Page 86 - Modern Spatiotemporal Geostatistics

Spatiotemporal Geometry 67

POSTULATE 2.11: The geometry assumed in a physical application can
have significant effects on the spatiotemporal map produced.
The term "map" in this case may include both spatiotemporal estimation and
simulation of natural variables (e.g., kriging estimation and turning bands sim-
ulation). Some practical demonstrations of Postulate 2.11 follow.
EXAMPLE 2.31: In Christakos et al. (2000a), a two-dimensional field with a
constant mean and an exponential covariance c x(h) = exp(—1.5 \h\) h =
(hi, h%) was considered. A metric should in principle be derived based on
a physical model of the field. For the sake of illustration, assume that the
physically appropriate metric for this field had the non-Euclidean form |/ii| +
|/i2 . Under these conditions, spatial estimates were generated on the basis of
a hard data set Xhard using a geostatistical kriging technique, thus leading to
the map in Figure 2.16a.
Practitioners of geostatistics often favor a theory-free approach that focuses
solely on the data set available and ignores physical models. By ignoring,
e.g., the underlying physics and using commercial geostatistics software (which
allows only for the Euclidean metric \//if + h% in covariance modeling and
kriging), the same data set Xhard as above results in the map of Figure 2.16b.
As was expected, the two maps display considerable differences. The map
generated by the standard means (Fig. 2.16b) is based on a convenient but
inadequate choice of metric, whereas the correct one (Fig. 2.16a) properly
accounts for the physical geometry.
Most mapping techniques of classical geostatistics have been developed
assuming Cartesian coordinate systems (e.g., Deutch and Journel, 1992). How-
ever, as is demonstrated in Example 2.32, this system of coordinates may be
inadequate for many physical situations and can lead to inaccurate maps.
EXAMPLE 2.32: Figure 2.17 presents variograms and maps of paleo-sea sur-
face conditions from a study by Schafer-Neth and Stattegger (1998). The vari-
ogram and map obtained using Cartesian coordinates are considerably different
than those produced using spherical coordinates. While analysis in terms of
spherical coordinates provided an adequate representation of spatial variabil-
ity, the Cartesian coordinates were shown to be inappropriate because they
caused a distortion of spatial dependencies in the case of large scales and led
to inaccurate predictions. Estimation in terms of spherical coordinates pro-
duced maps that were in better agreement with the data and offered a more
realistic description of physical reality than maps produced using Cartesian
coordinates.
The implications of producing an inaccurate map—as a result of making
the incorrect metric assumption—may be even more serious at subsequent
stages of BME analysis. For example, a map of porous geometry that is based
on a physically inappropriate metrical structure and that serves as input to
the subsurface laws could lead to erroneous predictions of subsurface flow and
contaminant transport processes.