shows a fairly regular oscillation (see Fig. 11-5). There is a relatively rapid

Fig. 11-5. Calculated discharge of a Devonian river of East Greenland. After H. Olsen (1990).

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variation that has an approximate wavelength of 20 m and there appears to be a larger megacycle with a thickness of 111 m. The megacycle is not only seen in the discharge data, but it is also reflected in the thickness variation of the sandstone bodies. The megacycles are recognisable on aerial photographs and they represent mappable units. The average thickness of such megacycles was found to be 101 m and the ratio of the 101 m cycle, divided by the average of the 20 m short cycle is nearly 5.0. Timing of the cycles, can only be done in the most general way. Assuming that the Famenian had a duration of 9 Ma, which is represented in the area by 4300 m of sediments, one obtains an average sedimentation rate of 480 mm/ka. The durations of the two cycles would therefore be 233.3 ka and 42.3 ka respectively. Olsen emphasises that these are only approximate time estimates and he suggests that considering the fairly well-established ratio of 1 : 5, one may think of eccentricity and precession cycles as being the cause of the sediment variation.

PRODELTA TURB IDITES

If climate variations controlled the flow of rivers, one ought to find this effect well-developed in marine delta and pro-delta environments. Indeed, Milankovitch cyclicity is thought (van lissel, 1987) to be responsible for the cyclicity in the Upper Devonian delta margin sediments of the Catskil Delta in Virginia (U.S.A.). The sediment siltstones and shale are turbiditic in nature and individual packets of sandstone and shales can be traced over several tens of kilometres. nKo to three of such packets, combine to make a larger upwards fining cycle of approximately 20 m thickness. Van Tassel differentiated 21 such cycles in the Foreknobs Formation at Scherr in West Virginia. He estimated the duration of the cycles to be 100 ka, suggesting that they are caused by the eccentricity cycle. As with the situation in Greenland, it is difficult to obtain reasonably accurate time estimates. A short series which is very well dated, is described by Weltje and de Boer (1993) from the Lower Pliocene of Corfu (Greece). The Corfu sequence consists of a 500 m thick series of regularly stacked depositional lobes, exposed along the north-west coast of the island. The series has been dated by magneto-stratigraphy and an average accumulation rate of 1000 mm/ka has been calculated. Each depositional lobe represents a time interval of 20 ka to 25 ka and it is thought that they are controlled by the precession cycle. Sedimentological analysis showed that there is a variation of turbidite frequency and a variation in the amount of terrestrial material in the turbidites. A lobe develops quickly at first, with a large number of turbidity currents and a maximum of land derived material. This is followed by a decline in the depositional lobe and pure marine sediment predominates. The pattern therefore reflects the development of a river system which was possibly some 150 km to 200 km south-east of the distal turbidites that have been studied. Weltje and

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de Boer assume that during climatic periods of high precipitation, an immature river system delivers a great deal of sediment that reaches maturity during periods of low precipitation. Although this model is very hypothetical, it is of general interest since it shows how the climate can interact with a given environment and how both the morphological development of the source area and the climate, combine to determine the sedimentation.

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But to begin with fossils, before the physical groups are determined, and through them to establish the nomenclature of a system, would be to invert the whole logic of geology, and could produce nothing but confusion and incongruity of language.

A. Sedgwick (185 1)

In this last chapter, an attempt will be made to summarise some of the basic problems of cyclostratigraphy, together with some of the findings which may have a bearing on solving these problems. Three questions will be discussed in particular. What are the prospects of using cyclostratigraphy as a practical tool in geology? How positive can we be about identifying Milankovitch cyclicity in pre-Pleistocene sediments? What has been learned so far, about the mechanisms which link astronomical variations with sedimentary cycles?

PRACTICAL CYCLOSTRATIGRAPHY

The sedimentary cycles which have been discussed in the previous sections are approximate time units. They can include strata which are the most basic units that give stratigraphy its name but they are shorter than most of the units of sequence stratigraphy. The latter are the physical groups referred to by Sedgwick and are the formations which were originally established by mapping. The Milankovitch cycles span a time interval of 20 ka to 2000 ka and according to the nomenclature of Vail et al. (1991), they would be third to sixth order cycles. As already mentioned, this terminology is not helpful when talking about Milankovitch cycles, because the orders in no way reflect the Milankovitch frequencies. Milankovitch cycles are the product of an oscillating system, according to our more specific definition. They are therefore fundamentally different from the “cycles” in Vail’s system. It seems utterly illogical to call Milankovitch cycles which are the product of a quite different process, the third or even sixth order of this unrelated process. Despite this disagreement regarding nomenclature and possibly interpretation, Milankovitch cycles can be used like “sequence cycles” in any stratigraphic exercise, regardless of whether they represent equal time intervals or not. Milankovitch cycles are like varves and are proper time units, at least within the accuracy of geological measurements. Some of these units, such as obliquity and

