Geographical & Environmental Modelling, Vo!. 6, No. 2, 2002, 147-169 .r~ Carfax Publishing11" T'ylo,&","mGm"p

1

The Incorporation of Model Uncertaintyin Geostatistical Simulation

P. A. DOWD & E. PARDO-IGUZQUIZA

ABSTRACT A growing area of application for geostatistical conditional simulation isas a tool for risk analysis in mineral resource and environmental projects. In theseapplications accurate field measurement of a variable at a specific location is difficultand measurement of variables at all locations is impossible. Conditional simulationprovides a means of generating stochastic realizations of spatial (essentially geologicaland/or geotechnical) variables at unsampled locations thereby quantifying the uncer-tainty associated with limited sampling and providing stochastic models for 'down-stream' applications such as risk assessment. However, because the number ofexperimental data in practical applications is limited, the estimated geostatisticalparameters used in the simulation are themselves uncertain. The inference of theseparameters by maximum likelihood provides a means of assessing this estimationuncertainty which, in turn, can be included in the conditional simulation procedure. Acase study based on transmissivity data is presented to show the methodology wherebyboth model selection and parameter inference are solved by maximum likelihood. Theauthors give an overview of their previously published work on maximum likelihoodestimation of geostatistical parameters withparticular reference to uncertainty analysisand its incorporation into geostatistical simulation.

Introduction

Mineral resource and environmental projects are designed on the basis of variablesthat are subject to extreme uncertainty. This uncertainty arises both because of thenature of the variables and the cost of obtaining information about them. Geologicaland geotechnical variables can only be assessed and quantified on the basis of sparsedrilling and sampling programmes. Such programmes provide data on a relativelylarge scale, which is invariably an order of magnitude greater than the scale requiredfor modelling, prediction and risk assessment.

In a gold mining project even at the (relatively advanced) mine planning stage, thegrades of 4 m x 4 m x 5 m selective mining units may be estimated from the gradesof samples taken from drillholes on a 30 m x 30 m grid; geotechnical design is basedon the geotechnical properties of sparse samples often not even collected for thepurpose at hand. Risk assessment of hazardous waste disposal sites requires spatial

FA. Dowd, Department of Mining and Mineral Engineering, University of Leeds, Leeds LS2 9fT. UK Fax:

+ 44-(0)113-246-7310; E-mail: p.a.dowd@leeds.ac.uk

E. Pardo-Igzquiza, Department of Mining and Mineral Engineering, University of Leeds, Leeds LS2 9fT. UK

1361-5939 printf1469-8323 online/02/020147-23

DOl: 10.1080/1361593022000029476

IQ2002 Taylor & Francis Lld

148 P A. Dowd & E. Pardo-Iguzquiza

1

I

I

modelsof relevantgeologicaland geotechnicalvariablessuchasporosity,permeabil-ity, fracture networks and transmissivity.

Ultimately, quantified risk analysis requires an estimate of the likelihood, orprobability, of an event occurring. It may be argued that in the case of trueuncertainty it is not possibleto determine probabilities.However,this is a simplisticview of probability and, in the context of most of the assessments required in mineralresource and environmental applications, it is an incorrect view. What is required isthe generation of possible states of nature based on process models and then anassessment of the likelihood of particular events occurring given these states ofnature.

The possible states of nature in these applications are values of geological andgeomechanical variables which are interpreted as spatial random variables (or, in thegeostatistical terminology, regionalized variables). Geostatistical simulation providesa means of generating stochastic realizations of spatial variables and these can formthe basis of quantitative risk assessment. In essence, these realizations are treated aspossible realities and risk assessmentis conducted by subjectingthem to responsefunctions and observingthe frequencywith which specifiedcriteria are exceededorfail to bemet. An exampleis providedby the assessmentof the risk of contaminationof the water tableby leakagefrom a proposedundergroundhazardous wastedisposalsite. Geostatistically simulated models of rock properties, including fracture networks,porosity and permeability, can be subjected to fluid flow models to determine theproportion of the simulated models in which contaminant pathways can be foundfrom the disposal site to the water table. Examples of risk assessment for mineralresource extraction projects are given in Dowd (1994a, 1997). The assumptions inthis approach to risk assessment are:

.the spatial models of variability used to generate the simulations adequatelyquantify the sources of variability on all relevant scales;. the number of geostatistical simulations is sufficient to represent the range ofpossibilities and that the frequency of occurrence of these possibilities reflects theiractual likelihood of occurrence.

Whilst conditional simulation provides a meansof generatingstochasticrealiza-tions of spatial variablesit is based on a model of spatial variability that can onlybe inferred from sparsedata and the model itself is, therefore,subjectto uncertainty.A major criterion for assessing the performance of a simulation (or a simulationmethod) is the extent to which the simulated values reproduce the specified modelof spatial variability. However, the significant uncertainty associated with the modelraises serious questions about the results and use of simulated values in riskassessment.

In general, simulation is required when data are sparse and variability is erratic.In such cases the spatial model of variability is uncertain and the uncertaintyincreases with the variability and the lack of data. The model partially drives thesimulation (the extent depends on the simulation algorithm) and reproduction ofthis uncertain model is no guarantee that the simulation is an adequate representationof reality.

In this paper the authors propose the use of maximum likelihood methods toquantify the uncertainty associated with models of spatial variability and demonstratehow this uncertainty can then be incorporated into geostatisticalsimulation.A casestudy is used to illustrate the effect of model uncertainty on geostatistically simulatedrealizations of transmissivity.

l

Model Uncertainty in Geostatistical Simulation 149

1

The Geostatistical Framework

Values of spatial variables are measured at specific locations x. These values, z(x), atlocations x are interpreted as particular realizations of random variables, Z(x), atthe locations. The set of auto-correlated random variables {Z(x), xED} defines arandom function. Spatial variability is then quantified by the correlations among therandom variables.

