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),

Watson (1984), and Cressie (1991).

one or more spatial surfaces (realizations) that are

more realistic in preserving global properties than

the surface produced by interpolation algorithms,

such as kriging. These realizations are produced

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*(x*i) 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

representations of reality.

chastic setting, the resulting predictor will be opti-

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 *x*1, *x*2, ...,

Gaussian (or if a transformation may be found that

makes the process Gaussian), the most common

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*

plicated, Gaussian simulation method that is par-

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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