## 1. Introduction

Hybrid data assimilation systems combine two approaches traditionally used in operational weather forecasting: ensemble Kalman filters (EnKF) and the 3D variational data assimilation (3D-Var) and 4D-Var methods. For example, a hybrid system based on the developmental work of Barker (1999), Hamill and Snyder (2000), Lorenc (2003), Buehner (2005), Buehner et al. (2010a,b), and Wang et al. (2007a,b, 2008a,b, 2013) has recently been implemented at the National Centers for Environmental Prediction (NCEP) for use in operational forecasting (Kleist 2012; Wang et al. 2013). 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, due to flow dependence and the corresponding improvement in multivariate covariance information. While such hybrid approaches have been shown to improve upon the existing operational variational systems, it is unclear how the static covariance matrix and the minimization procedure 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 start with an EnKF and use a variational approach to apply a correction within the model space to stabilize the EnKF.

## 2. Methodology

### a. Model

*m*= 40 grid points (Lorenz 1996),

*F*= 20 (as used by Wilks 2005, 2006; Messner 2009) and Δ

*t*= 0.01, for

*j*= 1, …,

*m*. Because this model’s internal doubling time varies strongly with forcing (Orrell 2003), we note that our results were qualitatively similar using

*F*= 8 with a forecast time step Δ

*t*= 0.05 (as originally used by Lorenz). Lorenz (2005) 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–Fehlberg orders 4 and 5, while forecast runs are performed with Runge–Kutta–Fehlberg orders 2 and 3 (Fehlberg 1970).

### b. Data assimilation methods

*J*over potential model states

**x**, where

**x**

^{b}is the background estimate,

**y**

^{o}is the observation vector, and

**x**from the model space of dimension

*m*to the observation space of dimension

*l*. The matrices

^{−1}

**x**=

^{−1}

**b**. The matrix

*r*

_{B},

We implement the local ensemble transform Kalman filter (LETKF) of Hunt et al. (2007), inspired by Bishop et al. (2001) and Ott et al. (2004), as our EnKF method. In LETKF, an analysis is computed for each grid point based on local observations within radius *r*. 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} for use in a 3D-Var step. A similar approach was shown by Wang et al. (2007b) to be equivalent to the control-variable method of Lorenc (2003) and Buehner et al. (2010a,b). We refer to this method as the Hybrid/Covariance-LETKF.

**y**

^{o}−

**x**

^{b}). Brett et al. (2013) indicate that it is the Kalman gain matrix that determines stability of the filter. We construct the following general hybrid gain matrix:

*β*

_{1}= 1,

*β*

_{2}=

*α*, and

*β*

_{3}= −

*α*, so that

^{B}given (

*α*allows a further manual scaling of

^{B}, which is equivalent under a monotonic mapping to applying a scalar inflation

*ρ*

_{B}(deflation for 0 <

*α*< 1) directly to the static

^{B}.

The values for these *β* coefficients are chosen specifically so that we can construct the following algorithm, which gives an algebraically equivalent result (see appendix A for proof). We refer to this algorithm as the Hybrid/Mean-LETKF, and it is first described in words: The standard LETKF is used first. The analysis mean from LETKF is then used as the “background” for 3D-Var, which is performed locally in model space after each grid point is analyzed by LETKF. An empirically chosen weighted average of the two analysis solutions is computed, and the LETKF analysis ensemble is recentered at the new solution. Kalnay and Toth (1994) performed a similar procedure using a single bred vector and 3D-Var. Another choice of coefficients—*β*_{1} = (1 − *α*), *β*_{2} = *α*, and *β*_{3} = 0—can be implemented similarly by using the ensemble forecast mean as the background for 3D-Var (see appendix B). For reference in the results section, we will call this the Hybrid/Mean-LETKF(b).

^{b}=

^{b}), the columns of

^{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,

Next we relocalize in model space. Greybush et al. (2011) discuss the impacts of localization in observation and model spaces, termed R-localization and B-localization, respectively. We define the local model dimension, *m*_{loc} = 2*r* + 1, and select the appropriate rows and columns of the full ^{b} matrices to define _{loc} and _{loc}.

