incorporating information about edge effects when simulating lithofacies

P1: FPY/FKQ/FFB P2: FNN

Mathematical Geology [mg] PL093-886 November 13, 1999 1:28 Style file version June 30, 1999

Mathematical Geology, Vol. 32, No. 3, 2000

Incorporating Information About Edge EffectsWhen Simulating Lithofacies1

Chris Roth2

Some years ago, earth scientists came to realize that knowing more about the geology of an orebodyor an oil reservoir makes it easier to make the appropriate decisions concerning mine planning orreservoir exploitation. Geostatistical techniques for simulating lithofacies—that is, the geometry ofthe geology—were developed as a result of this. These methods should be able to produce geologicalimages that respect not only the anisotropies of the different lithofacies but also their spatial layoutrelative to one another. While indicator variograms ensure that anisotropies are respected, another toolneeds to be incorporated in the simulation technique to reflect the relative spatial layout of the differentlithofacies. We propose to use the concept of edge effects that define the position of one lithofaciesrelative to another. Simple tests using direct and cross indicator variograms confirm the presence orabsence of edge effects. We investigate if and how edge effect information can be incorporated in thedifferent indicator simulation techniques—sequential indicator simulations, simulated annealing, thetruncated Gaussian method and plurigaussian simulations. Results show that the choice of simulationmethod must be guided by the edge effect characteristics of the experimental lithologic data.

KEY WORDS: indicators, random sets, sequential indicator simulation, simulated annealing, pluri-gaussian simulation, truncated Gaussian method.

INTRODUCTION

Simulating lithofacies, or the geometry of the geology, is the focus of continuingresearch as earth scientists have come to realize that knowing more about thegeology of an orebody or an oil reservoir makes it easier to make the appropriatedecisions concerning mine planning or reservoir exploitation. Directly simulatingthe metal grades or the permeability is often not enough to produce a realisticnumerical model of the orebody. The geology must also be taken into account.So the characterization procedure is a two-step procedure based on the simulationof, first, the different lithofacies that make up the domain, and then the relevantvariable within each of the different lithofacies. In this paper we are interested inthe simulation of lithofacies, or equivalently the simulation of random sets where

1Received 3 June 1998; accepted 17 November 1998.2Centre de G´eostatistique, 35 rue Saint Honor´e, 77305 Fontainebleau, France. e-mail: [email protected]

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each random set is made up of all those points in space that belong to a particularrock type. In particular, we wish to incorporate a maximum of information in thesimulation technique, not only the usual criteria of the proportion and continuity ofthe random sets but also their relative position, that can be defined by the presenceor absence of edge effects. The aim of this paper is to identify which simulationtechniques allow us to incorporate the experimental edge effect findings. We willsee in the next section that the test for edge effects is based on the ratio of thedirect and cross indicator variograms where each indicator defines the presence orabsence of a particular rock type.

There are two broad types of simulation techniques, based on either objectsor indicator variables. Theobject-based techniquesessentially consist of placingobjects on a given background. The resulting simulations depend on the parametersinputted into the model, which include the size and shape distribution of the objects,the hierarchy of the objects and their spatial interaction—repulsion, independence,or clustering. These different parameters, which define the particularities of eachsimulation, mean that it is not possible to develop a general expression for the cor-responding indicator variograms. Because we cannot develop theoretical results,we cannot provide a criterion for when object based simulations produce imageswith or without edge effects. So the object-based techniques will not be consideredfurther in this paper.

On the other hand, techniques based on the simulation ofindicator variablesshould lend themselves well to be investigated for its edge effects characteris-tics. The currently available methods for simulating indicators are Markov ran-dom fields, sequential indicator simulations, simulated annealing, the truncatedGaussian method, and plurigaussian simulations. We will not considerMarkovrandom fieldshere for the same reasons given for the object-based simulations.They are produced from algorithms that depend on several parameters (e.g., theclique, multivariate distribution), each of which determines to some extent theform of the resulting indicator variograms. No theoretical results can be deducedfor these variogram in the general case and so no conclusions can be drawn aboutthe algorithm’s suitability in incorporating the edge effect criterion.

Sequential indicator simulations (Alabert, 1987; Deutsch and Journel, 1992)are based on fitting direct and cross-variogram models directly on the experimentalindicator variograms. According to the authors, the local conditional probabilityof any point belonging to a given lithofacies can be estimated using these models.A uniformly distributed value is then randomly drawn and compared to estimateddiscrete probability density function and the point is assigned to the relevant litho-facies. Simulated annealing (Aarts and Korst, 1989; Deutsch, 1992) is based onthe permutation of an initially defined field to reproduce certain required spatialcharacteristics. The initial field is drawn to respect a predefined histogram. Thesevalues are then iteratively swapped until the resulting field reproduces the charac-teristics defined by a given objective function that could, for example, minimize



Information About Edge Effects 279

the difference between the experimental and the modeled variogram. Accordingto the truncated Gaussian method (Matheron and others, 1987; de Fouquet andothers, 1989; Beucher and others, 1993; Galli and others, 1994) indicator func-tions are obtained by truncating an underlying multigaussian random function atthreshold values that are determined by the proportions of the different lithofacies.The lithofacies simulation is obtained as the truncation of a multigaussian simu-lation. A Gibbs sampler algorithm is used for the conditioning step (Freulon andde Fouquet, 1993) to convert the experimental indicator data into a series of con-ditioning multigaussian values. The plurigaussian simulations (Galli and others,1994; Le Loc’h and others, 1994; Le Loc’h and Galli, 1997) are an extension of thetruncated Gaussian method. Instead of truncating just one multigaussian randomfunction, several such functions are used simultaneously.

In this paper we will investigate these simulation methods and their ability totake the experimental edge effect findings into account. We begin by a reminderon what is meant by edge effects with respect to the spatial distribution of randomsets.

