1. Introduction
Data assimilation (DA) is the process by which available observations are combined with dynamical models to constrain model predictions and determine the best possible estimate of a system’s state (Ghil and Malanotte-Rizzoli 1991). DA methods are generally grouped into two categories: 1) variational methods that aspire to fit the model’s trajectory to available observations through adjustments of a well-chosen set of control parameters (Le Dimet and Talagrand 1986) and 2) sequential methods that follow a probabilistic framework to split the Bayesian estimation problem into cycles of alternating forecast steps, to advance the distribution of the system state with the model, and analysis steps, to compute the posterior distribution based on incoming observations (Todling 1999).
The ensemble Kalman filter (EnKF) was introduced by Evensen (1994) as a Monte Carlo implementation of the Kalman filter (KF) (Kalman 1960) to tackle large-scale, sequential, nonlinear DA problems. It approximates the distribution of the system state by a collection of random realizations, or ensemble members, whose mean is the state estimate and sample covariance is the associated error covariance (Hoteit et al. 2015). The EnKF integrates the (analysis) ensemble members forward with the model in the forecast step and then updates them in the analysis step with incoming observations using a Kalman-like (best linear unbiased estimation) correction. Because of its reasonable computational cost, robust performance, and nonintrusive formulation, the EnKF has seen widespread use in many geophysical applications (Reichle et al. 2002; Carton and Giese 2008; Anderson et al. 2009; Hoteit et al. 2013; Nævdal et al. 2005; Triantafyllou et al. 2013).
The EnKF perturbs observations with random noise sampled from the observational error distribution before assimilation, such that its error covariance asymptotically matches that of the KF (Burgers et al. 1998). Later, this scheme became known as the stochastic EnKF (Tippett et al. 2003). This perturbation, however, introduces an additional source of sampling error and often underestimates the analysis error covariance, especially when the filter is implemented with small ensembles (Whitaker and Hamill 2002; Hoteit et al. 2015). Several deterministic EnKFs were proposed to update the ensemble’s sample mean and covariance without perturbing the observations, exactly as in the KF (Tippett et al. 2003), and then to resample a new ensemble to proceed with the next forecast step. Among these, we cite the ensemble transform Kalman filter (ETKF) (Bishop et al. 2001), the ensemble adjustment Kalman filter (EAKF) (Anderson 2001), and the ensemble square root Kalman filter (EnSRF) (Whitaker and Hamill 2002), which performs the ensemble resampling implicitly using different deterministic square root transformations. This method of resampling contrasts with the explicit stochastic resampling of the singular evolutive interpolated Kalman (SEIK) filter (Pham et al. 1998; Hoteit et al. 2002), which is based on orthogonal random transformations.
Accurate estimates of the forecast error covariances, also known as background covariances, are crucial to EnKF-like filters. In real-world, large-scale applications, however, EnKFs are implemented with relatively small ensembles to limit computational cost and with poor model error statistics (Song et al. 2010). This limits the representativeness of EnKFs’ backgrounds and, thus, the filters’ performance. Various auxiliary techniques have been proposed to mitigate this background limitation, such as inflation (Anderson 2001), localization (Houtekamer and Mitchell 1998), hybrid (Hamill and Snyder 2000) and adaptive formulations (Song et al. 2010), and robust filtering (Luo and Hoteit 2011). Here, we follow a different approach that exploits information in (future) observations to compute improved backgrounds in EnKFs. The idea is to follow the one-step-ahead (OSA) smoothing formulation of the Bayesian filtering problem (Desbouvries et al. 2011), which first applies a smoothing step with the future observation before integrating the resulting “smoothed” state with the model to compute the background, all within a fully Bayesian-consistent framework. This approach has been recently used by Ait-El-Fquih et al. (2016) to derive a Bayesian framework for state-parameter estimation for the well-known dual EnKF (Moradkhani et al. 2005). In the resulting EnKF-OSA scheme, the observation is exploited twice: first to constrain the sampling of the forecast ensemble with the future observation, and next to update the resulting ensemble with the data. This additional OSA-smoothing step provides an improved background for the next analysis cycle and is, thus, expected to enhance EnKFs’ results. Indeed, when implemented under challenging situations, such as strong nonlinearities, sparse observation networks, or large assimilation windows, the ensembles’ perturbations estimated by standard EnKFs may not well represent the uncertainties of the prior state, thereby limiting the impact of the observation. Conditioning the forecast ensemble by the observations through a smoothing step should provide more accurate estimates of the background uncertainties for better exploitation of the observation in the analysis step.