The Universal Model

The experimental data are assumed to have been generated by the so-called universalmodel (or generalized linear model):

Z(x) = m(x) + R(x) (1)

where x denotes location, Z(x) is a random function, m(x) is the mean or drift, andR(x) is the residual. The drift is the mathematical expectation of the randomfunction: E[Z(x)] = m(x) and it is modelled as a linear combination of known basisfunctions (monomials) multiplied by unknown coefficients. In matrix notation:

J1= XP (2)

where J1is an n x I vector of means, X is an n x p matrix of monomials, and p is ap x I vector of unknown coefficients. The residual is a zero-mean term:

E[R(x)] (3)

characterized statistically by its second-order stationary covariance function:

C(h) = E{[R(x) - m(x)][R(x + h) - m(x + h)]}. (4)

On the assumption of second-order stationarity the variogram is defined as:y(h) = C(O)- C(h). There are several commonly used variogram models each ofwhich is defined by three parameters: a small-scale, or nugget, variance, Co, due tovariability that occurs on a scale less than the sample volume or sample spacing(including measurement error); a larger scale variance, C, due to variability on ascale larger than the sample volume; and a range of influence, a, that defines thedistance within which variable values are auto-correlated; the total variance (knownas the sill value) is Co + C. In practice, the larger scale variance (C) may besubdivided into any number of sub-scales (Ci, i = I, ... , n) each with its own rangeof influence (ai, i= I,...,n).

In matrix notation the universal model is

z = Xp + E (5)

where z is an n x I vector of experimental data and Eis an n x I vector of residuals.The mathematical expectation is then: E[z] = Xp and the covariance of the

residual is

COV(E)= E(u') = V (6)

1

I

150 P. A. Dowd & E. Pardo-Iguzquiza

1

where V is the n x n variance-covariance matrix and prime denotes transpose of thevector.

The universal model is completely specified by the order of the drift, the coefficients,p, and the parameters, 9, of the covariance, or variogram, model. In applicationsnone of the parameters are known in advance and they must be estimated from theexperimental data. Once the covariance model and the order of the drift have beenspecified the most critical step in geostatistical applications is the inference of theparameters {p, 9}.

Generalized Increments

Generalized increments (Matheron, 1973), or error contrasts, are linear combinationsof the data expressed as

y=pz (7)

where y is an n x 1 vector of generalized increments and P is an n x n transformationmatrix.

The matrix P is chosen such that

PX=o (8)

and in terms of generalized increments the universal model is

y = PXP+ PE = PE (9)

and the drift has been filtered out (although the order of the drift must still bespecified).

One possibility for P (Kitanidis, 1983) is to use the projection matrix:

P = 1 - X(X'X) - 1 X' (10)

The relation in (7) can then be written as

y = Az = AE (11)

where the new transformation matrix A is derived by eliminating p rows from matrixP where p is the order of the drift and is equal to the rank of the matrix X. Thisoperation reflects the fact that p of the increments are linearly dependent on the restof the increments (Kitanidis, 1983).

The first two moments of the generalized increments are

E[y] = 0

E[yy'] = AE[zz']A'.

(12)

Within the framework of the universal model the second moment becomes

E[yy'] = AVA' (13)

and the parameters, 9, of the variance-covariance matrix, V,can be estimated withoutthe need to infer the drift coefficients p.

.,

Model Uncertainty in Geostatistical Simulation 151

Geostatistical Simulation

Geostatistical simulation (Dowd, 1992; Journel & Alabert, 1989, 1990; Journel &Huijbregts, 1978; Journel & Isaaks, 1984) is a generalization of the conceptsof Monte Carlo simulation to include three-dimensional spatial correlation. Ageostatistical simulation is one in which:

. at sampled locations the simulated values of each variable are the same as themeasured (observed) values of those variables;.all simulated values of a given variable have the same spatial relationships observedin the data values (spatial correlation);.all simulated values of any pair of variables have the same spatial inter-relationshipsobserved in the data values (spatial cross-correlation);

. the histograms of the simulated values of all variables are the same as thoseobserved for the data.

The methods can be extended to most descriptive or qualitative variables simplyby defining the variables in terms of presence/absence at sampled and simulationlocations (Dowd, 1994b). When natural, physical structures are a significant sourceof variability and/or exert a significant controlling influence on other variables (e.g.geological controls on mineralization, lithostratigraphic controls on porosity andpermeability, rock types and rock properties may be significant factors in the physicaldistribution of grade) they, or at least their effects,must be included in the simulation.In some cases the modelling of categorical or descriptive variables may be anintermediate stage that provides a means of accurately modelling a quantitativevariable (e.g. gold grades associated with quartz veins) in other cases they may bethe object of simulation (e.g. flow zones for the prediction of groundwater flows).

Geostatistical simulation is now widely used and accepted as a method ofgenerating stochastic models of mineral deposits, hydrocarbon reservoirs and geo-logical structures which can then be subjected to various operational procedures(David et al., 1974; Dowd, 1994a; Dowd & David, 1976) for design, analysis andrisk assessment (Dowd, 1997).There are many applications described in the literatureusing one or more of the range of methods (Dowd, 1992) now available.

Maximum Likelihood

Maximum likelihood (ML) estimation is used extensively for the estimation ofunknown parameters of hypothesized probability density function (pdf) modelsusing experimental data sets that are assumed to be outcomes of independent andidentically distributed (with the hypothesized pdf) random variables. Under theseassumptions the joint pdf of n experimental data may be expressed as

n

p(z;9) = p(z! ;9)'P(Z2;9).. .p(zn;9) = flp(Zi;9)i= 1

(14)

where 9 is an m x 1 vector of parameters that define the pdf and fez; 9) is the jointpdf defined by 9 for the data z. The ML function is simply the joint pdf in (14)viewed as a function of the unknown parameters 9 and containing the data z.