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precession, have become longer with the increasing age of the planet system and it is not known how stable the system was in the very remote past. Milankovitch cycles have been recorded as long ago as the early Cambrian and Precambrian (cf. Grotzinger, 1986) and they show no evidence of any unpredictable changes. The accuracy of the astronomical calculations which are based ultimately on present day observations, does not allow one to calculate the precise date of an eccentricity maximum 200 Ma ago, for example. Indeed, it may not be possible to state how many 100 ka maxima occurred in this time span. However, it has been possible to show that the 2 Ma eccentricity maximum at the Cretaceous Tertiary boundary which was 65 Ma ago, appeared exactly where it should have been (ten Kate and Sprenger, 1992). Work in the Tertiary (Hilgen, 1989) and Cretaceous does offer the possibility of a continuous cyclostratigraphic time scale, at least for part of the Cretaceous. This will depend to a great extent on our ability to recognise the longer eccentricity cycles and the 2 Ma cycle, in particular. The main difficulties in working with such long cycles are on the one hand, data collection and on the other hand, the disturbing influence of trends. Such trends are a property of sedimentary sequences and with an increasingly better knowledge of sequence stratigraphy, some of the difficultiescould be overcome by tuning the sections. However, it is clear that the development of a cyclostratigraphic time scale will need at least as much work as has been devoted over the last century to the development of a biostratigraphic system. Unless a similar effort is made, one cannot condemn the concept of a consistent cyclostratigraphy. Apart from the ultimate aim of cyclostratigraphy,one can use sedimentary cycles in a multitude of stratigraphical and sedimentological problems. For example, comparing the thicknesses of various cycles is an ideal way of obtaining sedimentation rates from different environments. Cycles can also be used to compare the lengths of stages and biozones. By comparing cyclostratigraphic time determinations with larger stratigraphic intervals which have been radiometrically dated, an estimate of the completeness of sedimentation can be made (Strauss et al., 1989; Weedon, 1989). Probably the most important use of sedimentary cycles lies in detailed mapping. Stratigraphers have made various statements concerning the incompleteness of the stratigraphic record and such statements could be quantified by producing either maps or profiles which show what types of cycles are missing and where they are missing. For example, if 100 ka cycles are used in such an exercise, one could show completeness or incompleteness to an accuracy of 100 ka.

THE RECOGNITION OF MILANKOVITCH CYCLES

We have called any sedimentary cycle which is controlled ultimately by changes in the earth’s orbit, a Milankovitch cycle. Are there any rules by which one can recognise such cycles? As has just been discussed, a positive identification of

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Milankovitch cycles makes it feasible to use them as equal time markers. If it is possible to identify specific Milankovitch periods, then such time markers can be associated with absolute time values. To obtain a clear understanding of the problems involved in the recognition of Milankovitch cycles, it is useful to discuss first the reasons why the Milankovitch theory was accepted for the Pleistocene sediments, after considerable discussion. Some of the more important reasons are the following. (1) Most of the data used in reconstructing the Pleistocene stratigraphy, comes from deep sea sediments which are relatively undisturbed and contain an isotope signal which makes an almost ideal proxy variable for climatic conditions. (2) The cyclicity of the isotope signal can be timed fairly accurately and the time periodicity can be tested by using magnetic reversals which have been fixed by 14Cdates and these are considerably more accurate than any obtained by other radiometric methods. (3) The isotope signal can be correlated over vast areas and therefore it records a global condition which excludes local tectonic or local climatic oscillations. (4) Spectral analysis has found a number of frequencies which coincide accurately with the frequencies predicted by the Milankovitch theory. It was the last point which led to the general acceptance of the theory, but perhaps a few more imaginative geologists would have regarded points two and three as sufficient evidence for orbital control. As was stated in the first chapter, the astronomical control theory is the only theory at the present time which can explain the cyclic sedimentation that represent time periods of 10 ka to several 100 ka. In this statement, the term cyclic sedimentation is defined as sedimentation which is the outcome of an oscillating process. This definition is more restricted than the one used by most geologists, but without such a restriction, the term becomes quite meaningless. As has been developed specifically in Chapter 2, oscillating systems must involve time, if they are real physical systems and such systems lead to time periodic or quasi-periodic processes. Since this concept of cyclicity is particularly important for the recognition of Milankovitch cycles, it will be discussed once more. In Fig. 12-1 the left hand side has been borrowed from a paper by Weltje and de Boer (1993) and shows a river system that is driven by climatic variations which are assumed to be astronomically controlled. During wet periods, the rivers deliver much sediment and during dry periods, the sediment transport is reduced and the system behaves like a mature river system. The right hand side of this cartoon shows the system without the climatic driving force. This system will produce precisely the same sediment sequence as the system under climatic control, during its natural development from the juvenile to the mature stage. Indeed, if rejuvenation does eventually take place, repeated and more or less identical sequences will be deposited. The two systems are not representatives of all0 and auto-cyclic systems, for the simple reason that the system on the right is not cyclic but episodic. The

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Fig. 12-1. Diagram to illustrate the difference between cylic and episodic sedimentation in a river system.

system would become an oscillating system, only if there was a link between the deposited material and the process of rejuvenation which means that rejuvenation depends on the amount of sediment deposited. The difference between episodic and cyclic is also the difference between the so-called first order “cycles” of sequence stratigraphy and the Milankovitch cycles of cyclostratigraphy and it would be helpful if this difference could be recognised. The fact that the two hypothetical systems of Fig. 12-1 can generate apparently the same sequence of sediments, leads to the question of how one can differentiate between the two. The recognition of time cyclicity always does depend on how accurately one can read time from the stratigraphic record. The problem is an old one (Sander, 1936; Schwarzacher, 1947). However, if it can be established that sedimentation was continuous within given statistical limits, then equal thickness intervals do represent equal time intervals. One can measure the variability of the assumed time or thickness cycles and compare it with data from different environments that have well-established cyclicity, for example varves. The final judgement as to how reliable a sequence has been in recording time, must always be based on geological observations and arguments. ‘This means that occasionally one will accept a hypothesis which cannot be strictly proved but which appears to be the most likely explanation. To illustrate this, consider the example of the two river systems. A basic difference between the two is that the system under climatic control must have been synchronous with any other river system formed in this way. This is unlikely to be the case for systems formed by the second method of episodic rejuvenation. However, if systems of the second kind are compared in areas which have the same tectonic history, then it is very possible that they can be correlated.