EDGE EFFECTS

A mathematical quantification of the influence of edge effects on random setswas introduced by Rivoirard (1990 or 1994 for the English version) in the casewhere the sets are obtained by truncating a continuous random variable. Let usdefine what is meant by edge effects and show how this concept can be linked toindicator variograms.

Definition. Let {Ai : i = 0, . . . ,m} denote a finite series of nested randomsets defined for each pointx of a closed domainD ∈ Rn wheren = 1, 2, or 3.By “nested” we mean that any point belonging to a random set also belongs to thefollowing ones:x ∈ Ai+1 ⇒ x ∈ Ai or Ai+1 ⊆ Ai ∀i = 0, . . . ,m− 1. The firstset includes all the points of the domain:A0 = {x:x ∈ D}with the subsequent setspotentially being made up of several disjoint areas or zones withinD.

Consider two such sets,Ai andAj for i 6= j such thatAj ⊆ Ai . We say thatthere areedge effectswhen going fromAi to Aj if there is a buffer zone, or in somecases, an edge around the setAj that belongs toAi . That is, if going from outsideAi to insideAj depends on the distance traveled. Consider an initial point outsidethe setAi . Moving away from this point along a straight line we check whetherthe current point passes throughAi ∩ A j before it is withinAj . The same test isrepeated starting from all points of the domain. If a tendency of passing throughthis buffer zone before reachingAj is found then we say that there are edge effectsgoing fromAi to Aj . In other words, the probability of being inAj , given we arealready inAi depends on the distance from the initial point considered. Similarlyit possible to have edge effects when going in the opposite direction, fromAj to



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Figure 1. Schematic example of random sets: A with and B without systematic edge effects.

Ai . This is tested by starting at any point within the setAj and moving away fromit. If there is a tendency to go throughAi ∩ A j before being out ofAi , then edgeeffects also exist in this direction. A simple example is presented in Figure 1 forthe nested setsA2 ⊆ A1 ⊆ A0, whereA0 is the entire domain. In this figure thelines represent the limits of the different sets that are colored so that the darkercolored zones also belong to the sets defined by a lighter color. This example isinspired by the presentation of edge effects in Rivoirard (1990).

There are edge effects in both directions in Figure 1A. The placement of thesets is such that we systematically pass through the light grey part ofA1 whengoing either fromA2 (dark grey) to the white part ofA0 or from the white part ofA0 to A2. But this is an extremely simple example of edge effects whereA1 ∩ A2

(the light grey) really forms an edge aroundA2. Figure 1B presents a more subtleexample of edge effects. Here, when going from the white part ofA0 to the darkgrey A2, we generally pass through a zone of light grey, not because they forman edge aroundA2 but because of the spatial distribution of the different zones.That is the probability of being in the dark grey, given we are already in light grey,depends on the distance travelled.

Of course the existence of edge effects in one direction does not imply theirexistence when going in the opposite direction. In Figure 1B for example there areno edge effects when going fromA2 to the lighter colored zones. When leavingA2 we are just as likely to go directly to the white part ofA0 as we are of going tothe light grey part ofA1. In this direction the probability of being in a white zonegiven we are outside the dark grey does not depend on the distance traveled fromthe initial point.

Probabilistic Formulation.Let us return to the finite series of nested randomsets{Ai : i = 0, . . . ,m}, and in particular, the case of going fromAj to Ai forj > i . Consider two pointsx andx + h separated by a distance ofh in 1, 2, or3 dimensions. Suppose that the pointx belongs to the setAj (and hencex ∈ Ai )




but that the second point does not:x + h /∈ Aj . Then, according to Rivoirard(1990), there are no edge effects fromAj to Ai if the likelihood that the pointx + h is also outside the larger setAi is the same for all distancesh considered.On the other hand, if the probability thatx+ h /∈ Ai given thatx+ h /∈ Aj varieswith distance then we say that there are edge effects going fromAj to Ai . Thisconditional probability can be written as

pi, j (h) = Pr{x+ h /∈ Ai | x+ h /∈ Aj ∩ x ∈ Aj }

= Pr{x ∈ Aj ∩ x+ h /∈ Ai }Pr{x ∈ Aj ∩ x+ h /∈ Aj }

(1)

because the nested character of the sets implies thatx + h /∈ Ai ⇒ x + h /∈ Aj .So the presence, or absence, of edge effects is determined by the behavior of theconditional probabilitypi, j (h) as a function of the separation distanceh. If pi, j (h)is constant for allh, then there are NO edge effects going fromAj to Ai ; otherwiseedge effects are said to exist.

Indicator Variograms.Let us formalize the presence or absence of edge ef-fects in terms of indicator variograms. Each random set can be defined in terms ofindicator variables that take the value 1 for points within the set and zero otherwise:

I i (x) ={

1 if x ∈ Ai

0 otherwise(2)

for i = 0, . . . ,m. It can be shown (see the appendix for details) that Equation (1)can be rewritten in terms of the direct and cross indicator variograms:

pi, j (h) = Pr{I j (x) = 1∩ I i (x+ h) = 0}Pr{I j (x) = 1∩ I j (x+ h) = 0} =

γi, j (h)

γ j, j (h)(3)

whereγi, j is the cross variogram betweenI i andI j andγ j, j is the direct variogramof I j . So the presence or absence of edge effects going fromAj to Ai is defined bythe behavior of the ratio of the cross and direct indicator variograms as a functionof h. Similarly, when going from the larger setAi to Aj , it is the ratio between thecross variogram and the variogram ofI i : pj,i (h) = γi, j (h)/γi,i (h) that defines thepresence or absence of edge effects.

