Monthly Weather Review The Hybrid Local Ensemble Transform Kalman Filter

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1 Monthly Weather Review The Hybrid Local Ensemble Transform Kalman Filter --Manuscript Draft-- Manuscript Number: Full Title: Article Type: Corresponding Author: Corresponding Author's Institution: First Author: Order of Authors: Abstract: Suggested Reviewers: The Hybrid Local Ensemble Transform Kalman Filter Expedited Contribution Stephen Gregory Penny, Ph.D. University of Maryland College Park, UNITED STATES University of Maryland Stephen Gregory Penny, Ph.D. Stephen Gregory Penny, Ph.D. Hybrid data assimilation methods combine elements of ensemble Kalman filters (EnKF) and variational methods. While most approaches have focused on augmenting an operational variational system with dynamic error covariance information from an EnKF, we take the opposite perspective of augmenting an operational EnKF with information from a simple 3D-Variational (3D-Var) method. We wish to determine which aspects of the variational methods are necessary for successful application of a hybrid method. To this end we have developed the Hybrid Local Ensemble Transform Kalman Filter (Hybrid-LETKF), which improves analysis errors when using small ensemble sizes and low observation coverage versus either LETKF or 3D-Var used alone. The results imply that for small ensemble sizes, allowing a solution to be found outside of the space spanned by ensemble members provides robustness in the hybrid method compared to LETKF alone. Finally, the simplicity of the Hybrid-LETKF design implies that this method can be applied operationally while requiring almost no modification to an existing operational 3D-Var system. Chris Snyder, Ph.D. Senior Scientist, NCAR chriss@ucar.edu Chris published one of the earliest papers on hybrid data assimilation. He has since co-authored a number of papers discussing hybrid methods in detail. Craig Bishop, Ph.D. Naval Research Laboratory Craig.Bishop@nrlmry.navy.mil Craig developed one of the original Ensemble Transform Kalman Filters. He has experience with hybrid methods and has co-authored a number of papers on the subject. Sebastian Reich, Ph.D. Professor, Universität Potsdam sreich@math.uni-potsdam.de Sebastian has expertise in the mathematical fundamentals of data assimilation, particularly mathematical issues in general Markov Chain Monte Carlo Methods. Ross N. Hoffman, Ph.D. Chief Scientist and Vice President, Atmospheric and Environmental Research rhoffman@aer.com Ross has experience using and implementing the Local Ensemble Transform Kalman Filter in a variety of model scenarios. He is very familiar with the inner-workings of the core algorithm. Jeff Anderson, Ph.D. Senior Scientist, Section Head of Data Assimilation Research Section, NCAR jla@ucar.edu Jeff has experience working with the Lorenz-96 model in numerous test environments. He has insights into the behavior of the system under various conditions. Peter Jan van Leeuwen, Ph.D. Powered by Editorial Manager and Preprint Manager from Aries Systems Corporation

2 Head of the Data Assimilation Research Centre, University of Reading Peter is an expert on data assimilation methods, is head of DARC, and an associate editor of the Monthly Weather Review. He would have valuable insights into the phenomena discusses in this paper. Jeffrey S. Whitaker, Ph.D. research meteorologist, NOAA Earth System Research Laboratory Jeff's research is focused on the use of ensembles in weather forecasting and data assimilation. In particularly relevant work, he has compared variational and ensemblebased data assimilation systems for reanalysis of sparse observations. Powered by Editorial Manager and Preprint Manager from Aries Systems Corporation

3 Cover Letter Click here to download Cover Letter: MWR_CoverLetter.pdf DR. STEPHEN G. PENNY University of Maryland, College Park, MD Tuesday, April 09, 13 Dear Editors, Please find enclosed a manuscript entitled: "The Hybrid Local Ensemble Kalman Filter" which I am submitting for exclusive consideration of publication as an article in the Monthly Weather Review. The paper demonstrates a new approach to hybrid data assimilation with greater stability than traditional hybrid methods. A hybrid data assimilation scheme was recently adopted at NCEP for the atmosphere, and the method within this paper is being developed for the ocean as an advancement of the existing Global Ocean Data Assimilation System. As such this paper should be of interest to a broad readership including those interested in both operational and research- oriented data assimilation of the atmosphere and ocean. Thank you for your consideration of my work. Please address all correspondence concerning this manuscript to me at the Department of Atmospheric and Oceanic Science at the University of Maryland and feel free to correspond with me by e- mail at: Steve.Penny@noaa.gov. Sincerely, Dr. Stephen G. Penny Research Associate, Department of Atmospheric and Oceanic Science, University of Maryland Visiting Scientist, National Centers for Environmental Prediction