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A very strong point in favour of orbital control, is the accurate timing which has been possible in the Pleistocene. The accuracy of chronometric dating, however, decreases considerably in pre-Pleistocene times when reasonably well-established dates are more and more widely spaced. The process of calculating the duration of cycles from long term accumulation rates, has been rightly criticised by Algeo and Wilkins (1988). (Surprisingly, they themselves use the method to obtain data for their “statistical tests ’’ on cyclicity). To be useful, sedimentary cycles should be first tested for spatial cyclicity and their duration should be determined if possible, by actually counting the number of cycles in a known time interval. Such procedures can be used to eliminate or at least recognise gaps in sedimentation, if parallel profiles are available. It cannot get around inaccuracies of chronometric timing. Discrepancies between different time scales that are applied to short intervals, may lead to estimates of cycle lengths which differ by as much as 50%. This means that either the 100 ka eccentricity cycle can be confused with the 41 ka obliquity cycle, or that the obliquity cycle could be confused with the precessional cycles, so that the identification of Milankovitch periods based solely on chronometric timing is uncertain (Schwarzacher, 1987). Cyclic sequences which are composed of several frequencies, provide the best chance of identifying Milankovitch cycles. Although the ratios between some of the Milankovitch frequencies have changed throughout geological time, these can be calculated and compared with the ratios found between the frequency peaks of the analysed stratigraphic sections. The method used for finding the different frequencies is usually power spectral analysis. In situations where stratification cycles are known only as a bundle consisting of a number of beds, simple counts of the numbers of beds per cycle give a useful description. Frequency distributions of such counts or frequency distributions, of bed thickness ratios and cycle thicknesses can sometimes be more revealing than spectral analysis (Schwarzacher, 1954). The interpretation of spectral analyses has to be done with great care and with regard to possible bias, which can be introduced either by sedimentation or by diagenesis and in particular by compaction. As an example, Fig. 12-2 shows a sine

Sianal

Response

.==F aE F cf Fig. 12-2. The sedimentary response to a modulated signal. a shows a complete record and linear time scale, b shows an incomplete record and linear time scale, c shows a record with a changing time scale.

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wave which has been modulated in a similar way to the astroclimatic index. It is assumed that a maximum which is an excursion to the right in the examples, causes the formation of a bedding plane. A master bedding plane is formed when the maximum exceeds a certain threshold and therefore this mechanism could explain the formation of a bundle of five beds. This is shown in Fig. 12-2a.The probability of obtaining such a regular response is extremely small and since the signal responsible for bed formation is variable, it may happen that not all the bedding planes are fully developed (see Fig. 12-2b).The repeatedly mentioned relationship of sedimentation rate and sediment type is possibly much more difficult to treat and implies that the modulation will determine not only the amplitude of the variation but also the frequency, via the sedimentation rate. In the hypothetical example of Fig. 12-2c, it is assumed that sedimentation slows down near a master bedding plane. An effect that is identical to changes in sedimentation rate, can be produced by compaction and other diagenetic changes in different lithologies. Variable sedimentation rates or compaction can be corrected for to some extent, but when both the lithology and the associated rates change, this correction becomes impossible. Most distortions of the “time” scale occur either through changes in sedimentation rates or compaction, or through the incomplete recording of an event that will change the relative position of the frequency maxima in the power spectra. In some cases, this can create spurious frequency maxima and make the practice of identifying Milankovitch cycles through frequency ratios very difficult, if not impossible. The process of tuning is an important method for recognising the longer Milankovitch cycles. One makes the hypothesis that a well-developed cycle represents an exact time period and then adjusts the sedimentation rates in such a way that equal thickness intervals result. This removes a certain amount of trend and makes it possible to recognise low frequency cycles. If such cycles are in agreement with the Milankovitch theory, then one can regard the hypothesis which formed the basis of the tuning as being justified. An important property of Milankovitch cycles is their persistence in time. This feature is not easily expressed in quantitative terms but nevertheless, it is an important argument for external control. In some sequences, cycles persist and are uninterrupted throughout several formations. On an even larger scale, it is found that many patterns of stratification in carbonate and non-carbonate sediments, have been the same since the Cambrian at least. The previously discussed examples from the Carboniferous, Jurassic and Cretaceous, look almost identical if they are seen in photographs of outcrops. However, the same pattern of cyclic sedimentation is also preserved when there is a gradual change in facies. Such observations indicate that cycle formation is a general feature of sedimentation which is independent of local conditions and although terrestrial conditions may change considerably, the pattern of cycles remains the same. Considering the difficulties in interpreting the time record of most stratigraphic sequences, it is not surprising that one cannot provide a fixed rule for the identifi-

