ETL 1110-1-175
30 Jun 97
( 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
d. Gaussian variogram. The Gaussian vario-
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.
e. Linear variogram. Parameters for a linear
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, hmax, 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-
b. Generally, the plot of the points of 8 from
(
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-
4-7. Additional Trend Considerations
buted to a bias resulting from the estimation error
in the drift form and its coefficients. Thus, this
a. If a drift in the data is indicated as in sec-
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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