8/14/2019 Dexter et al 2009

1/13

[The Journal of Geology, 2009, volume 117, p. 349361] 2009 by The University of Chicago.All rights reserved. 0022-1376/2009/11704-0001$15.00. DOI: 10.1086/599021

349

ARTICLES

Distinguishing Milankovitch-Driven Processes in the Rock Recordfrom Stochasticity Using Computer-Simulated Stratigraphy

Troy A. Dexter, Micha Kowalewski, and J. Fred Read

Department of Geosciences, Virginia Polytechnic Institute and State University,4044 Derring Hall, Blacksburg, Virginia 24061, U.S.A.

(e-mail: tdexter@vt.edu)

A B S T R A C T

Repetitive patterns of facies recurrence are frequently reported from the shallow-water sedimentary rock record andare postulated to have been driven by orbital forcing on eustatic sea level.Consequently, multiple statistical techniqueshave been developed to evaluate whether patterns of stratigraphic succession are more consistent with a periodic sealevel signal or are stochastic. Previous studies focused on development and/or application of such methods to testempirical geological records. However, the character of such records cannot be known a priori, as deposition, erosion,and preservation inuence the resultant composition, nor can those records be readily manipulated to explore thesensitivity, robustness, and overall validity of statistical methods. Here we simulate carbonate layers using computer-modeled successions generated by periodic sea level changes. The resulting stratigraphic records were then evaluatedstatistically. Thickness distributions of simulated lithofacies were compared to distributions predicted for Poissonprocesses, which by denition are not driven by cyclical sea level changes. Our results suggest that periodic processesproduce stratigraphic thickness frequencies that are difcult to distinguish from random frequencies except underhigh-magnitude sea level uctuations. Similarly, autocorrelation fails to correctly recognize cyclic patterns in suchsimulated records. Models with high-magnitude sea level uctuations (icehouse conditions) had thickness frequenciesthat are suggestive of orbital forcing, whereas low-magnitude sea level uctuations (greenhouse conditions) appearedindependent even though they were modeled using Milankovitch orbital forcing. The increasing evidence based onspectral data from real rock successions suggests that Milankovitch drivers are common in both icehouse and green-house periods. Because statistical approaches are unsuccessful in recognizing the cyclic driver of these simulatedrecords, we infer that it is difcult to disprove independence from real stratigraphy even when orbital forcing iscontrolling the rock composition. Even in the necessarily simplied world of computer simulations, the numerousfactors involved in depositing stratigraphic successions work to complicate or mask anyperiodic signal, thus generatingthe appearance of stochasticity in some successions.

Introduction

The idea that changes in earths orbital parameterscould affect not only climate change but also thesuccession of sedimentary facies dates back to themidnineteenth century (Adhemar 1842; Croll1864, 1867). Nearly a century later, Milankovitch(1941) rened the concept of how orbital forcingaffected climate. Sedimentary rock layers are oftenbelieved to be deposited in a manner suggestive ofperiodic changes in sea level or lake level (Fischer1964; Van Houten 1964; Goodwin and Anderson1985; Grotzinger 1986; Goldhammer et al. 1987;

Manuscript received June 6, 2008; accepted February 2, 2009.

Elrick and Read 1991; Balog et al. 1997; Meyers2008). Deeper-water pelagic deposits have also beenused to track climatic changes thought to be as-sociated with Milankovitch orbital forcing(Broecker et al. 1968; Fischer et al. 1991; Sagemanet al. 1997). A number of methods (e.g., autocor-relation, average spectral mist, Fischer plots, fre-quency modulation analysis, layer thickness in-ventory plots, Markov chain analysis, thicknessfrequency distribution) have been developed for ex-amining the presence or absence of a cyclic driverin the rock record (e.g., Wilkinson et al. 1996; Hin-nov and Park 1998; Hinnov 2000; Bailey and Smith

8/14/2019 Dexter et al 2009

2/13

350 T . A . D E X T E R E T A L .

2005, 2008; Meyers 2008). Much debate remains asto whether periodic signals are preserved in theshallow-water sedimentary record of carbonateplatforms (Algeo and Wilkinson 1988; Wilkinson etal. 1996, 2003; Diedrich and Wilkinson 1999). On

the other hand, orbitally derived cycles have beenused as a means to improve the resolution of thegeologic timescale (Hinnov and Ogg 2007).

If the rock record is dependent on orbital forcing,then a cyclic, repetitive pattern should be incor-porated into the stratigraphy. Otherwise, the recordmust be considered independent of orbital forcing.Some investigators have suggested that such in-dependence should produce a frequency distribu-tion that conforms to a Poisson distribution andthat correlation of real stratigraphic lithofacies tothis theoretical distribution then determines theprobability that the composition of facies succes-sion cannot be distinguished from independence(Diedrich and Wilkinson 1999; Wilkinson et al.2003). Autocorrelation has also been used as ameans to demonstrate order in the lithofaciesthicknesses that would be expected when a cyclicdriver is controlling stratigraphic successions. Cor-relating lithofacies thicknesses to a Poisson distri-bution or using autocorrelation of lithofacies thick-nesses have been tested to a limited extent againstrock layers on shallow platforms that are believedto have been driven by Milankovitch cyclicity.However, the sensitivity of these methods has notbeen evaluated using records a priori known to becyclic.

Here we employ simulated stratigraphic succes-sions generated under conditions of periodicallychanging sea level and evaluate whether such rec-ords can be recognized as nonrandom via these sta-tistical approaches. Moreover, by modifying sim-ulation parameters, the performance of statisticalmethods can be evaluated under varying circum-stances. Real rock layers that have been examinedfor a correlation to the Poisson exponential distri-bution have shown that some are condently cor-related, while others are probably not independent

(Burgess 2008). Thus, computer-generated strati-graphic layers not only can evaluate the overall sen-sitivity of the statistical approach, but also assesswhich environmental conditions are required topreserve a recognizable signature of a Milankovitchdriver in the rock record and which conditions tendto appear independent of any periodic driver.

