Exploring the Statistics of Sedimentary Bed Thicknesses ...

Exploring the Statistics of Sedimentary Bed Thicknesses –Two Case StudiesCarl Drummond and John CoatesDepartment of GeosciencesIndiana University/Purdue UniversityFort Wayne, Indiana [email protected]

ABSTRACTAnalysis of stratigraphic sections typically con-

sists of recognition and interpretation of lateraland vertical heterogeneities in sedimentary rock.Qualitatively, significant information is obtainedby careful observation of changes in various litho-logic components (grain composition, size, texture,sorting) as well as the presence or absence of awide range of sedimentary structures (ripples, cross-stratification, desiccation cracks). Taken together,these physical manifestations of conditions withinthe depositional environment allow for construc-tion of complexly detailed facies models of ancientsedimentary systems. However, modern stratigraphicanalysis is becoming increasingly concerned withmore than the construction of facies models. Suchtranscendent analytical effort represents a fur-ther refinement of past attempts at quantificationof processes of deposition. To date, the principalapproaches to quantitative stratigraphic analysishave been statistical – and the datatype upon whichthe most effort has been placed is bed thickness.As such, evaluation of several commonly used ana-lytical techniques provides an important backgroundto, and overview of, the statistical analysis of bedthicknesses. By providing students with an intro-duction to the statistical foundations of modernstratigraphy, it is possible to greatly enhance un-derstanding of both stratal architecture as well asthe relationships between depositional process andthe stratigraphic record.

Keywords: Education – geoscience; geology –teaching and curriculum; miscellaneous and mathe-matical geology; stratigraphy, historical geology,paleoecology.

INTRODUCTIONSince the days of Nicolaus Steno, stratigraphers

have evaluated vertical sequences of rocks throughanalysis of lithology and thickness. Measurement ofthe vertical extent of recognizably distinct layers ofsedimentary rock (beds) provides a simple and readilyobtainable database for quantitative study. Beyondease of acquisition, what is the utility of bed-thicknessanalysis? Unquestionably, the deposition of sedimen-tary rock occurs across a wide range of continuous totemporally discrete time spans. As such, thicknessesof sedimentary rocks are generally interpreted eitherin terms of the duration of a sedimentary event (rela-tively continuous deposition) or in terms of the mag-nitude or intensity of that event (episodic processes

of deposition). Stratal thickness can therefore impartimportant chronological information as well as pro-vide significant insight into the nature of the deposi-tional process. In either case, the quantitative studyof stratigraphic sections serves as a complementaryapproach to the more commonly conducted analysisof spatio-temporal variation in lithologic composition.Yet, it must be clearly represented to students thatbed thickness is never a perfect recorder of depositionalprocess. Infidelity in the coupling between thicknessand process, as well as diagenetic and compactionaloverprinting during lithification, frequently serve toalter bed thickness. Despite these complications, bedthicknesses provide the first best step to the quanti-tative analysis of stratigraphic architecture and, assuch, represent an important avenue through whichstudents can be exposed to the mathematical basis ofstratigraphy.

CASE STUDY ONE –COMPARATIVE STATISTICAL ANALYSIS

The middle to upper Ordovician Viola Springs For-mation, outcropping in the Arbuckle Mountains ofsouthern Oklahoma, serves as the stratigraphic datasource for this portion of the class experience. Fordetailed analysis, a spectacular exposure of thin-wavy-bedded subtidal micrite was measured along the westernside of I-35, north of Ardmore, Oklahoma (Finney,1988). This section consists of 859 individual beds,comprising 72 meters of continuous section, with anaverage bed thickness of 84 millimeters. Depositionis interpreted to have occurred along the southernmargin of the Anadarko basin at depths ranging from200 to 500 m (Finney, 1988). Conformable transitionfrom peritidal sediment of the Pooleville Member ofthe Bromide Formation to deep-water deposits of theViola Springs Formation has been interpreted to rec-ord a rapid increase in regional subsidence withinthe southern Oklahoma aulacogen. Shallowing withinthe Viola Springs is recorded by progressively moreaerobic conditions (Galvin, 1983) as vertical accumu-lation of subtidal sediment outpaced subsidence andgradually filled the available accommodation space.The Viola Springs Formation has been the subject ofboth detailed diagenetic (for example, Gao and others,1996) and paleontologic study (for example, Shaw,1991). The abundance of thin, subtidally deposited,laterally continuous beds exposed in the I-35 sectionalso provides an interesting source of high-resolutionthickness data from a deep carbonate shelf. As such,the Viola Springs Formation was chosen for classanalysis because of the exceptional quality of theI-35 outcrop, the ease with which collection of a large

Journal of Geoscience Education, v. 48, 2000, p. 487

quantity of high-resolution stratigraphic data wouldbe accomplished, and – unlike other stratigraphic in-tervals that could have been chosen to be the subjectof similar comparative analysis – various analyticaltechniques could be performed and evaluated in theabsence of significant debate over the interpretationof their results. That is, the beds of the Viola Springsprovide an exemplary case study for the comparisonof various techniques of quantitative stratigraphicanalysis, without the complications associated withdefending or attacking any particular stratigraphicparadigm. Thus, students can focus their attentionon learning the statistical techniques that can subse-quently be applied to the analysis of process.

