Turbidite bed thickness distributions: methods and ... · Turbidite bed thickness distributions are often interpreted in terms of power laws, even when there are signiﬁcant departures

Turbidite bed thickness distributions: methods and pitfallsof analysis and modelling

ZOLTÁN SYLVESTER1

Department of Geological and Environmental Sciences, Stanford University, Stanford, CA 94305-2115,USA (E-mail: [email protected])

ABSTRACT

Turbidite bed thickness distributions are often interpreted in terms of power

laws, even when there are significant departures from a single straight line on

a log–log exceedence probability plot. Alternatively, these distributions have

been described by a lognormal mixture model. Statistical methods used to

analyse and distinguish the two models (power law and lognormal mixture)

are presented here. In addition, the shortcomings of some frequently applied

techniques are discussed, using a new data set from the Tarcău Sandstone of

the East Carpathians, Romania, and published data from the Marnoso-

Arenacea Formation of Italy. Log–log exceedence plots and least squares

fitting by themselves are inappropriate tools for the analysis of bed thickness

distributions; they must be accompanied by the assessment of other types of

diagrams (cumulative probability, histogram of log-transformed values, q–q

plots) and the use of a measure of goodness-of-fit other than R2, such as the

chi-square or the Kolmogorov–Smirnov statistics. When interpreting data that

do not follow a single straight line on a log–log exceedence plot, it is

important to take into account that ‘segmented’ power laws are not simple

mixtures of power law populations with arbitrary parameters. Although a

simple model of flow confinement does result in segmented plots at the

centre of a basin, the segmented shape of the exceedence curve breaks down

as the sampling location moves away from the basin centre. The lognormal

mixture model is a sedimentologically intuitive alternative to the power law

distribution. The expectation–maximization algorithm can be used to

estimate the parameters and thus to model lognormal bed thickness

mixtures. Taking into account these observations, the bed thickness data

from the Tarcău Sandstone are best described by a lognormal mixture model

with two components. Compared with the Marnoso-Arenacea Formation, in

which bed thicknesses of thin beds have a larger variability than thicknesses

of the thicker beds, the thinner-bedded population of the Tarcău Sandstone

has a lower variability than the thicker-bedded population. Such differences

might reflect contrasting depositional settings, such as the difference between

channel levées and basin plains.

Keywords bed thickness, turbidites, power law distribution, lognormaldistribution, log–log plots, lognormal mixture, Tarcău Sandstone, Marnoso-Arenacea Formation.

1Present address: Shell International Exploration and Production Inc., 3737 Bellaire Blvd., P.O. Box 481, Houston,TX 77001-0481, USA.

Sedimentology (2007) 1–24 doi: 10.1111/j.1365-3091.2007.00863.x

� 2007 The Author. Journal compilation � 2007 International Association of Sedimentologists 1

INTRODUCTION

One of the most striking features of turbiditesequences is the rhythmic alternation of sand andshale over great thicknesses. This is due to thefact that in deep-water settings coarse clasticsediment deposition is dominated by discretesedimentation events and the background energylevels are low and, as a result, individual sedi-mentation events are often well preserved. Thefrequent interbedding of sand and shale stronglyinfluences the overall heterogeneity of the result-ing succession and the distribution of turbiditebed thicknesses along with the lateral thicknesschanges are important in modelling hydrocarbonreservoirs that were deposited in deep water (e.g.Flint & Bryant, 1993; Drinkwater & Pickering,2001; Tye, 2004; Pyrcz et al., 2005). A precisestatistical description, along with appropriategraphical and computational tools for the analysisof this basic stratigraphic parameter, is essentialfor further advancements in the field. In addition,some parameters of the bed thickness distributionmight prove to be useful in differentiating depo-sitional settings (Malinverno, 1997; Carlson &Grotzinger, 2001; Mattern, 2002; Sinclair & Cow-ie, 2003; Clark & Steel, 2006), even when workingwith data of limited lateral extent, such as wellsand smaller outcrops.

Four different types of statistical distributionshave been proposed to describe sedimentary bedthickness data: truncated Gaussian, lognormal,exponential and power law (or fractal) distribu-tions. According to an initial line of thought, theright-skewedness that seems to be ubiquitous inthickness data is the result of a truncated Gaussiandistribution, reflecting a balance between deposi-tion and erosion (Kolmogorov, 1951; Mizutani &Hattori, 1972; Muto, 1995). Muto (1995) has shownhow Kolmogorov’s truncated Gaussian modelmight lead to exponential distributions. Severalresearchers have suggested that the turbiditesequences are best characterized by lognormaldistributions (e.g. McBride, 1962; Enos, 1969;Ricci Lucchi & Valmori, 1980; Murray et al.,1996; Talling, 2001). Other studies have focusedon whether lognormal or exponential bed thick-ness distributions dominate the geologic record.Drummond & Wilkinson (1996) suggested that theexponential distribution might describe sedimen-tary thicknesses over a wide range of scales, but themeasurements are biased against very thin beds. Oflate, in parallel with increasing interest in fractalphenomena and power law scaling in nature(e.g. Mandelbrot, 1983; Turcotte & Huang, 1995;

Turcotte, 1997), the power law distribution hasgained popularity in studies of turbidite sequences(Hiscott et al., 1992, 1993; Rothman et al., 1994;Malinverno, 1997; Pirmez et al., 1997; Chen &Hiscott, 1999; Winkler & Gawenda, 1999; Awad-allah et al., 2001; Carlson & Grotzinger, 2001;Chakraborty et al., 2002; Sinclair & Cowie, 2003).

Power law bed thickness distributions havebeen linked to earthquakes with comparablydistributed magnitudes (Beattie & Dade, 1996;Awadallah et al., 2001) and to self-organizedcriticality (Rothman et al., 1994; Bak, 1996). Thetheory of self-organized criticality was introducedby Bak et al. (1988). This theory explains thewidespread occurrence of fractal structures and1/f noise by the tendency of large dissipativesystems to develop a state of criticality andgenerate events of all sizes. The simplest modelfor such a system is a sand pile: once it reaches acritical slope, avalanches of all sizes will begenerated. Obviously, turbidity current-initiatingmechanisms are diverse and probably more com-plicated than sand piles (e.g. Normark & Piper,1991). Not much is known about the sizedistribution of turbidity current initiating events,but the magnitudes of such events are likely tohave a skewed distribution and a heavy-taileddistribution is not unreasonable. Whether thisdistribution is power law, exponential, lognormalor best fits some other model cannot be deter-mined based on the available data. In addition,hundreds of kilometres might separate a slide onthe continental slope, canyon wall or delta frontand the final site of deposition and, as Talling(2001) has argued, ‘it is unlikely that there is sucha simple correspondence between the frequencydistribution of sediment volumes failing on thecontinental slope, or the magnitude of seismicshaking, and the thickness of turbidites depositedmuch further downslope’. Well-defined powerlaw distributions are relatively rare; most turbi-dite bed thickness data sets show significantdepartures from the ideal power law model.However, this has not prevented the power lawmodel from being applied. Rather, significantdepartures from a single power law have beencalled ‘segmented power law distributions’ andinterpreted as the results of modification of theinitial power law distributed input volumes byerosion, amalgamation, confinement or a combi-nation of these (Malinverno, 1997; Winkler &Gawenda, 1999; Carlson & Grotzinger, 2001;Sinclair & Cowie, 2003).

