1. Introduction
The ensemble Kalman filter (EnKF) (Evensen 2003) is widely used for data assimilation into geophysical models. The filter was introduced by Evensen (1994) as a variant of the Kalman filter (KF) (Kalman 1960), designed for large-scale nonlinear models. As any variant of the KF, it operates sequentially in two steps: a forecast step and an analysis step. In the forecast step, the model is integrated forward in time starting from the current estimate of the system state and the associated error covariance matrix to compute the forecast state and its error covariance. In the analysis step, incoming observations are used to update the forecast in order to obtain a more accurate estimate of the system state. This new (analysis) state vector and its error covariance are then used for the next forecast step. When the model is linear, the forecast step is straightforward and requires only standard matrix operations. In the nonlinear case, however, this step becomes more challenging, requiring multiple forward model integrations over the whole state space with respect to the analysis error distribution. The extended Kalman filter (EKF) avoids this difficulty by linearizing the model around the state estimate and then applying the KF on the resulting linearized system. In contrast, the EnKF uses a set of vectors in the state space (or ensemble members in the terminology of the EnKF) to represent the state estimate error distribution. This same ensemble framework is also used by other ensemble-based Kalman filters, including the singular evolutive interpolated Kalman (SEIK) filter (Pham 2001), the ensemble transform Kalman filter (ETKF) (Bishop et al. 2001), the ensemble adjustment Kalman filter (EAKF) (Anderson 2001), the ensemble square root filter (EnSRF) (Whitaker and Hamill 2002), the deterministic EnKF (DEnKF) (Sakov and Oke 2008), and the unscented Kalman filter (UKF) (Wan and van der Merwe 2000; Luo and Moroz 2009). This framework is expected to be more accurate than simple linearization, and we will not discuss it further as it was extensively investigated in several previous studies (e.g., Evensen 1994; Hoteit et al. 2002, 2005; Korres et al. 2007). Here, we mainly focus on the EnKF analysis step as presented by Burgers et al. (1998).
In the common case of linear observations, the analysis step is optimally, in the sense of minimizing the analysis error variance, implemented by the KF.1 The ensemble-based filters follow the KF to update the ensemble mean and covariance and then update deterministically (or resample in the case of SEIK) a new ensemble with a sample mean and covariance exactly matching those of the KF based on the ensemble-estimated background covariance. The EnKF, on the other hand, tries to match the same Kalman analysis mean and covariance by individually updating the ensemble members with stochastically perturbed observations using the KF gain matrix (still based on the ensemble-estimated background covariance). This, however, requires the observations to be perturbed by random noise that has exactly the same first two statistical moments as those of the observational error distribution (Burgers et al. 1998). Consequently, the EnKF was then referred to as the stochastic EnKF (Tippett et al. 2003). An important advantage of the EnKF analysis step is that it readily provides an analysis ensemble for forecasting randomly sampled from the assumingly Gaussian analysis distribution, avoiding the deterministic updating step that may distort the features of the forecast ensemble distribution as in the other ensemble-based KFs without perturbations (Lei et al. 2010).
The first two moments of the EnKF analysis may only asymptotically match those of the KF (Evensen 2003). In this sense, the EnKF analysis step always introduces noise during the filter’s analysis. This may become pronounced in the typical oceanic and atmospheric data assimilation applications where the rank of the observational error covariance matrix 
In this work, we introduce a new EnKF algorithm to mitigate the sampling error of the observational error distribution in the stochastic EnKF. Whitaker and Hamill (2002) and Sakov and Oke (2008), respectively, proposed two deterministic ensemble KFs, the EnSRF and DEnKF, which exactly yielded the first two moments of the KF analysis (based on the ensemble-estimated background covariance), while separately updating the ensemble perturbations using a modified Kalman gain. The DEnKF approximates the Kalman gain, and this may affect the distribution of the new sampled members. The EnSRF modification of the Kalman gain is just a scaling; it should not be particularly harmful to the distribution of the new sampled members, except insofar as it inherits the KF’s assumption that the distributions are Gaussian, so that they can be simply scaled rather than calculating the full Bayesian product (Lawson and Hansen 2004). Two methods were also previously proposed for mitigating the effects of perturbed observations. Anderson (2003) suggested pairing the analysis ensemble members with the ensemble forecast members in the observation space. By doing so, the same set of analysis states can be achieved while minimizing the analysis increments for each ensemble member (Bowler et al. 2013). Pham (2001) and Bowler et al. (2013) proposed schemes for removing spurious correlations between the observation and forecast perturbations. The modified (stochastic) EnKFs improved performance but did not provide better results than those obtained with some ensemble KFs without perturbations (Bowler et al. 2013).
