## 1. Introduction

Data assimilation addresses the problem of producing useful analyses and forecasts given imperfect dynamical models and observations. The Kalman filter is the optimal data assimilation method for linear dynamics with additive, state-independent Gaussian model and observation errors (Cohn 1997). An attractive feature of the Kalman filter is its calculation of forecast and analysis error covariances, in addition to the forecasts and analyses themselves. In this way, the Kalman filter produces estimates of forecast and analysis uncertainty, consistent with the dynamics and prescribed model and observation error statistics. However, the error covariance calculation components of the Kalman filter are difficult to implement in realistic systems because of (i) their computational cost, (ii) the nonlinearity of the dynamics, and (iii) poorly characterized error sources.

The ensemble Kalman filter (EnKF), proposed by Evensen (1994), addresses the first two of these problems by using ensemble representations for the forecast and analysis error covariances. Ensemble size limits the number of degrees of freedom used to represent forecast and analysis errors, and Kalman filter error covariance calculations are practical for modest-sized ensembles. The EnKF algorithm begins with an analysis ensemble whose mean is the current state estimate or analysis and whose statistics reflect the analysis error. Applying the full nonlinear dynamics to each analysis ensemble member produces the forecast ensemble; tangent linear and adjoint models of the dynamics are not required. Statistics of the forecast ensemble represent forecast errors; in its simplest form, the EnKF only accounts for forecast error due to uncertain initial conditions, neglecting forecast error due to model deficiencies. The forecast ensemble mean and covariance are then used to assimilate observations and compute a new analysis ensemble with appropriate statistics, and the cycle is repeated. The new analysis ensemble can be formed either stochastically (Houtekamer and Mitchell 1998; Burgers et al. 1998) or deterministically (Bishop et al. 2001; Anderson 2001; Whitaker and Hamill 2002). Deterministic methods were developed to address the adaptive observational network design problem and to avoid sampling issues associated with the use of “perturbed observations” in stochastic analysis ensemble update methods.

The EnKF and other ensemble data assimilation methods belong to the family of *square root filters* (SRFs), and a purpose of this paper is to demonstrate that deterministic analysis ensemble updates are implementations of Kalman SRFs (Bierman 1977; Maybeck 1982; Heemink et al. 2001). An immediate benefit of this identification is a framework for understanding and comparing deterministic analysis ensemble update schemes (Bishop et al. 2001; Anderson 2001; Whitaker and Hamill 2002). SRFs, like ensemble representations of covariances, are not unique. We begin our discussion in section 2 with a presentation of the Kalman SRF; issues related to implementation of ensemble SRFs are presented in section 3; in section 4 we summarize our results.

## 2. The Kalman SRF

Kalman SRF algorithms, originally developed for space-navigation systems with limited computational word length, demonstrate superior numerical precision and stability compared to the standard Kalman filter algorithm (Bierman 1977; Maybeck 1982). SRFs by construction avoid loss of positive definiteness of the error covariance matrices. SRFs have been used in earth science data assimilation methods where error covariances are approximated by truncated eigenvector expansions (Verlaan and Heemink 1997).

^{f}

_{k}

^{a}

_{k}

*n*×

*n*forecast and analysis error covariance matrices at time

*t*

_{k}; 𝗠

_{k}is the tangent linear dynamics; 𝗛

_{k}is the

*p*×

*n*observation operator; 𝗥

_{k}is the

*p*×

*p*observation error covariance; 𝗤

_{k}is the

*n*×

*n*model error covariance matrix and 𝗞

_{k}≡

^{f}

_{k}

^{T}

_{k}

_{k}

^{f}

_{k}

^{T}

_{k}

_{k})

^{−1}is the Kalman gain;

*n*is the dimension of the system state; and

*p*is the number of observations. The error covariance evolution depends on the state estimates and observations through the tangent linear dynamics 𝗠

_{k}. The propagation of analysis errors by the dynamics with model error acting as a forcing is described by Eq. (1). Equation (2) shows how an optimal data assimilation scheme uses observations to produce an analysis whose error covariance is less than that of the forecast.

The forecast and analysis error covariance matrices are symmetric positive-definite matrices and can be represented as ^{f}_{k}^{f}_{k}^{fT}_{k}^{a}_{k}^{a}_{k}^{aT}_{k}^{f}_{k}^{a}_{k}*matrix square roots* of ^{f}_{k}^{a}_{k}*m,* there is an *n* × *m* matrix square root 𝗭 satisfying 𝗣 = 𝗭𝗭^{T}; in low-rank covariance representations the rank *m* is much less than the state-space dimension *n.* This representation is not unique; 𝗣 can also be represented as 𝗣 = (𝗭𝗨)(𝗭𝗨)^{T}, where the matrix 𝗨 is any *m* × *m* orthogonal transformation 𝗨𝗨^{T} = 𝗨^{T}𝗨 = 𝗜. The projection ‖**x**^{T}𝗭‖^{2} = **x**^{T}𝗣**x** of an arbitrary *n*-vector **x** onto the matrix square root 𝗭 is uniquely determined, as is the subspace spanned by the columns of 𝗭.

