1. Introduction

In the first part of this study (Purser et al. 2003, henceforth referred to as Part I) we focused on the numerical methods that could be applied efficiently to the task of generating spatially homogeneous and horizontally isotropic smoothing kernels on a regular grid. We showed how carefully constructed high-order quasi-Gaussian recursive filters could overcome some of the well-known deficiencies of the simpler first-order recursive filters used in empirical data analysis schemes by Purser and McQuigg (1982), Hayden and Purser (1995), and discussed by Lorenc (1997). The most serious problem with the first-order filters is that, unless iterated many times, they show a pronounced spurious anisotropy aligned with the directions of the numerical grid lines. The high-order filters of Part I produce more closely circular contours of amplitude without the need for numerous costly iterations. In addition, we showed how to improve the treatment of boundaries and noted that Fujita and Purser (2001) used a generalization of the cyclic boundary treatment as the basis of one of the efficient parallelization strategies they tested in a related context.

The emphasis of this second part of the study is the generalization and refinement of the methods of Part I to enable a far richer variety of covariance operators to be generated with controlled geographic variations. Section 2 deals with specific proposals for the construction of non-Gaussian parameterized families of covariance models based on linear superposition of the quasi-Gaussian “building blocks” that the recursive filters of Part I provide. There are many reasons that Gaussian forms are convenient in this respect, and some of these are discussed in Part I. Gaspari and Cohn (1998, 1999) have successfully used efficient compact-support approximations to Gaussian-shaped covariances in their implementation of the physical-space statistical analysis system (PSAS) described in da Silva et al. (1995). Some of the techniques for synthesizing more general covariances from quasi-Gaussian building blocks apply equally to these direct applications of compactly supported covariance models. One covariance family that we have found to be extremely convenient to use and beneficial in applications comprises bell-shaped distributions with significantly fatter tails than the Gaussian. We discuss the efficient construction of approximations to these fat-tailed distributions that allow a broader dynamical range of scales in the analysis increments to be assimilated.

A variation of the characteristic horizontal scale of the filters allows analysis schemes to adapt to the geographic variations in the density and quality of the recent observational data, which clearly are reflected in the statistical characteristics of the errors of the forecast background field—the starting point for the new analysis. Also, for global, and other large-scale, domains, it is not possible to maintain a uniform resolution everywhere with an orthogonal coordinate grid, owing to the intrinsic curvature of the earth. Thus, even were we to require a covariance of perfectly uniform characteristic scale in units of true distance, this would necessarily translate to a requirement for a filter of varying scale in the grid units that are relevant to the numerical construction of the filter. These matters are dealt with in section 3 and a proposal for the approximation of the filter's amplitude in the case of inhomogeneous filters is provided in the appendix.

A further important generalization of the covariances involves deliberately relaxing the artificial constraint of local isotropy. The advantages in a variational analysis of not insisting on isotropic covariances are probably obvious to any practitioner and are discussed in the recent article by Otte et al. (2001). Studies of the covariances implied by four-dimensional variational methods (e.g., Thépaut et al. 1996) provide objective confirmation of the need for such covariances. When the analysis is performed in a favorably distorted coordinate system, such as the geostrophic momentum coordinates used in the study of Desroziers (1997), an anisotropy in the geographical metric occurs automatically even while the filter remains “isotropic” relative to the (distorted) grid. In a somewhat analogous way, a meteorological analysis carried out in an isentropic coordinate framework (e.g., Shapiro and Hastings 1973; Benjamin 1989) automatically provides enhanced vertical resolution in zones of above average static stability where it is very often beneficial to clarify the more intricate structures of frontal zones or inversion layers. However, it is also desirable to be able to acquire a more precise control over the form and degree of anisotropy assigned to the covariance kernels than one obtains by relying on the fortuitous properties of certain predefined coordinate choices. Section 4 describes a new approach to the construction of general anisotropic covariances, which we call the “hexad” algorithm. At any given location, the three-dimensional covariance is synthesized from combinations of six recursive filters that act along oblique lines of the analysis grid (in two dimensions there is an analogous “triad” algorithm). The symmetric “aspect tensor” prescribing the local second moment structure of the desired covariance can be shown to have a unique hexad (or triad) representation by line filters on the given grid. We note that the application of anisotropic filtering is analogous to a diffusion process generalized to posses a diffusivity in the form of an anisotropic symmetric tensor. The aspect tensor for a homogeneous diffusion process with diffusivity D acting for a duration, t, is then just twice the product of D and t. This explicitly diffusive approach is discussed by Weaver and Courtier (2001) in their extension of the earlier work of Derber and Rosati (1989), and is another valid way of considering the problem of generating anisotropic Gaussian smoothers. The main advantage of employing recursive filters is that they tend to require significantly fewer iterations to achieve the desired Gaussian form. Our concluding remarks are given in section 5.

