1. Introduction

There are many methods available for objectively analyzing the meteorological data required to initialize a numerical weather prediction model (e.g., see Daley 1991). Those methods based on formal statistical principles (e.g., Gandin 1963; Lorenc 1986, 1997; Parrish and Derber 1992; Courtier et al. 1998), which permit a proper accounting to be taken of multivariate aspects of the problem, have now largely superseded the overtly empirical methods of “successive corrections” (Bergthorssen and Döös 1955; Cressman 1959; Barnes 1964). Nevertheless, for specialized applications, the empirical methods continue to enjoy the advantages of greater computational efficiency and the ability to adapt more flexibly to the typically large inhomogeneities of density and quality of the available data. While the high efficiency of empirical methods becomes progressively less of a critical factor as available computational power continues to increase, adaptivity remains a factor of considerable importance in circumstances where the day-to-day variability of data quality and quantity are hard to predict beforehand, such as occurs in the processing of satellite sounding data. In this context Hayden and Purser (1995), following up on the work of Purser and McQuigg (1982), developed a numerically efficient and spatially adaptive analysis scheme using spatial smoothers. Each spatial smoother was built up of more basic numerical operators consisting of rather simple recursive filters acting unidirectionally upon the gridded data residuals.

The numerical efficiency of these basic operators can also be turned to advantage within a statistical analysis scheme, specifically in the synthesis of the effective covariance-convolution operators needed by the descent algorithms of the large-scale linear solvers involved (Lorenc 1992). The Statistical Spectral Interpolation (SSI) of the National Centers for Environmental Prediction (NCEP) is an example of an analysis scheme in which the spectral representation of the background error covariance is employed directly (Parrish and Derber 1992). Methods of this type are inherently limited in their ability to deal conveniently with geographical inhomogeneities. Although one motivation of the present two-part study was to develop the tool of recursive filters to allow the operational three-dimensional variational analysis (3DVAR) scheme to accommodate spatial inhomogeneities in the background covariance, the inhomogeneous and anisotropic aspects of the filtering technique will be reserved for the companion paper (Purser et al. 2003, referred to henceforth as Part II); the focus of the present paper is to demonstrate the ability of appropriately constructed recursive filters to achieve acceptably isotropic covariance functions of Gaussian form. Part II will extend this study to more general non-Gaussian profiles and explore the case of spatially adaptive covariances of either locally isotropic or generally anisotropic forms.

A brief review of the ideas that underlie 3DVAR is given in section 2 in order to clarify the points at which the recursive filter plays a part. In section 3 we set forth the relevant theory pertaining to the construction of basic recursive filters capable of being forged into convolution operators reasonably representing the qualities desired by modeled covariance convolutions within an adaptive analysis scheme with a uniform Cartesian grid and with homogeneous covariances. Like the Gaussian covariances of Derber and Rosati (1989), which are obtained by multiple iterations of a diffusion operator, the basic recursive filters are crafted to produce approximately Gaussian smoothing kernels (but in fewer numerical operations than are typical in the explicit diffusion method). Some of the technicalities discussed in this section are treated in greater detail in the appendixes. In another paper, Wu et al. (2002), we provide examples of the applications of some of the techniques presented here to global variational analysis of meteorological data.

2. 3DVAR

The statistical characteristics of the errors ϵ_b and ϵ_o are quite difficult to describe in complete detail owing to numerous complicating factors. Among the common simplifying assumptions, we normally assume unbiasedness:

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The covariance, 𝗥 ≡ 〈ϵ_oϵ^T_o〉, of observational error is assumed to be diagonal, equivalent to assuming the observational errors are statistically independent. The corresponding covariance, 𝗕 ≡ 〈ϵ_bϵ^T_b〉, of background error, however, is never assumed to be diagonal in its representation based on the state-space constructed from the gridded value components; the characteristically smooth form in space of background errors implies that neighboring points have errors, in fields of the same type, that are strongly positively correlated. Although the principles of variational analysis can accommodate strong nonlinearities if required, it is often numerically convenient to exploit the typically weak nonlinearity of H by approximating the effects on H(x) of small increments of x, using the linearization of H:

dxdHx(2.5)