The ML estimate of 9 is the value that satisfies all equality and inequalityconstraints for which the likelihood function attains its maximum value. As thelogarithm is a monotonic function, the value of 9 that maximizes feZ; 9) also

152 P A. Dowd & E. Pardo-Iguzquiza

~

maXImIzes In p(z; 9). The log-likelihood function is frequently used in order tochange multiplicative properties into additive ones. It is common practice to takethe negative of the log-likelihood function so as to change the maximization problemto one of minimization. Unless otherwise specified, the ML function will be takento be the negative log-likelihood function (NLLF):

1

I

I

L(z;9) = - L In {P(Zi;9)}i=1

(15)

and the ML estimates are the values of 9 that minimize equation (15).The heuristic argument for the ML estimator is that, amongst all sets of possible

values for the parameters, it yields the set that has the greatest possibility of givingrise to the observed sample (with the hypothesized pdf).

The attraction of ML estimation lies in its large-sample or asymptotic properties.Under certain regularity conditions (Norden, 1972) the ML estimator is consistent,asymptotically normally distributed and asymptotically efficient.

Maximum Likelihood in Geostatistics

In geostatistical applications the experimental data are spatially correlated and thusthe form of the joint pdf of the experimental data in (14) is inadequate. For reasonsgiven below the most convenient choice of an alternative model is the multivariateGaussian distribution (mGd):

p(z;9) = (2n)-n/2IVI-1/2 exp{ - ~(z - Jlyv-1 (z - Jl)}(16)

where 11denotes determinant and prime denotes transpose matrix.Although the method has been widely reported in geostatistical applications

(Dietrich & Osborne, 1991; Hoeksema & Kitanidis, 1985; Kitanidis, 1983, 1987;Kitanidis & Lane, 1985; Mardia & Marshall, 1984; Mardia & Watkins, 1989;Zimmerman, 1989; among others) it also has its detractors (Ripley, 1988, 1992;Warnes & Ripley, 1987).

There is a common misconception that ML is not applicable because of theassumption that the data come from a mGd, an assumption which, in practice, isimpossible to verify. A reasoned justification for the choice of the mGd is given inPardo- Iguzquiza (1998) but perhaps one of the best reasons, albeit empirical, is thatML with the mGd gives good results in practice.

In addition to the distributional assumption there are two further objections tothe ML estimation method:

. there are many instances where the ML estimator is biased;. the ML method is computationally more intensive than other methods.

The second objection is becoming increasingly irrelevant with the rapidly increasingpower and speed of computers. Moreover, a relatively new method-approximateML estimation (Pardo-Iguzquiza & Dowd, 1997; Vecchia, 1988) described here-significantly reduces the computational overhead of ML estimation. The referenceto bias cannot be considered a serious objection for several reasons:

1

Model Uncertainty in Geostatistical Simulation 153

1

.The bias tends to zero as the number of samples increases (in practice the bias issmall if the number of samples is large enough).

. On the basis of the mean square error (which is a trade-off between bias andvariance) the ML estimator may be better than many others..The bias can be corrected..It is possible to obtain unbiased estimators by a suitable transformation of theoriginal data (e.g. using generalized increments).

The negative log-likelihood function (hereafter referred to as the ML function)corresponding to the mGd of equation (20) may be written as

1 1 1L(z;e) = 2.nln(2n)+2.lnl VI+2.ln(z - Jlyv-I (z - Jl). (17)

The values of e that minimize (17) are the ML estimates. The covariance matrix canbe factored as

V = (J2Q (18)

where (J2is the variance and Q is the correlation matrix. Noting that

IVI = (J2n1VI (19)

and

V-I =(J-2Q-I (20)

the ML function (17) can be written as

nilL(P,(J2,e,z) = 2.ln(2n)+ n In(J+ 2.lnIQI+ 2(J2(z - XPYQ- I (z- XP) (21)

where e now represents the covariance parameters but no longer the variance.The ML estimate of P is obtained by minimizing (21) with respect to p. This

estimate, denoted by p, is identical to the generalized least squares estimate of P(Searle, 1971)

P = (X'Q-I X)-I X'Q-I Z. (22)

The value 62 that minimizes (21) is

62 = !(z - XPYQ-I (z - XP).n (23)

The ML estimates of the covariance parameters e are the values that minimize theexpressIOn:

. n nn 1 n . .L'(P, 62, e;z) = 2.ln(2n) + 2.- 2.ln(n)+ 2.lnIQI+ 2.ln[(z- XPYQ-I (z - XP)],

(24)

154 P A. Dowd & E. Pardo-Iguzquiza

""I

Restricted Maximum Likelihood Estimation (REML)

It has been argued that the simultaneous estimation of drift and covariance para-meters results in biased covariance estimates (Kitanidis, 1987; Matheron, 1971). TheREML method has been proposed (Kitanidis, 1987) as a means of reducing bias. InREML instead of working with the original data, one works with generalizedincrements (Matheron, 1973).

The ML function (21) with (11) takes the form

A m mm 1 mL(J2, 8;y) = 2In(2n) + 2 - 2 In(m) +2:lnIAQA'1 + 2In[y'(AQA')-ly]. (25)

The REML estimates of 9 are the values that minimize (25). REML is verysimilar to the ML estimation of generalized covariances in which case Q is replacedby the generalized covariance K. The estimator of the variance is

,2 y'(AQA')-ly(J= .n-p

(26)

Patters on and Thompson (1971) report the use of REML to estimate covariancecomponents although it is not clear why REML was preferred to ML (see Harville(1977) and the comment by Rao (1977)). In geostatistical applications, REMLhas been considered by several authors including Zimmerman (1989) and Dietrichand Osborne (1991), but attention has been focused on efficient algorithms withlittle or no emphasis on why REML should be preferable to ML.