2. Synthesis of a covariance in terms of Gaussians

Let G(x₁, x₂) denote the kernel of a (quasi-) Gaussian recursive filter operator of the kind defined by Eq. (3.23) of Part I. For a univariate covariance constructed directly from this Gaussian, the amplitude (i.e., the “variance”) may be made to vary geographically while preserving the important property of self-adjointness by modulating the otherwise homogeneous filter before and after application:

B₁₂w^1/2₁G₁₂w^1/2₂(2.1)

where w(x) is the total effective weight at x applied to the G. However, it has been recognized that objective analysis using the Gaussian shape to model the covariance severely hampers the ability of the analysis to assimilate the smallest scales of significant background error. In adverse configurations of the data, the problem is apt to manifest itself in excessive and damaging extrapolation effects at the edge of isolated data voids where the analysis strives to fit the surrounding more densely distributed data smoothly. Lorenc (1981) provides an illustration of this effect with idealized data. The small-scale analysis increments are inadvertently inhibited when, as with a Gaussian model, the presumed power spectrum values at moderately large wavenumbers become much smaller than the values that the data and experience indicate to be appropriate. Recall that the power spectrum for a spatially homogeneous covariance model is simply the Fourier transform of that covariance. For example, the two-dimensional Gaussian function of unit integrated weight, and scale parameter, a,

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is associated with the power spectrum,

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which is itself of Gaussian form and which, therefore, does possess a rapidly diminishing tail in the spectral region of total wavenumbers exceeding a few times the “characteristic wavenumber,” (1/a). Clearly, the ideal remedy would be the representation of the actual covariance operator (whatever that is); at least it should be possible, within the space of isotropic covariances, to find one possessing more appropriate tail characteristics, although this almost certainly necessitates exploring options other than the Gaussian family. In principle, however, by the methods of Laplace transforms, it should be possible to synthesize almost any practical isotropic covariance profile as a superposition of Gaussian components, as is noted by Schoenberg (1938) and more recently discussed by Gneiting (1999):

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In practice, we would wish to approximate the general superposition (2.4) by a discrete approximation involving many Gaussian constituents. That is,

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where M ≈ 5 typically. Given that the individual Gaussian components are positive definite, then B̂ is also, provided the weights w_p are all non-negative (in general, a positive superposition of positive definite operators is itself positive definite). We would also like to confine ourselves to constructions generalizing this type so that, in the spatial domain, the combination can be expressed in a way that does not jeopardize the self-adjointness that the iterative solution algorithms depend on. The appropriate form in the spatial domain is therefore

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Note that, although in practice we should expect the weights w_p to vary smoothly in space, the self-adjointness and positive definiteness of this construction hold regardless. The profile of weights must be regarded as samples, at the discrete scales selected, of an underlying continuous weight profile. The density of selected scales per “octave” (the term we shall use for a change by a factor of 2 in a scale or a wavenumber) is something we must determine according to the smoothness of the continuous weight profile in the log-scale domain. With an adequate scale resolution, it then becomes possible to change the overall scale progressively across the extent of a large domain without appreciably altering (except by linear contraction or expansion) the intended shape of the covariance profile.

One of the simplest general families of scale profiles accommodating the requirement that the sampled weights w_p all be non-negative is what we shall call the “hyper-Gaussian” family of functions. For the two-dimensional isotropic hyper-Gaussian functions normalized to have unit integrals, a generic member of this family is characterized by a scale parameter, σ, and a shape parameter, γ. One may regard each member function as being a continuous superposition of isotropic Gaussians of horizontal scales exp(s) according to a weighting profile in s that is itself of a Gaussian form, centered on log(σ) and with a dispersion parameter in s of γ:

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Note that, in the limiting case, γ = 0, this model reverts to pure Gaussian form.

