Page 162 - Modern Spatiotemporal Geostatistics

The Choice of a Spatiotemporal Estimate 143

EXAMPLE 7.5: Serre and Christakos (1999b) considered Darcy's law governing
one-dimensional groundwater flow,

The specific discharge q(s) was assumed to be deterministic; the hydraulic
head H(s) and the hydraulic conductivity K(s) were considered as random
fields; and the value of the hydraulic head was assumed known at the spatial
origin s = 0. Then, a K(s) profile was generated that had an asymmetric
distribution. On the basis of this profile and Darcy's law, the actual head
fluctuation profile H(s) — H(s) was calculated and compared to the estimated
profiles as follows. In Figure 7.3a the estimated head fluctuation profile was
derived from simple kriging (SK) using only hard head data; the actual head
fluctuation profile is also shown for comparison. Note the poor SK estimates at
unobserved locations. The head fluctuation profile in Figure 7.3b was obtained
from BME using hard and soft (interval) head data as well as Darcy's law. Note
that the BME estimated profile is a substantial improvement over the classical
SK method. Indeed, by being able to incorporate Darcy's law, the BME method
allowed us to account for hard and soft (interval) hydraulic conductivity data,
as well. Serre and Christakos (1999b) also considered the problem of estimating
the hydraulic resistivity profile using sparse head and resistivity measurements
(this is sometimes called the inverse problem; Kitanidis and Vomvoris, 1983).
Again, the BME approach led to a considerably more accurate estimate of the
hydraulic resistivity profile than the kriging techniques.

The West Lyons Porosity Field

Porosity data were collected in the West Lyons field in west-central Kansas
(Olea, 1999). This comprises a spatial data set on a reservoir occurring in
Mississippian (Lower Carboniferous) sediments deposited in the shallow epi-
continental seas that covered much of North America in the Late Paleo-
2
zoic. The study area is approximately 2.5 x 4.5 miles . A total of 76 data
values were available, as shown in Figure 7.4. The general knowledge consid-
ered consists of the porosity mean and covariance functions, also plotted in
Figure 7.4.

COMMENT 7.2 : I n some practical applications, hard data ma y b e involved

in both the prior stage (indirectly) and the meta-prior stage (directly) of
BME. Particularly in cases in which a physical model is not available, the

prior constraints (e.g. , mean an d covariance) ma y b e calculated from hard

data sets alone. Thus, at the prior stage, hard data provide a partial char-

acterization of the underlying random field —since there are several realiza-

tions sharing the same statistics. Hard data considered at the meta-prior

stage are used in the subsequent integration stage, this time with the pur-
pose of providing the desired posterior estimate. This double role of hard