As REML and ML are two different estimators, they should be comparedstatistically by comparing the sampling distribution of the estimates. One suchstudy is given in Pardo-Iglizquiza (1998). Of particular interest are the mean andthe variance of the sampling distribution:

9= E[O]

Var(O)= E[(O- 9)(0 - 9)'].

(27)

(28)

The bias b is defined as the difference between the mean of the samplingdistribution and the true value of the parameter:

b = 9 - 9. (29)

The variance of the estimator is the variance of its sampling distribution andquantifies the dispersion of the estimates around the mean. The standard error isdefined as the square root of the variance of the estimator.

The bias is related to the accuracy of an estimator and the variance to theprecision. In the evaluation of the performance of an estimator there is a trade-offbetween bias and variance. A badly biased estimator is as bad as an unbiasedestimator with a large estimation variance.

A true measure of the accuracy and precision of an estimator is provided by themean square error (mse). The mse is defined as the dispersion of the estimaterelative to the true value of the parameter rather than to the mean value of thesampling distribution:

mse = var(O) + bb'. (30)

In general the estimator with the lowest mse is preferred.

Model Uncertainty in Geostatistical Simulation 155

It can be shown (Kendall & Stuart, 1979) that the estimator of the variance (26)is biased by an amount equal to

b=_£(J2n (31)

where b is bias and n, p and (J2have already been defined.The negative bias leads to underestimation of the variance (on average). An

unbiased estimator may be obtained by multiplying the biased estimates by thefactor c defined by

nc=-

n-p(32)

Then

{f2= C.62 (33)

is the unbiased estimator with estimation variance

var ((f2) = C2 .var(&2). (34)

An unbiased estimator is obtained at the cost of increasing the sampling variance.The bias given by (31) decreases as the number of data increases, and increases

as the order k of the drift increases (in two dimensions, for k = 0, 1 and 2 thevalues of pare 1, 3 and 6, respectively). Thus, the amount of bias expected fordifferent values and for different orders of the drift can be assessed. For example,with k = 2 and n = 15, c in (32) is 1.666, as the expected bias is 40% of the valueof the parameter, i.e. on average, the variance is underestimated by 40% of its truevalue. Estimator (23) is seriously biased and can be corrected by (33) which impliesthat the sampling variance increases. The trade-off between bias and variance isgiven by the mse which is equal to the squared bias plus the variance. If the valuec is close to 1 the expected bias is small and the estimator may be consideredunbiased. In fact, the ML estimates are efficient and asymptotically unbiased.

An example of bias calculation and correction of covariance parameters estimatedby (24), for a simulated set of values, is given in Pardo-Iglizquiza and Dowd(1998c).

Minimization

The ML estimates of P and (J2 can be expressed analytically but the ML estimates

of the covariance parameters 9 require the numerical minimization of (24). Thisrequires an iterative procedure for minimization in an m-dimensional parameterspace.

The minimization procedure is the core of the ML estimation routine and thesuccess of the estimation is closely related to the performance of the minimizationprocedure. In addition, because each evaluation of (24) requires the inversion of ann x n matrix, rapid convergence of the minimization procedure is important forcomputational efficiency.A number of methods can be used to minimize (24), but inour experience five have been found to be particularly suitable: direct search, scoring

1

156 P A. Dowd & E. Pardo-Iguzquiza

,I

I

method, axial search, simplex method and simulated annealing. A description ofeach, together with a performance comparison and a description of a publicdomain program (MLREML) are given in Pardo-Iguzquiza (1996). An example ofminimization in a five-dimensional space using the simplex method can be found inPardo-Iguzquiza and Dowd (1998b). The conclusion from these studies is that thedirect search method is preferred when the number of parameters is less than three,and the simplex method in all other circumstances. The minimum can be verified byaxial search. If multiple extrema are expected, the simulated annealing methodshould be used for more than two dimensions in the parameter space.

Approximate Maximum Likelihood

The computational problems are caused by the evaluation of equation (21) and itsderivatives. In particular, the matrix Q of n x n data must be calculated and invertedas many times as are required for the minimization procedure to reach convergence.

The approximate maximum likelihood approach starts from the well-knownmultiplicative theorem which states that for any n events AI, Az, . . . , An' the followingrelation holds:

Pr (A 1nAzn... nAn) = Pr(AI)' Pr(AzIAI)" .Pr(AnIAI,Az,... ,An-d (35)

where Pr(A IB) is the conditional probability of A given B.Then, for the multivariate pdf:

n

p(y) = P(YI)' flp(YdYI ,Yz,... ,Yi-I)'i=Z

(36)

Using the argument that some information provided by the data is redundant(Vecchia, 1988), the following approximation can be used for the conditionalprobability:

p(YiIYi-I,Yi-Z," ',YI) ;:;;p(YiIYi-I,Yi-Z," ',Yi-m) (37)

with i-m> I.Thus, instead of minimizing the complete likelihood (21), the function minimized

is the NLLF derived from (36) and taking into account the approximation (37), i.e.the approximate negative log-likelihood function (ANLLF):

n

L(y) = -lnp(YI) - L Inp(YiIY)i=Z

forj=I,...,m. (38)

On the assumption that the experimental data {y} are muItivariate Gaussian, theconditional probability, P(Yi IYj),j = 1, . . . , m is also Gaussian for any i and any m,with mean vector (Graybill, 1976):

I!iU=Jli+VijVjjl(Yj-I!) (39)

1

Model Uncertainty in Geostatistical Simulation 157

and covariance matrix:

ViU = Vii - VijVjjl Vji (40)

where

., Yj is a m x 1 vector of experimental dataJ.lj is a m x 1 vector of meansJ.liis a 1 x 1 matrix of the means at each of the experimental locationsVii is a 1 x 1 matrix of the variances

Vij is a 1 x m vector of covariances between the ith point and the m points of vector YjVji is a m x 1 vector equal to the transpose of VijVjj is a m x m matrix of covariances between the points of vector Yjand themselves.

