1. Introduction

Work by Lorenc (2003), Buehner (2005), Wang et al. (2007), Buehner et al. (2010a,b), Liu et al. (2009), and others indicate a growing interest in using localized ensemble covariances in three- and four-dimensional variational data assimilation (3D-VAR and 4D-VAR, respectively) schemes. Buehner et al. (2010a,b) found that a 4D-VAR scheme that used localized 3D ensemble covariances for a background error covariance matrix and a tangent linear model (TLM) and adjoint, outperformed versions of operational 4D-VAR and ensemble Kalman filter (EnKF) data assimilation (DA) schemes. They also showed that an ensemble-4D-VAR scheme that only used localized 4D ensemble covariances and did not use a TLM or adjoint, outperformed a version of the operational 4D-VAR scheme both in the tropics and Southern Hemisphere, but not in the northern extratropics. They also mention that, without including the cost of generating the ensemble, ensemble-4D-Var could be performed at a fraction of the cost of a conventional 4D-Var scheme. However, the descriptions of how ensemble covariances can be incorporated into variational schemes given in Lorenc (2003), Buehner (2005), Wang et al. (2007), and Buehner et al. (2010a,b) give scant details or understanding of the factors that influence efficiency. This paper provides understanding vital to the design and implementation of fast ensemble covariance localization algorithms for variational DA. The fast algorithm derived pertains to Buehner et al’s (2010a,b) ensemble-4D-VAR in which the 4D ensemble covariances are nonadaptively localized in space alone, with no localization in time or modulation of covariances between variable types. The fast algorithm draws heavily from techniques that have been developed for nonensemble variational DA.

Variational DA algorithms perform a global minimization of a penalty function containing a background or prior term written in terms of a background error covariance matrix 𝗣₀^f. This is a nonsparse matrix defining the error covariance between every pair of points and variables in a forecast model, which for modern NWP systems may have a billion degrees of freedom. Not only is it computationally impossible to store and manipulate matrices of this size, but it is also impossible to know them (Dee 1991): there is not enough information in all the forecasts made in the lifetime of an NWP system to accurately estimate even a static covariance. Another problem is that the poor conditioning of the covariance prevents efficient minimization. In 4D-VAR algorithms, these problems are addressed together, by representing 𝗣₀^f in terms of a square root that is implied by a sequence of simple transformations. These simple transformations give the necessary cost savings and, just as importantly, can impart covariances based on a physical understanding of the dynamics governing forecast error. The transformations define the covariance, rather than being chosen to fit a known or hypothesized covariance. Traditional variational assimilation methods use a parameter transform to balanced and unbalanced variables that are expected to be uncorrelated, together with vertical and horizontal transforms that model the three-dimensional correlation structure in each of the transformed variables (e.g., Parrish and Derber 1992).

In traditional variational methods, the designer of the background error covariance matrix selects transform operators that implicitly define the background error covariance matrix. In contrast, designers of ensemble covariance localization methods such as Bishop and Hodyss (2007, 2009a,b) explicitly design localization functions with which to multiply explicit ensemble covariance functions. Apart from Kepert’s (2009) sophisticated formulation of balance constraints for ensemble Kalman filters, expressions for localized ensemble covariance matrices are, in general, trivial to write down in matrix form. In contrast, the background error covariance matrix implied by the sequence of transformations and balance operators that define the background error covariance matrices of traditional variational methods are nontrivial to write down in matrix form. This note shows how the easily written down matrix vector product associated with a nonadaptively localized ensemble covariance matrix can be reduced to a sequence of cost-efficient transform operators similar to those used in variational DA schemes.

Section 2 shows how a straightforward coding of the matrix-vector multiplications would be extremely expensive for forecast error covariance matrices based on localized ensemble covariance matrices. Section 3 derives a fast algorithm for performing this multiplication. Conclusions follow in section 4.

2. The problem

Consider the localized 4D ensemble covariance matrix given by 𝗣 ^f = 𝗣_K^f ȯ , where ȯ indicates the element-wise matrix product, and = ^1/21/2T is the correlation matrix used to localize the ensemble covariance matrix , where x_i is the ith ensemble perturbation (ensemble member minus the ensemble mean) outputted at a series of n_q times across the data assimilation time window, and ^1/2 is the left square root of .

