1. Introduction
Atmospheric data assimilation is the process of estimating the spatiotemporally evolving state of the atmosphere based on observations. The resulting state estimate at a given time is called analysis. Modern data assimilation algorithms obtain the analysis by a statistical interpolation process: the analysis is computed by updating an a priori estimate of the state, called background, based on the observed information assuming that the background and observation errors are random variables with known statistical parameters (Daley 1991; Kalnay 2003). In particular, the data assimilation schemes, which are able to handle the large number of state variables and observations in a realistic operational or research application, assume that the probability distribution of the background and observation errors is a multivariate normal distribution with a known covariance matrix. The focus of this paper is on the estimation of the covariance matrix that describes the distribution of the background error. This matrix is called the background error covariance matrix and we denote it by 
Sequential data assimilation schemes use a short-term forecast started from the analysis of the previous analysis time as background. Thus, 
Because of the large computational expense associated with an ensemble of model integrations, the computationally affordable sample size is limited; for example, the typical value of K in a practical application is between 20 and 100 (e.g., Houtekamer and Mitchell 2005; Hamill 2006). The small ensemble size K poses two important challenges for the statistical postprocessing algorithm that provides the final estimate 
Practical implementations of the EnKF address the aforementioned two problems by applying a physical-distance-dependent stationary (time independent) filter function to the sample covariances (e.g., Houtekamer and Mitchell 1998, 2001; Hamill et al. 2001; Anderson 2001; Ott et al. 2004). This approach is called localization, because it forces the covariance to zero beyond a prescribed distance d. Some localization functions do not change the sample-based estimate 
While there exist dynamical arguments to support localization (Yoon et al. 2010) and it may formally solve the problems it is designed to address,1 the particular shape of the tapering function is typically selected based on intuition. Moreover, if the true covariance structure of the system is complex and does not decrease monotonically with distance, the localized sample covariance may not be a good fit to the true covariance structure. The ad hoc nature of most localization algorithms used in the literature motivates us to start exploring the effects of covariance filtering on the performance of the EnKF in a more systematic way. In particular, we compare the results obtained with localization to the results obtained with an adaptive nonparametric method to estimate the entries of 
The rest of the paper is organized as follows. Section 2 is a formal description of covariance filtering in EnKF. Section 3 explains the design of the data assimilation runs that provide the data for our quantitative investigations. This section includes a brief description of the particular implementation of the serial square root EnKF scheme of Whitaker and Hamill (2002) on the Lorenz-96 model (Lorenz 1996; Lorenz and Emanuel 1998), which we use to carry out all data assimilation experiments described in this paper. In section 4, we explain the nonparametric statistical method we use to estimate the covariance, while in section 5 we compare the performance of a standard localization method to that of the nonparametric scheme in estimating the background covariance. Then, in section 6, we compare the performance of the different filtering strategies by numerical experiments. Our conclusions are drawn in section 7.
2. Covariance filtering in EnKF
In this section, we illustrate the process of estimating 
a. The EnKF algorithm of Whitaker and Hamill (2002)
b. Covariance filtering
In the computational algorithm defined by Eqs. (8)–(11), the background error covariance enters in Eq. (9) through 
A potential problem with covariance filtering is that it produces an analysis ensemble, {Xa(k): k = 1, … , K}, that is not fully consistent with the background error covariance used in the computation of the Kalman gain: while the Kalman gain is computed based on the filtered estimate 
c. Covariance inflation
In a square root EnKF scheme, such as the algorithm of Hamill and Whitaker (2005), the diagonal entries of the sample covariance matrix 
3. Experiment design
In this section, we briefly introduce the Lorenz-96 model and our approach to generate the time evolving “true” states and the simulated observations of the “true” states.
