determining groundwater elevations following a

local or regional gradient. In such studies sample

are specified and the distance between the two

most separated data points, *h*max, is subdivided

variogram computations need to be made using

according to these equal increments, or a *K *is

residuals obtained by subtracting the estimated

drift value at each location from the value of the

chosen that defines the bin width. For the Sara-

datum at the location.

toga data, a bin width of about 8 km established

8

2

binned Di, *j *values of Figure 4-2 are shown in Fig-

without considering the relative direction between

ure 4-3. The lag plotting positions are the average

2

the locations; that is, Di, *j *is isotropically com-

puted. A plot of Di, *j *versus *h*i,j for all *i,j (i>j)*,

2

where *h*i, *j *= * *x * i & *x*j * , produces a cloud of

this differentiation will be discussed in section 4-3.

8

points whose properties govern the behavior of (.

Although the sample variogram is still preliminary,

The central tendency of the cloud would generally

its general behavior at this stage is adequate to

increase with *h*. A substantial increase in the

indicate if nonstationarity needs to be addressed

central tendency that persists for large h can indi-

before sample variogram refinement is undertaken.

cate a nonstationary spatial mean. The cloud com-

puted for the Saratoga data, with groundwater

levels (*z*) in meters and distance (*h*) in kilometers,

is shown in Figure 4-2 and does show increasing

non-stationarity.

or drift in the spatial mean would be a parabolic

8

shape through all lags in a plot of (. This shape

occurs because differences between data contain

ter in these plots, as seen in Figure 4-2, and this

differences in the drift component that increase as

scatter can conceal the central behavior of *D*2 with

Equation 2-17, squaring the differences in

greatly amplifies the increase with *h*. In these

minimize the effect of aberrant data values is to

collect the *D*2 into *K *bins or lag intervals of width

cases of drift, generally a low-order (less than

8

(*)h)*k , k=1,...*K *and assign to ( the average of the

three) polynomial drift in (*u,v*) is fitted to the data

values of *D *in each bin. This process is similar to

and subsequently subtracted from the data to

the way data are placed in bins for obtaining histo-

obtain residuals. Trend surfaces are not neces-

grams. The expression for the kth average bin

sarily limited to polynomial forms. For example, a

value is

numerical model of groundwater flow may be used

to obtain residuals of groundwater head data.

j

1

2

( (*h*k) =

(4-2)

^

2*N *(*h*k)

slowly varying drift in the spatial mean and, as

such, one regional trend surface should be fitted to

where *N(h*k) is the number of squared differences

all the data. However, often the drift and residuals

that fall into bin *k*, and *h*k is the lag distance asso-

are obtained locally; that is, using moving neigh-

ciated with bin *k. I*k(hi,j) is an "indicator function"

borhoods of locations. Estimates of these values at

that has a value of one if the *h*i,j falls into bin *k *and

any point are thus made using a reduced number

2

zero otherwise (it only includes values of Di, *j *in

(usually between 8 and 16) of surrounding loca-

the calculation that have an *h*i,j that falls into the

tions. This is done because ultimately the kriging

bin). The lag value *h*k can be the midpoint of the

estimates are made using only the data values in

bin or it can be the average of the actual lag values

the given neighborhood. Manipulating the kriging

for the points that fall in the bin.

4-3

Integrated Publishing, Inc. |