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cation of Milankovitch cycles. However, the first problem is invariably to investigate the space cyclicity of a sequence and from there, one may consider how closely the cycles approach true time periodicity. In the examples which were given in the previous chapters, one may distinguish between the various levels of confidence with which one can regard the identification of Milankovitch cycles. It is interesting to review the evidence for this identification in a somewhat condensed form. In the example of the Irish Carboniferous cycles (Chapter 7), the time estimates are not accurate but the sedimentation rates that are calculated by interpreting the cycles as being of 100 ka duration are reasonable. In favour of the Milankovitch interpretation, is the persistency of the cycles in different facies through a long, uninterrupted time period. The frequency composition can be interpreted as a series with 400 ka 100 ka, 20 ka and possibly 2 Ma periods. This evidence is regarded as convincing. The English and American examples have not been equally well established in the Carboniferous. The Upper Triassic carbonate cycle (Chapter 8) shows good evidence for eccentricity and precessional cyclicity and the high sedimentation rates are to be expected within the environment in which they formed. The Milankovitch interpretation is supported further by the observation of similar cyclicity in the basinal facies outside the reef belts. The discrepancy between cyclostratigraphic and radiometric time estimates in the Latemar region has not yet been explained. The Jurassic examples from the English Lias and Kimmeridge (Chapter 9), show considerable differences in their interpretation and they demonstrate that the established chronometric time scales are not sufficiently accurate for a decision to be made between 100 ka and 40 ka cycles. For a more convincing interpretation of these cycles, more data and analyses are needed. The results obtained from the Aptian-Albian Scisti a Fucoidi (Chapter 10) provide by far the best established example of Milankovitch cyclicity in the 1/400 ka, l!lOO ka and 1/20 ka frequency bands. The success in this sequence is mainly due to the superior faunal and chemical data used in the analyses. The finding of two frequency peaks in the 1/100 ka region may or may not depend on the tuning applied to the series, but it makes the Scisti a Fucoidi Milankovitch cycles just as well-established as most of the Pleistocene cycles. The evidence for Milankovitch cyclicity in the Cenomanian Scaglia itself, is very meagre. However, since the cyclicity can be seen to develop directly from the underlying Scisti a Fucoidi, the rating of these cycles gains considerably. The top Cretaceous and Paleocene cycles described by ten Kate and Sprenger (1992) not only have the expected frequencies, but they can also be timed fairly accurately from magneto stratigraphic data. Once again, there can be little doubt about the correct identification of the cycles. The evidence from the lacustrine examples (Chapter 11) appears to be good in the Trias, where all the expected Milankovitch frequencies were found. Surprisingly, the evidence for the Tertiary Green River Formation is less good than Bradley orig-

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inally thought. It seems somewhat doubtful whether one can apply sedimentation rates which have been derived from varves, to the whole series which clearly, has gone through a wide spectrum of different environments. For a full interpretation of the sequence, one would like to see more analyses, and evidence from longer borehole sections. Finally, the Permian evaporite cycles are the best documented pre-Pleistocene precession cycles to date. The timing of the cycles is done by actually counting the individual varves and not by simply taking an average thickness. It is particularly interesting that Anderson (1988) obtained an average cycle length of 19.4 ka for the precession cycle which he thought should be 23 ka, but which in fact is a very good value according to modern calculations (Berger et al., 1989).

THE CAUSES OF SEDIMENTARY CYCLES

It is tempting to search for a common cause or mechanism which can explain all the Milankovitch cycles. However, such simplification is only possible if one goes back to the primary concept that Milankovitch cycles are the result of changes in the Earth’s orbit around the Sun. The different positions and distances that the earth’s surface takes up against the sun, determine the amount of insolation. Since insolation is practically the only energy input which runs the atmospheric system, its behaviour, which we call climate, is strongly determined by the insolation. The climate depends on a multitude of factors such as the absorption of energy, as well as heat transport in the atmosphere and hydrosphere. The amount of radiation that is returned from the earth back into space, is strongly dependent on the distribution of land and sea which have vastly different heat capacities, the albedo which is strongly determined by the distribution of snow and ice and the formation of polar ice caps. Some theoretical work (North and Crowley, 1985) suggests that the global climate has two stable states, either with large ice caps or with ice-free polar regions. Small ice caps are thought to be unstable and it would require a considerable energy input to shift one stable stage to the alternative stage. We know from the geological record, that the change from strongly glaciated to possibly ice-free conditions has taken place several times. The so-called ice-house and green-house states are a typical part of the climatic history of our planet. We still know very little about the reasons for changing from one system to the other, but possible causes include drastic changes in the distribution of land and sea and changes in the composition of the atmosphere with for example, the release of C02 by extensive weathering or volcanic emanation (Hay, 1992). The climate changes between glacial and nonglacial states would have been considerably larger than any variation that could have been generated by Milankovitch cyclicity. Nevertheless, orbitally controlled cycles can be detected during glacial as well as non-glacial states.

