Solving this system of equations gives the weights iMyMs (Z*(x) - ZQ)), called the "kriging variance,"is: j, and the resulting variance of the kriging error i V( | xi—x]) +4 For details on the derivation of these equations, and extensions to estimating area averages and to the case where E(Z(x)) is not constant, see Delfiner, 1975. Because the Var(Z*(x) - Z(x)) is expressed in terms of the variogram y(h), the weights \; do not depend on the data values Z(x;), but only on y(h) and the relative geometry of the xj. One advantage of this is that, for a given island, the same set of weights is applicable to every complete square array of data points used in estimating an area average. In other words, the set of weights could be calculated once, and would apply to most of the island area, with individual computations required only for estimates on the island edges. This resulted in a substantial saving in computer memory and time required to make the calculations. Although the weights do not depend on the Z(x;), they do depend on the variogram, which must be estimated from the data. Most of the variograms encountered in practice, including those observed in Enewetak, fit one of several common models. Figure 5-1 shows a few of these models. As shown by the spherical model] in Figure 5-1, the variogram may be bounded, that is, may attain a maximum value for y(n). The bound is called the "sill," and this value represents the general underlying variance of the population of sample points. The distance at which y(h) reachesits sill is called the "range" and this corresponds to the concept of the zone of influence of a data point. By definition (0) = 0, but y(h) may not be approaching zero as h gets small Such a discontinuity is called a "nugget effect," so named because the presence of a nugget of gold in a mine will cause a discontinuity in the variogram. A nugget effect can be caused by changes in the variogram structure at distances smaller than the smallest distance between observed data values, as in the gold nugget example. It can also be caused by uncertainty in the data measurements themselves. Most of the variograms on Enewetak data were linear and all had a nugget effect which was probably due to a combination of the two causes. Ratio Estimation. The cleanup criteria for Enewetak were expressed in terms of average TRU activity, but the data from the IMP were 241 am activities. The TRU activity was calculated using an estimated ratio of TRU to “4! Am. This ratio should theoretically be constant at a given time for fallout from a particular nuclear event. Many of the northern islands received fallout from several events, however, so the measured ratio represented composites from several fallout incidents. If an island was not the site of a nuclear event, the ratio was usually found to be fairly constant for that island. On ground zero islands, the effects from the various events appeared to influence the ratio for different parts of the island, so several ratio populations were present. However, these islands could usually be divided into several areas each having a single ratio population. The divisions were based on prior information such as known soil recontouring activities or on cluster analysis of data collected during the cleanup. The data for estimating ratios came from alpha~ and gamma-spectrometric analyses of soil samples. Soil sample locations were chosen in an attempt to get a representative sample of an island and the samples were collected in a consistent manner (see Section 4.2.1), A sample consisted of two composites of six subsamples each, with the subsample taken in a specific pattern. (See Procedure No. 4.) This was designed to roughly reflect the angular efficiency characteristics of the in situ detector, thereby increasing the comparability of IMP data and laboratory data from soil samples. In the early stages of the cleanup, the ratio of TRU to 241 am was estimated using the sample mean of the ratios from individual soil samples. The sample standard deviation was used to estimate the error in the ratio estimate. Use of these estimators assumes that the variance of the TRU value is proportional to the square of the corresponding 24lam value. As more soil data became available, they showed that it was more accurate to assume that the variance of the TRU was 140