Page 104 - Modern Spatiotemporal Geostatistics

Physical Knowledge 85

The knowledge base S in Equation 3.30 includes single-valued measurements
Xi (i = 1, • • • ,rrih) in space/time. Data of the form in Equation 3.30 usually
constitute the first act of a space/time analysis, and have been employed tradi-
tionally by classical geostatistics and spatial statistics techniques (e.g., Agter-
berg, 1974; Davis, 1986; Cressie, 1991; Kitanidis, 1997). In practice, hard data
may include measurement sets, meteorological surveys, remote-sensing obser-
vations, census data, etc.; and they are available on regular space/time grids,
lattices, arbitrary sampling networks, etc.

COMMENT 3.4 : I n most geostatistical applications i t i s presupposed that
the natural phenomenon under investigation has not been modified by the

experimental procedures leading to the data set (Eq. 3.30). If, however, one

is dealing with a situation in which the experiments modify certain features

of the natural variable, this effect should be taken into account by modern

geostatistical analysis.

Specificatory knowledge in terms of soft data
As we discussed in previous sections, observations presuppose a theoretical or
conceptual framework. Insofar as some of these theories are incomplete and
uncertain, the guidance they offer as to what kind of observations should be
made and in what manner could be incomplete and misleading (important
factors may be overlooked, etc.); see Shafer and Pearl (1990), for example.
An empirical feature of human knowledge is that it does not only rely on
sets of hard data and pure facts. It also includes incomplete or qualitative
data linked to experts' opinions, experience, intuition, etc. In the case of
such qualitative data, the impossibility theorem (Arrow and Raynaud, 1986)
shows that reconciling experts' opinions may imply a considerable amount of
uncertainty. The result of all this is the generation of an amount of uncertainty
about the observed variables.
In situations such as the above, the observation statements take the form
of soft data, which are assumed to be available at the remaining m s = m — mh
points, i.e.,

Soft data may represent varying levels of understanding of uncertain obser-
vations leading to the direct calculation of the probabilities or their indirect
estimation from accumulated experience. In fact, depending on the situation,
several types of soft data may be available to the geostatistician. A strategy
for evaluating the soft data types available in a particular situation would be
based on criteria such as consistency, completeness, and relevance to stated
objectives. A few specific examples are considered next.

s
EXAMPLE 3.14: Thex so/t ' often expressed in terms of intervals 7, of possible
values of the xt (* = fih + ,-••,m), i-e-,
!