Page 99 - Modern Spatiotemporal Geostatistics

80 Modern Spatiotemporal Geostatistics — Chapter 3

Spatial covariances, flow moments of higher order, etc. may be introduced
in a similar fashion.
EXAMPLE 3.9: Under certain assumptions, one-dimensional groundwater flow
may be represented by the differential equation (Bear, 1972)

where X is the random hydraulic gradient and Y is the random hydraulic
log-conductivity slope at a spatial location 5. Equation 3.21 leads to

where A = 1, 2 accounts for the moments considered, and the subscript a
denotes the location of interest. Equations for moments of higher order derived
from the flow law (Eq. 3.21) can be processed in a similar manner and the
analysis can be extended to the entire set of locations s map = (sj,..., s m, Sfc).
Suppose, e.g., that we seek to process knowledge about the covariance of X.
Then, Equation 3.21 gives rise to the system of equations

where the subscript a! accounts for the pair of locations s, and Sj under
consideration.

COMMENT 3.2: In some situations, th e h a functions ma y be formulatedin-

directly: the stochastic moment equations associated with the physical equa-

tions are derived and the h a functions are chosen so that they express the
statistical moments appearing in these equations (mean, variance, covari-

ance, etc.j. Se e also Chapter 5.

In many applications, it may be preferable to transfer a problem expressed
in terms of Class B physical laws into a Class A problem. This may be done in
the following manner.
Class B—>Class A: A Class B problem can be transferred to a
Class A problem by discretizing the differential equation represent-
ing the physical law, thus obtaining an algebraic equation of the
form given in Equation 3.5, i.e.,