Methodology

The stratigraphic columns were simulated usingthe process-based program PHIL 1.5, basin analysis

software developed by S. Bowman (1989; g. 1). Theprogram creates a two-dimensional platform basedon input variables specied by the user and is de-signed to reveal the processes that control the de-position of real rock records (Goldhammer et al.

1990; Vail et al. 1991; Bowman and Vail 1999). Theprogram was selected because all the input vari-ables can be manipulated to allow any type of sit-uation to be modeled (including unrealistic modelssuch as depth-invariant carbonate production), andit has been tested in the past as a means of mim-icking real rock records (Bowman and Vail 1999;Scheibner et al. 2003). The software allows the userto extract individual stratigraphic columns fromany location along the outcrop (g. 1). The simu-lations are determinant in that each combinationof parameters had only one possible outcome (i.e.,there was no randomness in any simulation thatwould allow for different results from the same in-put parameters). The program models carbonateplatform deposition (input data in table 1) by as-signing cells to the initial platform, which was de-ned on the basis of the models prole width andnumber of cells. It used a driving (tectonic) subsi-dence rate assigned to key cells (and interpolatedto intervening cells) and that is set to increaseacross the platform (table 1). Initial bathymetry isalso specied for key cells and interpolated for theintervening cells (table 1). The initial bathymetrydenes a depositional prole on which sediment isdeposited as the model runs (g. 1). The platformmorphology used in the model was a low-gradientramp, which steepened slightly seaward of a spot140 km offshore; slopes were all below 1 (g. 1).

Calculations of all the variables for each cell aredone for each time slice according to a user-denedtime step to allow modeling of Milankovitch bandsea level uctuations, the smallest of which were19 kiloyears (k.yr.). The model calculates the var-ious parameters in the simulation for each cell dur-ing each time step. As sea level rises and falls, theprogram determines the water depth for each cell(based on both the water level and the amount of

sediment already built up at that location) and in-puts the rate of sedimentation based on the param-eters set for carbonate deposition rate (g. 2). Sim-ulations were run for 800 k.yr. in time steps of 2.5k.yr. The program used a sea level curve denedby periods and magnitude of sea level uctuationassumed to be typical of a greenhouse or icehouseworld, with periods of 19, 23, 40, 100, and 400 k.yr.(table 1). A low-magnitude 800-k.yr. period wasalsoused in all models to create the typical low-stand,transgressive, and high-stand systems tracts (g. 1).Some sea level curves in the model used a saw-

8/14/2019 Dexter et al 2009

3/13

Journal of Geology C O M P U T E R - S I M U L AT E D S T R AT I G R A P H Y 351

Figure 1. Image of the model outcrop, lithologic unit key, sea level curve, and constituent stratigraphic columnsfor the greenhouse model simulated using PHIL 1.5 basin analysis software (Bowman 1989) and modied from thesoftwares output graphics. Slopes on the platforms are highly exaggerated and never exceeded a 1 angle. Lithologies

are determined in the simulation by water depth and bathymetric province (tidal at, lagoon, reef, etc.). The strat-igraphic columns were extracted from the simulation at increasing distance along the platform using the software.The triangles beside the stratigraphic columns are representative of missing time, with larger triangles equivalent toa greater amount of time missing. The sea level curve shows the change in water depth throughout the models entiresimulation run starting at 800 k.yr.

toothed pattern of sea level rise and fall while oth-ers used a sinusoidal pattern (table 1). The saw-toothed pattern splits the period allowing for arapid rise followed by a slower fall in sea level. Thesaw-toothed patterns used 15% of the period set forthe rise with 85% of the period set to fall, simu-

lating the rapid sea level rise from deglaciation fol-lowed by slow sea level drop from glacial buildup.All of the magnitudes and periods in an individualsimulation were summed together to create a sealevel curve that determined sea level position ateach time step.

Sediment (color-coded according to lithology)was added to each cell below sea level using a sed-iment production function (g. 1; table 1). This sed-iment production function was an exponentialfunction centered about a designated water depth(the depth of this center being the so-called width

of the production function, with rate falling off intothe tidal zone and rapidly falling off into deeperwater), and it was additionally dened by maxi-mum production rate and the water depth of max-imum production (g. 2; table 1). The width of thefunction determined where the production curve

reached zero deposition above and below the waterdepth of maximum production, and the depth ofmaximum production for the shelf margin was setat 5 m (Broecker and Takahashi 1966). The maxi-mum sedimentation rate was set at a value similarin magnitude to the subsidence rate to prevent toomuch or too little accumulation throughout thehistory of the simulation. For a subset of simula-tions, in order to compare the effects of rapidlychanging sediment production with depth, someruns were done with carbonate production kept rel-atively constant (with respect to the shallow plat-

8/14/2019 Dexter et al 2009

4/13

T a

b l e 1

.

I n p u t P a r a m e t e r s U s e d t o M o d e l t h e S i m u l a t i o n s i n P H I L 1 . 5 B a s i n A n a

l y s i s S o f t w a r e

M o d e l t y p e

V a r i a b l e m o d e l p a r a m e t e r s

S e a

l e v e

l m a g n i t u d e

( i n m a t n

k . y r . )

M a x .

p e l a g i c

s e d . r a t e

( c m /

k . y r . )

B i o g e n i c p r o d u c t i o n o n

s h e l

f m a r g i n

B i o g e n i c p r o d u c t i o n o n s

h e l f

1 9 a

2 3 a

4 0 1 0 0 a 4 0 0 8 0 0

M a x . s

e d .

r a t e

( c m

/ k . y

r . )

W i d t h o f

f u n c t i o n

( m )

D e p t h o f m a x .

p r o d u c t i o n

( m )

M a x .

s e d .

r a t e

( c m

/ k . y

r . )

W i d t h o f

f u n c t i o n

( m )

D e p t h o f m a x .

p r o d u c t i o n

( m

)

1 9 - k . y

r . g r e e n h o u s e

6

0

1

2

2

1 0

1

5 0

4 0

5

1 5

2 0

2

1 9 - a n

d 2 3 - k . y

r . g r e e n h o u s e

6

6

1

2

2

1 0

1

5 0

4 0

5

1 5

2 0

2

1 0 0 - k . y r .

