Page 153 - Modern Spatiotemporal Geostatistics

134 Modern Spatiotemporal Geostatistics — Chapter 6

(v) in order to account for hard and soft data, complicated combinations of
the various types of kriging techniques are considered, many of which are cum-
bersome and involve arbitrary approximations while lacking a sound theoretical
background; and (vi) variograms for very high and very low thresholds are
customarily difficult to model. BME does not need to make any of the above
assumptions and approximations and, thus, it does not suffer from any of the
limitations of the multi-Gaussian and indicator approaches. In addition, while
both of these approaches calculate local (i.e., single-point) probability distri-
butions, an important feature of BME is that it can calculate local as well as
global (i.e., multipoint) pdf. Instead of complicated combinations of various
types of estimation techniques, BME provides a unified general framework for
integrating and processing various kinds of hard and soft data.

COMMENT 6.3 : Some geostatisticicms ma y prefer t o concentrate o n th e

purely mathematical formulation of the BME space/time approach. In that
case, the five main steps involved in the formal BME approach are:

(i.) I n light o f th e general physical knowledge Q available, formulate th e

corresponding equations of the stochastic moments (see Eq. 3.1, p. 74).

(ii.) Assume apdfofthe general formo f Equation 5. 6 (p . 106) and (depend-

ing on the kind of moments involved in Eq. 3.1) select the g a-functions.

(iii.) Substitute Equation 5. 6 into Equation 3. 1 and solve for th e multipliers

fj, a. Insert these multipliers back into Equation 5.6 to the exact

form of the Q-based pdf of the map.

(iv.) I n light o f th e specificatory knowledge S available, develop th e cor-

responding hard and soft data parameters and operators (p. 82 and

Table 6.1, p. 133).

(v.) Insert th e S-operators o f (iv. ) togetherwith th e (f-based pd f o f (iii. )

into Equation 5.35 (p. 120) [or, Eq. 6.17, p. 132] to the Abased
pdf of the map. Select appropriate space/time estimates, depending on

the goals of the study (see also Chapter 7).

One may notice that the principle of maximum expected information (Pos-
tulate 5.2, p. 106) is not mentioned in the mathematical derivation of the

space/time equations according to steps (i-)-(v. ) above. I n fact, Equation

5.6 o f (ii. ) may b e considered as a reasonable mathematicalassumption con-

sistent with the ^-knowledge available. Thus, Postulate 5.2 may be viewed

by some geostatisticians a s optional, i.e. , t o b e used only i f they wish t o

provide an epistemic justification for the choice of the pdf form (Eq. 5.6).

In the following pages, the analytical results we obtained in the preced-
ing sections will be tested by means of synthetic examples in a controlled
environment and by real-world case studies. The distinction between the two
somehow resembles that between experimental and observational tests: While
observational data (or real-world studies) may be fraught with many uncon-
trolled variables, experiments (or synthetic examples, in this case) have the
crucial advantage that the scientist can control most of the variables except
the ones that are of particular interest.