1. Introduction

The fixed-lag Kalman smoother (FLKS) has been proposed by Cohn et al. (1994; CST94 hereafter) as an approach to perform retrospective data assimilation. The term retrospective data assimilation denotes a procedure to incorporate data observed well past each analysis time into each analysis, taking into account error propagation through dynamical effects. Since a goal of reanalysis efforts is to produce a long archive of best-possible analyses based on all available data, whereas current reanalysis projects (e.g., Gibson et al. 1997; Kalnay et al. 1996; Schubert and Rood 1995) incorporate only data observed up to and including each analysis time, retrospective data assimilation should be an ultimate goal of reanalysis efforts, as pointed out in CST94. Moreover, although retrospective data assimilation is studied in this article primarily as a means of improving analysis quality, it is foreseeable that such a procedure could also be adopted in numerical weather prediction to improve mid- to long-range forecasts, starting from a given retrospective analysis. The preliminary efforts of Gelaro et al. (1996) can be viewed as an approach to retrospective data assimilation with this purpose.

Cohn et al. (1994) gave a particular derivation of the optimal linear FLKS. In that work, it was pointed out that the same algorithm can be derived from the approach of “state enlargement,” or “state augmentation” as it is more commonly known, first suggested in the engineering literature by Willman (1969), to reduce the smoothing problem to a filtering problem. In the state augmentation approach, the state vector at each time is appended with the state vector at previous times when the desired smoother estimates are to be calculated. A Kalman filter (KF) problem can then be solved for the augmented system. The first derivation of a smoother algorithm via state augmentation was that of Biswas and Mahalanabis (1972) for the linear fixed-point smoothing problem. Subsequently, Moore (1973) derived a linear fixed-lag smoother via the same approach, which results in the same algorithm as that derived in CST94. Extension of the FLKS formulation to nonlinear systems can be achieved using the same technique of state augmentation, as indicated by Biswas and Mahalanabis (1973), for both the fixed-point and fixed-lag smoothing problems [see also Todling and Cohn (1996a) for an explicit derivation of the extended fixed-lag Kalman smoother]. The utility of state augmentation is that the resulting smoothers are often computationally less demanding than those arising from some other approaches (e.g., Sage and Melsa 1970, section 9.5). For instance, smoothers based on state augmentation avoid inversion of the filter error covariance matrices and of the tangent linear propagator (e.g., Ménard and Daley 1996; see also the appendix of the present article). These inversions are also avoided by an earlier smoother algorithm due to Bryson and Frazier (1963), which can be shown to reduce to the FLKS algorithm of CST94 for the case of linear systems. Algebraic equivalence between smoothers obtained by state augmentation and by methods such as maximum likelihood (Sage and Ewing 1970; Sage 1970) or conditional expectation (Leondes et al. 1970) exists in most cases. The interested reader is referred to Meditch (1973) and Kailath (1975) for detailed reviews of the literature on linear and nonlinear smoothing. The distinction among different types of smoothing problems, and the connection between fixed-interval smoothing and four-dimensional variational (4DVAR) analysis, is drawn in Cohn (1997).

Brute-force implementation of the (extended) FLKS to build an operational retrospective data assimilation system (RDAS) is not possible for the same reasons that a brute-force (extended) KF-based data assimilation system would be impractical: computational requirements are excessive, and knowledge of the requisite error statistics is lacking. Therefore, approximations not only must be employed but cannot be escaped from. Thus, in this article, we develop and evaluate the performance of potentially implementable approximate schemes. To provide an exact evaluation, we choose a barotropically unstable linear shallow water model as a test bed for this investigation. All of the approximate schemes evaluated here have relatively simple nonlinear equivalents.

In the sequel, we first briefly review, in section 2, the linear FLKS of CST94 and outline the performance evaluation technique employed to study the behavior of linear suboptimal filter and smoother algorithms. Section 3 gives a summary of the suboptimal filters and smoothers evaluated subsequently in section 4, in the context of the linear shallow water model. We draw conclusions in section 5.

