ETL 1110-1-175
30 Jun 97
kriging. However, the variogram that needs to be
mean-squared prediction error is smallest among
used in the kriging equations to make the kriging
all predictors that are linear in the measurements.
predictor equivalent to the spline predictor is
This optimality property is local, in that the mean-
determined by the basis functions selected for the
squared error of predictions at unsampled locations
spline. Because the type of basis functions used is
considered one at a time is minimized, without
subjective on the part of the user, the resulting
specific regard to preservation of global spatial
features. If, however, the actual realization z(x)
equivalent variogram may not be representative of
the true variogram of the data. Because kriging
could be compared to the kriged prediction surface
uses the data to indicate reasonable variogram
based on n measured values, the kriged surface
choices, kriging has an important advantage over
would be much smoother than the actual surface,
splines. Another advantage of placing the problem
especially in regions of sparser sampling. Thus,
in the kriging framework is the interpretation of the
the kriged surface will be a good and realistic
representation of reality in the sense that the n
smoothing parameter in terms of measurement
errors. In many cases, an objective estimate of the
measured values are honored, but it will be less
magnitude of measurement error can be obtained.
realistic with respect to global properties, such as
The connections between kriging and splines are
overall variability.
discussed further by Wegman and Wright (1983),
b. The purpose of simulation is to produce
Watson (1984), and Cressie (1991).
one or more spatial surfaces (realizations) that are
more realistic in preserving global properties than
7-7. Trend-Surface Analysis
the surface produced by interpolation algorithms,
such as kriging. These realizations are produced
a. Trend-surface analysis is the process of
by using numbers that are drawn randomly (Monte
fitting a function, such as that in Equation 2-43,
Carlo) to impart variability to the simulated sur-
using least squares to determine the coefficients
face, making the simulated surface more realistic in
that yield the best fit. Computationally, trend-
preserving the overall appearance of the actual
surface analysis is equivalent to universal kriging
surface. Generally speaking, simulation uses the
with an assumption that the Z*(xi) are uncorre-
idea that the true value of a random surface may be
lated. Thus, there is no need to estimate a vario-
expressed as the sum of a predicted value (which is
gram, and readily available regression packages
obtained by kriging) plus a random error, which
may be used for estimating the coefficients. As in
varies spatially and depends on the random
universal kriging, polynomial surfaces are the most
numbers drawn. Generally a number of indepen-
commonly used.
dent realizations will be generated, and these
realizations will be taken to be equally probable
b. When trend surfaces are applied in a sto-
representations of reality.
chastic setting, the resulting predictor will be opti-
c. A simulation algorithm is said to be condi-
mal if deviations from the surface are uncorrelated
and have a common variance.
tional if the resulting realizations agree with the
measurements at measurement locations x1, x2, ...,
xn. If the underlying process Z(x) is assumed to be
7-8. Simulation
Gaussian (or if a transformation may be found that
makes the process Gaussian), the most common
a. Consider again a regionalized random
method of conditional simulation is known as
variable Z(x), where x is a location in a two-
sequential Gaussian simulation (Deutsch and
dimensional study region R. Kriging is an inter-
Journel (1992), pp. 141-143). Another, more com-
polation algorithm that yields spatial predictions 8
Z
plicated, Gaussian simulation method that is par-
(x) that are best, or optimal, in the sense that has
ticularly useful for three-dimensional simulations
because of its computational efficiency is the
been discussed at some length in this ETL. The
7-5
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