Page 129 - Modern Spatiotemporal Geostatistics

110 Modern Spatiotemporal Geostatistics — Chapter 5

with the moment equations]. More specifically, the main steps of the approach
are as follows:

(a) Transformation: Obtain the stochastic moment equations (Eq. 3.1 or
3.2) associated with the physical law. The way the transformation is
made depends on the form of the physical law.
(b) Formulation: Derive the form of the prior pdf (Eq. 5.6) in view of the
stochastic moment equations of the previous step.

(c) Solution: Insert the pdf (Eq. 5.6) into the stochastic equations and solve
for the Lagrange multipliers. These multipliers are then substituted back
into Equation 5.6 to obtain the final form of the prior pdf.

The BME analysis of physical laws above deserves some additional comments
(an outline of the analysis is given in Fig. 5.1). In the case of physical laws
represented by algebraic equations (see also Class A in Chapter 3, p. 77), the
moment equations in the transformation step (a) are of the form given in Equa-
tion 3.6 (p. 77); for physical laws expressed in terms of differential equations
(ordinary or partial; see Class B in Chapter 3, p. 78), the moment equations
are of the form given in Chapter 3, Equations 3.6 and 3.13 or Equations 3.14
and 3.15 (p. 77-78).
The formulation step (b) depends on whether or not the moment equations
in step (a) can be solved explicitly for the mean, covariance, etc. of the random
field of interest X(p). Two possible methods are proposed for handling the
situation (Fig. 5.1):
Method A: In many cases explicit solutions of the moment equa-
tions are intractable or the moments of X(p) are not known. In
these cases we incorporate directly the moment equations into
BME analysis using the stochastic physical law representations
discussed in Chapter 3.
Method B: In some cases the physical law is such that the moment
equations can be solved. The solutions may be exact or they may
involve some approximations in terms of perturbation expansions,
diagrammatic analysis, etc. (Christakos et al., 1999). In these
cases, the BME equations are essentially reduced to the set in
Equation 5.9.

COMMENT 5.4 : Note that a n interesting advantage o f Method A over
Method B is that the moment equations are rigorously taken into consid-

eration without solving them for the specified moments (which is the case
with Method B); see also Comment 3.3, p. 81. This avoids approximations

involved in the experimental calculation of the moments (e.g., Stein, 1999)
and eliminates the well-known circular problem of geostatistics: the data are

first used t o calculate th e mean, variogram, etc . (which ar e inserted into

the estimation system to obtain the kriging weights), and then the same