This paper is organized as follows: Section 2 describes the EnKF analysis step and discusses in detail the observational error sampling noise in the EnKF. The new method to mitigate this noise is introduced in section 3, and the resulting algorithm is summarized in section 3c. Results of numerical experiments with the Lorenz-96 model are presented in section 4. A summary of the main results followed by a general discussion concludes the paper in section 5.
2. The analysis step in the EnKF
a. Analysis step and approximations
b. Discussion
The Kalman gain matrix depends on the exact knowledge of the forecast error covariance matrix 
The nonlinearity is the root cause of the problem; if the model is nonlinear, the true forecast covariance can be exactly computed only when we use an infinite number of ensemble members. In practice, this number is generally quite limited relative to the dimension of the state space, depending on the available computational resources. However, when the model is linear, then it suffices that the sample covariance of the analysis ensemble at the previous step matches the true analysis covariance matrix. This is the reason behind the use of second-order exact sampling schemes in some ensemble-based KFs to draw the new analysis ensemble members with the mean and covariance matrix matching those computed by Eqs. (2)– (7). As long as the model is weakly nonlinear, exact knowledge of the distribution of the ensemble members 
3. An EnKF with exact second-order observation perturbations sampling
a. Analysis
Suppose that the rank of the matrix 
In the general case where 
The observations’ perturbations in the proposed scheme cannot be Gaussian because of the involved constraints. Two examples of observations’ perturbation distributions as produced by the proposed scheme in the data assimilation experiments with the Lorenz-96 model presented in section 4a are plotted in Fig. 1 for all observed variables. These show reasonable deviations from the ideal (prescribed) Gaussian distribution of the observations’ errors. There are further good reasons to believe that it may be more important, from an ensemble Kalman filter performance point of view, to comply with the undersampling and independency constraints of the observations’ perturbations rather than Gaussianity. One can indeed argue that the Gaussianity of the observational error is not crucial for an EnKF. Without it, the Kalman filter update remains optimal, among linear filters (Gauss–Markov theorem). Moreover, EnKF calculations only involve the first- and second-order statistics and not the whole distribution. Our goal here is to make the ensemble members more representative of the forecast and analysis error distributions. In the EnKF, perturbations are introduced to “inflate” its analysis covariance matrix to the correct covariance. The effect of these on the ensemble distribution should be generally small, but this would introduce Monte Carlo sampling errors in the mean and covariance statistics. By contrast, other ensemble-based filters update the ensemble members (or redraw them in the SEIK) to match the Kalman analysis and covariance, which may distort the distribution of the forecast ensemble members. Our scheme follows the EnKF approach by retaining the original ensemble members’ distribution and introducing the perturbations to inflate its analysis covariance. The difference is that we constrain the perturbations so that this matrix matches the correct covariance, as in the other ensemble-based Kalman filters.