So a variogram ratio, like that in Equation (3), can be used to test for thepresence or absence of edge effects in the experimental lithofacies data. Theseexperimental findings should then be incorporated in any simulation of the litho-facies. However, rigorously respecting the presence of edge effects is by no meanstrivial. We must ensure not only that the theoretical variogram ratio depends on thedistance considered but also that it mimics the experimental findings. For example,does the experimental variogram ratio increase, decrease, or undulate about a given



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value as a function of the distance? To date no theoretical results defining how torespect the evolution of the experimental variogram ratio are available, due to thecomplexity of this problem. However, we can incorporate a less complex criterionbased only on whether edge effects are present or not. Such a criterion, a constantindicator variogram ratio, can be evaluated theoretically for the different indicatorsimulation techniques, thus allowing us to evaluate which technique is appropri-ate when experimental results confirm the absence of edge effects between litho-facies. By deduction, the remaining techniques only allow us to simulate thosecases where the experimental findings indicate the presence of edge effects, butwithout actually defining how the variogram ratio evolves with the distance. Thispaper is therefore dedicated to investigating the suitability of the different simu-lation techniques in reproducing the no edge effect criterion, without quantifyingany edge effects they may produce.

SEQUENTIAL INDICATOR SIMULATIONS

For the application of sequential indicator simulations, the lithologic data isfirst transformed into a discrete random variable and indicators are created fromthis random functions according to the different lithofacies. The indicators arecokriged to produce what is assumed to represent the estimation of the discreteprobability density function (pdf) of the underlying function conditioned by theknown surrounding values—that is, the probability of a certain point belongingto the different lithofacies. Random values are then drawn according to this localconditional pdf to supply the simulated value at the point considered.

Testing for Edge Effects. The direct and cross variogram model defined forthe indicators is obtained from the fitting of the experimental indicator variograms.However, this is not an easy task given the often large number of indicators used inpractice and the conditions that the fitted models must obey to be a valid indicatorvariogram model (Matheron, 1989; Rivoirard, 1993). The modeling procedure be-comes even more difficult if edge effect conditions, like those seen in Equation (3)need to be incorporated. This is because individual lithofacies—that is, disjointrandom sets—are modeled in the sequential indicator simulation algorithm andnot the nested ones of Equation (3). Consider a simple example of no edge effectsbetween the nested setsAi = Bk ∪ Bl and Aj = Bl , whereBk and Bl are twodisjoint sets. A simple calculation allows us to write the edge effect criterion inEquation (3) as

pi, j (h) = γi, j (h)

γ j, j (h)= γBl ,Bl (h)− E{I Bl (x)} + γBk,Bl (h)

γBl ,Bl (h)(4)

whereγBk,Bl is the cross indicator variogram for the setsBk andBl , and E{I Bl (x)}is the probability that the pointx lies in the setBl . Note that we have simplifiedthe notation by writing this probability as a function of the distance only. If there




are no edge effects,pi, j (h) is independent of the distanceh—that is, it equals aconstant,c say. In this case we find

γBl ,Bl (h) = 1

c− 1γBk,Bl (h)+ E{I Bl (x)}

1− c(5)

So this simple no edge effect criterion represents one more constraint that must berespected when inferring a valid direct and cross indicator variogram model forthe sequential indicator simulations. Moreover, the value ofc must be chosen soas to be consistent with the resulting variogram model.

This situation becomes more complex when considering other cases. Let ustake the example presented later in this paper where there are no edge effects whengoing fromAj = Bl ∪ Bm to Ai = Bk∪ Bl ∪ Bm. In this case the relationship seenin Equation (3) becomes

pi, j (h) = γi, j (h)

γ j, j (h)= γBl∪Bm,Bl∪Bm(h)− E{I Bl∪Bm(x)} + γBk,Bl∪Bm(h)

γBl∪Bm,Bl∪Bm(h)(6)

This can be expanded in terms of the means E{I Bl (x)} and E{I Bl (x)}, and all thedirect and cross indicator variograms ofBk, Bl , andBm. Again, setting the proba-bility to a correctly defined constant allows us to define the additional constraintsthat must be respected when inferring a valid model.

The difficulty in fitting such a valid indicator variogram has meant that asimplification (Deutsch and Journel, 1992) has been proposed. All direct and crossindicator variograms are supposed identical up to a multiplicative constant; that is,

γBk,Bl (h)

γBk,Bl (∞)= r (h) ∀k, l (7)

whereγBk,Bl (∞) is the sill of the cross variogram andr (h) the so-called normedvariogram structure. This assumption corresponds to a model of intrinsic correla-tion for which the cokriging of the indicators is equal to their kriging. While theindicator variogram modeling and estimation is thus much simplified, the approachis really only justified when all the lithofacies have the same spatial continuity.Applying the variogram model in Equation (7) to the expression in Equation (4)gives us

pi, j (h) = 1+ γBk,Bl (∞)

γBl ,Bl (∞)− E{I Bl (x)}γBk,Bl (∞)r (h)

(8)

which always depends on the distanceh, irrespective of the variogram structurer (h) initially fitted. So, when the simplification of Equation (4) is adopted, sequen-tial indicator simulations produce images for which there are always edge effects



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between all the lithofacies. The simplification is therefore not appropriate when theexperimental indicator variograms confirm that there are no edge effects betweensome lithofacies.

SIMULATED ANNEALING

Given that we are interested in simulating lithofacies, an initial field is drawnto respect the global proportion of each lithofacies. Pairs of values are then swappediteratively so that the resulting image respects the spatial continuity of the differentlithofacies. A swap is accepted if the objective function is lowered by it. Theobjective function could, for example, be the average squared difference betweenthe experimental and the modeled variogram. So for lithofaciesBk the objectivefunction, denoted byOBk,Bk , could take the form

OBk,Bk =∑

h

{γ ∗Bk,Bk

(h)− γBk,Bk (h)}2

γBk,Bk (h)2(9)

whereγ ∗Bk,Bkis the experimental variogram calculated after the swap andγBk,Bk is

the inputted variogram model. However, in practice, when several lithofacies aresimulated simultaneously the objective function becomes somewhat more com-plicated because it must incorporate all the direct and cross indicator variogramcalculations:

O =N∑

k=1

N∑l=1

{OBk,Bl } =N∑

k=1

N∑l=1

{∑h

{γ ∗Bk,Bl

(h)− γBk,Bl (h)}2

γBk,Bl (h)2

}(10)

whereN is the number of lithofacies that, as above, are denoted byBk for k =1, . . . , N. For each proposed swap the time needed to update the objective functionincreases with the number of lithofacies. This can then be compounded by the needto divide the objective function (Deutsch and Journel, 1992) so as to avoid localdiscontinuities in the resulting simulation.