As shown in the appendix and as discussed in Buehner (2005) and Bishop and Hodyss (2009b), the square root of 𝗣_K^f ȯ is given by [diag(x₁)^1/2, diag(x₂)^1/2, … , diag(x_K)^1/2], where diag(x_i) is the diagonal matrix whose diagonal is composed of the elements of x_i.

Variational algorithms that minimize a measure of the distance of the estimated state from observations and the first guess require the repeated evaluation of terms like

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and

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Here f_i has as many elements as there are columns in ^1/2 while h has as many elements as the 4D state vector. The meaning of the vectors f_i and h and the order in which the operations (1) and (2) are performed depends on whether the primal or dual forms of variational data assimilation are being used—see El Akkraoui et al. (2008) for details. For the purposes of this paper, f_i and h are simply symbols that denote the vector part of the matrix vector products defined by (1) and (2). The symbol p_i will be used as shorthand for [diag(x_i)^1/2]^Th.

If the sole component of the forecast error covariance matrix is localized ensemble covariances, the analysis correction x^a − x^f takes the following form: . In other words, the analysis correction is a weighted sum of the ensemble perturbations x_i. The vector of weights for the ith ensemble perturbation is given by the vector (^1/2f_i). The less the values in the vector (^1/2f_i) vary with space, time, and variable type, the more the global analysis approximates a simple linear combination of global ensemble perturbations. The more the values in the vector (^1/2f_i) vary with space, time, and variable type, the freer the global analysis is to fit observations. Here, however, we consider the nonadaptive, purely spatial localization used with Buehner et al’s (2010a,b) ensemble-4D-VAR scheme. In this scheme, the values of (^1/2f_i) depend only on the spatial location of the grid point to which they pertain and are independent of time and variable type. The scale of the correlation functions defining determine the penalty associated with allowing the ensemble weights to vary. Broadscale correlation functions in only permit broadscale fluctuations of the weights while short-scale correlation functions allow shorter-scale fluctuations of the weights. In this way, the localization correlation matrix controls the spatial variation of the ensemble weights defining the analysis correction. In contrast, in the local ensemble transform Kalman filter the variation of ensemble weights is controlled by the assigned sizes of observation volumes and distance-dependent observation error variance inflation (Hunt et al. 2007; Miyoshi and Yamane 2007; Szunyogh et al. 2008; Yang et al. 2009).

We assume that the columns of ^1/2 are separable functions; that is, each column is the product of a member of a set of basis functions of the horizontal coordinate and a member of a set of basis functions of the vertical coordinate. Under this assumption, the number of columns n_c in ^1/2 is given by the product of the number of horizontal basis functions L_h and the number of vertical basis functions M; in other words, n_c = ML_h. We will also assume that the model has ñ_h horizontal grid points and ñ_z vertical levels. For the sake of simplicity, we shall assume that the number of variable types at each model grid point is equal to the constant n_ϖ. The total number of variables is then ñ_T = ñ_hñ_zn_qn_ϖ (recall that n_q is the number of time levels). The cost of computing (nonsparse) matrix-vector products like those in (1) and (2) is order ML_hñ_T operations. Assuming [ñ_h, ñ_z, n_ϖ, n_q] ∼ O[10⁶, 10², 10, 10] so that ñ_T ∼ O(10¹⁰), and assuming that [L_h, M] ∼ O[10⁵, 10] so that ML_h ∼ 10⁶, it follows that ML_hñ_T ∼ O(10¹⁶) operations would be required for the matrix vector products in (1) and (2). This study shows how this cost can be reduced by many orders of magnitude.

3. Derivation of a rapid algorithm for ensemble covariance localization

A first step to a rapid algorithm is to recognize that parts of it can be performed on a reduced resolution grid. Consider a coarse-resolution grid that only has n_h × n_z grid points giving a total of just n_T = n_qn_hn_zn_ϖ variables instead of the ñ_h × ñ_z grid points and ñ_T variables of the high-resolution grid. Assume that the coarse-resolution grid has n_h = ñ_h/u grid points in the horizontal and n_z = ñ_z/s grid points in the vertical, where u and s are scalars.¹ Let 𝗟 be a ñ_T × n_T sparse rectangular matrix operator that interpolates coarse-resolution n_T vectors to high-resolution ñ_T vectors. When the horizontal and vertical length scales of the functions that describe the columns of ^1/2 are broad, ^1/2 ≈ 𝗟𝗖^1/2, and (1) and (2) can be approximated by