a. The Lorenz-96 model
b. Generation of the time series of “true” states and the observations
We solve Eq. (12) using a fourth-order Runge–Kutta time integration scheme and a time step of 0.05 dimensionless time unit. This time unit is defined by the e-folding time of the dissipation in the model (e.g., Lorenz and Emanuel 1998). Assuming that the e-folding time of dissipation in the atmosphere is about 5 days, the time of 0.05 dimensionless time unit is equivalent to about 6 h in real time. We carry out all numerical experiments under the perfect model hypothesis, generating the “true” state space trajectory with a long time integration of the model. The initial condition for this integration is obtained by adding small-magnitude random noise to the unstable steady-state solution xu = 0, u = 1, … , M; then, discarding the initial transient part of the trajectory (first 1000 time steps) and defining the true states xt(t) with the states along the remaining portion of the trajectory. Simulated observations are generated for each time step by adding normally distributed random noise with expectation zero and standard deviation 1 to each variable xu, u = 1, … , M. That is, the observation error covariance matrix 
4. Statistical methodology
We now introduce a nonparametric statistical method to estimate the covariance matrix, which is computationally feasible but alleviates some of the potential problems with the sample and the localized sample covariance matrices. We first review some terminologies in spatial statistics, then introduce the nonparametric scheme, gradually relaxing the assumptions we make about the covariances. We illustrate our main points with quantitative results for the Lorenz-96 model. In these calculations, we use a sample covariance matrix based on a K = 5000 member ensemble, which we denote by 
a. Terminology
We introduce the notation 
One option to estimate the covariance function is to use a parametric covariance model. There are various parametric covariance models that are isotropic, for example, the exponential function 
Sample covariances between the two locations u and υ using K ensemble members for u = 3, 30 and υ = 1, … , 40.
Citation: Monthly Weather Review 139, 9; 10.1175/2011MWR3577.1
Sample covariances between the two locations u and υ using K ensemble members for u = 3, 30 and υ = 1, … , 40.
Citation: Monthly Weather Review 139, 9; 10.1175/2011MWR3577.1
Sample covariances between the two locations u and υ using K ensemble members for u = 3, 30 and υ = 1, … , 40.
Citation: Monthly Weather Review 139, 9; 10.1175/2011MWR3577.1
We use a kernel smoothing approach as our nonparametric estimation method, which requires the selection of a kernel function and a bandwidth. A kernel is a nonnegative function, which is symmetric around zero, and decreases monotonically as |u − υ| goes to ∞; for example, G(u − υ) = exp[−(u − υ)2], is a Gaussian kernel function. There are various statistical methods to choose optimal kernels and bandwidths, and in many applications the effect of the selection of the kernel is insignificant compared to the selection of the bandwidth.6 In what follows, we first develop a nonparametric method to estimate the covariance function for the stationary case and then we extend the method to the nonstationary case.
b. Stationary estimate
Figure 2 shows the estimated covariance function obtained with Eq. (13) for u = 20 and υ = 1, … , 40. Because of the stationarity assumption, the shape of the estimated covariance function is the same for all u = 1, … , 40. Moreover, the estimated covariance function is symmetric around the center. Each row shows the result for a given bandwidth. The curves in Fig. 2 do not match the sample covariances in Fig. 1, further suggesting that the assumption of stationarity for the background error in the Lorenz-96 model is not appropriate.
Fitted covariance between the locations u = 20 and υ = 1, … , 40 using the nonparametric approach under stationarity assumption.
Citation: Monthly Weather Review 139, 9; 10.1175/2011MWR3577.1
Fitted covariance between the locations u = 20 and υ = 1, … , 40 using the nonparametric approach under stationarity assumption.
Citation: Monthly Weather Review 139, 9; 10.1175/2011MWR3577.1
Fitted covariance between the locations u = 20 and υ = 1, … , 40 using the nonparametric approach under stationarity assumption.