The application of simulated annealing presents the same problems as thatof sequential indicator simulations. A valid cross and direct indicator variogrammodel that also incorporates any existing edge effect constraints, like those seen inthe preceding section, must be defined. The use of simulated annealing is, however,even more difficult due to the delicate adjustment of numerous tuning parameterslike the acceptance/rejection criteria for each swap and the speed at which the“temperature” decreases to ensure the correct convergence of the objective func-tion under reasonable time constraints. So while theoretically simulated annealingcould be used to simulate lithofacies displaying given edge effect constraints,practical constraints limit its domain of application.




THE TRUNCATED GAUSSIAN METHOD

The truncated Gaussian method is just one of the numerous random-genericmodels (Matheron, 1969; or Jacod and Joathon, 1971) used to simulate randomsets. The advantage of this method is that conditional simulations can be per-formed. Practical applications of the truncated Gaussian method have focused onthe simulation of the essentially sedimentary layers so often encountered in the oilindustry.

Definition. The different random sets are obtained by truncating a standardmultigaussian random function, denoted here byY(x), at given thresholds. Thethreshold values, denoted byti , may or may not depend on the point in spaceconsidered. Each random set,Bi say, can be defined as

Bi = {x: ti ≤ Y(x) < ti+1} (11)

for i = 0, . . . ,m and wheret0 = −∞ andtm+1 = ∞. The setsBi are mutuallydisjoint because each point in space belongs to one and only one of them. Theirunion is of course the entire domain. The thresholdsti are determined experimen-tally from the proportion of each lithofacies in space that can be equated to theprobability that a certain point lies in a given lithofacies:

Pr{ti ≤ Y(x) < ti+1} = G(ti+1)− G(ti ) (12)

whereG(·) is the cumulative distribution function of a standard Gaussian vari-able. Without loss of generality, we suppose that the thresholds are order ranked:ti < ti+1.

Similar to Equation (2), an indicator function can be defined for each setBi , and hence in terms of the multigaussianY(x). This link between indicatorand Gaussian functions provides a unique relationship between their covariances.The covariance model forY(x) is then fixed as that which, via the correspondingindicator covariance model, leads to the best fit of the experimental indicator var-iograms. The lithofacies simulation is obtained by truncating a simulation ofY(x)that respects both the proportion of each lithofacies and the indicator covariances.

Let us consider an example where the domain is made up of 5 differentlithofacies, each of which takes up 20% of the total area. So the thresholds aret1 = −0.84, t2 = −0.25, t3 = 0.25, andt4 = 0.84, as shown in Figure 2A on theGaussian pdf. As seen in Equation (5), the different colored areas under this curverepresent the proportions of the five lithofacies (or setsBi ) that are equal here. Forthis synthetic example,Y(x) is assigned an isotropic Gaussian covariance modelwhose practical range is equal to about one seventh of the width of the domain.Figure 2B shows the resulting lithofacies simulation. The color of each lithofaciesmatches that used in Figure 2A, corresponding to a certain part of the Gaussian pdf.



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According to our notation, the setB0 corresponds to the white area,B1 to the lightgrey area, etc., up toB4, which is made up of the black “islands.”

Testing for Edge Effects. The edge effect test in Equation (3) is based onindicator variograms for a series of nested sets. We therefore transform the mutuallydisjoint setsBi in the following way:

Ai =m⋃

l=0

Bl = {x: x ∈ Bi ∪ · · · ∪ Bm} (13)

for i = 0, . . . ,m into a series of nested setsAi , where Ai−1 = Ai ∪ Bi−1. InFigure 2, each nested set is made up of those areas defined by a given color and allthe darker colors. For exampleA4 = B4 is made of the black “islands.” The nextset A3 = B3 ∪ B4 is then made of the black “islands” plus the dark grey zones.This leads to an image that is very similar to Figure 1A in that when going fromthe lightest to the darkest colors, we go through the middle tones of grey. That isthe middle tones tend to form an edge around the darkest grey. Soa priori thereappears to be edge effects when going from one set to the next, meaning that thefunctionY(x) tends to take different values (higher or lower) at the edge or borderof the setAi than toward the central zone.

To verify that the truncated Gaussian method always produces edge effects,we calculate the direct and cross indicator variogram ratio. The setAi can bedefined by an indicator function:

Ai = {x: ti ≤ Y(x)} and I i (x) ={

1 if ti ≤ Y(x)0 otherwise

(14)

So spatially each random set is made up of several areas or zones within which therandom function is greater than the corresponding threshold. If we then replace thesetsAi in Equation (1) by their definition in Equation (14) in terms of indicatorswe obtain

pi, j (h) = Pr{Y(x) ≥ t j ∩ Y(x+ h) < ti }Pr{Y(x) ≥ t j ∩ Y(x+ h) < t j } =

γi, j (h)

γ j, j (h)(15)

So the direct and cross indicator variograms are defined by the bigaussian pdf ofthe pair{Y(x),Y(x+ h)} and the threshold valuest j andt j :

γi, j (h) =∞∫

t j

du

ti∫−∞

g(u, v, ρ(h)) dv (16)

where ρ(h) = Cov(Y(x),Y(x+ h)) is the covariance of the functionY andg(· , · , ρ(h)) is the joint pdf of the bivariate normal pair (Y(x),Y(x+ h)). The



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direct variograms can obviously be calculated in a similar way. So the test for edgeeffects can be rewritten in the following way:

pi, j (h) =∫∞

t jdu∫ ti−∞ g(u, v, ρ(h)) dv∫∞

tidu∫ ti−∞ g(u, v, ρ(h)) dv

(17)

However, this expression can only be simplified to one that is independent ofhfor a few trivial cases. Whenρ(h) is a pure nugget effect covariance, we find thatpi, j (h) = (1− G(t j ))/(1− G(ti )). But we cannot simply assign a nugget effectcovariance toY so as to simulate no edge effects. This covariance is determinedby the experimental indicator covariance. For a general covariance model, exceptfor the nonsensical solutiont j = ti , pi, j (h) in Equation (17) always depends onh. The truncated Gaussian method necessarily produces lithofacies simulationsdisplaying edge effects in both directions. This method is therefore not suitable tothose cases where the experimental indicator variograms show that there are noedge effects between certain lithofacies.

