Page 97 - Modern Spatiotemporal Geostatistics

78 Modern Spatiotemporal Geostatistics — Chapter 3

where a' accounts for the pair of points Pt, PJ £ p map under consideration.
Other functions of Xi ar| d V'j resulting from the physical law (Eq. 3,7) may be
expressed in a similar manner, e.g.,

where a' = 1,..., N c now accounts for all combinations of points p i € p mop;
etc. (see also Chapter 9 on multipoint and multivariable BME analysis).
Class B: Physical laws that can be expressed in terms of the
general form

where D p is a differential operator in space/time, and G is an
algebraic function.

Equation 3.12 offers a powerful physical basis for relating X(p) with Y(p).
Two cases may be considered. First, assume that a solution to Equation 3.12
can be obtained so that we can write X = H(Y). Then, the representation of
Equation 3.6 is valid, where now the possible forms of the h a and g a functions
above include

i.e., the moments h a are expressed simply in terms of the known Y(p) statis-
tics. In the case that an explicit solution is difficult to obtain or does not exist,
the physical Equation 3.12 can be used in BME analysis by considering the
following representation

where the subscripts a account for all space/time points considered, and the
possible forms of the h a and g a functions include

The choice of the h a and g a functions above is not unique, since there is
a number of factors (mathematical and physical) that may influence such a
choice. It is possible, e.g., that one may need to consider functions that in-
volve derivatives of the pdf with respect to the space and time coordinates.
As it turns out, while the statistics of the random field X(p) are implicit in