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From the evidence that is available so far, we have to accept that Milankovitch cycles have been recorded from the Precambrian to the present. Changes in the frequency composition of the orbital variations must have occurred, but little is yet known about how this affected sedimentary cycles. So far, it appears that eccentricity and precession are far more effective than obliquity, in controlling sedimentary cycles. There are, however, two sources of bias which could account for this. The theory of astroclimatology suggests that the effect of eccentricity modulated insolation should be most noticeable in the lower latitudes, whereas obliquity variations should predominate in the higher latitudes. It is worth noting that carbonate sediments have been found to be particularly well-suited to the recording of climatic cycles and therefore localities with relatively low palaeolatitudes have been favoured. The predominance of eccentricity over obliquity could then be explained by the preferential low latitude distribution of carbonates. The second bias is introduced because one of the most valuable ways of recognising Milankovitch cycles is to find the various eccentricity maxima, together with the precession maxima. If a record is found to contain only one cycle, it cannot be identified unless the time period is very accurately known. It is known theoretically that independent eccentricity changes should have little effect on the insolation because the earth-sun distance variations are very small. However, eccentricity modulates the precession and therefore determines the differences between the seasons. This difference is called the seasonality and it plays an important role in the variation of the climate. For example, in the year 9000 B.P., solar radiation during January was approximately 7% less than it is at the present and during July of the same year, it was 7% higher than now (Kutzbach and Bliesner, 1982). This increased variation had little effect on the ocean temperatures but lead to considerable warming and cooling of the land, which is mainly in the northern hemisphere. The result of this, was an intensified monsoonal circulation which largely determined the climate at this time. This effect is strongly influenced by the land sea distribution which has to be reconstructed, if the model is applied to pre-Pleistocene times. Cretaceous land sea configurations have been used in a general circulation model (Baron and Washington, 1982). It was found that the subtropical high occurring in January and March and the intertropical convergence zone in July, were centred over the subtropical Tethys. These are very different positions from the present. This strongly influenced the general circulation pattern which was also different from the present. Further experiments with a circulation model (Park and Oglesby, 1991) attempted to evaluate the relative importance of the precessional effects, compared with the obliquity effects. Three regions in particular were examined: the early South Atlantic, the Tethys and the North American Western Interior. The South Atlantic in mid-Cretaceaous times consisted of a restricted basin that was open to the South and was connected to the Tethys, possibly only by a narrow gap. Precipitation, runoff and evaporation balance estimates show that this part of the Atlantic could have changed regularly under

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orbital control, from estuarine into anti-estuarine circulation. The latter is like the present day Mediterranean, where surface water enters through the strait of Gibraltar and deep water leaves from this basin. In estuarine circulation, deep water enters through an inlet and surface water leaves a basin, as for example in the present Black Sea. The estuarine circulation leads to a nutrient trap and this may lead to anoxic bottom waters. The anti-estuarine circulation on the other hand, may produce a nutrient deficiency and oxygenated bottom conditions can exist. The climatic simulations show that the change between the two states is determined principally by the precessional index; obliquity has very little effect on the process. Anoxic black shales alternating with lime muds, have indeed been found in the Cretaceous sediments of the South Atlantic. An examination of the Tethys showed that evaporation which is dependent on the precessional index, reaches a maximum in July. This produces high salinity surface water which sinks to create warm deep water. The downward transport includes large amounts of organic material from high productivity areas and this can lead to anoxia. Once again, the precessional cycle can lead to alternating oxidation and reduction conditions in the bottom waters. In the Western Interior of the United States, precipitation-evaporation balances indicate that they are predominantly determined by obliquity changes and only to a lesser degree, by precession. Here it is possible that the main factor in generating sedimentary cycles is a variation of runoff. Climate models are an important tool in the deductive analysis of the cycle problem but this approach often leads to conclusions which are not sufficiently detailed to explain the actual processes. For example, the black shale which is a typical phase of many Milankovitch cycles, has been explained as being the product of organic over-production due to the upwelling of nutrient-rich waters, or it was explained as being due to descending water in areas of high productivity. Alternative explanations are tnat they are the result of restricted circulation, perhaps in a stratified water body, or the result of an influx of non-marine organic material, either by river or wind transport. To decide which of these possibilities actually occurred, one needs specific geological information. The problem of marl-limestone cycle formation is similar. Three hypotheses to explain such cycles have been given and the productivity, dissolution and dilution theories have been discussed already. Dissolution cycles can be recognised sometimes, by examining the preservation state of the carbonate suppliers. It has been said that the distinction can be made between productivity and dilution couplets because fluctuating productivity will change the thicknesses of the carbonate layers, whereas the shale thicknesses remain constant. With dilution cycles, the opposite is said to be the case. However, this is a generalisation that is based on the assumption that the proportion of time spent in, let us say carbonate production, remains constant. De facto variations of productivity and dilution in cycles can only

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be recognised by regional studies and by the lateral tracing of cycles. These are inductive rather than deductive methods. Cycles in carbonate platforms that are related to sea level fluctuations are more convincingly established because they are based on deductive reasoning. Although it is difficult to estimate the amplitudes of such fluctuations accurately, they must have been of the order of several metres and they are most likely to have been of glacioeustatic origin. As has been pointed out, the fluctuation could have been small enough not to have left an isotopic signal that can be recognised by using present techniques and by assuming a similar ice volume sea level fluctuation relationship to that known from the Pleistocene. We do not know how persistent small-scale sea level fluctuations in the order of metre amplitudes have been throughout geological history. Nor do we know how effective such variations have been in determining the cyclicity of hemipelagic and even pelagic sediments. The example of Milankovitch cyclicity in lacustrine environments proves positively that eustatic fluctuations are not the unique answer for explaining sedimentary cycles but that they may have had a very much wider effect than is generally realised. The intermediate stages in the causal chain between climatic factors and actual sediment formation, are subjects about which still very little is known. Such intermediate stages are particularly important because they are probably responsible for the non-linear response of sedimentation to climate and because they can integrate climatic variables to create new conditions which may be very important in the sedimentary response. An example for such an intermediate factor is the accumulation of snow and ice in polar and high altitude positions. The climate is primarily responsible but it is the ice accumulation which causes the lowering of the sea level and which is the important factor as far as sedimentation is concerned. Similar intermediate stages can involve the biosphere, the morphology of the land which provides the source area for sediments and as already discussed, the ocean itself. The effect of the climate on the biosphere may be one of the most important factors determining sedimentation. This does not only apply to the sediment-forming organisms but also to any life form on land and sea that contributes to the environment. Particularly important is the vegetation cover on land since it is in a feedback relationship with the climate and also determines the supply of terrigenous sediments and nutrients to the sea. The same is true to a lesser extent, about the topography which, together with the climate, determines the erosion and therefore the sedimentation. The whole field of explaining sedimentary cycles in detail is still wide open but cyclostratigraphy can be practised even without knowing these details.