Talling (2001) has analysed bed thickness datafrom the Marnoso-Arenacea Formation in Italy

2 Z. Sylvester

� 2007 The Author. Journal compilation � 2007 International Association of Sedimentologists, Sedimentology, 1–24

and from three other locations and has shownthat turbidites with specific basal grain sizeclasses have lognormal distributions. Therefore,sequences that consist only of thick-beddedturbidites or of thin-bedded turbidites tend tofit lognormal models. Distributions that plot asconvex-up curves on log–log plots can resultfrom the mixing of lognormal populations, andthere are a number of sedimentological reasonswhy the lognormal mixture model is preferableto the segmented power law model. The resultspresented here build on and support thesefindings and the reader is referred to Talling(2001) for a detailed discussion of the origin andthe arguments in favour of the lognormal mixturemodel. This paper focuses on the problems andpotential solutions of deciding which distribu-tion model best fits the data, by: (i) presentingthe statistical methodology and the associatedpitfalls in analysing bed thickness distributions;(ii) re-examining the idea of ‘segmented powerlaw distributions’ from a statistical point of view;(iii) suggesting a methodology for estimating andmodelling the two main components of a log-normal mixture; and (iv) applying these tech-niques to a turbidite succession and comparingthe results with the Marnoso-Arenacea Forma-tion. It is not the purpose of this paper to reviewall the possible distribution models or to cover alarge number of turbidite bed thickness data sets.Rather, the focus is on the power law andlognormal models, using the bed thickness datafrom the Tarcău Sandstone as a case study.

GEOLOGIC SETTING OF THE TARCĂUSANDSTONE

The Tarcău Sandstone is one of the majorsand-rich formations of the flysch belt of theRomanian East Carpathians. It represents a turbi-dite system deposited during the Palaeocene toMiddle Eocene (Săndulescu & Săndulescu, 1973)between the active margin of a small continentalblock to the west (Fig. 1), termed the Tisza-DaciaBlock (Csontos, 1995), and the passive margin ofthe East European Craton to the east. Palaeocurrentdata suggest a longitudinal dispersal pattern(Contescu et al., 1967; Jipa, 1967). A microfaunaconsisting mainly of agglutinant foraminiferacharacteristic of deep-water flysch assemblages(Săndulescu & Săndulescu, 1973) and an abundantichnofauna typical of the Nereites ichnofacies(Buatois et al., 2001) suggest deposition at bathyalwater depths. Predominant lithologies are thick-

bedded pebbly sandstones and medium-bedded tothin-bedded sandstones and shales (Figs 2 and 3),largely corresponding to the Bouma (1962) andLowe (1982) turbidites. The presence of abundantamalgamation, erosion, debris flows and conglom-erate beds within the thicker-bedded intervalssuggests the presence of channels (Sylvester,2002), and most of the thin-bedded intervals arelikely to represent overbank deposits. The TarcăuSandstone shows a number of important differ-ences when compared with the well-studiedMarnoso-Arenacea Formation: (i) it has numerouscoarser-grained packages (not only pebbly sand-stones, mainly, but also conglomerates); (ii) manyof these coarse-grained deposits probably repre-sent channel fills, whereas most of the Marnoso-Arenacea is interpreted as basin plain deposits;and (iii) the lateral extent of beds in the TarcăuSandstone is limited and correlation from oneoutcrop to the next is impossible, whereas numer-ous beds in the Marnoso-Arenacea can be correla-ted across the whole basin (Ricci Lucchi &Valmori, 1980; Amy & Talling, 2006). Comparingbed thickness data from these two successionsmight give insights into the differences betweenbasin plain deposits and channel levée systems.

FIELD METHODS

The Tarcău Sandstone is exposed in a series ofroadcuts near the Siriu Dam in the Buzău Valleyarea of the East Carpathians (Fig. 1). More than700 m of stratigraphic sections were measured(Fig. 2; Sylvester, 2002) and thicknesses of indi-vidual Bouma divisions and other characteristics,such as maximum grain size, presence of erosion,mud clasts and sedimentary structures werenoted. The overall finer-grained and mainly thin-ner-bedded upper part of the Tarcău Sandstone isinformally referred to below as the Upper TarcăuSandstone and the bed thicknesses of this sectionhave also been analysed separately (excluding theuppermost thick-bedded pebbly sandstone pack-age; Fig. 2).

Precise measurement of bed thicknesses is onlypossible where abundant sandstone beds arepresent in the succession. Some relatively poorlyexposed, predominantly shaly sections weretherefore not included in the study. For thepurpose of this paper, ‘bed thickness’ is equivalentto ‘sedimentation unit thickness’. In parts of thesuccession where individual events consist of asandstone–shale pair, the combined thickness ofthe two represents one value in the bed thickness

Turbidite bed thickness distributions 3


series (Fig. 4). Working with sedimentation unitthicknesses as opposed to treating sandstone andmudstone layers separately is reasonable if one ofthe underlying assumptions of the analysis is thatbed thicknesses have a clear relationship withindividual event bed volumes (e.g. Malinverno,1997). In addition, if viewed separately, sandstoneand mudstone thicknesses still have the signa-tures of mixtures (see below), suggesting thatanalysing event bed thicknesses, grouped as afunction of their basal Bouma divisions or grainsize (Talling, 2001), is a better approach.

In coarser-grained and thicker-bedded parts ofthe succession, where amalgamation is frequent,

boundaries of individual event beds were care-fully identified (Fig. 4). This was possible mainlyby detecting subtle surfaces across which well-defined coarsening occurs. Breaking down amal-gamated sections into individual event beds (seealso Sinclair & Cowie, 2003) is in contrast withthe lumping approach adopted by Carlson &Grotzinger (2001). Note that bed thickness dis-tributions of the thick-bedded parts of the TarcăuSandstone would be entirely different if theamalgamation surfaces were ignored. Workingwith event bed thicknesses makes it possible tostudy how bed thickness relates to bed volumeand depositional processes, whereas the

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4 Z. Sylvester


thicknesses of amalgamated units convolve theeffects of bed volume, depositional process andfacies clustering.

