Interpolation algorithm ranking using cross-validation and the role of smoothing effect. A coal zone example $ Oriol Falivene a,n,1 , Lluı ´s Cabrera a , Raimon Tolosana-Delgado b , Alberto Sa ´ez a a Geomodels Institute, Group of Geodynamics and Basin Analysis, Dept. EPGM, Universitat de Barcelona, c/Martı ´ i Franqu es s/n, Facultat de Geologia, 08028 Barcelona, Spain b Maritime Engineering Laboratory, Technical University of Catalunya, c/Jordi Girona 1-3, 08034 Barcelona, Spain article info Article history: Received 17 February 2009 Received in revised form 28 September 2009 Accepted 30 September 2009 Keywords: Interpolation Cross-validation Smoothing effect Kriging Inverse distance weighting abstract For a property measured at several locations, interpolation algorithms provide a unique and smooth function yielding a locally realistic estimation at any point within the sampled region. Previous studies searching for optimal interpolation strategies by measuring cross-validation error have not found consistent rankings; this fact was traditionally explained by differences in the distribution, spatial variability and sampling patterns of the datasets. This article demonstrates that ranking differences are also related to interpolation smoothing, an important factor controlling cross-validation errors that was not considered previously. Indeed, smoothing in average-based interpolation algorithms depends on the number of neighbouring data points used to obtain each interpolated value, among other algorithm parameters. A 3D dataset of calorific value measurements from a coal zone is used to demonstrate that different algorithm rankings can be obtained solely by varying the number of neighbouring points considered (i.e. whilst maintaining the distribution, spatial variability and sampling pattern of the dataset). These results suggest that cross-validation error cannot be used as a unique criterion to compare the performance of interpolation algorithms, as has been done in the past, and indicate that smoothing should be also coupled to search for optimum and geologically realistic interpolation algorithms. & 2010 Elsevier Ltd. All rights reserved. 1. Introduction Interpolation algorithms aim to predict the value of a property at a location by using values of the same property sampled at scattered neighbouring points (Journel and Huijbregts, 1978; Jones et al., 1986; Davis, 2002). These algorithms yield a unique (though different for each method) property map honouring input data. Interpolation in geosciences is widely used for both predictive and visualization purposes. A variety of algorithms have been developed to carry out interpolations (Morrison, 1974), for example inverse distance weighting (IDW, Kane et al., 1982), Kriging, (Matheron, 1963), splines (Ahlberg et al., 1967; Mitasova and Mitas, 1993) or polynomial regression. The selection of optimal interpolation strategies for continuous variables is an important and ongoing subject of debate (Lu and Wong, 2008; Bater and Coops, 2009). Cross-validation (CV) has often been used to compare the performance of interpolation algorithms (Table 1). CV is based on calculating the value of the variable at locations where the true value is known, but has been temporally removed from the input data, and then measuring the CV error by comparing the estimated value against the true one (Davis, 1987; Isaaks and Srivastava, 1989). Past comparisons based on CV error have yielded a variety of results, not always consistent (Table 1). For instance, in comparison of two widely used algorithms such as Kriging and IDW, some authors have found that Kriging yields better interpolations (Weber and Englund, 1994; Zimmerman et al., 1999; Goovaerts, 2000; Teegavarapu and Chandramouli, 2005; Lu and Wong, 2008), some have not found any significant differences in the results (Dirks et al., 1998; Moyeed and Papritz, 2002; Gallichand and Marcotte, 1993), and others have found that IDW yields better interpolations (Weber and Englund, 1992; Lu and Wong, 2008). The disparity in the results obtained from existing interpola- tion algorithm rankings using CV error (Table 1) motivated this research. We demonstrate that the comparisons solely based on CV error are utterly flawed. Apart from the fact that rankings may depend on some specific characteristics of the particular dataset used for the comparison, we provide evidence that the size of the search neighbourhood plays a determinant role in algorithm rankings considering only CV error. The search neighbourhood is amongst the factors controlling the smoothing effect of each interpolation strategy. These findings challenge the practice of ranking and qualifying interpolation algorithms considering CV error (Table 1), and show that there is no absolute best ARTICLE IN PRESS Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/cageo Computers & Geosciences 0098-3004/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.cageo.2009.09.015 $ Supplementary material is available to access through the online version. n Corresponding author. Tel./Fax: + 31(0) 70447 2336. E-mail address: oriol.falivene@shell.com (O. Falivene). 1 Present address: Shell International Exploration and Production, Kessler Park 1, 2280 AB Rijswijk, The Netherlands. Computers & Geosciences 36 (2010) 512–519