Page 192 - Modern Spatiotemporal Geostatistics

Modifications of BME Analysis 173

EXAMPLE 9.4: As general knowledge we consider the groundwater flow law
(Eq. 3.25, p. 81). The flow equation offers a physical basis for relating the
stochastic moments of hydraulic head and hydraulic log conductivity. A pos-
sible discretization of this law is given in Equation 3.26. Specificatory data
include the hard data {Xi,j-i,k, X»-i,j,fc. X»,j+i,fc. V'i-iAj.fc. ^»j+i/2,fc} and
the soft (interval) data {Xi+i,j,k, ^i,j-i/2,fc. i>i+i/2,j,k}• An estimate of X(p)
is sought at the point p = (i,j, k). Then, a possible term of the corresponding
BME equation may be as follows

where the vector I denotes the interval data domains. As was mentioned in
Example 3.10 (p. 81), in a realistic flow analysis several terms of the form
of Equation 9.21 will be included in the BME equation. Depending on the
situation (objectives of the analysis, statistics available, etc.), other forms of
the ^-operator are also possible. The BME equation should be solved for
Xi,j,ki usually numerically.

The analysis above suggests an interesting approach of studying stochas-
tic algebraic and differential equations representing physical laws. Generally,
given that a natural variable X obeys an equation of the form D(X, Y) = 0,
where Y— (Yi,...,Yfc) are observed variables, we wish to find solutions of
the corresponding stochastic expectation equation, say D(X, Y) = 0. The
traditional approach is either to solve the original physical equation for X and
then take the expected value of the solution, or to solve the corresponding
expectation equation directly for the moment of interest. Alternatively, BME
analysis suggests another way, as follows: Include the expectation equation
in the BME mapping process together with any other form of general and
specificatory knowledge available. BME will search for solutions such that the
expectation equation is an identity. A similar approach can be used in the case
of a series of physical equations

Transformation laws

Now we consider another interesting application of multivariable BME analysis.
Assume that a natural variable X(p) can be expressed in terms of a secondary
variable Y(p) by means of a transformation law of the form