Normalization of the diffusive filters that represent the inhomogeneous covariance operators of variational assimilation, using asymptotic expansions and techniques of non-euclidean geometry; Riemannian geometry and the generic parametrix expansion method; Part II Riemannian geometry and the generic parametrix expansion methodalgorithms

This note is the second part of a study which develops the geometrical ideas that underly the 'parametrix expansion' asymptotic method for estimating the shape of the impulse response function of the diffusion equation in a smoothly curved non-Euclidean space. This part of the study develops the algebraic tools required to treat the general n-dimensional case in Riemannian geometry. Notations are developed both to facilitate the hand calculations for the low-order expansion terms, and to enable the bulk of the algebraic manipulations to be automated, should it become desirable to carry out the asymptotic expansion to a higher order. The general procedure is illustrated by exhibiting, in moderate detail, the algebraic operations required to obtain the first two coefficients for the amplitude adjustment quotient in the n-dimensional case, and for the important special cases of two and three dimensions where the curvature terms upon which the formulas depend take on simpler forms. R. James Purser. "June 30, 2008." "This is an unreviewed manuscript, primarily intended for informal exchange of information among the NCEP staff members." System requirements: Adobe Acrobat Reader. Includes bibliographical references (pages 54-55). 2008 NWS (National Weather Service) NCEP (National Centers for Environmental Prediction) Library Public Domain 2285