g r e e n h o u s e

6

6

1

7

2

1 0

1

5 0

4 0

5

1 5

2 0

2

H i g h - m a g n i t u d e g r e e n h o u s e

6

6

1

7

5

1 0

1

5 0

4 0

5

1 5

2 0

2

L o w - m a g n i t u d e g r e e n h o u s e

5

5

1

3

3

1 0

1

5 0

4 0

5

1 5

2 0

2

I c e h o u s e

7

7

1 0

6 0

5

1 0

. 2 5

7 0

4 0

5

3 0

4 0

2

D e p t h - i n v a r i a n t i c e h o u s e

7

7

1 0

6 0

5

1 0

. 2 5

7 0

3 0 0

2 0 0

3 0

3 0 0

2 0 0

D e p t h - i n v a r i a n t g r e e n h o u s e

6

6

1

7

5

1 0

1

5 0

3 0 0

2 0 0

1 5

3 0 0

2 0 0

P r o

l e p a r a m e t e r s

S t a t i c

m o d e l p a r a m e t e r s

V a l u e

I n i t i a l b a t h y m e t r y

S u b s i d e n c e

C a r

b o n a t e s e

d i m e n t

p r o

l e

D i s t a n c e

( k m

)

D e p t h

( m )

D i s t a n c e

( k m

)

R a t e

( c m

/ k . y

r . )

L o c a t i o n

D e g r e e s

T o t a l t i m e s e t t i n g

8 0 0 k . y r .

0

4 5

4 0

0

S a b k h a

. 0 2 3

P r o

l e w i d t h

2 0 0 k m

4 0

2 0

8 0

0

T i d a l

a t

. 1 1 5

N u m

b e r o f c e

l l s

3 2 0

8 0

1 2

1 2 0

1 . 5

L a g o o n

. 0 2 3

T i d a l r a n g e

2 m

1 2 0

1 0

1 6 0

2

B a c k r e e f

. 0 0 6

O f a p

b r e a

k

1 0 m

1 3 0

8

2 0 0

3

F o r e s l o p e

2 . 2 9

O f a p r o

l l o v e r w i d t h

1 0 0 m

1 4 0

5

E f f e c t i v e

l i t h o s p h e r e t h i c k n e s s

2 2 . 6

8 k m

1 5 0

1 0

1 6 0

4 5

1 8 0

1 0 0

2 0 0

1 1 5

N o t e :

T h e v a r i a b

l e m o d e l p a r a m e t e r s w e r e u s e d t o c r e a t e d i s t i n c t s i m u l a t i o n s .

T h e s t a t i c m o d e l p a r a m e t e r s w e r e u n c h a n g e

d

t h r o u g

h o u t a l

l s i m u l a t i o n s

( s e e t e x t f o r

f u r t

h e r p a r a m e t e r e x p l a n a t i o n s

) .

a

P e r i o d s u s e a s a w - t o o t h p a t t e r n s e t a t a 1 5 % : 8 5 % r a t i o .

8/14/2019 Dexter et al 2009

5/13

Journal of Geology C O M P U T E R - S I M U L AT E D S T R AT I G R A P H Y 353

Figure 2. Carbonate production curves modeled by PHIL 1.5 basin analysis software (Bowman 1989). Curves arecalculated by the parameters width of function, depth of maximum production, and maximum sedimentationrate (see table 1). These are the actual functions used by the software in all the simulations. A , Basic carbonateproduction curve dependent on water depth. B, Carbonate production curve modied to be nearly independent ofwater depth for depth-invariant simulations.

form). These depth-invariant models were createdby changing the carbonate production functionwidth to 300 m (compared to 50 m for the othermodels) and the depth of maximum production to200 m (compared to 5 m for the shelf margin pre-viously; g. 2 B). This kept the carbonate produc-tion on the shallow platform relatively constantwith the exception of the very rapid increase insedimentation rate for the tidal to subtidal transi-tion (g. 2 B). Deep-sea pelagic sediment was inputas a rate dependent on thickness of the water col-umn. Subsidence related to sediment and waterloading was calculated using a exural beam modeldependent on the thicknessof the lithosphere (table1). The shape of the carbonate prole of the plat-form was determined by using user-dened gradi-ents for each of the bathymetric provinces (e.g.,tidal ats, lagoon, back reef, slope; table 1). Anysediment deposited above the slope of the stableprole for that particular bathymetric province was

redeposited seaward to the next cell (and so forthuntil reaching a cell whose surface was below thestable slope prole). Missing time is denoted vi-sually in the stratigraphic columns as triangles (seeg. 1) and digitally as time steps with layers of zerothickness. These periods of missing sediment de-position occur whenever the cell wasbrought abovesea level or when the critical slope angle for thestable prole wasexceeded andsediment was trans-ported downslope (table 1). The sediment depositedat each time step was given a lithologic type de-termined by the water depth and the province to

which it was deposited (e.g., algal laminites in thetidal ats, carbonate boundstone in the fore reef,etc.; g. 1).

The model output was in the form of strati-graphic cross sections, from which stratigraphiccolumns from designated cells can be extracted.Data for the stratigraphic columns were takenalong the depositional gradient of the simulatedshelf at 100, 120, and 140 km from the shore. Whenthe model generated a cyclic stratigraphic recordfurther seaward (as was the case in the icehousemodels), data were also taken at 160 km from theshore. The lithofacies thickness data (based on thethickness of an individual lithology, typically con-sisting of several time steps) were extracted fromthese columns. PHIL 1.5 Utility software usedthe model data from the PHIL Analysis softwareto create digital output for each stratigraphic col-umn at every time step, and this was imported intoMicrosoft Ofce Excel 2003. The output recorded

the simulations data for a selected column, in-cluding the age, sediment type (lithology), waterdepth, position of sea level, and the thickness ofeach layer. In the simulations, the thickness of eachlithofacies was measured to generate the data setsfor the statistical analyses from each simulatedstratigraphic column. Lithofacies thicknesses areexpressed in meters and are the sum of sedimentdeposition plus sediment transported into the cellminus subaerial erosion (when applicable). The dig-ital output is arranged in time steps, one or moreoften spanning a single lithofacies. Where a litho-

8/14/2019 Dexter et al 2009

6/13

354 T . A . D E X T E R E T A L .

facies spanned several time steps, the individualthicknesses of each time step were summed to ob-tain the lithologic layer thickness. Similar litho-facies were split only when the unit was separatedby an unconformity (marked by periods of erosion

or breaks in sediment deposition). This allowed theresulting data on lithofacies thicknesses to beequivalent to those measured by geologists in realstratigraphic successions.

