3d1405140406_cc9daafb6adee4c1016b8a88ad605806

.Earth-Science Reviews 50 2000 5175www.elsevier.comrlocaterearscirev

Repetitions and cycles in stratigraphyW. Schwarzacher

School of Geosciences, Queens Uniersity, Belfast, BT7 1NN, Northern Ireland, UKReceived 14 December 1998; accepted 4 November 1999

Abstract

Stratigraphic sections show repetitions of similar or identical conditions on all scales from millimeters to many hundredsof meters, representing time intervals from possibly seconds to many millions of years. Any such repetitions have beencalled cycles. To understand the reason why cycles occur, a time scale is essential. Beds can be with or without timeinformation: the latter are known as event stratifications. The event, which triggers the formation of beds or any successionof sediments, could be a random effect, or it could be part of a mechanism in which cyclic behaviour is due to an oscillatingphysical system. Such systems are dynamic systems, which means that they involve the transport of masses over distancesand in definite time intervals. Cycles generated by such systems can be used as time units. A special example is the earthsolar system which can generate cycles from daily periods to orbital variations of thousands and millions of years. Orbitalvariations are known as Milankovitch cycles and they are the subject of cyclostratigraphy proper. The recognition ofMilankovitch cycles or of any other cycle associated with oscillating systems, rests on the regularity or time periodicity ofsuch patterns. The correct ways of analysing such cycles are the methods of time series analysis. Repetition patternsrepresenting time intervals of millions and possibly many hundreds of millions of years, are known from sequencestratigraphy. We know as yet very little about the mechanisms which generate such long periods of repetition. It has been

suggested that the earths crust has reached a state of self organized criticality Bak, P., 1996. How nature works. Springer,.New York, 212 pp. , in which case sequence formation could be part of the complexity which undoubtedly is an attribute of

the earths crust and history. q 2000 Elsevier Science B.V. All rights reserved.

Keywords: repetitions; cycles; stratigraphy

1. Introduction

Stratigraphic texts tell us that geological historywas written in chapters which repeated themselvesover and over again. Every fixed point on the globein the course of geological history, has at one timebeen either a source or a receiver of sediments. Theenvironments of deposition from the Precambrianonwards have been similar and repeat themselves;apart from the fortunate exception of the biosphere,there are very few indications of a progressive devel-

opment in geological processes during the last 1000Ma. Indeed, based on our present observations, onecould easily believe that most sedimentation andtherefore stratigraphy should have ended long ago.All basins should have been filled and all mountainseroded. This is not the case and leads us to believethat tectonic events must interfere and revitalize thesedimentation systems. Tectonic is used here in itswidest sense, meaning structural changes in theearths crust as well as possibly deep seated pro-cesses.

0012-8252r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. .PII: S0012-8252 99 00070-7

( )W. SchwarzacherrEarth-Science Reiews 50 2000 517552

However, the tectonic intervention is not the onlymechanism which leads to repetitive and therefore inthe widest sense, cyclic processes. In particular, cli-matic variations, which can be either part of fluctuat-ing processes in the atmosphere hydrosphere system,

.or variations in solar radiation Milankovitch cycles ,have a profound effect on sedimentation and itscyclicity. Very complex systems may develop as aresult of the interaction between the biosphere, cli-mate and sedimentation.

The repetitive change in sedimentation andstratigraphy takes place at all scales of time and therepetition can be either regular or quite irregular. Theterm cycle implies some sort of regularity and inorder to establish this regularity, cycles have to berepeated at least several times. Unfortunately, thiscondition is not always true for sediment sequenceswhich geologists call cyclic. Indeed, the termcycle has been applied to so many unrelatedphenomena that it has become useless, unless itsmeaning has been clarified in specific cases.

During the last few decades, the word cycle hasbecome strongly associated with sequence stratigra-phy and it has become common practice to refer to

the duration of cycles as their order Vail et al.,.1991 . According to this system, a first order cycle

has a duration of over 50 Ma, a second order cyclehas a duration of 50 to 3 Ma, 3rd order cycles havedurations of between 3 and 0.5 Ma and higher ordersrange from 500 to 10 ka in length. This system ofordering cycles is widely used but has serious short-comings. By pretending that there exists a hierarchyof cycles, repetitive patterns, which have nothing incommon except their duration, are mixed together.Such a classification is just as meaningless as group-ing elephants and fleas into an order based on theirsize.

The regularity of repetition, which leads to peri-odic or near periodic sequences, is an importantproperty of cycles that has been largely neglected in

the cycle terminology which is used in sequence.stratigraphy . Periodic cycles are the result of oscil-

lating physical systems which differ fundamentallyfrom systems in which the time intervals betweenrepetitions are not necessarily part of the mechanismwhich causes the cycles. Periodicity is particularlyimportant in the recognition of astronomically con-

.trolled cycles Milankovitch cycles that provide the

basic units for cyclostratigraphy. The relationship ofcyclostratigraphy to sequence stratigraphy will bediscussed in later paragraphs. For the present, it maybe noted that many cycles as used in cyclostratigra-phy fully conform to the sequences treated in se-quence stratigraphy but that not all sequences can beregarded as cycles in the sense of cyclostratigraphy.Time, however, is an important problem in bothstratigraphic approaches.

We could find out a great deal more about howcycles formed, if we could know how much time itrepresents. On the other hand, if we knew somethingabout the formation of cycles, we may also learnsomething about the time they represent.

2. Sediment and time

The well-known law of superposition, which statesthat the younger strata rest on older strata, is at-tributed to Hutton who recognised this importantstratigraphic principle as a consequence of his funda-mental discovery that sedimentary rocks haveformed as sediments. Because sediments are piled ontop of each other, we can have absolute confidencein the superposition law.

Sedimentation always takes place at the interfacebetween the sediment and the sedimenting mediumand this represents a surface which connects pointsof identical geological age. Such a surface is called a

time plane. We can use the z axis which is at right.angles to the interface to fix the stratigraphic posi-

tion of successive time planes. Measurements takenalong the z axis are made in centimeters or meters.However, our ultimate aim in stratigraphic analysisis to replace such measured z values by time values

. .which will be expressed in years a , kilo years ka .or millions of years Ma .

To investigate the connection between time andsediment accumulation, one needs some insight intothe mechanism of sedimentation. We shall consider avery simple model. Because sedimentation is a highlycomplex process, the model will be based on arandom process and in this way, it can be made verygeneral and can be applied to a wide range ofsedimentation processes.

Let us assume that at each consecutive moment oftime, one of three processes can take place. Either a

( )W. SchwarzacherrEarth-Science Reiews 50 2000 5175 53

.sedimentary particle is deposited deposition , or .nothing happens, nondeposition , or a particle is

.removed erosion . Which of these events actuallyhappens is determined by some probability distribu-tion.

The position of the sediment surface will move asa function of time in an irregular way describing a

path which is called a random walk curve A in Fig..1 . Provided that the probability of deposition ex-

ceeds the probability of erosion and nondeposition,sedimentation will increase more or less linearlywith time and any deviation from this will be nor-mally distributed. The tendency towards a normaldistribution of deviations, is quite independent of theprobability distributions of the increments. Thesecould be the previously mentioned particles or forexample, successive laminae or strata. If the cumula-tive curves of such incremental steps plot as approxi-mately straight lines, then this provides evidence thatthe mechanism of sedimentation has not changed andthat the sedimentation system can be regarded asstationary.

As an example, cycle thicknesses in the Trias- .sic Latemar limestone Goldhammer et al., 1990

have been plotted in Fig. 2. The graph shows tworegions in which the cumulative thickness increasesapparently linearly with increasing bed numbers. Thelower part from 0 to 180 beds, has a slope of about40 mr100 beds, followed by a distinct increase toabout 110 mr100 beds. A more detailed plot of thelowermost 100 beds shows that the successive endpoints of this sedimentation are within the limitspredicted by a random walk model, which is basedon the assumption that the statistical properties of the

.step formation cycles have not changed.

Fig. 1. Realisation of a random walk. Curve A shows the positionof the sediment surface as a function of time. Curve B shows thestratigraphic record. Erosional stages are not recorded.

Fig. 2. Cumulative cycle thicknesses from the Latemar Limestonein the Dolomites. Two more or less linear trends can be seen. Anylinear trend indicates unchanged conditions of sedimentation,irrespective of the type of cycle used in the exercise.