4 Manuscript (non-latex) Click here to download Manuscript (non-latex): Penny_HybridLETKF_MWR_FINAL_ pdf The Hybrid Local Ensemble Transform Kalman Filter Stephen G. Penny 1,2 April 21, 13 Corresponding Author: Dr. Stephen G. Penny, Steve.Penny@noaa.gov 1 Applied Mathematics and Scientific Computation, University of Maryland, College Park, Maryland, USA 2 Department of Atmospheric and Oceanic Science, University of Maryland, College Park, Maryland, USA

5 Abstract Hybrid data assimilation methods combine elements of ensemble Kalman filters (EnKF) and variational methods. While most approaches have focused on augmenting an operational variational system with dynamic error covariance information from an EnKF, we take the opposite perspective of augmenting an operational EnKF with information from a simple 3D-Variational (3D-Var) method. We wish to determine which aspects of the variational methods are necessary for successful application of a hybrid method. To this end we have developed the Hybrid Local Ensemble Transform Kalman Filter (Hybrid-LETKF), which improves analysis errors when using small ensemble sizes and low observation coverage versus either LETKF or 3D-Var used alone. The results imply that for small ensemble sizes, allowing a solution to be found outside of the space spanned by ensemble members provides robustness in the hybrid method compared to LETKF alone. Finally, the simplicity of the Hybrid-LETKF design implies that this method can be applied operationally while requiring almost no modification to an existing operational 3D-Var system.

6 Introduction Hybrid data assimilation systems combine two approaches traditionally used in operational weather forecasting: ensemble Kalman filters (EnKF) and variational methods such as 3D-Var and 4D-Var. For example, a hybrid system based on the developmental work of Barker (1998), Hamill and Snyder (00), Lorenc (03), Buehner (05, a,b), and Wang et al. (07a, 07b, 08a, 08b, ), has recently been implemented at the National Centers for Environmental Prediction (NCEP) for use in operational forecasting (Kleist, 12), and another at the Met Office (Clayton et al., 12). Most of the justification given for the improved performance of the traditional hybrids over the variational methods has been that the background error covariance is better defined with an ensemble, either due to flow dependence or to the better defined multivariate covariance information. While such hybrid approaches have been shown to improve upon the existing operational variational systems, it is unclear which aspects of the variational systems benefit the EnKF. We examine the impacts that a simple 3D-Var has on an EnKF in order to determine the source of these benefits. In an operational environment, the choices of ensemble size and observation coverage are limited by costs of computational facilities and observing equipment. Thus, it is important to identify the preferred algorithmic approach when these parameters are prescribed. We introduce a new hybrid using an EnKF combined with a simple 3D-Var and demonstrate its effectiveness from this perspective. Traditional hybrids start with a variational approach and incorporate the ensemble information through the ensemble-derived covariance matrix. Here we instead

7 start with an EnKF and use a variational approach to apply a correction to the EnKF within the model space Methodology a) Model For the forecast model, we use the Lorenz-96 model on m = grid points (Lorenz, 1996), dx dt = x j+1! x j!2 ( ) x j!1! x j + F, (1) with F= (as used by Wilks 05, 06a; and Messner, 09) and Δt = 0.01, for j=1..m. Because Orrell (03) states that this model s internal doubling time varies strongly with forcing, we note that our results were similar using F=8 with a forecast time step Δt = 0.05 (as originally used by Lorenz). Lorenz (05) discusses further implications of varying F. In this model, the first term represents advection constructed to conserve kinetic energy, the second is damping, and the third is forcing. The boundaries are cyclic, such that x m+1 = x 1, and x 0 = x m. The truth or nature run is performed with Runge- Kutta order 4-5, while forecast runs are performed with Runge-Kutta order b) Data Assimilation Methods We solve the data assimilation problem by minimizing the traditional cost functional (Kalnay, 03), J ( x) = (x! x b ) T B!1 (x! x b )+ (y o! Hx) T R!1 (y o! Hx). (2) We minimize J over potential model states x, where x b is the background estimate, y o is the observation vector, and H is an operator transforming x from the model space of