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and

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Notice that for the sake of conciseness, in (3) and (4) and throughout the rest of the paper, we drop the ensemble index subscript i from f, x, and p (except in section a of the appendix where it is needed again). The cost of mapping from high- to low-resolution grids and vice versa is order ñ_T operations. Without any further simplification, the use of the coarse grid reduces the cost of (1) and (2) to order ML_hñ_T/(su) operations. However, as we will see below, there are even more profound gains in computational cost that can be obtained by judicious use of the separable nature of the functions that define the columns of 𝗖^1/2.

When each column of 𝗖^1/2 is proportional to a product of vertical and horizontal structure functions, we can use the modulation product with symbol “”—defined in the appendix and in Bishop and Hodyss (2009b)—to write

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where each column of the n_T × L_h matrix 𝗪^h consists of a single horizontal structure (such as a scaled spherical harmonic in a global model) replicated at all vertical levels, all time levels, and all of the n_ϖ variable types. Each column of the n_T × M matrix 𝗪^υ consists of a single vertical structure (such as a scaled eigenvector of a vertical correlation matrix) that is the same for all variable types and has no horizontal or temporal variation. In other words, the single vertical structure is replicated in the horizontal and through time for all variables. According to the definition of the modulation product given in the appendix, (5) means that 𝗖^1/2 has n_c = L_hM columns.

To define 𝗪^h algebraically, let 𝗥^h be a n_T × n_h replication matrix and let 𝗚 be a n_h × L_h matrix listing the L_h horizontal basis functions defining a square root of the horizontal correlation matrix associated with 𝗖^1/2. We then have

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where the replication matrix 𝗥^h is defined by the submatrices for all vertical levels m = 1, 2, … , n_z, all times q = 1, 2, … , n_q, and all variable types r = 1, 2, … , n_ϖ, where 𝗜 is the n_h × n_h identity matrix. Note that in (6), we have assumed that the n_hn_zn_qn_ϖ elements of the state vector have been listed such that the first n_h elements of the state vector correspond to the horizontal field defining the variation of the first variable at the first time at model level 1. The second n_h elements of the state vector correspond to the horizontal field defining the variation of the first variable at model level 2 and so on, until all n_hn_z variables defining the 3D state of the first variable at the first time level have been defined. The next n_hn_z variables define the state at the second time level until all n_hn_zn_q variables have been listed that define the 4D state of the first variable. The next n_hn_zn_q define the 4D state of the second model variable, and so on until all n_hn_zn_qn_ϖ elements defining the 4D state of all model variables have been listed.

Here 𝗪^υ is completely analogous to 𝗪^h, but its algebraic form changes to accommodate the noncontiguous location of the vertical indices. To define 𝗪^υ algebraically, let the n_z × M matrix 𝗩 = [v₁, v₂, … , v_j, … , v_M] be the square root of the vertical correlation matrix associated with 𝗖^1/2. The jth column of the n_zn_h × M matrix

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then gives a 3D function with vertical columns equal to v_j and no horizontal variation. It then remains to replicate this 3D structure function for all variable types and all output times using the replication matrix 𝗥^υ defined by

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Using (5) in the RHS of (3) gives

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The elements of f are coefficients of the columns of 𝗪^υ 𝗪^h. Hence, [𝗪^υ 𝗪^h]f is a mapping from coefficient space to physical space. In section b of the appendix, we prove that , where f_j^T is a L_h vector and f^T = [f₁^T, f₂^T, … , f_j^T, … , f_M^T]. Using this, (6) and (8) in (9) gives

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To rephrase (10) in a form that avoids the explicit use of the replication matrices 𝗥^υ and 𝗥^h, let denote a vertical column of the vector corresponding to the rth variable type, qth time level, and lth horizontal location. One may then deduce from (10) and (6)–(9) that

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where g_l^T is the lth row of 𝗚.

The following list of tasks enables x ȯ [𝗟(𝗖^1/2f)] to be computed in a computationally efficient way. Each task is associated with an operator or subroutine identified with capitalized script letter.