The cumulative curve of sediment increments canbe used to relate accumulated sediment to time. Thisrelationship is only exact when the increments alongthe x axis represent equal time intervals. If this isnot known, then the slope of such curves is deter-mined by the spacing of the time intervals as well asby the amount of sediment in each increment. It istherefore not permissible to conclude automaticallythat an increase of slope in a cumulative curve,indicates an increase in sedimentation rates. Indeed,in the above example, it is thought that the increaseof slope is largely due to the more frequent fusion of

two cycles. Therefore the increments units along the.x axis in the lower half are not the same as in the

upper part.Many studies of the sedimentation rates of real

sections have shown that such rates apparently de-pend on the lengths of time over which such esti-

.mates are made Gilluly, 1949; Sadler, 1981 . It was .pointed out by Tipper 1983 that geological profiles

can only record a sedimentation history in which all .the negative erosional steps are missing. The ran-

dom walk of Fig. 1 that represents the true history ofsedimentation is therefore represented in the geologi-

.cal column as an accumulation curve B in Fig. 1 .The end point of this curve is determined by theconditional probability of a sum of steps, provided

that accumulation occurred. This probability like theprevious unconditional probability of reaching an

.endpoint in curve A , tends towards normality withlinear increasing mean and variance. However, inthis case, the statistics of the individual steps cannotbe obtained from the strongly biased data in thepreserved record. The theoretical analysis of this


process shows that with a decreasing number of .steps length of section , the apparent rates of sedi-

mentation decrease in the same way as has been .found in empirical data Strauss and Sadler, 1989 .

The discrete random walk can be replaced by acontinuous model called the Wiener process, whichis determined by a drift that corresponds to a meansedimentation rate as well as a variance. The modelshows that the probability of the time record beingpreserved, at first rapidly decreases, but reaches aconstant value after a long time. The completeness ofa section therefore depends on its length. This lengthdetermines the short term accumulation rates whichwhen plotted on a log log scale against estimatedtime, show an almost linear decline but eventually

.reach a constant value Sadler, 1981 .The analyses by Tipper and Strauss lead to two

very important concepts: stratigraphic resolution andcompleteness. The resolution is given by the shortesttime interval that can be recognised and it is clearlyrelated to the elementary steps of the discrete ran-dom walk. Completeness on the other hand, is thepercentage of the time intervals that can be recog-nised in a given time.

As will be found in practically all stratigraphicproblems, the concept of completeness is strictlyrelated to a predefined scale. For example, we canexpress completeness on a 100 ka scale, meaningthat we have evidence of a certain number of 100 katime intervals in a given section. Such scales do notnecessarily have to be expressed in time. For exam-ple, we can also express completeness of a sectionby counting the numbers of beds or any other recog-nisable event. The important point is that withoutreference to a given scale, completeness in stratigra-phy loses its meaning.

3. Sedimentary beds

The bed is the official smallest stratigraphic unitbut regardless of its wide usage, it is very difficult togive an accurate definition of this basic term. Ratherthan attempting a perfect definition, we will regardbeds as repeated units that are separated from eachother by bedding planes. The thicknesses of beds aremeasured along the z axis and the units can be either

homogeneous or composite, depending on the scaleused in their description.

The question of scale enters most definitions ofbeds, and this is well illustrated by the almost ubiqui-tous graded bed that seems to indicate a continuouschange in the sedimentation conditions within a sin-gle bed. If the same change takes place on a largerscale, a series of sandsiltshale beds will be gener-ated and we would probably refer to a group of suchbeds as a cycle. The terms bed, composite bed, andcycle can be logically applied to any scale and whichterm is used is often simply a question of conve-nience.

The same dependency on scale also applies if oneconsiders the time intervals associated with beds.

.According Seilacher 1991 , beds represent the onlytruly coherent paragraphs in a record which is full oferosional gaps. However, what we consider ascoherent is once again determined by the scale onwhich we choose to examine the bed. On a verydetailed scale, beds also may be full of gaps. Therelevant question is whether the thickness of a bed ora group of beds reflects the time it took to form. Tobe meaningful, the times represented by the bedshave to be compared with the average accumulationrate of a larger stratigraphic interval. It is quitepossible that beds are formed practically instanta-neously and that most time is spent in the beddingplanes when no deposition takes place. Under suchconditions, single beds do not contain any timeinformation. Nevertheless, beds can still be regardedas the elementary steps in a random walk and it is ofinterest to question how many beds are needed toform an increment that contains time information.Since the model is applicable to any step fromlaminae to beds and groups of beds, one can increasethe step size until the steps are related to time.Naturally with increasing step size, the scale willbecome less accurate and its statistical significancewill diminish rapidly.

To answer the problem of time information, onevery much needs the sedimentological data. Sedi-mentological and taphonomic studies can provideevidence about sedimentation rates. For example,turbidites consist of units that can be regarded ashaving been instantaneously deposited and the sameis true for many beds that are formed by eventstratification. Such beds not only form rapidly but


also occur in an unpredictable pattern Seilacher,.1991 . Also, there is of course considerable evidence

for slow and extremely slow sedimentation, includ-ing evidence for nondeposition, all of which are animportant source of time information.

Since the nature of sediments is largely controlledby the environment, changes in lithology lead almostinvariably to changes in sedimentation rates. Thesequence of materials that make up a bed or a seriesof beds is therefore an important factor in the timethickness relationship and will have to be discussedin some detail.

4. Bedding planes as time planes

Time planes exist for every moment of time andfor the whole globe but we cannot recognise them.The interface between the sediment and the sedi-menting medium is by definition a time plane andsedimentation therefore determines whether timeplanes are recorded and are separated along the zaxis, or whether they merge in areas of nondeposi-tion. Individual stages of sedimentation can only berecognised by changes, such as changes in the de-posited material or changes in the rate of sedimenta-tion. Changes in sedimentation lead to bed formationand it is therefore important to know how closelybedding planes can be identified with time planes.Since practically all sedimentation processes involvea certain amount of lateral transport, there will al-ways be a horizontal component in the sequence ofbedding planes as well as in the position of succes-sive time planes. Surfaces which represent intermit-tent periods of nondeposition, something whichprobably applies to many bedding planes, can beaccurate time planes even if their age history may bedifferent in different directions. It has been a re-

peated theme of stratigraphers in particular biostrati-.graphers to stress that all lithological units are time

transgressive. A typical example of this emphasis onthe time transgression of a lithological unit can be

.found in the interpretation by Shaw 1964 of theMississippian cyclothems in the mid-continent re-gion of the United States. The cyclothems are thoughtto have originated from the transgression of a shal-low sea on to a slightly inclined shelf. The sequencesof basal sandstone and possibly non marine shales

and coals which are followed by marine shales andlimestones, are produced by migrating facies beltswhich are continuously shifted by the advancingtransgression. It follows from this hypothesis thateach lithological unit is time transgressive.

Shaws interpretation follows very closely theprinciple that was first established by JohannesWalther and which states that corresponding to anyvertical sequence of facies types, there exists a lat-eral succession of facies belts which by their migra-tion, have caused the vertical succession. The princi-ple is a geometrical necessity as long as one isdealing with a strictly depth related classification offacies, for example in marginal basin settings. If themigration of the facies belts is discontinuous, itbecomes possible that the resulting vertically stackedfacies units are practically time parallel. In such aninterpretation, transgressions and regressions are fast,compared with the accumulation.

The changes in sedimentation, which are neces-sary to recognise time planes, are not solely facieschanges and do not always have to be depth related.Stratification of sediments is usually due to the muchmore rapid changes of either sedimentation rates ormaterial, than to the more slowly moving faciesbelts. Indeed, bedding can be frequently traced fromone facies into its neighbour, which often leads to aninterdigitation of adjoining facies. Evidence for fa-cies changes, which have been so rapid that Waltherslaw was apparently disobeyed, is found in the stratig-raphy of the Mediterranean salinity crisis, for exam-ple.

The fact that a slight time transgression of thelithological boundaries due to lateral transport isunavoidable, applies equally of course to time planesderived from biostratigraphic events. Not only doesthe preservation of fossils depend on the history ofthe sediment which contains them, but biosystemstoo are not created on the spot and have to migrate.

The severity of time transgression can be esti-mated by comparing the rates of lateral transportwith vertical sedimentation rates. Attempts to make

.such comparisons Schwarzacher, 1975 came to theconclusion that lateral rates are 108 to 1010 timesfaster than vertical rates. For example, the timeinterval corresponding to a cyclothem in the Missis-sippian example is now believed to be 400 ka .Heckel, 1990 and one can estimate that the trans-


gression that covered large parts of Kansas, pro-ceeded with roughly 200 kmr400 ka or 50 cmryearat a minimum. The average sedimentation rate ofPennsylvanian cyclothems is 25 mmmryear and itfollows that the maximum gradient of a time planewithin a cyclothem is only 5)10y4. If it is assumedthat the flooding has been interrupted by intervals ofnondeposition, then any error involved in makingindividual beds time parallel, becomes very small.Unfortunately, data for such estimates are very sparsebut will become better with increased high resolutionstratigraphy.

Accuracy in estimating the synchronism of geo-logical events is obviously important in stratigraphiccorrelation, but of similar interest is the question ofhow far and over which areas we can correlate thesynchronous time planes. Distance and time reso-lution clearly go together. Within a thin section, it isoften possible to recognise two events which musthave occurred within seconds. On the other hand,sychronism of two events that may have occurredthousands of kilometers apart, can only be estab-lished by reference to a geological time scale whichmay have a time resolution of several Ma.