A very similar idea, known as “running in place” (RIP), was proposed in Kalnay and Yang (2010) as a heuristic way to accelerate the spinup of the ETKF. It iteratively applies the two-stage update step based on criteria that ensure a reasonable fit to the data. Using the observations more than once in each assimilation window during spinup is justified by the need to maximize the initial extraction of information (Kalnay and Yang 2010). The RIP iterations may be interpreted as some kind of iterations on the OSA-smoothing step to deal with the nonlinear character of this step—a very common approach in the subsurface community (Emerick and Reynolds 2013; Luo et al. 2015; Ma et al. 2017; Chen and Oliver 2013). This method was successfully tested with a series of realistic models (Penny et al. 2013; Yang et al. 2013). By ignoring the underlying criteria, we found that RIP with one iteration is very similar to the proposed OSA-smoothing-based filtering scheme. This means that RIP with one iteration is Bayesian consistent and, thus, not heuristic, as has been suggested.
The goal of our work is twofold. Realizing the benefit of an improved background covariance for state estimation with the EnKF, we first demonstrate the relevance of the OSA-smoothing-based approach for state estimation. The performance of EnKF-like methods is known to level off after enough members are used (about the rank of the covariance matrices) (e.g., Hoteit et al. 2002; Evensen 2003). The idea here is to better exploit the observations in order to enhance the (approximate) ensemble background using future information. We then derive a new deterministic OSA-smoothing-like ensemble filter, the SEIK-OSA, which is suitable for data assimilation into large-scale oceanic and atmospheric models. This should improve the filtering results, compared to the EnKF-OSA, when the ensemble size is small, compared to the number of assimilated observations, a common scenario in geophysical applications (Hoteit et al. 2015). We demonstrate the efficiency of the SEIK-OSA with extensive numerical experiments with the strongly nonlinear Lorenz-96 (L96) model under various assimilation settings and scenarios.
The remainder of this paper is organized as follows. Section 2 describes the concept of the filtering problem with OSA smoothing and presents a (stochastic) EnKF-based solution for general nonlinear systems. The SEIK-OSA algorithm is derived in section 3. Results of numerical experiments with the Lorenz-96 model are then analyzed in section 4. The main conclusions are summarized and discussed in section 5.
2. OSA-smoothing-based filtering
a. Problem formulation
The filtering problem consists of estimating, at each time 
Diagram depicting the KF and KF-OSA algorithms, where the subscript n stands for time, 
Citation: Monthly Weather Review 146, 2; 10.1175/MWR-D-17-0175.1
Desbouvries and Ait-El-Fquih (2008) and Desbouvries et al. (2011) derived KF and particle filter (PF)-like algorithms based on the OSA-smoothing formulation [see also Pitt and Shephard (1999) and Luo et al. (2014), who introduced an auxiliary PF involving OSA-smoothing, but in a different way]. Lee and Farmer (2014) also followed a similar strategy and presented a number of algorithms for estimating both the state and system noise. More recently, EnKF-OSA schemes were also proposed by Ait-El-Fquih et al. (2016) and Gharamti et al. (2015) to derive a dual-filter-like algorithm for state-parameter estimation of large-scale nonlinear systems. It is important to realize that the OSA-smoothing-based schemes are not fixed-lag smoothers (see, e.g., Simon 2006; Anderson and Moore 1979). The main difference between an OSA-smoothing-based scheme and a fixed-lag scheme is in the computation of the analysis distribution. The former computes it from the OSA-smoothing distribution, while the fixed-lag scheme computes it from the forecast distribution, following exactly the standard filtering update. The smoothing distribution/estimate is the final goal of the fixed-lag algorithms, while in our OSA-smoothing-based scheme, this distribution is only used to compute the analysis distribution, which is our final solution. Below, we present the EnKF-OSA algorithm for state estimation based on Ait-El-Fquih et al. (2016).