1. Use the operator _h to compute 𝗚f_j for j = 1, 2, … , M at a cost of order n_hL_hM operations. Note that the operation 𝗚f_j returns the scalar value g_l^Tf_j at the lth horizontal grid column for l = 1, 2, … , n_h, ready for use in (11).

2. Use the operator _z to compute the right-hand side of (11) for all horizontal grid points. This operation provides the ensemble weight for the ith ensemble perturbation at each grid point and costs order n_hn_zM operations.

3. Use the operator _tr to replicate the weights at each grid point for all variable types and all output times to obtain the full coarse resolution multivariate weight n_T vector (𝗖^1/2f) at a cost of order n_qn_hn_ωn_z = n_T operations.

4. Use the operator to interpolate the weight vector from the coarse resolution grid to the high-resolution grid (using 𝗟) and perform the element-wise product x ȯ [𝗟(𝗖^1/2f)] at a cost of order ñ_T operations.

In operator form, the algorithm is x ȯ [𝗟(𝗖^1/2f)] ≡ _rtz𝗪_h(f). Using parameter values from section 2, and assuming that the coarse resolution domain has about an order of magnitude fewer horizontal grid points and about the same order of magnitude of vertical grid points gives u ∼ O(10) and s ∼ O(1), its cost is order n_hL_hM + n_hn_zM + n_T + ñ_T ∼ O(10¹¹ + 10⁸ + 10⁹ + 10¹⁰) ∼ O(10¹¹) operations. This is five orders of magnitude fewer operations than the approach described in section 2.

It now remains to describe how similar approaches can be used to speed the adjoint operation defined by (4). Defining y = 𝗟^T(x ȯ h) and using (5) in (4) gives

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Section c of the appendix shows that

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Now from the definition of 𝗪^h given in (6):

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where y_mqr and are each n_h vectors describing horizontal fields of the qth model variable type on the mth vertical level at the qth time level. To see the connection between (14) and vertical functions 𝗩, let denote the lth horizontal element of the n_h vector . From (8), . Consequently,

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where the n_z vector d_l^T is given by

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Using (16) to define the n_z × n_h matrix 𝗗 gives

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it follows that

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and using (18) in (14) gives

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We can then use (19) and (14) in (13) to obtain the following:

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The above analysis allows us to see that the operation in (4) may be computed by applying the following list of operators.

1. Use the operator 𝗟^T to compute y = 𝗟^T[x ȯ h] at a cost of order ñ_T operations.

2. Use the operator to compute the arithmetic sum over variables and time [see (16)] at each model grid point defining the 3D grid and hence derive the n_z × n_h matrix 𝗗 [see (17)] at a cost of order n_hn_zn_qn_ϖ = n_T operations.

3. Use _z^T to compute 𝗗^Tv_j for j = 1, 2, … , M with order n_hn_zM operations.

4. Use _h^T to compute 𝗚^T(𝗗^Tv_j) for j = 1, … , M and build the vector p in accordance with (20) at a cost of order ML_hn_h operations.

In operator form, the above algorithm may be summarized as (𝗖^1/2)^T𝗟^T(x ȯ h) ≡ _h^T_z^T_rt^TT(x ȯ h). Unsurprisingly, the cost for these adjoint operations is the same order of magnitude as it was for the forward algorithm at O(10¹¹) operations and is again five orders of magnitude fewer operations than the approach described in section 2.

4. Conclusions

A fast algorithm for including nonadaptively localized ensemble covariances in variational algorithms has been derived and explained. This algorithm requires several orders of magnitude fewer computations than a simple-minded evaluation of the equations that were originally given in Lorenc (2003), Buehner (2005), and Buehner et al. (2010a,b). Elements of the fast algorithm were actually implemented by both Lorenc and Buehner in their work on localized ensemble covariances. However, their papers did not describe or explain the numerous nontrivial aspects of creating a fast algorithm. This paper serves to provide this information.

Some but not all of the techniques introduced here to speed nonadaptive ensemble covariance localization can also be applied to adaptive localization methods such as those discussed in Bishop and Hodyss (2009b). These include (i) defining the adaptive localization correlation matrix on a coarser resolution grid, and (ii) summing over variable type if the same adaptive localization functions are being applied to all variable types.