Figure 1a illustrates the radial profile of the correlation implied by this covariance model for γ = 0, 0.1, 0.2, and 0.3. The parameter γ provides control over what we may refer to, following statistical parlance, as the “kurtosis” of the distribution, which is a quality of the response shape sensitive primarily to the nondimensionalized fourth moment, as defined below. Suppose we define unidirectional moments of an isotropic distribution, H(x):

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Thus, for the normalized Gaussian, G_a, we have the moments, μ₀ = 1, μ₂ = a², and μ₄ = 3a⁴. Since the construction (2.7) is a linear superposition of Gaussians, and each moment is a linear functional of the distribution, we find that, for the hyper-Gaussian with shape parameter γ, the corresponding moments are μ₀ = 1, μ₂ = σ² exp(2γ) and μ₄ = 3σ⁴ exp(8γ). Then, if we adopt the definition of kurtosis to be the nondimensional quantity,

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we find that the kurtosis for the hyper-Gaussian of shape parameter γ is

κ_γγ

As a generic shape parameter, the kurtosis has its limitations. In particular, it is generally not appropriate to define the kurtosis of a distribution that has regions of negative values, for example, covariances with negative sidelobes such as those defined below.

A further consequence of the superposition property is that the power spectra of the hyper-Gaussian, and its negative Laplacian, are expressible as simple integrals:

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These power spectrum families are shown, for the same range of shape parameters as before, in Figs. 2a (for Ĥ) and 2b (for Ĥ′). Note the dramatic effect on the power at small scales (large wavenumber) that results from even small positive values of the shape parameter, γ.

The natural question that now arises is how fine a resolution in the log scale, s, is required to adequately represent these covariance models by the approximations that replace the integral representations (with respect to s) by discrete summations. This can be answered by observing how far the discrete approximations' power spectra depart from the exact integral representations' power spectra. In practice, we find that, for γ = 0.3, about three scales per octave appears to be adequate. For smaller γ it is prudent to increase this density of discrete scales. In the context of a multigrid construction (discussed in Part I), it is clearly convenient numerically to have an integer number of discrete smoothing scales of the basic Gaussians in each octave, or, in other words, to have the same whole number of smoothing scales per grid of the multigrid hierarchy.

3. Inhomogeneous generalizations

In this section we treat cases in which the grid remains orthogonal and smooth in terms of its resolution, but not necessarily uniform or without curvature. At the same time, we treat the case in which the filter remains locally isotropic, but whose smoothing scale is permitted to vary geographically. Polar grids, such as plane polars or global latitude and longitude grids, possess special rotational symmetries that can be exploited in the case of the spatially homogeneous smoothing filters that respect those symmetries. But polar grids also present unique difficulties involving the polar singularities themselves, which then require special corrective measures to be applied to the filters. We pay attention to these problems in this section and suggest some of the remedies that are possible by exploiting the analogy between recursive filtering and a general diffusion process. In this regard, the generalized recursive filters can be regarded as accelerated numerical solvers for approximating the action of inhomogeneous diffusion. These methods can therefore be considered consistent alternatives to the direct application of diffusion operators proposed by Derber and Rosati (1989) and recently generalized by Weaver and Courtier (2001).

a. Inhomogeneities of grids or filter scales

One should not be led to believe that our construction of quasi-Gaussian filters is necessarily restricted to perfectly uniform Cartesian grids. On a smoothly varying nonuniform grid in one dimension, the tridiagonal discretization of the differential operator d²/dx², and polynomials of the discretization by which the requisite powers of d²/dx² are approximated, still lead to banded matrices that, as shown below, can be rendered symmetric by a similarity transformation with a diagonal matrix related to the metrical properties of the grid. A power, i, of the simplest tridiagonal representation of a second derivative operator results in a band matrix of 2i + 1 diagonals. Also, we can generalize the conditions of homogeneity of the smoothing scales to incorporate the effects of a scale that can vary smoothly across the grid, again, without invalidating the property of self-adjointness. However, this additional generalization requires that, in all appearances of the operator, [−(a²/2)d²/dx²], in the counterpart to the polynomial (3.8) of Part I, a form of the second derivative factor is substituted that is self-adjoint even when a is a function of x. Of the qualifying possibilities, the one that is most convenient in practice and that leads to a substance-conserving filter is the one most closely identified with the operation of a diffusive process:

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The operator, (3.1), would be appropriate when the grid lines along which x varies are all parallel but, in a general orthogonal curvilinear grid, this is no longer true. The final generalization we add in this section is the accommodation of grids with converging or diverging grid lines. We do this by including a metric term, τ, whose reciprocal is the density of x-grid lines so that τ itself may be thought of as the line or area measure (according to whether the grid is two- or three-dimensional) of the interface orthogonal to the grid line and attributed to it in finite-difference operations. Using partial derivatives to emphasize the implied multidimensionality, the operator we need to generalize (3.1) is

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which is self-adjoint in the sense of an inner product defined

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Let x_i be the main grid coordinates for integers i and let the intermediate staggered grid of points such as x_i+1/2 be a smooth interpolation from it. Likewise, by smooth interpolation, we assume a and τ to be available at the main and staggered grids. Define

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Redefining the scale parameter σ = a/δx as in Part I, the simplest consistent discretization of the operator (3.2) is

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where

ν_iτ_iδx_i(3.7)

defines the local grid cell measure (area or volume). We can relate this operator to a tridiagonal matrix, 𝗞, that serves to generalize the K of Eq. (3.1) of Part I. A representative row, i, of 𝗞, is defined by

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in terms of which, the finite-difference operator of (3.6) is obtained:

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where σ and ν are the diagonal matrices formed from the values σ_i and ν_i. The components of the matrix 𝗞 obey the approximation

K_i,i−1K_iK_i,i+1(3.10)

very closely when a, δ_x, and τ are all smooth and slowly varying in x, tending to these values in the limiting case of constant a, δx, and τ. Having found a consistent self-adjoint, but low-order-accurate numerical approximation to the appropriate second derivative, the suggested refinement of accuracy available through the use of the coefficients b_1,j defined in Part I is

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Taking the Taylor series for the exponential of this operator, but truncating all the terms composing matrices of half-bandwidth exceeding n (as discussed in Part I, n is typically four to obtain results closely approximating the Gaussian shape), we obtaining the sought-for generalization of D^*_(n) in Eq. (3.12) of Part I:

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While the coefficient-finding method of appendix A of Part I is no longer applicable in the general inhomogeneous case, Cholesky factorization is still possible, since at least the matrix sandwiched between diagonals 1/ν and ν of (3.12) remains symmetric. This factorization provides the means to construct the associated advancing and backing recursive filters. However, these filters now have coefficients varying in space and so are slightly more complicated to apply. Also, the method of setting end conditions described in appendix B of Part I can no longer accurately simulate the indefinite continuation of the grid beyond a boundary in general. In practice, we apply those techniques as if the smoothing parameters at the boundary point in question are the values everywhere and we then find that the imperfections that result are often barely noticeable.

In order to control the amplitude of the covariance synthesized from inhomogeneous filters of the kind we have described we need to estimate, for each point in the domain, the amplitude of the result of applying the sequence of basic filters to a unit impulse located at this same point. The homogeneous case conforms to the Gaussian model. The Gaussian model with constant scale parameter a for one direction corresponds to diffusion in this direction with “diffusivity” D for “time” t, when

Dta²(3.13)