Citation: Monthly Weather Review 139, 9; 10.1175/2011MWR3577.1
c. Nonstationary estimate
Fitted covariance between the locations u and υ using the nonparametric approach for the nonstationary case. The two chosen u values are the same as in Fig. 1. The black curve gives 
Citation: Monthly Weather Review 139, 9; 10.1175/2011MWR3577.1
Fitted covariance between the locations u and υ using the nonparametric approach for the nonstationary case. The two chosen u values are the same as in Fig. 1. The black curve gives
Citation: Monthly Weather Review 139, 9; 10.1175/2011MWR3577.1
Fitted covariance between the locations u and υ using the nonparametric approach for the nonstationary case. The two chosen u values are the same as in Fig. 1. The black curve gives
Citation: Monthly Weather Review 139, 9; 10.1175/2011MWR3577.1
d. Adaptive bandwidth
As seen before, both the stationary and nonstationary estimates of the covariance require the selection of a bandwidth h: a larger value of h results in a smoother estimate of the covariance, because it involves a stronger averaging of the sample errors over the neighboring locations. On the one hand, when h goes to zero, the estimate provided by Eq. (14) becomes similar to the estimate provided by the sample covariance estimate, except that in Eq. (14) we divide the sums by K instead of K − 1 and filter the background error perturbation 
Because we expect the signal-to-noise ratio to be lower at larger distances |u − υ|, we introduce a bandwidth h = h(|u − υ|) that increases with the distance.7 In particular, to make h smoothly varying with the distance, we let h = h1 exp{(|u − υ|/h2)2}. Thus, we make the bandwidth adaptive at the price of replacing the single bandwidth parameter h with a pair of parameters, h1 and h2. Figure 4 shows examples of the dependence of the adaptive bandwidth on the two parameters (Fig. 4, top panel) and a comparison of the corresponding covariance estimates with the sample covariance using 20 ensemble members (Fig. 4, bottom panel). The two figures use the same color scale. For the bottom panel, we also display sample covariances using 20 and 5000 ensemble members for comparison. When h1 = 10−7 and h2 = 1, the adaptive bandwidth is about 10−7 at all distances; thus, the nonparametric estimate becomes similar to the sample covariance. The results for h1 = 0.01 and h2 = 2 are fairly similar. For the other choices of h1 and h2 the bandwidth increases with the distance and the corresponding fitted covariance values are close to zero after a short distance (Fig. 4).
A few adaptive bandwidth examples and the corresponding covariance fits with 20 ensemble members. (top) The shape of four adaptive bandwidths against the distance for the four combinations of h1 and h2 values. (bottom) The corresponding fitted covariance values between the locations u = 3 and υ = 1, … , 40 with 20 ensemble members. Same scale is used for the two figures for the adaptive bandwidth case. For the bottom panel, we have two additional curves displaying sample covariances with 20 and 5000 ensemble members for comparison.
Citation: Monthly Weather Review 139, 9; 10.1175/2011MWR3577.1
A few adaptive bandwidth examples and the corresponding covariance fits with 20 ensemble members. (top) The shape of four adaptive bandwidths against the distance for the four combinations of h1 and h2 values. (bottom) The corresponding fitted covariance values between the locations u = 3 and υ = 1, … , 40 with 20 ensemble members. Same scale is used for the two figures for the adaptive bandwidth case. For the bottom panel, we have two additional curves displaying sample covariances with 20 and 5000 ensemble members for comparison.
Citation: Monthly Weather Review 139, 9; 10.1175/2011MWR3577.1
A few adaptive bandwidth examples and the corresponding covariance fits with 20 ensemble members. (top) The shape of four adaptive bandwidths against the distance for the four combinations of h1 and h2 values. (bottom) The corresponding fitted covariance values between the locations u = 3 and υ = 1, … , 40 with 20 ensemble members. Same scale is used for the two figures for the adaptive bandwidth case. For the bottom panel, we have two additional curves displaying sample covariances with 20 and 5000 ensemble members for comparison.
