• One of the foundation stones of geostatistics is the assumption that there is a spatial semivariogram function which is invariant under spatial translation. In
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One of the foundation stones of geostatistics is the assumption that there
is a spatial semivariogram function which is invariant under spatial
translation. In many geological situations of interest (I might even say
'most'), this assumption is patently false. For example, in a typical
hydrothermal gold deposit, the central part of the ore body will have high
gold grades - with high variance, and short ranges. On the margins of the
deposit, grades are lower, variance is lower, and the range will be longer.
In unmineralised rocks around the deposit, grades wil be very low, so will
the variance, and the range will be long as it is likely to be controlled
principally by lithological variation.

The geostatistician conventionally would give one of three answers to cope
with this:
(1) you should separate the data into different zones on the basis of
geological criteria, and model each zone separately.
(2) you shouldn't use linear geostatistics based on Matheron's principles,
but try one of the many other flavours of geostatistics which allow for the
observed effects
(3) don't worry about it: linear geostatistics is robust enough to cope with
such violations of its assumptions

Unfortunately, none of these is a satisfactory answer. Below are my comments
on each.

(1) Certainly you should always do this if it is possible. But who said that
there were geological boundaries in every case that allow definition of
zones ? If there are no boundaries then any zonation is artificial and the
geostatistical models produced will be incorrect. If there are boundaries,
and zones are defined, is it really likely that the semivariogram properties
will truly be homogeneous within each zone ? Possibly, but only if the zones
are drawn small enough - and then the resulting model will owe its estimated
values far more to the geological interpretation than to the geostatistical
modelling. Any simpler method of interpolation (eg weighted moving average,
or even polygonal) within such zones will give results which are just as
good as kriging.

(2) As soon as you move away from Matheron's linear geostatistics, which
does indeed have some nice statistical properties, you are entering a
twilight mumbo-jumbo world. Invoking 'higher powers' immediately junks most
of the mathematical elegance, especially in such abominations as indicator
kriging. Furthermore, with most such modified geostatistical variants, the
field of validity is even more restricted than with linear geostatistics. A
classic example of this, recognised even by most geostatisticians, is
lognormal kriging, which is invalid under even modest departures from true
lognormality.

(3) The relative robustness of linear geostatistics has been a factor in its
longevity as a method. However, this argument is also an acceptance that, in
situations where its assumptions are false, it cannot honestly be claimed to
be better than any other modelling method.

The fact that in many projects so-called advanced geostatistical modelling
methods have been used with apparent success is probably due more to the
fact that in these projects ANY method applied with sufficient care would
have been at least as successful. Especially if you stay with linear
geostatistics, based as it is on the weighted moving average, you are
unlikely to go too far wrong, because such methods are themselves fairly
robust with respect to the weightings used. At least the weighting method of
linear kriging incorporates an automatic de-clustering method.

However, it should seriously be asked whether the time and expense of using
geostatistical methods really is justified. In particular, it has always
bothered me that kriging uses a two-stage process in which subjective
interpretation of dreams (sorry, semi-variograms) is required before you can
move on to the modelling. And even more, that with a vast array of methods
to choose from, all blessed by the term 'geostatistics' and an array of
intensely mathematical published papers, you have complete freedom to select
the one which gives a client the answers they want - and it is always
possible to put together a justification for selecting that particular
method.

There are simpler and more objective methods and procedures which could be
used. For example - distance-weighted moving average combined with a
cross-validation procedure to select iteratively the best weighting
function. If there is a serious problem with clustered data this could be
preceded by a de-clustering step, or alternatively a set of de-clustering
weights can be applied to the data. If you specifically want a more robust
model (to deal with badly behaved data - noisy, or with highly skewed
distributions) then you could try a nonparametric approach (such as proposed
in my book 'Nonparametric Geostatistics' 1981 - see
http://www.silicondale.com/articles.htm)

Having said all that, I must still admit that over the past 30 years I have
used geostatistics (even, occasionally, the abominable indicator kriging) on
many real projects for real clients. However, this has always been with both
eyes firmly on the credibility and robustness of the model and usually one
eye on a comparison with models produced by simpler and more objective
methods like IPD moving average: and all three eyes on the geology. (yes,
today's geologist needs not just three hands, but three eyes too).

- Steve Henley
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