Kriging in the Presence of Locally Varying Anisotropy Using Non-Euclidean Distances
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Kriging in the Presence of Locally Varying Anisotropy Using Non-Euclidean Distances J.B. Boisvert · J.G. Manchuk · C.V. Deutsch
Received: 17 August 2007 / Accepted: 3 April 2009 / Published online: 10 June 2009 © International Association for Mathematical Geosciences 2009
Abstract A stationary specification of anisotropy does not always capture the complexities of a geologic site. In this situation, the anisotropy can be varied locally. Directions of continuity and the range of the variogram can change depending on location within the domain being modeled. Kriging equations have been developed to use a local anisotropy specification within kriging neighborhoods; however, this approach does not account for variation in anisotropy within the kriging neighborhood. This paper presents an algorithm to determine the optimum path between points that results in the highest covariance in the presence of locally varying anisotropy. Using optimum paths increases covariance, results in lower estimation variance and leads to results that reflect important curvilinear structures. Although CPU intensive, the complex curvilinear structures of the kriged maps are important for process evaluation. Examples highlight the ability of this methodology to reproduce complex features that could not be generated with traditional kriging. Keywords Geostatistics · Variogram · Stationarity · Geological structures · Multidimensional scaling
1 Introduction A decision of stationarity is required for geostatistical techniques. If there are drastic differences in statistical parameters the study area could be subdivided into independent domains, each modeled separately. The chosen domains may be further subdivided by rock-type or facies. In this case, geostatistical modeling is done on a by-facies and by-domain basis. We will present a technique to explicitly deal with J.B. Boisvert () · J.G. Manchuk · C.V. Deutsch Centre for Computational Geostatistics, Department of Civil and Environmental Engineering, University of Alberta, Edmonton, Alberta, Canada e-mail: [email protected]
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Math Geosci (2009) 41: 585–601
Fig. 1 (A) Simple folding showing locally varying directions only at the crests of the folds (modified from Bennison and Moseley 1997). (B) More complex folding showing locally varying directions throughout (modified from Groshong 2006). (C) A simple fluvial channel deposit showing locally varying directions in the channels (modified from Reading 1996). (D) A more complex fluvial channel deposit showing erratic locally varying directions in a single channel (modified from Reading 1996)
multiple directions of continuity within a domain. Spatial features such as anisotropy, range of correlation, and secondary variables can be exploited to increase the accuracy of modeling. If these characteristics are well understood, they can be transferred into the numerical geological model to improve subsequent performance predictions. Consider a vein type deposit where the strikes of the veins are known and vary over the deposit. It would be difficu
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