Page 252 - Modern Spatiotemporal Geostatistics
P. 252
Popular Methods in the Light of Modern Geostatistics 233
The following sections present analytical results and numerical applications
in which various sorts of geostatistical kriging are compared to BME-based
techniques, and by means of which the BME theory is amply confirmed.
Kriging Estimators
As we saw above, the BME theory preserves the referents of earlier geostatistical
results which are derived as its limiting cases. This powerful feature of BME
is demonstrated in this section by showing that it can easily reproduce certain
well-known kriging results. Of course, BME can lead to improved maps in
situations in which additional sources of knowledge become available (higher
order moments, multiple-point statistics, scientific theories, etc.).
Simple and ordinary kriging
We start with some basic results that are valid in the restrictive case in which
our knowledge consists of hard data and low-order space/time statistical mo-
ments.
PROPOSITION 12.2: When the general knowledge is limited to the
mean and (centered) covariance and the specificatory knowledge in-
cludes only hard data, the BMEmode estimate is
where, as before, x, denotes the mean value and c it} is the z'fc-th element
of the inverse (centered) covariance matrix c~^ ap. Equation 12.5 coin-
cides with the simple kriging estimate.
Proof: Under the conditions described in the proposition, the BME equation
reduces to
or
where and After
derivation, Equation 12.7 reduces to Equation 12.5.
Working along the same lines, the following proposition can easily be
proven.