Drift and residuals

If the surface is not stationary, the kriging equations can be expanded so the drift can be estimated simultaneously, in effect removing the drift and achieving stationarity. Kriging will estimate the drift and these residuals from the drift at every grid node so they may be mapped. A best estimate of the original surface is created by combining the surfaces representing the estimated drift and the estimated residuals. Perhaps most importantly, kriging yields estimates of the likely error (in the form of standard errors or error variances) at every node in the grid. These error estimates can be mapped to give a direct assessment of the reliability of the contoured surface. As an example, Figure 19 is a map of the elevation of the water table in the Equus Beds aquifer in south-central Kansas, prepared by universal kriging. Contours are given in feet above sea level. Figure 20 is a map of the associated standard error, given in feet within one standard error. If the errors of estimation are assumed to be normally distributed, the probability is 95% that the true value of the surface lies within the interval defined by the kriged estimate twice the standard error. In most applications, the assumption of normality of the errors is reasonable.

In theory, no other method of grid generation can produce better estimates (in the sense of being unbiased and having minimum error) of the form of a mapped surface than kriging. In practice, the effectiveness of the technique depends on the correct specification of several parameters that describe the semivariogram and the model of the drift. However, because kriging is robust, even with a naive selection of parameters the method will do no worse than conventional grid estimation procedures.

The price that must be paid for optimality in estimation is computational complexity. A large set of simultaneous equations must be solved for every grid node estimated by kriging. Therefore, computer run times will be significantly longer if a map is produced by kriging rather than by conventional gridding. In addition, an extensive prior study of the data must be made to test for stationarity, determine the form of the semivariogram, set the neighborhood size, and select the proper order of the drift if it exists. These properties are not independent, and because the system is underdetermined, trial-and-error experimentation may be necessary to determine the best combination. For this reason, to warrant the additional costs of analysis and processing, kriging probably should be applied in those instances where the best possible estimates of the surface are essential, the data are of reasonably good quality, and estimates of the error are needed.

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