Generally, the mean is assumed to have a func-

j *w*jDij %

j 8 k f k (*x*i)

1

tional dependence on spatial location of the form

(2-46a)

(*u*, *v*) = j $j fj (*u*, *v*)

= Di0, *i*=1, 2, ..., *n*

(2-43)

where the *f*j(u,v)'s are known deterministic func-

j *w*j fk (*x*j)

tions of *x*=*(u,v) *(that is, these functions serve as

(2-46b)

independent variables) and the $j's are regression

coefficients to be estimated from the data. For

= *f*k (*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 *f*1(*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 *w*iDi0

2

Fk

0

(2-45)

(2-47)

& j 8k fk (*x*0)

and two regression coefficients ($1 and $2). The

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-

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-

surements. However, there may be applications

2-13

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