2. Review and performance evaluation equations

Before we summarize the linear FLKS algorithm and performance evaluation equations, let us recall that the linear Kalman filter equations, in the notation of CST94, are

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Here (1a) is the expression for the forecast n vector w^f_k|k−1, obtained through evolution of the analysis n vector w^a_k−1|k−1 between two consecutive analysis times t_k−1 and t_k via the propagator A_k,k−1; (1b) is the corresponding expression for the n × n forecast error covariance matrix P^f_k|k−1, where Q_k is the n × n model error covariance matrix. The state estimate w^f_k|k−1 is updated using (1d), as p_k observations w^o_k become available at each time t_k:the difference between the observations and their predicted values H_kw^f_k|k−1, expressed via the p_k × n observation matrix H_k, is added to the forecast after being weighted by the n × p_k Kalman gain matrix K_k|k. At each observation time, the gain matrix is computed according to (1c), where

Γ_kH_kP^f_k|k−1H^T_kR_k(2)

is the p_k × p_k innovation covariance matrix and R_k is the p_k × p_k observation error covariance matrix. The resulting analysis error covariance matrix P^a_k|k is given by (1e), which completes the Kalman filter cycle.

The subscript notation utilized here is common in estimation theory, and is particularly important when considering smoothing problems. Specifically, the forecast vector w^f_k|k−1 is the conditional mean of the true state at time t_k, where the conditioning is on all observations up to and including those at time t_k−1, hence the double time subscript. Similarly, the analysis vector w^a_k|k is the conditional mean of the true state at time t_k conditioned on data up to and including time t_k. Analogously, the forecast and analysis error covariance matrices carry a second time subscript to indicate the set of observations upon which they are conditioned. A more comprehensive explanation of the Kalman filter equations, including the probabilistic assumptions from which they are derived, can be found elsewhere (e.g., Jazwinski 1970; Gelb 1974; Cohn 1997).

The FLKS update equation (3) is only applicable for l ⩽ k. If the (fixed) total number of lags is L, meaning that (3) is to be applied in general for l = 1, 2, . . . , L, then for k = 0, 1, . . . , L − 1, the condition l ⩽ k is not satisfied for all l. Therefore, the update (3) is applied only for l = 1, 2, . . . , min(k, L), which is a restricted range of l when k = 0, 1, . . . , L − 1. In the language of estimation theory, this restriction corresponds to computing fixed-point smoother results for all k up to k = L − 1, after which the fixed-lag smoother starts operating. This is an initialization procedure for the fixed-lag smoother (e.g., Gelb 1974, 173–176). In practice, because the FLKS algorithm employed by CST94 already has the structure of a fixed-point algorithm, this procedure simply amounts to controlling the ending points of certain loop statements in a computer code.

In the optimal FLKS algorithm of CST94, the n × p_k retrospective analysis gain matrix K_k−l|k is given by

K_k−l|kP^fa_{k,k−l|k−1}^TH^T_kΓ⁻¹_k(4)

where the innovation covariance matrix Γ_k is thesame as that used to calculate the filter gain matrix K_k|k in (1c), since the retrospective analysis update (3) of the FLKS is based on the same innovation vector as KF. The n × n matrix P^fa_{k,k−l|k−1} is the forecast–retrospective analysis error cross-covariance matrix, and evolves according to the following set of equations:

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Here the n × n matrix P^aa_{k−1,k−l|k−1} is the filter analysis–retrospective analysis error cross-covariance matrix, and the n × n matrix P^a_k−l|k is the retrospective analysis error covariance matrix. Equations (1)–(5) complete the description of the FLKS algorithm of CST94, with (1)–(2) giving the filter portion and (3)–(5) giving the retrospective analysis portion. There are a number of advantages to this FLKS algorithm. In particular, model error is incorporated implicitly in the retrospective analysis portion: no model error terms appear in (3)–(5). This point is clarified in the appendix, where the algebraic equivalence of this algorithm with a more well-known alternative formulation is exploited.