b. The (stochastic) EnKF-OSA algorithm
Let 
1) Smoothing step
2) Analysis step
3. A deterministic EnKF-OSA algorithm (SEIK-OSA)
To derive a deterministic EnKF-OSA, we first present the KF-OSA algorithm that was introduced by Desbouvries et al. (2011), based on which we then derive SEIK-OSA.
a. KF-OSA
In the linear Gaussian framework [i.e., 
1) Smoothing step
2) Analysis step
b. SEIK-OSA algorithm
We follow the formulation of the SEIK filter in Pham (2001) and Hoteit et al. (2002) to derive SEIK-OSA from the KF-OSA algorithm.
1) Smoothing step
2) Analysis step
c. Summary of the SEIK-OSA algorithm
The SEIK-OSA algorithm can be summarized as follows, starting from a given analysis ensemble 
1) Smoothing step
2) Analysis Step
The pseudoforecast ensemble
The resulting
The new analysis members
The proposed SEIK-OSA scheme, therefore, involves two forecast steps with the model: one to compute the forecast ensemble and the other to compute the pseudoforecast ensemble. It further performs two update steps: one to smooth the previous analysis estimate and the corresponding error covariance matrix using the “future” observation and another to update the pseudoforecast estimate and its error covariance using the same observation. This is, therefore, roughly twice more computationally demanding than the standard SEIK. As we will demonstrate with the numerical experiments presented in the next section, however, exploiting the information in the observation more efficiently, enhances the ensemble filter’s performance, even when implemented with only half the ensemble of the classical filtering scheme—a situation in which both schemes require comparable computational cost.
4. Numerical experiments
a. Experimental setting
b. Results and discussion
To test the filters under realistic and challenging situations, we consider three observational scenarios: full (i.e., all model variables are observed), half (i.e., every second variable is observed), and quarter (i.e., every fourth variable is observed). In each scenario, a series of sensitivity experiments is performed with various ensemble sizes, spatial and temporal observation frequencies, and observational error variances. Results of these experiments are presented with two-dimensional plots of the RMSE as a function of the inflation factor and the localization radius. The localization support radii vary from 2 (strong localization) to 40 (weak localization) grid points. The inflation varies from 1 to 1.3 when no bias is introduced in the forecast model and from 1 to 2.5 otherwise.
1) On the approximation 
The SEIK-OSA algorithm was derived assuming that 
Plots of the covariance terms 
Citation: Monthly Weather Review 146, 2; 10.1175/MWR-D-17-0175.1
2) Sensitivity to the ensemble size
We first examine the sensitivity of the filters to the ensemble size. In realistic large-scale applications, one would be restricted to small ensembles to avoid prohibitive computational burden. We conduct the experiments using five different ensemble sizes: 
Time-averaged RMSE as a function of the localization radius (x axis) and inflation factor (y axis). All filters were implemented with 10 members, and observations were assimilated from (top) all, (middle) half, and (bottom) one-quarter of the model variables at every fourth model time step (or 24 h in real time). A logarithmic color scale is used to emphasize the low RMSE values. The minimum-averaged RMSEs are indicated by asterisks, and their associated values are given in the title. White boxes indicate divergence of the filter.