Citation: Monthly Weather Review 139, 9; 10.1175/2011MWR3577.1
5. Quantitative comparison
Next, we compare the covariance estimates provided by the nonparametric scheme with those based on the sample and the localized sample covariance at a fixed time step. For this comparison, we again use 
Results are shown for various values of the parameters h1 and h2 in Fig. 5. The “optimal” pair of values is h1 = 0.05 and h2 = 0.8 in terms of the median, and h1 = 0.1 and h2 = 1.5 in terms of the mean. For localization, we use the fifth-order kernel function defined by Eq. (4.10) of Gaspari and Cohn (1999), which is the most widely used localization function in the EnKF literature. This function has a single parameter c, which controls the localization length: while the function formally becomes zero at distance 2c, the filtered covariances are already nearly zero at distance c. Figure 6 shows the median and mean of the Frobenius norm for different values of c. For a 20-member ensemble, the “optimal” localization length, with respect to the Frobenius norm, is smaller than c = 24, the value that was reported to be optimal with respect to the analysis accuracy for a 10-member ensemble in Whitaker and Hamill (2002). Also, there is a noticeable difference between the “optimal” localization length with respect to the median, c = 7.5, and to the mean, c = 5.
Frobenius norm of the (left) median and (right) mean of the differences between the fitted covariances from the nonparametric scheme and 
Citation: Monthly Weather Review 139, 9; 10.1175/2011MWR3577.1
Frobenius norm of the (left) median and (right) mean of the differences between the fitted covariances from the nonparametric scheme and
Citation: Monthly Weather Review 139, 9; 10.1175/2011MWR3577.1
Frobenius norm of the (left) median and (right) mean of the differences between the fitted covariances from the nonparametric scheme and
Citation: Monthly Weather Review 139, 9; 10.1175/2011MWR3577.1
Frobenius norm of the differences between the fitted covariances from localization and 
Citation: Monthly Weather Review 139, 9; 10.1175/2011MWR3577.1
Frobenius norm of the differences between the fitted covariances from localization and
Citation: Monthly Weather Review 139, 9; 10.1175/2011MWR3577.1
Frobenius norm of the differences between the fitted covariances from localization and
Citation: Monthly Weather Review 139, 9; 10.1175/2011MWR3577.1
The covariance estimates provided by the nonparametric scheme and the localization are compared for the “optimal” values of the parameters of the two schemes (Fig. 7). Overall, with respect to the mean, the nonparametric scheme provides the most accurate estimate except when the ensemble size is 5. The advantage of this scheme over the sample covariance and the localization with c = 24 is particularly large for the small ensembles (K ≪ 50). The advantage of the nonparametric scheme over localization with c = 5 is much more modest. In addition, with respect to the median, the localization with c = 5 outperforms the nonparametric scheme when K < 50. The system used here gives a highly nonstationary covariance structure for the background errors. The nonparametric scheme would be superior if the true covariance structure was stationary or isotropic [Eq. (13)].
Frobenius norm of the differences between the fitted covariances and 
Citation: Monthly Weather Review 139, 9; 10.1175/2011MWR3577.1
Frobenius norm of the differences between the fitted covariances and
Citation: Monthly Weather Review 139, 9; 10.1175/2011MWR3577.1
Frobenius norm of the differences between the fitted covariances and
Citation: Monthly Weather Review 139, 9; 10.1175/2011MWR3577.1
6. Analysis experiments with filtered covariances
In section 5 we compared the accuracy of the filtered covariance estimates for a single analysis time. In this section, we carry out analysis experiments with a K = 20 member ensemble using the filtered estimates of the background covariance in the computation of the Kalman gain. The filter is run for 1500 time steps and statistics are computed based on the last 1000 time steps.
