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Geology; March 2008; v. 36; no. 3; p. 235-238; DOI: 10.1130/G243326A.1
© 2008 Geological Society of America
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The nature of shallow-water carbonate lithofacies thickness distributions

Peter M. Burgess1

1 Shell International Exploration and Production, PO Box 60, 2280AB Rijswijk, Netherlands


   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 DATA AND ANALYSIS
 DISCUSSION
 CONCLUSIONS
 REFERENCES CITED
 
Carbonate lithofacies thickness distributions are of fundamental importance to understanding shallow-water carbonate deposystems because they record evidence of the lateral distribution and migration of lithofacies elements, as well as evidence for the various other intrinsic and extrinsic controls on stratal geometries and accumulation rates. Previous analyses of lithofacies thickness data led to the suggestion that exponential distributions are ubiquitous in the ancient record. This has been interpreted to indicate deposition by stochastic Poisson process lithofacies mosaics. To further investigate these ideas statistical analysis of 56 outcrop and core examples was performed. The Kolmogorov-Smirnov test was used to identify the degree to which measured lithofacies thicknesses are well represented by a theoretical exponential distribution. Results from this analysis show that 16 of the 56 examples can be confidently shown to be exponential, while 28 are very probably not exponential. This indicates that stochastic Poisson processes are a plausible explanation for many carbonate successions, but they do not explain all of those tested here, suggesting that other non-Poisson processes, either stochastic or deterministic in nature, or both, must also be important. Thus lithofacies planform geometries, and the processes controlling vertical stacking in ancient carbonate platform top deposystems, were likely more diverse than has been suggested, requiring significant further quantitative analysis and numerical forward modeling to properly understand.

Key Words: carbonate • facies • bed thickness • exponential distribution


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
DATA AND ANALYSIS
 DISCUSSION
 CONCLUSIONS
 REFERENCES CITED
 
The nature of carbonate lithofacies thickness distributions is of fundamental importance to understanding carbonate deposystems because they record evidence for both the lateral distribution and migration of facies elements, and the various other intrinsic and extrinsic controls on carbonate strata, such as variations in production rate and oscillations in relative sea level. This information is critical to our understanding of depositional processes in shallow-water carbonate strata, both in terms of understanding their controlling processes, and also because it potentially provides a link between vertical and lateral extent of carbonate lithofacies elements. Previous analysis of Paleozoic and Neoproterozoic lithofacies thicknesses has suggested that many platformal carbonate outcrop examples exhibit an exponential distribution of lithofacies thicknesses (Wilkinson et al., 1997, 1999) with many thin units, and unit frequency decreasing at a particular rate as thickness increases. This is significant because exponential distributions can be interpreted to indicate operation of a stochastic phenomenon known as a Poisson process, which has some important implications for predicting how horizontal lithofacies distributions translate into vertical lithofacies distributions, and vice versa.

In a Poisson process, waiting time for an event to occur is random, or memoryless, so that the waiting time for the next event to occur does not depend at all on previous waiting times, nor does it have any influence on the waiting time for any subsequent events. In terms of lithofacies thicknesses in platformal carbonate strata, a Poisson process may lead to random accumulation of lithologies, where deposition of a particular lithology persists for a certain time and then is terminated randomly by onset of deposition of a new lithology. Lithofacies thickness would depend on duration of deposition, which in turn would depend on the waiting time for onset of deposition of a new lithology. Waiting time would depend on the size, geometry, and rate of migration of lithofacies. Thus a mosaic of different lithology elements of random size, or perhaps a mosaic of similar-sized elements but with different migration rates, could stack vertically to generate the observed exponential lithofacies thickness distributions (Wilkinson and Drummond, 2004). The term mosaic can be defined tentatively in this context as a planform arrangement of multiple different lithological elements, lacking simple trends in spatial arrangement, but showing some statistical relationship between element size and frequency of occurrence (e.g., Wright and Burgess, 2005).