Citation: Monthly Weather Review 146, 2; 10.1175/MWR-D-17-0175.1
As in Fig. 3, but for 20 ensemble members.
Citation: Monthly Weather Review 146, 2; 10.1175/MWR-D-17-0175.1
As in Fig. 3, but for 40 ensemble members.
Citation: Monthly Weather Review 146, 2; 10.1175/MWR-D-17-0175.1
The OSA-smoothing-based filters clearly perform better than the standard filters; this becomes more and more pronounced when less data are assimilated and when the filters are implemented with small ensembles, as shown in Fig. 3, in which the filters used 
Figures 3–5 suggest that the overall performances of the different schemes are sensitive to the choice of the filtering parameters, such as inflation factor and localization radius. For instance, the SEIK-based algorithms seem to require less localization than the EnKF ones, mainly when the ensemble size is smaller than the number of observations, a case in which the stochastic perturbations in the EnKF may introduce noise in the filter error covariances. When the ensemble size is relatively large, compared to the number of observations (see, e.g., Fig. 5, bottom panels, where 
In terms of inflation, OSA-smoothing-based schemes seem to require larger inflations in general, compared to the standard schemes, and this becomes more pronounced as less data are assimilated. OSA smoothing tends to decrease the ensemble spread because of its two-stage update, thereby requiring large inflation to preserve enough spread for the following assimilation cycle.
To investigate more closely the impact of the additional smoothing step, we plot in Figs. 6 and 7 the time evolution of two model variables (observed variable 19 and nonobserved variable 20) over a 30-day period (i.e., between days 366 and 395) as estimated by SEIK and SEIK-OSA, respectively. The EnKFs exhibit similar behavior and are, thus, not shown. We present the box plots in which the whiskers are set using 0 and 95th percentiles. The dashed green line represents the filter estimate (the mean of the ensembles), the stars are the observations, and the circles are the true state. Plots are shown for the challenging case with only 10 ensemble members and assimilation of one-quarter of the observations. We use a localization radius of 4 and an inflation factor of 1.15. Overall, the forecast and analysis of SEIK-OSA are closer to the reference states than those of SEIK. This is more pronounced for the nonobserved variable. Consider, for example, the period ranging from day 382 to 387 in Fig. 7, where SEIK fails to accurately recover the reference states. SEIK-OSA, on the other hand, better tracks the true trajectory. This confirms our expectations related to the effect of the smoothing step on the pseudoforecast ensemble. From a probabilistic point of view, the forecast members are, indeed, samples from the forecast distribution 
Time evolution between days 366 and 395 of the true state (circles), observations (stars), mean of analysis ensemble members (green dashed line), and boxplots of 10 ensemble members for the observed model variable 19, as resulting from (a) SEIK forecast, (b) SEIK analysis, (c) SEIK-OSA forecast, (d) SEIK-OSA analysis, (e) SEIK-OSA pseudoforecast, and (f) SEIK-OSA smoothing.
Citation: Monthly Weather Review 146, 2; 10.1175/MWR-D-17-0175.1
As in Fig. 6, but for the nonobserved model variable 20.
Citation: Monthly Weather Review 146, 2; 10.1175/MWR-D-17-0175.1
Figure 8 presents bar plots of the minimum RMSEs resulting from the best configuration of each filter, including also results from 80 and 160 members, as a function of the ensemble size, for the full, half, and one-quarter observation scenarios. Overall, the OSA-smoothing-based filters outperform the standard ones for all ensemble sizes. Furthermore, as expected, for large enough ensembles (
Minimum average RMSE for all tested filters as a function of ensemble size. (left) All, (middle) half, and (right) one-quarter of the observations were assimilated every four model steps.
Citation: Monthly Weather Review 146, 2; 10.1175/MWR-D-17-0175.1
Summary of the run times (s) of EnKF, EnKF-OSA, SEIK, and SEIK-OSA for different ensemble sizes. All filters were implemented without inflation and localization, and all data are assimilated every four model steps.