For linear systems, filter performance evaluation can be accomplished following the procedure of Todling and Cohn (1994), and here is extended to incorporate the retrospective analysis performance evaluation equations as well. These equations can be obtained from the derivation of the FLKS of CST94 [cf. Eqs. (2.33), (2.39), and (2.45) in CST94], and are valid for general (filter and retrospective analysis) gain matrices K̃_k|k and K̃_k−l|k:

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and

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Together with (6) and (7) calculated exactly, these equations give the update and evolution of all of the actual filter and retrospective analysis error covariances. Expression (8a) is the well-known Joseph formula, and gives the performance of the filter analysis for a general gain matrix K̃_k|k, and (8b) gives the performance of the retrospective analyses for general gains K̃_k−l|k. Notice that the performance evaluation equations [(6), (8a)] for the filter are independent of those [(7), (8b), (8c)] for the retrospective analysis, whereas the converse is not true. This is simply a consequence of the fact that the optimal linear filter is independent of the optimal linear retrospective analysis. This independence does not carry over to some nonlinear extensions, for example, to the globally iterated smoother (Jazwinski 1970, 280–281).

3. Summary of suboptimal filters and smoothers

We now summarize the suboptimal schemes to be evaluated here in the context of the linear shallow water model of the next section. The following are the suboptimal schemes considered in this article for the filter portion of the fixed-lag smoother algorithm (see Cohn and Todling 1996, CT96 hereafter; Todling et al. 1996;Todling and Cohn 1996a,b).

4. Results for a shallow water model

To evaluate the performance of the suboptimal schemes described above, we use the barotropically unstable model of CT96, a shallow water model linearized about a meridionally dependent squared-hyperbolic secant jet (Bickley jet; Haltiner and Williams 1980, 175). We refer the reader to Fig. 1 of CT96 for the shape, extent, and strength of the jet. The model domain is shown in Fig. 1 here. The assimilation experiments employ the observing network of CT96: 33 radiosonde stations observing winds and heights every 12 h and distributed outside the strongest part of the jet. The tick marks in the figure indicate the 25 × 16 model grid. In the experiments referring to a trailing error covariance matrix we construct it, exactly as in CT96, using the slow eigenmodes of the autonomous unstable dynamics of our shallow water model.

Before evaluating the performance of a few suboptimal RDASs, we discuss results obtained for the optimal FLKS. The performance of the optimal filter and fixed-lag smoother can be seen in Fig. 2, which shows the domain-averaged expected root-mean-square (erms) analysis error in the total energy as a function of time. This quantity is calculated as a weighted average of the analysis error variances of the three variables of the model, which are extracted from the main diagonal of the actual analysis error covariance matrix. The top curve in the figure corresponds to the filter analysis every 12 h, whereas successive retrospective analysis results are given by successively lower curves, which refer to analyses including data 12, 24, 36, and 48 h ahead in time—that is, lags l = 1, 2, 3, and 4. The filter curve is the same as that in Fig. 2 of CT96 (shown, here, only up to 5 days). The most relevant results are those for the transient part of the assimilation period, before the filter and smoother begin to approach steady state. Incorporating new data into past analyses reduces the corresponding past analysis errors considerably. The largest impact is on the initial analysis, which would not be the case if a significant amount of model error were present.

Further illustration of the behavior of the optimal FLKS is given in Fig. 3, where we display maps of the analysis error standard deviation in the height field at t = 0.5 days. The panels are for the filter analysis errors (Fig. 3a), and for the retrospective analysis errors for lags l = 1 (Figs. 3b) and l = 4 (Fig. 3c). Thus, in Figs. 3b,c and the analysis errors are reduced by incorporating data 12 and 48 h ahead of the current analysis time (t = 0.5 days), respectively. We see not only the overall decrease in error levels from Fig. 3a to Fig. 3c, as expected from Fig. 2, but also that within each panel errors are largest in the central band of the domain, where there are no observations and where the jet is strongest. Furthermore, the error maximum in the center of the domain moves westward and diminishes as more data are incorporated into the analysis through the smoothing procedure (from Figs. 3a to 3c). This property of the FLKS of propagating and reducing errors in the direction opposite of the flow has already been observed in the experiments of CST94 and Ménard and Daley (1996).

We now study the behavior of suboptimal RDASs. We start with schemes that approximate the filter and retrospective analysis portions similarly. In this category, we investigate the behavior of the RRF and RRRA corresponding to expressions (12) and (16), respectively, as well as the behavior of the PSF and PSRA corresponding to expressions (10) and (13), respectively.