PLURIGAUSSIAN SIMULATIONS

As an extension of the truncated Gaussian model, the plurigaussian simu-lations are much more powerful in being able to reproduce complex geologicalcharacteristics seen in practice. While theoretically any number of multigaussianrandom functions can be incorporated, to date the use of just two such functionsadds enough flexibility to the methodology to handle a very wide range of appli-cations including the complex task of modelling multiphase tectonic results (Rothand others, 1998) often encountered in the mining industry.

Introduction. In this paper we will consider the two independent standardmultigaussian functions, denoted byY1 and Y2. By extension of the truncatedGaussian method in Equation (11), we will consider a given disjoint set as result-ing from the rectangular thresholding of the two Gaussian functions:

Bi = {x: ti,1 ≤ Y1(x) < ti,2 ∩ si,1 ≤ Y2(x) < si,2} (18)

for i = 0, . . . ,m defining the (m+ 1) different lithofacies. Again the thresholdsti,1, ti,2, si,1, andsi,2 may depend on the point considered if the proportions are notmodeled as being constant in space. The experimentally known proportion of eachlithofacies throughout the domain defines the link between the different thresholds:

Pr{ti,1 ≤ Y1(x) < ti,2 ∩ si,1 ≤ Y2(x) < si,2} = (G(ti,2)−G(ti,1))(G(si,2)−G(si,1))

(19)




However, unlike the truncated Gaussian method, this does not uniquely definethe values of the thresholds as we have (m+ 1) known proportions and 4(m+ 1)unknown thresholds.

To illustrate this, consider 5 lithofacies that each take up 20% of a givendomain. There are several possible ways of truncating the Gaussian functionsY1

andY2 so as to satisfy this condition. Four of them are shown in Figures 3B to3E where the partition of bigaussian space (Y1,Y2) is presented in plan view:Y1

is plotted on the horizontal axis andY2 on the vertical axis. Each disjoint setBi

is defined by a particular shade of grey. For example, in Figure 3C the white setcorresponds to all those points whereY1 is less than the threshold value, irrespectiveof the value ofY2. This representation of the bigaussian partition is somewhatmisleading as it does nota priori appear to respect the equal proportion of alllithofacies. This is because it is the volume under the bell-shaped bigaussian pdf(Fig. 3A) that determines the different thresholds and not the area of the rectanglesin plan view. In practice, once we have established in which way the bivariatespace will be partitioned, the threshold values are uniquely defined. Then, afterdefining the relevant indicator functions, the covariance models forY1 andY2 arefixed as those that lead to the best fit of the experimental indicator variograms.The lithofacies simulationi is obtained by truncating a simulation ofY1 andY2

according to the initially fixed bigaussian partition.So the partitioning ofY1 andY2 is a vital step in the plurigaussian simulation

process. This partitioning step will of course be guided by the existing physicalconstraints of the domain like the touching or non touching constraints (describedin more detail by Le Loc’h and Galli, 1997). For example, if for some mineralogicalreason the white lithofacies cannot touch the black one, then only Figures 3B, 3C, or3E can be considered as the lithofacies simulations resulting from these partitionswill respect the nontouching constraint. Similarly, the presence or absence ofedge effects represents more information about the lithofacies that needs to beincorporated in the partitioning step.

Testing for Edge Effects

Let us first consider two disjoint sets that are defined by a rectangular partitionof bigaussian space (Y1,Y2). Let us denote them byBi andBj such that:

Bi = {x: ti,1 ≤ Y1(x) < ti,2 ∩ si,1 ≤ Y2(x)si,2}Bj = {x: t j,1 ≤ Y1(x) < t j,2 ∩ sj,1 ≤ Y2(x)sj,2}

(20)

The corresponding nested sets are defined asAi = Bi ∪ Bj and Aj = Bj . Totest for edge effects, we require expressions for the direct and cross indicatorvariograms for indicators defined on the setsAi and Aj . If γ j, j (h) denotes the



290 Roth

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indicator variogram ofAj , then we find that

γ j, j (h) = Pr{x ∈ Aj ∩ x+ h /∈ Aj }

=t j,2∫

t j,1

dux

sj,2∫sj,1

dvx

t j,1∫−∞

∞∫−∞+

t j,2∫t j,1

sj,1∫−∞+

t j,2∫t j,1

∞∫sj,2

+∞∫

t j,2

∞∫−∞

× g1(ux, ux+h) dux+hg2(vx, vx+h) dvx+h (21)

where g1 and g2 are the bigaussian pdf’s of (Y1(x),Y1(x + h)) and (Y2(x),Y2(x+ h)), respectively, that also depend on the covariance models:ρ1(h) ={Y1(x),Y1(x+ h)} andρ2(h) = {Y2(x),Y2(x+ h)}. The bracketed part of Equation(19) defines the Pr{x+ h /∈ Aj }.