The distance over which correlation is possibledepends, however, not only on the paleogeographyof the time in question, but also on the agent ordictator which determined the historical develop-ment at the time. The dictator is a word which

.was first used by Sander 1936 and means simplythe cause or causes that generate certain geologicalprocesses. The usefulness of the term is the verygeneral way in which it can be applied. One can forexample, speak of a climatic or tectonic dictatorwithout specifying anything else. A dictator is fur-ther associated with a domain. The latter is an areaover which a dictator is active and one can thereforehave local, basinwide or worldwide dictators. It isthis concept of a domain in particular that can giveus clues about the reasons why sedimentation changestook place.

5. The repetition of stratigraphic events

One of the most important differences betweenbiostratigraphy and lithostratigraphy is that the for-mer deals largely with unique events, whereas the

latter is mainly concerned with frequently repeatedevents. If one considers geological history as a seriesof specific events, repetition is unavoidable simplybecause we are limited by the number of differentevents we can specify. The events we consider areeither beds or lithologies or on a large scale, succes-sions and system tracts.

Repetition can occur on many different scales,sometimes relatively regularly and sometimes quiteirregularly and at random. Repetition may be eitherthe repeated occurrence of events in the stratigraphicsection, or it may be a repetition in time. The latterof course implies some interpretation. The termcycle, which has been universally used for de-scribing repetitive patterns, has been used in so manyways that we will specify them as geometric cycleswhen they refer to measured thicknesses betweenrepetitions, and to time cycles when this is relevant.

For the study of stratigraphic variation, it is usefulto consider observed stratigraphic data as a func-tion of z:

s z . .The stands for a variable that may be either abiostratigraphic or a sedimentological observationand z is its position in the section or well. The

.ordered sequence z , zs1, 2, 3, . . . is called atime series, a term which is generally used for anyordered series and does not imply that z has to haveany relation to time. The series can be either continu-ous or, as is more often the case with stratigraphicdata, discontinuous. In any case, it is possible toassociate certain events with a particular value of .

In theory, there are two extremes with whichrepetition can occur. Repeated events can be ex-tremely regular and after a constant interval P,which is called the period, identical sediments areobserved. This can be expressed as:

z s zqP ) r rs1,2,3 . . . . .The other extreme is a situation in which the se-quence of events is determined by a series of uncor-related random events. The distribution of intervalsunder such circumstances is a probability distributionwhich is given by the negative exponential. Thestochastic process that leads to this type of sedimen-tation is called the Poisson process. Since randomevents are an integral part of sedimentation, distribu-


tions of bed thicknesses and thickness distributions .of specific lithologies phases of beds or cycles,

frequently resemble negative exponential distribu-tions.

However, one should not expect to find eithertruly periodic regularity or complete disorderliness inany stratigraphic section. Even if there exists somemechanism that produces repeated sedimentationconditions, the process of sedimentation itself, to-gether with subsequent changes such as diagenesis ortectonics, would introduce errors which at the bestwill give us a record of equal repetition periods witha statistical error. Complete unpredictability in sedi-mentation systems is equally unlikely since progres-sive sedimentation very much depends on its prede-cessor and therefore is most likely to depend on acombination of random events.

It is possible to analyse bed thickness distribu-tions in terms of combined random events and drawsome conclusions from models which generate simi-

.lar distributions Schwarzacher, 1975 . For example,the two thickness distributions of shale limestonebeds obtained from two consecutive formations inthe Carboniferous from NW Ireland provide a good

.example see Fig. 3 . Such distributions can bederived by the convolution of different randomprocesses and the distributions can range from theexponential to the nearly normal distribution. Theincreasing symmetry of the distributions in the twoexamples could be interpreted as an increasing num-ber of random events involved when the beds wereformed. Indeed, in the case of the Glencar Lime-stone, one can show by very detailed analysis thatthe bed formation very often involved at least three

random pulses of sedimentation Schwarzacher,

Fig. 3. Frequency distributions of bed thicknesses for two exam-ples from the Irish Carboniferous. It is suggested that the highersymmetry of the Dartry Limestone frequency distribution indi-cates a bed formation which was more complex than that of theGlencar Limestone.

.1975 . On a detailed scale, such beds are thereforecomposite. However, during field work this wouldnot be obvious.

The bed thickness distribution is also importantwhen assessing the time information provided by

.beds see earlier . The more symmetrical distribu-tion, which consists of many events, is likely toindicate relatively continuous and therefore time rep-resentative sedimentation. Unfortunately, it is possi-ble to show that quite different models of bed forma-tion can lead to the same thickness distribution andalthough a certain distribution may indeed arise froma model, this does not in itself prove that the modelprovides a valid explanation for an observation. Afurther difficulty is that some distributions, whicharise from theoretical models, are difficult to tellapart. For example, distributions like the gammadistributions, the compound Poisson distributions andthe Lognormal distribution, all look very similar andcan only be differentiated when the tail ends of thedistributions are known. Unfortunately, extreme val-ues are always rare and therefore they are usuallymissing from observations.

The nature and completeness of the observed datavery much determine their methods of analysis. Un-der the most favourable conditions, observationsconsist of measurements which can be expressed as acontinuous function of the stratigraphic position z.Examples of such data are chemical analyses takenat equal intervals, geophysical values like conductiv-ity, gamma counts, or any count of sedimentologicalor paleoecological significance. Less favourable foranalysis are discrete data, such as a series of succes-sive rock types and least informative are data whichsimply record the recurrence of a similar event. Anexample of the latter are bed thickness measure-ments, which of course only record the distances atwhich bedding planes reappear.

The statistics which are used in the analysis ofstratigraphic time series need to recognise that theobservations have been taken in an ordered se-quence. Such statistics therefore do not only refer tothe quantity which has been observed but also to itsposition. Two statistics which are important in thisanalysis are the autocorrelation and the power spec-trum. The latter has been found particularly useful instratigraphic analysis. The use of spectral analysiscan be illustrated by the stratigraphic record shown


Fig. 4. A typical stratigraphic record. Nominal colour values weredetermined from a freshly cut core of Pliocene Trubi Marls .Sicily .

in Fig. 4. This record shows the colour of a freshly .cut core through the Pliocene Trubi marl Sicily

using a nominal colour index which ranges between0 and 10. Inspection of the record shows that thevariation, although it is quite irregular also showssome systematic variations. The power spectrum ofthis series is given in Fig. 5 which is a plot of power .variance along the vertical axis against a frequencyscale along the horizontal axis. The analysis hasdetermined how much power, that is variation, iscaused by fluctuations with individual frequencies.For example, if the colour distribution would have

.been completely random white noise , each fre-quency in the investigated range would have con-tributed and this would have produced a constantspectrum. The analysis of the colour record, how-ever, shows quite distinct peaks or maxima at vari-ous frequencies. The frequency unit in this particularcase is cycles per centimeter. A more familiar unit isthe Hertz or cycles per second. The wavelength,which is the reciprocal of the frequency, tells us thatin the example at intervals of 60, 118, 171 and 827cm, we have repetitions of similar colours. Thisresult clearly needs some geological explanation.

6. The causes of repeated sedimentation

Section 5 has shown that repetition of strati-graphic events is unavoidable but that the repetitionthat follows a definite pattern needs an explanation.There are many processes in geology about whichwe know so little that we cannot predict when andwhere events connected with them will occur. In thiscase we usually assume that such repetitions are atrandom. The random events in such systems maytrigger developments in the environment which leadfor example to definite sequences of lithologies. Intime series analysis, this type of generating mecha-

nism is known as a stochastic process. Stochasticprocesses can be made responsible for the oftenobserved unpredictability of stratigraphic repetitions.However, it is also possible that the stratigraphicsystem has entered a state of deterministic chaos. Inthe description of sections, it makes little differencewhether we regard the stratigraphic history as beingunpredictable, stochastic or unpredictable chaotic,but for the interpretation of observed records, itwould indeed be interesting to know if the origin ofa complex stratigraphic history can be explained by asystem that involved relatively few degrees of free-

.dom. In a phase space see later of low dimensions,models which can be based on non-linear but rela-tively compact systems lead to a much better under-standing than models which have to refer to aninfinity of random variables. Unfortunately, themethods which can be used to find the dimensions of

non-linear chaotic systems Nicolis, 1987; Mudelsee.and Stattegger, 1994 are not very successful when

applied to observations and processes which alsocontain a large stochastic element see also Vautard

.et al., 1992 .When one is concerned with the causes of re-

peated sedimentation, one is talking about real pro-cesses, or in other words we are concerned with thephysics of sedimentation. Physics involves time, notthe comfortable relative time the geologist uses as anescape, but real time which is measured in seconds,minutes, years and millions of years. Absolute timevalues are very rare in stratigraphic analyses and anymeasure of time can only come from the spacing ofevents in the section. The z values somehow have tobe transformed into t values.