a. Verification methods
We emphasize that Eqs. (19) and (20) do not provide information about the accuracy of the estimates 
b. Dependence of the results on the filtering parameters
To establish a baseline of the analysis error, against which we can measure the effectiveness of covariance filtering, we first carry out an analysis experiment using the unfiltered sample covariance matrix 
First, we filter the sample covariance estimates using localization. Whitaker and Hamill (2002) studied, in detail, the sensitivity of the analysis error to the localization parameter c and the variance inflation coefficient ρ for K = 10. They found that the root-mean-square error in the analysis mean 
Finally, we investigate the sensitivity of the analysis error to the parameters h1 and h2 using the nonparametric scheme with adaptive bandwidth to filter the covariance. We show results for ρ = 1.025, the value of the variance inflation we found to provide the smallest value of Δ. The results are summarized in Fig. 8. In this figure, the white area indicates the parameter range where the filter fails, indicated by a value of Δ, which is larger than one, the root-mean-square of the observation error. The typical value of Δ in this region is between 3.0 and 4.0, which is similar to the standard deviation of the temporal changes in the model variables. An interesting feature in Fig. 8 is the sharp boundary between the parameter ranges where the filter fails and where the analysis error is the smallest. The parameter range where the performance of the EnKF is nearly optimal (marked by blue shades) is wide: a large value of h2 cannot be used with a small value of h1, but once h1 is larger than about 0.03, the analysis error becomes insensitive to the choice of h2. In essence, an increase of the bandwidth with distance is important only when the bandwidth at zero distance is small. In the blue region, the root-mean-square error is about Δ = 0.2, lower than that for localization with c = 5, but slightly worse than that for localization with c = 24.
The grayscale shades show the value of Δ for ρ = 1.025 as a function of the parameters h1 and h2.
Citation: Monthly Weather Review 139, 9; 10.1175/2011MWR3577.1
The grayscale shades show the value of Δ for ρ = 1.025 as a function of the parameters h1 and h2.
Citation: Monthly Weather Review 139, 9; 10.1175/2011MWR3577.1
The grayscale shades show the value of Δ for ρ = 1.025 as a function of the parameters h1 and h2.
Citation: Monthly Weather Review 139, 9; 10.1175/2011MWR3577.1
c. Consistency between the estimated and the true errors
Figure 9 illustrates the level of consistency between the sample covariance and the covariance for the true background error. While the general shape of the covariance function is captured well by the sample covariance, at short distances, |u − υ| ≤ 2 (18 ≤ υ ≤ 22), the absolute value of the covariance is somewhat overestimated. This overestimation can be easily corrected by reducing the variance inflation, but reducing the variance inflation quickly leads to a loss of the stability of the EnKF. At longer distances, beyond |u − υ| ≥ 5, the time mean of the sample covariance is about zero, while the time mean of the true error is small, but not zero at most distances. We note that the near-zero values at the larger distances for the sample covariance are due to the filtering effect of time averaging, as we observe relatively large instantaneous estimates of the covariance (results not shown) at those distances. This result suggests that the relatively large instantaneous values at the larger distances are dominated by statistical fluctuations.
The components 
Citation: Monthly Weather Review 139, 9; 10.1175/2011MWR3577.1
The components
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The components
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We show results on the consistency of the estimates of the covariance for c = 5 (Fig. 10) and c = 24 (Fig. 11). In these figures, both the sample and the localized estimates slightly overestimate the variance. As in the case of the results shown in Fig. 9, this overestimation can be corrected by reducing the variance inflation, but reducing the variance inflation makes the filter unstable. At distances |u − υ| ≤ 2, the consistency is clearly better for c = 24 than for c = 5, as for the latter choice of c, the magnitude of the negative covariance at |u − υ| = 2 is underestimated. The difference in the accuracy of the estimates at |u − υ| = 2 may explain the better performance of the EnKF for c = 24 than for c = 5. For c = 5, at distances |u − υ| > 10, the filtered estimate is zero, while the sample covariance shows some distance-dependent variation. For c = 24, the sample and the filtered covariance shows a high level of consistency with each other. This is an indication that the localized covariance with c = 24 leads to a better consistency between the localized covariance estimate and the ensemble perturbations.
