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Designing Optimal Sampling Configurations with Ordinary and Indicator Kriging
LLOYD, C.D. (email@example.com)and ATKINSON, P. M., University of Southampton, Department of Geography, Southampton, SO17 1BJ, U.K.
Key Words: ordinary kriging, indicator kriging, stationarity, sampling
The objective of this paper is to examine the applicability of two geostatistical approaches, ordinary kriging (OK) and ordinary indicator kriging (IK), to the design of optimal sampling strategies. A by-product of OK is the OK variance. The OK variance is a measure of confidence in estimates. It is a function of (i) the form of spatial variability of the data (modelled, for example, by the variogram), and (ii) the spatial configuration of the samples. The disadvantage of the OK variance is that it is independent of the magnitude of data values locally. For a given sampling configuration, the OK variance will be the same irrespective of the data values; thus, if data are measured on a regular grid, the maximum OK variance will be identical; however, much of the data vary locally. An approach that is conditional on the data values would be more suitable in such cases. This paper uses the conditional variance of the conditional cumulative distribution function (ccdf) derived through IK to assess local uncertainty in estimates. Since the conditional variance of the ccdf is conditional on the data values, the problem of OK variance data independence is overcome. Previously, to determine an acceptable sample grid spacing, investigators have plotted the maximum OK variance for a range of sample spacings and used the plot to select a sample spacing that achieved a given precision of estimation; however, where the spatial variability is not stationary across the region of concern, the OK variance will be biased as it is independent of the data values locally. The maximum conditional variance was used in the same way to account for the magnitude of data values as well as the form of spatial variability and the spatial configuration of the data. A photogrammetically derived digital terrain model (DTM) was sampled on a regular grid, and the success of the OK and IK approaches in ascertaining optimal sampling intervals was examined and compared with reference to the DTM. Once the variogram and indicator variograms were computed for the sample data, mathematical models were fitted and the model coefficients were used for kriging. The performance of the two approaches was assessed in three separate ways: (i) The model coefficients were used to ascertain the maximum OK and conditional variance for several sampling intervals; (ii) The DTM was then sampled at several (progressively smaller) spacings, and estimates were made from the samples. The differences between the estimates and the population (that is, the complete DTM) were then computed and the errors using OK and IK were related to the maximum error that was predicted by the OK variance or the conditional variance. The proportion of estimates that fell outside the estimation variances were quantified and the different results compared; (iii) The estimation errors for each grid cell were plotted against the OK variance and conditional variance, and the form of the relationships was assessed. Finally, the implications of using the two approaches were discussed.