Fig. 5. Power spectrum of the record shown in Fig. 4. Thewavelengths of the most important maxima are given in centime-ters and the lower 10% significance limits are indicated byvertical lines.


Let us first assume that there exists a section withperfect sedimentary repetition, for example, a seriesof beds with exactly the same thickness or some veryregular geometric cycles. This could logically beinterpreted in two ways. Either there was a mecha-nism which at irregular times delivered sedimentpulses with equal volume, or sedimentation wassteady but changes occurred at constant time inter-vals. A combination of the two processes cannot be

expected to result in precisely equal beds see ear-.lier .The assumption of absolutely equal beds is unre-

alistic but the possibility of sedimentation mecha-nisms which deliver nearly constant sediment vol-ume or which operate at nearly equal time intervals,must be seriously considered. It is clear from whathas been said earlier, that constant bed thicknesstogether with irregular time intervals implies that thebeds have no time information and formed instanta-neously. One associates such sedimentation withcatastrophic events such as slumps, tsunamis, stormsand deluges. There is increasing evidence that theeffects of such events follow a scale invariant power

.law Hsu, 1983 . Direct confirmation of this is pro-vided by the thickness measurements of turbidites .Rothman et al., 1994 which in the range of 1300cm thickness, follow a log log survivor function. Ifthe mechanism suggested by these data is one whichis controlled by self organised criticality, this wouldrule out any tendency towards the formation of bedswith equal thickness.

It is possible to think of situations in specificenvironments which generate equal amounts of sedi-ment at irregular times. For example, in a givenfluviatile environment, crevasse formation and chan-nel diversions could lead to cycles of similar thick-ness and geometry, without the necessity of timecyclicity. Such sedimentation sequences can be verysuccessfully modelled by stochastic processes suchas the moving average process and auto regressive

.processes Vistelius, 1949 .Processes which generate time related beds of

nearly equal thicknesses are called oscillating. Oscil-lating systems are dynamic systems, which meansthat they invariably involve the movement of masses.A description of such systems is given by the phasediagram which represents the status of the system atvarious time points. In a very simple oscillating

system as for example a pendulum, we need onlytwo variables, the position of the point mass and its

.speed. In the phase diagram Fig. 6a , we use the xaxis to plot the position and the y axis to plot speed.The position is the distance from the resting pointand reaches its extrema at the turning points of thependulum where the speed is zero. The speed reachesits maximum when the point mass passes through theoriginal resting position.

If one plots successive states when the pendulummoves from right to left in the upper half of theco-ordinate system and the reverse movement in thelower half, one obtains a closed path. A closed pathin a phase diagram always indicates a periodic sys-tem. Since the ideal pendulum involves no friction,the system is nondissipating and the path remainsunchanged and is known as the attractor. In a realpendulum, friction will introduce damping and thephase trace will spiral towards the resting position,which in this case represents a one dimensional pointattractor. The dimension of the phase space dependson the degrees of freedom of the system and forexample, a so-called quasi periodic system cannot be

.Fig. 6. a The phase portrait of a simple oscillating system, forexample, a pendulum. The system can be represented by twovariables: speed and position of the plummet. The phase path is

.closed and the process is therefore periodic. b A phase portraitof a system which tends towards a periodic movement. If thesystem is disturbed it will revert to the limit cycle.


represented in a two dimensional space and needsthree or more dimensions, depending on its complex-ity.

Oscillations in the real world can only be main-tained if energy is supplied to the system. Such asupply of energy has to be in phase with the oscilla-tion and this is achieved by feedback mechanisms.Systems with feedback are essentially non-linear andtheir development in time can become unstable. Someinstabilities can lead to chaotic behaviour, where thesystem fluctuates within a well-defined range, butwith an infinite number of random values. Suchsystems represent an intermediate state between os-cillating and random and complete divergence. Theattractor of chaotic systems is fractal, and for exam-ple, a two dimensional phase space consists of acurve which fills an area but never touches itself andexactly identical states are never visited again. Therate with which neighbouring paths diverge or con-verge is measured by the Lyaponov exponent. Apositive exponent indicates exponential divergenceand therefore chaos.

Nearly periodic oscillations can be maintained inself oscillating systems in which the attractor is

.given by the limit cycle Fig. 6b , which is a periodicpath towards which the system will tend, even afterit has been disturbed. Many man-made machinesfrom clocks, steam engines and petrol engines toelectrical circuits, represent such self oscillating sys-tems. Environmental cycles which are responsiblefor sedimentation changes, follow exactly the sameprinciples and are caused by dynamic systems.Therefore one has to ask: what are the masses whichare moved and over which distances and at whatspeeds are masses transported? What are the regulat-ing processes and how is energy supplied to thesystem?

Oscillating models for natural processes have beenexamined by climatologists in particular. Relativelyshort cycles with periods of 4 to 8 years have beenassociated with the El Nino Southern Ocean oscilla-

.tion ENSO and the North Atlantic oscillations .NAO , which are the result of large scale atmo-sphere ocean interactions. Longer cycles of 1.5 ka,which are also attributed to changes in the oceaniccirculation, have been found in the records repre-

sented by the Greenland ice cores Grootes and.Stuiver, 1997 . The length of cycles which can be

generated by atmospheric or oceanic circulation islimited by the time lag which such systems canprovide. For example, it has been estimated that thetotal mixing time for the worlds oceans is 1 to 2 kaand this appears to be the longest time which anoscillation that is exclusively based on the atmo-spherehydrosphere system can have.

In order to generate longer cycles, climatic mod-els include various feedbacks from the results ofclimatic variations, which include marine and landice cover and tectonic responses to loading with ice.Such models can generate cycles with periods of 5

.ka to 10 ka Le Treut and Ghil, 1983 and cycles ofthis length have been observed in the late Pleistocene

.sediments the Heinrich events as well as in the icecores.

If it is true that climate moves in steps Berger,.1982 that represent episodes of several million years,

then any exclusively terrestrial oscillating systemwould be upset by such changes. The same argu-ments apply to possible oscillations in the biosphere,which may involve interactions between populationsand evolutionary steps which would be highly sensi-tive to environmental changes.

The slow mass transports of either tectonic orerosional and depositional processes could providelonger time lags but here again, it is difficult tovisualise undisturbed oscillations without damping,in an environment which undergoes continuouschanges.

Mass transports on and in the sun are probablymuch faster and this possibly is the reason why solarcycles with periods of 11 years and 210 years areconsiderably shorter than possible internal oscilla-tions on earth.

7. Autonomous and autocyles

In a discussion of sedimentary cycles, Brinkman .1932 differentiated between two groups which hecalled autonomous and induced cycles. The au-tonomous cycle refers to a cyclic process whichtakes place within a definite and size restricted envi-ronment. Such autonomous cycles are generated byoscillating processes and consequently can lead totime cyclicity. For example, the seiches in a lake arean autonomous system which is strictly determined


by the shape, size and depth of the lake. Similarlyoscillations which can develop in an ecological sys-tem for example, involving a predator prey relation-ship, are contained within a restricted domain, de-fined by the biotope of the organisms concerned.Brinkman contrasted such cycles with induced cycleswhich are the product of an outside control. Pro-cesses which are influenced by the climate or theseasons or the tides are examples for induced cyclicprocesses. Brinkmans classification is not much dif-ferent from classifying cycles according to the sizeof their domain, which can be restricted or large orworldwide.

.Beerbower 1964 introduced the two terms auto-cycles and allocycles which at first sight seem identi-cal with Brinkmanns. This is only the case with theterm allocycle which corresponds exactly toBrinkmanns induced cycle. However, autocyclemeans something different from Brinkmanns au-tonomous cycle. Beerbowers study is exclusivelyconcerned with processes on alluvial plains and heexamined simply the sequences of lithologies whichare deposited in such an environment. An autocycleaccording to his definition, is a sequence in which alllithologies originate from the environment in whichthe cycle is deposited. In other words the sedimentsof an autocycle are simply redistributed. The physi-cal processes which lead to this type of deposition influviatile environments are channel migrations andcutoff, crevassing and channel diversion. Insofar asthese processes are not controlled by allocyclicalseasonal variations, they are predominantly deter-mined by random events. Autocycles sensu Beer-bower are therefore not connected with oscillatingsystems and they are strictly geometrical and nottime related. The thickness distributions of suchcycles have not been examined and it is difficult toobtain large data sets from such environments.

The term autocycle has been taken up by geolo-gists in the belief that many sedimentation processeshave the inherent property of forming cyclic de-posits. Insofar as all sedimentation processes arerepetitive and involve stochastic processes, practi-cally all sediments should be autocyclic. Withoutexamining why repetition does occur, the term be-comes quite meaningless. Very few geologists haveattempted to investigate the mechanism which couldhave led to the so-called autocycles.