The “primal” variational principle in 3DVAR seeks the minimum over x of the penalty function L₁(x) defined by

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which may be justified by minimum-variance arguments. The solution, x = x_a, then obeys

⁻¹x_ax_b^T−1yHx_a(2.7)

and, hence, must define an increment of the form

x_ax_b^Tf(2.8)

where

f⁻¹yHx_a(2.9)

The inherent smoothness of the background field errors ϵ_b, and hence that of the covariance 𝗕 of these errors, is therefore imprinted on the analysis increments themselves.

Treating this dual form of the analysis problem is essentially the approach adopted at the Data Assimilation Office of the National Aeronautics and Space Administration (NASA) Goddard Space Flight Center and described by da Silva et al. (1995) and by Cohn et al. (1998). This “physical-space statistical analysis system” (PSAS) may be advantageous when using sophisticated preconditioning methods based on careful grouping of data, as discussed by Daley and Barker (2000). However, when only the simplest preconditioning strategies are employed, Courtier (1997) shows that the primal and dual forms imply essentially identical condition numbers for the linear inversion problems. Regardless of which form of 3DVAR is adopted, one must rely on iterative methods to converge toward a practical solution. The most costly part of each iterative step of a solution algorithm is the multiplication of some grid-space vector v by the covariance matrix 𝗕. This is required once per iteration whether treating the primal or the dual form of the problem. Even a single multiplication of this form becomes prohibitively expensive if carried out explicitly with the full matrix having the dimensionality of x. One remedy, adopted by Gaspari and Cohn (1998, 1999) and extending earlier work of Oliver (1995), is to constrain the allowed covariance models to those possessing compact support. They show how families of approximately Gauss-shaped compact-support covariance models may be formulated and applied with much greater numerical efficiency than would be possible for more general functions. Our own approach to achieving reasonable efficiency without restricting the covariances to being of the compact-support type is to assume that an operation of the form 𝗕v may be factored into smaller, less costly, factors.

In the first step, the multivariate structure of 𝗕 is broken apart by the judicious selection of a set of nonstandard analysis variables for which the contributions from 𝗕 naturally separate out. For example, a single variable representing the quasigeostrophically balanced combination of mass and rotational wind fields can be attributed a univariate spatial covariance for its background errors quite independently of the corresponding spatial covariance for the residual unbalanced rotational wind component. Meanwhile, the divergent wind field can be treated independently of either. Further steps in the program of reducing the operator 𝗕 might be, next, to carry out a crude separation of a few additive components of the operator on the basis of their characteristic spatial scales. If this can be done to render the resulting operator components into Gaussian forms, then, in the absence of anisotropies obliquely oriented with respect to the grid, the Gaussians themselves may be factored into the three respective coordinate directions. [A more general discussion of radially symmetric covariance functions can be found in the work of Oliver (1995).] Finally, along each single dimension, a computational advantage may be gained by employing a spatially recursive filter, carefully constructed to mimic the required Gaussian convolution operator, but at a fraction of the still considerable cost of applying directly the explicit Gaussian convolution operator itself. It is the objective of the following sections to reveal precisely how such a recursive filter may be fabricated and applied. We do not wish to imply that the Gaussian form is inherently desirable in data assimilation. On the contrary, careful investigation of the spatial profiles of forecast background error (Thiébaux 1976; Thiébaux et al. 1986; Hollingsworth and Lönnberg 1986) reveals covariance functions that cannot be reconciled with the Gaussian shape alone. However, by treating the two- or three-dimensional quasi-Gaussian filter combination as a relatively cheap “building block,” a far larger range of possible profile shapes becomes accessible, by the superposition of appropriately weighted combinations of quasi Gaussians of different sizes and by the application of the negative Laplacian operator to such components in order to induce the negatively correlated sidelobe characteristics of some components of background error. These aspects are dealt with in Part II. Thus, the motivating consideration for using recursive filters in this context is predominantly that of computational efficiency together with the recognition that much more general forms become available through the exploitation of superposition.