The results of Todling and Cohn (1996b) showed that the RRF described above, with resolutions 13 × 16 and 13 × 12, provides good filter performance in our shallow water model context. This was attributed mainly to the fact that at these resolutions the barotropically unstable jet is fairly well resolved. As a matter of fact, the meridional jet is fully resolved at resolution 13 × 16. In Fig. 4 we show results of the performance evaluation for the RRF and RRRA algorithms at these resolutions (Fig. 4a for 13 × 16; Fig. 4b for 13 × 12). No trailing error covariance models are taken into account here, that is, T^"_k|k−1 = 0 and X^"_{k,k−l|k−1} = 0 for the RRF and RRRA, respectively. As in Fig. 2, the upper curve in each panel is for the performance of the filter analysis, and the lower curves in each panel are for the performance of the successive retrospective analyses. Comparison of Fig. 4a with the optimal FLKS results of Fig. 2 shows remarkable agreement between the filter and retrospective analysis results when the jet is fully resolved. The agreement with the coarse meridional resolution result in Fig. 4b is still quite good, especially during the transient part of the assimilation. Asymptotically, the analysis error levels for the case of 13 × 12 resolution are somewhat higher than those at 13 × 16 resolution.

The results of the RRF can be improved by modeling the trailing error covariance matrix T^"_k|k−1 to account for scales that are neglected in the RRF approximation. The impact of modeling this covariance matrix is greatest in experiments where the jet is least well resolved by the RRF scheme (results not shown). In these cases, the retrospective analysis results also improve due to improvement of the filter results. We have not investigated the impact of modeling the RRRA trailing error cross-covariance.

Along similar lines, we investigate the performance of an RDAS using the PSF algorithm for the filter portion, and the PSRA algorithm for the retrospective analysis portion. From the experiments of CT96, we know that using the first 54 singular modes of the 12-h propagator of the linear shallow water model—those with singular values greater than or equal to one—is enough to produce a stable suboptimal filter. Moreover, we learned in CT96 that adaptively tuning a modeled trailing error covariance matrix T_k|k−1 improves the filter results; we use the same procedure here. However, we do not model the trailing error cross-covariance matrix for the retrospective analysis portion, that is, we take X_{k,k−l|k−1} = 0 at all times.

Figure 5 shows performance results for the PSF–PSRA suboptimal RDAS when the first 54 modes are used for both approximations (out of a total of 325 slow modes). The filter results, when compared to the optimal results of Fig. 2, are once again quite good—the reader is encouraged to compare the top curve of Fig. 5 with the curve labeled S54 in Fig. 11 of CT96; results now are better due to the adaptively tuned trailing error covariance matrix. The PSRA results, on the other hand, are not nearly as good as those for the optimal smoother (Fig. 2), with little difference among results for lag l = 1 and those for higher lags l = 2, 3, and 4 in Fig. 5. The next two experiments demonstrate that this poor smoother performance can be attributed mostly to neglecting the trailing forecast–retrospective analysis error cross-covariance matrix X_{k,k−l|k−1} in the PSRA algorithm. A further experiment later in this section, where the PSF scheme is combined with the exact retrospective analysis algorithm, also shows much better smoother performance than that seen in Fig. 5.

