modelling of cyclical stratigraphy using markov

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International Journal o f Min ing and Geological Engineering, 1987, 5, 121-130

SHORT COMMUNICATION

Modelling of cyclical stratigraphy using MarkovchainsSummaryState-of-the-art on modelling of cyclical strat igraphy using first-order Markov chains is reviewed.Shortcomings of the presently available procedures are identified. A procedure which eliminates allthe identified shortcomings is presented. Required statistical tests to perform this modelling aregiven in detail. An example is given to illustrate the presented procedure.

IntroductionProper characterization of subsurface stratification is importan t in many disciplines, such asgeotechnical engineering, petroleum engineering, mining engineering, mineral sciences,hydrology and water resources. Some stratigraphic sections show evidence of cyclical orrecurrent sedimentation. Markov chains have been applied to model such stratigraphicsequences (Krumbein and Dacey, 1969; Ethier, 1975; Ali et aI., 1980). Two parameters, stateand time, are needed to describe a Markov chain. In the application of Markov chains toanalyse stratigraphy, the state parameter is used to identify different lithology types, and thetime parameter is changed to a location parameter in space and is used to record the transitionsamong different lithologies in space. For a stratigraphic sequence to possess the Markovproperty, the lithology type observed at a location in space should depend probabilistically onthe states the sequence occupied at previous locations. At this point it is important to identifythe two extreme models which lie on either side of Markov models: ( t) if the state of the systemat any poin t in space can be predicted with 100% certainty, it is a deterministic model, and (2) ifthe state at any point in space is independent of previous states, it is known as a Poisson process.If the transition of a Markov chain depends only on the immediately preceeding state, the chainis a first order Markov chain. If the transition depends on more than one previous state, then it isa higher order model. If the transition model of a Markov chain does not vary with the spatialparameter, it is called a homogeneous or stationary chain.Homogeneous first order Markov chains have been used in modelling stratigraphy in thevertical direction. The papers by Krumbein and Dacey (1969) and Yu (1984) form the state-of-the-art of modelling stratigraphy in one dimension. Two types of Markov chains have beenemployed in these studies. The first approach considers the stratification at discrete points thatare spaced equally along a vertical profile. The points are numbered consecutively, and the useKeywords: Cyclical stratigraphy; Markov chains; geomathematics;computer modelling.0269-0316/87 $03.00+ 12 1987Chapman and Hall Ltd.


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122 Kulatilakeof the M arko v chain is based on the assumption that the l i thology or s tate at point n dependsupo n the l i thology at the preceeding point (n - I) . Because the same l i thology ma y be observedat successive points , the transi tion matrix that gives the p robabil i ty o f going from o ne l i thologyto another generally has non-zero de me nts on the main diagonal . This type of Ma rkov chain isknow n as a conventional or ordinary Ma rko v chain. If s trat igraphy fol lows a f irs t orderconventional Ma rkov chain, then the thicknesses of l i thologies should fol low geometricdistr ibutions (Krumbein and Dacey, 1969). This important property can be used in test ingwhether a s trat igraphy fol lows a f irs t order conventional M arko v chain.

The second approach considers only the succession of l i thologies, and because eachtransition is to a different lithology within th e system, the diagon al eleme nts are all zero. ThisMarkov chain is known as an embedded Markov chain. In this case, the distr ibutions forlithologic thicknesses need no t follow geom etric distributions. Thu s, som e stratigraphicsequences may be modelled by using an embedded Markov chain to describe the transi t ionsbetween different lithologies, and using different probability distributions to describethicknesses of different lithologies. Such a process is know n as a sem i-M arko v process.Invest igations performed so far on m odell ing strat igraphy in 1-D contain on e or several of thefollowing shortcomings: (a) have assumed homo geneity withou t performing suitable stat ist icaltests to check the applicabil i ty of homogen eity, (b) have used conve ntional M arko v chainswith out satisfying the require men t tha t the thicknesses of tithologies follow geom etricdistr ibutions, (c) have not considered semi-M arkov chains wh en they are mo re suitable thanconventional M ark ov chains, or (d) have not u sed proper statis tical tests to check the Ma rko vprope rty of emb edded M arko v chains, The strat igraphy modell ing procedure suggested in thispaper eliminates all these shortcomings. The sections which follow describe the modellingproced ure. An exam ple in the 'Applicatio n Section' illustrates the use of the modellingprocedure.

