Stochastic Simulation

If one analyzes a regular grid of kriging estimates for a spatial attribute such as that shown in Figure 9.1, one finds that there is an uneven smoothing in the grid of estimates—smoothing that is inversely proportional to the data density. Such distortion can be visualized in several ways: (a)The experimental semivariogram of the estimates is different from the sampling experimental semivariogram. As illustrated in Figure 9.2, the experimental semivariogram for the grid has a smaller sill and a larger range than the experimental semivariogram for the sampling, denoting an exaggerated continuity in the estimated values.(b)The histogram of the sampling is different than the histogram of the estimated values. Relative to the sample histogram, the histogram for the estimated values has fewer values in the tails and a larger proportion close to the mean. In Figure 9.3, the quartile deviation of the sampling is 36 ft, while that for the grid values is only 26.1 ft.(c)Crossvalidation of the sampling reveals that there is a tendency of kriging to underestimate values above the sample mean and to overestimate those below the mean, which results in a regression line such as that shown in Figure 9.4. Its slope is less steep than the ideal slope of 1.0 for the main diagonal—a distortion called conditional bias in the estimation.