Page 125 - Modern Spatiotemporal Geostatistics

106 Modern Spatiotemporal Geostatistics — Chapter 5
that fo cannot be determined in a convenient way for a specific mapping

application, the discrete-domain formalization of BME, which does not in-

volve a noninformative prior, can be used.
General knowledge-based map pdf

In BME analysis, the shape of the pdf f g at the prior stage is determined on
the basis of the following postulate.
POSTULATE 5.2: The prior (£-based) pdf f g of the map is obtained
by maximizing the expected prior information (Eq. 5.2) subject to the
physical constraints introduced by Equation 3.2 (p. 75).
The prior pdf suggested by Postulate 5.2 is associated with a certain
knowledge base and a method of using it. This ^-based pdf will be used in
subsequent BME stages. The application of Postulate 5.2 requires the introduc-
tion of the mathematical method of Lagrange multipliers (the basic concepts
of the method may be found in Ewing, 1969).
Lagrange multipliers method: The maximum value of the
integral

with respect to the function fg and subject to the conditions

can be found from the Euler-Lagrange equation

where fi a are Lagrange multipliers. The p, a are determined by
introducing the function f § found from Equation 5.5 into the con-
straints (Eq. 5.4).

In the case of BME analysis, ®[x, f§(x)] = - f$ (x) tog f @ (x) and
<Pa[x, /<?(*)] = 9a(x)f@(x)- The shape of the prior pdf f g(x map) is de-
rived by maximizing Equation 5.2 with respect to fg(Xmap)< subject to the
constraints imposed by general knowledge j. This is a straightforward exer-
(
cise of the Lagrange multipliers method, leading to the following proposition.
PROPOSITION 5.1: The mathematical implementation of Postulate
5.2 produces the prior pdf