8 dimension m to the observation space of dimension l. The matrices B and R are the background and observation error covariance matrices, respectively. For 3D-Var we use the preconditioned conjugate gradient (PCG) minimization algorithm. In PCG, a preconditioner matrix M is used to solve M -1 Ax = M -1 b. The matrix B is used as the preconditioner where, A = I + BH T R!1 H, and (3) b = x b + BH T R!1 y o (4) The B matrix is constructed using an exponential decay function with maximum radius r, B i, j =! 2 b e! i! j, for row i, column j, and i-j r. (5) We implement the Local Ensemble Transform Kalman Filter of Hunt et al (07), inspired by Bishop et al. (01) and Ott et al. (04), as our EnKF method. The analysis ensemble generated by LETKF identifies directions of strongest error growth within the ensemble subspace (as with bred vectors (Toth and Kalnay, 1997)). In LETKF, an analysis is computed for each grid point based on local observations. Each analysis is formed within the linear space spanned by the ensemble members. We implement two hybrid algorithms using LETKF as a basis. First, a hybrid inspired by traditional methods computes a linear combination of B and the ensemble background error covariance matrix P b for use in a local 3D-Var step. A similar approach was shown by Wang et al. (07b) to be equivalent to the control-variable method of Lorenc (03) and Buehner (a,b). We refer to this method as the Hybrid/Covariance- LETKF. The second algorithm we refer to as the Hybrid/Mean-LETKF, and is first described in words: The standard LETKF is used first. The analysis mean from LETKF is

9 then used as the background for 3D-Var, which is performed locally in model space after each grid point is analyzed. A weighted average of the two analysis solutions is computed, and the LETKF analysis ensemble is re-centered at the new solution. Kalnay and Toth (1994) performed a similar procedure using a single bred vector and 3D-Var. The Hybrid/Mean-LETKF algorithm is detailed as follows: We calculate the LETKF analysis following Hunt et al (07), first computing the analysis error covariance in ensemble space,!p a = " (k!1)i /! + Y #$ b ( ) T R!1 Y b % &'!1, (6) where Y b = H(X b ), the columns of X b are ensemble perturbations from the mean state, and ρ is the local inflation parameter. The symmetric square root of this matrix is computed to determine the weights for the analysis ensemble, W a = " #(k!1) P! a $ %1/2. (7) To transform from ensemble space back to model space we multiply these weights with each of the background ensemble members, X a = X b W a. (8) Finally, the analysis mean is computed as, w a =! P a ( Y b ) T R!1 Y y o! y b ( ), (9) x a = X b w a + x b. () At this point, the LETKF algorithm is complete. Next we re-localize in model space. Greybush et al. (11) discuss the impacts of localization in observation and model spaces. We define the local model dimension, m loc = 2r+1, and select the appropriate rows and columns of the full B matrix.