A model which is frequently quoted, is the Gins-burg model for carbonate accumulation on a size

restricted shelf which undergoes subsidence Gins-.burg, 1971 . This model, however, is incapable of

oscillating and does not even lead to repeated deposi-tion unless one either introduces critical thresholdvalues or accepts that it is driven by random distur-

.bances Schwarzacher, 1993 . Critical threshold val-ues undoubtedly play an important role in environ-mental developments but if they are to have anyexplanatory value, they must be understood as a partof a dynamic system which regulates the sedimenta-tion. For example, a model which has been proposed

.by Drummond and Wilkinson 1993 assumes that acritical depth is necessary before sedimentation canoccur. Once this depth is reached, sedimentation isextremely rapid and compared with the accommoda-tion space creating subsidence, can be regarded asinstantaneous. Without a further explanation of sucha critical depth, the model leads to the conclusionthat not only are such cycles without any time

information as the rate of subsidence does not enter.the model but it also predicts that all sediments are

cyclic with cycles of equal thickness.A general feature of all autocycles is that they are

specific to the environments in which they form. Ifthe Ginsburg mechanism works, one would expectdifferent cycles for different shelves and similarlyone would expect different critical depths in theDrummond Wilkinson model for different environ-ments. Allocycles on the other hand, if they areassociated with oscillating processes, may be modi-fied in different environments but will show similari-ties for example, in frequency composition. The timeinformation of cycles is most important and ulti-mately will remain a question of geological interpre-tation. The practice of explaining repetitive sedi-ment sequences by calling them autocyclic withoutidentifying a proper mechanism, which causes theobserved cyclicity, is highly unsatisfactory.

8. Milankovitch cycles

The theory that variations in the earths orbitdetermine the amount of radiation and therefore theclimate, was first fully developed by Milankovitch .1941 . The solar radiation or insolation at a specific


locality on the earths surface is determined by thedistance of this point from the sun and by the anglewith which the radiation hits the surface.

The solar system can be regarded as an oscillatingsystem which is almost nondissipative and which canbe treated almost like a linear system. The positionof a planet in this system at different times is givenby its orbit which is determined by six parameters.

.Lagrange 1871 formulated six differential equa-tions corresponding to the six parameters. The solu-tions to the orbital equations can be developed into a

.trigonometric expansion Bretagnon, 1974 and fromthese, one can obtain the important paleoclimatologicvariables: eccentricity, obliquity and precession.

The eccentricity determines the distance betweenthe earth and the sun. The obliquity is the inclina-tion of the earths axis against the pole of the eclipticand the precession is the longitude v of the vernal

.node relative to the perihelion Fig. 7 .The three parameters are quasi-periodic and each

contains a large number of periods which are theresult of the complicated interaction of the planets inthe solar system. The most important wavelength ofthe eccentricity cycle is at approximately 400 ka andfurther strong cycles are found between 128 and 94ka. These are often referred to as the 100 ka eccen-tricity cycle. The obliquity has cycles between 40and 50 ka and the main precession cycles are at 23and 19 ka and are often referred to as the 21 kacycle.

.Laskar 1988 used numerical integration to solvethe orbital equations and these very accurate solu-

Fig. 7. The sunearth system. A and P are the aphelion andperhelion. g is the vernal equinox. The longitude v of theperihelion is measured from the vernal equinox and the obliquity is the angle between the terrestrial north pole N and the celestialpole.

tions begin to diverge when calculated over longperiods. After approximately 30 Ma, the divergencebecomes exponential with a positive Lyaponov expo-nent, indicating that parts of the solar system arechaotic and cannot be approximated by quasi-peri-odic solutions.

This is important for the geologist because it setsdefinite limits to the time in either the past or futurefor which the orbital elements can be accuratelycalculated. It is found that precise solutions for theearths movement can only be given for about 100Ma and therefore an absolute time scale of not morethan approximately 20 Ma.

The precession and obliquity, which are the resultof the sunearthmoon interactions, depend on therotational momentum and rotational speed of theearth. Small variations of speed are caused by tidaldissipation but the effects are small. Laskar et al. .1993 calculated obliquity values with and withouttidal dissipation and found only a small shift ofphase between the two. Whether the tidal effectscould become more important when the configura-tion of ocean basins is completely different, is notknown. The changes, which is determined by themass distribution on the planet, in the dynamicalellipticity of the earth seem to be more important.Changes of dynamical ellipticity could be caused forexample, by ice accumulations during ice ages or bymajor tectonic events. Such changes would be ex-pected during an ice age. There could be a relativelylarge effect because the change in ellipticity mayinduce a resonance with the JupiterSaturn Perturba-tions and this could lengthen the obliquity period by100 years and cause a noticeable increase of ampli-tudes.

Also important are the changes which have oc-curred over much longer periods and in particular the

increasing length of the day due to the slowing of.the earths rotation and the increasing earthmoon

.distance. Berger et al. 1989 tried to calculate thiseffect and according to their model, obliquity cycleshave increased by about 45% since Silurian time andthe precession cycles showed an increase of about20%.

The amount of solar radiation which reaches theearth is clearly the most important factor regulatingclimatic conditions. An average annual insolation forthe whole earth depends exclusively on the distance


between it and the sun and the solar constant whichis believed to have been practically unchanged evenin geological time. Integrated distance variations overa full year depend only on the eccentricity and theresulting variations in insolation are extremely small.Much more important are seasonal changes in solarradiation and, as already recognised by Mi-lankovitch, the summer insolation is particularly im-portant for the development of ice ages. The summerinsolation may vary by as much as 22% betweenextrema. The calculation of daily or monthly insola-tion for any given latitude is without any majorproblem; algorithms to calculate insolation were

.given by Berger 1978a; b .The calculated insolation values give the radiation

that hits the earths surface when a completely trans-parent atmosphere is assumed. However, radiationwill be partly transformed into heat when it passesthrough the atmosphere and this heat, together withthe reflected heat from the surface, is the only energywhich keeps the weather engine going. Clearly, vari-ations in this energy supply will change the climate,but how this variation is transformed into climaticchanges and how this eventually influences changesin sedimentation is far from being completely under-stood. This is particularly the case because the cli-matic conditions in the past are very incompletelyknown. For the more recent Pleistocene, some paleo-climatic information is available and one can studymodels which at least give some indications aboutpossible mechanisms of climatic changes. For olderformations where not even the configuration of landand sea is completely known, and for which thedistribution of climatic belts is hardly known, thereconstruction of climate changes is much more dif-ficult.

Even without precise knowledge of the transfor-mation from insolation into climate and eventuallyinto sediments, one must expect phase shifts to occurduring the transmission. Some frequencies could becompletely removed, others may be enhanced ifthere are self-oscillating processes in the environ-ment which could resonate with the astronomicalsignal. Frequency doubling is also possible if thereaction of the environment depends on thresholdvalues or reacts to the rate of change in the signal.Without doubt, any stratigraphic record will bestrongly contaminated with noise which originates

either from the insolation to climate, or climate toenvironment, or environment to sediment transmis-sion. Clearly, the recognition of Milankovitch cyclesin the stratigraphic record will not be easy.

9. The Milankovitch cycles in stratigraphic sec-tions

Oxygen isotope data which became available inthe 1950s provided a new and powerful tool for

.reconstructing ancient climates. Emiliani 1955 , whoinvestigated the distribution of isotopes in deep seasediments, recognised that the Pleistocene tempera-tures deduced from oxygen isotopes followed thecycles formulated by the Milankovitch theory very

.closely. An important paper by Hays et al. 1976showed by spectral analysis that eccentricity, obliq-uity and precession cycles were indeed present in theisotope record of deep sea sediments and this pro-vided direct support for the Milankovitch theory.

It is useful to examine what makes the Pleistocenerecords such convincing evidence for the presence ofMilankovitch cycles. Possibly most important is thefact that the oxygen isotopes provide an exception-ally good proxy variable which could be directlylinked to the climatic variations which we knowexisted in the Pleistocene. Secondly, sedimentationrates in the deep sea are relatively undisturbed andtogether with an independent magneto stratigraphy,permit a relatively good absolute dating of the ob-served variations. Thirdly, not only the frequencyratios but also the observed frequencies of the iso-tope variations conform to the Milankovitch theory.Finally, it is important that it is possible to correlateisotope stratigraphies throughout the oceans andtherefore we have direct evidence that one is dealingwith a worldwide system. No other geological sys-tem, including the Pliocene, provides such com-pelling evidence for the existence and detailed natureof climatic cycles.