BED-THICKNESS ANALYSISQuantitative evaluation of bed thicknesses com-

monly is focused upon establishing descriptions of twoimportant statistical parameters: the central tendencyof a sample and the dispersion of the sample aboutthat central tendency. Efforts to completely charac-terize the quantitative aspects of stratigraphic sec-tions must naturally consider the statistical structure

of the observed distribution of bed thicknesses. Thatis, researchers have recently begun to evaluate thefunctional relationships between stratigraphic thick-nesses and frequency of occurrence (see, for example,Rothman and others, 1994). Inasmuch as any givenstratigraphic study typically considers samples of bedthickness drawn from a population produced by asingle process of deposition (for example, turbidity-current deposition, tidal-flat accumulation, or tractionor suspension deposition during over-bank flooding),the central tendency of the sample is generally takenas an important measure of some component of thecausative depositional process. However, stratigra-phers have long realized that samples of bed thick-nesses display marked positive skewness, whereinthe most commonly occurring thickness (mode) of thesample is shifted towards thinner beds relative tothe average thickness (arithmetic mean) of the sam-ple (Schwarzacher, 1975) (Figure 1A). The modernstatistical approaches to bed-thickness analysis, whileperhaps daunting at first glance, are not significantlymore complex than the traditional characterizationof grain-size distributions (see, for example, Till, 1974;


Exploring the Statistics of Sedimentary Bed Thicknesses – Two Case Studies

Figure 1. Thickness/frequency distributions of beds of the Viola Springs Formation. Panel A illustrates the stronglyskewed character of the distribution, wherein the modal thickness (31-40 mm) is shifted to the left of the average bedthickness (arithmetic mean, 84 mm). Recasting the thickness/frequency distribution using bin sizes that increase ac-cording to a geometric series results in a significant increase in the symmetry of the distribution (Panel B). Corre-spondence between the lognormal mode (46-64 mm) and the geometric mean (61 mm) could allow for interpretation ofthese subtidal beds in terms of either a temporally or energetically recurrent depositional process.

Grace and others, 1978) that comprise a commoncomponent of sedimentology classes at the under-graduate level.

Stratigraphic sections displaying positively skewedbed thicknesses present a major interpretive challengeto the student of quantitative stratigraphy. To thedegree that bed thickness is in some way a propor-tional proxy of the depositional system – either dura-tion or intensity of accumulation – the presence of asignificant difference between the most frequentlyoccurring thickness and the average thickness in thesample indicates a level of complexity in the nature ofthe depositional process warranting further evalua-tion. Complex natural systems, such as subtidal car-bonate deposition, commonly display characteristicsthat provide opportunities for the development of agreater understanding of nature. The skewness dis-played in beds of the Viola Springs is a prime exampleof an information-rich form of stratigraphic complexity.

In a simple world, one might envision a deposi-tional process that exhibits symmetrical variation inintensity about some central tendency (for example,Kolmogorov, 1951; Mizutani and Hattori, 1972). How-ever, bed-thickness distributions displaying suchGaussian-like form are rare or non-existent, and stra-tigraphers have developed various other analyticaltechniques to apply to their quantitative studies ofstratigraphic sections. These include evaluation of bed-thickness distributions through comparison with log-normal, Poisson, and gamma distributions, as well asthe more recently popularized exceedence-probability-analysis technique. The following exercise is an il-lustrative example of the application of these variousstatistical approaches to a single continuous stra-tigraphic sequence, through which it is possible forstudents to evaluate the relative strengths andweaknesses of each approach. The goal of this effortis not to establish a direct and definitive linkage be-tween measured bed thickness and depositional pro-cess but rather to foster student understanding ofhow various statistical approaches can lead to differ-ent interpretations of a single dataset.

Lognormal DistributionTo initiate the study of the bed thicknesses, a

simple histogram of the thickness/frequency distri-bution was constructed using a bin interval of 10millimeters (Figure 1A). Beds of the Viola SpringsFormation range from 6 to 399 millimeters, with anaverage thickness of 84 millimeters and a standarddeviation of 68 millimeters, and display a strong de-gree of positive skewness found to be typical of manydifferent stratigraphic systems (Schwarzacher, 1975;Davis, 1986). The most frequent class of thicknessesspans the 31 to 40 millimeter range. Thus, the modalthickness is found to be approximately half the meanthickness of the sample (Figure 1A), with only 30percent of the beds thinner than 40 millimeters. Suchstrong positive skewness makes this sample ideal forrecasting in the form of a lognormal distribution(Schwarzacher, 1975).

The lognormal transformation of a stratigraphicdata set consists of constructing a sequence of thickness

intervals that increase in size according to a geomet-ric series, such as:

B(z)=KZ ,

where K is a constant and z is an element in a linearsequence (for example, 1, 1.5., 2, 2.5, 3, 3.5 …). Thenumber of samples falling within each interval isplotted against a logarithmic thickness axis (Figure1B). Recasting the data in this form results in a sig-nificant increase in the symmetry of the distribution.The geometric mean (Gm) of such a sample is thencalculated according to the formulation:

� �Gm=2 log( )/ log(2)t n� / ,

where t is the thickness of any individual bed and nis the number of beds in the sample. In the case ofthe Viola Springs data, the geometric mean is foundto be 61 millimeters, which falls within the modalclass of the lognormal distribution (46 to 64 millime-ters, Figure 1B). Clearly, application of the lognor-mal transformation has produced a marked increasein symmetry of the data, as well as inducing a sig-nificantly greater correspondence between the sam-ple’s mean and modal values (Figure 1B).

Yet for students, the question remains, what ad-vantage is gained by this transformation? Beyond theincrease in ease with which further statistical analy-ses can be conducted using a normally distributedsample (for example, see Davis, 1986; Mendenhalland Sincich, 1984), bringing the average bed thick-ness into agreement with the most common thick-ness gives students a single thickness value that canbe interpreted in terms of both the most abundantand most likely stratigraphic product of an underlyingdepositional process. To the degree that one mightwish to interpret deposition of subtidal carbonate interms of a process that either recurred with nearconstant intensity or with near constant periodicity(see, for example, Elrick and Read, 1991; Osleger,1991), the stratal thickness of the 61 millimeters cal-culated to be the geometric mean of the Viola SpringsFormation data could be held as a representativestratigraphic product of such a presumed recurrentprocess. While such a periodic interpretation is onlyone of several conceivable for the deposition of theViola Springs Formation, it is interesting for stu-dents to consider how this dataset could be statisticallydescribed in terms of a cyclic depositional process,resulting in a modal stratigraphic product.

