ETL 1110-1-175
30 Jun 97
In short, the use of stochastic techniques provides
and makes explicit the background assumptions
the investigator with a way of objectively quanti-
that are being made.
fying errors and determining weights. In practice,
b. Important geostatistical concepts. Below
spatial predictions obtained using kriging are
almost always accompanied by a measure of the
are some of the key ideas in geostatistics that will
associated error. Most kriging practitioners
be given detailed attention in this ETL. They are
consider such an error evaluation to be an integral
introduced in much the same order that they are
part of the analysis, and point to error analysis as
discussed in Chapter 2, where more detail is
one of the principal advantages of using kriging (or
presented.
stochastic techniques in general) over other
procedures.
(1) Variograms.
(6) Nonstochastic techniques, on the other
(a) A central idea in geostatistics is the use of
hand, are typically applied strictly empirically,
spatial correlation to improve spatial predictions,
with no assumptions concerning the existence of an
or interpolations. The variogram is the principal
underlying random process and with no theoretical
tool used to characterize the degree of spatial
framework with which to evaluate statistically the
correlation present in the data and is fundamental
performance or optimality of the techniques.
to kriging. The correlation between measurements
When they are applied in such a manner, it is not
at two points is usually assumed, as described
possible to evaluate in advance whether such a
above, to depend on the separation between the two
procedure would be expected to yield results that
points. Values for all possible pairings of sample
are satisfactory. Two techniques that are com-
points can be examined by squaring the difference
monly applied in a nonstochastic setting are simple
between the values in each pair. The squared
averaging, mentioned above, and trend analysis,
differences are then categorized according to the
which is a least- squares method for fitting a
distance separating the pair. For small separa-
smooth surface to the data. Even though these
tions, or lags, the squared differences are usually
techniques are usually applied nonstochastically, it
small and increase as the lag increases. A plot of
is still possible to assess their performance if a
the squared differences per sample pair as a func-
stochastic setting is assumed. Loosely speaking
tion of lag is referred to as the sample variogram.
(these ideas are discussed more precisely in Chap-
ter 7), simple averaging would perform well if
(b) The general behavior of the sample vari-
there is no trend and no spatial correlation, and
ogram points relates to the spatial correlation
trend analysis would perform well if there is a
between sample sites and can provide investigators
trend that can be modeled, but no spatial correla-
with qualitative information about the spatial pro-
tion. Lack of correlation in the observations is one
cess, but in order to use this information in a math-
assumption that is made in ordinary statistical
ematically rigorous manner as a basis for inter-
regression analysis, and in fact trend analysis, if it
polation, a function with specific properties must
is placed in a stochastic setting, is actually one
be fit to the sample variogram points. The fit, as
special type of regression. The stochastic method
with all curve-fitting procedures, takes the scat-
of kriging explicitly incorporates the spatial corre-
tered points and passes a smooth curve through the
lations which are ignored in trend analysis. In
points. The curve, which can be represented by a
Chapter 7, a few other common techniques that are
mathematical expression or function, is called a
usually applied in a nonstochastic setting will be
model. Several named models with characteristic
discussed briefly. Most of these techniques are
features introduced in Chapter 2 are commonly
designed to incorporate the notion of spatial con-
used in geostatistics. The resultant variogram
tinuity, but the way it is incorporated may be
model is used to determine kriging weights for use
subjective. Kriging provides an objective means of
in interpolation.
incorporating the presence of spatial correlation
1-6
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