Page 269 - Modern Spatiotemporal Geostatistics

250 Modern Spatiotemporal Geostatistics — Chapter 12

Also, the soft data available at estimation points are usually not taken
into account.
(c) Kriging techniques do not offer multipoint mapping.
(d) Most kriging techniques involve linear estimators (see ordinary, simple,
intrinsic kriging, etc.).
(e) Additional constraints on kriging techniques are often imposed on the
form of the estimator (unbiasedness, etc.).
(/) Kriging techniques are mainly interpolative (e.g., extrapolation is not
reliable beyond the range of the data).
(g) Specialized forms of kriging (indicator kriging, e.g.) do not account for
the monotonic cdf property, may lead to unfeasible probability values,
involve large numbers of kriging systems and variograms (some of them
difficult to model), etc.

(h) Standard practice in geostatistics does not address in a satisfactory man-
ner the circular problem (i.e., covariance or variogram models are esti-
mated empirically from the same data set that is used for kriging).

In contrast, none of these limitations apply to the BME approach. BME,
in fact, rigorously takes into consideration many forms of physical knowledge,
that improve the accuracy and scientific content of space/time mapping and
also provide the means to avoid the circular problem of empirical geostatis-
tics. General knowledge includes scientific laws, multiple-point statistics, and
empirical relationships. Soft data at neighboring points or at the estimation
points, themselves, are incorporated. Both single-point and multipoint map-
ping are allowed. Kriging estimators are based on the MMSE criterion that
may fail in the case of heavy-tailed random fields with large variances (Painter,
1998). In contrast, BME permits more flexible estimation criteria (e.g., pos-
terior pdf maximization) that are well-defined even for heavy-tailed fields. In
general, BME is a nonlinear estimator. No constraints are imposed on the esti-
mator being sought, non-Gaussian laws are automatically incorporated, and by
taking into account physical laws, BME possesses global estimation features.
These are significant improvements. Indeed, as emphasized by Stein (1999),
the linear estimators commonly used in spatial statistics can be highly ineffi-
cient compared to nonlinear estimators associated with non-Gaussian random
fields.
One may symbolically represent the application domains of the methods of
modern spatiotemporal geostatistics as in Figure 12.15, where: Epistemic D
BME D Traditional kriging. More specifically, one could write