A significant drawback of the lognormal transfor-mation to be discussed with students is the fact thata disproportional level of statistical significance isgiven to thin beds relative to that given the thickerbeds comprising a larger proportion of the total stra-tigraphic section. This interpretational issue centersupon the question of which is more important, occur-rence frequency or stratigraphic thickness. Thin bedsare commonly more frequent, yet the less commonthick beds make up a larger portion of measured sec-tions. The specific nature of the scientific inquirywill determine the utility of the lognormal technique.Thus, while perhaps applicable and useful in some



contexts, the lognormal transformation of thickness/frequency data tends to induce a loss of informationabout the statistical structure of the thickness samplerather than presenting a universally useful metricwith which to evaluate bed-thickness distributions.As such, the use of lognormal transformations of bedthickness to contribute information about the natureof depositional processes has not been a major com-ponent of recent statistical studies of bed thickness.Exploring the reasons for this with students pro-vides an important opportunity to discuss both thepersonal biases and the mathematical rationale forthe selection of specific statistical techniques.

Poisson Frequency DistributionOne of the central questions to be raised with stu-

dents of stratigraphy as they study complex systemsis the role of recurrence within the depositional environ-ment. Those stratigraphic sections found to have re-current lithologic structures are said to be “Markovian”(for example, see Gingerich, 1969), that is, strati-graphic sections displaying Markov properties arethose where the composition of a given stratal ele-ment is at least partly dependent on the compositionof preceding elements. Thus, Markovian systems ex-hibit cyclical ordering; the presence of which can beeasily tested for against the null hypothesis of no order(Carr, 1982; Davis, 1986). Conversely, stratigraphicsystems described as Poissonian lack any temporalor stratigraphic order – that is, the thicknesses or li-thology of any particular stratal element is independ-ent of the thicknesses or compositions of precedingelements (Wilkinson and others, 1997a).

The waiting times between events described byPoisson processes are described by the negative ex-ponential distribution. Interestingly, thickness/frequencydistributions from a large number of stratigraphicsystems have been found to possess this general statis-tical form (for example, see Drummond and Wilkinson,1993; Drummond, 1999); and importantly, the fre-quency distribution (F) of thicknesses (T) produced bya Poisson process is characterized by the formulation:

F=(BN L)e2 -TN/L/ ,

where B is the bin size, N is the number of beds inthe sample, and L is the total thickness of the section(Wilkinson and others, 1997a). Thus, the statisticalform of an exponential distribution of bed thicknessesis only dependent upon the average thickness (L/N)exhibited by that sample and the total length of sec-tion measured (Davis, 1988; Swan and Sandilands,1995).

Thickness/frequency data derived from the ViolaSprings Formation can be compared to models withPoisson characteristics by plotting the logarithm ofthe frequency against thickness (Figure 2B). Markedlinearization of the data by this style of logarithmictransformation indicates that a significant propor-tion of the sample variance is explained by this modelof accumulation. Comparison of the distribution cal-culated according to the model formulation above (Fig-ure 2B) with a simple least squares linear regression

of the log-transformed data (Figure 2A) indicates thateither approach results in similar model slopes and anear identical degree of fit to the data r = 0.925 inboth cases).

However, visual comparison of these results withthe thickness/frequency distribution of the ViolaSprings Formation indicates significant deviation be-tween the model with Poisson characteristics andthe observed frequency of bed thicknesses at smallbin sizes. That is, the smallest bins (0-10 and 11-20millimeters) fall below both the linear regression andmodel lines, while the next several bins are found tofall significantly above both lines (Figure 2A and B).The presence of this “hook” in the log-transformeddata is expected given the strongly lognormal char-acter of the thickness/frequency distribution as illus-trated in Figure 1.

Clearly, the model based upon Poisson character-istics successfully characterizes a large proportion ofthe variance in the distribution of thicknesses, but itfails to adequately account for under-representationof very thin beds and the associated over-representationof several of the larger intervals of bed thickness asobserved in the Viola Springs bed thicknesses. Thepresence of this and other forms of deviation fromthe Poisson model have been interpreted to be theproduct of a failure to recognize thin beds due to theinadvertent lumping of those beds into adjacent stra-tigraphic elements (Drummond and Wilkinson, 1993),the result of mixing of populations of beds of differentaverage thickness and thus perhaps different origin(Wilkinson and others, 1997a), or the erosion of thethinnest elements. Neither of these interpretationsseem to be directly applicable to the subtidally de-posited, largely monolithologic, and extremely wellexposed beds of the Viola Springs Formation. How-ever, one must always acknowledge the possibility ofmodification of stratal architecture by diagenetic orcompactional processes, especially in fine-grained sub-tidal sediments such as those comprising the ViolaSprings Formation. Presenting students with this typeof analytical problem provides an opportunity for thedevelopment of a greater understanding of both thestatistical models and the role of process in the depo-sition of populations of bed thicknesses.

It is important to illustrate to students that thepurpose of comparative statistical analysis goes be-yond simply finding the “best-fit” regression of thedata. Rather, the goal of statistical comparison is toaid in the identification of an underlying depositionalmechanism (process) that produces a model distribu-tion statistically similar to that observed in the rockrecord. That is, interpretations must make sense geo-logically as well as statistically. The presence of sig-nificant deviations between data and model lines oversmall thickness intervals clearly indicates that a de-positional process with Poisson characteristics is in-adequate to fully explain the observed thicknessdistribution. As such, deviations from these statisticalmodels allow for a more detailed understanding ofprocess by demanding a more complete mathemati-cal and theoretical characterization of the data. Thatis, a more compete statistical description can ultimately



result in greater geological understanding. This link-age between statistical analysis and geological inter-pretation is perhaps the most difficult for students tomaster. It is, however, one of the most important forstudents to understand.

