ETL 1110-1-175
30 Jun 97
Generally, the mean is assumed to have a func-
j wjDij %
j 8 k f k (xi)
n
p
1
tional dependence on spatial location of the form
s k=1
(2-46a)
j=1
(u, v) = j $j fj (u, v)
p
= Di0, i=1, 2, ..., n
(2-43)
j=1
where the fj(u,v)'s are known deterministic func-
j wj fk (xj)
n
tions of x=(u,v) (that is, these functions serve as
(2-46b)
independent variables) and the $j's are regression
j=1
coefficients to be estimated from the data. For
= fk (x ), k = , 1, 2, ..., p
example, suppose Z(x) is hydraulic head in an
0
aquifer. If the flow is in a steady state, it might be
reasonable to assume, in a given case, that the
where, in contrast to the ordinary kriging equa-
mean of Z(x) has a unidirectional groundwater
tions (2-35a and b), there are now p coefficients
81, ..., 8p resulting from the unbiased condition on
gradient that is given by
the predictor. The first term in the mean (Equa-
(u, v) = $1 % $2 u
tion 2-43) will usually be a constant, or intercept,
(2-44)
for which f1(x) = 1. Therefore, the universal krig-
In this example, there are two independent
ing model includes ordinary kriging as a special
case. The universal kriging variance is given by
variables:
(x ) = s 1 & j wiDi0
n
f1 (u, v) = 1
2
Fk
0
(2-45)
i=1
(2-47)
f2 (u, v) = u
& j 8k fk (x0)
p
and two regression coefficients ($1 and $2). The
k=1
mean can include other independent variables
besides simple algebraic functions of u and v. For
These equations can be easily solved to obtain
example, if the aquifer is not of uniform thickness,
universal kriging predictors and kriging variances
an independent variable that involves the aquifer
for any desired location. The estimated trend
thickness at location (u,v) could be included.
surface does not actually need to be computed to
obtain the universal kriging predictor. If a particu-
(2) The form assumed for the mean in Equa-
lar application needs an estimate of the trend sur-
tion 2-43 is also generally used in standard linear
face, then generalized least-squares regression can
regression analysis. In regression, ordinary least-
be used to estimate the coefficients ($j's) in the
squares is generally used to solve for the coeffi-
regression equation.
cients; when this is done, it is assumed that the
residuals are independent and identically distribu-
d. Block kriging.
ted. Universal kriging is an extension of ordinary
least-squares regression that allows for spatially
(1) Up to this point, the problem of predicting
correlated residuals. Assuming that Z(x) is a
the value of a regionalized random variable at a
regionalized random variable with a mean as in
given location in the region over which the variable
Equation 2-43 and residual correlation function as
is defined has been considered. Implicit in this
in Equation 2-28, the best linear unbiased predictor
analysis is the assumption that the support of the
(Equation 2-10) can be obtained from the follow-
variable being predicted is defined in exactly the
ing n+p equations, called the universal kriging
same way as the variables that make up the mea-
equations (Journel and Huijbregts 1978):
surements. However, there may be applications
2-13
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