( when the data are spatially correlated. But (

used models for real valued spatial studies, it

seems to be a predominant model for indicator

cannot be estimated until a drift equation is

values at various cutoff levels as, for example, in a

obtained to yield the residuals. Therefore, obtain-

study of lead contamination (Journel 1993).

ing a sample variogram and a subsequent theoreti-

cal variogram from drift residuals of a specified

drift form is an iterative process (David (1977),

gram parameters (Equation 2-25) are a nugget

pp. 273-274) framed by the following steps:

value *g*, and a sill *s*, and this variogram also has a

practical range *r*. The Gaussian variogram is hori-

(1) An initial variogram is specified and drift

zontal from the nugget, becomes a concave upward

coefficients are computed to obtain residuals. For

function at small lags, inflects to concave down-

this step, a pure nugget (i.e. constant) variogram

ward, and asymptotically approaches a sill value

can be used to compute the initial estimates of the

(Figure 2-3). After a nugget value and sill value

drift coefficients. This is an ordinary least-squares

are specified based on the points, the variogram

estimate of the drift yielding a first-iteration sam-

value at a lag of one-half the estimated practical

ple variogram of residuals.

range will be two-thirds of the sill value. Again,

this fitted variogram needs to be supported by the

(2) A theoretical variogram is fitted to the

8 points to a reasonable degree. As will be

(

sample variogram of the residuals and is used to

described in the example using the Saratoga data,

obtain updated drift coefficients.

the Gaussian variogram often is used where the

variable analyzed is spatially very continuous,

(3) The residuals from the drift obtained in

such as a groundwater potentiometric surface.

step b are used to compute an updated sample

variogram.

variogram (Equation 2-26) are a nugget value *g*,

(4) The sample variogram computed at the end

and a slope *b*. Sample points indicating a linear

of step 3 is compared to the sample variogram of

variogram would increase linearly from the nugget

step 2. If the two sample variograms compare

value and fail to reach a sill even for large lags

favorably, then the theoretical variogram from

(Figure 2-3). With the nugget as the intercept, the

step 2 is accepted as the variogram of residuals for

slope is computed for the line passing through the

subsequent kriging computations. If the sample

8 points. A pseudosill *s *can be defined as the

(

variogram from step 3 differs markedly from the

value of the line at the greatest lag, *h*max, between

sample variogram of step 2, steps 2-4 are repeated

any two locations. This lag becomes the defacto

using the sample variogram of the most recent

range *r *for a linear variogram. Examples of the

step c.

usage of the linear variogram occur in hydrogeo-

(

chemical studies of specific conductance and in

a set of residuals will initially increase with *h*,

studies of trace elements such as barium and boron

(Myers et al. 1980).

reach a maximum, and then decrease as seen in

Figure 4-4. This typical haystack-type behavior,

discussed by David (1977, pp. 272-273), is attri-

buted to a bias resulting from the estimation error

in the drift form and its coefficients. Thus, this

behavior in the variogram of the residuals gen-

tion 4-3, the theoretical variogram of residuals

erally would more readily occur with a higher

that has been fitted thus far is used to update the

degree of drift polynomial. This behavior should

drift equation. Although ordinary least squares

not prohibit acceptable variogram determination

often suffices for computing a polynomial drift

because the initial points of the sample variogram

equation, drift determination itself is a function of

of residuals are still indicative of the theoretical

4-13

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