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Normalization of the diffusive filters that represent the inhomogeneous covariance operators of variational assimilation, using asymptotic expansions and techniques of non-euclidean geometry; Analytic solutions for symmetrical configurations and the validation of practical algorithms; Part I
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    Analytic solutions for symmetrical configurations and the validation of practical algorithms
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    A versatile way of synthesizing the spatial covariance linear operators of a variational assimilation scheme is by using basic building blocks that are either of pure Gaussian form or, more generally, conform to the distributions that result from a finite-time diffusive process whose effective diffusivity is non-trivially tensorial and spatially varying. Such a synthesis is exemplified by the so-called 'recursive filter' approach to assimilation and by the explicitly diffusive syntheses proposed by Derber and Rosati and generalized by Weaver and Courtier. An outstanding problem associated with the practical implementation of these methods in the important cases where there is smooth spatial variability of the diffusive tensor is that of anticipating the local amplitude and derivatives of the result of diffusing a unit impulse. It is on the basis of such an estimate that the diffusion operator or filter must be symmetrically modulated by an amplitude profile in order to ensure that the background covariance is being applied in the assimilation with the amplitude (i.e., the variance) intended. In the spatially homogeneous case, there is no difficulty because the result of a diffusive process acting over a half-unit of effective time leads to a pure Gaussian with second moment tensor equal to the diffusivity and an amplitude calculable precisely from the standard formula for such a Gaussian. In the inhomogeneous case, the amplitude is altered from the Gaussian value by an adjustment factor that depends in a complicated way on the local variations of the diffusivity tensor. This note and its sequel describe an approach, based on a reformulation of the variable-diffusivity problem into an equivalent problem of uniform diffusivity, but in a non-Euclidean geometry, which has the advantage of forcing the amplitude adjustment factor to depend only upon the particular combinations of higher derivatives of the induced metric that define 'curvature' and its derivatives. The proposed approach leads to the systematic construction of successive asymptotic approximations for the adjustment factor in terms of the relevant `curvature' diagnostics of the local diffusivity variations. In this Part I, the problem is discussed in the restricted case where the geometry of the transformed representation happens to possess certain symmetries. In the special case where the implied curvature is uniform (spheres and hyperbolic spaces), analytic methods can be brought to bear and lead to benchmark asymptotic series for both the two- and three-dimensional cases. A more general approach, based on a systematic iteration to recover successive 'slabs' of the asymptotic parameters, is introduced and exhibited, for the two-dimensional case, in the only slightly less restrictive case of a geometry with variable, yet still axi-symmetric, distribution of curvature. It is demonstrated that the proposed procedure reproduces the analytic coefficients that correspond to the uniform curvature cases but that additional terms involving derivatives of curvature emerge in the more general geometry. Treatment of the diffusion problem in the most general smooth manifolds of two and three dimensions requires the full resources provided by the tensorial machinery of Riemannian geometry. This topic, and the demonstration that the corresponding versions of essentially the same iterative algorithm apply also to general manifolds, are deferred to the sequel, where they are discussed thoroughly.
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