Two examples of observations perturbations distributions [fitted by a Kernel density (KDE) estimator] as produced by the proposed filter (EnKFesops) for all observed variables at a given analysis step in data assimilation experiments with the Lorenz-96 model and compared with the ideal observation errors distribution. (left) Filter run with no inflation, no localization, 30 members, and assimilation of all model variables, and (right) filter run with inflation (
Citation: Monthly Weather Review 143, 7; 10.1175/MWR-D-14-00088.1
Note that the scheme described above is meant only for the case where the ensemble size N minus 1 (
b. Initialization and forecast
We initialize the filter from an analysis step. If an initial forecast ensemble is readily available, one may directly start by removing one rank from the ensemble following the singular value decomposition (SVD) procedure described below and then go to the analysis step. One may also generate the initial forecast ensemble based on some prior knowledge (without using the available observation at the initialization time). Appendix A describes a procedure to draw 
The “first” forecast ensemble members satisfy the rank 
c. Algorithm
This section presents in detail the implementation of the proposed algorithm, which we refer to as the EnKF with exact second-order observation perturbations sampling (EnKFesops). The algorithm is basically very similar to that of the serial EnKF, with just two additional operations: (i) an SVD to the forecast perturbations matrix after every forecast step to reduce its rank by one and compute the first 
Initialization: One may initialize the filter from a readily available forecast ensemble or generate an ensemble from a Gaussian prior given a mean and a
Analysis (serial) step: Once an observation
- For
End the loop on j.
- Forecast step: The terms
where
In the unlikely situation where
The implementation of the EnKFesops is therefore straightforward and can be very easily included in any readily available serial EnKF code. In term of complexity, the EnKFesops algorithm has almost the same computational cost as that of the serial EnKF, with only the additional cost of the two operations for iteratively updating the vector 
4. Numerical experiments
a. Experimental setting
To generate the filters’ initial (forecast) ensemble, the model was first integrated forward without assimilation for several years in real-world time. The starting forecast ensemble members were sampled by adding independent Gaussian random perturbations with unit variance to each variable of the mean state of this long model run. With this choice, the initial ensemble members are off the model attractor and have no clear relationship between spread and error. The spinup period was, however, long enough to remove any detrimental impact. All filters were implemented with covariance inflation and covariance localization as described by Whitaker and Hamill (2002). Localization was implemented using the fifth-order function of Gaspari and Cohn (1999), which behaves like a Gaussian function but reaches zero at finite radius. For a given length scale, the correlation between two grid points becomes zero when the distance between two points is greater than twice the length scale (Hamill et al. 2001).
The experimental setup presented below is very similar to that of Whitaker and Hamill (2002), but we ran experiments assimilating the observations at every time step and, less frequently, at every fourth time step, which is equivalent to 1 day in real time, to mimic more common and challenging situations. Two different observational strategies were also considered: observations sampled from all model variables and every other model variable, using constant sampling intervals. Observations were perturbed with independent random noise of mean zero and unit variance. Accordingly, the observational error covariance 
Experiments were performed over a period of 5 yr (or 7300 model steps), excluding an early spinup period of about 20 days. The results were averaged over 10 runs to reduce statistical fluctuations. Several longer (more than 
b. Experiments results
In this section, we present and analyze assimilation results obtained using the proposed EnKFesops algorithm with the Lorenz-96 model. We also compare the performance of the EnKFesops with the EnKF when implemented with serial assimilation of the data EnKFser and with the regular (batch) updating scheme EnKFreg, EnSRF, and DEnKF. Sensitivity experiments were also conducted using different ensemble sizes and different spatial and temporal observation frequencies to study the behavior of the filters under different experimental settings. To visualize the results, we use two-dimensional plots of the RMSE as a function of the inflation factor, varying between 1 and 1.2, and localization length scale, varying between 1 (strong localization) and 40 (weak localization).
In the first experiment, we implement the EnKFreg, EnKFser, and EnKFesops with 10 members, and we assimilate observations at every model step (i.e., every 6 h). As shown in Fig. 2, when all 40 variables are observed, both serial and regular EnKFs exhibit similar performances, in term of 
Time-averaged RMSE as a function of the localization length scale (x axis) and inflation factor (y axis). The (left) EnKFreg, (middle) EnKFser, and (right) EnKFesops are implemented with 10 members and assimilation of observations from (top) all model variables and (bottom) half of the variables at every model time step (or 6 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 maps. White boxes indicate divergence of the filter.