There is a link between the cross indicator variogramγi, j (h) and the directvariogram seen in Equation (21):

γi, j (h) = Pr{x ∈ Aj ∩ x+ h /∈ Ai } = γ j, j (h)− Pr{x ∈ Aj ∩ x+ h /∈ Bi } (22)

because of the disjoint nature of the setsBi andBj . The second term of the right-hand side can be rewritten in terms of these sets as

Pr{x ∈ Aj ∩ x+ h /∈ Bi }(23)

=t j,2∫

t j,1

dux

sj,2∫sj,1

dvx

ti,2∫ti,1

si,2∫si,1

g1(ux, ux+h) dux+hg2(vx, vx+h) dvx+h

According to Equation (3), there are no edge effects fromAj to Ai if pi, j (h) =γi, j (h)/γ j, j (h) does not depend onh. Given the relationship (22) this conditionis satisfied if the ratio of Equation (23) to Equation (21) is independent ofh. Allother cases are therefore imply the existence of edge effects between the setsAi

andAj .Let us concentrate on the no edge effect case, ignoring the nonsensical solution

t j,1 = ti,1 andt j,2 = ti,2 andsj,1 = si,1 andsj,2 = si,2, which implies thatAi = Aj

or Bj = [. Similarly, we will not consider pure nugget effect covariance modelsfor ρ1(h) andρ2(h). In the general case, there are no edge effects only if we cansimplify the four quadruple integrals in Equation (21) into the one in Equation (23).The only possible solutions are given in Table 1.

The constantsr, s, u, andv can take any value such thatr < s andu<v.These four different cases are presented graphically in Figure 4, where the setBi

is presented in light grey andBj in dark grey. Because of the symmetry of the



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Table 1. Threshold Values Leading to No Edge Effects WhenGoing fromAj to Ai

Y1(x) Y2(x) Y1(x+ h) Y2(x+ h)

Case t j,1 t j,2 sj,1 sj,2 ti,1 ti,2 si,1 si,2

1 −∞ ∞ −∞ r u v s ∞2 −∞ ∞ s ∞ u v −∞ r3 −∞ u −∞ ∞ u ∞ r s4 v ∞ −∞ ∞ −∞ v r s

Figure 4. Partitions of the bigaussian space that satisfy the no edge effect conditionwhen going fromBj (the dark grey set) to{Bi ∪ Bj } (the union of the dark and lightgrey sets).




Gaussian pdf, case 1 is the same as case 2 whens = −r . Likewise, case 3 isidentical to case 4 whenv = −u. This basic configuration is of two rectanglesforming a T-shape in bigaussian space. This T-shape could also correspond to anL-shape when the thresholds defining the setBi are defined up to±∞. That is,whens= ∞ or r = −∞ in cases 1 and 2 or whenv = ∞ or u = −∞ in cases 3and 4. It is interesting to note that the two lithofacies must touch for there not tobe edge effects between them.

Incorporating information about the presence or absence of edge effects im-proves the resulting plurigaussian simulation by limiting the different ways ofpartitioning the bigaussian space (Y1,Y2) to only those that will reproduce theedge effect characteristics of the experimental data. For example, from the par-titioning examples seen in Figure 3, only Type II or Type IV are applicable ifthe experimental geological data confirmed the absence of edge effects betweencertain lithofacies.

PRACTICAL APPLICATION

Consider the idealized cross-section (Kerswill, Henderson, and Henderson,1996) of the ore unit of a Canadian stratiform banded iron formation (BIF) hostedgold deposit shown in Figure 5. According to the authors, “the figure is drawn toscale for the stratiform portions of the centre zone and is based on detailed surfaceand underground mapping, drill core logging, and petrographic investigations.”Much of the gold is uniformly disseminated in thin, laterally continuous units ofsulphur-rich BIF and nodules of arsenic-rich sulfide-BIF while the rest is containedin the late quartz veins that have overprinted the folded BIF. Mine managementwould like to simulate different possible realizations of the ore bearing lithofaciesfor mine planning purposes. The two distinct orientations of these gold-bearinglithofacies, down dip for the late veins and horizontally for the sulfide-BIF, makethis ore unit an excellent candidate for plurigaussian simulation technique basedon two independent Gaussian variables likeY1 andY2.

From the original image, we extract five lithofacies of interest: the three ore-bearing lithofacies, the alteration zones around the late quartz veins, and the hostrock. These lithofacies correspond to four disjoint setsB0 to B4 shown in Figure 6,where the higher the index of the set the darker its color is:B0 corresponds towhite to B4 in black. The alteration zonesB3 have a very similar spatial behavioraround the late quartz veinsB4 in that they systematically encompasses the quartzveins, forming a layer between the veins and other lithofacies. There are thereforeobvious and systematic edge effects between the quartz veins and the alterationzone. Because of the interdependence between them, these two sets will be treatedas one, called the extended veins and denoted by{B4∪ B3}, when testing for edgeeffects with the other lithofacies.

A similar edge effect analogy can be made for the setB1, the nodules ofarsenic-rich sulfide-BIF. While their form is related to the horizontal extension



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Figure 6. Spatial distribution of the five lithofacies of interest.

of the BIF and host rock sets,B0 and B2, respectively, the spatial distributionof the nodules is necessarily associated with the location of the extended veins.It is therefore almost certain that there are edge effects when going from theextended veins{B4∪ B3} to the extended veins plus nodules{B4∪ B3∪ B1}. On theother hand, as the quartz veins seem to have randomly overprinted the underlyingstratiform BIF, it is much more likely that there are no edge effects between theextended veins and either the sulfur-rich BIF or the host rock; that is, when goingfrom {B4∪ B3} to {B4∪ B3∪ B2} or to {B4∪ B3∪ B0} respectively. Of course,there are many other possible combinations to test, but the last two seema priorithe most likely not to display edge effects.

These different pairs of nested of sets are coded as indicator variables, as inEquation (2). Calculating the relevant ratio of cross to direct indicator variogramsallows us to test for edge effects when going from one set to the next. Some of theexperimental results obtained are presented in Figure 7 where the variogram ratio isplotted against the calculation distance. The maximum calculation distance is equalto about 1/4 of the width of the rectangular field. Four calculation directions areshown: horizontally and vertically that correspond to the main axes of anisotropyof the BIF and host rock, and dipping at 45◦E (along dip) and 45◦W (across dip)as the directions of anisotropy of the extended veins.