Overview of the suggested procedureThe suggested procedure is applicable only to strat igraphy data which show cyclicalsedimentat ion. The f irst s tep in the procedure is to check the hom ogeneity o f the strat igraphiccolumn. The test suggested by Anderson an d G oo dm an (1957) for s tat ionari ty can be used inchecking the homogeneity of conventional Ma rkov chains. How ever, this test is notapprop riate to check the homog eneity of emb edde d Ma rkov chains. At present, no suitable testis available to check the stat ionari ty of emb edded Ma rko v chains. Unti l such a test becomesavailable, the abov e mentioned test may be used as the test for homogeneity. If the test resultindicates non-homogeneity, then further invest igat ions should be carr ied out to separate theentire s trat igraphy into regions where the homogeneity property is applicable. Then eachhomogeneous strat igraphic sect ion should be analysed for the other propert ies explainedbelow.

Strat igraphic data generally fall into o ne of the following groups: (a) the observed d ata have afirst order conventional Ma rko v d ependency in the succession of l i thologies, and geometricdistr ibutions for al l l i thologic thicknesses, (b) the observed data have a f irs t order e mbed dedM arko v dependen cy in the succession of l i thologies, but they do not have geometric


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Modelling o f cyclical stratigraphy using Marko v chains 123distributions for all lithologic thicknesses, (c) the observed data have neither first orderconvent ional Markov dependency nor embedded Markov dependency in the success ion oflithologies, but they do have geometric distributions for all lithologic thicknesses, or (d) theobserved data nei ther sat isfy M ark ov dependen cy in the succession of l i thologies, nor do the yhave geo metric distr ibutions for l ithologic thicknesses. If com bination (a) holds, then f irst orderconventional Ma rkov chains should be used for modell ing. If comb ination (b) holds, then theapprop riate m odel is the semi-M arkov chain. For comb ination (c) , an independent event modelsuch as the mult inomial m odel is sui table. Com binatio n (d) cannot be modelled appropriatelyby ei ther the conventional or the semi-Markov model .

Tests given by Anderson and Goodman (1957) and Billingsley (1961) for the Markovprope rty can be used to test the independence against dependence of s tates for the conventionalM ark ov transi t ion m atrix. If the results of these two tests lead to the rejection of the nullhypothesis , then i t only implies the presence of some depe ndence betw een state t ransi t ions. Ifone wants to prove that the transi t ion m atrix fol lows a f irs t order conven tional M arko v chain,then i t is necessary to show that the distr ibution of thickness of each l i thology in the strat igraphysequence fol lows a geometric distr ibution. This can be checked b y performing goodness-of-f i ttests (Ang and Tang, 1975) on the thickness data. To check the presence of an em bedd edMarkov chain, tests given by Yu (1984) seem the best . These tests originate from Goodman(1968) in the conte xt of incom plete conting ency tables.

Stratigraphy modelling using firs t order conventional Markov chainsBasic conceptsIn the case of a s trat igraphic colum n, ob servations of the state are usually made start ing at thebo tto m a t discrete intervals of vertical distance. Ea ch interval represents o ne step in the indexspace of the conventional M ark ov chain. A f irst order M arko v chain is one where the transi t ionfrom state i to s ta tej depends o nly on the previous state of the chain. If the num ber of observedtransitions from state i to statej isf~j, then the tally matrix, F, is given by

F = ~ j ] ; i , j = l , 2 , . . . m (1)where m is the total n um ber of s tates. The transi tion probabil i ty from state i to s tate j , p~j, isgiven by

Pij =fij j (2)J

The transi t ion probabil i ty matrix, P , is defined byP= [P if]; i , j= l , 2 . . . . m (3 )

If the transi tion matrix doe s not vary with the location of the space parameter , then the Ma rkovchain is s tat ionary or ho moge neous. Th e diagonal probabil i t ies , p~, are related to the relativethicknesses of the l ithologic units (H arbau gh and Bonha m-Ca rter , 1970). The transi tionprob ability m atrix is sensitive to the interval em ploye d. If the interval is too small, the resulting