To investigate the PSRA scheme further, we compare performance results between two RDASs using the KF for the filter portion, with the retrospective analysis portion given by either the PSRA scheme or the PSRA2 scheme, both with 54 singular modes retained. Thus, the suboptimality in these two RDASs is solely in the retrospective analysis portion. Figure 6 shows the erms errors in the total energy for these two cases: Fig. 6a corresponds to the RDAS using the PSRA scheme and Fig. 6b corresponds to the RDAS using the PSRA2 algorithm. The filter curves in both panels (topmost curves) are identical to one another, as well as to the filter curve in Fig. 2 for the optimal FLKS case. Comparison between the lower curves in the two panels of Fig. 6 shows the superiority of the PSRA2 scheme: beyond lag l = 1 little is gained in the PSRA scheme, even when based on the KF (cf. Fig. 6a with Fig. 5), whereas successively higher lags do have a significant impact in the PSRA2 scheme (Fig. 6b). The poor performance of the PSRA scheme indicates that its neglected trailing part contains a large amount of cross-(co)variance when retaining just the 54 singular modes of the propagator with singular values larger than or equal to one. The PSRA2 scheme with 54 modes, on the other hand, captures most of the cross-(co)variance, as comparison of Fig. 6b with the optimal result in Fig. 2 indicates. We conclude from these experiments that the trailing cross-covariance matrix is more significant in some approximate retrospective analysis schemes than in others. Moreover, the singular modes of the propagated filter analysis–retrospective analysis error cross-covariance matrix (employed in the PSRA2 scheme), rather than the singular modes of the propagator itself (employed in the PSRA scheme), contain most of the information relevant for retrospective analysis. This distinction between the PSRA and PSRA2 schemes is completely analogous to the distinction between the PSF and PEF schemes drawn in CT96, where somewhat better performance of the PEF scheme over the PSF scheme was demonstrated. This distinction is even more pronounced in the retrospective analysis context, as seen in Fig. 6.

The good performance of the PSRA2 scheme when combined with the KF suggests evaluating the performance of two hybrid RDASs. Figure 7a shows results for the combined PSF–PSRA2 algorithm, with 54 modes, and Fig. 7b shows results for the combined PEF–PSRA2 algorithm, with 54 modes as well. Both the PSF and PEF filtering strategies include an adaptive tuning procedure for the modeled trailing error covariance matrices T_k|k−1 and T^′_k|k−1, respectively, following CT96. Because the performance of the PEF is only slightly better than that of the PSF (top curve in each panel), both with 54 modes, the RDASs using these suboptimal filters yield retrospective analyses differing only slightly from each other, as a comparison of the two panels in Fig. 7 indicates. Comparing Fig. 7a (PSF–PSRA2) and Fig. 7b (PEF–PSRA2) with Fig. 6b (KF–PSRA2) shows that employing a suboptimal filter degrades the performance of the retrospective analyses, especially beyond lag two. Comparing Fig. 7a (PSF–PSRA2) to Fig. 5 (PSF–PSRA), however, demonstrates again the superior performance of the PSRA2 scheme, especially for the first two lags.

We evaluate next the performance of schemes that approximate only the filter portion and carry out the retrospective analysis calculations exactly. We start with an RDAS in which the adaptive CCF scheme is used for the filter part. Figure 8 shows the evolution of the actual erms errors up to day 5 (same as Fig. 3 of Todling et al. 1996). While the performance of the CCF scheme (top curve) is worse than that seen in Fig. 2 for the optimal KF, it is significantly worse only beyond day one; adaptive tuning of more than a single parameter would likely improve this filter result. As a consequence of suboptimality of the CCF scheme, the performance of the CCF-based retrospective analyses shown in Fig. 8 is also suboptimal. However, a comparison between Figs. 2 and 8 indicates that retrospective analysis based on a suboptimal filter can be viewed as a way of improving suboptimal filter performance toward optimal filter performance. For instance, notice that by day 2.5, the lag-2 suboptimal retrospective analysis of Fig. 8 has about the same error level as that of the optimal filter analysis of Fig. 2.

We also see in Fig. 8 that the retrospective analysis at 12 h for lag-4 is worse than the retrospective analyses for smaller lags. This is an indication that we can expect to extract only so much information from the data when using the rather crude approximate forecast error covariance matrix of the CCF. Even though the correct dynamics is still used to propagate filter information from 60 h back to 12 h, the approximate forecast error covariance used by the CCF is largely different from the true forecast error covariance at 60 h, and we expect that the retrospective analysis results can actually degrade during the transient period, the more lags we include in the retrospective scheme.

When comparing the RDAS using the CCF scheme (Fig. 8) with the RDASs using the RRF–RRRA of Fig. 4 and the PSF–PSRA of Fig. 5, we see that the performance of the CCF scheme itself is not much different than that of the RRF with 13 × 12 resolution and that of the PSF with 54 modes (top curve in each figure). The performance of the CCF-based retrospective analysis, however, exceeds that of the 13 × 12 RRF-based RRRA scheme and the 54-mode PSF-based PSRA scheme, for every lag, beyond the initial transient assimilation period. During the transient period, the RRF–RRRA algorithm shows better performance, for high lags, than either the CCF-based retrospective analysis algorithm or the PSF-based PSRA algorithm.