Figure 7A shows that there are indeed edge effects going from the quartzveins B4 to the extended veins{B4 ∪ B3}. Additionally, we note that variogram



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Figure 7. Edge effect test: ratio of cross to direct indicator variograms.

ratio reaches a sill of one. This means that all pairs of points considered for thatcalculation distanceh are either in both sets or in neither set. So, whatever theposition ofx we find that (x+ h) /∈ {B4 ∪ B3} ⇔ (x+ h) /∈ B4. So using thedefinition in Equation (2), we find

p3,4(h) = Pr{x ∈ B4 ∩ x+ h /∈ {B4 ∪ B3}}Pr{x ∈ B4 ∩ x+ h /∈ B4} = Pr{x ∈ B4 ∩ x+ h /∈ B4}

Pr{x ∈ B4 ∩ x+ h /∈ B4} = 1

(24)

This result is of course specific to the spatial distribution of the veins and alterationzone of the case study treated. Edge effects are also seen in Figure 7B when goingfrom the extended veins{B4 ∪ B3} to the nodules of arsenic-rich sulfide-BIF{B4 ∪ B3 ∪ B1}. Here the behavior of the variogram ratio is similar to that of




the preceding figure. Figure 7C, however, presents a markedly different exampleof edge effects: when going from the extended veins{B4 ∪ B3} to the host rock{B4∪ B3∪ B0}, the variogram ratio now decreases with the distance. In fact thesethree examples, by showing how much the behavior of the variogram ratio can varyas a function of the calculation distance, increasing or decreasing and at differentrates for different directions, highlight the difficulty in quantifying the amount andtype of edge effect.

On the other hand, the precise case of no edge effects as presented in Fig-ure 7D when going from the extended veins{B4∪ B3} to the sulphur-rich BIF{B4∪ B3∪ B2} has been defined as corresponding a certain partitioning of thebigaussian space, to one of those presented in Figure 4. For the case study, theextended veins in Figure 7C correspond to the setBj and extended veins plussulphur rich BIF to{Bi ∪ Bj } in Figure 4. As mentioned earlier however, the fourcases of Figure 4 can be reduced to two by the symmetry of the Gaussian pdf, andthese two are equivalent if the variablesY1 andY2 are exchanged. We are thereforeleft with only one valid partition of bigaussian space as shown in Figure 8 for(a) the general case and for (b) the simplified case where one of the thresholds ofY2 has been set to negative infinity:r = −∞.

In practice, once we know this “imposed” partition that respects the lackof edge effects, the remaining steps of the plurigaussian simulations follow in alogical order. First, the threshold valuev of Y1 is determined by the total proportionof quartz veins and alteration zones in the study field, which is the mean indicatorvalue for the set defined by the extended veins{B4∪ B3}. The variographic analysisof this indicator allows us to determine the covariance model of the GaussianvariableY1. The threshold values of Y2 is chosen so as to respect the proportionof sulphur rich BIF, the setB2. Given these thresholds and the covariance ofY1,we can then determine the covariance of the second Gaussian variableY2 as the

Figure 8. Partition of bigaussian space: no edge effects from{B3 ∪ B4}to {B2 ∪ B3 ∪ B4}.



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one leading to the best fit of the indicator variogram of the setB2. Then all thatremains is to define the subdivision of our predefined partition in Figure 8 for theindividual disjoint sets. That is, the division of{B4∪ B3} into B4 andB3 and thatof the white zone of Figure 8 intoB0 andB1. This secondary partitioning is doneby taking the geological characteristics of the ore unit into account. We know thatthe alteration zone forms a layer between the quartz veins and the backgroundstratiform BIF and that the arsenic-rich nodules of sulfide-BIF are always attachedto the extended vein structures.

CONCLUSION

The presence or absence of edge effects between different lithofacies rep-resents supplementary information that must be taken into account when tryingto reproduce realistic images of the lithofacies via geostatistical simulation tech-niques. Experimental geological data can be used to characterize edge effectsby observing how the ratio of cross to direct indicator variograms evolves withdistance. The existence of edge effects is confirmed when this ratio depends onthe distance without specifying its behavior. It turned out that the ratio may in-crease or decrease and at varying rates. So further work will be required to in-corporate this type of information on edge effects into simulations. On the otherhand, no edge effects means that the variogram ratio is independent of the dis-tance and thus represents a precise condition that the lithofacies simulation shouldrespect.

The indicator simulation techniques studied here all displayed different qual-ities with respect to the reproduction or not of edge effects. Simple experimentaltests for edge effects can be used to identify the appropriate simulation techniquefor a given data set. Theoretically, sequential indicator simulations and simulatedannealing are able to incorporate edge effect constraints. In practice, however, theirdomain of application is somewhat limited because of the difficulty in inferring avalid direct and cross variogram model for the former, and because of the delicateadjustment of tuning parameters for the latter. In its simplified form, however,sequential indicator simulations are not able to reproduce the presence of edgeeffects between lithofacies. This is somewhat disappointing given the very com-mon occurrence of edge effects. In contrast to this the truncated Gaussian methodassumes that there are always edge effects and as such would not be suitable whenit is shown that there are none. Last, it was shown that the plurigaussian simula-tion technique is much more flexible and can accommodate both the presence andabsence of edge effects between lithofacies.

Plurigaussian simulations respect experimental findings because the edgeeffects constraints between the lithofacies are integrated in the methodology viathe partitioning of the bivariate space. So practically speaking this methodologyis the most general of the lithofacies simulating techniques, being able to produce




geological images that respect not only the proportions and anisotropies of thedifferent lithofacies, but also their spatial layout relative to one another.

REFERENCES

Aarts, E., and Korst, J., 1989, Simulated annealing and boltzmann machines: John Wiley & Sons, NewYork, 328 p.

Alabert, F., 1987, Stochastic imaging of spatial distributions using hard and soft data: unpublishedmaster’s dissertation, Stanford University, California, 197 p.