^{b}in the local model space with dimension

*m*

_{loc}, similar to that performed by Hamill and Snyder (2000),

*=*

_{loc}and

_{loc}, but other choices are possible. The preferred choice for

_{loc}. Because this approximation overestimates the analysis error, we adjust the analysis mean back toward the LETKF analysis with a scaling parameter

*α*and recenter the analysis ensemble to this mean,

**v**= (1 1 … 1 1)

^{T}is a column of

*k*ones used to add the mean to each column of

^{a}, resulting in the final analysis ensemble having the hybrid-derived analysis as its mean. Finally, as with the standard LETKF, we update the single grid point at the center of the local region with the hybrid solution. For both hybrid methods,

*α*is chosen empirically based on the ensemble size (

*k*) and observation coverage (

*l*).

We note that while the observations have been procedurally reused, the results using the Hybrid/Mean-LETKF algorithm are algebraically equivalent to the previous hybrid gain form given in (9) and (10) in which the observational increments are used only once. The form that defines the Hybrid/Mean-LETKF algorithm has the advantage that existing operational systems can be used without significant modification to generate the hybrid analysis.

## 3. 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 (2–40) and observation coverage (1–40) for each method, and examine variations in the localization radius.

Observations are generated randomly in space from a uniform distribution on the interval [0, 40] with errors from a normal distribution using a prescribed standard deviation of *σ*_{r} = 0.5. We assume these observation statistics are known. A linear interpolation scheme is used to construct the observation operator *σ*_{b}^{2} = 1.0 centered on the diagonals with a local radius of *r*_{B} = 5. The Lorenz-96 model is spun up over 14 400 time steps, as per Lorenz (1996), to ensure convergence to the attractor. An additional 600 time steps are run with Δ*t* = 0.01 to form a nature run **x**^{t}(*t*). The experiment initial conditions are sampled from a Gaussian distribution, *N*[**x**^{t}(0), 0.1]. Unless otherwise noted, we use a constant multiplicative background covariance inflation of *ρ* = 1.1 (10%) and a local radius of *r* = 5 for LETKF and the associated hybrid methods.

## 4. Results

We show in Fig. 1 that the standard LETKF algorithm performs well with a large ensemble size (e.g., 20, given that 40 would be full rank), but it fails due to catastrophic filter divergence (Harlim and Majda 2010) when using smaller ensembles (e.g., *k* = 5). This filter divergence is typical in our experiment setup for *k* ≤ 5 and is dependent on the observation locations and Δ*t*. We see that for a large ensemble size, 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* = 20.

The energy for this system, *s*^{2} as defined by Lorenz (2005, 2006), 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 40 to 90 and is tracked well by the standard LETKF analysis for ensemble sizes *k* > 5. Bishop and Satterfield (2013) showed that as the variance of the underlying true distribution increases, the effective random sample ensemble size decreases. In the standard LETKF (*k* = 5), *s*^{2} “blows up” when the underlying variance increases, while both the Hybrid/Mean-LETKF and Hybrid/Covariance-LETKF using the same *k* = 5 members track closely with the standard LETKF (*k* = 20) (see Fig. 2).

We next examine the impact of varying observation coverage and ensemble size. We begin with the standard LETKF in Fig. 3, for which *k* = 1 represents the pure 3D-Var results. This figure contains results from 40 × 40 = 1600 runs similar to 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, and 3) 3D-Var outperforms the ensemble–hybrid method. Our goal is to maximize the parameter space of regime 1 while simultaneously minimizing analysis error.

As shown in Fig. 4, the Hybrid/Covariance-LETKF is successful at stabilizing the filter for small ensemble sizes. For improved stability of PCG, we required an initial guess of **x** = **x**^{b}. With smaller values of *α* (e.g., *α* = 0.0–0.2) for all ensemble sizes, the mean absolute analysis errors are close to that of the standard LETKF (*k* > 5). However, as the observation coverage decreases, the analysis errors increase. For larger values of *α* (e.g., *α* = 0.5), the mean absolute analysis errors increase throughout regime 1 and converge toward the 3D-Var solution (*k* = 1) as *α* increases to 1.0.