The following relations are obtained by taking into account the factorization (18):

J.liU=/li + QijQjjl(Yj-J.l) (41)

(42)Qilj=Qii-QijQjjlQji= l-QijQjjlQji'

The Gaussian conditional probability function is then

p(Y11Y2"",Ym)=P(Y1Iy) (43)

= (2n) -1/2CT-1IQI-1/2 exp[ - 2~2 (Yi - J.liurQil} (Yi - J.liU)]and the Gaussian pdf for the first data location is

P(Yl) = (2n)-1/2CT-lexp[ - 2~2(Yl-/ll)2J(44)

Introducing the notation

£i=Yi-J.li (45)

(46)

(47)

(48)

(49)

£j = Yj - J.lj

hi = Yi - J.liU= Yi - J.li- QijQjj 1(Yj - J.l) = £i - QijQjj 1£j

J.li=XiP

J.lj= XjP

where

£i is a 1 x 1 vector of the residual at the ith location£j is a m x 1 vector of residuals at the m locationshi is a 1 x 1 vector of the conditional residual at the ith locationXi is a 1 x P vector of basis functions at the ith location, where P is the number of

basis functions that depend on the order of the driftXj is a m x p matrix of basis functions at the m experimental locations.

158 P A. Dowd & E. Pardo-Iguzquiza

The ANLLF can be written as

2 I n I n2 n SI" " 2 -I

L(P,(j ,9Iy) = 2In(2n) +nln(j) + 2(j2 + 2i~ InlQiul + 2(j2i~ hi QiU'(50)

1

I

Taking the partial derivatives of the ANLLF with respect to the different para-meters and setting the resultingequations to zero givesthe approximatelikelihoodequations which can be solvedto givethe followingestimates:

n

P'=

YIXI + L Qil](Yi-QijQj}IY)(X.i-Qij%.~/X)i=2

(51)

X'I XI + " Q '-ll(X.- Q ..Q :-:1X.)

'(X..- Q ..Q :-:1X.)L... L} L L}}} } L L}}} }

i=2

2 " h2Q

-lSI + L... i iU

i=282 =

n (52)

The estimation of the covariance parameters 9is done numerically by minimizingequation (50) after substituting the estimates given by equations (51) and (52) for Pand (j2, respectively.

The uncertainty of the estimated parameters is assessed by the inverse of the Fisherinformation matrix which gives the variance-covariance matrix of the estimates. Thesquare root of the diagonal elements of this matrix is the standard error of eachestimate.

For the drift coefficients:

[

n

J

-l

Var(p) = (j2 X'1Xl + .~ Qii] (Xi - QijQj) 1Xl (Xi - QijQj) 1X)L-2(53)

and for the variance:

2(j6 n - 1

Var(82) = ~. [si +t hfQii] J - (j2n L-2

(54)

In practice, P and (j2 are unknown and are replaced by pand 82, then

Var(a-2) = 2(j4n . (55)

It may come as a surprise that the sampling variance given by (55) is the same asthat for a Gaussian variable when the samples are independent (and thus uncorre-lated), but it should be noted that the estimation of the variance (52) takes intoaccount the correlation among the data.

,

Model Uncertainty in Geostatistical Simulation 159

1

Var(9) is evaluated numerically by fitting a quadratic surface to the ANLLF atthe estimated minimum. The minimum of (52) is found by a minimization procedurewhich requires the evaluation of the equation at each step of the process. The mainadvantages of using the approximate likelihood instead of the complete likelihood are:

.Computational time saving. Each step of the minimization procedure requires theinversion of an n x n matrix (where n is the number of experimental data) andmatrix inversion is an n3 process. For example, n = 1000 data requires 109 opera-tions. The ANLLF method with an approximation of m = 10 requires n inversionsof m x m matrices or 106 operations achieving a reduction factor of 1000.

. Memory space saving. The working matrices are of size m x m in the approximatelikelihood instead of n x n if the complete likelihood is used.

A description of the method and a computer program is given in Pardo- Iguzquizaand Dowd (1997).

An Example- TransmissivityData

This example has been chosen because the data are in the public domain; thecomplete model inference case study can be found in Pardo-Iguzquiza and Dowd(l998b) and only the results are presented here. The original application (Gotway,1994) was for nuclear waste site performance assessment, where uncertainty in thegroundwater travel time of a particle is assessed through its probability densityfunction (pdf). This pdf is estimated by running groundwater flow and transportprograms with different transmissivity field inputs. These inputs are generated byconditional simulation which generates possible images of the spatial variability oftransmissivity that honour the experimental data and reproduce the model of spatialvariability inferred from the experimental data.

In this case study we have used the spectral decomposition method of simulation.This method assumes multivariate normality and the conditioning data were trans-formed to normally distributed values by interpolation with a standard normaldistribution followed by an inverse transform of the simulated values. The techniquesdescribed in this paper could, however, be used with most methods of geostatisticalsimulation.