No attempt has been made to separate turbiditemud from hemipelagic shale. However, truehemipelagic or pelagic deposits seem to be rare

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Fig. 3. (A) Thin-bedded, fine-grained turbidite package overlainby a thick-bedded, amalgamatedpebbly sandstone unit. (B) Twomedium-bedded, laminated sand-stone packages separated by a thin-bedded, mudstone dominated unit.For the stratigraphic position ofthese sections, see Fig. 2.



in the succession and mainly present in themiddle of shale-dominated sections. These de-posits are unlikely to strongly affect the measuredbed thickness distributions.

POWER LAW BED THICKNESSDISTRIBUTIONS

A power law distribution, called Pareto distribu-tion by economists, is characterized by an exceed-ence probability (i.e. the probability that a value

is greater than or equal to x) described by a powerlaw:

P½X � x� ¼ mx

� �bð1Þ;

where m is the minimum value of x. The exceed-ence probability is often called complementarycumulative distribution function (CCDF). Thecumulative distribution function (CDF) is

P½X < x� ¼ 1� mx

� �bð2Þ;

and the probability density function (PDF) is:

P½X ¼ x� ¼ kmbx�ðbþ1Þ ð3Þ:

Note that the PDF of a Pareto distribution is alsoa power law, but the exponent is )(k + 1), asopposed to )k in the case of the exceedenceprobability (which equals 1 ) CDF; Fig. 5). Thismeans that, although both the PDF and the1 ) CDF plot as straight lines on log–log plots,the PDF line is steeper than the exceedence line(Fig. 5B). A power law distribution is defined bytwo parameters: the location parameter m and theshape parameter b. Both parameters must bepositive. For comparison, the PDF, CDF andCCDF curves are illustrated for the lognormaldistribution as well (Fig. 5).

The problems with least squares fitting

Most studies dealing with turbidite bed thick-ness distributions rely on the fact that power lawdistributions plot as straight lines on log–logexceedence probability diagrams. The best dis-tribution model is chosen through visual inspec-tion of these plots. The power law exponent isdetermined using least squares fitting on the log-transformed values of bed thickness and exceed-ence probability. In some cases, the coefficient ofdetermination R2, which is the ratio of theexplained sum of squares to the total sum ofsquares, is used as a measure of goodness-of-fiteither to support or to reject the power lawmodel (Carlson & Grotzinger, 2001; Sinclair &Cowie, 2003).

Least squares fitting is a simple method forestimating the power law exponent. However,as noted by Goldstein et al. (2004), suchestimates must be qualified by some assessmentof their goodness-of-fit. R2 is inappropriate forthis purpose, especially when working withexceedence probability plots or cumulative plotsin general. By definition, the exceedence

Fig. 4. Examples of measured sections from a thick-bedded, amalgamated pebbly sandstone unit and amedium-bedded, laminated sandstone unit. Event bedboundaries, used in most of the bed thickness calcu-lations, are shown. Note different vertical scales.

6 Z. Sylvester


probability is decreasing as bed thicknessincreases (Fig. 5), and R2 will be relatively largeeven when the curve is clearly non-linear. Inaddition, the significance of this statistic cannotbe tested because some assumptions of classicalregression are violated. The validity of theregression statistic depends on the distributionof the residuals (e.g. Swan & Sandilands, 1995):they must be homoscedastic, that is, therecannot be any trend in the distribution ofvariance; and they must be normally distri-buted. These assumptions are not satisfied bythe residuals of a typical exceedence probabilityanalysis. The Kolmogorov–Smirnov and thechi-square goodness-of-fit tests are more appro-priate measures; they will be described brieflylater in the paper.

The ‘thin bed problem’

Another characteristic of the exceedence prob-ability log–log plots is that they give emphasis tothe upper tail of the distribution, that is, to thethick beds and tend to hide the shape of thedistribution at the lower tail, where the muchmore numerous thin beds are located (see alsoDrummond & Coates, 2000). Departures from thepower law model often occur at the thin tail of thedistribution, but these departures are visuallysuppressed on these plots. For example, in thecase of the Upper Tarcău Sandstone, the depar-ture from a power law at the thin end of thedistribution seems relatively insignificant on theexceedence plot (Fig. 6A). Based on this diagram,it looks like the data are relatively well described

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by a power law with a slope of about 1Æ08; R2 is0Æ964. However, there are several problems withsuch a model. Firstly, plotting the histograms ofthe log-transformed data for the real data set and asimulated distribution reveals that the departurefrom the power law at the thin beds is much moresignificant: the shapes of the two histograms arestrongly dissimilar (Fig. 6). Secondly, a locationparameter has to be chosen, that is, a minimumbed thickness for the power law distribution. Abetter visual fit can be achieved if this is set at3 cm, the point where the departure starts. Thenagain, there are clearly a number of beds in theTarcău data that are thinner than 3 cm. If thelocation parameter is set to 1 cm, the thinnest bedmeasured, the model exceedence probabilitiesdrop significantly below the Tarcău data, againsuggesting the mismatch in the number of thinbeds. It may be calculated what proportion of bedsshould be thinner than 3 cm (Fig. 6A) in order tosatisfy this model: almost 70% of the beds shouldbe less than 3 cm thick. This means either thatalmost 3000 thin beds were not measured duringthe field work, or that the power law model is notappropriate in this case. Although it is true thatthe thin beds probably are under-represented inthe data set, mainly because of the less-than-perfect exposure of some of the shaliest sections,such a degree of mismatch at least raises ques-tions about the practicality of the power lawmodel. On the one hand, it is only in the rarestcircumstances that it is possible to measurecorrectly all bed thicknesses throughout a wholestratigraphic unit (e.g. Talling, 2001); on the otherhand, the requirement to measure all the thinbeds focuses the attention on the facies thatcommonly forms the non-reservoir parts of hydro-carbon accumulations. Compared with the ‘thinbed problem’, departures from a single straightline in the mid-range of the distribution probablyrepresent a more significant challenge to thegeneral applicability of the power law model.

Segmented power law distributions

In many cases, turbidite bed thickness distribu-tions show significant departures from a straightline on the log–log plot and these departures aredescribed frequently in terms of ‘segmentedpower laws’ (Rothman & Grotzinger, 1995; Pirmezet al., 1997; Chen & Hiscott, 1999; Awadallahet al., 2001; Sinclair & Cowie, 2003). Of 18 plotsof bed thickness data in the literature, 14 areinterpreted in terms of segmented power laws,two as impossible to fit with straight lines, and

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Fig. 6. (A) Log–log exceedence probability plot of bedthicknesses in the Upper Tarcău Sandstone (thick blackline). Two power-law models with the same power-lawexponent (b ¼ 1Æ08) but different location parameter m(or minimum bed thickness) are also shown. (B) His-togram of bed thickness data from the Upper TarcăuSandstone. (C) Histogram of a synthetic bed thicknessseries that fits a power law with b ¼ 1Æ08 and m ¼ 3.The histograms clearly indicate that the actual data setshould have a lot more thin beds in order to fit a powerlaw distribution, whereas the log–log plots tend to hidethis disparity.