The Hybrid/Mean-LETKF algorithm (Fig. 5) using *α* = 0.5 retains much of the accuracy of the standard LETKF (with *k* = 20) 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, then 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 solution: the mean absolute analysis errors decrease throughout regime 1 while the minimum observation count required for filter stability for 2 < *k* < 5 steadily increases from *l* = 3 to 7 (*α* = 0.5) to *l* = 5 to 18 (*α* = 0.2), and finally to *l* = 40 (*α* = 0.0). We note that these results obtained with a localized 3D-Var are similar when the 3D-Var correction is instead applied globally after LETKF or after a (nonlocalized) ensemble transform Kalman filter (ETKF), with comparable accuracy (not shown). The Hybrid/Mean-LETKF(b) has qualitatively similar results (Fig. 6).

While the *α* parameter scales the analysis errors with an apparent monotonic relationship between the standard LETKF (*α* = 0.0) and 3D-Var (*α* = 1.0) for each hybrid method, the exact *α* values are not directly comparable across methods. All of the methods give a range of mean analysis errors varying from those found with the standard LETKF to approximately those found with 3D-Var. Therefore an appropriate set of parameter values should allow one to tune to any desired mean analysis error accuracy in that range. For example, the Hybrid/Covariance-LETKF with *α* = 0.1 (Fig. 4) appears to have similar analysis errors to the Hybrid/Mean-LETKF (Fig. 5) with *α* = 0.5 (with the exception of the cases with very low observation counts, e.g., *l* < 4, for which the Hybrid/Mean-LETKF has lower mean analysis errors).

Based on their experiments with an extended Kalman filter, Trevisan and Palatella (2011) hypothesized that in an ensemble approach, when observations are sufficiently dense and accurate so that error dynamics are approximately linear, the necessary and sufficient number of ensemble members is equal to the total number of positive and null Lyapunov exponents. Our experiments indicate that for LETKF fewer ensemble members are needed. This is due to localization and is in agreement with Ott et al. (2004). Using parameters similar to Trevisan and Palatella—*F* = 8, Δ*t* = 0.05, *l* = 20, 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. Using the Hybrid/Mean-LETKF, as few as two ensemble members can be employed.

The Lorenz-96 system is extensive, which implies that the number of positive Lyapunov exponents grows linearly with the system size (Pazo et al. 2008; Karimi and Paul 2010). Holding the number of observations fixed at *l* = 20, we vary the localization radius *r* and the ensemble size *k* from 1 to 20 (Fig. 7). The minimum ensemble size for stability of LETKF is *k* = 15 for *r* = 20 and decreases linearly to *k* = 2 for *r* = 1. Assigning the covariance localization in 3D-Var to match LETKF (i.e., *r* ≡ *r*_{B}), the Hybrid/Mean-LETKF eliminates the catastrophic filter divergence for all localization distances but exhibits increased mean errors that exceed the observation errors for short localization distances. However, when fixing the B-localization at *r*_{B} = 5 and applying 3D-Var globally to ensure a minimum dimension for the solution space, the Hybrid/Mean-LETKF has consistent results regardless of the localization radius *r* used in LETKF.

Multiplicative covariance inflation gives limited benefit to the standard LETKF. As shown in Fig. 8, only a few cases with smaller ensemble sizes are afforded improvements in mean analysis error. When using sufficient observations (*l* = 20), the Hybrid/Mean-LETKF renders inflation unnecessary. However, when using fewer observations (*l* = 4), a small inflation (5%–10%) has either a positive or neutral effect on reducing the mean analysis error.

## 5. 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 LETKF, 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 used a new approach combining gain matrices from LETKF and 3D-Var, and was implemented via the Hybrid/Mean-LETKF algorithm, for which a simple 3D-Var is applied after completion of LETKF to adjust the ensemble mean in model space and to add stability to the filter for small ensemble sizes. While we demonstrated this approach with LETKF and 3D-Var, it is generally applicable to other data assimilation methods as well. A larger value of a tuning parameter *α* enhanced the stability of LETKF at the cost of accuracy. Thus, in practice, a reasonable approach may be to begin with a larger *α* value and gradually decrease *α* as long as diagnostic metrics continue to improve. The optimal value for *α* is dependent on problem specifications, such as the ensemble size, observation coverage, and localization radius.