Citation: Monthly Weather Review 146, 2; 10.1175/MWR-D-17-0175.1
Table 1 reports the percentage of improvements computed using the minimum average RMSEs resulting from EnKF-OSA and SEIK-OSA when implemented with only half of the number of members of EnKF and SEIK, respectively. SEIK-OSA outperforms the standard SEIK for all tested scenarios, while EnKF-OSA improves upon the classical EnKF starting at 
Percentages of improvements, in terms of RMSE, computed using the minimum average RMSEs resulting from the OSA-smoothing-based filters and the standard filters using only half of the ensemble size. The notation 
3) Sensitivity to the frequency of observations
In the second set of experiments, we fix the ensemble size to 
As in Fig. 4, but with assimilation of observations every model time step.
Citation: Monthly Weather Review 146, 2; 10.1175/MWR-D-17-0175.1
As in Fig. 4, but with assimilation of observations every six model time steps.
Citation: Monthly Weather Review 146, 2; 10.1175/MWR-D-17-0175.1
When the observations are assimilated frequently in time (every model step), filtering with or without OSA smoothing leads to similar results. With small assimilation windows, the system is well constrained by the data, and the filters are implemented under favorable conditions, with a large enough ensemble and weakly nonlinear conditions. With fewer observations assimilated in time, a situation in which the model becomes more nonlinear within the forecast period, the accuracy of the background approximation degrades, and the benefit of the OSA smoothing becomes clearly pronounced. The OSA-smoothing-based schemes are found to be more robust and provide more accurate estimates, compared with those of EnKF and SEIK (Figs. 4, 11). When the observations are assimilated every six model time steps, EnKF-OSA leads to about 15%, 20%, and 16% improvement over EnKF when all, half, and one-quarter of the observations are assimilated, respectively, while SEIK-OSA leads to 18%, 22%, and 21% improvement over SEIK for the same observational scenarios.
Figure 12 illustrates the evolution of the minimum RMSEs for different frequencies of assimilated observations, ranging between every model step and every 20 model steps. The percentages of improvements resulting from the OSA-smoothing-based schemes with respect to the standard ones are also reported. As stated above, the benefit of the OSA-smoothing step becomes pronounced only when the data are not assimilated very frequently; the largest relative improvements generally correspond to an assimilation frequency of six or eight model steps. This is, however, true only up to a certain frequency (2 to 12 or 16, depending on the observational density), beyond which the performance of the OSA-smoothing-based schemes starts to converge toward that of the standard ones. For large assimilation windows, the model becomes strongly nonlinear, and the linear cross correlations become irrelevant to conducting the smoothing step (Song et al. 2013). One ad hoc way that might improve the results with large assimilation windows would be to apply the smoothing on the model state between the previous and current analysis times, as suggested by (Yang et al. 2013).
(bottom) Minimum average RMSE for all tested filters as a function of the frequency of observations, defined as the ratio between the filter step (fstep) and the model step (mstep), and (top) percentages of relative improvements, respectively, brought by EnKF-OSA and SEIK-OSA with respect to EnKF and SEIK. (left) All, (middle) half, and (right) one-quarter of the observations were assimilated every four model steps using 20 ensemble members.
Citation: Monthly Weather Review 146, 2; 10.1175/MWR-D-17-0175.1
4) Sensitivity to the variance of the observational errors
In this set of experiments, we fix the ensemble size to 
As in Fig. 4, but for an observational error variance equal to 0.1.
Citation: Monthly Weather Review 146, 2; 10.1175/MWR-D-17-0175.1
As in Fig. 4, but for an observational error variance equal to 2.