The Poisson model is especially important when compared to many sequence stratigraphic models of carbonate strata because it offers a radically different interpretation of ancient deposystems. Sequence stratigraphic models have tended to invoke simple, ordered, and therefore predictable patterns of lateral migration and stacking, all driven by periodic oscillations in some external forcing mechanism, such as relative sea level (e.g., Lehrmann and Goldhammer, 1999). In contrast, the Poisson process model invokes a more complicated, indistinguishable from random, and essentially memoryless process of stacking facies mosaic elements in which external forcing mechanisms operate and may confer some order, but only on a large multidecameter scale (Wilkinson et al., 1997, 1999; Wilkinson and Drummond, 2004). However disquieting the latter model may be to geologists who want to make simple deterministic predictions, the Poisson facies mosaic model seems to have the advantage that it identifies and explains examples of carbonate strata with exponentially distributed lithofacies thicknesses, which sequence stratigraphic interpretations generally have failed to do. However, is the assertion that Poisson processes are ubiquitous in shallow-water carbonate deposystems supported by sufficient lithofacies thickness evidence? Are exponential lithofacies thickness distributions predominant in ancient carbonate strata? The purpose of this paper is to demonstrate that a more diverse range of processes may be required to explain observed lithofacies thickness distributions.


   DATA AND ANALYSIS
TOP
 ABSTRACT
 INTRODUCTION
 DATA AND ANALYSIS
DISCUSSION
 CONCLUSIONS
 REFERENCES CITED
 
Lehrmann and Goldhammer (1999) used a database of 56 outcrop and cored well examples from dominantly shallow-water platform-top strata, ranging in age from Proterozoic to Neogene, to examine secular variations in parasequence and high-frequency sequence development. The database also includes data on lithofacies thickness for each outcrop and well section. Examples of database lithofacies classes are mud-cracked cryptalgal laminate, oolitic and/or skeletal grainstone, burrowed peloidal wackestone, lime mudstone, and laminated grainstone, illustrating that the classification schemes used are not overly interpretive, and therefore probably suitable for reasonably objective application. Note that they are also similar to those used in previous similar studies (e.g., Wilkinson et al., 1999), and that where environmental interpretations are made, they are usually combined with sufficient lithological information to permit some objective assessment of the interpretation.

These lithofacies data have been used here to test for incidence of exponential thickness distributions. Testing is carried out using the Kolmogorov-Smirnov (K-S) statistical test for comparing distributions (Press et al., 1992). Outcrop thickness data are plotted as a normalized cumulative frequency distribution. The comparable theoretical cumulative exponential distribution F(t) is then calculated using the same total thickness and number of lithofacies units, so


Formula 01

where t is lithofacies thickness, N is the number of lithofacies units in the section being considered, and L is the total section thickness. The observed and theoretical curves are plotted on the same normalized axis, and the maximum offset or difference D between the two distributions can then be calculated (Fig. 1). This value D forms the basis for the K-S test of the null hypothesis that the distribution being investigated is indistinguishable from an exponential distribution. The significance probability p of an observed value of difference D is calculated via


Formula 02

where


Formula 03

and the function parameter {lambda} is given by


Formula 04


Figure 01
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  		Figure 1. Cumulative frequency plots. A: Cambrian–Ordovician strata from measured section in Nopah Range, Utah, USA (section 56; see the GSA Data Repository [see footnote 1]). B: Cretaceous Aptian–Albian strata logged at Sierra de el Abra, San Louis Potosi, Mexico (section 11; GSA Data Repository). Both show sampled cumulative normalized frequency distribution plotted against theoretical exponential cumulative normalized frequency distribution calculated for the same thickness L and number of lithofacies units n. In each case, maximum difference between two distributions, D, is marked by vertical line terminated by two squares. Distributions in B are broadly similar, leading to a lower value of D relative to example in A, where the curves show a marked difference in shape. Significance level p, calculated from the Kolmogorov–Smirnov test, is also shown in each case, along with µ, the sample mean lithofacies unit thickness.

 
This significance probability p is the probability that values of D at least as extreme as that observed would occur just by chance sample variation if the distribution was an exponential. Hence values of p close to zero indicate that the observed distribution is highly unlikely to be exponential, equivalent to rejecting the null hypothesis at a >99% significance level. Values of p ≥0.10 provide insufficient evidence to reasonably reject an exponential interpretation; in these cases an exponential distribution can be considered a good model to represent the observed thickness data. For values of p between 0.1 and 0.01, interpretation is more difficult, depending on what significance value is chosen, so these cases are considered here as indeterminate. Note also that in cases where the sample size is large, the K-S test can return a significantly low value of p even for relatively small departures from an exponential curve, since such departures are highly unlikely to have occurred by chance sampling.

