Mathematical Geology, Vol. 30, No. 8, 1998

Is Lognormal Kriging Suitable for LocalEstimation?1

Chris Roth2

Lognormal kriging was developed early in geostatistics to take account of the often seen skeweddistribution of the experimental mining data. Intuitively, taking the distribution of the data intoaccount should lead to a better local estimate than that which would have been obtained whenit is ignored. In practice however, the results obtained are sometimes disappointing. This papertries to explain why this is so from the behavior of the lognormal kriging estimator. The estimatoris shown to respect certain unbiasedness properties when considering the whole working field usingthe regression curve and its confidence interval for both simple or ordinary kriging. When examinedlocally, however, the estimator presents a behavior that is neither expected nor intuitive. These resultslead to the question: is the theoretically correct lognormal kriging estimator suited to the practicalproblem of local estimation?

INTRODUCTION

Lognormal kriging took its place among the early developments in geostatisticsin the 1970s. The technique was developed because experimental mining dataoften showed a right skewed distribution that could more or less be fitted by alognormal distribution characterized by two or sometimes three parameters. Inthat situation, practitioners found that standard linear kriging methods lead toimportant local differences between real and estimated results when a few highexperimental values were smeared over a too large an area by linear interpola-tion. The idea behind lognormal kriging was to take advantage of the distributionof the data to reduce the influence of these few high values. So the data werefirstly transformed logarithmically and then kriged. The kriged estimates togetherwith a corrective factor are then back transformed to obtain the so-called log-normal kriging estimate. Numerous papers have been written about the theoryapplied to the different variants of the so-called lognormal kriging method. Just

1Submitted 14 May 1997; accepted 25 March 1998.2Centre de Geostatistique, 35 rue Saint Honore, 77305 Fontainebleau, France, e-mail: roth@cg.ensmp.fr

KEY WORDS: conditional expectation, conditional bias, regression, simple kriging, ordinarykriging, lognormal distribution.

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0882-8121/98/1100-0999$15.00/1 © 1998 International Association for Mathematical Geology

a few of the more important works are Aitchison and Brown (1957), Marechal(1974), Matheron (1974), Wainstein (1975), Parker, Journel and Dixon (1979),Rendu (1979), Journel (1980), Krige (1981), Dowd (1982), Magri (1983), orMiller (1983). Rivoirard (1990) provides a review of the different lognormalestimators commonly used in practice. The proof of a certain number of resultspresented here can be found in these works.

Intuitively, incorporating the lognormal experimental distribution in theestimation process should lead to a more appropriate estimate. In practice, how-ever, the lognormal kriging results have sometimes been disappointing. So themethodology has fallen out of favor with a certain section of the geostatisticalcommunity as being too sensitive or not robust enough. Despite this, lognormalkriging is still currently used, in South Africa, for example, and practitionerscontinue to battle to explain some strange local estimates obtained.

This paper examines the behavior of the lognormal kriging estimator. It willbe reviewed how the estimator is built so as to provide a conditionally unbiasedestimator for simple kriging and an unbiased estimator in the ordinary krigingcase. The unbiasedness will be shown to come from a majority of overestimatedreal values compensated by the underestimation of a few higher values. Thiscombination will be shown to produce local estimations that, while consistentwith lognormal kriging theory, do not correspond with what the practitioner intu-itively imagines from the kriging process. A simple example is used to highlightthe type of local estimations that can be obtained. The results seen there maywell explain why the lognormal kriging estimator is used cautiously in practice.

LOGNORMAL KRIGING

Given the large number of variants going by the name of lognormal krig-ing, let us begin by defining what we mean by lognormal kriging in this paper.So as to limit the need for supplementary hypotheses, the problem of changeof support is not treated here. Only point support estimation is considered fora supposedly multivariate lognormal distribution. The underlying hypothesis isthat the experimental data represent points from a second order stationary region-alized random function, denoted here by Z(x), whose multivariate distributionis lognormal. It follows that:

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where x is the coordinate vector in Rn, for n = 1, 2 or 3, L(x) ~ N(m, a2) is asecond-order stationary multivariate normal (or gaussian) random function andF(x) ~ N(0, 1) is a standard normal random function. The parameters m and a2

are called the logarithmic mean and variance respectively of Z(x). Equation (1)can be rewritten in a more convenient form that includes the mean of Z(x):

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The relationship between the covariance of Z(x) and the covariance of L(x) canbe deduced from the bivariate normal distribution of the pair (L(x), L(x + h)):

where h is the distance vector between two points, and CL(h) = Cov(L(x), L(x+h)) is the covariance of L(x) where Ct(0) = a2.

The Corrective Factor

The lognormal kriging (LK) estimator is obtained as the exponential of thekriged (K) estimator of L(x) multiplied by a corrective factor. This correctivefactor, denoted by K0(x), is calculated so as to ensure the unbiasedness of thefinal estimator: £(Z(x)Lk) = E(Z(x)).