Stratigraphic column data were entered into SAS9.1, and all time steps with zero sediment deposi-tion (zero thickness) were deleted, leaving a datamatrix containing only the sediment layers presentin the stratigraphic column. To evaluate the dis-tribution of facies in the simulated records in termsof a periodic driver, the following methods havebeen employed.

1. A theoretical independent distribution ( F)equivalent to a Poisson process was calculated us-ing the lithofacies thickness frequencies of the sim-ulated records according to the following formuladerived by Diedrich and Wilkinson (1999) and Wil-kinson et al. (2003):

2N [ t # (N / L )]F p bin size # # e .L

In this formula, N is the total number of lithologicunits or layers in the simulation, L is the totalthickness of the stratigraphic section, and t is therange of lithofacies thicknesses up to the thickestlayer size in the section. Bin size determines thewidth of each frequency bin and is predened. Thethickness frequencies of the simulated data werethen compared to this theoretical distribution us-ing Pearsons product-moment correlation.

2. An autocorrelation analysis was run on thelithofacies thicknesses for individual simulationsto locate repeated patterns of sedimentation rate.Autocorrelation correlates time series data to itselfin order to test whether patterns are repeated (pro-ducing a sine wave appearance). The expectation ofthe autocorrelation analysis is that cyclically

driven deposition should have repetitive thicknesspatterns in which similar repeated thicknessesshould be highly positively correlated (and dissim-ilar thicknesses that should be highly negativelycorrelated). The preserved lithofacies thicknessesof the simulated successions were analyzed in SASusing an IML (interactive matrix language) codethat reproduced the column and compared it to theoriginal column. The reproduced column was thenshifted a step to align the next layer with the pre-vious one, and the columns were compared again.This procedure was iterated until only ve layers

were left for the comparison (i.e., too few layerswere left for a robust statistical test). The columnswere compared using Pearsons product-momentcorrelation and the Spearman rank correlation, andthe correlation was recorded at each iterative step.

Signicance was evaluated at using thea p 0.05sample size N based on the number of layers undercomparison for each iteration step.

3. Autocorrelation was also conducted on the li-thology of each layer (boundstone, coarse grain-stone, etc.) in an individual simulation with theexpectation that these rock types should be re-peated if the data are periodically produced. ThePHIL Analysis software outputgave the lithologyof each layer (based on bathymetric province andwater depth) as well as numerically coding eachlithology for water depth and location. The nu-meric coding used the likely water depth of thefacies in order to rank the different facies in orderfrom shallowest to deepest as well as by the likelydistance from shore. Autocorrelation was then runin SAS on these numeric values to look for repet-itive patterns. Spearman rank correlation was usedto compare the layer lithologies at each iterativestep, and signicance was evaluated at .a p 0.05The Spearman rank correlation coefcient was theappropriate test because the numeric values usedas a proxy for lithology were not continuous vari-ables but rather ranks based on the likely waterdepth.

Results

Eight simulations were conducted with data col-lected on three extracted stratigraphic columnsfrom each of the stratigraphic simulations. Six sim-ulations were modeled using greenhouse worldconditions, and two were modeled representing ice-house worlds (table 1). Various periods in the green-house simulations were selected to have dominantmagnitudes to determine whether this inuencedthe result; a 19-k.yr. dominant model, a 19- and 23-k.yr. dominant model, and a 100-k.yr. dominant

model. One greenhouse simulation had all periodswith high magnitudes, and one had periods withall low magnitudes (table 1). The icehouse simu-lations had high magnitudes for 40- and 100-k.yr.periods. Two depth-invariant simulations wereconducted, one for a greenhouse world and one foran icehouse world (table 1). All of the results pre-sented herein use bin sizes for the thickness fre-quencies that allow for a minimum of ve lledbins in order to compute the Pearsons product mo-ment correlated to the theoretical distribution. Themaximum number of lled bins for the frequency

8/14/2019 Dexter et al 2009

7/13

Journal of Geology C O M P U T E R - S I M U L AT E D S T R AT I G R A P H Y 355

Figure 3. Simulated carbonate stratigraphic thickness frequencies compared to a theoretical distribution of inde-pendent thicknesses. The dark lines are the expected distribution of thicknesses if they are independent of extrabasinalforcing, and the points are the actual layer thickness frequencies of the simulated columns. All graphs were composedfrom stratigraphic columns taken at 120 km into the basin except the icehouse simulation, which was taken at 140km. All graphs were calculated using a bin size of 0.35 m. Note that all graphs are highly correlated to the theoreticaldistribution except the icehouse simulation.

calculation of a simulated column was 22, and theaverage number of lled bins for all the simulationswas nine.

All the greenhouse simulations were signi-cantly correlated to values predicted by the theo-

retical distribution for independence; in otherwords, they appeared random (g. 3; table 2). Thelocation of the simulated stratigraphic columns rel-ative to the shoreline had no effect on the green-house simulations, and all columns were signi-cantly correlated to the theoretical distribution forindependence or randomness (table 2). In mostcases, the icehouse simulations were not signi-cantly correlated to a Poisson distribution andcould not be considered independent from periodicdrivers (g. 3; table 2). The distance from the shore-line appeared to have a discernible effect on theoutcome of the icehouse simulations, where the

more offshore, rapidly subsiding areas (160 km dis-tance onto the platform) correlate to the distribu-tion expected for independence (i.e., they appearrandom).

