ETL 1110-1-175
30 Jun 97
b. An accurate estimate of a variogram is
Chapter 4
needed from a kriging perspective because the cor-
Practical Aspects of Variogram
relation matrix used to obtain the kriging weights
Construction and Interpretation
is constructed from the variogram values. Even
more directly, the variogram affects the computa-
tion of the kriging variance (Equations 2-36 and
4-1. General
2-47) through the product of the kriging weights
and variogram values. An accurate variogram also
a. Chapter 2 presented the mathematical
has utility outside the strict context of kriging. For
foundation for geostatistics and the kriging tech-
example, in augmenting a spatial network with new
nique. One theme that pervades the technique is
data collection sites, the range parameter of the
the importance of the theoretical variogram. The
variogram could be used as the minimum distance
theoretical variogram, or what we will often refer
of separation between the new sites and between
to simply as the variogram, is a mathematical
new and existing sites to maximize overall
function or model which is fitted to sample-
additional regional information. In another non-
variogram points obtained from data. Permissible
kriging-specific application, the variogram is used
models, which include those given in Chapter 2,
in dispersion variance computations in which the
belong to a family of smooth curves having par-
variance of areal or block values is estimated from
ticular mathematical properties and are each speci-
the variance of point-data values (e.g., Isaaks and
fied by a set of parameters. Chapter 4 will
Srivastava (1989), p. 480).
describe a sequence of stages for estimating and
investigating sample variogram points and a cali-
c. The stages of variogram construction are
bration procedure for specifying the parameters of
described using an example data set of ground-
the variogram model eventually fitted to the sample
water elevations measured near Saratoga, WY
points. Although the calibration procedure is
(Lenfest 1986), that are summarized in Table 4-1
largely an objective means for evaluating theoreti-
and whose relative locations are shown in
cal variograms, the process of obtaining sample
Figure 4-1.
variogram points and finalizing a theoretical vari-
ogram remains an art as much as a science. An
d. The sequence of steps in computing sample
understanding of the material presented in Chap-
variogram points depends on the stationarity prop-
ter 2 as well as professional judgment achieved
erties of the regional variable represented by the
through experience in geostatistical studies is
data. If the mean of the regional variable is the
important in effectively using the guidelines pre-
same for all locations, then it is said to be spatially
sented in this section.
Table 4-1
Univariate Statistics for Example Data Sets1
Standard
Example
Number of
Minimum
Maximum
Mean
Median
Deviation Skewness
Identifier
Measurements Transformation (Base units) (Base units) (Base units) (Base units) (Base units) (Dimensionless)
Saratoga
Drift
2,016.6
2,254.3
2,119.25
2,104.35
56.79
0.45
W ater level A
83
Drift
25.6
65.68
42.30
38.54
10.13
1.03
W ater level B
74
Drift
25.6
65.68
42.85
38.71
10.59
0.87
Bedrock A
108
None
22.64
80.48
44.42
42.82
10.76
0.89
Bedrock B
89
None
24.53
69.22
43.67
43.17
8.58
0.26
W ater quality A
66
Natural log
2.08
8.01
5.19
5.59
1.75
-0.42
1
Base unit for Saratoga, water levels, A and B, and Bedrock A and B is feet; base unit for water quality A is log concentration,
concentration in micrograms per liter.
4-1
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