Page 237 - Modern Spatiotemporal Geostatistics

218 Modern Spatiotemporal Geostatistics — Chapter 11

a theoretical construction which encompasses the mathematical and physical
features that geostatisticians seek. Due to space limitations, theoretical for-
mulations for only a few selected general and specificatory knowledge bases are
considered below (some of these formulations have been used in applications
discussed in previous chapters). There are infinite possibilities, however, limited
only by the availability of physical knowledge bases in practice. As emphasized
throughout the book, the theoretical richness of the BME construction is a
powerful development in modern spatiotemporal geostatistics.
Table 11.1. The basic BME equations.

Equation* Eq.no./

* Equations appear on p. 1 75-1 76.

Ordinary Covariance — Hard and Soft Data

We start with the following fundamental proposition [the proof is very similar
to that of Proposition 10.1 (p. 198) and is not included here].

PROPOSITION 11.1: Let x hor. d be a vector of hard data at points
Pi (i = 1, 2, ..., irih) and \ soft be a vector of soft data (of various
possible forms; see Table 6.1) at points p t (i = m^ + 1, . . . , m). General
knowledge includes the mean and the (centered) ordinary covariance.
Then, the BME posterior pdf is given by Equation 9.32 (p. 176) with

and
where now

is the mean vector for points p it i = 1, ..., m, fci, . . , k p (notice the
.
difference compared to Eq. 10.3, p. 199), and

is the centered covariance matrix between all the points. The BMEmode