Gamma DistributionThe negative exponential distribution character-

istic of Poissonian processes is a special case of themore general gamma probability-density function(Mendenhall and Sincich, 1984). Gamma distribu-tions are described in terms of two parameters: �,

the shape controlling variable, and �, the scaling pa-rameter. Both of these statistical variables are di-rectly calculable from the mean (�) and variance (�2)of the sample:

� � �� 2 2/ .

� � �� 2 / .

The probability-density function of a gamma-typerandom variable F(t) is given by:

F( )= e-1 -t/t t� � �� / ( ) ,



Figure 2. Thickness/frequency representations of beds from the Viola Springs Formation. Least-squares linear regres-sion (Panel A) produces a model slope (m) of -0.005, a frequency intercept (b) of 93.3, and a correlation of 0.925. Atheoretical Poisson model of bed thickness/frequency (Panel B) is calculated from the number of beds in the sample(N = 859) and the total thickness of the section (L = 72,033 mm), resulting in a slope of -0.012 and a correlation of0.925. Note the lack of agreement between the model and data exhibited by both the linear regression and Poissonanalyses over the first few thickness intervals. The general gamma model of bed thickness/frequency distribution(Panel C) is calculated from the shape parameter (� = 1.5) and the scale parameter (� = 56) derived from the mean andvariance of the sample (see text for details). This distribution more closely matches the under-representation of thinbeds relative to those in the next larger thickness classes. As such, this statistical model reflects a slightly higher de-gree of correlation with the thickness data (r = 0.937) than either the linear regression or Poisson models and impliesthat while Poisson-like with respect to recurrence, the deposition of these subtidal carbonates likely occurred via aprocess defining a weak modal intensity or periodicity.

where t is the bed thickness and (�) is the gammafunction derived from statistical tables (for example,see Pearson, 1956) for a particular value of �.

When the shape parameter (�) is 1, the distribu-tion takes on the form of the standard Poisson nega-tive exponential; however, for values of � greaterthan 1, the distribution becomes skewed similar tothat illustrated in Figure 2. Thus, any given sampleof bed-thickness data can be compared to the gammadistribution through the calculation of shape and scaleparameters and subsequent generation of the appro-priate model frequency distribution based upon thosevalues. From the mean and variance of the ViolaSprings bed-thickness data, values for � and � wereestimated to be 1.5 and 56, respectively. When scaledto the number of beds measured, the general gammadistribution model produces a thickness/frequencycurve that closely follows the pronounced “hook” oversmall bed thicknesses (Figure 2C). Despite the factthat the model curve fails to account completely forthe very low abundance of thicknesses in the range 0to 10 millimeters (importantly, at or near the limit ofphysical measuring resolution in this study), clearlythe gamma technique does a significantly better jobof characterizing the observed thickness data thaneither the simple linear regression or Poisson models.A critical point to stress with students is the impor-tance of elementary statistical parameters (mean andvariance) to the description of a complex statisticalsample using the gamma approach.

What are the geological implications of choosingthe gamma over the restricted Poisson distributionto model stratigraphic data? Selection of the nega-tive exponential function implies that an infinite abun-dance of infinitely thin beds must occur in any strati-graphic section defined to be Poisson. This is obviouslyan extrapolation to the absurd. Geological processesoccur only over finite ranges of magnitude and, assuch, are likely to produce stratal elements spanninga finite range of thickness. At the extreme, one wouldnot expect to find a bed of sedimentary rock thinnerthan an individual grain, nor thicker than the litho-sphere. Illustrating these less obvious implicationsof various statistical models allows students to de-velop a more intuitive understanding of the naturallimits of scale over which such models are applicable.

As evidenced by the Viola Springs data, limita-tions on the ranges of expected bed thicknesses mightbe more adequately described in terms of the generalgamma function rather than the negative exponential.Importantly, selection of the gamma function as adescriptor of stratal-thickness distributions in no wayinvalidates the notion of an absence of stratigraphicdependence. Indeed, stochastically based gamma modelshave long been recognized to be adequate represen-tations of a wide range of complex natural processes(Mendenhall and Sincich, 1984). Direct interpretationallinks between the gamma shape factor and the char-acter of the depositional process have as yet not beenfully established for any particular facies associa-tion, thus providing fertile ground for dialog withstudents regarding the potential relationships betweencausative process and resultant bed thickness.

Gamma distributions with shape factors (�) greaterthan 1 provide a mechanism by which to model atemporally random recurrent process that results ina modal frequency at a value greater than zero. Thatis, as the shape factor � becomes larger than 1, themode of the population of beds shifts progressivelyaway from zero. This generally agrees with the no-tion that geological processes are often irregular withrespect to both their magnitude and their temporalrecurrence but that the range of variance exhibitedby any given process is limited and typically displayssome most-frequent or most-common magnitude. Amode-based philosophical conception of Earth-surfaceprocesses is pervasive throughout stratigraphy (see,for example, Grotzinger, 1986; Osleger and Read, 1991),and as such, the statistical analysis of bed thick-nesses provides an opportunity for students to evalu-ate the strengths and weaknesses intrinsic to suchgeneral models.

Exceedence-Probability AnalysisAn important concern associated with standard

techniques of bed-thickness frequency analysis is theimpact that the size of a bin interval can have on theshape of the resultant distribution (Figure 3). Selec-tion of a coarse bin size results in a loss of stra-tigraphic information due to excessive lumping ofelements into classes, while choice of too small a binsize spreads the samples over a large number of binslikewise resulting in a loss of information. In the ab-sence of simple integer data, there is no clear-cutmethod of choosing the proper bin size. When facedwith this dilemma, a student will generally try sev-eral different interval sizes before selecting the onethat presents the stratigraphic data in the most un-derstandable fashion. However, such arbitrarinesscan lead to intentional or accidental misrepresenta-tion of stratigraphic data. Therefore, studies have



Figure 3. Calculation of thickness/frequency distribu-tions of Viola Springs bed thicknesses at various binsizes. As bin size increases from 5 to 40 millimeters, theform of the distribution takes on less of a lognormal andmore of an exponential character. As such, the way inwhich a distribution of bed thicknesses is interpretedcan be highly dependent upon the bin size used to dis-play the data.

begun to utilize the exceedence-probability techniqueto evaluate statistical structure in stratigraphic data(for example, Hiscott and others, 1992; Rothman andothers, 1994).