10 For the Hybrid/Covariance-LETKF, a linear combination is formed with the static B matrix and the ensemble-generated P b in the local model space with dimension m loc and the analysis solution replaces the LETKF ensemble mean, J ( x a ) = (x a! x b ) T (!B loc + (1!!)P b loc )!1 (x a! x b )+ (y o! Hx a ) T R!1 loc (y o! Hx a ). (11) For the Hybrid/Mean-LETKF we minimize the cost function, J ( x a ) = (x a! x a ) T ˆB!1 (x a! x a )+ (y o! Hx a ) T ˆR!1 (y o! Hx a ). (12) Here we use ˆB = B loc and ˆR = R loc, but other choices are possible. We then update the analysis mean as a weighted combination of the 3D-Var and LETKF solutions and re-center the analysis to this mean, a x Hybrid =!x a + ( 1!! ) x a, (13) 161 a X Hybrid = X a a + x Hybrid v T, (14) where v = ( ) T is a column of k ones used to add the mean to each column of X a, resulting in the final analysis ensemble having the hybrid-derived analysis as its mean. Finally, we update the single grid point at the center of the local region with the hybrid solution Experiment Design We first examine special case scenarios using limited observations and a small ensemble size: l=4 observations per time step and ensemble size k=5. We show the nature run, and compare free run forecast error and data assimilation analysis error for LETKF, 3D-Var and the Hybrid/Mean-LETKF. We then generalize the results across the full range of ensemble sizes (2-) and observation coverage (1-) for each method.

11 Observations are generated randomly in space from a uniform distribution on the interval [0,] with errors from a normal distribution using a prescribed variance of σ 2 r = 0.5. We assume these observation statistics are known. A linear interpolation scheme is used to construct the observation operator H. The B matrix used for all methods is constructed as a double exponential distribution function with maximum σ 2 b =1.0 centered on the diagonals with a local radius of r=5 grid points. The Lorenz-96 model is spun-up over 14,0 time steps (as per Lorenz (1996)) to ensure convergence to the attractor. An additional 600 time steps are run with a forecast time step of Δt=0.01 to form a nature run. The experiment initial conditions are sampled from a Gaussian distribution, N(x t, 0.1), with mean equal to the truth and standard deviation equal to 0.1. For LETKF, we use a constant multiplicative background covariance inflation of ρ = 1.1 (%) Results We show in Figure 1 that the standard LETKF algorithm performs well with a large ensemble size (e.g. k=), but fails due to filter divergence when using smaller ensembles (e.g. k=5). This filter divergence is typical in this experiment setup for k 5 and is dependent on the observation locations and forecast time step Δt. We see that for a large ensemble size (e.g. given a full-rank of ), the standard LETKF algorithm is quite accurate. However, as the ensemble size decreases, the analysis solution degrades until the filter eventually diverges from the nature run. When implementing the Hybrid/Mean- LETKF, using k=5 ensemble members, the filter recovers stability and has comparable accuracy to the standard LETKF with k=.

12 The energy for this system, s 2 as defined by Lorenz (05; 06), is simply the mean square of the system state across all grid points. For longer time periods, the total energy oscillates chaotically in a range from to 90 and is tracked well by the standard LETKF analysis for ensemble sizes k > 5. In the standard LETKF (k=5), s 2 blows up toward 5 while the Hybrid/Mean-LETKF using the same k=5 members tracks closely with the standard LETKF (k=) (see Figure 2). We next examine the impact of varying observation coverage and ensemble size for the standard LETKF in Figure 2, for which k = 1 represents the pure 3D-Var results. This figure contains results from x=1600 of the previously described special case scenarios. We define three regimes within the parameter space: (1) the ensemble/hybrid method outperforms 3D-Var, (2) the ensemble/hybrid method fails, (3) 3D-Var outperforms the ensemble/hybrid method. Our goal is to maximize the parameter space of regime 1 while minimizing analysis error. Trevisan and Palatella (11) showed that the number of positive and null Lyapunov exponents for the Lorenz-96 system is a monotonic function of model dimension. Based on their experiments with an Extended Kalman Filter they hypothesized that in an ensemble approach, when observations are sufficiently dense and accurate so that error dynamics are approximately linear, then the necessary and sufficient number of ensemble members is equal to the total number of positive and null Lyapunov exponents. Our experiments indicate that a smaller set of ensemble members representing only the dominant positive Lyapunov exponents is necessary for the standard LETKF to track the nature run. This is due to localization and is in agreement with Ott et al (04). Using a configuration matching Trevisan and Palatella, F=8,