One of the issues with carbonate facies that may

act to mask cyclicity from periodic forcing is thedependence of depositional rate on water depth. Inorder to explore the affect of depth dependence oncarbonate depositional rates, simulations were runholding carbonate deposition relatively invariant towater depth (g. 2 B). When carbonate depositionwas nearly depth invariant, the greenhouse simu-lation remained relatively unchanged and againhighly correlated to the theoretical independentdistribution mimicking randomness (table 2). How-ever, the depth-invariant icehouse simulation ap-peared even less correlated to the independent dis-tribution than the depth-variant icehouse simu-

8/14/2019 Dexter et al 2009

8/13

356 T . A . D E X T E R E T A L .

Table 2. Correlation of the Simulations to a Theoretical Distribution Indicative of Independence from Cyclicity

Model type, distance (km)

Bin size selected for calculating distribution (m)

.10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60

19-k.yr. greenhouse:100 .6015 .7072 .6980 .8626 .9023 .9747 .9718 .9732 .9811 N/A N/A120 .7278 .7373 .8879 .8920 .9352 .9290 .9254 .9917 .9542 N/A N/A140 .6153 .7281 .8401 .9155 .9154 .9057 .9691 .9799 .9646 N/A N/A

19- and 23-k.yr. greenhouse:100 .7680 .7853 .8242 .8417 .8587 .9067 .8919 .9370 .9199 N/A N/A120 .7889 .8498 .8688 .8498 .8778 .9135 .9025 .9405 .9090 N/A N/A140 .7054 .8139 .8778 .9053 .9003 .8866 .9681 .9605 .9709 N/A N/A

100-k.yr. greenhouse:100 .5982 .7263 .7125 .8200 .8313 .8181 .8019 .8513 .7883 N/A N/A120 .6191 .7148 .7130 .7128 .7855 .7476 .7709 .7707 .7867 N/A N/A140 .6880 .8407 .8827 .9063 .9072 .9450 .9653 .9510 .9494 N/A N/A

High-magnitude greenhouse:100 .5111 .5912 .6800 .7312 .7014 .8628 .7687 .7357 .8268 N/A N/A120 .6699 .7731 .7855 .8320 .8592 .8060 .9052 .8889 .9428 N/A N/A140 .8192 .8540 .8975 .8036 .9056 .8702 .9416 .8347 .9382 N/A N/A

Low-magnitude greenhouse:

100 .6661 .8159 .8104 .8790 .8228 .8718 .8809 .8902 .9384 N/A N/A120 .7743 .8219 .8449 .8613 .9123 .8403 .9183 .8983 .9596 N/A N/A140 .7843 .8516 .9392 .8973 .9698 .8880 .9508 .9807 .9943 N/A N/A

Depth-invariant greenhouse:100 .7254 .7529 .8640 .8717 .8952 .9015 .8864 .8872 .8738 N/A N/A120 .7297 .7516 .7898 .8244 .8978 .8517 .8854 .9208 .8866 N/A N/A140 .6838 .6725 .6612 .6463 .7275 .8005 .8005 .8050 .8701 N/A N/A

Icehouse:120 N/A N/A .5394 .4976 .5425 .5681 .5732 .8171 .6899 .8402 .8573140 N/A N/A .4173 .6458 .5550 .6557 .6060 .7779 .8397 .8098 .9067160 N/A N/A .6782 .5151 .6828 .7235 .7716 .5353 .5616 .6856 .6444

Depth-invariant icehouse:120 N/A N/A .4303 .6213 .6165 .6790 .7647 .7567 .8320 .7621 .8070140 N/A N/A .4387 .9459 .6863 .6326 .5649 .8183 .9316 .8496 .9491160 N/A N/A .4016 .4371 .5364 .5082 .5599 .6779 .8031 .8451 .7992

Note: Boldface indicates value is not signicantly correlated to a stochastic distribution at . N/A indicates samples wherea p 0.05too few bins were present to properly assess the statistical signicance (i.e., either fewer than ve lled bins total or a large numberof empty bins in between the lled bin).

lation, suggesting the involvement of a periodicdriver (table 2). The depth-invariant icehouse sim-ulation was not correlated at greater water depths(160 km out onto the platform) unlike the depth-dependent icehouse simulation.

One of the previously underexplored effects onthe outcome of the test was the effect of the some-what arbitrary choice of bin size used to compute

thickness frequencies for the comparison to thecorresponding Poisson distribution for that binsize.In the case of the data simulated here, altering thebin size resulted in substantial changes in Pearsonsproduct-moment correlation ( r) used to measure thet of the simulated data to theoretical independentdistribution. Namely, varying bin size within a rea-sonable range (note that at least ve frequency binswith data were used as an acceptable minimumhere) produced a range of outcomes where r for thesame simulated data with different bin sizechanged by as much as 0.66 (g. 4). Consequently,

the change of bin size can change the statisticaldecision from signicant to insignicant, and viceversa. When a range of bin sizes (0.10.6 m) wasused for the greenhouse simulation, all values weresignicantly correlated with the theoretical inde-pendent distribution (g. 4). When this range wasused for the icehouse simulation, ve selected binsizes were not signicantly correlated to the in-

dependent distribution (g. 4).Autocorrelation was conducted on the layerthicknesses of each simulation to evaluate whetherrepeated patterns in sedimentation rate existed (g.5 A5F). Cyclically repeated data analyzed by au-tocorrelation should possess high positive and highnegative correlations (as similar and dissimilar pat-terns are compared), producing an output with asine wave appearance. When autocorrelation wasconducted on the layer thicknesses, it was difcultto distinguish any pattern that would be distinctfrom noise (g. 5 A5F). Only a small number of