It is in the nature of any historical record thatgoing further back in time, information is less accu-rate. As has already been discussed, this also appliesto the calculations of planetary orbits as well as toany method of absolute dating. Isotopes which havebeen so useful in the Pleistocene and Pliocene areless used in older sediments mainly because it is


more difficult to find clearly defined sources such asspecific foraminifera tests, which have not beenchanged by diagenesis. In addition much less isknown about the distribution and controlling factorsof isotopes in the older environments. There are,however, a large number of environmental indicatorswhich are used in the study of sediments and whichcan be used as a basis for cyclostratigraphic studies.Good examples of such proxy variables are counts ofspecific microfossils, sediment petrographic andchemical analyses and also physical properties likemagnetic susceptibility and gamma ray counts. Suchvariables provide a continuous record but in mostcases, it is not clear how this relates to the astronom-ical time series of either their orbital data, or theinsolation record which has been deduced from suchdata. If there is a connection between the environ-mental history and the assumed astronomical signal,then it is reasonable to expect that the frequencycontent of both curves would be similar and underfavourable circumstances even the amplitudes couldbe comparable. However, the general shape and thephase almost certainly changed in the transformationfrom signal to record. Such changes make the identi-fication of Milankovitch cycles in sedimentary se-quences difficult and they are also very important inattempts to match or even tune stratigraphic records

.to astronomical time series see later .One of the difficulties of obtaining good continu-

ous time series is the time and cost of collecting andanalysing a sufficiently long series of samples andobservations. For this reason, many cyclostrati-graphic studies are based on coded descriptions ofrocks and measurements which can be made in thefield. The analyses of such data is important, particu-larly when it is remembered that it was the regularityof bedding and groups of beds which first attractedthe attention of geologists and which is a characteris-tic feature of sediments needing an explanation.

The most elementary field measurement is the bedthickness. Such measurements can be graphicallydisplayed in two ways, either by a series where thespike height is proportional to the bed thickness, orby a continuous curve which gives the local density

.of bedding planes at evenly spaced intervals Fig. 8 .The latter method does involve a certain amount ofinterpolation which can introduce a bias in the lateranalysis. If it is known that each bed represents a

Fig. 8. Two methods of treating bed thickness measurements. Thelower plot shows the bed thickness in centimeters by the height ofspikes which are located along the line of the section. The upperplot gives the density of bedding planes at equal intervals. Theconstant interval in this example is 20 cm. Data from Carbonifer-

.ous Limestone N.W. Ireland .

definite time interval as for example in varve analy-.sis , bed thicknesses can be directly plotted at equal

.intervals. In this case, the thickness height of spikerepresents the environmentally highly significant ac-cumulation rate per time unit. Usually nothing isknown about the time interval represented by bedsand so a mode of plotting which preserves thestratigraphic position of the measurements is moreappropriate. In this case, the height of the spikes isbetter interpreted as the local density of beddingplanes.

If the cumulative curve of bed thicknesses shows .a linear increase as for example in Fig. 2 , then the

aerage bed density and by implication the sedimen-tation rate, is constant. Locally, the densities canshow variations which may or may not be connectedwith changes in sedimentation rates.

The distribution of bedding planes can be exam-ined by power spectral analysis. Such an analysiscannot determine whether the individual beds repre-sent a Milankovitch cycle. It can only bring outpatterns in the repetition of beds. I have called suchpatterns stratification cycles or bundles .Schwarzacher, 1952 . In the more recent Americanliterature, they are also referred to as stacking pat-terns. Apparent bundling of beds can be the result ofa variety of sedimentation processes and it is perhapsmost prominent in systems where the lithologiesfluctuate between two phases. Good examples arelimestone marl sequences, where beds of limestonealternate with layers of marl and with a more or lessconstant ratio of thickness between the two litholo-gies. However, at regular intervals the marl for ex-


ample, may be increased and in this way generatesbundles or units of higher order. Similar effects canalso be created by repeated regular thickness changesof beds, in sequences with only one lithology. Strati-fication cycles which contain a constant number of

beds, are a particular type of bundle Schwarzacher,.1947 . Such cycles are very difficult to explain if not

by the interaction of two periodic processes andtherefore they point directly towards the Mi-lankovitch theory. The same argument is true, butless compellingly, for stratification cycles of equal

thickness but which do not have an absolutely con-.stant number of beds .

.Drummond and Wilkinson 1993 have arguedthat the use of stacking hierarchies should not beused as evidence for orbital forcing. The authorsshow that the interplay between such variables assedimentation rate and available accommodation,which is partly due to subsidence and partly due toeustatic sea level changes, can modify the develop-ment of stratification cycles. They completely fail toshow how bundles with a constant number of bedscan develop without involving at least one periodicprocess.

Bundles which contain a constant number of bedscan be understood as the response to a composite

.periodic signal Fig. 9 . One could assume that abedding plane is formed whenever the signal reachesa certain threshold. The critical value in this systemdetermines the number of beds in a bundle and thiscan vary according to the sensitivity with which thesediment reacts to the signal. Evidence to support

Fig. 9. The formation of bedding planes as a response to anenvironmental signal. It is assumed that a bedding plane onlyforms when the signal reaches a certain threshold. More noisysignals will produce irregular bedding patterns.

this simple model was found in the alpine Trias,where the number of recorded subdivisions per bun-dle always increased when an identical cycle wastraced from south facing to north facing exposures .Schwarzacher, 1954 . This is due to different weath-ering which is stronger in north facing exposures andconsequently brings out more bedding planes.

A modification of bed numbers in cycles can alsobe due to limits in the accommodation space, a

.process which was examined by Pittet 1994 using .simple simulations and Schwarzacher 1993 using

analytical methods.The very common observation that bedding planes

disappear laterally and cause the merging of twobeds or that bedding planes appear causing a split ofbeds can be used to classify beds into differentorders, according to their lateral persistency. Thishas led to the term master bedding plane .Schwarzacher, 1958 , which is a bedding plane thatcan be traced over large areas. Most bundle bound-aries are such master bedding planes. If the masterbedding plane is taken as a first order bedding plane,one can also have second and third order bedding

planes according to their persistency Schwarzacher.and Fischer, 1982 . Such terms are of course relative

and need a regional study of cycles which is alwaysmore profitable than the study of a single profile.

10. Cyclostratigraphy and the geological time scale

The interest in sedimentary cycles has for a longtime been associated with the hope that they couldprovide a unit with which to measure geological

.time. Gilbert 1895 counted cycles, which he be-lieved to be precession cycles and estimated that itmay have taken 20 Ma to accumulate the Cretaceoussediments along the base of the Rocky Mountains.Estimates of this type were made by a number ofgeologists but the first serious attempt to obtainabsolute dates using the astronomical theory weremade by Milankovitch himself in his history of thePleistocene.

The construction of a geological time scale basedon cyclostratigraphy depends on several conditions.The most important of course is a valid compilationof astronomical parameters for the past. Assumingthat this is available, it is also essential to have a


stratigraphic record which is capable of recordingMilankovitch cycles and which can be traced withpreference continuously from the present into thepast. This continuity is necessary, since the dating ofcyclic records is essentially done by counting cyclesof a given order. Fortunately, the complexity of theorbital history with periods of 20 ka to 2000 kamakes it possible to breach some gaps. Because thequasi-periodic nature of the Milankovitch cyclesgives various intervals a characteristic signature,even disconnected records can be identified as longas their position within the absolute time scale isapproximately known.

Stratigraphic records which react to the Mi-lankovitch cyclicity do so because they are sensitiveto paleoclimatic variations. Often, however, it is notknown which of various possible astroclimatic pa-rameters is the best representative for a particularclimatesedimentation system. Such a representativetarget curve is necessary for tuning an observedseries. Basic astronomical functions such as theobliquity and precession index are an obvious choice

for target curves and these were used Imbrie et al.,.1984 for the construction of a time scale for the late

Pleistocene. The stratigraphic time series in this caseconsisted of a series of d 18 O values as a function ofdepth in sediment cores. From this series, whichinitially was calibrated using radiometric dates, anobliquity and a precession component was extractedby filtering. These were compared with the astro-nomical curves and the positions of the maxima andminima were adjusted by compressing and stretchingthe positions in such a way that a constant phaserelationship between the two records was obtained.

An alternative approach is to use calculated inso-lation curves as the target, such as a summer insola-tion curve for high latitudes.

The tuning consists of adjusting the stratigraphicsequence in such a way that there is a good corre-spondence between it and the target. Martinson et al. .1987 have compared various tuning strategies: theconstant phase method, tuning to insolation curvesassuming both linear and nonlinear response, andtarget curves consisting of the pure componentsand their harmonics.

Any tuning procedure risks the possibility of in-corporating variance that is not connected to the timedependent signal and this can lead to errors in find-

ing a correct phase relationship between signal andresponse. Furthermore, a time scale has to be trans-ferable to become useful. This means that it has todate stratigraphic events which can be recognised ina stratigraphic type section as well as in other sec-tions. The recognition of such events again involveserrors, which have to be included when judging theaccuracy of a time estimate.