The exceedence-probability technique comparesthe thickness of any individual bed (ti) to the thick-nesses of all other beds in the sample. The number ofbeds thicker than the given bed (n > ti) is divided bythe total number of beds in the sample (N) to pro-duce the probability (P) that any particular bed willbe thicker than the given bed.

P=(n>t Ni) /

The calculated exceedence probability is then plottedagainst bed thickness and the resulting distributionis analyzed in linear, log-linear, or log-log thickness/frequency space (Drummond, 1999).

Results of exceedence-probability analysis as con-ducted on beds of the Viola Springs Formation are il-lustrated in Figure 4. There are some obvious andsignificant differences between this representationof the bed-thickness structure and those illustratedin Figure 2. First, there is a sharp reduction in scat-ter about the model line. Second, a linear regressionof the exceedence probability data returns a substan-tial improvement in the correlation coefficient (r =0.999, as opposed to 0.937 and 0.925). Third, the im-portant hook-like deviation from the exponential trendat small bed sizes illustrated by lognormal, Poisson,and gamma techniques (Figures 1 and 2) is greatlyminimized by exceedence-probability analysis. Thus,the concentration of data points at the upper left-hand portion of the distribution renders meaningfulinterpretation of variations at thin bed sizes nearlyimpossible. Fourth, the exceedence-probability tech-nique illustrates a small but significant deviation

from the exponential trend at very thick bed sizes(greater than about 300 millimeters, Figure 5). Thisdeviation is not obvious in any of the representationsof bed-thickness distributions previously described.The implication of this observation is that, at thick-nesses greater than about 300 millimeters, beds withinthe Viola Springs Formation are thinner than pre-dicted by an exponential model of the entire sample.The depositional or post-depositional origin of such athinner-than-expected anomaly within the Viola Springsis not obvious. However, it can be pointed out to stu-dents that it is just this type of unexpected deviationfrom a statistical model that serves as the focal pointfor the interpretation of depositional process. That is,any theory proposed for the origin of bed-thicknesseswithin the Viola Springs Formation must reproducethis observed thick-bed deviation.

As with the other approaches considered, theexceedence-probability-analysis technique exhibitsseveral strengths and weaknesses. The reduction insample variation and associated increased correlationproduced by this technique could possibly lead toover-confidence in an analytically based interpreta-tion of depositional process (see, for example, Rothmanand others, 1994; Beattie and Dade, 1996). Conversely,the lack of reliance upon arbitrary bin sizes and thepresence of a significant deviation from the exponen-tial trend at large bed thicknesses within the ViolaSprings data represent examples of the analytical



Figure 4. Representation of Viola Springs data using theexceedence-probability technique. Note the significantreduction in scatter about the least-squares regressionline (slope -0.006 and intercept of 1.238) and greater degreeof correlation (r = 0.999) compared to standard binningtechniques (Figure 2). Compression of thin bed data maskstheir under-representation within the sample. Significantdeviation of very large beds from the regression trend isfound to occur beyond 300 millimeters thickness.

Figure 5. Enlargement of the thick bed portion of theexceedence-probability distribution illustrated in Fig-ure 4. Beds thicker than 300 millimeters define anexceedence-probability trend steeper than that describedby the entire bed-thickness sample (m = -0.012 vs. m =-0.006). Such a transition indicates that the thickest bedsof the Viola Springs Formation are thinner than expected,relative to the portion of the sample between 0 and 300millimeters thickness. This observation, coupled with theunder-representation of thin beds illustrated in Figure 2,defines the bed-thickness limits over which exponentialscaling can be interpreted for this thickness/frequencydistribution. As such, it is only through a combination ofseveral analytical techniques can subtleties within athickness/frequency distribution be fully identified andsubsequently interpreted.

strengths of this technique. The power of a statisticalanalysis of bed thickness lies not in the descriptionof the data but rather in the identification of previ-ously unknown characteristics of the stratigraphicsection. It is, therefore, the exception or deviation be-tween data and theory that proves to be most insight-ful. Learning to appreciate the importance of exceptionsis one of the most significant lessons students cantake from their study of quantitative stratigraphy.

CASE STUDY ONE – SUMMARYWhat has been learned by this comparative assess-

ment of methodologies common to modern quantita-tive analysis? First, each analytical procedure presentsits own unique characteristics. Therefore, the stu-dent of quantitative stratigraphy must fully considerthe strengths and weakness of all techniques whenselecting any particular approach for implementa-tion. Second, statistical aspects of a single datasetcan be highlighted by the application of different tech-niques. Lognormal and Poisson/gamma approachesare shown to be well suited to describing deviationsfrom the model at small stratigraphic thicknesses.Conversely, exceedence-probability analysis seems tomask deviations at the small end of the distributionwhile accenting them at the larger end of the thicknessrange. Given the importance of accurate interpreta-tion of bed-thickness statistics, comparative evaluationssuch as provided by this case study clearly describethe risks associated with drawing conclusions froman incomplete analysis – a critical point for studentsto grasp.

As with all statistical approaches, the techniquechosen for stratal-thickness analysis is as much de-pendent upon the goals and biases of the researcheras it is upon the nature of the data under considera-tion. Therefore, students should be strongly encour-aged to evaluate stratigraphic data using all possibletechniques and to highlight the one that most clearlyillustrates the stratigraphic features deemed most sig-nificant, while providing a complete – multi-statistical– analysis. Finally, it should be clearly stated to stu-dents that no procedure of thickness/frequency analy-sis directly comments upon the presence or absenceof stratigraphic organization beyond the scale of in-dividual beds. In order to evaluate fully the criticallyimportant position-in-sequence aspects of a strati-graphic data set, it is necessary to combine thickness/frequency analysis with positional techniques suchas Fischer-plot analysis, autocorrelation analysis, orFourier spectral analysis. Only then can a clear andcomplete understanding of both the large- and small-scale stratigraphic structure of a sedimentary sequencebe achieved. It is this relationship between position-in-sequence analysis and bed-thickness analysis thatis the subject of Case Two.