3. Homogeneous recursive filtering theory

The theory of digital filtering was initially developed in the context of time series analysis. However, many of the same techniques are equally applicable in two or more spatial dimensions when the numerical grid is of a sufficiently regular configuration, as it usually is in numerical weather analysis. While we attempt to keep the technical discussion of this section self-contained, other related aspects of the topic of digital filter design are well covered by standard texts such as Otnes and Enochson (1972) and Papoulis (1984).

a. Quasi-Gaussian recursive filters in one dimension

Let K/δx² denote the finite-difference operator:

Kψ_iδx²ψ_i−1ψ_iψ_i+1δx²(3.1)

approximating the differential operator, −d²/dx², on a line grid of uniform spacing δx. The spectral representation of the operator at wavenumber k (wavelength 2π/k) is

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Inverting this relationship, we obtain a formula for k² in terms of K̂:

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Clearly, the same formula relates operator −d²/dx² to operator K; in fact, the algebraic manipulations we set forth here can be regarded as an application of the “calculus of operators” (Dahlquist and Björck 1974, p. 311). Using the standard expansion,

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where

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we may obtain a power expansion for k²δx², and thence, the expansions for the term (k²δx²)ⁱ:

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From (3.2) and (3.3) we find that

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and other coefficients then follow by the application of the inductive step:

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The coefficients, b_i,j, which are all positive and rational, are listed in Table 1 for j ≤ 6.

Consider the differential operator, D_(n),

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whose spectral representation is

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or, with σ = a/δx,

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Since

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the substitution of each power series (3.5) up to degree n for the powers of k² into (3.9) gives us a way of approximating this exponential function in terms of K̂:

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Correspondingly, there is a finite-difference operator, D^*_(n), composed of the nth-degree expansion of K implied by this approximation, which, following a rearrangement of terms, we may write

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Note that, owing to the positivity of all the coefficients b_i,j, this operator is positive definite and therefore possesses a well-defined inverse. Note also that, for σ ≫ 1, the only coefficients in (3.12) that remain significant are the “diagonal” ones, b_i,i = 1, yielding simply the truncated Taylor series for the exponential function of σ²K/2. Shortly, we shall examine the practical impact of the off-diagonal components, b_i,j, j > i, but first we describe the process of extracting from the above algebraic developments a practical class of smoothing filters.

The reciprocal of the function exp(a²k²/2) in (3.10) is a Gaussian function in k and is the Fourier transform of a convolution operator (on the line x) whose kernel is also of Gaussian form. Provided we can find a practical way to invert the operator equation,

D^*_(n)sp(3.13)

for a given input distribution, p, the resulting output, s, will be an approximation to the convolution of p by the Gaussian function whose spectral transform is the reciprocal of the right-hand side of (3.10). The approximation, (D^*_(n))⁻¹, to this convolution is what we refer to as a quasi-Gaussian filter. The common centered second moment of operator D_(n) and its approximation, D^*_(n), is exactly −a², so a is a convenient measure of the intrinsic distance scale of the smoothing filter implied by the inversion of (3.13). A useful fact is that the square of the intrinsic scale of the composition of sequential smoothing filters is the sum of squares of the scales of the individual components. Also, as a consequence of the statisticians' “central limit theorem” (e.g., Wilks 1995) applied to convolutions in general, the effective convolution kernel of such a composition of several identical filter factors resembles a Gaussian more closely than does the representative factor. Thus, provided it becomes feasible to invert (3.13), we possess the means to convolve a gridded input distribution with a smooth quasi-Gaussian kernel, at least in one dimension.