The corrective factor depends on the difference between the variance of L(x)and L(x)K, which in turn depends on whether simple (SK) or ordinary kriging(OK) has been used:

where n denotes the Lagrange multiplier required in ordinary kriging. Note that,despite our simplified notation, the Lagrange multiplier and the kriging variancedo in fact depend on the point x being estimated and the data configurationretained.

In the simple kriging framework, the lognormal kriging estimator in (4)is optimal because minimizing the simple kriging variance Var(L(x) - L(x)sk)implies that the lognormal kriging variance Var(Z(x) - Z(x)LK) is also mini-mized. This is not the case for ordinary kriging: minimizing the ordinary krig-ing variance does not necessarily imply that the lognormal kriging variance isalso minimized. In this case, the optimal lognormal kriging estimator (Matheron,1974) can be obtained from a non linear system of equations. In this paper, how-

The width of the CI is depends only on the simple kriging variance. Becausethe median of the gaussian Lc(x) is equal to its mean L(x)SK, there is as muchchance of overestimating the real value as of underestimating it.

What happens in lognormal space where the value of Z(x)LSK limits therange of possible values taken by Z(x)? If Zc(x) denotes this restricted range ofvalues of Z(x) then:

Let us present these results graphically for the case of a unit logarithmic vari-ance and assuming asK = 0.25. In Figure 1, the lognormal regression (the thincontinuous line along the diagonal) is plotted against Z(x)LSK. The width of theasymmetric CI (the dashed lines) increases with the value of the LK estimator:the greater the value of Z(x)LSK, the greater the dispersion of the underlyingreal values is around it. The conditional probability density function (pdf) ofZ(x) (the thick continuous line) has been plotted across the CI to show the dis-tribution of real values for a given value of the estimator. Because the median ofthe lognormal distribution is less than its mean, the real value is overestimated(the area under the pdf below the diagonal) more often than it is underestimated(the area under the pdf above the diagonal). In our moderately skewed example

ever, we will limit our discussion to those estimators given in (4) because theyare the ones most commonly used in practice.

SIMPLE KRIGING CASE

To perform simple kriging we suppose that the mean of the underlying ran-dom function is known. It is well known that in gaussian space simple krigingprovides a conditionally unbiased estimator: E{L(X)\L(x)SK} - L(x)SK becausethe estimation error is independent of the estimator. So the value of the estimatorL(x)SK limits or conditions the range of possible values taken by L(x). If Lc(x)denotes the variable L(x) restricted by the value of the simple kriging estimatorthen Lc(x) ~ N(L(x)SK, a(x)sK). Using this fact we could, for example, definethe 95% confidence interval (CI) for Lc(x):

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where f/(x) - W(0,l) is a standard normal variable independent of L(x)SK.So for a given estimator Z(x)LSK the real value is lognormal with a mean:E{Z(x)\Z(x)LSK} = Z(x)LSK. The conditionally unbiased lognormal estimatorequals the conditional expectation. The 95% lognormal CI is defined as:

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(coefficient of variation of 1.31) there is almost a 70% probability that a givenpoint is overestimated. If we consider a slightly more skewed underlying distri-bution (ff2 = 1.5) then this probability could reach up to 90% given the moreasymmetric resulting CI. We could even produce an example of a very highlyskewed distribution where the conditional expectation lies above the upper limitof its CI.

Repercussions on Local Estimation

In practice, however, we deal with one realization of the underlying ran-dom function known at a few experimental points. The ensemble distribution isassumed to be invariant from point to point in the working field and that theexperimental distribution is assumed to match the ensemble distribution. So theensemble distribution can be replaced by the spatial distribution over the ergodicworking field. The average real value of those points with the same estimatedvalue is equal to the estimated value itself. A few highly underestimated pointsare enough to counterbalance the more common overestimation and reproducethe mean when considering all estimated points of the deposit. This balancingact between overestimation and underestimation allows the unbiased evaluationof the overall content in the working field even when considering only thosevalues above a given cut-off value.

Locally, however, the conditionally unbiased estimator leads to some sur-prising individual estimated points as will be shown using a simple one-dimen-sional example. Consider four regularly spaced data points with the same value,denoted by z0. In practical terms, this could be part of a deposit where the sam-ple values do not vary much. Suppose that Z(x) =exp(L(x)) where L(x) ~ N(0,1).

Figure 1. Lognormal SK regression curve ( )and its 95% confidence interval ( - - - - ) . The condi-tional pdf of Z(x) is traced (^^~) across the con-fidence interval.

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Figure 2. Profile of the gaussian SK estimator L(x)SK plotted against the position of the estimatedpoint. The sample locations are given by (•) and the gaussian mean E(L(x)) by ( - - - - ) .

