Estimating data (cont.)

ERDC TN-DOER-C15

July 2000

where

v = the data point being estimated

wi = a weighting factor

vi = a data value

In estimating, adjustments are made to the sample weighting factor for distance from the point being

estimated and clustering of data points. Samples closest to the data point being estimated will be

given more weight than those at a greater distance. Data points that are clustered close together

rather than uniformly distributed over an area will be given less weight because they are not

representative of the larger area and may unduly influence the value of estimated global parameters.

Isaaks and Srivastava (1989) describe a number of two- and three-dimensional declustering

approaches. Additionally, closely spaced samples having similar values contain redundant infor-

mation, and sample weights should be adjusted for this factor as well.

An important point to note is that distributions estimated from data points are volume dependent

(Isaaks and Srivastava 1989); that is, if a homogenized 0.3-m (1-ft) section of core constitutes a

single data point, the distribution estimated with this and like data points constitutes the distribution

of parameter values for homogenized 0.3-m (1-ft) core sections. For this application, however,

parameter values are desired for larger volumes of a scale that can be practically and economically

excavated. Estimates of recoverable materials based on core analysis may not reflect the averaging

that occurs when the material is excavated in larger volumes. Correcting for the error introduced

by extrapolating small volume estimates to large volumes is a difficult problem, but some effort

should be made to evaluate the potential effect of this factor. One approach might be to examine

the standard deviation of the means of individual core sites.

Geostatistical estimating methods require identification of a model upon which the estimates are

based (Isaaks and Srivastava 1989). A deterministic model can be used if enough is known about

the process effects being measured to quantify them. For example, Stokes' law might be used to

model the expected distribution of grain sizes across a portion of a CDF based on settling velocities,

and extrapolate between data points to estimate the location of transitions across a certain grain size

threshold. (Note that Stokes' law applies only to discrete settling of individual, nonflocculating

particles. Discrete settling of fines does not normally occur in a CDF; thus the model would be

applicable only to the coarse material in the CDF.) However, it is unlikely that most CDFs have

been operated in a manner consistent enough to use this approach. Probabilistic models are used

when no suitable deterministic model is available; in this approach the sample data are viewed as

the result of some random process (Isaaks and Srivastava 1989). The most common parameters

used in probabilistic approaches are the mean, or expected value, and the variance.

Global estimation is the determination of mean parameter values for large areas. Point estimation

is the estimation of parameter values for small areas, or specific locations (Isaaks and Srivastava

1989). Declustering methods are used in both global and point estimation when samples are

clustered rather than distributed over the entire area of interest. Point estimation methods also

require weighting of sample values to reflect the relative distance from the point or area being

estimated. Two point estimation methods, polygons and the local sample mean method, are