The experimental data consist of 41 values of transmissivity measurements in theCulebra Dolomite formation in New Mexico, USA (Gotway, 1994, Table 1). Thedata are the decimal logarithm of transmissivity in units of m2 s -1. The spatial

Table 1. Estimated covariance model parameters (range and sill) using ML andREML

Estimator Sill Range

Drift order Estimate Standard error Estimate Standard error

ML 0 3.98 0.880 8.14 2.050REML 0 5.98 1.337 12.82 3.131ML 1 1.28 0.284 1.99 0.667REML 1 1.61 0.368 2.69 0.865ML 2 1.18 0.260 1.76 0.610REML 2 2.22 0.530 3.99 1.179

160

35

P A. Dowd & E. Pardo-Iguzquiza

30 I :""0' . . .:.. . . . . . ~ ~.. .. . . . . :. . . . . . . . ... . . .. . . . .

251 ~ : ;..0 ~ q.:.........

20l.. .. .. ..:.. .. .. . a:.. .. .. ?.:. q 0 ~ .

. 'oAJ"""""" .>- : 0 : 0 ~t18 0 : ":"""'"

15 : :...0...0.'. 00: 0 :. : : a 00" ci ':.. .. . .. ..;... .. .. ..

101""""':""""+?"""~""""~""""':'o'"...

51 ~ ?: ~..o.o..j :.........

iI

05

: a.10 302515

X20

Figure 1. Spatial locations of experimental data.

locations of the data are shown in Figure 1 where the x and y coordinates are inkm. The cluster of points in the 5 km x 5 km central area contains a1most halfthe data.

The histogram of the data values is shown in Figure 2 and the omni-directiona1variogram is shown in Figure 3.

ML and REML were used to estimate the parameters of an exponential covariancemodel for drift of orders 0, 1 and 2 and results are summarized in Table 1. Theresults in Table 1 show that the estimates of the covariance parameters obtained byREML are larger than the ML estimates. Table 1 also shows the standard errors(square root of the estimation variances) of the estimates. This parameter quantifiesthe uncertainty of the estimates and can be used to construct confidence intervalsprovidedthat a model is assumedfor the samplingdistribution of the estimates.The

14

12

~10c:Q)

5- 8~~ 6'5

g 4.Cl«

2

0

-12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2

log T

Figure 2. Histogram of experimental data.

1

I

Model Uncertainty in Geostatistical Simulation 161

2 4 6 8 10 12 14 16 18 20

Distance (Iag)

Figure 3. Omni-directional variogram.

uncertainty associated with REML estimates is higher than that of ML estimates,especially for drift orders 0 and 2. The variograms of the residuals for k = 0, 1 and2 are shown in Figure 4. The variogram of the residuals for k = 0 is the omni-directional variogram shown in Figure 3; the variograms of the residuals for k = 1and 2 are much more easily reconciled with those of second-order stationary randomfunctions.

The Akaike information criterion (Akaike, 1974) was used to select the mostappropriate drift order. The order chosen was k = 1. The variogram of the residualsfor k = 1 is shown in Figure 5 together with the model fitted by ML using theparameters givenin Table2.

E 2.5co

.~ 2.0tu> 1.5

l-+-k=0--8-k=1-.-k=214.0

3.5

3.0

1.0

0.5

0

2 4 6 8 10 12 14 16 18

Figure 4. Variograms of residuals for drift orders 0, 1 and 2.

Distance (Iag)

4.0

3.5

3,0

2.5Eco

g. 2.0'C:co> 1.5

1.0

0.5

0.0

0

162 P A. Dowd & E. Pardo-Iguzquiza

1.6

1.4

1.2

1

II

Ef:! 1.0Cl0

.~ 0.8~

.~ 0.6IJ)

0.4

0.2

0.0

2 4 6 8 10 12 14 16 18

Distance (Iag)

Figure 5. Residual variogram for drift of order k = 1 and model fitted.

Table 2. ML estimates and standard errors offirst-order drift coefficients

The ML estimates and associated standard errors for the k = 1 coefficients areshown in Table 2. Although the drift is a deterministic component in the universalmodel, in practice the coefficients are estimated from the experimental data and theyare thus random variables with the means and standard deviations given in Table 2.This means that the drift is also uncertain and the information in Table 2 can beused to construct confidence intervals to quantify the uncertainty associated withthe estimated drift.

The model adopted is a universal model with drift order k = 1with drift coefficientsgiven in Table 2 and a zero-mean residual with isotropic exponential covariance withsill 1.28 [log(m2s- t )]2 and range 1.99 km (practical range approximately6 km).There is an uncertainty associated with this model, part of which is difficult toevaluate and involves the model itself; the other part is merely a statistical uncertaintydue to the inference of the parameters from a limited number of data and has beenassessed by the standard errors of the estimates. This latter uncertainty can bequantified; for instance interval estimates may be constructed assuming a model forthe estimation error, for example Gaussian. In this way the 95% confidence intervalsfor sill and range are [0.71, 1.85] and [0.66, 3.32], respectively, and are obtained asthe estimates::!::twice their standard deviation.

In this case, as there is no nugget variance, range and sill are estimated indepen-dently by ML. The correlation between range and sill is thus zero and anycombination of values of the parameters inside their respective intervals is inside the

Parameter Estimate Standard error

{3t - 1.6062 0.8653

{3z -0.2245 0.0426

{33 -0.0141 0.0323

1

I

Model Uncertainty in Geostatistical Simulation 163

l. A

E

C

0.5 5.0

Figure 6. 95'Yoconfidence region for sill and range.

95% confidence region as shown in Figure 6. This is useful when using conditionalsimulation in uncertainty analysis. Instead of using only the estimated parameters(the centroid of the rectangle in Figure 6), the extreme cases (though still inside the95% confidence region) represented by the combination of parameters (sill, range)given by the corners in the rectangle of Figure 6 may be used. For example, the upperright corner represents greater continuity (range 3.32 km) and greater variability (sill1.85[log(m2 s- 1)]2), that may produce spatial variabilitypatterns of transmissivitydifferent to those using the estimated values. The same may be said for the rest ofthe values in the 95% confidence region.