We consider four cases where Z0 = 0.43, 0.78, 1.28, or 2.32 (or equivalentlyl0 = -0.84, -0.25, 0.25, and 0.84 where I0 = In(z0)). These are just the 20%,40%, 60%, and 80% quantiles of the respective distributions. Two sphericalvariogram models are used. The first, called Model A, has a range equal to 2times the sample spacing. For the second variogram model, Model 8, the rangeis halved to equal the distance between the sample points, which leads to a lessprecise estimation than for Model A.

Let us firstly consider the estimation of L(x). Figure 2 shows the profile ofthe L(x)SK plotted against the position of the estimated point x for both ModelsA and B. The four continuous lines correspond to the estimator for the four setsof sample values. The mean of L(x) is shown as a dashed line and the positionof the samples is given by the dots. The rightmost data point is considered asthe edge of the sampled area: the deposit lies to its left and extends beyond theleftmost point shown. As expected, the estimator tends to the mean as soon asthe estimated point does not coincide with a data value: 0 < \L(x)SK\ < |l0|because of the increasing weight Xm assigned to the mean. This is accentuatedfor Model B and for points lying outside the data zone when the estimator equalsthe mean beyond the range of the variogram.

Figure 3 presents the evolution of Z(x)LSK as a function of the position ofx after back transforming to lognormal space. The lognormal mean e1/2 = 1.65is shown as a dotted line and the samples are represented as dots. We find that:

• The estimator is greater than the sample value: Z(x)LSK > z0 for all z0.This is contrary to what one may have imagined for high values of z0

for which the estimator is further from the mean than the samples them-selves.

• For well estimated points (Model A) the size of the bounce that the esti-mator takes increases proportionally with the value of the samples. The

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Figure 3. Profile of the lognormal SK estimator Z(x)LSK plotted against the position of the estimatedpoint. The sample locations are given by (•) and the lognormal mean E(Z(x)) by ( - - - - ) .

estimator of the highest valued data goes further from the mean than theestimator of the lowest valued samples is drawn to it.

• For Model B, however, the estimator of the lowest valued samples showsa greater bounce toward the mean while the estimator of the highest val-ued samples only take a smaller leap away from it. This means howeverthat the estimator can bounce above the mean even though all samplesare far below it.

So, somewhat surprisingly, the lognormal kriging estimator does not always tendto the mean. Only once the ratio <2

SKf\m reaches a certain value compared withthe value of L(x)SK does Z(x)LSK tend to the mean, as is the case for points tothe right of the data zone. While this result is compatible with lognormal krigingtheory, there being no reason why Z(x)LSK should be convex, it does not corre-spond with what one intuitively expects from a simple kriging type estimator.

ORDINARY KRIGING CASE

Ordinary kriging is used when the mean of the underlying randomfunction is unknown. In gaussian space, the distribution of the estimator isL(x)OK ~ N(m, a2 - a2

OK + 2/i). Using the bigaussian distribution of the cou-ple (L(x),L(x)OK), we can obtain an expression for the real value conditionedby the ordinary kriging estimator:

where r = n/(a2 - o2OK + 2/i) is the ratio between the Lagrange multiplier and

the variance of the L(\)OK and l/(x) is a standard normal variable independent

of L(x)OK. The conditional distribution is therefore gaussian with a variance ofVar{Lc(x)} = a2

OK - pr and a mean of:

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So the OK estimator is conditionally biased with the amount of bias being deter-mined by the value of r. The linear gaussian regression is therefore tilted aboutthe point L(x)OK = m so that the underestimation of the low gaussian valuescompensates for the same degree of overestimation of the higher ones.

When backtransforming into the original lognormal space, we find:

which defines a conditional lognormal distribution whose 95% CI can be writtenas:

The mean of Zc(x), the lognormal regression, is given by:

So £{Zc(x)} > Z(x)LOK if Z(x)LOK < MeCov(L(x),L(x)OK) and E{Zc(x)} < Z(x)LOK

otherwise. Figure 4 shows the lognormal regression and 95% CI when a2 = 1,a2

OK = 0.25 and r = 0.1. The dotted line along the diagonal shows what wouldbe a conditionally unbiased estimator. The distribution of Zc(x) around its meanis lognormal (the thick continuous line traced across the CI). Compared with thelognormal SK result, there is now an even greater probability of overestimatingthe highest real values while the probability of overestimating the lowest realvalues is now lower.

Repercussions on Local Estimation

Ordinary kriging means that while globally the mean is reproduced, theaverage real value of those points with the same estimated value is not equalto that estimated value even when considering all points over the working field.This means that even the estimation of the overall content in the working fieldabove a given cut-off value is biased because the highest estimated values sys-tematically overestimate the reality.