and the impulse-response value of the result is just (4πDt)^−1/2. In more than one dimension, the diffusivity D generalizes to a tensor (Weaver and Courtier 2001) and the appropriate generalization of the impulse response, |4πDt|^−1/2, involves the determinant of D. What we have referred to as the aspect tensor is simply the tensorial generalization to these higher dimensions of the combination 2Dt of (3.13). But inhomogeneity of scale, which we may interpret as inhomogeneity of the effective diffusivity in the diffusion analog of our filters, leads to impulse-response functions that differ slightly from the profile calculated on the basis of the Gaussian model. An asymptotic analysis of this difference, which is outlined in the appendix, provides us with a valuable practical refinement to the Gaussian amplitude approximation. It emerges from this analysis that, to a good approximation, the impulse response at a given point of simulated inhomogeneous diffusion acting for time t is the same as the amplitude obtained by diffusing for duration t with an alternative homogeneous diffusion process whose constant diffusivity D is a local weighted average of D. The appropriate weighted average can be obtained by applying to D the original diffusion process, but for only half the usual time, t/2, and evaluating the result of the smoothed diffusivity field D at the point under consideration. The refinement is valid to first order in the magnitude of the modulation of D, so it is actually sufficient (and more practical) to simply smooth the single field of |4πDt|. Note that the diffusion operation acting for duration t/2 is equivalent to the application of the “square root” (in the convolution or linear operator sense) of the total filter. (Note that, for homogeneous filters, the square root filter is expressible, by way of the convolution theorem, as the inverse transform of the square root of the Fourier transform of the original filter.) In many practical applications of the recursive filter method to data analysis, this square root filter, or at least a very good approximation to it, is already available owing to the manner in which the total filter is synthesized from simpler components. Thus, the amplitude refinement imposes no significant extra computational burden in typical cases where, once constructed, the same smoothing operator is subsequently applied over and over again.

The generalizations of the recursive filters we have described in this subsection work on a wide variety of grids provided each grid itself contains no singularity. But this restriction unfortunately precludes the use of the methods on a polar grid in the immediate vicinity of the pole. In order to treat such a case, the next subsection discusses some of the special techniques that can be brought to bear.

b. Polar grids

We shall only treat in detail the special case of filters with homogeneous filtering scale, a. On a plane-polar grid or on a global grid of latitudes and longitudes, the recursive filter method can be adapted in conjunction with discrete Fourier transforms applied azimuthally or longitudinally to data, providing that the longitudes are uniformly spaced and of a number factorable into small primes, as required for the efficient application of the FFT algorithm (e.g., Press et al. 1992). Fourier transformation separates the two-dimensional smoothing problem into independent one-dimensional filtering operations in the radial or meridional directions applied to the zonal Fourier coefficients.

Suppose θ and λ are latitude and longitude, respectively. From initial data χ(θ, λ) we may apply the zonal transform:

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The idea is to redefine the operator D^*_(n) of Eq. (3.9) of Part I in a fully two-dimensional way as the polynomial of −(a²/2)∇² instead of −(a²/2)d²/dx². But we shall see that, through the process of Fourier transformation, the action of this operator is reduced to a set of one-dimensional operators acting on the various wavenumber components separately. For constant a, the operator corresponding to, −(a²/2)∇², takes the following form in the semispectral domain (latitude and zonal wavenumber):

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where the metric term, τ, is now defined for an earth of radius R simply as

τθRθ(3.16)

Note that, for each separate zonal wavenumber m, this operator has its simplest numerical representation in the discrete latitude grid as a tridiagonal matrix, but high-order corrective terms can be added using the method involving coefficients b_1,j described in section 2a of Part I. As in the context of a nonuniform one-dimensional grid discussed above, this matrix operator is transformable to symmetric form through a similarity relation. In fact, the second term of the operator on the right of (3.15) applied to a scalar s may be identified exactly with either side of (3.9) when we use the substitutions

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together with the same 𝗞 defined as in (3.8a)–(3.8c). Following the previous example, the Taylor series of the exponential function of the discrete matrix representation of −(a²/2)∇² is taken, and again, only component matrices of half-bandwidth not exceeding n are retained, in order to obtain the representation, at each wavenumber m, of D_(n). By analogy to (3.12), the numerical approximation D^*_(n,m) to the wavenumber m components of this nth-order operator, D_(n), for the polar grid may be expressed:

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From the derived Cholesky factors, the coefficients of the advancing and backing basic filters are then extracted separately for each wavenumber component.

These filters are more expensive to apply globally than the doubly recursive filters of section 3a because they require a zonal FFT to be applied to the input data at each latitude and an inverse FFT applied to the final output data. But they do provide a satisfactory solution to the “polar problem” in the case of homogeneous smoothing scale, a. In an earlier phase of this study, two of us (RJP and NMR) investigated filters of this semispectral form for a global analysis and devised methods for constructing hybrid filters in which only the polar caps are treated by the FFT, the data elsewhere being dealt with by the methods of section 3a. A discussion of a proposed extension to this technique to accommodate geographically inhomogeneous scale can be found in Purser et al. (2001), but perhaps a more relevant generalization is the accommodation of fully anisotropic functions, discussed below.