Citation: Monthly Weather Review 143, 7; 10.1175/MWR-D-14-00088.1
When every second variable is assimilated (for a total of 20 variables), the performance of the filters slightly degrades and divergence eventually occurs (indicated by the white boxes in the figure), mainly with very small (and large) localization length scales and large inflation factors (Fig. 2). Overall the results are consistent with those resulting from the assimilation of all model variables. The EnKFesops is more sensitive to the choice of inflation and localization but again achieves the minimum average RMSE of 0.42, compared to 0.46 and 0.44 for the EnKFreg and EnKFser, respectively. Using only 10 ensemble members affects the robustness of the EnKF algorithm, especially when less data are assimilated. Moreover, removing one rank from its sample covariance matrix means that the EnKF will be practically operating with nine members. This may counterbalance any improvement from the new proposed algorithm EnKFesops when the number of assimilated observations is not much larger than the ensemble size, as is the case here, especially when only half of the model variables were assimilated. This behavior of the EnKFesops is further analyzed below.
Based on the same setup but using 30 members instead of only 10 members, the performance of the EnKFesops clearly outperforms the EnKFreg and EnKFser as shown in Fig. 3. The minimum average RMSEs achieved by the EnKFesops is respectively 0.18 for the experiment assimilating all model variables, compared to 0.2 for both EnKFreg and EnKFser, and 0.29 for the experiment assimilating half of the model variables, compared to 0.33 for both EnKFreg and EnKFser. The EnKFesops requires less inflation and localization to achieve these minima. The EnKFesops further exhibits more robust behavior with respect to the choice of localization and inflation, generally achieving better performances than the EnKFreg and EnKFser with weaker inflation and localization.
As in Fig. 2, but for 30 ensemble members.
Citation: Monthly Weather Review 143, 7; 10.1175/MWR-D-14-00088.1
The clearly better behavior of the EnKFesops with 30 members compared with 10 members can be explained by the rank-minus-1 approximation of the sample covariance of the forecast ensemble that is required after every forecast step of the EnKFesops (as discussed in section 3b). The rank-minus-1 approximation should indeed have much less impact on a matrix of rank 29, which is the rank of the sample covariance of an ensemble of 30 members than a matrix of rank 9, which is the covariance matrix rank of an ensemble of 10 members. This was verified by monitoring the ratio between the smallest singular value and the sum of all the singular values for both EnKFreg and EnKFser forecast ensemble covariance matrices, which provide a measure of the significance of the SVD approximation of this matrix by a rank-minus-1 matrix. This ratio is almost the same for the EnKFreg and EnKFser but about one order of magnitude less in the runs with 30 ensemble members compared to the runs with 10 members.
To evaluate the performance of the filters in a more challenging setting, we decreased the temporal availability of the observations to 1 day, instead of 6 h, and continued using 30 ensemble members (Fig. 4). This setup eventually makes the system more nonlinear and thus less accurate estimates are expected. On top of providing the lowest RMSEs, the EnKFesops is again shown to be more robust to less inflation and localization as can be seen from the two-dimensional (2D) RMSE plots compared to the regular and serial EnKFs, which experience divergence more often, particularly when implemented with large localization length scales. In terms of accuracy of the state estimates, when all 40 variables are assimilated, the average improvements (over all assimilation runs with different localization length scales and inflation factors) of the EnKFesops over the EnKFreg and EnKFser are about 26% and 46%, respectively. This reduces to 12% and 23%, respectively, when only half of the model variables are assimilated.
As in Fig. 2, but for 30 ensemble members and assimilation every four model time steps.
Citation: Monthly Weather Review 143, 7; 10.1175/MWR-D-14-00088.1
We further compared the performance of the proposed EnKFesops against the EnSRF and DEnKF using 30 ensemble members with assimilation of all model variables every 6 h. The resulting average RMSE values using the EnSRF and the DEnKF are generally smaller than those obtained with the EnKFreg and EnKFser, but they are larger than those obtained with the EnKFesops. The proposed EnKFesops algorithm, as shown in Fig. 5, performs slightly better than the EnSRF and the DEnKF for all tested settings of inflation and localization respectively leading to about 4% and 8% improvement on average and in terms of RMSE, when all variables are assimilated. When only half of the variables are assimilated, the EnKFesops and the EnSRF/DEnKF performances become more comparable, but again the EnKFesops produces the lowest RMSEs (about 2% and 3% improvements on average compared to the EnSRF and DEnKF, respectively). The EnKFesops is further found to be more robust with respect to the choice of inflation and localization, especially compared to the DEnKF.