As a matrix, D^*_(n) is banded and, for an infinite domain, symmetric. Conventionally, the linear inversion of a system such as (3.13) might be effected by employing an LU factorization (Dahlquist and Björck 1974) of D^*_(n),

D^*_(n)(3.14)

with lower-triangular band matrix, 𝗔 and upper-triangular band matrix, 𝗕, allowing the solution to proceed as two steps of recursive substitution. On an infinite grid, the same principle pertains, but with the guaranteed simplification of (i) a translational symmetry ensuring that every row of 𝗔 is identical (allowing for the trivial translation) and every row of 𝗕 is identical and (ii) ordinary matrix symmetry by which we can ensure that 𝗕 is simply the transpose of 𝗔. In this case, the LU decomposition of D^*_(n) is also of the symmetric, or Cholesky, type (Dahlquist and Björck 1974).

In the two stages of solution,

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the explicit recursions of the back substitutions are the following basic recursive filters:

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which are conveniently referred to as the advancing and backing steps, respectively, since, in the first, index i must be treated in increasing order while, in the second, it must be treated in decreasing order, in order that the terms on the right are already available at each step. Note that the correspondences between notations of (3.15), (3.16) and of (3.17), (3.18) are

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Defining the total “substance” implied by the distribution p to be Σ_i δxp_i, the operator D^*_(n), and hence its inverse, preserve this quantity. By symmetry, the factor operators, 𝗔 and 𝗕 must therefore also preserve substance, implying that

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The task of distilling the coefficients α_j from the parameters defining D^*_(n) is somewhat technical and is relegated to appendix A. Filters may be constructed at different orders, n, and with or without the refinements implied by the off-diagonal coefficients, b_i,j, for j > i. For n = 1 the filter response comprises back-to-back decreasing exponential functions, which Fig. 1a shows (dashed curve) in comparison to the Gaussian function (solid curve) of the same width of one grid unit. Better approximations to the Gaussian are obtained after application of the second-order filters, as shown in Fig. 1b, and fourth-order filters, shown in Fig. 1c, for the case of the filter with only the diagonal coefficients b (short-dashed curves) and with all of the b coefficients (long-dashed curves). We see that the advantage of keeping all the coefficients is greater at higher order, where they make the resulting filter response a significantly better approximation to the intended Gaussian function. However, the alternative treatments of the b coefficients are virtually indistinguishable at smoothing scales of a few grid units, as the truncation errors of the component numerical derivative operators become insignificant in comparison to the error resulting from the finite truncation of the series for the Gaussian employed in the construction of the filter operator. The cost of applying the filters with or without the off-diagonal b coefficients is the same; therefore, we always adopt the more accurate formulation that includes the off-diagonal coefficients.

We have described the idealized case of operators acting on data extending indefinitely in both directions. In practice, we are confronted with geometrical constraints, either in the form of definite lateral boundaries to the domain, or as periodic conditions appropriate to a cyclic domain. Fortunately, it is possible to generalize the application of the advancing and backing recursive filters to both of these situations. Appendix B treats the case of lateral boundaries and shows how the effect of a continuation of the domain to infinity can be simulated by the imposition of appropriate “turning” conditions at the transition between the advancing and backing stages. Appendix C treats the case of periodic boundary conditions. In both of these special cases the main part of the filtering algorithm and the basic filter coefficients employed are the same as in the case of the infinite domain. By a generalization of the treatment used in the cyclic case, one may efficiently distribute the recursions across multiple processors of a massively parallel computer. This is discussed in detail in a related context by Fujita and Purser (2001).

b. Quasi-Gaussian filters in two dimensions

Let x and y be horizontal Cartesian coordinates, and k and l the associated wavenumber components. Then in two dimensions, we can exploit the factoring property of isotropic Gaussians:

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where ρ = (k² + l²)^1/2 is the total wavenumber. In terms of basic one-dimensional Gaussian smoothing filters, D^(x)_(∞) and D^(y)_(∞), operating in the x and y directions, a two-dimensional isotropic filter, G_a, also of Gaussian form, results from the successive application of the one-dimensional factors, D^(x)_(∞) and D^(y)_(∞). For example, an input field, χ, is smoothed to produce the output field, ψ, by the convolution:

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where

G_aD^(y)_(∞)D^(x)_(∞)

The crucial significance of the Gaussian form for the one-dimensional filters is that this form is the only shape that, upon combination by convolution in the x and y directions, produces an isotropic product filter. In order to generalize our filters to alternative shapes, while preserving two-dimensional isotropy, we shall always attempt to base the construction of the more general filters on the building blocks supplied by the quasi-Gaussian products of the approximations, D^(x)_(∞) and D^(y)_(∞), to the true Gaussian smoothers. But we must first establish what is the minimum order of such a filter that will preserve the isotropy of the product combination, at least to a degree that ensures that any residual anisotropies are not obtrusively obvious.

