Ooce Note Series on Global Modeling and Data Assimilation Construction of Correlation Functions in Two and Three Dimensions and Convolution Covariance Functions
Ooce Note Series on Global Modeling and Data Assimilation Construction of Correlation Functions in Two and Three Dimensions and Convolution Covariance Functions
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通讯作者：
Richard B Rood;Head;G. Gaspari;Stephen E. Cohn
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作者：
Richard B Rood;Head;G. Gaspari;Stephen E. Cohn
Section 2 was divided into four subsections. Much of the material diiers only slightly from the original. The material in Section 2.2 signiicantly extends the development in OOce Note 96-03. To maintain consistency with the notation used in the refereed publication related to this document, the subsection numbering was modiied from that of OOce Note 96-03. Theorem 3.1.5 (now Theorem 3.a.5) was strengthened. Consequently, the proof given is more involved than the original one. changes were made to several proofs to improve the clarity of presentation. The rst three examples of Section 4 diier only slightly from those given in OOce Note 96-03. The fourth example (Section 4.4) is new. An Appendix was added. Appendix A.1, A.2, and A.3 contain the proofs described above. Appendix A.4 is new. Four additional gures (Figures 9-12) were added. Sharper images of Figures 1-8 replace those given in OOce Note 96-03. Abstract This article focuses on the construction, directly in physical space, of simply pa-rameterized covariance functions for data assimilation applications. A self-contained, rigorous mathematical summary of relevant topics from correlation theory is provided as a foundation for this construction. Covariance and correlation functions are deened, and common notions of homogeneity and isotropy are clariied. Classical results are stated, and proven where instructive. Included are smoothness properties relevant to multivariate statistical analysis algorithms where wind/wind and wind/mass correlation models are obtained by diierentiating the correlation model of a mass variable. The Convolution Theorem is introduced as the primary tool used to construct classes of co-variance and cross-covariance functions on R 3. Among these are classes of compactly supported functions that restrict to covariance and cross-covariance functions on the unit sphere S 2 , and that vanish identically on subsets of positive measure on S 2. It is shown that these covariance and cross-covariance functions on S 2 , referred to as being space-limited, cannot be obtained using truncated spectral expansions. Compactly supported and space-limited covariance functions determine sparse covariance matrices when evaluated on a grid, thereby easing computational burdens in atmospheric data analysis algorithms. Convolution integrals leading to practical examples of compactly supported covari-ance and cross-covariance functions on R 3 are reduced and evaluated. More specii-cally, suppose that g i and g j are radially symmetric functions deened on R 3 such that g i (x) = 0 for kxk > d i and g j (x) = 0 for kxk > d j …
