On the Validity of Commonly Used Covariance and Variogram Functions on the Sphere
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On the Validity of Commonly Used Covariance and Variogram Functions on the Sphere Chunfeng Huang · Haimeng Zhang · Scott M. Robeson
Received: 29 June 2009 / Accepted: 10 May 2011 / Published online: 9 June 2011 © International Association for Mathematical Geosciences 2011
Abstract Covariance and variogram functions have been extensively studied in Euclidean space. In this article, we investigate the validity of commonly used covariance and variogram functions on the sphere. In particular, we show that the spherical and exponential models, as well as power variograms with 0 < α ≤ 1, are valid on the sphere. However, two Radon transforms of the exponential model, Cauchy model, the hole-effect model and power variograms with 1 < α ≤ 2 are not valid on the sphere. A table that summarizes the validity of commonly used covariance and variogram functions on the sphere is provided. Keywords Power variogram · Spherical covariance · Stable model · Variogram models
1 Introduction Global-scale processes and phenomena are of utmost importance in the geosciences. Data from global networks of in situ and satellite sensors are used to monitor a wide array of geophysical processes. Most methods and models in spatial statistics, however, are developed in Euclidean spaces Rn and these methods have not been investigated as intensively in spherical coordinate systems (Robeson 1997). Among the most C. Huang () Department of Statistics, Indiana University, Bloomington, IN 47408, USA e-mail: [email protected] H. Zhang Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS 39762, USA S.M. Robeson Departments of Geography and Statistics, Indiana University, Bloomington, IN 47405, USA
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Math Geosci (2011) 43:721–733
commonly used covariance and variogram functions in the geosciences are the exponential, Gaussian, spherical, hole-effect, and power models (Kitandis 1997; Isaaks and Srivastava 1989; Webster and Oliver 2001). Haylock et al. (2008), for instance, compared these five models when developing a high-resolution data set of daily precipitation over Europe. Other geoscientists have used the exponential variogram model on the sphere to develop quantitative estimates of atmospheric carbon dioxide concentrations, global fire distributions, and oceanic mixed layer depths (Alkhaled et al. 2008; Carmona-Moreno et al. 2005; de Boyer Montégut et al. 2004). The Gaussian model on the sphere has been used to reconstruct paleoceanographic temperature (Schäfer-Neth et al. 2005) while Janis and Robeson (2004) used a power model with spherical distances to analyze errors in minimum temperature data. Clearly, a wide range of geostatistical models are being used with databases that are coded in spherical coordinates. Before adapting Euclidean covariance and variogram models to the sphere, one must first evaluate their properties to ensure their validity. A random process is stationary on the sphere if its covariance function depends solely on the spherical angle. To be valid, the covariance function must
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