Estimating data (cont.)

ERDC TN-DOER-C15

July 2000

Total area cov ered by samples

Average spacing between samples =

(3)

Number of samples

At the same time that one must be concerned with having a sufficient number of samples for

estimating, too many samples can be problematic. Computations for estimating procedures such

as kriging become cumbersome with too many samples. This can be addressed by compositing

samples outside the immediate area of the point being estimated. This procedure is further described

in Isaaks and Srivastava (1989).

Ordinary kriging is an unbiased estimating method that is intended to minimize the mean residual

mR, or error, and the variance σ2 of the errors. A probability model in which the bias and the error

variance can be calculated is used, and nearby samples weighted to give mR = 0 and minimize σ2 .

The sample weights will change as unknown values are estimated (Isaaks and Srivastava 1989).

The weighting matrix w is derived by multiplying two matrices (C and D) constructed from a selected

random function model and parameters. The mathematical development of this procedure is

somewhat complicated and the relationship to the physical problem not readily apparent. Simply

described, the matrices are composed of the covariances between sample data points and the point

being estimated. The D matrix "provides a weighting scheme similar to the inverse distance

methods" (Isaaks and Srivastava 1989); the covariance between any sample and the point estimated

decreases as the distance between them increases. The difference between the D matrix and inverse

distance weights is that the covariances can be calculated from a larger family of functions, rather

than being limited to a single form |h|-p (where h is the distance between the points and p is an

arbitrarily selected exponent, as previously described). In effect, the kriging distance can be

considered a statistical distance, rather than the geometric distance of the inverse distance methods

(Isaaks and Srivastava 1989). The C matrix takes into account spatial continuity and redundancy,

automatically providing an adjustment for clustering of data points. Ordinary kriging is therefore

less adversely affected by sample clustering than other estimating methods, although it is compu-

tationally more difficult.

One characteristic of ordinary kriging is that, for selected functions, some of the sample weights

may be negative, although the sum of the sample weights will always be 1, a necessary condition

of unbiasedness. The result is that the procedure can yield estimates larger than the largest sample

value and smaller than the smallest sample value. Since the data set is unlikely to contain the most

extreme values, this is advantageous. A disadvantage is that negative estimates may also result.

These may be arbitrarily set to zero when negative values do not make physical sense, as in the case

of concentrations (Isaaks and Srivastava 1989). Selection of an appropriate model and model

parameters requires fitting available data with a suitable function. This procedure and the result of

varying the function parameters are extensively discussed in Isaaks and Srivastava (1989). Addi-

tionally, the random function model can be selected to reflect the degree of anisotropy of the site.

Obviously, judgment and experience are requisite to using this procedure.

Other methods of point estimation can be found in Isaaks and Srivastava (1989). While ordinary

kriging provides a method of obtaining point estimates, block kriging is a procedure for estimating

an average value within a prescribed block. The previously described estimating methods use the

spatial continuity of a single variable to provide estimates for unsampled points. Cokriging is a