8 Z. Sylvester


there are only three that seem to fit single powerlaws. The two segments with different power lawexponents are treated as two different popula-tions, purportedly well characterized by theexponent estimated from the exceedence plot.The implicit assumption is that the data are bestdescribed as a mixture of two power lawdistributions. A similar interpretation of theTarcău Sandstone bed thickness data is shownin Fig. 7. Beds with thicknesses

It is hard to think of a relatively simple processresulting in a power law mixture that satisfiesthese criteria for generating a segmented powerlaw on a log–log plot. Yet there is a mathematicalmodel of confinement of turbidite beds that canindeed result in segmented power laws if a seriesof assumptions are valid. This model will be thesubject of the next section.

The Malinverno (1997) model for segmentedpower laws

Assuming a power law distribution of bed vol-umes, Malinverno (1997) derived a simple rela-tionship among the scaling between bed lengthand bed thickness c, the exponent of the bedvolume distribution c, and the spatial distributionof the bed depocentres relative to the verticalsampling line, characterized by the exponent d:

b ¼ c þ cð2c � dÞ ð5Þ:

Malinverno (1997) also developed a model ofbasinal confinement of turbidites and showedhow the confinement can alter the bed thicknessdistribution at the basin centre. Beds of cylindri-cal shape are placed randomly into a circularbasin and the bed thicknesses are measured at avertical sampling line at the basin centre. Malin-verno (1997) interpreted the resulting change inthe shape of the exceedence probability curve as atransition to a segmented power law, and derivedrelationships for the exponents (or slopes) of thetwo segments or populations. Beds with a radiusmuch less than the basin diameter will be distri-buted throughout the basin so that d � 2, and theoriginal relationship becomes:

bsmall ¼ cð1þ 2cÞ � 2c ð6Þ:

Beds with diameters larger than the radius of thebasin will be intersected by a sampling line at thebasin centre, therefore d � 1, and:

Blarge ¼ cð1þ 2cÞ ð7Þ:

Some beds from both these categories are incontact with the basin margin; they can bethought of as deposits of flows that locallyinteract with the basin margin (e.g. they aredeflected or reflected). In contrast, ‘fully ponded’beds should cover the whole basin floor and theyform a third category that can be distinguished ona log–log exceedence plot if modelled appropri-ately.

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Fig. 8. Log–log exceedence probability curves of the-oretical mixtures of two power law distributions. Of allthese possible curve shapes, only one specific curve,which belongs to case (C), results in a segmented powerlaw. Numbers near the curves indicate the proportionof the lower-exponent (b1 ¼ 1) population in the mix-ture. (A) Neither of the distributions is truncated; bothhave a minimum bed thickness of 1 cm, but they havedifferent exponents. (B) The thicker-bedded populationis truncated at its lower end: the minimum bed thick-ness is m2 ¼ 100 cm. (C) The thinner bedded popula-tion is truncated as well, at its upper end.

10 Z. Sylvester


In order to assess the general applicability ofthis ‘segmentation through confinement’ model, aseries of numerical experiments comparable withthose of Malinverno (1997) have been run.Although Malinverno (1997) used a bed volumedistribution with the largest bed diameter equal tothe basin size, in the present experiments there isno interdependence between the largest bedvolume and the size of the basin. The basin sizeis fixed and beds that would exceed the basindiameter using the thickness–length scaling rela-tionship are considered ‘ponded’, their diameteris set equal to the basin diameter and theirthicknesses are increased accordingly. This setupmakes it possible to run millions of flows with apower law distribution of volumes without hav-ing to change the basin size. The resulting bedthickness series can contain several tens ofthousands of data points, an important advantagethat helps visualize the subtle variations in theshapes of the exceedence probability curves.

Figure 9 shows the results of two experimentsrun with similar parameters, the only differencebeing the value of the bed volume distributionexponent c. Based on these results, the followingobservations can be made.

1 At the basin centre, the Malinverno modeldoes generate exceedence probability curves withstraight segments, although the transition be-tween the first two segments is smoothly curvedand does not show a well-defined breakpoint.

2 The value of bsmall is overpredicted by Eq. 6(Fig. 9) and is better approximated by Eq. 4. Thereason for this is that the original bed thicknessexponent (which, for d ¼ 2, is the same as bsmall)decreases as the oversampling of thick bedspushes up the first segment of the curve. For allbed thicknesses other than the thickest beds, theexceedence probability increases compared withthe curve unaffected by confinement becausemore thick beds are intercepted by the samplingline. If Eq. 6 holds, for given c and c values, thesmaller-than-predicted bsmall would require a dvalue larger than 2 which is impossible by defi-nition. Because only a few beds exceed thethicknesses characterized by blarge, the value ofblarge is affected to a much smaller degree and thesecond straight segment of the numerical experi-ment has a slope similar to the predicted value.

3 As suggested by Malinverno (1997), extremelylarge bed volumes result in a third segment with aslope equal to the bed volume exponent c. Thesevolumes would be the equivalent of flows thatentirely cover the basin floor and are fully ponded.

This is a relatively simple model that can resultin straight segments on log–log exceedence prob-ability plots of bed thickness data. However,before interpreting most departures on log–logplots from a single straight line as due toconfinement, it is important to consider thelimitations of the Malinverno (1997) model:

1 The input bed volumes must have a powerlaw distribution. Although recent estimates fromthe Marnoso-Arenacea Formation suggest thatbed volumes have a lognormal distribution (Tal-ling et al., 2007), it is possible that the power lawassumption is valid in other systems. However,even if the power law input volume distributionis accepted, this initial range will be strongly

Fig. 9. Results of two numerical experiments that usethe assumptions of the confined basin model of Ma-linverno (1997). The black lines represent bed thick-nesses measured at the centre of the circular basin; thered lines correspond to the curves predicted by theequations of Malinverno (1997). The only differencebetween the setup of experiment 1 and experiment 2 isthe power law exponent of the bed-volume distribu-tions c; m1 is the minimum bed thickness; m2 is thethreshold bed thickness that separates small-volumebeds from beds with a diameter larger than the radius ofthe basin; and m3 is the threshold bed thickness abovewhich beds are ‘fully ponded’ (beds with a diameterequal to the basin diameter). The other parameters are:c ¼ exponent of bed volume distribution; c ¼ scalingexponent between bed length and bed thickness;a ¼ scaling constant between bed length and bedthickness (l ¼ ahc, where l is the bed diameter and hthe bed thickness); n beds ¼ total number of bedsdeposited in the basin; min bed vol ¼ minimum bedvolume; basin size ¼ basin diameter; ntotal ¼ numberof beds intercepted by sampling line; n1 ¼ number ofbeds in the first population; n2 ¼ number of beds in thesecond population; n3 ¼ number of beds in ‘fullyponded’ population.



modified if there are well-developed channellevées. In such systems, most of the flows areguided by channels, and often deposit a signifi-cant part of their sediment in the overbank areaand, as a result, go through an efficient modifi-cation of volumes before reaching the lesschannelized site of deposition.

