Page 131 - Modern Spatiotemporal Geostatistics

112 Modern Spatiotemporal Geostatistics — Chapter 5

The conceptual and technical issues introduced in the preceding discussion
are best clarified with the help of a few simple examples. Complex situations
can be studied as extensions of the simple, well-understood solutions to these
examples. Frequent references to Chapter 3 are made in the following examples.

EXAMPLE 5.3: Consider the empirical law of Example 3.6 (p. 77) which relates
standard penetration resistance X and vertical stress Y for a cohesionless soil.
In light of constraints (Eqs. 3.8 and 3.9), a ^-based prior pdf is given by

where
the subscripts a and a' account for all points considered. Other constraints
involving the multivariate pdf may also be imposed. Equation 5.16 is sub-
stituted into the BME equations (Eqs. 3.8 and 3.9) which together with the
normalization constraint can be solved for the Lagrange multipliers /ZQ, ^ a and
/W- This process will give us the ^-based prior pdf f s. Note that it is not
necessary to calculate the cross covariance between the penetration resistance
and the vertical stress which is, nevertheless, taken into consideration implicitly
in the BME equations above.

EXAMPLE 5.4: Let us study the physical law expressed by Equation 3.16 (p.
79). From the set of equations in Equation 3.17 we obtain the system of
moment equations below

where the subscript a = 1,..., m,k accounts for all times t a of interest. In
light of Equation 5.17 and taking ho(t a) =~go(t a) = 1 into consideration, the
9^-operator in Equation 5.7 is given by

From Equation 5.17 we can write in terms of the pdf

with / ?(Xma P; tmap) = exp [/i 0 + %}• By substituting Equation 5.18 into
Equation 5.19, we get