4. Anisotropic covariances

In addition to spatial inhomogeneity, we would like to be able to stretch the shape of a local representative contour surface of the covariance function into the form of an ellipsoid (or an ellipse, if in two dimensions). Except in the unnatural special cases where the principal axes for the stretching exactly coincide with the coordinate grid directions, we cannot achieve the desired stretching without including nonstandard grid lines among the set of directions along which recursive smoothing operators apply. In three dimensions, the description of a general linear stretching involves six independent components of the symmetric aspect tensor defining the spatial second moments. The essentially additive property of second moments under composition by spatially unbiased filters (which is an exact result in the case of spatially homogeneous smoothers, and a good approximation in most other cases) allows the six independent aspect tensor components to be resolved into a hexad of generalized grid lines and their associated one-dimensional second moments of dispersion. A special convention for choosing this hexad, which we will briefly describe, ensures that this resolution of the aspect tensor is essentially unique.

a. Definition of a feasible hexad

On a grid represented by any 3-vector of integers, the directions of a feasible hexad are collectively the set generated by the integer displacements, or “generators,” g_p, for p = 1, … 6, where the triple product

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In the context of such a hexad it requires only linear analysis to resolve the aspect tensor components into the additive components associated with each of the six generators of a given hexad. For example, suppose the intended aspect tensor at a particular grid location is given by some symmetrix matrix, 𝗔, and, adopting and extending the notation established in Part I, let the six grid-relative smoothing scales associated with the given hexad of generators, g_p, be denoted σ_p. A unit value for such a smoothing scale assigned to a direction defined by a generator g would result in a contribution to the aspect tensor of gg^T, and therefore, by the additive property of the contributions from the sequence of six smoothing operations with scales σ_p, we shall find that the final aspect tensor obtained at the end of this sequence will be

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Formally, since there are six independent components of 𝗔, this tensor equation can be rearranged as a set of six linear equations for the six unknowns σ²_p forced by the “right hand side” vector, A, of the independent components of 𝗔 through the action of the appropriate 6 × 6 matrix constructed from the corresponding components of the terms g_pg^T_p. For example, suppose we are given the hexad whose generators are defined by the components of the set g listed in Table 1, and an aspect tensor:

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Putting the independent components of 𝗔 into the 6-vector, A, according to

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we resolve the aspect tensor into this hexad's smoothing quantities, σ²_p, through the following matrix equation:

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Each column of the square matrix in (4.6) comes from the six independent components, taken in the same order as in (4.5a)–(4.5f), of the tensor g_pg^T_p for the corresponding generator. For example, the independent components of

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are gathered into the vector [1, 1, 0, 0, 0, −1]^T forming the first column of the matrix in (4.6). For this example, the solution of (4.6) is

σ²_p(4.8)

However, for the hexad to have validity as a smoother, all six of these projected components σ²_p must also be non-negative, which is evidently not the case for the hexad of generators in the present example, since the first component σ²₁ < 0. Although the proof is beyond the scope of this paper, it can be shown that for every positive-definite aspect tensor there is always, and only, one way to associate positive projected components with the directions of a feasible hexad. The following subsection describes the procedure by which the required hexad is found.

b. The hexad algorithm

Geometrically, any feasible hexad of generators, together with their antipodes, constitute a set of 12 points whose convex hull forms a linearly transformed cuboctahedron, two examples of which are shown in Fig. 3. [A cuboctahedron is the semiregular polyhedron formed by slicing off the corners of a cube between its edge midpoints to leave a boundary of eight equilateral triangles and six squares; e.g., see Coxeter (1973, 18–19).] In seeking the valid hexad for a given aspect tensor, when an invalid trial hexad (Fig. 3a, say) projects a negative component of this tensor onto one of its six diameters, we step closer (e.g., Fig. 3b) to the sought-after valid hexad by replacing only the generator associated with the offending line with a generator of the only possible alternative line that the defining hexad rules permit. The stepwise replacement of one trial hexad configuration by another therefore has a nice geometrical interpretation, which is illustrated in Fig. 3 by the mutation from panel a to b. We see that the pair of vertices at the extreme right and left of Fig. 3a are absent in the configuration of Fig. 3b, while the extreme upper and lower vertices shown in Fig. 3b are absent in Fig. 3a. If the vertex at the extreme right of Fig. 3a had the designation, g₁, with other vertices assigned so as to conform to the rules, (4.1) and (4.2), then one way of assigning labels to the vertices of the configuration in Fig. 3b, also consistent with these rules, is to define the new generators (primed) as follows:

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Corresponding rules are easily found for the other five transition possibilities. In the case of the numerical example introduced in the last subsection, where it was indeed found that generator g₁ led to a corresponding σ²₁ with negative value, the replacement of g₁ by the alternative, g^′₁, in accordance with these prescribed rules leads to the new hexad defined in the right column of Table 1. In this case, when we resolve the aspect tensor into the components σ²_p we obtain

σ²_p(4.10)

whose components, all being positive, confirm that the new hexad in this example is a valid one. Note that Fig. 3 may be regarded as a pictorial representation of our one-step hexad iteration viewed approximately from the direction of the vector, (1, 1, 1).

In practice, a short chain of such iterations will usually suffice to locate the given aspect tensor's unique valid hexad (and there is an analogous “triad algorithm” for the two-dimensional case). The resolution of a given aspect tensor into its equivalent hexad is carried out at every analysis grid point and the six non-negative values of σ²_p that are obtained at each point can then be used to prescribe the coefficients of the recursive filter that produces this degree of dispersion along the generalized grid line that the corresponding hexad generator defines. Since the generators at one grid point are typically largely the same ones at the neighboring grid points, it becomes possible to link together the smoothing step in each particular generalized grid direction to form continuous nontrivial line segments. It then becomes desirable to define a procedure for smoothing along such segments in an efficient way that avoids the application of the filter along a line in one direction interfering with the application along another intersecting line. This matter is addressed in the following section.

5. Discussion

We have extended the methods of Part I to show how recursive filters can be used to construct covariances with shapes more general than purely Gaussian. The problem of efficiently accommodating approximately isotropic but spatially inhomogeneous covariance functions in a variational analysis has also been solved using recursive numerical filters. This approach may be regarded as an alternative to that of Gaspari and Cohn (1998, 1999), who use direct calculation of the covariances but achieve numerical efficiency by restricting the approximations to Gaussian forms to having compact support. In our case, the covariances are never explicitly computed; instead, their effects as convolution operators are represented, through a sequence of applications of carefully designed recursive filters operating along the various lines of the appropriately chosen computational grids. In a regional analysis, there is no reason not to use the grid of the intended numerical prediction model. In a global context, where the usual latitude and longitude grid possesses polar singularities, we may either adopt the special methods for polar grids discussed in section 3b or, by invoking additional interpolations, cover the global domain in overlapping maps, each of which being furnished by an appropriate Cartesian grid. For example, we can adopt square Cartesian grids embedded in the respective polar stereographic projections for the polar cap regions and a Mercator grid elsewhere, in order to preserve the property of local isotropy, and use the multi-Gaussian methods of synthesis to provide the necessary control over the horizontal scale (needed to compensate for the map-scaling factor, if nothing else). Experiments reveal no evidence that the analysis results are significantly degraded by adopting grids that differ from the model grid, as long as the conversions between them are by high-order accurate interpolations. However, this synthetic method results in non-Gaussian covariances, even when Gaussians are preferred, and, since we must account for the cost of the additional grid-to-grid interpolations, it can be more expensive than adopting the special procedures of section 3b. For a global analysis, the user must choose the method best adapted to his or her requirements. These matters are dealt with in greater detail in Wu et al. (2002), which successfully applies these recursive filter methods to a global analysis of real meteorological data.

An additional development that we have described in section 4 is the further generalization of the covariance operators to accommodate fully anisotropic effects. Recent approaches to three-dimensional data assimilation where it is not assumed that the covariances must be locally isotropic have been reported by Desroziers (1997) and by Riishøjgaard (1998), and objective statistical methods for estimating the parameters of anisotropy from the data themselves are suggested by the work of Dee and da Silva (1999) and Purser and Parrish (2003). We expect that anisotropic covariances will play a much more significant role in variational data assimilation in the future.