8/14/2019 Dexter et al 2009

9/13

Journal of Geology C O M P U T E R - S I M U L AT E D S T R AT I G R A P H Y 357

Figure 4. Effect of bin size on the calculated Pearsonscorrelation for the 19-k.yr. greenhouse simulation andicehousesimulation. The X -axis is the bin sizes that were selected for the formula in order to calculate the theoreticalindependent distributions and compare them to the simulated frequency thicknesses. The greenhouse simulationwas calculated using a stratigraphic column at 100 km into the basin, while the icehouse simulation was from 120km. By altering the bin size in the formula, the Pearsons correlation ranges from 0.5678 to 0.9811 for the greenhousesimulation and from 0.1940 to 0.8573 for the icehouse simulation. All samples had at least ve lled frequency binsin order to run the correlation statistic. All values for the greenhouse simulation were signicantly correlated to thetheoretical distribution. For the icehouse simulation, 12 out of 30 were signicantly correlated to the theoreticaldistribution. As the formula bin size was increased, the number of lled frequency bins decreased, which broughtabout an increase in the statistical correlation.

peaks could be considered signicantly correlated

at an a of 0.05, with the 19- and 23-k.yr. greenhousesimulation having the highest number of signi-cant peaks (g. 5 A5F). When sedimentation ratewas kept depth invariant, there appeared to be littledifference in the results from the autocorrelationanalysis, and few peaks reached statistically sig-nicant correlation values (g. 5 B, 5F). When au-tocorrelation was conducted on the lithology ofeach unit (ranked numerically on the basis of prob-able water depth rank and distance from shore), thepattern produced was no different from that mea-sured using layer thicknesses (g. 5 G vs. 5 J ). Theoutcomes of depth-invariant simulations also re-mained unchanged when lithology (water depthrank) was used instead of lithologic thickness (g.5H vs. 5 J ).

Discussion

These stratigraphic simulations indicate thatthicknesses of lithofacies units produced synthet-ically to simulate periodic change in sea level cre-ate a pattern that was difcult to distinguish fromlithologic successions due to randomchanges when

using the thickness frequencies and autocorrela-

tion methods. The modeling software has beendemonstrated to approximate real stratigraphy, andthe simulations for this project were specicallydesigned to represent actual carbonate successions(Bowman and Vail 1999; Scheibner et al. 2003).Thus, using these tests, actual shallow-water car-bonate successions that accumulated under peri-odic sea level changes may appear to have formedunder the inuence of a nonperiodic driver. Themost apparent difference in the simulated fre-quency distributions exists between greenhouse(low, precessionally dominated magnitudes) andicehouse (high 100-k.yr. magnitude) simulations.Greenhouse simulations were exponentially dis-tributed (which would be expected if they were in-dependent of orbital forcing), whereas icehousesimulations were distinctly different and suggest aperiodic driver. This indicates that when using au-tocorrelation and thickness frequency distribu-tions, the inuence of high-magnitude sea levelchanges in periodically driven carbonate succes-sions is potentially recognizable; this inuence isgreater where rates of carbonate accumulation wereunrelated to water depth. The offshore columns of

8/14/2019 Dexter et al 2009

10/13

8/14/2019 Dexter et al 2009

11/13

Journal of Geology C O M P U T E R - S I M U L AT E D S T R AT I G R A P H Y 359

Figure 5. Autocorrelation analysis of the lithologic unit thicknesses and lithologic type for the stratigraphic columns. AF, Graphs of the autocorrelation on the unit thicknesses using Pearsons product-moment correlation. GJ , Graphsof the autocorrelation for the type of lithology in the stratigraphic column (numerically ranked for water depth andthe distance of the bathymetric province from shore) using the Spearman rank correlation. Graphs AD , G , and Hwere calculated using stratigraphic columns taken at 120 km along the platform. Graphs E, F, I , and J were calculatedusing stratigraphic columns taken at 160 km along the platform. Total sample number is the number of layers beingcompared to one another at each iterative step. The solid line is the Pearsons correlation coefcient or Spearmancorrelation coefcient at each iterative step. The dashed line is the signicance level at a of 0.05 as determined bythe number of layers under comparison at each iterative step.

the icehouse simulations tended to correlate to thePoisson distribution, unlike the columns nearshore. The more rapid subsidence of these locationsprevented subaerial exposure, and they possessedthe widest range in lithofacies thicknesses (from

0.41 to 11.48 m in the 160 km column of theicehouse simulation). This lack of subaerial expo-sure allowed for the preservation of a greater num-ber of thin lithofacies and relatively few thick lith-ofacies, which produced a distribution that had asignicant t to the exponential Poisson distribu-tion. Thus, even with high-magnitude sea leveluctuations, a periodic driver of sediment deposi-tion can be concealed.

These simulations closely resemble real shallow-watercarbonate successions. Resulting exponentialthickness frequencies are similar to those reportedfrom many actual carbonate successions that havebeen interpreted as reecting an absence of uni-modal or polymodal thickness frequencies thatwould result from periodic forcing (e.g., Diedrichand Wilkinson 1999). Given that our simulationsdo indeed incorporate periodic changes in sea level,one might expect that resulting thickness frequen-cies would not be exponential. In all likelihood, thecomplexity of the simulations (e.g., a complex sealevel signal composed of one or more periods andmagnitudes, differential subsidence across theshelf, seaward sediment transport, depth-depen-dent carbonate production rates, and cessation ofaccumulation during exposure) mayserve to distortthe periodic sea level signal, except in those incor-porating the high magnitudes of sea level change.Indeed, simpler models incorporating uniform sub-sidence, one or two cycles of sea level variation,and no sediment transport or erosion yield thick-ness frequencies that are easily differentiated fromexponential Poisson distributions. In addition, thethickness frequency distributions are strongly de-pendent on choice of bin size (g. 4)

This study focuses on whether a Poisson distri-bution of lithofacies thickness frequencies or au-tocorrelation of lithofacies provides a rigorous test

of stochasticity versus periodic sea level drivers incyclic successions. Real stratigraphic successionsappear to have a depositional and preservationalbias that can distort the original cyclic signal pro-duced from orbital forcing. This signal loss is fur-

ther demonstrated by the autocorrelation tests.Fewof the peaks are signicantly correlated above an aof 0.05 for the autocorrelation of layer thicknesses,and the results appear very noisy (g. 5 A5F).