The filtering solution derived with LETKF is highly accurate when applied to the Lorenz-96 system when allowed a sufficient number of ensemble members. However, when using few members, there is catastrophic filter divergence. Because LETKF computes the analysis in the ensemble subspace, its stability is highly dependent on the ensemble dimension. It was shown that for the primary configuration used in this study, regardless of observation coverage, five ensemble members were insufficient to prevent filter divergence. For a specified level of observation coverage (*l* = 20), this minimum ensemble size was shown to be a linear function of localization radius. While decreasing the localization radius helped stabilize LETKF for small ensemble sizes with Lorenz-96, in practice the minimum localization radius is constrained by the fidelity of the observing network and computational model. A larger localization radius is typically needed for continuity of the analysis field. 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 (2011), we suspect the minimum ensemble size for the standard LETKF is directly related to the local dimensionality of the error growth.

Bishop and Satterfield (2013) found that if an ensemble-based estimate of covariance is undersampled, then a superior estimate can be obtained by combining that with a climatological estimate. Our results support that finding: the Hybrid/Mean-LETKF approach generated solutions that outperformed both 3D-Var and the standard LETKF for observation coverage with 2 < *l* < 10 and ensemble size 1 < *k* < 5. As has been reported for the traditional hybrid methods (Hamill and Snyder 2000; Wang et al. 2007a), we conclude that it is the computation of the analysis in the higher-dimensional model space that stabilizes the Hybrid/Mean-LETKF when using small ensemble sizes.

In operational settings, all practical ensemble sizes are small relative to the dimension of the state space. Both hybrid LETKF methods are well suited for applications using small ensemble sizes and limited observation coverage, the typical situation for global ocean data assimilation (Penny 2011; Penny et al. 2013) and coupled atmosphere–ocean data assimilation. In practice, selection of the Hybrid/Covariance-LETKF versus the Hybrid/Mean-LETKF would depend upon the ease of implementation based on the design of existing operational software.

Success with the Lorenz-96 model does not guarantee success with all models. However, because a similar approach by Kalnay and Toth (1994) was effective on both a Lorenz-63 model and a T62 National Meteorological Center (NMC) model, this gives a positive outlook for use of the Hybrid/Mean-LETKF with more realistic models. As one example, there has been success in preliminary applications of the Hybrid/Mean-LETKF algorithm to NCEP’s operational Global Ocean Data Assimilation System (GODAS). As 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 algorithm is a simple extension that can be adopted to enhance the performance of these LETKF-based data assimilation systems.

## Acknowledgments

I gratefully acknowledge the thoughtful comments from unofficial reviewer Eugenia Kalnay, official reviewer Ross Hoffman, and two anonymous reviewers. Also, I thank Jim Carton, Steven Greybush, Brian Hunt, and Daryl Kleist for discussions that led to this work; David Behringer and NCEP for motivating the Hybrid-LETKF; and Craig Bishop for the helpful discussions on the general role of hybrid methods in data assimilation. This material is based upon work supported by the National Science Foundation under Grant OCE1233942.

## APPENDIX A

### Equivalence of the Hybrid Gain Matrix (Using *β*_{1} = 1, *β*_{2} = *α*, and *β*_{3} = −*α*) and the Hybrid/Mean-LETKF Algorithm

**x**

^{a}is defined by minimizing the variational equation,

*J*equal to zero, we can solve for

**x**

^{a}:

^{B}and

## APPENDIX B

### Equivalence of the Hybrid Gain Matrix (Using *β*_{1} =(1 − *α*), *β*_{2} = *α*, and *β*_{3} = 0) and the Hybrid/Mean-LETKF(b) Algorithm

*J*equal to zero, we can solve for

**x**

^{a}:

^{B}into (A16) to get

*β*

_{1}= (1 −

*α*),

*β*

_{2}=

*α*, and

*β*

_{3}= 0 that

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