A comparison of different tuning methods showedthat they all led to time scales that were very close toeach other. The error for the high resolution stratig-raphy for the last 300 ka is estimated to be "3.5 ka.An overall accuracy of "5 ka is estimated for the

.slightly longer time scale of Imbrie et al. 1984 .This also includes errors which are made in transfer-ring the time scale from one isotope record to an-other, as well as the possibility that the sedimenta-tionclimate system does not respond to the orbitalforcing as a linear system with an essentially con-stant phase lag.

The extension of the cyclostratigraphic time scaleto the Pliocene was helped largely by the land-basedsections of Neogene sediments in the Mediterranean,

.as studied by Hilgen 1987; 1991a; b . Well-devel-oped cyclic sediments are found in the lower Pleis-tocene and Pliocene of Calabria and Sicily and adetailed magnetostratigraphy and biostratigraphy isavailable. Of particular importance is the occurrenceof distinct horizons containing sapropelites whichenable a bed by bed correlation of different sectionsand provide a link to sapropelites in the Mediter-ranean sediments. The sapropelite formation is prob-ably due to an increased sea surface productivity andthis in turn is clearly related to strong precession

.minima. Lourens et al. 1992 showed that sapro-pelite beds occurred in clusters and that these clus-ters coincide with intervals in which the processionalindex has high amplitudes. The clusters thereforereflect the 400 ka and 100 ka modulation of theprecession. In the lower Pliocene, sapropelites are

.replaced by dark marls Hilgen, 1991b which corre-spond to precession minima. The carbonate produc-tion is clearly related to the eccentricity signal andcan be directly matched to it.

Using the Berger 1990 solution, Hilgen con-structed a time scale which essentially attributed anage to each individual bed in a composite section.The dating was achieved by matching marl and


sapropelite layers to precession peaks and by assum-ing a constant time lag. The time scale of Hilgen .1991a,b was slightly revised by Lourens et al. .1996 using an insolation curve based on the Laskar1990 astronomical solution as target curve. The newages differ by about 1 to 4 ka in the range of 2.53.5Ma bp and by more than 20 ka in the range of3.55.3 Ma bp. The divergence is largely due to thereinterpretation of a sedimentary cycle which re-sulted from the merging of two precession minima.Furthermore, it is also possible that the phase lagrelationships have changed. These were originallycalibrated using very accurate 14C dates, before ex-tensive glaciations began.

The accuracy of an astronomical time scale willvary in different parts of a section. Clearly, theaccuracy will be better in time intervals with well-defined maxima and minima. If there is continuity ofcycles from the present to the past, dating is straight-forward. In the case of the Calabrian and Siciliansediments, continuity is achieved by a very accuratecorrelation with the established Pleistocene astro-nomical time scale. Errors can arise if not everyclimatic cycle is recognised in the sedimentaryrecord. This missed beat effect leads to the fusionof cycles and can upset the dating of course. It canbe minimised to some extent, by the study of parallelsections. Such errors would be cumulative, if it werenot for the complex nature of the astronomical andclimatic signals which are compounded from a num-ber of frequencies. For example, not only couldHilgen recognise 400 and 100 ka cycles in the recordof calcium carbonate production in the Pliocene marlsof Sicily, but he could also recognize its characteris-tic shape, which is the result of superimposed longer . .2 Ma cycles. Similarly, Lourens et al. 1992 foundthat the formation of sedimentary cycles is deter-mined by the interference between the precessionsignal and the obliquity signal, which generates pat-terns in the insolation curve that are characteristic forcertain time intervals.

Expanding the Mediterranean time scale into theMiocene involved the Messinian interval, for whichthere are no readily recognisable Milankovitch cy-cles. Well-developed cycles are again found in thepre-evaporite sections of Sicily and some Greek

. .islands. Hilgen et al. 1995 and Krijgsman 1996showed that the cyclicity here is very similar to that

in the lower Pleistocene and Pliocene sediments. Inparticular, sapropelite layers again show a clear pat-tern of clustering which corresponds to eccentricitymaxima. Both the 400 and 100 ka eccentricity cyclesare represented by large and smaller scale clusters.The sections were initially dated using magneticreversals. Matching to the correct 400 ka cycle isagain made possible by recognising the effect of the2 Ma eccentricity signal. The astronomical recordshows that the 100 ka maxima at 8.3 and 8.7 Ma bpare well-developed and a low 100 ka modulation isfound at 7.5 and 9.5 Ma bp. This signature ismatched by the distribution of the small scale sapro-pelite clusters. Using a method of progressive match-ing from the 400 ka to the 100 ka and the 21 ka

.Milankovitch cycles, Hilgen et al. 1995 could de-termine the length of the Messinian as 1.91 Ma anddate the Messinian Tortonian boundary as 7.240 Ma.

The accuracy of all astronomically dated strati-graphic data is always at least as accurate as theshortest cycle in the target curve. This ensures thatthe minimum accuracy of the Miocene dates is "10ka. If additional information about the phase rela-tionships is available, this accuracy can be improved.The accuracy of the astronomical solutions for thelast 10 to 20 Ma is likely to be very much higherthan the above minimum accuracy. Comparison of

.the solutions by Laskar 1988 , Berger and Loutre . .1991 and Quinn et al. 1991 shows that the differ-ences between the positions of extrema is only of theorder of 1 ka. Furthermore, Hilgens astronomicaldates are in good agreement with argon dates fromvolcanic ashes in the Mediterranean.

11. Dating beyond 30 Ma bp

The discovery of chaotic motion in the solarsystem means that orbital movement can no longerbe approximated by quasi-periodic solutions in the

.more distant past. Laskar 1988 has been able toestimate the effect which this non-linear chaoticbehaviour could have on the fundamental frequen-cies of the planets over a time of 200 Ma and foundthat variations in the rate of movement in the innerplanets could be as high as 0.2 arsecryear. Usingthis figure, one can estimate that the 100 ka eccen-tricity period could have an uncertainty range from


87 to 113 ka or that the 2 Ma period could havevaried in length from 3 to 1.5 Ma. These values arevery approximate and should only be taken as arough estimate of possible variations. Experimentscalculating orbital elements which are based onslightly altered fundamental frequencies, show thatthe relative amplitudes of all Milankovitch periodschange and that therefore the shape of the signalbecomes uncertain and it is then no longer possibleto identify a paleoclimatic record which has nocontinuity with the present by comparing it to thecharacteristic shape of a calculated curve.

Both the precession and obliquity are more likelyto undergo variations compared with eccentricity andthe longer eccentricity cycles can perhaps be used toextend the dating range. In particular, the 2 Ma cycleis within the accuracy of radiometric dating andtherefore could be identified in theory. This was thebasis of a very brave attempt to date the KrPgboundary in limestone marl sequences in northern

.and southern Spain Ten Kate and Sprenger, 1992 .The authors collected a very complete stratigraphicseries of calcium carbonate analyses and magneticsusceptibility measurements which provided a timeseries which showed clear evidence of Milankovitchcyclicity. Two conspicuous minima in these recordswere identified and correlated with two strong min-ima due to the 2 Ma modulation of the calculatedeccentricity curve. According to the extrapolatedBerger 1978 solution, two such minima and an inter-val of low 100 ka modulation appear between 64.74and 65.2 Ma bp. Since the KrPg boundary age isfairly accurately known, they could identify the twominima and date the KrPg boundary age as 64.56Ma.

However, this apparently very good accuracy hasto be taken together with what has been said aboutthe irregularities introduced by the chaotic move-ments of the inner planets and with the inaccuracy ofthe astronomical solution when extrapolated overthis length of time. The accuracy can be judged tosome extent when one compares the Berger 1978

solution in which the 2 Ma cycle is 2,035,441 years.long with the Berger 1990 solution in which the

cycle is 2,379,077 years long M.F. Loutre pers..com . Time scales based on the two assumed cycle

lengths will differ from each other by a full cyclealready after about 15 Ma. To put it differently,

assuming that the second solution is correct andassuming that the KrPg boundary occurred at 65 Maand assuming that it was possible to count all the 2Ma cycles to the present, then the older solutionwould have been out by 9.2 Ma!

It is now certain that the existence of chaoticintervals within the solar system cause limitations toabsolute dating with Milankovitch cycles, but rela-tive dating is probably possible over much longerperiods.

12. Milankovitch cycles in older sediments

In assessing the potential of Milankovitch cyclesto date older geological records, one has to remem-ber that geologic ages are given in years. This is atime unit which of course is derived from presentday observations. In geology, it is conventional tobase the unit on the ephemeris second of the tropical

.year 1900 Harland et al., 1990 . To tie the geologictime scale in this way to the present is necessarybecause all radioactive decay rates used to calculategeologic ages are linked to the present day timesystem. Geologic ages expressed in years are there-fore expressed in geologic standard years and not theprecise number of rotations of the earth around thesun in the relevant time interval. The differencesbetween the two time systems are, however, toosmall to have an effect on the crude chronostrati-graphic scale used by geologists. Changes in theastronomical time units are, however, important inthe orbital behaviour of the earth.