Details of the outcrop examples used and the results of the statistical analysis are given in the GSA Data Repository.1 Figure 2A is a frequency plot showing counts of the outcrop examples that fall into each interpretation category according to the calculated p values. It is clear from Figure 2A that 28 of the 56, or half of the lithofacies thickness distributions in this data set, where p ≤0.01, very probably do not have exponential lithofacies thickness distributions. It is also clear that 16 of the examples, with p ≥0.1, are well represented by an exponential model. The nature of the remaining 12 is more uncertain; the K-S test is not able to show that they are not exponential to a high enough level of significance to support confident interpretation.


Figure 02
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  		Figure 2. A: Frequency of Kolmogorov–Smirnov test significance probability p values, categorized by degree to which they refute the null hypothesis that sample lithofacies thickness distribution is indistinguishable from an exponential distribution. Plot shows that just more than half of the 56 outcrop sections deviate markedly from an exponential lithofacies thickness distribution, that 16 of the examples are well matched by an exponential model, and that the nature of the remaining 12 is more uncertain. B: Frequency plot of the non-exponential cases classified according to how observed curve differs from an exponential. Type 1 has relatively few thin and intermediate thickness lithofacies units, and too many thick units. Type 2 has too few thin units, and too many intermediate and thick units. Type 3 has too many thin units, and too few thick. Type 4, of which there is only one case, has too few thin and thick units, and too many intermediate thickness units.

 
Among the nonexponential lithofacies distributions, certain patterns are present in the way they deviate from a theoretical exponential. Four types of distribution can be identified, and their frequencies are shown in Figure 2B. Type 1 curves have too few thin and intermediate beds, and too many thick beds (Fig. 3A). Type 2 curves have too few thin and too many intermediate and thick beds (Fig. 3B). Type 3 curves have too many thin beds and too few thick beds (Fig. 3C). Type 4 has too many intermediate, and too few thin and thick beds, but only one example of this type occurs in the data set (Fig. 2B). It is not obvious at this point how to interpret these nonexponential distributions. Further work is required to understand what process might have created them.


Figure 03
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  		Figure 3. Examples of Type 1, Type 2, and Type 3 observed lithofacies unit thickness distributions. A: Type 1 curves have too few thin and intermediate units, and too many thick units. B: Type 2 curves have too few thin and too many intermediate and thick units. C: Type 3 curves have too many thin units and too few thick units. Lithofacies thicknesses are plotted as cumulative frequency curves, with a matching theoretical exponential distribution in each case, and the point of maximum difference D for each marked with two squares joined by a vertical line. See Figure 1 for explanation of variables.

 

   DISCUSSION
TOP
 ABSTRACT
 INTRODUCTION
 DATA AND ANALYSIS
 DISCUSSION
CONCLUSIONS
 REFERENCES CITED
 
The results show that 16 of the 56 successions considered here can be reliably reproduced with exponential lithofacies thickness distributions, but 28 of the 56 show thickness distributions that are probably not best modeled as expo nential. This supports the suggestion of Wilkinson et al. (1999) that Poisson processes are important in explaining shallow-water carbonate deposystems, but also suggests that other processes, either stochastic or deterministic or both, may be required to explain the more diverse range of distribution types observed. This in turn suggests a number of additional questions that need to be addressed.

How objectively can lithofacies be determined when observing core or outcrop, and therefore how accurately can a succession be logged? Clearly this will affect the accuracy of measurement of the lithofacies distribution, and may hinder accurate classification. Significant factors here include the difficulty of recognizing and accurately measuring thin units, grouping of thin units to form thicker measured units, and the problems of delineating lithofacies when transitions are subtle or gradational, or when subtly different lithofacies are grouped. It is possible that variation in measurement method has some influence on the sampled bed thickness distributions, for example via grouping of thin lithofacies units, and this may contribute to apparent divergence from an ideal exponential distribution.

Scrutiny of the results suggests some dependence on lithofacies thickness. Two-thirds of the maximum discrepancies between observed lithofacies thickness frequency and theoretically predicted frequencies occur for unit thickness of <0.5 m. Of these, 80% have an observed frequency less than the predicted theoretical frequency. This apparent pattern in the discrepancy between the observed and theoretical exponential may be interpreted in at least two ways. Either the effect is real, and an exponential model consistently tends to predict too many thin lithofacies units, or the observed frequencies may reflect measuring bias that tends to group thin lithofacies units. Suitable data to resolve this issue are not available. This issue should be kept in mind when interpreting these results, and requires future work to address.

