Page 232 - Modern Spatiotemporal Geostatistics

Single-Point Analytical Formulations 213

where K map is the generalized covariance matrix. Then,

and, hence,

From Equation 10.38 we get The
posterior pdf is given by

where the A, B, D and E s depend on the form of the soft data available
(Chapter 6). If, e.g., the soft data are of the interval form, Equation 10.39
reduces to

where the 9§ is given by Equation 10.37. In this case the BME equation
becomes

where Equation 10.41 can then

be solved with respect to the BME estimate
The choice of the S/TRF operator Q involved in the BME calculations
above should be made in a way that is mathematically rigorous as well as in-
ternally consistent (the Q-operator and the various physical theories and laws
governing the natural variables involved in a specific application must be in-
terrelated and corroborative). It is possible, e.g., that Q represents the finite
difference scheme obtained from the discretization of the differential equa-
tion law governing a physical phenomenon. In many cases, the form of the
S/TRF operator Q may change from one space/time neighborhood to another
(Christakos, 1992).

Some Non-Gaussian Analytical Expressions

The preceding analytical results are concerned about general knowledge bases §
that involve second-order space/time moments (e.g., covariance or variogram
functions). This essentially implies that the ^-based operator 9£ has one of
the familiar quadratic expressions, and the resulting prior pdf f s of Equation 5.6
(p. 106) has a Gaussian form. If the ^-base includes higher order space/time