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221

SUBJECT INDEX

Adnet limestone, 157 Agost, 179 Albedo, 22, 43, 204 Algal mats, 126-128 Allocycles, 4, 5 Ammonitico rosso, 157, 158 Amplitude modulation, 66, 68, 71, 152 Anglo-Paris basin, 176 Anisian, 135 Anoxic, 149, 151, 166, 170, 175, 206 Apennines, 159 Aphelion, 29, 46 Aptian-Albian, 162, 203 Asbian, 109, 117-122 Astronomical climate, 30, 43, 66 Atlantic, 45, 159, 172, 173, 186, 205, 206 Attractor, 16-18, 22 Autocorrelation, 18, 52, 55, 56, 58, 60, 72, 86, 95 Autocycles, 4, 5, 25 Autocyclic, 4-7, 25, 117, 140 Bahamas, 145 Bakony mountains, 131, 133, 134 Bandwidth, 62,63 Barremian, 159, 162 Bed, 3, 5-7, 50, 51, 73, 81, 82, 93-105, 108, 109, 111-114, 116, 117, 121, 126, 128-131, 133, 138, 145, 149, 155, 157, 161-163, 166-172, 175-177, 182,201,202 Bed thickness distribution, 81, 98 gamma distributed, 79 polymodal, 98 Bedding cycles, 155, 176, 177 Bedding planes, 49, 93, 95-103, 105, 109, 112, 113, 119, 126-129, 133, 155, 161, 166, 168, 177,202 Belemnite marls, 151 Benbulbin shale, 108, 109, 113-118 Bifurcation, 16, 22, 23, 47 Biogenic component, 165

Bioturbation, 73, 85, 88-90, 109, 149, 172, 175, 191,192 Blake-Bahama Formation, 173 Blue Lias, 149-152 Bonarelli horizon, 166, 167, 169-171 Boom clay, 190-192 Bottom turbulence, 191 Breggia Gorge, 157 Bretagnon’s equations, 33 Bundle, 7, 66, 93, 132 ,157 of five, 98,202 Calciturbidites, 145 Caravaca (Spain), 173, 174 Carbon isotopes, 3,45 Carbonate platforms, 26, 79, 107, 125, 135, 139, 145,207 Carbonate production, 26, 83, 94, 125, 165, 167, 185,206 Carboniferous, 101 coal measures, 108 Dartry limestone, 108, 116 Lower, 107, 108, 117,119,121 North West Ireland, 100 Sligo and Leitrim, 108 Yorkshire and North Wales, 119, 120 Carnian, 125, 186 Castile Formation, 183, 184 Catskil Delta, 194 Cenomanian, 50,99 chalk marls, 176 Chaotic systems, 17 Circulation, 44, 183,205,206 anti-estuarine, 206 estuarine, 206 Climate, 2-4, 9,21,29, 30, 40,43, 46, 48, 73, 126, 150, 151, 183, 188,193-195,204,205,207 models, 21, 22, 27, 47, 206 Coherence, 59, 185 Colorado, 2, 188 Compaction, 90, 95, 97, 139, 201, 202

222

CYCLOSTRATIGRAPHYAND THE MILANKOVITCH THEORY

Completeness, 79, 80, 84, 105, 158, 198 Complex demodulation, 70,71, 116 Confidence limits, 60,65, 76, 77 Corfu, 194 Cospectral analysis, 58 Cretaceous, 2, 39, 91, 125, 159, 171-173, 175, 177, 180, 182,198,202,203,205,206 Atlantic sediments, 172 Blake-Bahama Formation, 173 Marne de Puaz, 162 Scaglia bianca, 50,98,99, 166-168 Scisti a Fucoidi, 159, 162, 163, 165-167, 178, 203 Tbrtiary boundary, 159, 171,177, 198 Cross-correlation, 58 Cross-plots, 150 Cumulative plot, 51, 103, 104, 178 Cycle boundaries, 67, 99, 100, 105, 113, 114, 117, 119, 131, 161, 162, 169, 177, 178 correlation, 112, 120, 150, 151, 169, 182 Cycles astronomical, 2, 73 autonomous, 4 complex, 7, 136 composite, 84 externally induced, 4 geometry, 142 time periodic, 27, 81 Cycloids, 82, 83 Cyclostratigraphy, 1,4, 7-9, 65,67, 138, 144, 149, 188, 190, 197, 198,200,207 and Milankovitch cycles, 197 and spectral analysis, 65 practical, 197 Sligo sequence, 117 Dachstein limestone, 126,128-132, 136, 137,145, 146 cyclicty, 128 Dartry limestone, 109, 116-1 18 Delaware basin, 183, 185 Densitometer, 163-165 Derbyshire, 119, 120 Devonian, 38, 39, 125, 193, 194 Diagenesis, 9, 84,90, 93, 95, 97, 149, 201 Digne, 172 Dilution, 206 model, 151, 175

