(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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