Citation: Monthly Weather Review 146, 2; 10.1175/MWR-D-17-0175.1
Figure 15 plots the minimum RMSE values for all filtering schemes, considering the four tested values of observational error variances (0.1, 1, 2, and 4) along with the percentages of relative improvements resulting from the OSA-smoothing-based schemes with respect to the standard ones. Overall, OSA-smoothing-based filters always outperform the classical ones, particularly when the variance of the noise in the observations is larger than 1, suggesting better ability to filter out the noise in the data using the improved background. For instance, when the observational error variance is set to 2 (which is the case of Fig. 14), the estimates are improved by about 12%, 12%, and 8% when, respectively, all, half, and one-quarter of the observations are assimilated using EnKF-OSA, and 14%, 11%, and 9% using SEIK-OSA.
(bottom) Minimum average RMSE for all tested filters as a function of the observational error variance and (top) percentages of relative improvements, respectively, brought by EnKF-OSA and SEIK-OSA with respect to EnKF and SEIK. (left) All, (middle) half, and (right) one-quarter of the observations were assimilated every four model steps using 20 ensemble members.
Citation: Monthly Weather Review 146, 2; 10.1175/MWR-D-17-0175.1
5) Sensitivity to bias in the forecast model
We test the filters’ performance with a biased forecast model. The L96 model is integrated with 
As in Fig. 4, but assimilation was carried out with a biased forcing 
Citation: Monthly Weather Review 146, 2; 10.1175/MWR-D-17-0175.1
To eliminate the ensemble undersampling error and focus on background limitations due to the bias, we conducted an experiment using a large ensemble of 200 members without localization. We found that the EnKF-OSA estimates are 10%, 12%, and 13% more accurate than those of the standard EnKF when, respectively, all, half, and one-quarter of the observations are assimilated, whereas SEIK-OSA estimates are 13%, 13%, and 14% more accurate than those of SEIK for the same scenarios. This suggests that exploiting more the observations may also help to mitigate bias in the forecast model.
5. Discussion and summary
We investigated the relevance of an ensemble filtering strategy that replaces the classical filtering path, which involves a forecast step between two successive analyses, with an alternative path that exploits more information in the observations. The proposed filtering scheme, which introduces one-step-ahead smoothing of the state between two successive analyses, is an ensemble implementation of a fully Bayesian filtering approach. The new smoothing step involves more information in the estimation process, which provides more reliable background. This is shown to improve the ensemble filter’s performance when the filter is implemented under imperfect conditions. The resulting filters bear strong resemblance to the running in place ETKF scheme, which was introduced as a heuristic approach.
Our goals were to evaluate the relevance of the OSA-smoothing strategy and to propose a new deterministic ensemble variant that is suitable for data assimilation into large-scale oceanic and atmospheric models. We constructed such a filter based on the SEIK filter, which we called SEIK-OSA.
Numerical experiments with the Lorenz-96 model demonstrated the relevance of the proposed OSA-smoothing strategy. Even though OSA-smoothing-based ensemble filters are computationally about twice more demanding than are standard ensemble filtering schemes for the same ensemble size, these schemes generally lead to more accurate estimates using only half the ensemble size of the standard schemes. The assimilation results also suggested that the proposed OSA-smoothing-based filters are clearly more efficient and more robust than the standard filters under all tested experimental settings and scenarios. This was more pronounced particularly when the filters were implemented in the challenging situations of low spatial and temporal data coverage, large sampling errors, noisy observations, and biased forecast model.
The proposed ensemble-OSA schemes are straightforward to implement and require only a few modifications to an existing ensemble filtering code. They were further found particularly efficient in situations depicting realistic scenarios. Future work will focus on testing the new OSA-smoothing schemes with large-scale oceanic and atmospheric data assimilation problems. Another future direction we will take is to consider using an OSA-smoothing-based strategy in the context of nonlinear, non-Gaussian filtering, such as the particle and Gaussian mixture filters.
Acknowledgments
This work is supported by King Abdullah University of Science and Technology Award CRG3-2156.
APPENDIX
Proof of (25)
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