Isotropic Variogram Matrix Functions on Spheres
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Isotropic Variogram Matrix Functions on Spheres Juan Du · Chunsheng Ma · Yang Li
Received: 21 June 2012 / Accepted: 7 January 2013 / Published online: 29 January 2013 © International Association for Mathematical Geosciences 2013
Abstract This paper is concerned with vector random fields on spheres with secondorder increments, which are intrinsically stationary and mean square continuous and have isotropic variogram matrix functions. A characterization of the continuous and isotropic variogram matrix function on a sphere is derived, in terms of an infinite sum of the products of positive definite matrices and ultraspherical polynomials. It is valid for Gaussian or elliptically contoured vector random fields, but may not be valid for other non-Gaussian vector random fields on spheres such as a χ 2 , log-Gaussian, or skew-Gaussian vector random field. Some parametric variogram matrix models are derived on spheres via different constructional approaches. A simulation study is conducted to illustrate the implementation of the proposed model in estimation and cokriging, whose performance is compared with that using the linear model of coregionalization. Keywords Absolutely monotone function · Cross variogram · Direct variogram · Elliptically contoured random field · Gaussian random field · Gegenbauer’s polynomials · Positive definite matrix J. Du () Department of Statistics, Kansas State University, Manhattan, KS 66506-0802, USA e-mail: [email protected] C. Ma Department of Mathematics, Statistics, and Physics, Wichita State University, Wichita, KS 67260-0033, USA e-mail: [email protected] C. Ma School of Economics, Wuhan University of Technology, Wuhan, Hubei 430070, China Y. Li Department of Statistics, Iowa State University, Ames, IA 50011, USA e-mail: [email protected]
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Math Geosci (2013) 45:341–357
1 Introduction Geophysical and atmospheric study often requires the modeling of large scale data with multiple attributes observed at each location recorded in spherical coordinates, such as the impact of soil greenhouse gas fluxes (CO2 , NO2 , etc.) on global warming potential, temperature-precipitation relationships in climate change, multilevel forecast-error representation in operational atmospheric-data assimilation, and correlation modeling of different layers of ocean or solid earth (AdvietoBorbe et al. 2007; Gaspari and Cohn 1999, Gaspari et al. 2006; Sain et al. 2011; Trenberth and Shea 2005; Tebaldi and Lobell 2008, among others). The data involved in those applications may be viewed as the realizations of vector random fields on spheres. For this purpose, potential dependence structure developments are in great demand for vector random fields on spheres with second-order moments or with second-order increments. Such structures are expected to model not only the autocorrelation of each component at different geophysical sites, but also the cross correlation between different components at the same or different locations (Du and Ma 2011, 2012; Haas 1998; Huang et al. 2009; Le and Zidek 2006; Sain and Cre
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