CASE STUDY TWO –COUPLING OF PROCESS AND RESPONSE

Hierarchical stratigraphic organization has becomea widely recognized indicator of allocyclic depositionalprocesses (see, for example, Goldhammer and others,1987, 1990; DeBoer and others, 1989; Kvale and others,

1989; Osleger and Read, 1991; Borer and Harris, 1991;Archer and others, 1991; Lanier and others, 1993). Con-versely, absence of stratal organization has been putforward as evidence for the significance of stochasticcontrols of deposition (for example, see Drummondand Wilkinson, 1996; Smith, 1994; Wilkinson andothers, 1997a). Yet, irrespective of the eventual inter-pretation presented to students, stratigraphic organi-zation is typically defined and analyzed either in termsof lithologic cycles (Koerschner and Read, 1989) orpatterns of bed thickness (Goldhammer and others,1987).

The analysis of lithologic pattern is centrally con-cerned with description of the Markovian propertiesof a stratigraphic section (Gingerich, 1969). Is therea statistically significant propensity for one particu-lar lithology or facies to follow another? Analysis ofthis type is highly dependent upon subjective de-scription and interpretation of stratigraphic sections,and central to this dependence is the notion of sub-stitutability. That is, do two lithologically independentfacies represent equivalent positions within a Walthe-rian sequence? Stratigraphic substitutability of thistype significantly complicates efforts to conduct mean-ingful Markov analyses. Clearly, the interpretationof facies sequences is dependent upon the goals andbiases of each individual researcher and, as such,analysis of lithologic patterning has proven to be anextremely complex endeavor. Giving students an op-portunity to explore that complexity is the goal ofCase Two.

Collection of thickness data represents one of themost objective of all stratigraphic measurements. Inmost cases, complications associated with lateralvariation in thickness, gradational lithologic con-tacts, and structural deformation can be minimizedby selecting high-quality stratigraphic sections andapplying careful, and detailed data-collection tech-niques. For students to fully grasp these factors, theymust experience the frustration of measuring stra-tigraphic sections in the field.

Bed thickness is an important stratigraphic pa-rameter because of its direct relationship to intensitiesand durations of sediment accumulation. The thick-ness of a bed is generally thought to be controlled bythe nature of the underlying depositional process. Forthis reason, analysis of bed thickness has proven tobe an important component of stratigraphic researchacross a wide range of depositional systems. Stra-tigraphic sections displaying pronounced hierarchicalthickness organization have often been interpretedin terms of temporal repetition of their causativeprocesses (Kvale and others, 1989; Goldhammer andothers, 1990; Osleger and Read, 1991). Description ofthese and various other similar stratigraphic modelscommonly constitutes a significant portion of stra-tigraphy classes. Interpretation of coupling betweendepositional process and stratigraphic response is oneof the most rapidly advancing areas of our under-standing of the role of allocyclicty in the develop-ment of stratigraphic organization.

Still, stratigraphic sections often fail to show litho-logic or thickness organization at statistically significant



levels. In those sections displaying apparently disor-ganized structure, how is one to approach questionsof depositional process? Students must be challengedto address stratigraphic complexity of this sort.

In some instances, insight is gained through analy-sis of the statistical characteristics of a population ofbed thicknesses. For example, as discussed previ-ously, beds displaying a pronounced modal distribu-tion might be interpreted to be the product of a singleprocess recurring with near-constant intensity; con-versely, beds with exponential populations have beeninterpreted to record Poisson-like stochastic processes(Wilkinson and others, 1997b), whereas the presenceof populations of bed thicknesses with power-law dis-tributions represents an interesting, but not-as-yetfully understood, type of stratigraphic organization(Hiscott and others, 1992; Rothman and others, 1994;Beattie and Dade, 1996).

This case study explores how the nature of process-response coupling within a depositional system canbe extracted from both the hierarchical structure ofstratigraphic organization and, in some cases, fromthe characteristics of a bed-thickness distribution.Recognition of depositional process from a bed-thicknessdistribution, even in the absence of position-in-sequencestratigraphic organization, implies that further ad-vances in our understanding of complex stratigraphicsystems and the processes that form them can beachieved through consideration of the origin of bedthicknesses. This is a powerful lesson for students –the ability to look at a single set of data in differentways often results in startlingly more complete in-terpretations of that data.

STRATIGRAPHIC DATA SOURCETo evaluate stratigraphic expressions of allocyclic

depositional processes, a micro-stratigraphic analysisof tidal laminites from the Pennsylvanian Mansfieldformation of southern Indiana has been undertaken.This case study utilizes these well described (Archerand Maples, 1984; Maples and Archer, 1987; Kvaleand others, 1989; 1994) clastic tidalites to illustratethe complex interplay that can exist between deter-ministic and stochastic controls of deposition, the re-sultant variability in stratigraphic architecture thatdevelops from such interplay, and how an under-standing of depositional process is best achieved throughstratigraphic analysis.