2 If confinement was the cause of most devia-tions from a single power law of bed thicknessdata, at least in some cases the third segment ofthe plot should be seen as well, that is, a few‘megabeds’ that cover the whole basin or con-tainer. No such outliers are reported in the lit-erature. In a high-quality data set, even a smallnumber of such data points would plot along adifferent trend than the second population char-acterized by blarge (Fig. 9). Many of the beds in theMarnoso-Arenacea Formation extend across theentire basin (Ricci Lucchi & Valmori, 1980; Amy& Talling, 2006), yet no third segment is observedon the log–log exceedence plots.

3 In channel levée systems, beds deposited inthe overbank areas are likely to be characterizedby different length–thickness relationships anddifferent degrees of confinement than bedsdeposited in the channels. In addition, the geo-metric configurations of these settings are obvi-ously more complicated than those of a circularbasin.

4 The individual segments corresponding toblarge and bsmall can easily be identified with largedata sets collected exactly at the basin centre, butthis task becomes increasingly difficult as thesampling location moves away from the centreand as the number of data points is reduced. Fiveexperiments have been run with the sameparameters but different sampling locations(Fig. 10). Although it is possible to derive rea-sonable estimates for blarge and bsmall in the case ofsampling location 1 and possibly location 2 aswell, the clear segmented character of the ex-ceedence plot disappears and determining blargeand bsmall becomes impossible at location 3. Thedashed circle in the centre of the basin shows theapproximate area characterized by more or lesssegmented plots. This is

lognormal distribution might be the most appro-priate for characterizing turbidite bed thicknessesfrom a number of successions. Talling (2001) alsodrew attention to the bimodality of many turbi-dite bed thickness data sets, and pointed out thattwo lognormal populations may result in ‘seg-mented power laws’ on exceedence probabilityplots. Although the two populations seem tocorrespond to lithologic and facies differences, itwould be useful to identify the two componentsof the lognormal mixture in a quantitative way.The questions addressed here are the following:(i) how can the parameters of the two (or more)lognormal components be extracted from a log-normal mixture?; and (ii) what is the appropriatestatistical methodology for comparing the good-ness-of-fit of the lognormal mixture model withthe power law model?

Finding the lognormal components of amixture

Separating components of a Gaussian mixture isan important subject in statistics that has beendiscussed in one of the early statistical papers(Pearson, 1894). Since the 1980s, in parallel withthe increase in computational power, the numberof publications on mixture modelling has ampli-fied considerably (e.g. McLachlan & Peel, 2000,and references therein). Before the widespreadavailability of computers, identifying and separ-ating the components of a grain-size mixture wasnot a simple exercise (e.g. Spencer, 1963; Tanner,1964). However, today there are several algo-rithms available to address this problem, and theexpectation–maximization algorithm is one of thebest candidates for separating components bothin grain size and in bed thickness data. Figure 11illustrates the shapes of the cumulative probabil-ity plots for two types of lognormal mixtures: inthe first type, the thinner-bedded population hasa smaller standard deviation than the thickerbedded population; in the second type, thethicker-bedded group has a smaller variance.

The expectation–maximization algorithm(Dempster et al., 1977) is a popular algorithm inmachine learning and statistical computing thatis often applied in parameter estimation prob-lems, for example, in estimating parameters ofGaussian mixture models (e.g. Kung et al., 2004).K-means clustering is an alternative approach,but the expectation–maximization algorithm hasthe advantage that the parameters of the twodistributions are derived using maximum like-lihood estimation during the iterations and the

classification results are probabilistic. In the caseof bed thicknesses, this means that the probabil-ities of each bed belonging to a thick-bedded orthe thin-bedded population may be found,whereas class memberships in K-means cluster-ing are either 0 or 1.

Using a Matlab� implementation of the EMalgorithm (Nabney, 2001), the bed thickness datafrom the Tarcău Sandstone and the Marnoso-Arenacea Formation have been analysed (Marn-oso-Arenacea Formation data from Talling, 2001)and the two most probable lognormal compo-nents separated. The results are shown in Figs 12and 13. In addition to the log–log exceedenceprobability plot, other diagrams are also presen-ted: the histogram of the log-transformed values,the PDFs of the model distributions and thecumulative probability plot of the log-trans-formed values. Using a range of diagram typesguarantees that all potential departures from thestatistical model are appropriately highlighted.

In both cases, the visual fit between thelognormal mixture model and the actual data isbetter than in the case of single or segmentedpower law models. Segmented power law inter-pretations applied to the log–log exceedence plotswould ignore obvious non-linear variations in thecurve shape that are remarkably well described bythe lognormal mixtures (Figs 12D and 13D). Inaddition, there is a relatively good, although notperfect match, between the thin-bedded andthick-bedded lognormal components and bedsstarting with Bouma sequences Tc or Td and Ta orTb respectively (Fig. 14). This relationship couldbe used to estimate proportions of differentturbidite facies where no data other than bedthicknesses are available (e.g. when bed thicknessis estimated from high-resolution image logs).

If plotted separately, thicknesses of Boumadivisions also seem to have lognormal distribu-tions (Fig. 15). In fact, Ta and Tb intervals matchthe lognormal model better than the thick-beddedpopulation identified as the beds starting with Taor Tb divisions (Fig. 14A). It is not clear howthese divisions tend to combine to form the thick-bedded population, but it does seem that thebasal Bouma division is one of the definingparameters of the thickness of the whole eventbed. In fact, the slight divergence from thelognormal model, when beds starting with Ta orTb divisions are grouped together, is reducedwhen beds starting with Ta and beds starting withTb are plotted separately. Thus, it is possible thatthe Tarcău bed thickness data comprise morethan two lognormal populations. However, the



two-component model gives a good match to thedata (Fig. 12) and adding more populationswould significantly – and probably unnecessarily– complicate the model. Sandstone bed thicknessis also plotted on Fig. 15; this curve is clearly nota single lognormal population, but a mixture ofseveral different groups.

Figure 16 shows the results of the classificationof the Tarcău Sandstone bed thickness series intothin-bedded and thick-bedded groups. In the caseof the Tarcău Sandstone, the limit between thethin-bedded and thick-bedded populations is ca30 cm, whereas the segmented power law modelwould predict a limit of ca 200 cm. The 30 cm

limit is notably similar to that in the Marnoso-Arenacea Formation (Talling, 2001). The classifi-cation results of the bed thickness series using thetwo different approaches show that the lognormalmixture model gives a more intuitive outcome:the lognormal thick-bedded population largelycorresponds to the thick-bedded clusters in thesuccession, whereas the 200 cm limit only dis-tinguishes a few very thick beds that are notclustered spatially.

