ETL 1110-1-175
30 Jun 97
e. To establish bins, either equal bin widths
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, hmax, 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
K=12 bins for (. The ( points computed from the
c. The data in Equation 4-1 are differenced
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
h values in the bin. The symbol x indicates that
2
the locations; that is, Di, j is isotropically com-
puted. A plot of Di, j versus hi,j for all i,j (i>j),
N(h) is less than 30 pairs for the particular bin and
2
where hi, j = * x i & xj * , 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,
4-3. Nonstationarity
is shown in Figure 4-2 and does show increasing
D2 with increasing h, indicating potential
a. An indication of substantial nonstationarity
non-stationarity.
or drift in the spatial mean would be a parabolic
8
shape through all lags in a plot of (. This shape
d. Generally, there is a large amount of scat-
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 D2 with
h increases. If Equation 2-16 is inserted into
h. One way to estimate the central tendency and to
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 D2 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
2
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
( (hk) =
(4-2)
Di, j Ik (hi, j)
^
b. In theory, the polynomial trend reflects a
2N (hk)
slowly varying drift in the spatial mean and, as
such, one regional trend surface should be fitted to
where N(hk) is the number of squared differences
all the data. However, often the drift and residuals
that fall into bin k, and hk is the lag distance asso-
are obtained locally; that is, using moving neigh-
ciated with bin k. Ik(hi,j) is an "indicator function"
borhoods of locations. Estimates of these values at
that has a value of one if the hi,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 hi,j that falls into the
tions. This is done because ultimately the kriging
bin). The lag value hk 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
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