To see that the variance estimates are independent of the range estimates note:

. the factoring, in equation (18), of the covariance as the product of the varianceand the correlation;.equation (23) for the variance estimator is derived by setting the partial derivativeof the negative log-likelihood function to zero;. the range estimator is obtained in a similar way to the variance estimator althoughthe solution is numerical rather than analytical.

The drift estimates are also independent of the variance and the range but thethree drift coefficients are not estimated independently of each other. As the estimateddrift coefficients are correlated, not every combination of the three parameters isequally reliable, i.e. values that are inside the 95% confidence interval of eachparameter when taken together may not be inside the 95% confidence region for theparameters. The confidence region is not a parallelepiped but an ellipsoid defined bythe vector W = (/31'/32'/33)that verifiesthe relationship(Draper & Smith, 1981):

(~- prX'v - 1X (~- P) = [n ~ p ] Y' (Y - X~) Fp,n - Po1 - a

(56)

where Fp,n - p, 1- a is the 1 - a point of the F distribution with p and n - p degrees offreedom and a is the significance level.

3.0

2.5

2.0D

i ,+;g!

1.0

BI

0.5

1

164 P A. Dowd & E. Pardo-Iguzquiza

0

0.05

1

I

ba

-0.05

-0.10

-0.40 -0.35 -0.30 -0.25 -0.20b2

-0.15 -0.10

Figure 7. 95'10 confidence region for drift parameters [32and [33with [31= -1.6062.

Figure 7 shows the confidence region for the drift coefficients ([32' [33) when thethird coefficient [31in the model:

drift(x,y) = [31+ [32X + [33Y

is fixed to the estimate 131 given in Table 2.For illustrative purposes we have used the conditional simulation of the universal

model with drift: - 1.43324- 0.1393x + 0.00763y and variogram of the residual:y(h)= 1.85exp(- h/3.32).

The parameters used are not the estimates but they are inside the 95% confidencelevels. A contour plot of one simulation chosen at random is shown in Figure 8.Areas with log-transmissivitygreater than - 0.2 log(m2 s-1) (which represent hightransmissivity values) were shown by the darkest colour and highlight paths of hightransmissivity suggested by the simulation.

To assess the effects of model uncertainty on the simulation outputs six simulationshave been generated for each pair of values denoted by A, B, C, D and E in Figure6. These points denote the mid-point and the extremities of the 95% confidence

Figure 8. Simulated values from one simulation chosen at random.

1I

Model Uncertainty in Geostatistical Simulation 165

Figure 9. Output from six simulations using the estimated variance and range para-meters denoted by A in Figure 6 (centroid of rectangle).

region for the sill and variance. Each set of six simulations was started with the samerandom number seed. The simulation outputs are shown in Figures 9-13.

The differences in the six simulation outputs for each of the points A, B, C, Dand E is due entirely to the changes in the model and these changes reflect the rangesof uncertainty associated with the model parameters. Running groundwater flow andtransport programs using each of the simulations in Figures 9-13 as inputs wouldprovide an assessment of the effects of model uncertainty in risk assessment andallow these effects to be incorporated in the risk assessment.

Conclusions

A major deficiency in the use of geostatistical simulation is the common failure totake into account the uncertainty of the geostatistical models inferred from experi-mental data. The failure to do so can render invalid uncertainty models used for riskanalysis. The uncertainty of the covariance or variogram parameters, when estimatedby the classical non-parametric method, is difficult to evaluate. However, parametricinference methods, such as ML, estimate the variogram/covariance parametersdirectly and their uncertainty can be readily quantified. Geostatistical conditionalsimulation generates images of reality that model the uncertainty at non-sampledlocations. By including the estimation variance of the estimated variogram parametersit is possible to generate images that model both the uncertainty due to limitedsampling and the uncertainty of the variogram itself (which is also due to limitedsampling). An overview of ML methods has been given and the methodology hasbeen illustrated by application to a set of transmissivity data. In applications inwhich there are large numbers of data the approximate maximum likelihood method

166 P A. Dowd & E. Pardo-Iguzquiza

1

Figure 10. Output from six simulations using the extreme case variance and rangeparameters denoted by B in Figure 6.

..

Figure 11. Output from six simu1ations using the extreme case variance and rangeparameters denoted by C in Figure 6.

l

Model Uncertainty in Geostatistical Simulation 167

1

Figure 12. Output from six simulations using the extreme case variance and rangeparameters denoted by D in Figure 6.

Figure 13. Output from six simulations using the extreme case variance and rangeparameters denoted by E in Figure 6.

168 P A. Dowd & E. Pardo-Iguzquiza

can be used instead of the complete maximum likelihood; a case study illustratingthis methodology is given in Pardo-Iguzquiza and Dowd (1998a).

i

Acknowledgement

This work was supported by EPSRC (Engineering and Physical Sciences ResearchCouncil) grant number GR/M72944.

References

Akaike, H. (1974) A new look at the statistical model identification. IEEE Transactions on Automatic

Control, AC-19(6), 716-723.

David, M., Dowd, PA & Korobov, S. (1974) Forecasting departure from planning in open pit design andgrade control. In: 12th Symposium on the Application of Computers and Operations Research in the

Mineral Industries (APCOM), Vol. 2. Golden, CO: Colorado School of Mines, pp. F13I-FI42.

Dietrich, CR. & Osborne, M.R. (1991) Estimation of covariance parameters in Kriging via restrictedmaximum likelihood. Mathematical Geology, 23(7), 655-672.

Dowd, PA. (1992) A review of recent developments in geostatistics. Computers and Geosciences, 17(10),1481-1500.

Dowd, PA (1994a) Risk assessment in reserve estimation and open-pit planning. Transactions of the

Institution of Mining and Metallurgy (Section A: Mining Industry), 103, AI48-AI54.

