Page 185 - Modern Spatiotemporal Geostatistics

166 Modern Spatiotemporal Geostatistics — Chapter 9

the outcomes of BME investigations (predictive maps, uncertainty measures,
etc.) are the input parameters to subsequent steps of scientific decision making,
engineering design, etc.

Functional BME Analysis

There is a plethora of applications involving some kind of functional analysis.
This is, e.g., the case of ore mining that involves the spatial estimation of
large mining blocks from smaller core samples. Waste-site characterization
also depends on the analysis of contaminant processes at various scales and the
establishment of suitable quantitative connections between the results obtained
at each one of these scales. In this section we study functional analysis from
the perspective of modern spatiotemporal geostatistics.

General formulation

In natural sciences we are often seeking a spatiotemporal map of the following
general functional

where A is a space/time domain, and the form of the functional f may depend
on the physics or the economics of the problem. As usual, general knowledge
as well as specificatory data Xdata are available at points pi (i = 1, . . . , m), and
the data available are considered as point samples. Physically, the meaning of
the term "point sample" is that its size is much smaller than the space/time
distances considered in geostatistical analysis and, certainly, much smaller than
the size of A. Accordingly, the X(p) in Equation 9.1 is usually called a "point"
natural variable.
Important special cases of Equation 9.1 in Earth sciences and environmen-
tal health engineering include:
(a) The V-block average of the natural variable X(p) is given by

where and

(b) Th e temporally averaged exposure

where X(p k) is the exposure rate, A = r e denotes the exposure duration, and
f e is the exposure frequency (i.e., the fraction of total exposure time during
which the receptor is actually exposed; in %).