An important difference between the TarcăuSandstone and the Marnoso-Arenacea Forma-tion is as follows. In the Tarcău Sandstone, thelog-transformed thin-bedded population has a

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A B

C D

Fig. 11. Cumulative probability and log–log exceedence probability curves of theoretical mixtures of two lognormaldistributions. Note that the exceedence curves shown in (D) could be misinterpreted as segmented power laws.Numbers near the curves indicate the proportion of the thinner bedded population in the mixture. (A) Cumulativeprobability curves of mixtures of two lognormal populations, the thinner bedded having a lower variance than thethicker bedded one. (B) Cumulative probability curves of mixtures of two lognormal populations, the thinner beddedhaving a higher variance than the thicker bedded one. (C) Exceedence probability curves of mixtures of two log-normal populations, the thinner bedded having a lower variance than the thicker bedded one. (D) Exceedenceprobability curves of mixtures of two lognormal populations, the thinner-bedded population having a higher vari-ance than the thicker-bedded population. m and s are the mean and the standard deviation of the 10-base logarithmof the bed thicknesses.

14 Z. Sylvester


smaller standard deviation than the thick-beddedpopulation; in the Marnoso-Arenacea Formation,the thicker-bedded cluster has less variabilitythan the thinner-bedded cluster. These two dif-ferent mixture types produce cumulative curvesof different shapes on both probability ‘paper’ andon log–log exceedence plots (Figs 11 and 14).Note that ‘Marnoso-Arenacea type’ mixtures, inwhich the thicker-bedded population has a smal-ler standard deviation, tend to generate log–logexceedence plots that could easily be misinter-preted as segmented power laws. Unless stronglydominated by the thinner-bedded population,‘Tarcău-type’ mixtures result in smoothly curvinglog–log exceedence plots that would be moredifficult to interpret in terms of segmented powerlaws (but not impossible; see Fig. 7A).

Choosing between the power law andlognormal models using statistical testing

The Kolmogorov–Smirnov and the chi-square teststatistics were used for testing both the singlepower law and the lognormal mixture models inthe case of the Upper Tarcău Sandstone.

The most widely used goodness-of-fit tests fornon-normal distributions are the chi-square testand the Kolmogorov–Smirnov test (e.g. Swan &Sandilands, 1995; Davis, 2002). The chi-squaretest compares the observed frequencies of valueswith the expected frequencies that correspond tothe distribution model. The main disadvantage ofthe chi-square test is that the data must begrouped into bins and the result is influencedby the number of bins used. In contrast, the

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6·3,

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. dev

. = 0

·23

mean

= 54

.5, st

d. de

v. = 0

·48

A B

C D

n = 1817

Fig. 12. Interpretation of the bed thickness data from the Tarcău Sandstone as a mixture of two lognormal popu-lations. (A) Histogram of the log-transformed bed thicknesses, with the two model distribution PDFs shown. (B)Model PDFs and their sum shown on a linear x-axis. (C) Cumulative probability plot of the data and the model curvesof the two lognormal distributions. Mean is the geometric mean, std. dev. is the standard deviation of the 10-baselogarithm of the bed thickness series. (D) Log–log exceedence probability plot of the data and the model curves.



Kolmogorov–Smirnov test compares the cumula-tive relative frequencies of the data with thecumulative distribution function (CDF) of theassumed underlying distribution model. Theresult does not depend on the way the data aredivided into groups. The Kolmogorov–Smirnovtest statistic corresponds to the largest differencebetween the cumulative relative frequency curveand the CDF of the model distribution.

Tables for critical values of the conventionalKolmogorov–Smirnov test statistic are valid onlyif the distribution parameters are not estimatedfrom the data (Davis, 2002; Goldstein et al., 2004).As in this study, both the location parameter andthe shape parameter of the power law distributionare estimated from the data, the critical valuesmust be recalculated. Largely based on the sug-

gestions of Goldstein et al. (2004), the followingmethodology has been adopted and implementedin Matlab�: (i) using maximum likelihood esti-mation, the power law exponent of the data set isassessed; (ii) based on the CDF of a distributionwith this exponent, the Kolmogorov–Smirnovstatistic is found; (iii) a large number of artificialpopulations with the same number of values andthe same parameters as the dataset are generatedand steps (i) and (ii) are repeated for each tocreate a distribution of KS statistics; and (iv)critical values can be read from this distributionfor different levels of significance. The nullhypothesis states that the data come from adistribution with the estimated shape and loca-tion parameters. If the significance level is chosenas 5% and the Kolmogorov–Smirnov statistic of


0 50 100 150 200 250 300 3500

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43

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. dev

. = 0

·21

A B

C D

n = 1529

Fig. 13. Interpretation of the bed thickness data from the Marnoso-Arenacea Formation as a mixture of two log-normal populations. Bed thickness data from Talling (2001). (A) Histogram of the log-transformed bed thicknesses,with the two model distribution PDFs shown. (B) Model PDFs and their sum shown on a linear x-axis. (C) Cumu-lative probability plot of the data and the model curves of the two lognormal distributions. Mean is the geometricmean, std. dev. is the standard deviation of the 10-base logarithm of the bed thickness series. (D) Log–log exceedenceprobability plot of the data and the model curves.

16 Z. Sylvester


the data set is larger than the 95th percentile ofthe values derived through randomization, thenull hypothesis can be rejected. Alternatively, thedifference between the actual and the expectedKolmogorov–Smirnov statistic, measured instandard deviations, can be used as a quantitativemeasure of goodness-of-fit to the distribution.

In a manner comparable with the procedureused to apply the Kolmogorov–Smirnov test to apower law model, the following steps were usedto calculate test statistics for the lognormal mix-ture model: (i) using the expectation–maximiza-tion algorithm, the parameters of the twolognormal populations are estimated from the

data; (ii) the test statistic (Kolmogorov–Smirnovor chi-square) of the original data is calculated;(iii) a large number of artificial populations withthe same number of values and the same para-meters as the data set are generated, the para-meters are estimated and then the test statisticsare calculated for each to create a distribution oftest statistics; and (iv) critical values can be readfrom this distribution for different levels ofsignificance.

The Upper Tarcău Sandstone bed thicknessdata plot close to a straight line on an exceedencelog–log diagram, and have been previously inter-preted as a good example of a power law distri-bution (Sylvester, 2002). The results of theKolmogorov–Smirnov and chi-square tests areshown in Table 1. As a result of the large numberof measurements, the null hypothesis can berejected in the Kolmogorov–Smirnov test for boththe power law and the lognormal mixture model.Using the chi-square statistic, the lognormalmixture model cannot be rejected at the 5%significance level, whereas the power law alter-native fails the test. More importantly, the devi-ations from the expected test statistics are almostfive times as large in the case of the power lawmodel as in the case of the lognormal mixturemodel. This result strongly suggests that the latterprovides a better description of this data set.