13 Δt=0.05, l=, and σ r =0.01, we obtain that 9 ensemble members are required for our configuration of LETKF compared to their hypothesized 14 for a general EnKF. Our configuration of the localized Lorenz-96 system with F= and Δt=0.01 has 5 such dominant Lyapunov exponents in a well-observed system. LETKF shows superior accuracy for ensemble sizes k > 5 and observation coverage l > 5 compared to 3D-Var. However, for small ensemble sizes there is a barrier below which LETKF fails and thus is always outperformed by 3D-Var. This ensemble size barrier increases as the observation coverage decreases for l 5. At l=1, both 3D-Var and LETKF fail. As shown in Figure 4, with a sufficient number of observations the Hybrid/Covariance-LETKF is successful at stabilizing the filter for small ensembles, but increases errors for larger ensembles and destabilizes the filter for low observation coverage. With lower values of α, (e.g. α = 0.2), the mean absolute analysis errors in regime 1 are smaller, but the total area of regime 1 in the parameter space decreases. The converse is true for larger values of α (e.g. α = 0.8). For stability of the 3D-Var step, we use initial guess x b in PCG, but note that use of the analysis ensemble mean from LETKF improves accuracy for cases with a low observation coverage. The Hybrid/Mean-LETKF algorithm (Figure 5) using α = 0.5 retains much of the accuracy of the larger k= standard LETKF, while still using a small (k=5) ensemble size and few (l=4) observations. These results are found to hold even when driving the ensemble size down to k=3 members. If the number of observations decreases further (e.g. to l=3, with k=3) however, this hybrid undergoes the same filter divergence as the standard LETKF. For this hybrid method, as α decreases there is a gradual adjustment back to the standard LETKF results: the mean absolute analysis errors decrease

14 throughout regime 1 while the minimum observation count required for filter stability for 2 < k < 5 steadily increases. We note that these results obtained with a localized 3D-Var remain the same when 3D-Var is instead applied globally after LETKF or ETKF is completed, with comparable accuracy (not shown) Conclusions This research began with an investigation into the source of benefits arising from hybrid methods when variational techniques are added to an EnKF. We compared solutions from 3D-Var with an EnKF, and showed the standard LETKF broke down when using small ensemble sizes. We then introduced two hybrid approaches. The first was the traditionally motivated Hybrid/Covariance-LETKF. The second was the Hybrid/Mean- LETKF for which a simple 3D-Var is applied after completion of LETKF to adjust the ensemble mean in model space and add stability to the filter for small ensemble sizes. LETKF is highly accurate when applied to the Lorenz-96 system if allowed a sufficient number of ensemble members. The Hybrid/Mean-LETKF approach generated solutions that outperformed both 3D-Var and LETKF for observation coverage with 2 < l < and ensemble size 1 < k < 5. From our results, we conclude that it is the computation of the analysis in local model space that stabilizes the Hybrid/Mean-LETKF at low ensemble sizes. Of interest is the local dimensionality of the unstable Lyapunov vectors relative to the size of the ensemble k and the local model dimension m loc. Based on the work of Trevisan and Palatella (11), we suspect the minimum ensemble barrier for the standard LETKF is directly related to the local dimensionality of the error growth, though like Ott et al (04) we found fewer ensemble members were required when using localization.

15 The hybrid methods enhance the filter by allowing solutions to be influenced by observations that cannot be reached in the linear span of ensemble perturbations. Both hybrid LETKF methods are well suited for applications using a small ensemble size due to computational limitations. However the Hybrid/Mean-LETKF has better accuracy overall and is better suited for applications that also have limited observation coverage, such as global ocean data assimilation (Penny, 11; Penny et al., 13) and coupled atmosphere/ocean data assimilation. As the standard LETKF is already being used or prepared for use in operational environments in Italy, Germany, Brazil, Argentina, Japan and the United States, the Hybrid/Mean-LETKF is a simple extension that could be adopted in an operational environment as well Acknowledgements I would like to acknowledge Eugenia Kalnay, Jim Carton, Steven Greybush, Brian Hunt, Kayo Ide and Daryl Kleist for discussions that led to this work, David Behringer and NCEP for motivating the Hybrid-LETKF, and Craig Bishop for helpful discussion on the general role of hybrid methods in data assimilation. This material is based upon work supported by the National Science Foundation under Grant No. OCE References Bishop, C. H., B. J. Etherton, and S. J. Majumdar, 01: Adaptive sampling with the ensemble transform Kalman filter. Part I: Theoretical aspects. Mon. Wea. Rev., 129, Buehner, M., 05: Ensemble-derived stationary and flow-dependent background error covariances: Evaluation in a quasi-operational NWP setting. Q.J.R. Met. Soc., 131,