As in Fig. 2, but for EnSRF, DEnKF, and EnKFesops.
Citation: Monthly Weather Review 143, 7; 10.1175/MWR-D-14-00088.1
As a final assessment of the filters’ performances in terms of RMSE, we show in Fig. 6 bar plots of the minimum RMSE resulting from the separately optimized configuration of each filter, as a function of the ensemble size 
Minimum average RMSE for all tested filters (EnKFreg, EnKFser, EnKFesops, EnSRF, and DEnKF) as a function of the ensemble size. (left) All variables and (right) every other variable are observed at every model time step (or 6 h in real time).
Citation: Monthly Weather Review 143, 7; 10.1175/MWR-D-14-00088.1
We finally compared the evolution of the ensembles’ spreads of all filters in time when implemented with 30 members and assimilation of all/half observations at every model/four model time steps. For that, we arbitrarily selected three pairs of localization length scales and inflation values and plotted the spread of the corresponding analysis ensembles (from one single assimilation run) in Fig. 7. For the 12 displayed (and actually all tested) cases, the EnKFesops spread is larger than the spreads of the regular and serial EnKFs, close but slightly smaller than that of the EnSRF, and always smaller than that of the DEnKF. The random sampling of the ensemble members in the analysis step of the regular EnKF estimates a small analysis error variance, as compared to the other ensemble filters, and the new proposed scheme brings it closer to that of the EnSRF. The DEnKF spread is the largest for all tested cases, and the EnKFser spread is generally lower than that of the regular EnKF.
Time evolution of the average ensemble spread as it results from EnKFreg, EnKFser, EnKFesops, EnSRF, and DEnKF implemented with 30 ensemble members as function of (from bottom to top) localization length scales (lc) and inflation factor (ic) and (left to right) spatial and temporal observation network densities. Full time and sparse time correspond to the assimilation of observations at every model time step and every four model time steps, respectively.
Citation: Monthly Weather Review 143, 7; 10.1175/MWR-D-14-00088.1
We further analyzed the time evolution of the spread of the different filters in this same setup but for their best achieved performances in terms of RMSE with respect to the choices of inflation and localization. The results are shown in Fig. 8 for the first year only, a shorter but “typical” time range, for better clarity of the plots. The filters seem to achieve their best performances with more or less the same spread, except for the DEnKF. The differences become more pronounced with fewer assimilated observations, and, in general, the EnKFesops produces the lowest spread, which is consistent with its lowest RMSE. To analyze the filters’ spreads in relation to their corresponding RMSEs, we plotted in Fig. 9 the time evolution of the spreads and RMSEs (as they result from the same single run) for the best performance of each filter with the different observations’ scenarios. The spreads and RMSEs are well comparable for all filters, except for the DEnKF spread that seems to overestimate the corresponding RMSE.
Time evolution of the average ensemble spread over the first-year assimilation period as it results from EnKFreg, EnKFser, EnKFesops, EnSRF, and DEnKF implemented with 30 ensemble members for their best performances in term of RMSE with respect to the choices of inflation and localization. Full time and sparse time correspond to assimilation of observations every model time step and every four model time steps, respectively.
Citation: Monthly Weather Review 143, 7; 10.1175/MWR-D-14-00088.1
Time evolution of the average ensemble spread (dark, thick colors) and the corresponding RMSEs (light, thin colors) over the first-year assimilation period as it results from the same single run for the best performance (with respect to the choice of inflation and localization) of each filter implemented with 30 ensemble members with the different observations scenarios. Full time and sparse time correspond to assimilation of observations every model time step and every four model time steps, respectively.
