ETL 1110-1-175
30 Jun 97
(2) Directional variogram and anisotropy. It
locations near points with measurements tending to
is often the case that spatial correlation depends
be smaller. One can then associate with any spa-
not only on distance between points, but also on
tial prediction a variance, which gives an indi-
direction. For example, measurements at pairs of
cation of the uncertainty in that predicted value.
points 100 m apart with the line between them
As mentioned before, this measure of uncertainty
oriented in a north-south direction may have a
gives kriging one of its principal advantages over
different correlation than measurements at points
many other techniques.
the same distance apart but with the line joining
them oriented in an east-west direction. The
(4) Trends and universal kriging. Special
spatial process is said to exhibit anisotropy, and
attention must be given in kriging to the question
what is known as a directional variogram must be
of whether there are spatial trends in the data. A
used for the geostatistical analysis.
trend in this case is usually any detectable ten-
dency for the measurements to change as a func-
(3) Kriging and kriging variance.
tion of the coordinate variables but can also be a
function of other explanatory variables. For
(a) Kriging yields optimal spatial estimates at
example, aside from random fluctuations, measure-
points where no measurements exist in terms of the
ments of groundwater elevations may exhibit a ten-
values at points where one does have data. As
dency to increase in a consistent manner the farther
discussed above, placing the problem in a sto-
one proceeds in a certain direction. A kriging
chastic framework permits precision-defining
analysis in which there is no spatial trend is known
optimality. In kriging, the restriction is first
as ordinary kriging; when a trend does exist, uni-
imposed that the predicted value at any point is a
versal kriging should be considered. In universal
linear combination of the measured values; that is,
kriging, one attempts to account for the trends
the kriging estimate is a linear predictor. Given
present. For example, it might be assumed that the
this restriction, the values of the coefficients in this
trend can be represented as a linear function of
linear function are chosen so as to force the pre-
coordinate variables. The form of the trend model
dictor to be optimal.
is then incorporated into the universal kriging
equations to obtain the optimal weights.
(b) The first criterion imposed is that the
estimate be unbiased, or that in an average sense
(5) Block kriging. What has been discussed in
the difference between the predicted value and
the preceding paragraphs is usually known as
actual value is zero. The second optimality cri-
point, or punctual, kriging. In point kriging, the
terion is that the prediction variance be minimized.
goal is to predict the value of a variable at discrete
This variance is a statistical error measure defined
locations. By contrast, in block kriging the goal is
to be the average squared difference between
to predict the average value, over a specified
predicted and actual values. Because the kriging
region, of a variable. As in point kriging, the opti-
estimate minimizes this variance, it is known as the
mal predictor is a linear combination of the mea-
best (minimum variance) unbiased linear predictor.
sured data values, and degree of uncertainty is
This minimization is performed algebraically and
indicated by a block kriging variance. Block
results in a set of equations known as the kriging
kriging variances tend to be smaller than point
equations, which give an explicit representation of
kriging variances because averages tend to be less
the optimal coefficients (weights) in terms of the
variable than individual values.
variogram. The form of these equations is pre-
sented in Chapter 2.
(6) Prediction intervals and normality.
(c) Also given in Chapter 2 is an expression
(a) A standard kriging analysis will give two
for the kriging variance. This variance depends on
values for any location: the optimal kriging esti-
geometry of the data sites, with the variance at
mate and the kriging variance. The variance
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