Accepting that these results robustly indicate that lithofacies thickness data sets are a mixture of exponential and nonexponential distributions, pending further testing, two things are required. First, one or more process explanations are needed for the nonexponential distributions. Second, it is necessary to consider if a stochastic Poisson process is the only mechanism that can generate the observed exponential distributions. Note that Wilkinson et al. (1999, p. 346) made this clear when they stated in reference to the Poisson model that "other scenarios of peritidal accumulation might result in equally attractive models" to explain the exponential distributions.

Scale is an important issue in derivation of any depositional model, and especially so in the case of carbonate facies mosaic models (Wright and Burgess, 2005). As stated previously, a mosaic of elements of random size, or perhaps a mosaic of similar elements but with different migration rates, could operate via a Poisson process to stack vertically and generate exponential lithofacies thickness. Wilkinson and Drummond (2004) showed two examples, one in the Persian Gulf, where they claim a mosaic exists over a depositional strike distance of ~1000 km, and one developed around the island of Antigua that is much smaller, developed over a scale of a few tens of kilometers. It is relatively easy to envisage how mosaic elements in the Antigua and Florida case could migrate and stack randomly, but more difficult to envisage how this would work across the complete Persian Gulf mosaic, where migration distances would be much larger, and bathymetry variations in both a dip and a strike direction might inhibit free migration of depth-sensitive facies (Purkis et al., 2005) across the entire area. Thus scale of mosaic development seems likely to be an issue in determining what kind of stratal record is preserved.

Over smaller areas, the nature of the lithofacies thickness distribution will presumably depend, among many other potential controls, on some ratio between lateral migration rate and vertical accumulation rate (Wilkinson et al., 1999). In other words, when facies lateral migration rates are rapid relative to the rate at which strata accumulate, we might expect to observe a higher frequency of thin units. In this case thicker units will be rare, perhaps tending to favor an exponential distribution. In the data set described here the opposite pattern seems to be observed in the majority of the distributions; thin units tend to be less frequent than predicted by a theoretical exponential. If this is a real effect and not an artifact of lithofacies grouping, it perhaps suggests relatively slow migration creating more persistent lithofacies units. The nature of the lithofacies distribution may also depend on stratigraphic completeness and the degree of reworking. Strata with much missing time may well appear more memoryless, showing less of a link between successive units, and successions deposited under slow rates of accommodation creation may be extensively reworked, creating palimpsest strata.

It is important to consider what effect external forcing and nonstationarity, or changes in the parameters of a Poisson process, might have on lithofacies thicknesses. Wilkinson et al. (1999) showed how low-frequency departures from homogeneous Poisson processes can be detected in carbonate strata at scales of tens to hundreds of meters. This kind of nonstationary behavior may be due to external forcing, such as changes in subsidence and accumulation rates, or changes in the frequency and amplitude of relative sea-level oscillations. Any depositional model trying to account for observed litho facies thickness distributions should include these kinds of forcing effects.

Taken together, the above points suggest that there is a requirement for new quantitative depositional models, building on recent work (e.g., Wilkinson and Drummond, 2004) to explain both how these nonexponential distributions might have come about, and also to demonstrate the range of Poisson and non-Poisson processes that might explain the exponential cases. While it is straightforward to loosely define models that might be assumed to account for observed lithofacies thickness distributions, proving the predicted behavior of the model and testing the model product against observation require a more rigorous, quantitative approach. One possible avenue of investigation is to construct numerical forward models of carbonate deposystems, both stochastic and deterministic, or combinations of both, noting also that the boundary between the two may be blurred (Burgess, 2006). These models can then be used to further explore this issue by representing different lithofacies planform geometry, migration styles, and types of external forcing, to determine what types of thickness distributions result, and compare these with the lithofacies distribution types described here. Much work remains to be done to fully understand both how we measure lithofacies thickness in carbonate successions, and what those thicknesses mean in terms of depositional processes and lateral geometries.


