Page 126 - Modern Spatiotemporal Geostatistics

Formulation of
the
Mathematical
Mathematical Formulation of the BME Method 107
BME
107
Method
where
where
Z
is the
given
by
partition
function
and
and Z is the partition function given by
In view of Proposition 5.1, the general knowledge equations (Eq. 3.2) are
written as follows

Equation 5.9 is part of the basic BME system of equations that we will develop
below for our spatiotemporal mapping purposes. The solution of the system of
N c + 1 equations (Eq. 5.9) determines the Lagrange multipliers /j, a.

C O M M E N T 5.2: Normalization o f the pdf leads t o th e following expression

for the partition function

Then, the general knowledge constraints (Eq. 5.9) are also written
Then, the general knowledge constraints (Eq. 5.9) are also written as as

Hence, th e functional lo g Z ca n b e viewed a s th e generator o f th e general

knowledge equations. The solution of the system of equations (Eqs. 5.10

and 5.11) also determines the Lagrange multipliers /j, a. Note that in this

system th e Lagrange multiplier ^o ha s been replaced b y Z, an d the number

of constraint equations is N c; the constraint for a = 0 has been replaced by

Equation 5.10.
General knowledge in the form of random field statistics
(including multiple-point statistics)
As we saw in a previous chapter, very often in geostatistical applications
the available knowledge has the form of random field statistics of any or-
der in space/time. In the language of geostatistics, the latter term includes:
space/time moments of any order (1,..., A) involving up to two points at a
time; and space/time moments involving more than two points at a time—also
called multiple-point statistics.
This sort of general knowledge is associated with a distinct class of 9£-
operators, which is best illustrated by means of examples. The first example