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124 Kulatilakep, tend to be one for any finite sample, with zero in the off-diagonals; if too large, someimportant layers may be missed. Therefore, one should carefully inspect the stratigraphiccolu mn be fore choosing an app ropria te size for the interval. Thicknesses of lithologies can beconsidered as ei ther discrete rand om variables (Krumbein an d Dacey, 1969) or co ntinuousrandom variables (Ethier, 1975). If they are treated as discrete random variables, then theyshould follow geom etric distributions for stratigraphy which can be m odelled by conven tionalMark ov chains (Krumbein a nd Dacey, 1969). If they are t reated as continuous rand omvariables, then they should follow expone ntial distributions.So far we have co nsidered on ly single-step transition probabilities. Multiple-step transitionprobabilities can be obtained by powering a single-step transition probability matrix. If amatrix of transition probabilities is successively pow ered with th e result that each row is thesame as every other row, the resultant matrix is termed a regular or steady-state transitionmatrix (H arbau gh an d B onh am -Carte r, 1970; Ang and Tang , 1984). The fixed probability rowvector of this matrix provides the pro port ion of each l i thology.Testing for homogeneityThis is based on the test suggested by Anderson and Goodman (1957) for stationarity. Thestratigraphic colu mn is divided into T subintervals. Then the following statistic is compute d.

S1=2 L ~ ifq(t)lg~FPiJ(t)~ (4), = 1 i = ~ j = ~ k PU _]

where t refers to the tth subinterval and Pu is the transition proba bility for the whole sequence. Ifthe null hypothe sis of homog eneity exists, the n S1 is asym ptotically chi-square distributed w ith( T - 1 ) m ( m - 1 ) degrees of freedom (DF). T he significance level at w hich S1 equ als thetheoretical value given in the chi-square table provides the maximum significance level,(Benjamin and Cornell, 1970), at w hich the null hypothesis can be accepted.Tests for Mark ov property of a conventional Marko v chainA test, reco mm end ed by Bitlingsley (t961), is the Pea rson statistic, given by

$ 2 = ~ ~ (fij-ein)2/eij (5)i= 1 j= l

where eij is the expected num ber o f i to j transitions un der the null hy pothesis that the statetransit ions com e from ind ependent m ult inomial t r ials. The max imum likelihood est imate of e uis given by

eij = (f iRfc j)/N (6a)where

f g = ~ fu for i= 1, 2 . . . m (6b)j= l


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Modelling of cyclical stratigraphy using M arkov chains 125fc~ = ~ fij for j = 1, 2 . . . . m (6c)

i=1and N is the total num ber of transit ions. Another test , recommended b y Anderson andGoodman (1957), is the likelihood-ratio statistic

$3 = 2 ~ ~ f~j loge(f/j/e,j . (7)i=1 j= l

Und er the null hypothesis of independence both these stat ist ics bec ome asymptoticallychi-square distrib uted with ( m - 1)z DF . I f the results of these two tests leads to the rejection ofthe null hypothesis, then it only implies the presence of some dependence between statetransitions. T o show that this depend ency is a conventional first order Ma rko v depend ency it isnecessary to pr ove tha t the thicknesses of lithologies follow geome tric distributions. Chi-squa reand Kolmo gorov-Sm irnov goodness-of-fit tests (Ang and Tang, 1975) can be performed tocheck this.

Stratigraphy modelling using first order semi-Markov chainsIn structuring an embedded transit ion probabil i ty matrix from observational data, only thelithology transitions are tallied. Hence, the nu mbe r o f entries in the em bedde d matrix is smallerthan for the equal interval matrix. As a result, the off-diagonal probabilities have differentnumerical values, bu t the relative probabilities for p~ where i j are the sam e in the two type s ofmatrices. F or s emi- Ma rkov chains, the thicknesses of lithologies need not follow geometricdistributions. In order to use the semi-Markov model, the stratigraphy data should satisfy theMarkov p roper ty fo r embedded Markov chains .Tests for Marko v property o f an embedded Marko v chainIn this case the Pe arso n statistic takes the following form (Yu, 1984)

s4= (8a)i= 1 j= lj--/:i

where eij= aib for i ~ j (8b)0 for i= jValues of ai and bj are compu ted using the iterative scheme given below (Yu, 1984)

go _ / ~ (9a)tep 1: 1 -J i R/ ~ 6~j fo r i= l , 2 . . . m[ j = l= 0 for i = jwhere 6~j = 1 for i-j (9b)


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126 Kulatilake_ 6 a 2"-z f o r j = l , 2 , . . m (9c)tep 2n(n > 1 ): b " - ~ =f~j ~j i

i

mS te p 2 n + l ( n > l ) : a2"=fis ~ 6, pf "-1 (9d)/ j = 1

The iteration c an be repeated until the required accu racy is obtained. The m aximu m likelihoodestimate in this case is given by (Yu, 1984):

$ 5= 2 ~, ~ f01Ogef~j--2 ~ f/R1og~ai--2 ~ fcjlog~b~ (10)f = 1 j = l j = l j = ljOi

Un der the null hypothesis that the state transitions com e from independent m ultinomial trials,both statist ics approximately follow the chi-square distr ibution with (m -1 ) 2- m degrees offreedom. Re sults should indicate rejection of the null hypothesis in order to satisfy the M ark ovproperty .