   CONCLUSIONS
TOP
 ABSTRACT
 INTRODUCTION
 DATA AND ANALYSIS
 DISCUSSION
 CONCLUSIONS
REFERENCES CITED
 
Previous workers have suggested that stochastic lithofacies mosaics were ubiquitous in ancient carbonate deposystems, and the results presented here show that 16 of the 56 outcrop and core successions analyzed can reliably be considered to show the required exponential vertical lithofacies thickness distributions. This demonstrates that Poisson processes are an important element in explaining shallow-water carbonate deposystems. However, half of the observed distributions show systematic deviations from an exponential distribution, and these may require a different, as yet unknown, process explanation. Further quantitative analysis and numerical forward modeling is required to determine the range of horizontal lithofacies distributions and stacking processes, ranging from deterministic to stochastic, that can explain the observed vertical lithofacies thickness distributions.


   ACKNOWLEDGMENTS
 
Special thanks are due to David Emery for help with implementation of the Kolmogorov–Smirnov test, and interpretation of the results; he was always an inspiration and his contributions will be sadly missed. Dan Lehrmann kindly provided the lithofacies thickness data used in this study, and David Pollitt provided additional data on the Honaker Trail section. I thank Bruce Wilkinson, Paul Wright, Noel James, and Gene Rankey for prompting this work; Gene, Paul, and David Pollitt for reading draft versions; and Dan Lehrmann, Dan Bosence, and Bernhard Riegl for careful reviews. I am grateful to all these learned gentlemen for much fruitful discussion.


   FOOTNOTES
 
GSA Data Repository item 2008058, details of the outcrop examples and the results of the statistical tests, is available online at www.geosociety.org/pubs/ft2008.htm, or on request from editing{at}geosociety.org or Documents Secretary, GSA, P.O. Box 9140, Boulder, CO 80301, USA. Back


   REFERENCES CITED
TOP
 ABSTRACT
 INTRODUCTION
 DATA AND ANALYSIS
 DISCUSSION
 CONCLUSIONS
 REFERENCES CITED
 

Burgess, P.M., 2006, The signal and the noise: Forward modeling of allocyclic and autocyclic processes influencing peritidal carbonate stacking patterns: Journal of Sedimentary Research, v. 76 pp. 962-977.[Abstract/Free Full Text][CrossRef][ISI][GeoRef]

Lehrmann, D.J., and Goldhammer, R.K., 1999, Secular variation in parasequence and facies stacking patterns of platform carbonates: A guide to application of stacking pattern analysis in strata of diverse ages and settings: in Harris, P.M., et al., eds., Advances in carbonate sequence stratigraphy: Applications to reservoirs, outcrops, and models: Society for Sedimentary Geology Special Publication 63, pp. 187-225.

Press, W.H., Teukolsky, S.A., Vetterling, W.T., and Flannery, B.P., 1992, Numerical recipes in C: The art of scientific computing (second edition): Cambridge, Cambridge University Pressp. 994 p.

Purkis, S.J., Riegl, B.M., and Andrefouet, S., 2005, Remote sensing of geomorphology and facies patterns on a modern carbonate ramp (Arabian Gulf, Dubai, U.A.E): Journal of Sedimentary Research, v. 75 pp. 861-876 doi: 10.2110/jsr.2005.067.[Abstract/Free Full Text][CrossRef][ISI][GeoRef]

Wilkinson, B.H., and Drummond, C.N., 2004, Facies mosaics across the Persian Gulf and around Antigua—Stochastic and deterministic products of shallow water sediment accumulation: Journal of Sedimentary Research, v. 74 pp. 513-526.[Abstract/Free Full Text][ISI][GeoRef]

Wilkinson, B.H., Drummond, C.N., Rothman, E.D., and Diedrich, N.W., 1997, Stratal order in peritidal carbonate sequences: Journal of Sedimentary Research, v. 67 pp. 1068-1082.[Abstract/Free Full Text][ISI][GeoRef]

Wilkinson, B.H., Drummond, C.N., Diedrich, N.W., and Rothman, E.D., 1999, Poisson processes of carbonate accumulation on Paleozoic and Holocene platforms: Journal of Sedimentary Research, v. 69 pp. 338-350.[Abstract/Free Full Text][ISI][GeoRef]

Wright, V.P., and Burgess, P.M., 2005, The carbonate factory continuum, facies mosaics and microfacies: An appraisal of some of the key concepts underpinning carbonate sedimentology: Facies, v. 51 pp. 17-23 doi: 10.1007/s10347–005–0049–6.[Medline]

Received for publication 3 August 2007

Revised manuscript received 9 November 2007

Manuscript accepted 10 November 2007





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