Block Kriging for Lognormal Spatial Processes
- PDF / 418,787 Bytes
- 31 Pages / 432 x 648 pts Page_size
- 87 Downloads / 229 Views
Block Kriging for Lognormal Spatial Processes1 Noel Cressie2 Lognormal spatial data are common in mining and soil-science applications. Modeling the underlying spatial process as normal on the log scale is sensible; point kriging allows the whole region of interest to be mapped. However, mining and precision agriculture is carried out selectively and is based on block averages of the process on the original scale. Finding spatial predictions of the blocks assuming a lognormal spatial process has a long history in geostatistics. In this article, we make the case that a particular method for block prediction, overlooked in past times of low computing power, deserves to be reconsidered. In fact, for known mean, it is optimal. We also consider the predictor based on the “law” of permanence of lognormality. Mean squared prediction errors of both are derived and compared both theoretically and via simulation; the predictor based on the permanence-of-lognormality assumption is seen to be less efficient. Our methodology is applied to block kriging of phosphorus to guide precision-agriculture treatment of soil on Broom’s Barn Farm, UK. KEY WORDS: geostatistics, MSPE, permanence of lognormality, phosphorus, precision agriculture, spatial prediction.
INTRODUCTION There have been quite a few publications in the past on geostatistics for lognormal data. The themes of these papers (Dowd, 1982; Journel, 1980; Marechal, 1974; Matheron, 1974; Rendu, 1979; Rivoirard, 1990; Roth, 1998) draw on the very best traditions of geostatistics: determine types of variogram models for lognormal data; decide whether to do inference on the original scale or the log scale; choose an optimality criterion for kriging; derive the kriging equations according to the optimality criterion; consider the cases of known or unknown mean (on the log scale); and consider whether knowing just the variogram (on the log scale) is enough to do kriging. The purpose of this article is to take a fresh look at geostatistics for lognormal data, build on the results of the earlier papers, and develop new results in light of the statistical literature on linear models and transformations. We shall 1Received
10 September 2004; accepted 23 August 2005; Published online: 30 August 2006.
2Department of Statistics, The Ohio State University, 1958 Neil Avenue, Columbus, Ohio 43210-1247;
e-mail: [email protected] 413 C 2006 International Association for Mathematical Geology 0882-8121/06/0500-0413/1
414
Cressie
vigorously pursue two of the many possibilities, chosen based on the following principles: • The original scale is for optimality criteria (including unbiasedness) but the log scale is for linear statistical analysis. • Some form of stationarity is needed for estimation of spatial dependence but it is not needed for spatial prediction (i.e., kriging). • Kriging is an empirical-Bayes methodology that requires efficient estimators of unknown parameters to be “plugged into” kriging equations. Notice that “permanence of lognormality” (e.g., Rivoirard, 19
Data Loading...