ApplicationTable 1 provides stratigraphy d ata from the Oficina Form atio n o f eastern Venezuela (Scherer,1968). These data were used to illustrate the modelling procedure given in the paper. Thestratigraphy column given in Table 1 was divided into five equal subintervals to perform the testfor homogeneity. The following results were obtained:

Degrees of freedom =4 8; S1 = 55.5; ~ = 0.22.This show s that the hom ogene ity can be ac cepted at a fairly high significant level. This allows usto treat the w hole strat igraphy column under one hom ogeneity set .Next, the tally matrices, transition probability matrices, and chi-square statistics for anordinary Markov chain were computed for interval sizes of 60, 120 and 240 cm. Results aregiven in Table 2. The results c learly show t he influence of interval size on the tra nsitionprobab ility matrix and on the chi-square values. As the interval size increases, Pu decreases. Inthis particular example, the influence is most pronounced on lignite and siltstone. A carefulinspection of Table 1 shows that both lignite and siltstone have pretty high frequencies forthicknesses less than 120 and 240 cm. F or this example, interval size of 60 cm seems a prettygood choice. Values obtained for ct dearly show the strong rejection of the null hypothesis ofindependence. The transition probab ility matrix o btaine d for the interval size of 60 cm was usedto compute the regular transition matrix. The results provided the following proportions oflithologies for the stratigraphic column:

Sandstone=0 .27 ; Shale=0 .49 ; S i l t s tone=0 . t2 ; L ign i te=0 .12Frequency distributions for lithology thicknesses are given in Fig. 1. Thickness data weresubjected to chi-square and K olm ogoro v-Sm irnov (K & S) goodness-of-fit tests for geometric


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128 KulatilakeTable 2. Transi tion probability matrices and chi-square statistics for modelling stratigraphyby conventi onal Markov chains.Interval sizeused forobservation 60 cm 120 cm 240 cm

A B C D A B C D A B C DA 104 3 8 9 84 22 14 10 31 18 11 7Tally B 5 74 8 14 26 171 21 25 21 72 7 19matrix C 5 6 13 7 9 21 13 12 4 10 3 6D 10 18 2 21 11 29 7 15 11 19 2 4Transition Aprobability Bmatrix CD

A B C D A B C D A B C D0.84 0.03 0.06 0.07 0.64 0.17 0 .11 0.08 0.46 0.27 0. t6 0.110.05 0.73 0.08 0.14 0.11 0.70 0.09 0.10 0.18 0.60 0.06 0.160.16 0.19 0.42 0.23 0.16 0.38 0.24 0.22 0.17 0.44 0.13 0.260.20 0.35 0.04 0.41 0.18 0.47 0.11 0.24 0.30 0.53 0.06 0.11

$2 247.6 171.4 33.4DF 9 9 9< 0.005 < 0.005 < 0.005$3 251.0 162.3 32.9DF 9 9 9< 0.005 < 0.005 < 0.005

~(a)

~' o.3o~ S a n d s t o n e ( s t a t e A) (c)o.zs] - - o b s e r v e di=r /w 0.20 ~r ~ 0.60,,'- theoretical Siltstone.= O,Sli !1" (geom etric) 05 5. (state C)

t = o , s -ell', , i ~ l . l ~ i i ' , ' , T , , , . r n . r n _ ~ "o ,zs 25~o 37,5 5o.o ~.~ ~,.o e'k5 ~ .o ,~.5,2~,.o - ~ o.,,o-

T h i c k n e s s - c m I~. 0 .35 -o.so-' (b) .~ o.3o-

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0.30.n."0.25-0`20"0.I 5"

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J0 .05 - ~ I , 0.05 - 0.05- ~ i ,

0 I2.5 25.0 37, 5 50 0 62.5 750 87.5 tO0.O 112,5 12.5,0 0 12.5 25 0 3"/5 50.0 0T h i c k n e s s - e r a T h i c k n e s s - e r a T h i c k n e s s - c m

Fig. 1. Observed relative frequencies and geometric distribution fittings on li thology thicknessdata from Table 1.