The autocorrelation of lithofacies thicknesses ap-peared to capture a second overlapping frequencyfor the 19- and 23-k.yr. greenhouse simulation thatstarts negative, rises to positive, and falls again tonegative values (g. 5 A). This may represent theoverarching 800-k.yr. period with its depositionalpattern of low-stand, transgressive, and high-standsystems tracts (g. 5 A). This same 800-k.yr. patternwas likely stied by the magnitudes of the otherperiods in the other simulations. For time seriesanalyses, there can be no missing time steps, butin the autocorrelation of stratigraphic units, miss-ing time is an unknown. With simulations, it ispossible to track the missing time steps and to leavethem in the analyses (as zero thicknesses in thesimulations output le). But the unrealistic inclu-sion of zero thickness units in the analyses stillfailed to produce a more signicant pattern.

The autocorrelation of the water-depth-rankedlithology would be expected to have a stronger sig-nal than the thickness of each layer since multipleadditional factors may be involved in controllinglayer thickness. But this analysis also failed to de-tect any pattern distinguishable from noise (g.5G 5J ). The simulations themselves are determi-nant and produced by a periodic driver, so onewould not expect much noise in the autocorrela-tion. However, the lithologic record resulting fromthese simulations was extremely noisy, whichmasked the original cyclic driver. The lack of sig-nicance in the autocorrelation of layer thicknessand lithology seems to indicate that autocorrela-tion is too crude a method to analyze for cyclicdrivers in carbonate successions.

8/14/2019 Dexter et al 2009

12/13

360 T . A . D E X T E R E T A L .

The incompleteness of the carbonate strati-graphic record may act to conceal cyclic drivingforces, in turn making it difcult to assess the qual-ity of methods developed to measure cyclicity;spectral analysis on such computer simulations

also produces a noisy record with numerous peaksin which the main driving frequencies are still rec-ognizable (Balog et al. 1997). The methods for test-ing for the presence or absence of a Milankovitchdriver in ancient successions must demonstratepatterns that are distinct from what would be ex-pected if the rocks were deposited independent oforbital forcing. One of the problems with many ofthe methods for detecting cyclicity is that they testa single series (e.g., a stratigraphic column). Thistends to miss lateral substitution of facies that oc-curs at similar water depths in real settings. Thebenet of using simulations is their ability to cap-ture information such as periods of no depositionor gaps in deposition from erosion that would oth-erwise be difcult to quantify in real successions.These simulations can be applied to test many ofthe other methods used in the past that do appearto nd instances of periodic deposition, includingMarkov chain analysis, Fischer plots, and spectralanalysis.

The method of analyzing lithofacies frequenciesdeveloped by Wilkinson et al. (1996, 2003) andDiedrich and Wilkinson (1999), which suggests thatstratigraphic packages are independent of orbitalforcing, has sparked an enormous amount of re-search in stratigraphy. New methods are continu-ally being developed to test for periodic drivers inthe rock record (Bailey and Smith 2005, 2008; Mey-ers 2008). Such methods include bilogarithmicthickness/frequency plots of layer thickness inven-tories (LTI) of Bailey and Smith (2008) and the av-erage spectral mist (ASM) of Meyers (2008). Themethod using LTI plots posits that stratigraphy in-dependent of orbital forcing should illustrate apower-law relationship and in periodically depos-ited stratigraphy this power law relationship is ab-sent (Bailey and Smith 2008). Average spectral mis-

t is another method that utilizes the nullhypothesis that layers are independent of periodic-ity and quanties the probability of rejecting thishypothesis (demonstrating that real rock packagesoften reject the hypothesis of independence; Mey-ers 2008). Testing for cyclic drivers in the rock rec-ord is of great importance because, if present, thedrivers are a predictive tool in stratigraphy. If the

sedimentary successions are indeed stochasticallyformed, then their predictability is negligible.

Conclusions

The simulations used a cyclic Milankovitch driverto produce cyclic stratigraphy, but the lithofaciesthickness frequencies and autocorrelation methodsused to analyze the resultant rock successionsfound that these records often appeared indepen-dent of periodic orbital forcing. This indicates thatthe factors involved in depositing cyclic sedimen-tary layers, as simulated in the model, tend to maskthe original periodic signal (such as Milankovitchorbital forcing) and produce the appearance of in-dependence or stochasticity. The hypothesis is thatthe rocks are independent of extrabasinal forcing,and these simulations indicate how difcult it isto disprove such independence. Real rock succes-sions are very likely to have been historically morecomplex than our simulations governed by merelya few basic parameters. This poses a challenge toeven most cleverly designed quantitative methodsused to test for stratigraphic patterns, with theirstatistical outcomes being inherently ambiguous:does a given outcome indicate that the record wasnot formed in a cyclic fashion, or does it merelyreect the fact that an original cyclic driver hasbeen masked by the complexity of depositional pro-cesses? It is important, therefore, to have controlsby which these methods can be tested. The use ofsimulations can provide such controls by producingsynthetic data with known Milankovitch cyclicdrivers and thus providing an independent assess-ment of statistical methods applied to test real em-pirical records.

A C K N O W L E D G M E N T S

This work was made possible by support from theByron Cooper Geosciences Graduate Student Re-search Award and the Leonard and Melva HarrisGraduate Student Award. The authors would like

to thank James D. Schiffbauer, Peter J. Voice, andJohn W. Huntley for their comments on early drafts.The authors would also like to thank Bruce Wil-kinson and an anonymous reviewer, whose gen-erous and extensive input greatly improved qualityof this project. The authors also wish to thank ScottBowman, who developed the modeling softwarethat made this project possible.

8/14/2019 Dexter et al 2009

13/13

Journal of Geology C O M P U T E R - S I M U L AT E D S T R AT I G R A P H Y 361

R E F E R E N C E S C I T E D

Adhemar, J. 1842. Revolutions de la mer, deluges period-iques. Paris, Carilian-Goeury & V. Dalmont, 184 p.

Algeo, T. J., and Wilkinson, B. H. 1988. Periodicity ofPhanerozoic sedimentary cycles and the role of Mil-ankovitch orbital modulation. J. Geol. 88:313322.

