What's wrong with Indicator Kriging ?
Indicator kriging has become an accepted standard method for resource modelling since it was first proposed by Journel 18 years ago (Journel, 1983). It was developed largely because of the difficulty of dealing properly with non-ideal data distributions. During the 1970s methods had been developed, such as disjunctive kriging (DK) and multigaussian kriging (MG), to handle highly skewed or otherwise difficult distributions. The original such method, lognormal kriging, had been found sensitive to small deviations from true lognormality, while these new methods were difficult to use, and required deep understanding of the mathematical operations involved. Journel thus posed the question:
Is it possible to design nonparametric distribution-free techniques that can do the job of either DK or MG ? The job is essentially twofold:
- handle highly variant phenomena without having to trim off important high-valued data
- provide risk-qualified estimates of unknown values of z(x) and more generally of spatial distributions of these values z(x) within delimited areas.
The method he developed to answer this question was indicator kriging. The data are pre- processed. For a set of different cutoff grades, new sets of ‘indicator' values are defined: for each data point, at each cutoff, an indicator is set to 1 if the data value is below the cutoff grade, and zero otherwise. Variograms are then computed and modelled for each set of indicators, and used to krige the indicators for each cutoff. The resulting set of point or block indicator estimates can then be combined for each point or block to give interpreted grade estimates.
Although it has been applied very widely, being (apparently) almost as easy to use as ordinary linear kriging, nevertheless a number of problems have been found.
- Journel himself admits that if different variograms are used for different cutoffs, then internally inconsistent results might be obtained (the so-called ‘order relation' problem). He suggests a workaround for this problem, which is to use the median indicator variogram for all indicators. However, this would nullify one of the most important potential advantages of IK, that different variograms can be used for different indicators, reflecting the situation in which the spatial behaviour of a variable is dependent on its value. This is indeed the true ‘nugget' effect - that a high-grade (or ‘anomalous') sample is likely to be typical of a much smaller surrounding neighbourhood than a low-grade (or ‘background') sample. Journel's other suggested workaround, to eliminate order-relation problems by sequentially adjusting the estimates for successive indicators, is commonly done but is quite clearly an empirical fix without any theoretical justification.
- The stationarity assumption needed for indicator kriging is stronger than for ordinary kriging, and similar to the strong stationarity required for nonlinear methods such as DK and MG. This means that great care must be taken to define homogeneous geological zones within which to carry out the IK procedure - and even then such stationarity may not be assured.
- Even more seriously, the indicator variable for location x and cutoff grade z, i(x;z) is defined as a binomial [0,1] variable, and by definition every point x in the space being estimated has an indicator value i(x;z) of either 0 or 1 (in other words, the underlying grade value is either below the defined cutoff, or it isn't). Yet the application of variogram computation and kriging equations makes no use of this property but rather treats the indicator as if it were a measure on a continuous interval scale. The result is that kriged estimates of indicators are values of a continuous variable between 0 and 1. Numerically they bear no relationship with the underlying true field of indicator values. A kriged model of indicators cannot under any circumstances be considered a good model of the indicator field. While block estimates obtained from indicator kriging Ai(x;z) can be used to estimate (A;z), the proportion of grades below cutoff z within block A, and these estimates will naturally be expressed as continuous variables in the range 0 to 1, nevertheless the point estimates which are being integrated cannot themselves be other than [0,1] binomial values as by definition they are drawn from a field of such values.
It is perhaps unfortunate that - because of its relative ease of use - indicator kriging has become almost the standard method for modelling precious metal resources. It is unlikely that most users of the method are fully aware of its limitations and problems. The indicator estimation concept is indeed an interesting approach, but a valid method if using indicators to derive grade estimates must recognise the peculiar nature of the indicator population, in particular its discrete distribution, rather than simply using the same numerical methods which are used in ordinary kriging on data drawn from continuous distributions. The semivariogram function is a measure of the continuity of a variable at different spatial separations. the unknown population of indicator values is absolutely continuous at a constant value of either 0 or 1 - except along lines (in 2D) or surfaces (in 3D) where there is a step change between the two states. It is clear that a semivariogram is a quite inappropriate type of statistic to compute from such data.
Is there a way round this problem ? The answer is a definite ‘maybe'. I think that in any event a variogram-based method, imposing as it does rather onerous stationarity requirements, is perhaps not the best approach. There is not space here to outline possible alternatives, but if you have been following Silicon Dale you will expect to see some more on this question very soon.
Journel, A.G., 1983: Nonparametric estimation of spatial distributions, Mathematical Geology, v.15,no.3, p.445-468.
Copyright © 2001 Stephen Henley
What's wrong with indicator kriging ?: Earth Science Computer Applications, v.17,no.1,p.1-2