Dinantian, 117-1 19 Discharge, 193, 194 Dissolution, 94, 95, 165, 206 Disturbing function, 33 Dolomia Principale, 126, 137 Dolomites, 125, 126, 132, 135, 137, 138, 141, 162 Domain of cycle generating system, 6 Dorset, 150-152 East Greenland, 193 Eccentricity, 30, 32, 34-36, 46, 47, 58, 61, 66, 71, 75, 114, 117, 118, 120-124, 134, 137, 145-147, 150, 151, 154, 155, 158, 161, 162, 164, 166, 169-182, 184, 185, 188, 190, 192, 194,198,201,203,205 Ecliptic, 29, 32, 40 Emersion, 119, 120, 126, 127, 135, 137, 140 Ergodicity, 52 Error, observational, 86 superimposed, 86 Eustatic sea level fluctuation, 1, 47, 192 .Evaporite, 6, 183, 184, 188, 204 Event stratigraphy, 4 Fenestral fabric, 126 Fertility, 166, 176 Field observations of cycles, 105 Filter, 67,68, 70, 71, 89, 113, 169 Filtering, 67-70, 72, 89, 105, 171 Fluviatile environments, 193 Foraminifera counts, 50, 165 Fourier series, 53, 54,91 Fourier transform, 18, 54-58, 65, 67, 68, 89 Frequency, 52 Frequency modulation, 86 Gamma distribution, 81, 98 Gault clay, 176 Geological oscillators, 24 Ghil oscillator, 21, 23, 47 Ginsburg model, 26, 140 Glacial lowstands, 145 Glaciation, 27, 44,47, 48, 107, 123, 141 Glencar limestone, 100, 109-116, 118 Great Scar limestone, 119, 121 Green-house, 204 Green River Formation, 2, 188, 189, 203 Greenhorn Formation, 175

SUBJECT INDEX Gubbio, 50, 69, 75, 76, 82, 98, 99, 159, 161, 162, 167, 171, 176-178 Gulf of Mexico, 172 Hauptdolomit, 126 Hauterivian, 172, 173 Hiatus, 84,85, 120, 121, 151 Hungary, 131, 133, 134, 136 Ice-house, 204 Insolation, 22, 40-43, 46, 47, 92, 183, 185, 188, 204,205 Insolation signature, 42 Isotopes Pleistocene, 45, 93 Jurassic, 68,94, 149, 150, 152-155,157-159,202, 203 Britain, 149 Germany, 154, 156 Tethyan region, 157 Kepler’s law, 31 Kimmeridge clay, 152 Kolmogorov model, 132 Lacustrine environments, 186, 207 Ladinian, 125, 135, 137, 138 Lagrange’s model, 33 Latemar, 135, 137, 144, 145, 203 Latemar limestone, 135-137 Limit cycle, 16, 22, 23 Line spectrum, 53, 54, 59, 60, 86, 87 Linear systems, 11, 40 Lofer, 101, 126-130, 134, 136, 138, 140 Lokatong Formation, 186 Low-pass filter, 68, 89 Maastrichtian, 178, 179 Magnetic reversals, 3, 179, 199 Magneto-stratigraphy, 159, 194 Maiolica, 159, 161, 162 Malm, 154-157 Mapping function, 73,83,90-92, 104 method by Martinson et al., 91 methods, 90 Marche, 159, 160 Marl-limestone, 94, 96, 99, 100, 102, 105, 113, 145, 155, 157, 159,165, 172, 174, 175,206 coding of a two-state system, 113

223 sedimentation, 94 Marne de Puaz, 162 Master bedding plane, 96,99,202 Meenymore Formation, 109, 118 Megacyclothem, 123 Mesothems, 118, 119, 121 Micarbs, 166 Milankovitch, 2, 3 curve for the Pleistocene, 44 frequency ratios, 61 Milankovitch cycles, 1, 43 persistence, 202 recognition, 198-200 Milankovitch theory, 3, 21, 30, 40, 45, 46, 61, 66, 73, 146, 199,202 climate, 43 in Piassic limestones, 146 Miravete Formation, 173 Mixed spectrum, 60 Monsoon, 205 Monte Petrano, 167, 169 Morbio Formation, 157, 158 Namurian, 122 Nannofossil counts, 166 New Jersey, 186 New Mexico, 183 Nicolis model, 23, 47 Niobrara Formation, 175 Non-carbonate cycles, 183 Non-stationarity, 69, 70 Norian, 125, 132, 136, 138, 139, 145, 146, 186 North Wales, 119-122 Northern Calcareous Alps, 125, 126, 132, 137, 139 Northern England, 107, 119, 120 Numerical description of bedded sequences, 101 Nyquist frequency, 62 Obliquity, 29, 34-39, 41, 44, 46,58, 61, 71, 117, 118, 134, 140, 145, 150, 158, 164, 173-177, 179, 184, 185, 188, 192, 197, 201,205, 206 Oil shale, 2, 152, 154, 188, 189 Oscillators stochastic, 17 Oxygen isotopes, 3,45,50 Palaeoclimatic indicator, 45 Palaeokarst, 119-121