Bailey, R. J., and Smith, D. G. 2005. Quantitative evidencefor the fractal nature of the stratigraphic record: resultsand implications. Proc. Geol. Assoc. 116:129138.

. 2008. Quantitative tests for stratigraphic cyclic-ity. Geol. J. 43:431446.

Balog, A.; Haas, J.; Read, J. F.; and Coruh, C. 1997. Shal-low marine record of orbitally forced cyclicity in aLate Triassic carbonate platform, Hungary. J. Sedi-ment. Res. 67:661675.

Bowman, S. A. 1989. PHIL user manual. Houston, MarcoPolo Software (PetroDynamics), 97 p.

Bowman, S. A., and Vail, P. R. 1999. Interpreting the stra-tigraphy of the Baltimore Canyon section, offshoreNew Jersey with PHIL, a stratigraphic simulator. InHarbaugh, J. W.; Watney, W. L.; Rankey, E. C.; Slin-gerland, R.; Goldstein, R. H.; and Franseen, E. K., eds.Numerical experiments in stratigraphy: recent ad-vances in stratigraphic and sedimentologic computersimulations. SEPM Spec. Publ. 62:117138.

Broecker, W. S., and Takahashi, T. 1966. Calcium car-bonate precipitation on the Bahama Banks. J. Geo-phys. Res. 71:15751602.

Broecker, W. S.; Thurber, D. L.; Goddard, J.; Ku, T.; Mat-thews, R. K.; and Mesolella, K. J. 1968. Milankovitchhypothesis supported by precise dating of coral reefs

and deep-sea sediments. Science 159:297300.Burgess, P. M. 2008. The nature of shallow-water car-

bonate lithofacies thickness distributions. Geology36:235238.

Croll, J. 1864. On the physical cause of the change ofclimate during geological epochs. Philos. Mag. 28:121137.

. 1867. On the change in the obliquity of the eclip-tic, its inuence on the climate of the polar regionsand on the level of the sea. Philos. Mag. 33:426445.

Diedrich, N. W., and Wilkinson, B. H. 1999. Depositionalcyclicity in the Lower Devonian Helderburg Group ofNew York State. J. Geol. 107:643658.

Elrick, M., and Read, J. F. 1991. Cyclic ramp-to-basincarbonate deposits, lower mississippian, Wyomingand Montana: a combined eld and computer mod-eling study. J. Sediment. Petrol. 61:11941224.

Fischer, A. G. 1964. The Lofer cyclothems of the AlpineTriassic. Kans. Geol. Surv. Bull. 169:107149.

Fischer, A. G.; Herbert, T. D.; Napoleone, G.; Silva, I. P.;and Ripepe, M. 1991. Albian pelagic rhythms (Piob-bico-core). J. Sediment. Petrol. 61:11641172.

Goldhammer, R. K.; Dunn, P. A.; and Hardie, L. A. 1987.High frequency glacioeustatic sea level oscillations

with Milankovitch characteristics recorded in middleTriassic platform carbonates in northern Italy. Am. J.

Sci. 287:853892.. 1990. Depositional cycles, composite sea-levelchanges, cycle stacking patterns, and the hierarchy ofstratigraphic forcing: examples from Alpine Triassicplatform carbonates. Geol. Soc. Am. Bull. 102:535562.

Goodwin, P. W., and Anderson, E. J. 1985. Punctuatedaggradational cycles: a general hypothesis of episodicstratigraphic accumulation. J. Geol. 93:515533.

Grotzinger, J. P. 1986. Cyclicity and paleoenvironmentaldynamics, Rocknest platform, northwest Canada.Geol. Soc. Am. Bull. 97:12081231.

Hinnov, L. A. 2000. New perspectives on orbitally forcedstratigraphy. Annu. Rev. Earth Planet. Sci. 28:419475.

Hinnov, L. A., and Ogg, J. G. 2007. Cyclostratigraphy andthe astronomical time scale. Stratigraphy 4:239251.

Hinnov, L. A., and Park, J. 1998. Detection of astronomicalcycles in the stratigraphic record by frequency modu-lation (FM) analysis. J. Sediment. Res. 68:524539.

Meyers, S. R. 2008. Resolving Milankovitchian contro-versies: the Triassic Latemar Limestone and the Eo-cene Green River Formation. Geology 36:319322.

Milankovitch, M. 1941. Canon of insolation and the Ice-Age problem. Special Publication 132 (Israel Programfor Scientic Translations, Jerusalem, 1969). Belgrade,Serbian Academy of Sciences and Arts, 634 p.

Sageman, B. B.; Rich, J.; Arthur, M. A.; Bircheld, G. E.;and Dean, W. E. 1997. Evidence for Milankovitch pe-

riodicities in Cenomanian-Turonian lithologic andgeochemical cycles, western interior U.S.A. J. Sedi-ment. Res. 67:286302.

Scheibner, C.; Kuss, J.; and Speijer, R. P. 2003. Strati-graphic modelling of carbonate platform-to-basin sed-iments (Maastrichtian to Paleocene) in the EasternDesert, Egypt. Palaeogeogr. Palaeoclimatol. Palaeo-ecol. 200:163185.

Vail, P. R.; Audemard, F.; Bowman, S. A.; Eisner, P. N.;and Perez-Cruz, C. 1991. The stratigraphic signaturesof tectonics, eustacy and sedimentology: an overview.In Cycles and events in stratigraphy. Einsele, G.;Ricken, W.; and Seilacher, A.,eds. Springer, Berlin, 955p.

Van Houten, F. B. 1964. Cyclic lacustrine sedimentation,Upper Triassic Lockatong Formation, central New Jer-sey and adjacent areas. Kans. Geol. Surv. Bull. 169:497531.

Wilkinson, B. H.; Diedrich, N. W.; and Drummond, C.N. 1996. Facies successions in peritidal carbonate se-quences. J. Sediment. Res. 66:10651078.

Wilkinson, B. H.; Merrill, G. K.; and Kivett, S. J. 2003.Stratal order in Pennsylvanian cyclothems. Geol. Soc.Am. Bull. 1115:10681087.