Continuity for Kriging with Moving Neighborhood
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Continuity for Kriging with Moving Neighborhood Jacques Rivoirard · Thomas Romary
Received: 29 March 2010 / Accepted: 10 September 2010 / Published online: 19 March 2011 © International Association for Mathematical Geosciences 2011
Abstract By definition, kriging with a moving neighborhood consists in kriging each target point from a subset of data that varies with the target. When the target moves, data that were within the neighborhood are suddenly removed from the neighborhood. There is generally no screen effect, and the weight of such data goes suddenly from a non-zero value to a value of zero. This results in a discontinuity of the kriging map. Here a method to avoid such a discontinuity is proposed. It is based on the penalization of the outermost data points of the neighborhood, and amounts to considering that these points values are spoiled with a random error having a variance that increases infinitely when they are about to leave the neighborhood. Additional details are given regarding how the method is to be carried out, and properties are described. The method is illustrated by simple examples. While it appears to be similar to continuous kriging with a smoothing kernel, it is in fact based on a much simpler formalism. Keywords Penalized kriging · Continuous kriging · Kriging weights 1 Introduction Kriging is an optimal linear estimator that aims at estimating the value of a regionalized variable, either at a punctual location or over a larger support such as a block or a domain (Cressie 1991; Chilès and Delfiner 1999). It is based on a model of spatial structure, generally consisting of a spatial covariance or a variogram with a possible trend hypothesis. The kriged value at a target point is an average of the data values, weighted by the kriging weights. When the data are numerous, the inversion of the covariance matrix—and consequently, the computation of kriging weights—are J. Rivoirard () · T. Romary Centre de Géosciences, Mines-ParisTech, 35 rue Saint-Honoré, 77300 Fontainebleau, France e-mail: [email protected]
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Math Geosci (2011) 43: 469–481
made numerically impossible. Both global and local approaches exist, using either all data or the nearest ones. Among global approaches, the covariance tapering method makes use of the computational advantage of sparse matrix algebra (Furrer et al. 2006). Other methods aim at reducing the dimension of the matrix to be inverted, as in fixed rank kriging (Cressie and Johannesson 2008) or in the predictive processes method (Banerjee et al. 2008). In local approaches, only the data present in a given neighborhood of the target point are used. This amounts to kriging with a moving neighborhood when mapping a regionalized variable: the domain is discretized into grid points and the neighborhood moves with the target grid point. In the simplest cases (considered later), the moving neighborhood is defined by a limit distance from the target point. However, more sophisticated neighborhoods are also currently used (for instance, based on oc
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