Page 189 - Modern Spatiotemporal Geostatistics

170 Modern Spatiotemporal Geostatistics — Chapter 9
point-block indicator covariance involved in indicator kriging; Journal, 1989).
BME analysis also provides a method for studying change-of-scale problems, as
well as applications related to effective parameters of random media (Christakos
and Hristopulos, 1998).

Multivariable or Vector BME Analysis

The BME analysis can be easily extended to include several natural variables
represented by a vector S/TRF. The primary natural variables are related to
certain secondary variables by means of a physical law, a theory, or an empirical
relationship (Chapter 3). In geochemical exploration, e.g., the grade of an
element (Fe^Os or Ag, etc.; a primary variable) could be related to geologic
characteristics of the region (lithological factors, etc., secondary variables).
Hard and/or soft data are usually available for some of these variables. This
sort of BME analysis is called multivariable BME, or vector BME or co-BME
analysis.

General formulation

The general formulation of the multivariable BME is as follows: Assume that
the primary natural variable X(p) is related to the N — 1 secondary variables
Y= (Y 2(p), . .,Y N(p)). We seek to estimate X(p) at point p = p k. The
•
basic BME equations for the situation are formulated as follows. The Lagrange
multipliers fi a are solutions of the system of moment equations

where and
The BMEmode estimation equation is

and the posterior pdf is given by

The g a are the important functions that incorporate general knowledge
in terms of Xdata ar|d * data (like in other kinds of BME analysis, the general
knowledge of vector BME may involve multiple-point statistics). To understand
these equations it is better to work out some special cases.
For illustration consider only two fields X(p) and Y(p). Assume that hard
data (Eq. 3.30, p. 84) and soft (interval) data (Eq. 3.32, p. 85) are available
for X(p) and Y(p) at points p i (i = 1,... , m; data for the two fields may be
available at the same or at different sets of points). An estimate of X(p) is