Analogously to Fig. 3, we show in Fig. 9 maps of the actual height analysis error standard deviation at day 0.5, for the experiment of Fig. 8. The panels are arranged as before: (a) filter analysis; (b) lag l = 1 retrospective analysis; and (c) lag l = 4 retrospective analysis. Comparing panels (a) and (b) with the corresponding panels in Fig. 3, we see that the CCF scheme and the resulting lag l = 1 retrospective analysis perform remarkably well. However, the retrospective analysis for lag l = 4 (Fig. 9c) is not significantly better than for lag l = 1 (Fig. 9b), as one might expect from Fig. 8 at day 0.5, and in fact compares poorly with the optimal case (Fig. 3c), particularly over the data-void central band.

Finally, we examine the performance of the more sophisticated PSF and PEF suboptimal filters and the corresponding suboptimal RDASs, using the exact retrospective analysis formulas. In both cases we retain only 54 leading modes and we adaptively tune the trailing error covariance matrices as before. In Fig. 10a the top curve refers to the performance of the PSF, whereas that in Fig. 10b refers to the performance of the PEF. The PSF result is identical to that displayed in Fig. 5 since the filter here retains the same number of modes as before. A comparison of the PSF-based retrospective analyses of Fig. 10a, which use the exact retrospective analysis formulation, and the PSF-based retrospective analyses of Fig. 5, where this formulation was approximated by the PSRA algorithm, shows clearly the superior performance of the exact formulation. The PSF-based RDAS (Fig. 10a) performance is similar to, and the PEF-based RDAS (Fig. 10b) performance is superior to, that of the CCF-based RDAS of Fig. 8. The RDAS using the PEF (Fig. 10b) presents very good long-term performance, with its results being fairly close to those of the optimal FLKS in Fig. 2, and only slightly inferior to those of the 13 × 16 RRF–RRRA scheme of Fig. 4a.

In Fig. 11 we show maps of the actual height analysis error standard deviations for the experiment using the PSF of Fig. 10a. Performance relative to the optimal case (Fig. 3) tends to worsen with increasing lag number, particularly over the data-void central band. Compared to the maps of Fig. 9, however, there is improvement in the analyses over specific regions of the domain. In particular, the lag l = 4 retrospective analysis in Fig. 11c shows a considerable error reduction beyond that of Fig. 9c over the central part of the domain and the Atlantic Ocean.

5. Conclusions

In this article we evaluated the performance of approximate (suboptimal) RDASs based on the FLKS formulation of Cohn et al. (1994). This formulation has several practical advantages over more commonly known smoother formulations. In particular, it avoids a number of large matrix inversions. This formulation also separates naturally into a filter portion and a retrospective analysis portion, enabling a variety of suboptimal implementations. Model error is incorporated implicitly in the retrospective analysis portion, because the filter portion is based directly on the Kalman filter, which already takes model error explicitly into account. Thus, a version of the retrospective analysis portion could be implemented operationally and remain unchanged while improvements in the filter portion, such as accounting for model error, take place.

For linear dynamics and observing systems, performance evaluation equations for approximate RDASs based on the FLKS formulation follow directly from the approach of state augmentation and the usual performance evaluation equations for linear filters utilizing general gain matrices. In this way, we examined the performance of a variety of suboptimal RDASs for a barotropically unstable shallow water model. We concentrated on evaluating the performance obtained when using approximate expressions for the error covariance propagation in the filtering portion of the RDAS, as well as for the error cross-covariance propagation in the retrospective analysis portion of the RDAS. Our experiments indicate that successful retrospective data assimilation schemes can be designed by approximating either the filter portion alone or by approximating both the filter and retrospective analysis portions simultaneously. An important conclusion from these experiments is that a few lags of suboptimal retrospective analysis may accomplish the performance of an optimal filter analysis. Sophisticated approximate filters that take dynamics of error covariances into account present the best suboptimal retrospective analysis performance. It would be interesting to verify how well these conclusions hold in more realistic data assimilation environments.