Of prime importance, is the slowing of the earthsrotation which has led to a decrease in the number of

days per year. This together with the gradual in-.creasing distance earthmoon is responsible for a

decrease in the length of precession and obliquity .periods with geological age see earlier . Since the

climatic precession is modulated by the various ec-centricity cycles which are not immediately deter-mined by the lunar effect and the rotational speed ofthe earth, the ratio between precession and eccentric-ity periods must have decreased with time. An aver-age 100 ka eccentricity cycle compared with anaverage precession cycle gives a ratio of 1:4.76 atthe present time. In the Silurian, using data given by

.Berger and Loutre 1991 , this ratio was 1:5.8.


The approximate ratio of 1:5 has been observed ina number of sections which show the development ofstratification cycles or bundles and this ratio shouldincrease with increasing geological age if it is due tothe eccentricity precession relation ship. To demon-strate this simply by counting bedding planes inbundles is difficult since beds do not by any meansalways correspond to precession cycles. However,spectral analysis based on an average bed thickness,representing an estimate of the average length of theprecession, appears to give results which are inrough agreement with the theoretically expectedchange in length of the precession periods. For ex-

.ample, Bond et al. 1991 found a peak of 0.14cyclesraverage bed in the Middle Cambrian. If thisaverage represented the precession and if for simplic-ity the spectral maximum represented a 100 ka ec-centricity, then bundles which are formed under suchconditions should have a 1:7.12 ratio. In the lateTriassic, a similar calculation gave the value of 5.2

.bedsrbundle Schwarzacher and Haas, 1986 . Possi-bly the most impressive data so far, come from the

.work of Anderson 1984 who measured over200,000 evaporite varves in the Permian Castileformation of New Mexico. This record shows awell-developed cycle with an average length of 19.2ka which comes very close to the theoretically ex-pected precession cycle.

The study of Milankovitch cycles at least as farback as the Cambrian, has shown that cyclic varia-tions, which can be attributed to orbital variations,have not changed in any dramatic way. The geologi-cal record therefore indicates that throughout theProterozoic and probably far beyond that, the annualrotation around the sun has been fairly regular, evenif the rotational speed of the earth has decreased andeven if perhaps some mild chaotic conditions exist insome parts of the system.

Although it is not possible to give the precise ageof an ancient event even if it were possible to count

the astronomical years which separates it from the.present , astronomically controlled cycles provide a

very useful time scale. Milankovitch cycles can beused to measure intervals within the stratigraphicframe work and it should be possible eventually tobuild up a cyclostratigraphy which runs parallel andcompliments existing biostratigraphies and chronos-tratigraphies.

13. Sequences and cycles

A sequence can be recognised because its begin-ning and its end are marked by an unconformity.Unconformities only form if a depositional phase isfollowed by an erosional interval. Changes fromdeposition to erosion, as previously discussed, takeplace at any scale from the short fluctuations in thecurrent velocities of a stream, causing laminations towidespread transgressions and regressions.

To be useful, unconformities which terminate se-quences have to be recognised over large areas andsequences, and therefore are selected from a widerange of unconformity defined stratigraphic unitssuch as laminae, beds, bed sets and so on. Theselection makes it unavoidable that all sequenceboundaries are caused by relative changes of waterdepth. This change of relative water depth is eitherthe result of local crustal movement or of a globaleustatic sea level fluctuation. The sequence stratigra-

.phy as developed by Vail 1988 and the geologistsof the Exon group have chosen the latter as the thesison which sequence stratigraphy rests. The sequencestratigraphic methods and the often extensive termi-nology, which has been introduced to describe thestratigraphy of sequences, has been discussed in

several papers Posamentier and Vail, 1988; Posa-.mentier et al., 1988 . A modern and also critical

discussion of sequence stratigraphy is given by Mial .1997 .

Basic to the interpretation of sea level curves isthe stratal architecture which develops during trans-gressions, highstands, regressions and lowstands. Thegeometrical facies pattern, which is produced bysuch changing sedimentation, has been called the

.sequence stratigraphic model Carter, 1998 . Thismodel and very extensive stratigraphical observa-tions lead to the development of a global sea levelcurve which provides the basis for the stratigraphicapplications of sequence analysis.

To prove eustatic sea level fluctuations demandsvery accurate stratigraphic correlations on a world-wide scale in order to demonstrate the synchronismof sea level changes. This difficulty has led to somedoubts about whether sequence stratigraphy can beapplied with as much optimism as originally envis-aged. Nevertheless, the principle of sequences asstratigraphic units has proved to be very useful.


Milankovitch cycles, which have been classifiedby Vail as fourth and fifth order sequences, arefrequently unconformity bound stratigraphic unitsand therefore sequences by definition. The sea levelfluctuations which cause such sequences are mostlikely to be glacio-eustatic and therefore are directlyrelated to climatic changes. Glacio-eustatic changescan have amplitudes from 1 to 2 m, for example, incarbonate platform environments and up to severalhundreds of meters in some PlioPleistocene sedi-ments. In the latter case, some proper sequencestratigraphic architecture of systems tracts can de-

.velop Naish et al., 1998 . Although Milankovitchcycles can conform to the definition of sequences,they have the added property of periodicity or nearperiodicity and furthermore their ultimate extraterres-trial origin is known. Milankovitch cycles, however,do not have to be represented by unconformitybounded units and they could develop without anysea level fluctuation.

The relationship between cyclostratigraphy andsequence stratigraphy therefore is quite clear: somesequences can be cyclic in the sense of cyclostratig-raphy but not all sequences are cycles. A hierarchicalclassification scheme which orders repeated strati-graphic events simply by the recurrence times ofsuch events is quite meaningless and suggests ge-netic similarities which do not exist. It is clearlyvaluable to know the frequency of a repetition orcycle but it is not helpful to hide this informationbehind some terminology. If a cycle has a period of100 ka, then it is just as easy to say so, rather thancalling it a cycle of 5th order. If the period is notprecisely known, one may even call it a cycle of theorder of 100 ka.

14. Non periodic changes of the sedimentation attime intervals of several millions of years

Sequences which represent time intervals exceed-ing 3 Ma have been called third, second, and firstorder cycles. It is the time order or absence of orderwhich will be examined in the following in the hopethat the analysis of repetition times may give someindications as to their origin.

.The global cycle chart Haq et al., 1987 providestwo different types of data. On the one hand are

given ages of sequence boundaries and downlapsurfaces, while on the other are given curves whichindicate the relative change of coastal onlap and sealevel, which are essentially derived from the changesof inferred shore-lines.

The curves can only be regarded as semiquanti-tive and express more or less diagrammatically theconditions which have led to the development of thesequences. A time series analysis of the curves wouldreflect more the draftsmanship of the authors thangive any useful information.

The lists of sequence boundary ages are moreobjective; of course they too are not without errors.Apart from the ever present uncertainty of absoluteages, all listed sequences are based on a subjectiveselection and the distribution of the sequence bound-aries therefore depends on the unconformities ofminor and major importance, introducing conceptslike parasequences, parasequence sets and so on, butthe decision as to what constitutes a sequence bound-ary is always subjective. A second difficulty is thatthe cycle list is apparently more complete in theyounger past compared with the older ages. Theapparent increase in cycle length with increasing age .Dickinson, 1993 may be responsible for this. Selec-tion has also influenced the frequency distribution ofsequence durations. Fig. 10 shows this distributionplotted as a survivor function, which is a cumulativecurve of all sequences that are longer than a speci-

.fied age which is plotted along the x axis . Using alogarithmic scale for both x and y axes, one obtainsan approximately straight line for sequences longerthan 10 Ma. The deviations from this line are mostlikely to be due to the under representation of theshorter sequences, which in the cycle chart havebeen designated as sequences of higher order.

Fig. 10. Using data from the global cycle chart, the durations ofsequences have been plotted as a cumulative survivor function.


The time series information available from theglobal cycle chart is just as limited as measurementsof bed thicknesses and must be treated in the sameway. One can either plot the durations of successivesequences as a discrete variable or one can plot thedensity of sequence boundaries in a given time inter-val. The latter has the advantage of providing valuesat equally spaced intervals. The series of sequencedurations together with their frequency distribution,are in fact the only quantitative data which can beobtained from the cycle chart.

Although several theories trying to explain se- .quence formation exist see Mial, 1997 , there is no

general agreement about the mechanisms which havegenerated either relative or eustatic sea level varia-tions. Indeed, it is by no means certain that there is auniversal explanation for all sea level fluctuations,which must have been the immediate cause of all thesequences.

The cumulative distribution of time intervals Fig..10 suggests that sequence formation follows a scale

independent power law. Such a distribution is acharacteristic feature of highly complex systemswhich may have reached self organised criticality.This state seems to apply to terrestrial as well asextra terrestrial processes, and in particular, the

earths crust may hav