In addition to goodness-of-fit tests, quantile–quantile (q–q) plots are also useful in choosingthe distribution model that provides a better fit.On a q–q plot, the quantiles of the data are plotted

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beds starting with Tab: n = 552 mean = 67·6 cm std· dev· = 0·405

beds starting with Tcd: n = 1258 mean = 6·6 cm std· dev· = 0·248

model: n = 1259 mean = 6·4 cm std· dev· = 0·236


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beds starting with Tab: n = 448 mean = 71·48 cm std· dev· = 0·233


beds starting with Tcd: n = 1081 mean = 9·1 cm std· dev· = 0·382


all beds

all beds

A BTarcau Sandstone Marnoso-Arenacea Formation

Fig. 14. (A) Cumulative probability plot of the data and the model curves for the Tarcău Sandstone, with plots ofthicknesses of beds starting with Bouma divisions Ta or Tb and Tc, Td or Te. (B) Cumulative probability plot of thedata and the model curves for the Marnoso-Arenacea Formation, with plots of thicknesses of beds starting withBouma divisions Ta or Tb and Tc, Td or Te. Mean is the geometric mean, std. dev. is the standard deviation of the 10-base logarithm of the bed thickness series.



against the quantiles of the model distribution(Fig. 17); the better the fit between the model andthe data, the closer the plot comes to the x ¼ ystraight line on the diagram. The q–q plots for theUpper Tarcău Sandstone support the results ofthe goodness-of-fit testing: compared with thelognormal mixture model, the departures fromthe power law model are significantly larger atboth the thick-bedded and thin-bedded ends.

Undeniably, the lognormal mixture model hasfive parameters (two mean, two standard devi-ation values and the proportion of the twopopulations), whereas the power law model hasonly two (the exponent and the minimum bedthickness); therefore, the better fit of the lognor-mal mixture does not necessarily qualify it as the‘correct’ model. More parameters mean largerflexibility and easier fit to any data set. However,the lognormal mixture model is preferred herebecause it is sedimentologically meaningful and

probably more useful in reservoir characteriza-tion, as explained below.

DISCUSSION

Power law distributions: the rule or theexception?

As noted by Mitzenmacher (2004), the questionwhether a particular empirical data set fits betterthe power law or the lognormal distribution hasgenerated discussion in a number of differentfields such as economics, biology, chemistry,information theory and computer science. Thetwo types of distributions can result from similarbasic generative models; and a significant portionof a lognormal distribution with a large variancebehaves approximately as a straight line on a log–log exceedence plot (Mitzenmacher, 2004). Turbi-

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Probabilistic classification based on lognormal model

Classification based on lognormal model

Classification based on power-law model

0 1Probability of belonging to

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Bed

thic

knes

s (c

m)

Log scaleLinear scale

Bed number

Classification based on basal Bouma division (Tab or Tcd)

Fig. 16. Bed thickness vs. bed number plots for the Tarcău Sandstone and results of different classification methods.The classifications based on the lognormal mixture model give a better match to the sedimentological classificationthan the analysis based on the segmented power law model.

18 Z. Sylvester


dite bed thickness series that allow precisemeasurement of all bed thicknesses and indeedappear to be linear on such a plot remaininteresting subjects of study. However, publishedturbidite bed thickness diagrams suggest that thepower law distribution is the exception ratherthan the rule and explaining clearly non-powerlaw distributions through various processes thatmodify an initial power law input remains spe-culative.T

able

1.

Resu

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Power law model

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A

B

n =1245

n =1245

Fig. 17. (A) Quantile–quantile (qq) plot of the UpperTarcău Sandstone bed thickness data, assuming a log-normal mixture model. (B) Quantile–quantile (qq) plotof the Upper Tarcău Sandstone bed thickness data,assuming a power law model.



Bimodality of turbidite bed thicknesses

As pointed out by Talling (2001), the commonlyobserved bimodality of turbidite bed thicknessdistributions probably reflects the differencebetween deposition from more dilute and moredense suspensions, or the low-density and high-density turbidity currents of Lowe (1982). Thethickness of a bed at any location depends on twoimmediate factors: duration of deposition andrate of sedimentation. It is probable that it is therate of sedimentation that causes the bimodality:once it starts, deposition from high-concentrationsuspensions is likely to be rapid, as reflected bythe common suppression of lamination or strati-fication and the relatively poor sorting in thick-bedded turbidites (Lowe, 1982; Hiscott, 1994;Sylvester & Lowe, 2004). From a slightly differentperspective, the bimodality of bed thicknessreflects the difference between competence-dri-ven and capacity-driven sedimentation (Hiscott,1994; Kneller & McCaffrey, 2003). Whether thickbeds are usually related to larger originatingevents or strictly reflect local depositional condi-tions is a different and more difficult question. Inany case, it is probable that bed thickness iscontrolled only in part by initial discharge(Pirmez & Imran, 2003) and flows depositingthick sand beds in channels can also deposit thinbeds on levées or in more distal locations.Therefore, it is unlikely that bed thicknessdistributions of turbidite systems with well-differentiated morphologic and stratigraphic ele-ments can be related directly to originating eventslike earthquakes; rather, they probably reflect thelocal depositional conditions (Talling, 2001).Taking into account the degree of complexity ofmany turbidite systems that can be observed onseafloor images and in high-quality seismic data(e.g. Deptuck et al., 2003), it is also unlikely that afew numbers or curves derived from bed thick-ness data alone can give general guidance aboutdepositional setting, degrees of confinement, ero-sion, bypass and other important characteristicsof the turbidite system.

The two different types of deposition (fromhigh-density and low-density suspensions) canoccur probably from the same flow. Recent stud-ies in the Marnoso-Arenacea Formation suggestthat the bimodality in bed thicknesses can berelated to a characteristic bed shape: beds that arerelatively thick in proximal settings undergorapid thinning towards more distal areas (Tallinget al., 2007). Any random vertical sampling line islikely to capture fewer transitional thicknesses

compared with the laterally more extensive prox-imal thick and distal thin zones.

An important implication of this study is thatsegmented power laws and the correspondingcharacteristic b exponents on exceedence prob-ability plots cannot be explained by differentdepositional processes leading to two or moredifferent populations. For example, it is tempt-ing to think that, in some cases, the twosegments of the exceedence plot could be theresult of deposition from turbidity currents vs.debris flows. If this was the case, one wouldexpect to see a range of curve types representingpower law mixtures, as shown in Fig. 8. There isno reason why different bed types could not mixin different proportions. However, there is noindication of ‘mixed’ curve shapes in the exist-ing data sets; all the segmented plots can onlycorrespond to one specific curve type of thefamily of curves illustrated in Fig. 8. At themoment, the only reasonable explanation for thiscurve type is the Malinverno (1997) model forconfinement and, therefore, any claims madeabout the presence of segmented power lawdistributions should consider the applicability ofthe assumptions and implications of this mathe-matical representation to the data under inves-tigation. In contrast, the lognormal mixturemodel is compatible with the two populationsoriginating from different transport and/or depo-sitional processes: the curve types seen on thecumulative plots of real data (Figs 12–14) cor-respond well to the various theoretical mixturesof two lognormal populations (Fig. 11). Thismodel requires no relationship between thepopulation parameters and the proportion be-tween the two groups.

