It is not necessary to have data on a regular grid to use the kriging method, but a grid pattern was
used because it has several advantages. First, the kriging theory shows that for a fixed number of
data values and any of the common variogram forms, a regular grid pattern will result in smaller
standard deviation of the kriging error than other patterns. A regular grid is also easier to set up in
the field, and it is easier to find the same location again than with a pattern such as random
sampling. Finally, by using a regular grid and limiting the total number of data values used in each
weighted average, the computations were simplified enough to be within the capability of the
microprocessor on-island. The validity of the results from the simpler program was verified by using
the same data in a general-purpose kriging program on a large computer. There were no significant
differences between the results of the two programs, so the results from the on-island program were
used throughout the project.
The mathematical assumption made in deriving the kriging estimator is that the observed data values
are samples from a realization of a random function Z(x) with the following properties:
a)
b)
E(Z(x)) = m
Var (Z(x+h) - Z(x)) = 2y(n),
where m is a constant, x is a two-dimensional location vector, and h is a vector distance.
function y(h) is the variogram function mentioned previously.
The
In practice, these assumptions need
hold only locally, where “local" means for h less than or equal to the maximum radius of the
neighborhood of points used in making an estimate. In the case of the Enewetak cleanup, the
maximum radius was about 70 m. Thus if the expected TRU activity did not change much in a 70 m
distance, and a reasonably good estimate of yh) could be made for h <70, then the kriging estimate
could be considered valid. Both these conditions were sufficiently fulfilled by the surface TRU data.
Under the assumptions above, the kriging estimator is the best linear unbiased estimator where
"best" is the sense of minimum variance. The linear condition means the estimator, Z*, is of the
form:
where \j are weights and Z(x;) is the observed data value at location xj. The unbiasedness condition
E(Z* (x)) = Z(x) =m,
leads to the constraint that,
Then minimizing Var(Z*(x) - Z(x)) under this constraint leads to the system of linear equations:
n
z PYAR
j=
th
=
Y(Ix—xl)i=1,2,...0
where |x;-x;| is the Euclidean distance between x} and Xj and ,.is the Lagrange multiplier used to
satisfy the constraint on the sum of the Aje
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