Stationary and Isotropic Vector Random Fields on Spheres
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Stationary and Isotropic Vector Random Fields on Spheres Chunsheng Ma
Received: 9 February 2012 / Accepted: 16 May 2012 / Published online: 6 June 2012 © International Association for Mathematical Geosciences 2012
Abstract This paper presents the characterization of the covariance matrix function of a Gaussian or second-order elliptically contoured vector random field on the sphere which is stationary, isotropic, and mean square continuous. This characterization involves an infinite sum of the products of positive definite matrices and Gegenbauer’s polynomials, and may not be available for other non-Gaussian vector random fields on spheres such as a χ 2 or log-Gaussian vector random field. We also offer two simple but efficient constructing approaches, and derive some parametric covariance matrix structures on spheres. Keywords Absolutely monotone function · Cross covariance · Covariance matrix function · Direct covariance · Elliptically contoured random field · Gaussian random field · Gegenbauer’s polynomials · Positive definite matrix 1 Introduction It is common to observe multiple components or variables at the same spatial location in various geophysical or environmental systems. For most applications, it is the joint behavior among these multiple variables that are of extreme importance. For example, in studies of variability and trends in surface temperature and precipitation, Trenberth and Shea (2005) focused on temperature-precipitation relationships. Tebaldi and Lobell (2008) showed how rigorously quantified uncertainties in shortterm climate change, in the form of PDFs of temperature and precipitation changes, 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
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Math Geosci (2012) 44:765–778
can be propagated into an impact model of crop yield changes. In mining applications, the silver concentration at a location may be observed together with the lead and zinc concentration levels and other minerals. Other examples may be found in Mardia (1988), Haas (1998), Zidek et al. (2000), Le and Zidek (2006), Calder (2007), Sain and Cressie (2007), Choi et al. (2009), Huang et al. (2009), Jun (2011), and Sain et al. (2011). Many geophysical or environmental observations are further measured on a global scale, and data sets are collected over a large portion of the Earth such as satellite data. These data may be treated as the realizations of scalar (univariate) or vector (multivariate) random fields on the sphere. Johns (1963a) studied the stochastic process on a sphere, with its applications to meteorology in Johns (1963b) and Cohen and Johns (1969). Research about stochastic processes on a sphere may also be found in Hannan (1966, 1969), Roy (1972, 1973, 1976), McLeod (1986), and Gaspari and Cohn (1999). A practical algorithm for modeling a large class of twoand three-dimensional scalar correlation functions on the sphere was
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