The local behavior of the OK lognormal estimator is investigated using the

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Figure 4. Lognormal OK regression curve ( )and its 95% confidence interval ( - - - - ) . The condi-tional pdf of Z(x) is traced (^^—) across the con-fidence interval.

example presented in Figures 3 and 4 for the simple kriging case. In gaussianspace, the OK estimator equals the sample value: L(x)OK s l0 without respect tothe distance to the data or the sample value, as presented in Figure 5, for bothModels A and B. In lognormal space, however, applying the corrective factorleads to a different type of estimator:

The lognormal kriging estimator does equal not the sample value Z0 except atthe data points. Figure 6 shows that the estimator again bounces between thedata points such that:

Figure 5. Profile of the gaussian OK estimator L(x)OK plotted against the position of the estimatedpoint. The sample locations are given by (•).

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• The amplitude of the rebound depends on the value of the sample as indi-cated by Equation (14). The greater the sample value the further the esti-mator bounces away from it.

• This tendency is accentuated in the case of badly estimated points (ModelB) with the extreme case being those estimated points to the right of thelast sample. As the point gets further away, the estimator increases untilit reaches a given plateau.

So locally the characteristic behavior of the OK estimator to tend to thelocal mean of the samples is not respected after back transformation to lognor-mal space. The application of the corrective factor that respects the overall meanimplies that Z(x)LOK is always greater than the local sample mean. This is some-what surprising even though we know that the estimator need not lie within theconvex hull of the data.

CONCLUSIONS

The simple lognormal kriging estimator, equivalent to conditional expec-tation, allows the unbiased evaluation of the entire working field whatever thevalue of the estimator considered. This is because a majority of overestimatedpoints is compensated by the underestimation of a few of the higher real val-ues to reproduce the lognormal mean. Locally this balancing act means that theestimator does not necessarily tend to the underlying mean of the lognormalvariable.

If an ordinary kriging approach is preferred, then the practitioner mustaccept the conditional bias of the estimator. Any global estimation based on acut-off value will be overvalued because of an even greater overestimation bythe highest estimator values and less so for the lower estimator values. Locally

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Figure 6. Profile of the lognormal OK estimator Z(x)LOK plotted against the position of the esti-mated point. The sample locations are given by (•).

we find that forcing the global mean on the estimator implies that it does nottend to the local sample mean; in our example, it is always above. Moreover,when extrapolating, the estimator increases steadily until the estimated point isoutside the range of the variogram.

In view of these results detailing the estimation of individual points, thepractitioner must decide whether the lognormal kriging estimator is suitable forlocal estimation. Perhaps another type of estimator based, for example, on unbi-asedness with respect to the median or mode of underlying variable given thesurrounding samples, would be more appropriate. In this way the role of thecorrective factor can be limited or even eliminated to produce an estimator thatis globally biased but locally more consistent with the experimental values. Asa final word of warning, it must be stressed that the examples presented herehave only moderately skewed distributions. The anomalies would be much moremarked for more skewed distributions.

REFERENCES

Aitchison, J., and Brown, J. A. C., 1957, The lognormal distribution: Cambridge University Press,Cambridge, 176 p.

Dowd, P. A., 1982, Lognormal kriging—The general case: Math. Geology, v. 14, no. 5, p. 474-500.Journel, A. G., 1980, The lognormal approach to predicting local distributions of selective mining

unit grades: Math. Geology, v. 12, no. 4, p. 285-301.Krige, D. G., 1981, Lognormal-De Wijsian geostatistics for ore evaluation—Geostatistics I: S. Afr.

Inst. Min. Metall. Monograph Series, Johannesburg, 51 p.Magri, E. J., 1983, Improved geostatistics for South African gold mines: unpubl. doctoral disserta-

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Geostatistique, Ecole des Mines de Paris, Fontainebleau, 10 p.Matheron, G., 1974, Effet proportionnel et lognormalite ou le retour du serpent de mer: Technical

Report N-374, Centre de Geostatistique, Ecole des Mines de Paris, Fontainebleau, 43 p.Miller, S. L., 1983, Geostatistical evaluation of a gold ore reserve system: unpubl. MSc thesis, Uni-

versity of South Africa, Pretoria, 142 p.Parker, H. M., Journel, A. G., and Dixon, W. C., 1979, The use of the conditional lognormal prob-

ability distribution for the estimation of open-pit ore reserves in stratabound uranium deposits:A case study, in O'Neil, T. J., ed., Proceedings of the 16th APCOM, 16-19 October, Tuscon,Arizona: SME, Littleton, p. 133-148.

Rendu, J.-M., 1979, Normal and lognormal estimation: Math. Geology, v. 11, no. 4, p. 407-422.Rivoirard, J., 1990, A review of lognormal estimators for in situ reserves: Math. Geology, v. 22, no.

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