The components 
Citation: Monthly Weather Review 139, 9; 10.1175/2011MWR3577.1
The components
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The components
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The components 
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The components
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We show results for two different choices of the bandwidth parameters of the nonparametric estimation scheme: h1 = 0.05, h2 = 1.5 (Fig. 12) and h1 = 0.11, h2 = 1.5 (Fig. 13). A common feature of these figures is that the consistency between the sample and the filtered estimates is lower than for the localization-based filtering for c = 24. (We recall that localization with c = 24 provides the most accurate analysis among all filtering schemes tested here.) Thus, we conclude that the scheme that provides the most accurate analysis, on average, is the scheme for which the sample and the filtered estimates are most consistent with each other (localization with c = 24). This is also the scheme that provides the best consistency with the true errors for short distances. The results of this section suggest that a better metric of covariance filtering skill would be one that combined a measure of closeness to the sample covariance matrix for a very large ensemble with a measure of similarity between the climatological averages of the filtered and sample covariance.
The components 
Citation: Monthly Weather Review 139, 9; 10.1175/2011MWR3577.1
The components
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The components
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The components 
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The components
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The components
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7. Conclusions
In this paper, we investigated the effects of filtering the sample covariances used in the computation of the Kalman gain in an EnKF on the accuracy of the estimates of the background error covariance. We considered two particular approaches to filter the sample covariance: the more traditional approach of covariance localization and a new kernel smoothing method with variable bandwidth to obtain nonparametric estimates of the covariances.
We found that for a single analysis time, the nonparametric scheme provided the overall most accurate estimate of the covariances and that localization provided more accurate estimates with short localization length than with long localization length. We also found, however, that when the analysis is cycled, localization with long localization length provided more accurate analysis in the root-mean-square sense than the nonparametric scheme or localization with short localization length. We explained this result with the better consistency between the covariance estimates, the true background error covariance, and the ensemble perturbations that represent the background uncertainty. Our results suggest that preserving such consistency is important.
Finally, we note that the Lorenz-96 model, with its rapidly changing background covariance between locations, poses a considerable challenge to the covariance estimation methods. It is plausible that some of our findings would not hold for a more realistic model in which the background error covariance changes in a smoother way. In such a model, distinguishing between the spatially rapidly changing sampling noise and the smoother changing true covariance should be easier, which we expect to benefit the new kernel smoothing method more than localization.
Acknowledgments
Mikyoung Jun, Marc G. Genton, and Fuqing Zhang acknowledge the support from the National Science Foundation (ATM 0620624). Mikyoung Jun’s research is also supported by NSF Grant DMS-0906532. Istvan Szunyogh acknowledges the support from NSF (ATM 0935538) and ONR (N000140910589). Marc Genton’s research is supported by NSF DMS-1007504. Fuqing Zhang acknowledges the support from ONR Grant N000140410471. Craig H. Bishop acknowledges support from ONR Project Element 0602435N, Project Number BE-435-003. The authors are grateful to Herschel Mitchell (the Editor), Andrew Tangborn, and one anonymous reviewer for valuable comments.
APPENDIX
Proof of the Positive Definiteness of Eq. (14)
We give a brief sketch of the proof for the positive definiteness of the covariance function in Eq. (14).
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Localization also makes the data assimilation algorithms more suitable for implementation on massively parallel computer architectures.
The set of analysis perturbations that satisfies this condition is not uniquely defined.
This assumption is solely made to simplify the notation and has no effect on the generality of our results.
This shortcoming of the model was successfully corrected by a modification of Eq. (12) in Lorenz (2005). Unfortunately, this improvement of the model was achieved at the expense of increasing the number of variables, which limits the appeal of the improved model as a low computational cost alternative to a more realistic atmospheric model.
We also carried out experiments in which we observed every third location, but because the results were qualitatively similar to those for observing all locations, we do not report the results for that setting.
For more details on kernel estimation methods, see Fan and Yao (2003).
Localization of the sample covariance is equivalent to using an h, which is nearly zero within the localization length and large outside the localization length.
For a matrix 
The correct figure to support their result can be found in the corrigendum of Whitaker and Hamill (2002).