Beucher, H., Galli, A., Le Loc’h, G., Ravenne, C., and the HERESIM Group, 1993, Including a regionaltrend in reservoir modelling using the truncated Gaussian method,in Soares, A., ed., GeostatisticsTroia ’92: Kluwer, Dordrecht, p. 555–566.

de Fouquet, C., Beucher, H., Galli, A., and Ravenne, C., 1989, Conditional simulation of randomsets. Application to an argilaceous sandstone reservoir,in Armstrong, M., ed., Geostatistics:Proceedings of the Third International Geostatistics Congress, Avignon, France, 5–9 Sept. 1988:Kluwer, Dordrecht, p. 517–530.

Deutsch, C. V., 1992, Annealing techniques applied to reservoir modeling and the integration ofgeological and engineering (test well) data: unpublished doctoral dissertation, Stanford University,California, 306 p.

Deutsch, C. V., and Journel, A. G., 1992, GSLIB: Geostatistical software library and user’s guide:Oxford University Press, New York, 340 p.

Freulon, X., and de Fouquet, C., 1993, Conditioning a Gaussian model with inequalities,in Soares, A.,ed., Geostatistics Troia ’92: Kluwer, Dordrecht, p. 201–212.

Galli, A., Beucher, H., Le Loc’h, G., Doligez, B., and the HERESIM Group, 1994, The pros and cons ofthe truncated Gaussian method,in Armstrong, M., and Dowd, P., eds., Geostatistical Simulations:Proceedings of the Geostatistical Workshop, Fontainebleau, France, 27–28 May 1993: Kluwer,Dordrecht, p. 217–233.

Jacod, J., and Joathon, P., 1971, Use of random-generic models in the study of sedimentary processes:Math. Geology, v. 3, no. 3, p. 256–279.

Kerswill, J. A., Henderson, J. R., and Henderson, M. N., 1996, Distribution of gold and sulphides atLupin, Northwest Territories—A discussion: Econ. Geology, v. 91, no. 5, p. 957–963.

Le Loc’h, G., Beucher, H., Galli, A., Doligez, B., and the HERESIM Group, 1994, Improvementin the truncated Gaussian method: combining several Gaussian functions:in ECMOR IV: 4thEuropean Conference on the Mathematics of Oil Recovery, Røros, Norway, 7–10 June 1994,unpubl. conference proceedings.

Le Loc’h, G., and Galli, A., 1997, Truncated plurigaussian method: theoretical and practical points ofview, in Baafi, E. Y., and Schofield, N. A., eds., Geostatistics Wollongong ’96: Kluwer, Dordrecht,p. 211–222.

Matheron, G., 1969, Les processus d’Ambarzoumian et leur application en géologie, unpubl. internalreport, Centre de Géostatistique, ENSMP, Fontainebleau, 169 p.

Matheron, G., 1989, The internal consistency of models in geostatistics,in Armstrong, M., ed., Geo-statistics: Proceedings of the Third International Geostatistics Congress, Avignon, France, 5–9Sept. 1988: Kluwer, Dordrecht, p. 21–38.

Matheron, G., Beucher, H., de Fouquet, C., Galli, A., Guerrillot, D., and Ravenne, C., 1987, Condi-tional simulation of the geometry of fluvio-deltaic reservoirs,in SPE 16753, SPE 62nd AnnualConference, Dallas, Texas, September 27–30, p. 571–599.

Rivoirard, J., 1990, Introduction au krigeage disjonctif et à la geostatistique non linéaire, unpubl. coursenotes, Centre de Géostatistique, ENSMP, Fontainebleau, 84 p.



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Rivoirard, J., 1993, Relations between the indicators related to a regionalised variable,in Soares, A.,ed., Geostatistics Troia ’92: Kluwer, Dordrecht, p. 273–284.

Rivoirard, J., 1994, Introduction to disjunctive kriging and nonlinear geostatistics: Clarendon Press,Oxford, 181 p.

Roth, C., Armstrong, M., Galli, A., and Le Loc’h, G., 1998, Using plurigaussian simulations to repro-duce lithofacies with contrasting anisotropies,in Proceedings of the 27th International SymposiumAPCOM’98: Institution for Mining and Metallury, London, p. 201–214.

APPENDIX

Here we present the demonstration of the relationship (3) in the text, thatbegins with the definition of the cross indicator variogram:

γi, j (h) = 1

2E{(I i (x)− I i (x+ h))(I j (x)− I j (x+ h))} (A1)

where the indicatorsI i andI j , defined by Equation (2), are based on two nestedsetsAi andAj such thatAj ⊆ Ai . The cross variogram will only be nonzero whenthe two factors that make it up are both nonzero. However because of the nestedcharacter of the sets, we haveI j (x) = 1⇒ I i (x) = 1 andI i (x) = 0⇒ I j (x) = 0,thus limiting the occurrence of nonzero factors making up the cross variogram totwo possible events:

Pr[{I i (x) = 1∩ I i (x+ h) = 0} ∩ {I j (x) = 1∩ I j (x+ h) = 0}]or

Pr[{I i (x) = 0∩ I i (x+ h) = 1} ∩ {I j (x) = 0∩ I j (x+ h) = 1}] (A2)

These events can be further simplified to

Pr{I i (x) = 1∩ I j (x+ h) = 0} or Pr{I i (x+ h) = 1∩ I j (x) = 0} (A3)

These two events have the same probability of occurring given that the probabilitiesare symmetric in space. In the second event, we putx∗ = x+ h and consider thedistancek = −h. So by summing the two probabilities, the cross variogram canbe written as

γi, j (h) = Pr{I i (x) = 1∩ I j (x+ h) = 0} (A4)

which corresponds to the relationship (3) in the text of the paper. This result, pre-sented here for the cross variogram, is equally applicable to the direct variogramsγi,i (h) or γ j, j (h).

incorporating information about edge effects when simulating lithofacies

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