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Modellin# o f cyclical strati#raphy usin# Markov chainsTable 3. Results of goodness-of-fit ests on thickness of lithologies.

129

Lithology type Sandst one Sha le Sil tstone LigniteChi-square value 3.78 8.05 6.36 0.99DF 7 9 4 20.80 0.50 0.19 0.62K & S value 0.04 0.06 0.05 0.03DF 15 18 7 5> 0.95 > 0.95 > 0.95 > 0.95

Using the tally matrix, $4 and $5 were computed according to Equations (8) and (10),respectively. The associated degrees of freedom and ~ values were also determined. Results aregiven below.

$4= 14.3; DF =5 ; ~=0.015$5= 14.8; DF=5; ct=0.012

The results show a rejection of the null hypothesis of independence. However this rejection isnot as strong as the rejection indicated by Table 2. Therefore, it can be concluded that theconventional Markov chain is better than the embedded Markov chain to model the consideredstratigraphy data.

ConclusionsThe paper provides a procedure to analyse cyclical s tratigraphy data . If lithology transitionssatisfy the Markov property, then the stratigraphy can be modelled using either first-orderconventional Markov chains or first-order embedded Markov chains. The Markov chain typewhich should be used depends on the structure of the stratigraphy. Statistical tests, whichshould be performed to choose the proper type of Markov chain, are given in detail. If lithologytransitions do not show any Markov dependence, then the stratigraphy should be modelled byan independent multinomial model.Once the model is constructed, then it can be used to generate stratigraphy using a Monte-Carlo simulation. Characterization of stratigraphy is an essential element in any geological orgeotechnical engineering analysis or design.

AcknowledgementsUSAE Waterways Experiment Station provided financial assistance for this study. This supportis gratefully acknowledged. Any opinions, findings, conclusions, or recommendationsexpressed in this paper are those of the au thor and do not necessarily reflect the views of theWaterways Experiment Station. Sue Wiedenbeck, a graduate student in Systems Engineering atthe University of Arizona, assisted with most of the calculations. The writer also gratefully


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13 0 Kulatilakeacknowledges John B. Pa lmer ton o f the Wate rw ays E xper imen t S ta t ion fo r h i s ass i stance andin te res t ove r the course o f the s tudy .

References

Ali, E.M., Wu, T.H . and Chang, N.Y. (1980) Stochastic model of flow through stratified soils, Journal ofthe Geotechnical Engineering Division, AS C E 106, 593-610.Anderson, T.W. and Goo dm an, L.A. (1957) Statistical inference abou t M arkov chains, Ann. Math.Statist. 28, 89-110.Ang, A.H-S. and Tang, W.H. (1975) Probability Concepts in Engineering Planning and Design 1, JohnWiley and Sons.Ang, A.H-S. and Tang, W.H. (1984) Probability Concepts in Engineering Planning and Design 2, JohnWiley and Sons.Benjamin, J.R. and C ornell, C.A. (1970) Probab ility, Statistics, and Decision or Civil Engineers,McGraw-Hill.Billingsley, P. (1961) Statistical m etho ds in Ma rko v chains, Ann. Math. Star. 32, 12--40.Ethier, V.G. (1975) Application of Ma rkov analysis to the B anff Form ation (Mississipian), Alberta,Mathematical Geology 7, 47-61.Go odm an, L.A. (196 8) Th e analysis of cross-classified data: independence, quasi-independence, andinteractions in contingency tables with and without missing entries, Jour. Amer. Statist. Assoc. 63,1091-131.Harbaugh, J.W. and Bonham-Carter, G. (1970) Computer Simulation in Geology, John Wiley and Sons.Krumbein, W.C. and Dacey, M.F. (1969) Markov chains and embedded Markov chains in geology,Journal of Mathematical Geology 1, 79-96.Scherer, W. (1968) Application o f Markov chains to cyclical sedimentation in the Oficina Formation, easternVenezuela, unpublished M S thesis, Northw estern University, Evanston, Illinois.

Yu, J. (1984) Tests for quasi-independence of emb edded M ark ov chains, Journal of Mathematical Geology16, 267-82.Depa rtment of Min ing and Geological Engineering,Universi ty of Ar izona,Tuscon,Arizona 85721, USA

PINNADUWAH H.S.W. KULATILAKE

Received 30 May 1986

modelling of cyclical stratigraphy using markov

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