224

CYCLOSTRATIGRAPHY AND THE MI LAN KOVITCH THEORY

Passaic Formation, 186 Pennsylvanian cyclothems, 108, 124, 139 Perihelion, 29, 32, 34, 40, 46 Periodic, 5, 6, 8, 10, 13, 14, 16-22, 27, 33, 37, 40, 46,47, 53, 54, 58, 59,64-66, 73,83-86, 126, 137, 146, 192 Periodic process, 47 Periodicity, 7, 8, 44, 65, 129, 199, 203 definition, 8 Peritidal, 125, 137 Permian, 39, 135, 183-185, 188,204 Phase modulation, 87 Phase plot, 12 Piobbico, 50, 162-164, 166, 167, 174 Planetary system, 2, 30, 37-39 Platform-basin interaction, 145 Playa Lake, 189 Pleistocene climate, 44, 45 stratigraphy, 3, 44, 61, 199 Polar ice caps, 204 Potschen Kalk, 145, 146 Precambrian, 40, 110, 198, 205 Precession, 2, 34-40, 58, 61, 66, 71, 112, 125, 137, 145, 150, 154, 155, 158, 161, 162, 164, 171, 174, 177, 185, 188, 190, 192, 194, 198, 204-206 Precessional index, 34, 38, 45, 206 Presrrvation of cycles, 143 Process autoregressive, 17 chaotic, 18 moving average, 17 periodic, 65 stochastic, 17, 51 Prodelta turbidites, 194 Productivity, 151, 206 model, 175 Proxy variable, 49, 171, 199 Quasi-periodic, 17, 37, 40, 54, 58, 65 Ramsbottom cycles, 119 Random walk, 74-81,86,87, 104 generalized, 80 Ratios of Milankovitch frequencies, 61 Realisation, 51, 52, 55 Redox state, 149, 163, 165 Regressive cycles, 128

Relleu, 179 Reversals, magnetic, 3 Rhaetic, 125, 132, 136, 145, 146 Rupelian, 190 Sabkha, 137 Saltzman oscillator, 22 Salzburg, 101, 126, 146, 157 Sander’s rule, 8, 155 Scaglia bianca, 50, 98, 99, 166-168 Scaglia rossa, 177 Scis!i a Fucoidi, 159, 162, 163, 165-167, 178, 203 Sea level fluctuations, 48, 108, 123, 125, 128, 138, 140, 142-146,207 glacio-eustatic, 123, 192 Seasonality, 188,205 Sedimentary cycles causes, 204 definition, 4 Sedimentation completeness, 79, 198 discontinuous, 73 error, 86 marl-limestone, 94 rate, 69, 79, 128, 150, 170, 172, 176, 177, 179, 187, 190, 192, 194 step, 74 Sensitivity experiments, 171 Sequence stratigraphy, 1, 2, 144, 197, 198, 200 Series of events, 49, 102 Sligo, North West Ireland, 82, 100, 109-111, 113, 114, 117, 118 Solar constant, 40 Spectral analysis, 45, 52, 53, 55, 57-59, 61, 6367, 69, 70, 72, 102, 105, 114, 115, 117, 131, 133-135, 145, 146, 150, 152, 155, 158, 161, 169, 174, 176, 184, 186, 189, 190, 199, 201 Spectrum Benbulbin shale Glencar limestone, 115 Blackman-%key, 56,61, 62, 191 Blue Lias (Lower Jurassic), 150 Dachstein limestone (Lofer), 131 Dartry limestone (Carboniferous), 117 German Jurassic, 157 Kimmeridge clay, 153 line, 59, 60 maximum entropy, 56, 57,66 Miravete Formation (Beriasian), 174 Scisti a Fucoidi, 164

SUBJECT INDEX Spectrum (continued) Thomson, 57,66, 138, 164 Upper Triassic, Hungary, 134 Walsh, 63 Stability of astronomical solutions, 36, 37 Stationarity, 52, 54 Stochastic process, 17, 51 Storm frequency, 192 Stratification, 93 cycles, 7, 93, 99, 132, 155, 162, 166, 168 pattern, 99, 111, 149, 172 Stratigraphic data, 50, 63, 66, 68, 93, 101, 171, 203 Stratigraphic mapping function, 73, 104 Stratigraphic trends, 104 Stylolites, 96, 161 Sub-bedding planes, 109-1 12 Subaerial exposure, 135, 136 Subsidence, 25, 26, 104, l z i , 125, 139, 140, 142144 equilibrium with sedimentation, 142 rates, 128, 139 Subtidal, 125, 135, 137 System autocyclic, 25 conservative, 13 dissipative, 5, 16 dynamic, 11 non-linear, 14 oscillating, 4, 11 self-oscillating, 5, 16, 21, 24, 27, 47 Tectonic movement, 192 Terrestrial, 2, 5, 9, 27, 29, 73, 94, 111, 175, 186, 194,202 Tethis, 125, 172 Texas, 183

225 Thickness-time relationship, 76 Threshold, marl-limestone weathering, 94 Time series, 3, 17, 18, 26, 37, 45, 49-53, 58, 63, 64, 67, 69, 92, 95, 154, 157, 164, 173, 175, 179, 184, 187, 191 Tithonian, 159 Trans-Danubian Central Range, 126, 131, 137 Transfer function, 67 ~ansgression-regression cycles, 122 Trends, 52, 104, 109, 169, 192. 198 Tbkey window, 55,62 Tbrbidites, 73, 179, 194 lkronian, 170 chalk, 177 Umbria (Italy), 159, 160 Upwelling, 206 Vail cycles, 1, 197 Valangian, 172, 173 Varve analysis, 75 Varves, 2, 5, 74, 184, 186, 188, 189, 197, 200, 204 Vegetation, 207 Vernal equinox, 29, 32, 40 Virginia, 194 Visean, 108 Vocontian basin, 159, 172-174 Walsh filter, 68 Walsh function, 63 Walsh spectrum, 63, 65, 149 Warlingham well, 152 Wyoming, 188 Yoredale facies. 119 Zumaya, 178,179,182

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