As part of this study, it is important to introducestudents to previous analyses of tidalite stratigraphy.Recognition of tidally deposited sediments of late Pro-terozoic (Williams, 1989a, 1989b) through Holoceneage (Visser, 1980) has facilitated calculation of changesin the orbital dynamics of the Earth-Sun-Moon system(Sonett and others, 1988; Deubner, 1990). Previousstudies of tidal systems have focused on establishingcriteria for recognition of tidal deposition (DeBoer andothers, 1989), estimating tidal range (Klein, 1971),and establishing hierarchical tidal periodicities (Kreisaand Moiola, 1986; Tessier and Gigot, 1989; Millerand Eriksson, 1997). In central North America duringthe Pennsylvanian, interaction between high-amplitudeeustatic variation and laterally dynamic large-scale

fluvial systems draining the Appalachian highlandsled to the development of tidally dominated estuariesthat recorded periodic tidal deposition with great fi-delity (for example, see Kvale and others, 1989; Kvaleand Archer, 1990; 1991; Lanier and others, 1993;Kvale and others, 1994). Of these, the early Pennsyl-vanian Hindostan Whetstone beds of the MansfieldFormation display particularly striking tidal laminae(Archer and Maples, 1984; Maples and Archer, 1987;Kvale and others, 1989; 1994; 2000) and as such, havebeen studied in detail as a high-resolution record oftidal activity during the Pennsylvanian (Kvale andothers, 1989; Archer and others, 1991).

Several levels of stratigraphic organization havebeen previously recognized within the Mansfield (forexample, see Kvale and others, 1989). Discrete silt-stone laminae are interpreted to have been depositedduring individual tidal rises. Laminae are typicallyorganized into couplets displaying thick-thin alter-nation that is taken as evidence for a semi-diurnaltidal system. Bundles of these couplets show thick-ness deviations that vary systematically from pro-nounced asymmetry to near equality, which has beeninterpreted to represent neap-spring modulations oftidal amplitude.

To evaluate the stratigraphic expression of thisallocyclic hierarchical depositional system, two handspecimens of Mansfield Formation laminated siltstonewere selected from samples of the Dishman quarryin Orange County, southern Indiana. Samples wereslabbed perpendicular to the bedding and polishedfor analysis. Thicknesses of individual laminae werethen measured by students under a magnifying worklamp using a hand-held digital caliper. The sampleschosen for study were selected because they displaya significant degree of stratigraphic organization and,as such, provide an interesting opportunity to compareand evaluate the potential significance of bed-thicknessdistributions to the interpretation of depositional process.

ANALYSIS OF THICKNESS STRUCTUREAn interpretation of tidal origin for the Mansfield

Formation has been, in part, based upon the obser-vation of hierarchical bundling of laminae, wherein amixed amplitude semi-diurnal tidal signal is thoughtto have been modulated by neap-spring periodicitiesas recorded by short and long-term variation in lam-ina thickness. However, this highly organized micro-stratigraphy is not universally present within theWhetstone beds of the Mansfield; some samples ap-pear to record tidal processes with a significantlylesser degree of fidelity.

These differences in stratal organization are wellillustrated by the two samples chosen for this casestudy. Sample 1 consists of 45 laminae comprisingone full, and two partial neap-spring cycles (Figure6A). In this case, the hierarchical organization is dis-tinct, with thick-thin alternation present throughoutmost of the sample and thinner, more equally sized,laminae defining crossover points between long-termcycles. Such multi-level stratigraphic organizationhas been interpreted to be the product of a stronglyperiodic, hierarchical, tidally dominated depositional



process. Conversely, Sample 2 shows very little stra-tigraphic organization (Figure 6B). Almost no thick-thin pairings are recognized within its 43 laminae, andwhile a general increase and decrease in lamina thick-ness occurs through the sample, the strong modulationassociated with neap-spring cycling is not evident.

The differences in the degree of stratigraphic or-ganization within these samples become even moreapparent when autocorrelation analyses are conducted.Autocorrelation compares a set of data to itself througha series of sequential offsets, or lags (Davis, 1986).When beds of similar thicknesses are compared, thevalue of the autocorrelation coefficient is high (closeto 1), and when beds of dissimilar thicknesses areevaluated, the coefficient takes on a low value (closeto 0). As such, differences in correlation at differentlag values can be used to quantitatively describe thatqualitative structure so apparent within the thick-ness data of Figure 6A and B. Importantly, stronglycyclical data have high values for the coefficient atlags equivalent to, and at multiples of, the principlestratigraphic periodicity. Exposing students to thissimple, yet powerful, analytical technique allows fora greater understanding of the processes by whichstratigraphic cyclicity is evaluated.

The autocorrelogram calculated for Sample 1 de-scribes an alternation in values of the correlation co-efficient between odd- and even-numbered lags (Figure6C). At lags of 2, 4, 6, and 8, the value of the coeffi-cient is high relative to those found for the interven-ing odd-numbered lags 1, 3, 5, 7 and, as such, clearlydescribes the semi-diurnal couplets present withinthe sample. Likewise, the fundamental frequency ofthat thick-thin bundling is illustrated by a very highvalue for the autocorrelation coefficient at the secondlag. Conversely, a significantly smaller amount ofstratal organization is displayed by Sample 2 (Fig-ure 6D). In this case, only a very weak correlation isfound at a lag of 2 and little or no correlation athigher lag values. Additionally, values of correlationfrom Sample 2 are generally lower at all lags thanthose calculated from Sample 1.

Autocorrelation analysis provides students with aquantitative reconfirmation of what is visually ap-parent – hierarchical stratigraphic organization ispresent in only one of the two tidal samples. Clearly,Sample 1 displays readily recognized, and easily in-terpreted, stratigraphic structure, while Sample 2 issignificantly more disorganized. This variable qualityof stratigraphic organization calls into question a tidal