16 Buehner, M., P. L. Houtekamer, C. Charette, H. L. Mitchell, and B. He, a: Intercomparison of variational data assimilation and the ensemble Kalman filter for global deterministic NWP. Part I: Description and single-observation experiments. Mon. Wea. Rev., 138, Buehner, M., P. L. Houtekamer, C. Charette, H. L. Mitchell, and B. He, b: Intercomparison of variational data assimilation and the ensemble Kalman filter for global deterministic NWP. Part II: One-month experiments with real observations. Mon. Wea. Rev., 138, Clayton, A.M., A.C. Lorenc, D.M. Barker (12), Operational Implementation of a Hybrid Ensemble / 4D-Var Global Data Assimilation System at the Met Office. Q.J.R. Met. Soc. (Submitted Feb. 12). Greybush, S. J., E. Kalnay, T. Miyoshi, K. Ide, and B. R. Hunt, 11: Balance and Ensemble Kalman Filter Localization Techniques, Mon. Wea. Rev., 139, Hamill, T.M., C. Snyder, 00: A hybrid ensemble Kalman filter-3d variational analysis scheme. Mon. Wea. Rev., 128, Hunt, B. R., E. J. Kostelich, and I. Szunyogh, 07: Efficient data assimilation for spatiotemporal chaos: a local ensemble transform Kalman filter. Physica D, 230, Kalnay, E., 03: Atmospheric Modeling, Data Assimilation and Predictability. Cambridge University Press. Chapter 5.

17 Kalnay, E and Z. Toth, 1994: Removing growing errors in the analysis. Preprints, th Conf. on Numerical Weather Prediction, Portland, OR, Amer. Meteor. Soc., Kleist, D.T., (12), An Evaluation Of Hybrid Variational-Ensemble Data Assimilation For the NCEP GFS. University of Maryland College Park, (Doctoral Dissertation). Lorenc, A.C., 03: The Potential of the Ensemble Kalman Filter for NWP a comparison with 4D-Var. Q.J.R. Meteorol. Soc., 129, Lorenz, E. N., 1996: Predictability A problem partly solved. In Proceedings of the Seminar on Predictability, volume 1. ECMWF: Reading, UK Lorenz, E. N., 05: Designing Chaotic Models. J. Atmos. Sci., 62, Lorenz, E. N., 06: Regimes in Simple Systems. J. Atmos. Sci., 63, Lorenz, E. N., and K. A. Emanuel, 1998: Optimal sites for supplementary weather observations: Simulation with a small model. J. Atmos. Sci., 55, 399. Messner, J., 09: Probabilistic forecasting using analogs in the idealized Lorenz96 setting. Diploma thesis, Department of Meteorology and Geophysics, University of Innsbruck. Orrell, D., 03: Model error and predictability over different timescales in the Lorenz 96 Systems. J. Atmos. Sci., 60, Ott, E., B. R. Hunt, I. Szunyogh, A. V. Zimin, E. J. Kostelich, M. Corazza, E. Kalnay, D.J. Patil, J. A. Yorke, 04: A Local Ensemble Kalman Filter for Atmospheric Data Assimilation. Tellus, 56A,