Figure 2 depicts the results obtained by smoothing a delta function placed at the center of a square grid. Figure 2a shows the result of a single application of the first-order filter, D₍₁₎, in the x and y directions. This result is clearly neither smooth nor even approximately isotropic. Figures 2b and 2c show the results obtained by using the filters of orders two and four. We see that the appearance of isotropy is not adequately attained until the order exceeds two, but the fourth-order filter shown in Fig. 2c seems to provide an excellent approximation to the isotropic Gaussian. For applications in data assimilation, it is usually worth the cost of applying a filter of at least fourth order if the filter is to be applied only once in each of the orthogonal grid directions. For a roughly equivalent cost, one may also apply the simple first-order filter four times in succession (but with a scale only a half as large in each instance). This has been the approach employed in earlier applications of recursive filtering to data analysis, such as Purser and McQuigg (1982), Lorenc (1992), and Hayden and Purser (1995). The result is shown in Fig. 2d, but is clearly inferior to the use of the single fourth-order filter.

Very often, the physical variables of interest in an analysis are derivatives of the variables it is convenient to base the covariance model on. For example, covariances of the steamfunction or velocity potential (scalars) are often more convenient to handle than the derived covariances among velocity components at two locations. Since we may wish to employ the results of our filters as building blocks of such differentiated covariances, it is as well to examine the derivatives of fields analogous to those of Fig. 2. In order to permit any departures from isotropy to stand out more clearly, we take the Laplacian of the result of smoothing the delta function. Figure 3 shows three such results (with a slightly smaller scale than was used in Fig. 2), involving single applications (in x and y) of D_(n) with n being 2, 4, and 6 in panels a, b, and c, respectively. Even more so than in Fig. 2, we see that it is not until we adopt at least fourth-order filtering that we obtain an acceptable degree of isotropy. For reference, the “right answer” obtained using the Laplacian of the true Gaussian, G_a, is shown in Fig. 3d.

4. Discussion

The problem of efficiently accommodating accurate approximations to isotropic covariance functions in a variational analysis has been solved using recursive numerical filters. The covariances are never explicitly computed; instead, their effects as convolution operators are represented. This is achieved 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. While the building blocks of a covariance operator generated by recursive filtering are always of quasi-Gaussian form as required by the factorization property (3.22), the final covariances themselves may be given more general radial profiles through the superposition of varyingly scaled component Gaussians, or by combining these components with the application of Laplacian operators, as we discuss in Part II.

A global model requires additional refinements to the technique we have described here for purely geometrical reasons. For one thing, the presence of polar coordinate singularities makes the generation of even approximately isotropic covariances impossible by the method of sequentially applying the quasi-Gaussian recursive filters sequentially along the two principal grid directions. But even if we adopt a patchwork of overlapping rectangular grids [which is essentially the approach we have adopted for the global analysis described in Wu et al. (2002)], the use of the present filters, which are homogeneous in scale only relative to the grid to which they are applied, translates to the application of smoothing operations, which, on grids of global extent, cannot be everywhere even approximately of uniform scale in the degree to which they smooth the given data. In order to address this need to generalize the filters to obtain control over their spatial smoothing scale from one geographical location to another, Part II will also describe technical extensions that not only allow geographical control of the characteristic scale and amplitude of the covariance, but that also allow geographically adaptive three-dimensionally anisotropic stretching of the quasi-Gaussian building blocks from which the covariances are constructed.