Figure 6. Left Panels – Stratigraphic organization of tidal laminae, up-section to the right. A) Highly organized laminaeshowing distinctive asymmetry produced by a semi-diurnal tidal system. Magnitudes of deviation between pairs oflaminae are modulated by longer neap-spring oscillations. B) A general absence of hierarchical organization, with agradual increase and decrease in lamina thickness through the sample. Dashed lines at 2 mm (A) and 1.5 mm (B)mark thickness breaks between sub-populations of laminae. Middle panels – Autocorrelograms of the three samplesshown in Figure 1. C) Pronounced even-odd alternation in the value of the correlation function records thick-thin cou-plets of Sample 1. D) Lack of even-odd alternation and generally lower values of the correlation function providequantitative evidence for a poorer quality stratigraphic record. Values of the autocorrelation function are significantlylower at all lags. Slightly higher value at a lag of 2 reflects a very weak thick-thin alternation within the data. Right panels –Thickness/frequency distributions of laminae as exceedence-probability plots. Lamina thickness is plotted on thehorizontal axis, while the base 10 log of the proportion of the population of laminae thicker than that particular laminais plotted on the vertical. Samples show a sharp break in their populations between shallowly sloped thin laminae andsteeply sloped thick laminae. The presence of such breaks in the distribution is taken as evidence for the underlyingcontrol of a semi-diurnal tidal process, even in the absence of significant hierarchical organization. Regression statis-tics from each sub-population are reported as the slope (m), intercept (b), and correlation (r). Vertical dashed lines at2 mm (E) and 1.5 mm (F) mark breaks between sub-populations and are shown in panels A and B.

interpretation for non-cyclic intervals of the Mans-field. What process was responsible for deposition oflaminae such as found in Sample 2? Was the tidalperiodicity overprinted by other, non-cyclic, fluvialprocesses? And more importantly, is it possible tolearn anything about depositional process from suchapparently disorganized portions of the stratigraphicrecord? These are fundamental questions that stu-dents are faced with when conducting real-world stra-tigraphic analyses.

Despite variation in the position-in-sequence struc-ture of these hand samples, both exhibit similar andvery distinctive laminae thickness/frequency distri-butions. Characterization of the statistical distribu-tion of laminae thicknesses has been accomplishedby means of exceedence-probability analysis as dis-cussed above. This technique plots the thickness ofan individual stratal element against the fraction ofthicker beds within the sample (Figure 6E and F)and has recently received wide use in the analysis ofstratigraphic data (for example, see Rothman andothers, 1994; Drummond, 1999).

Thickness structures of Mansfield tidalites dis-play an unusual break in their distribution – sub-populations of laminae from both samples plot alongnon-overlapping trends with a pronounced break inslope defined by groupings of thick and thin laminae.This is unexpected given that populations of elementthicknesses from most stratigraphic intervals are typi-cally found to exhibit single trends defining eitherexponential (Wilkinson and others, 1997b) or power-law populations (Hiscott and others, 1992; Rothmanand others, 1994).

In both of these samples, thickness populationsdisplay an abrupt change in slope from relativelyshallow slopes for thin laminae to relatively steepslopes for thick laminae. That is, the thick laminaedefine a sub-population of thicknesses distinctly dif-ferent from the thin laminae not only in absolutethicknesses but also in their thickness/frequency re-lationship. What is the origin of this break betweenthe distributions? In the case of Sample 1, the divisionis largely within paired semi-diurnal tidal couplets(Figure 6A and E). That is, the large break betweensub-populations at a thickness of 2 mm clearly sepa-rates most thick laminae from their thinner counter-parts. Conversely, Sample 2 exhibits a break betweensub-populations at a thickness of approximately 1.5mm (Figure 6F), yet that thickness fails to corre-spond to any obvious stratigraphic structure withinthe disorganized laminae (Figure 6B). However, thenature of deposition in a tidally influenced estuarystrongly suggests that the thickness of an individuallamina is directly proportional to the maximum rateof tidal rise (Kvale and others, 1989). That is, thicklaminae record high-amplitude tidal excursions, whilethin laminae record relatively minor tidal rises. Thus,the breaks in slope in these laminae thickness/frequency distributions are interpreted to be the di-rect result of a semi-diurnal origin for all tidal rhyth-mites irrespective of the lack of position-in-sequenceorganization.

A PROCESS-BASED STRATIGRAPHIC LESSONTidal laminae of the Mansfield Formation provide

an interesting, and perhaps unique, opportunity todirectly decipher the nature of a depositional processfrom the thickness/frequency distribution of stratalelements. In this case, samples display distinct sub-populations of thin and thick laminae. Because thelaminae are divisible into discrete groups, defined byindependent thickness/frequency relationships and non-overlapping populations, they are readily interpretedto be the product of two independent depositionalprocesses, a critical conclusion for students to drawfrom these data. This interpretation is, of course,fully in keeping with previous analysis of the Mans-field tidalites, wherein the two depositional processeshave been interpreted as the unequal amplitudes ofsemi-diurnal tidal rises. For students, the point ofgreatest significance to be extracted from this studyis that, even in the absence of clearly hierarchicalstratigraphic organization, the semi-diurnal tidal sig-nal is recognizable in the thickness/frequency struc-ture of the laminae.

This observation carries potentially wide-reachingimplications for the analysis of stratigraphic sections.Few depositional environments record allocyclic pro-cesses with the fidelity of tidal laminites and, assuch, most typically provide significantly more com-plex interpretative stratigraphic challenges. The factthat a semi-diurnal tidal signal is recognizable evenin poorly organized laminae implies that thickness/frequency analysis has greater interpretational utilitythan previous studies have acknowledged. The com-plex interplay of allocyclic, autocyclic, and stochasticvariables present within most depositional systemsdemands that stratigraphy students put to use anyand all analytical techniques that aid in the deconvo-lution of depositional process.

ACKNOWLEDGMENTSThis work was inspired in part by the comments

of Gail Ashley, Michelle Kominz, and Beverly Saylor.Special thanks go to Nathan Diedrich for providingfield assistance and Scott Argast for supplying handsamples. Bruce Wilkinson has provided invaluablesupport and continual criticism throughout the de-velopment of this offering. The analysis of patternsof sediment accumulation has been graciously fundedby the American Chemical Society – Petroleum Re-search Fund Grants 29737-B8 and 350001-B8 as wellas the Purdue University Faculty Summer ResearchFund. From these primary research dollars has comethe ability to expand the content of undergraduatesedimentology and stratigraphy classes. This manu-script was substantially improved by the commentsof Peter Sadler and Mark Harris.

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