18 Penny, S.G., 11: Data Assimilation of the Global Ocean using the Local Ensemble Transform Kalman Filter and the Modular Ocean Model. University of Maryland College Park, (Doctoral Dissertation). Penny, S.G., E. Kalnay, J. A. Carton, B. R. Hunt, K. Ide, T. Miyoshi, and G. Chepurin, 13: The Running-in-Place algorithm applied to a Global Ocean General Circulation Model. Submitted to Nonlin. Proc. Geophys., Special Issue: Ensemble methods in geophysical sciences. Toth, Z. and E. Kalnay, 1997: Ensemble forecasting at NCEP and the breeding method. Mon. Wea. Rev., 125, Trevisan, A., and L. Palatella, 11: On the Kalman Filter error covariance collapse into the unstable subspace. Nonlin. Proc. Geophys., 18, Wang, X., T.M. Hamill, J.S. Whitaker, C.H. Bishop (07a), A Comparison of Hybrid Ensemble Transform Kalman Filter-OI and Ensemble Square-Root Filter Analysis Schemes. Mon. Wea. Rev., 135, Wang, X., C. Snyder, T.M. Hamill (07b), On the Theoretical Equivalence of Differently Proposed Ensemble 3D-Var Hybrid Analysis Schemes. Mon. Wea, Rev., 135, Wang, X., : Incorporating Ensemble Covariance in the Gridpoint Statistical Interpolation Variational Minimization: A Mathematical Framework. Mon. Wea, Rev, 138, Wilks, D.S., 05: Effects of stochastic parametrizations in the Lorenz 96 system. Q. J. R. Meteorol. Soc., 131,

19 16 Wilks, D.S., 06: Comparison of ensemble-mos methods in the Lorenz 96 setting. 352 Figures: Truth / Nature Free Run error vs. Nature/Truth ï ï2 0 ï nodes 30 Analysis error from Nature/Truth data ï2 0 ï4 nodes 30 ï6 0 ï8 50 ï Analysis error from Nature/Truth data nodes 30 ï Analysis error from Nature/Truth data ï2 0 ï ï2 0 ï4 150 ï6 0 ï8 50 nodes 30 ï timesteps 0 timesteps 350 ï8 50 ï ï6 0 ï ï4 150 ï timesteps timesteps Analysis error from Nature/Truth data 600 timesteps Meteorological Applications, 13, timesteps ï2 0 ï4 150 ï6 0 ï8 50 nodes 30 ï ï6 0 ï8 50 nodes 30 ï Figure 1. Nature run for Lorenz-96 over 600 time steps with dt=0.01 (top left). Free Run error (top center). The following analyses are performed with l=4 observations per time step: Analysis error for 3D-Var (top right), LETKF, k= (bottom left), LETKF, k=5 (bottom center), and the Hybrid/Mean-LETKF, k=5 (bottom right).

20 17 1 Total Energy 1 0 s 2 (mean square) time step Figure 2. The total energy s 2 is plotted for 00 time steps (0 days) for the ensemble mean state in an analyses using l=4 observations per time step. Standard LETKF (k=) is shown in black and (k=6) in cyan. Four different cases of standard LETKF (k=5) are shown in red, each blowing up at a different time due to randomness in observation locations. The Hybrid/Mean-LETKF (k=5), shown in blue, recovers the stability and accuracy of the standard LETKF (k > 5). Mean absolute analysis error for standard LETKF Observation count (l) Ensemble size (k) Figure 3. Mean absolute analysis error for the standard LETKF using ensemble sizes k=2.., and observation coverage l=1.. randomly chosen throughout the domain. Results at k=1 correspond to the standard 3D-Var. Empty squares indicate cases in which the Runge Kutta ODE solver could not reach the required tolerance.

21 18 Mean abs analysis error for Hybrid(Covariance) LETKF alpha=0.2 (L96:F=,dt=0.01) 1 Observation count (l) Ensemble size (k) Figure 4. Mean absolute analysis error for the Hybrid/Covariance-LETKF, α=0.2. When α=0.5, the accuracy in regime 1 (dark blue) decreases, but the total area of regime 1 increases. As α goes to 1, all members converge to the pure 3D-Var solution (k=1). Mean abs analysis error for Hybrid(Mean) LETKF alpha=0.5 (L96:F=,dt=0.01) Observation count (l) Ensemble size (k) Figure 5. Mean absolute analysis error for the Hybrid/Mean-LETKF, α=0.5. As α goes to 0, the peak of regime 2 (dark red) at (k=2) gradually increases, as does the accuracy in regime 1 (dark blue), until all members converge to the standard LETKF solution in Figure 3. As α goes to 1, all members converge to the pure 3D-Var solution (k=1).