How important is erosion?

Pebbly sandstone packages with abundant amal-gamation surfaces are common in the TarcăuSandstone. If the erosional flows removed signi-ficant parts of the underlying beds, the bedthickness distribution was affected as well. Acomparison of the distribution of truncated andnon-truncated beds that start with Ta (Fig. 18)shows a discernible shift of the thick bedsaffected by erosion compared with non-erodedbeds. The geometric mean values of the twodistributions are 100 cm (134 non-truncatedbeds) and 75 cm (260 truncated beds). Thissuggests that, on average, most of the bed-scaleerosion seen in the Tarcău Sandstone is restrictedto ca 20–30% of the total bed thickness and does

20 Z. Sylvester


not alter the lognormal nature of the originalevent bed thickness distribution.

Significance of thickness variability

It has been shown here that the Tarcău Sandstonebed thickness data represent a lognormal mixturedifferent from the one that describes the Marnoso-Arenacea Formation. The log-transformed thin-ner-bedded population has a relatively smallstandard deviation, whereas the thick-beddedpart shows more variability (Fig. 14). A possiblecause for this is that the thin beds in the TarcăuSandstone probably were deposited as levées andturbidites on levees of modern submarine fanshave relatively uniform bed thicknesses (e.g.Pirmez & Imran, 2003). In contrast, the thin bedsof the Marnoso-Arenacea Formation are not levéedeposits; many thin beds have been correlatedover most of the outcrop area, and they weredeposited on a basin plain (Amy & Talling, 2006).Further studies, preferably of modern systemswhere the depositional settings are clearlyknown, are necessary to see whether the var-iances of the log-transformed bed thickness data,in conjunction with other facies characteristics,have some general implications for the interpreta-tion of the deposits.

Practical advantages of the lognormal model

From a reservoir characterization perspective, itis worth noting that in most areas of researchwhere power law distributions are of great

interest (e.g. the study of networks, economics,natural hazard prediction), it is the ‘heavy tail’that receives most of the attention, because it isimportant to predict rare but huge events. How-ever, in reservoir characterization, the very large‘events’, if any, are often already known fromwells and seismic data. It is the subseismic scale,in areas unpenetrated by wells, that is largelyunknown. On the other hand, very thin beds thatusually cause the deficiencies in the bed thick-ness measurements commonly are not reservoirsand, from this practical point of view, the diffi-culties of measuring correctly the thinnest bedscan probably be ignored. So a range of bedthicknesses, from a few centimetres to a fewmetres, are of most interest in reservoir charac-terization, and this range seems to be bestdescribed by the lognormal mixture model.

Regardless of what the theoretical implicationsand interpretations might be, the lognormal mix-ture model proposed by Talling (2001) is arelatively simple and intuitive model that offersa better description of the data analysed here thanthe power law models. Certainly, the statisticalanalysis of bed thicknesses can be refined furtherand other distribution models could be investi-gated as well. However, it is important to find thebalance between the precision of the statisticalmodels and their applicability: more complicatedmodels are likely to give better fits, but they maynot be practical for description and modelling ofbed thicknesses. The lognormal mixture modelmay often fail to pass the goodness-of-fit statisti-cal tests, especially with large data sets, but thisshould not preclude its use unless a better-fittingand relatively simple, sedimentologically insight-ful model is found.

CONCLUSIONS

1 Log–log plots and least squares fitting bythemselves are inappropriate tools for the analy-sis of bed thickness distributions. Least squaresfitting gives a reasonable estimate of the powerlaw exponent, but says little about how well thedistribution fits the power law model. The asso-ciated R2 is unsuitable as a measure of goodness-of-fit. Log–log plots and least squares fittingshould be accompanied by the assessment ofother types of diagrams (cumulative probability,histogram of log-transformed values, q–q plots)and the use of a measure of goodness-of-fit otherthan R2, such as the chi-squared and the Kol-mogorov–Smirnov statistics.

10 100 1000

1

10

20

304050607080

90

99

Cum

ulat

ive

freq

uenc

y (%

)

5005020 30 40 70 200 300 400tru

ncat

ed

non-

trunc

ated

Bed thickness (cm)

Non-truncated beds:n = 134Geom. mean = 100 cm

Truncated beds:n = 260Geom. mean = 75 cm

Fig. 18. Cumulative probability plot of thicknesses ofbeds starting with the Ta divisions, separated as afunction of whether they were affected by erosion at thetop (truncated) or not (non-truncated). Such a com-parison gives an estimate of the magnitude of the bed-scale erosion.



2 Segmented power laws on exceedence plotsare not simple mixtures of power law distribu-tions with arbitrary parameters. Although amodel of confinement does result in segmentedplots at the centre of the basin (Malinverno,1997), the segmented shape of the exceedencecurve breaks down as the sampling locationmoves away from the basin centre. Identifyingtwo or more relatively straight segments in ex-ceedence probability plots is subjective, andshould be used in the analysis of turbidite bedthickness data only if it is probable that theassumptions of the Malinverno (1997) model aresatisfied.

3 The lognormal mixture model for turbiditebed thicknesses (Talling, 2001), mainly reflectingthe differences in sedimentation rates betweenhigh-density and low-density turbidity currents,is a logical alternative to the power-law model.Identifying the parameters of the componentdistributions is difficult unless a detailed sedi-mentologic description of the internal structuringis available. It is suggested here that the expec-tation–maximization algorithm can be used toestimate the parameters and thus to model suchbed thickness mixtures.

4 The bed thickness data from the TarcăuSandstone of the East Carpathians (Romania) arebest described by a lognormal mixture modelwith two components. In contrast with theMarnoso-Arenacea Formation, the thinner-bed-ded population has a lower variability than thethicker-bedded population; this could be the re-sult of deposition of these beds on levees of sub-marine channels.

ACKNOWLEDGEMENTS

Part of this study was carried out with fundingfrom the Stanford Project on Deepwater Depo-sitional Systems (SPODDS), a consortium ofsponsors that included Anadarko, BP, Chevron,Conoco, ENI-AGIP, ExxonMobil, JNOC, Mara-thon, Occidental, Perez Companc, Petrobras,Rohöl-Aufsuchungs AG, Shell, Texaco and Un-ocal. The publicly available notes of CosmaShalizi on log–log plots have been influential insome of the discussions and conclusions of thispaper. I am thankful for discussions withDonald Lowe, Carlos Pirmez, Stephan Graham,William Normark, Maarten Felix, Anikó Sándor;and for reviews and comments by Peter Haugh-ton and Peter Talling.

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