Efficient Implementation of an Optimal Interpolator for Large Spatial Data Sets

Abstract

Abstract. Interpolating scattered data points is a problem of wide ranging interest. One of the most popular interpolation methods in geostatistics is ordinary kriging. The price for its statistical optimality is that the estimator is computationally very expensive. We demonstrate the space and time efficiency and accuracy of approximating ordinary kriging through the use of covariance tapering combined with iterative methods.

Estimating Coastal Digital Elevation Model Uncertainty

“…Thus, the terrain is not a stationary process, and one model will not accurately capture the spatial structure of the entire DEM (Maune et al, 2007). Computational limitations and varying morphologies can necessitate dividing the region of interest into smaller sections; however, this approach can cause abrupt vertical offsets along the borders of the sections in the final, composite DEM (Memarsadeghi and Mount, 2007;Meyer, 2004). Another limitation of using kriging to develop coastal DEMs with accompanying uncertainty surfaces is the treatment of measurement uncertainty when integrating multiple data sets of disparate quality and age.…”

Efficient Kriging via Fast Matrix-Vector Products

“…Ensuring unbiasedness of the error imposes a constraint on this function. Forrmalizing this objective function with its constraint results in the following system [3,5,12].…”

“…wr, Therefore, the minimization problem for a points reduces to solving a linear system of size (n + 1)2, which is impractical for very large data sets via direct approaches. It is also important that matrix C be positive definite [3,12]. Pairwise covariances are often modeled as a function of points' separation.…”

“…Recently an approximation approach to kriging has been proposed based on a technique called covariance tapering [11,12]. However, this approach is not efficient when covariance models have relatively large ranges.…”

2010 Rietz lecture: When does the screening effect hold?

“…The screening effect is the geostatistical term for the phenomenon of nearby observations tending to reduce the influence of more distant observations when using kriging (optimal linear prediction) for spatial interpolation [Journel and Huijbregts (1978), Chilès and Delfiner (1999)]. This phenomenon is often invoked as a justification for ignoring more distant observations when using kriging [Memarsadeghi and Mount (2007), Emery (2009)]. Only in some very limited special cases is the effect exact in the sense that the more distant observations make no contribution to the krig-This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Statistics, 2011, Vol.…”