Dowd, P.A. (1994b) Geological controls in the geostatistical simulation of hydrocarbon reservoirs. TheArabian Journal for Science and Engineering, 19(2B), 237-247.

Dowd, PA. (1997) Risk in minerals projects: analysis, perception and management. Transactions of theInstitution of Mining and Metallurgy (Section A: Mining Industry), 106, A9-AI8.

Dowd, PA. & David, M. (1976) Planning from estimates: sensitivity of mine production schedules toestimation methods. In: M. Guarascio, M. David & C Huijbregts, Eds, Advanced Geostatistics in theMining Industry, NATO ASI Series C: Mathematical and Physical Sciences, Vol. 24. Dordrecht: Reidel,

pp. 163-183.

Draper, N.R. & Smith, H. (1981) Applied Regression Analysis, 2nd edn. New York: John Wiley.

Gotway, CA. (1994) The use of conditional simulation in nuclear waste-site performance assessment.Technometrics, 36(2), 129-141.

Graybill, FA. (1976) Theory and Application of the Linear Model. North Sutuate, MA: Duxbury Press,704 pp.

Harville, H. (1977) Maximum likelihood approaches to variance component estimation and to relatedproblems. Journal of the American Statistical Association, 72(358), 320-388.

Hoeksema, R.1. & Kitanidis, PK. (1985) Analysis of the spatial structure of properties of selected aquifers.Water Resources Research, 21(4), 563-572.

Journel, AG. & Alabert, F (1989) Non-gaussian data expansion in the earth sciences. Terra Nova, I,123-134.

Journel, A.G. & Alabert, F (1990) New method for reservoir mapping. Journal of Petroleum Technology,February, 212-218.

Journel, A.G. & Huijbregts, C (1978) Mining Geostatistics. New York: Academic Press.

Journel, A.G. & Isaaks, EH. (1984) Conditional indicator simulation: application to a Saskatchewanuranium deposit. Mathematical Geology, 16(7),685-718.

Kendall, M. & Stuart, A (1979) The Advanced Theory of Statistics. Vo!. 2: Inference and Relationship,4th edn. London: Charles Griffin.

Kitanidis, PK. (1983) Statistical estimation of polynomial covariance functions and hydrologic applica-tions. Water Resources Research, 19(2), 909-921.

Kitanidis, P.K. (1987) Parametric estimation of regionalised variables. Water Resources Bulletin, 24(4),671-680.

Kitanidis, PK. & Lane, R.W (1985) Maximum likelihood parameter estimation of hydrologic spatialprocesses by the Gauss-Newton method. Journal of Hydrology, 79(1-2), 53-71.

Mardia, K.V & Marshall, R.J. (1984) Maximum likelihood estimation of models for residual covariance

in spatial regression. Biometrika, 71(1), 135-146.

Mardia, K. V & Watkins (1989) On multimodality of the likelihood in the spatial linear model. Biometrika,76(2), 289-295.

-:

Model Uncertainty in Geostatistical Simulation 169

Matheron, G. (1971) The theory of regionalized variables and its applications. Ecole Superieure des Minesde Paris, Fontainebleau, France.

Matheron, G. (1973) The intrinsic random functions and their applications. Advances in Applied Probability,5, 439-468.

Norden, R.H. (1972) A survey of maximum likelihood estimation. International Statistical Review, 40(3),329-354.

Pardo-Iglizquiza, E. (1996) MLREML: a computer program for the inference of covariance parametersby maximum likelihood and restricted maximum likelihood. Computers and Geosciences, 23(2), 153-162.

Pardo-Iguzquiza, E. (1998) Maximum likelihood estimation of spatial covariance parameters. Mathema-tical Geology, 30(1), 95-108.

Pardo-Iguzquiza, E. & Dowd, P.A. (1997) AMLE3D: a computer program for the statistical inference of

covariance parameters by approximate maximum likelihood estimation. Computers and Geosciences,23(7), 793-805.

Pardo-Iguzquiza, E. & Dowd, P.A. (1998a) Maximum likelihood inference of spatial covariance parametersof soil properties. Soil Science, 163(3), 212-219.

Pardo- Iguzq uiza, E. & Dowd, P.A. (1998b) A case study of model selection and parameter inference by

maximum likelihood with application to uncertainty analysis. Nonrenewable Resources, 7(1), 63-73.Pardo-Iguzquiza, E. & Dowd, PA. (1998c) The second order stationary Universal Kriging model re-

visited. Mathematical Geology, 30(4), 347-378.

Patterson, H.D. & Thompson, R. (1971) Recovery of interblock information when the block sizes areunequal. Biometrika, 58, 545-554.

Rao, 1.N.K. (1977) Comment to D.A. Harville (1977), Maximum likelihood approaches to variance

component estimation and related problems. Journal of the American Statistical Association, 72(358),338-339.

Ripley, B.D. (1988) Statistical Inference for Spatial Processes. Cambridge: Cambridge University Press.Rip]ey, B.D. (1992) Stochastic models of the distribution of rock types in petroleum reservoirs. In: AT

Walden & P. Guttorp, Eds, Statistics in the Environmental and Earth Sciences, Ch. ]2. New York: JohnWiley, pp. 247-282.

Searle, S.R. (1971) Linear Models. New York: John Wiley.

Vecchia, A. V. (1988) Estimation and model identification for continuous spatial processes. Journal of theRoyal Statistical Society (B), 50,297-312.

Warnes, 1.1. & Ripley, B.O. (1987) Problems with the likelihood estimation of covariance functions of

spatial Gaussian processes. Biometrika, 74(3), 640-642.

Zimmerman, D.L. (1989) Computationally efficient restricted maximum likelihood estimation of gen-eralised covariance functions. Mathematical Geology, 21(7), 655-672.