Page 100 - Modern Spatiotemporal Geostatistics

Physical Knowledge 81

The function <fr may have the form of a finite difference of any
order, etc. One such case is discussed in the following example.

EXAMPLE 3.10: The partial differential equation (pde) governing subsurface
flow in two dimensions (Bear, 1972) is

where p = (si,s%,t), X is the random hydraulic head, Y is the random conduc-
tivity field, and S is the storativity. The flow equation offers a physical basis for
relating the stochastic moments of hydraulic head and hydraulic conductivity.
For the flow law (Eq. 3.25), a possible discretization is as follows

where Xi,j,k and t/Jij,k are, respectively, the head and conductivity values at
the space/time grid node (i, j, k) associated with (si, 52, t); Asi and As?
are spatial discretization steps along the si and s^ directions, respectively; and
A t is the time step. Equation 3.26 is an algebraic equation of the form given in
Equation 3.5 and, hence, we can proceed as in the case of the Class A problems
above. In a realistic study, discretizations (Eq. 3.26) at various sets of points
are considered.

COMMENT 3.3: A s w e shall see in Chapter 5 ("General knowledge i n th e

form of physical laws," p. 109), the fact that in many cases we do not need to

solve the differential equations for the space/time moments of X(p) —which

are implicit in these equations —has a special significance in the mathe-

matical formulation of BME. In certain applications, the calculation of the
~g^ statistics is based on a theory that may include correlation modeling

(Deutsch and Journel, 1992), stochastic pde's, and entropy maximization

(Christakos, 1992). Specific observational data sets may be involved in an

indirect way, e.g. , i n th e estimation o f th e parameters o f th e theoretical

model. This way is justified by means of an empirical process that moves

from a set of particulars to general knowledge.
Some other forms of general knowledge
Depending on the application of interest, several other sorts of general knowl-
edge may be incorporated into the BME analysis. In most real-world problems,
the development of useful general knowledge bases involves a constant balanc-
ing of theory and practicality.
In some cases a geostatistician may feel comfortable considering the uni-
variate pdf of the natural variable as part of the available general knowledge.