Citation: Monthly Weather Review 143, 7; 10.1175/MWR-D-14-00088.1
We have also compared the ensemble rank histograms of the filters, but the results were not very conclusive in the sense that the rank histograms resulting from the EnKFesops were not significantly different from those of the standard EnKF. We therefore decided not to show them here for brevity.
5. Summary and discussion
The stochastic ensemble Kalman filter (EnKF) applies the Kalman filter (KF) analysis step to update each ensemble member with a different perturbed observation, directly generating a random analysis ensemble from the presumed analysis Gaussian distribution. This is different from other ensemble-based KFs, which deterministically update the analysis ensemble by matching the first two moments of the KF analysis. It has been argued that the random sampling of the ensemble allows the EnKF to better preserve the features of the forecast ensemble distribution. The EnKF analysis, however, only asymptotically matches the first two moments of the KF. In realistic applications, computational requirements often strongly limit the size of the ensemble. In this case, the observational error covariance will be undersampled, especially when the size of the observation vector is (much) larger than the ensemble size, which is a common situation in atmospheric and oceanic applications. This introduces errors to the filter estimates (and their covariance matrices), which may degrade the filter performance.
In this paper, we introduced a new serial EnKF algorithm in which we retained the idea of separately updating each ensemble member with perturbed observations, but we did this in such a way as to exactly reproduce the first two moments of the KF equations, given the ensemble-based forecast error covariance. For an ensemble size 
We investigated the behavior of the new EnKF algorithm with the strongly nonlinear Lorenz-96 model. We also evaluated its performance against those of the standard EnKF (when implemented in batch or serial mode), the ensemble square root filter (EnSRF), and the deterministic EnKF (DEnKF). These last two filters were introduced as ensemble Kalman filters that do not require perturbing the observations before assimilation. Our numerical results suggest that the new scheme improves the behavior of the EnKF and significantly enhances its robustness, alleviating its dependency on the choices of localization and inflation. We also found that with large enough ensembles (i.e., ensembles that are not much affected by the rank-minus-1 approximation of the forecast ensemble covariance matrix), the new EnKF algorithm is at least as good as the EnSRF and DEnKF, and in most cases it provided the best performance in our particular, but very common, experimental setting.
Our results suggest that the EnKF may deserve further reevaluation in large-scale atmospheric and oceanic data assimilation problems. This will be one of the topics of our future work. We will also explore the generalization and impact of the proposed scheme in the context of ensemble Kalman smoothing.
Acknowledgments
We thank three anonymous reviewers for their valuable comments and suggestions. The comments of one of the reviewers on the distinction between the different ensemble Kalman filters were very useful in revising our manuscript, which we gratefully acknowledge. Research reported in this publication was supported by the King Abdullah University of Science and Technology (KAUST).
APPENDIX A
Filter Initialization from a Gaussian Prior
APPENDIX B
Computation of the Smallest Eigenvalue of 
An eigendecomposition of 
The vector
The matrix
a. Algorithm for the computation of the eigenvector of the smallest eigenvalue
Let 
The Lanczos algorithm (Golub and Van Loan 1996) generates a sequence of orthonormal vectors 
The Lanczos method might, however, suffer from numerical instabilities. As k increases, the rounding error accumulates and may result in the loss of orthogonality of the Lanczos vectors, which in turn causes a loss of accuracy. To avoid this problem, we can apply the s-step Lanczos procedure of Karush (1951), which consists of stopping the Lanczos algorithm after s steps, computing the approximate eigenvector, and then starting the algorithm again from this vector and so on until convergence. Hereafter, we focus on the simplest case (
b. Algorithm
The algorithm can be summarized as follows.
Initialization: Start from a vector
- Iterations: If
according to whether or not
before proceeding with a new iteration.
The stopping criterion 
By construction, the criterion is decreased after every iteration of the algorithm, and hence it converges to some limit, say λ. Hestenes and Karush (1951) have shown that if the initial vector 
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In fact it is only optimal in the Gaussian case or if we restrict the optimality to linear updates.
The equation 
The kernel of a matrix 
The use of localization would remove this equivalence.