Covariance matrix square roots are closely related to ensemble representations. The sample covariance ^{a}_{k}*m*-member analysis ensemble is given by ^{a}_{k}^{T}/(*m* − 1), where the columns of the *n* × *m* matrix 𝗦 are mean-zero analysis perturbations about the analysis ensemble mean; the rank of ^{a}_{k}*m* − 1). A matrix square root of the analysis error covariance matrix ^{a}_{k}^{a}_{k}*m* − 1)^{−1/2}𝗦.

^{f}

_{k}

^{a}

_{k}

_{k}is neglected, (1) can be replaced by

^{f}

_{k}

_{k}

^{a}

_{k−1}

^{a}

_{k−1}

^{f}

_{k}

^{f}

_{k}

^{a}

_{k}

_{k}𝗛

_{k}) to the forecast perturbation ensemble

^{f}

_{k}

_{k}𝗛

_{k})

^{f}

_{k}

_{k}𝗛

_{k})

^{T}and includes analysis error due to forecast error; the Kalman gain 𝗞

_{k}depends on the relative size of forecast and observation error, and the factor (𝗜 − 𝗞

_{k}𝗛

_{k}) shows how much forecast errors are reduced. However, in this procedure the analysis ensemble does not include uncertainty due to observation error and so underestimates analysis error. A

*stochastic*solution to this problem proposed independently by Houtekamer and Mitchell (1998) and Burgers et al. (1998) is to compute analyses using each forecast ensemble member and, instead of using a single realization of the observations, to use an ensemble of simulated observations whose statistics reflect the observation error. This method is equivalent to the analysis perturbation ensemble update

^{a}

_{k}

_{k}

_{k}

^{f}

_{k}

_{k}

_{k}

_{k}is a

*p*×

*m*matrix whose

*m*columns are identically distributed, mean-zero, Gaussian random vectors of length

*p*with covariance 𝗥

_{k}/

*m.*The perturbed observation analysis equation (4) gives an analysis perturbation ensemble with correct expected statistics:

*deterministic*algorithm for transforming the forecast ensemble into an analysis ensemble with consistent statistics. The “Potter method” for the Kalman SRF analysis update (Bierman 1977) is obtained by rewriting (2) as

*m*×

*p*matrix 𝗩

_{k}≡ (𝗛

_{k}

^{f}

_{k}

^{T}and the

*p*×

*p*innovation covariance matrix 𝗗

_{k}≡

^{T}

_{k}

_{k}+ 𝗥

_{k}. Then the analysis perturbation ensemble is calculated from

^{a}

_{k}

^{f}

_{k}

_{k}

_{k}

_{k}

^{T}

_{k}

_{k}

^{−1}

_{k}

^{T}

_{k}

_{k}is an arbitrary

*m*×

*m*orthogonal matrix. As formulated, the updated ensemble

^{a}

_{k}

^{f}

_{k}

*p*×

*p*matrix 𝗗

_{k}and computing a matrix square root 𝗫

_{k}of the

*m*×

*m*matrix (𝗜 − 𝗩

_{k}

^{−1}

_{k}

^{T}

_{k}

## 3. Ensemble SRFs

### a. Analysis ensemble

*n*and the number of observations

*p*are large, and the method for computing the matrix square root of (𝗜 − 𝗩

_{k}

^{−1}

_{k}

^{T}

_{k}

^{a}

_{k}

*direct*approach is to solve first the linear system 𝗗

_{k}𝗬

_{k}= 𝗛

_{k}

^{f}

_{k}

*p*×

*m*matrix 𝗬

_{k}, that is, to solve

_{k}

^{f}

_{k}

^{ T}

_{k}

_{k}

_{k}

_{k}

^{f}

_{k}

*m*×

*m*matrix 𝗜 − 𝗩

_{k}

^{−1}

_{k}

^{T}

_{k}

_{k}

^{f}

_{k}

^{T}𝗬

_{k}is formed, and its matrix square root 𝗫

_{k}computed and applied to

^{f}

_{k}

*p*is large, is practical when the forecast error covariance has a low-rank representation and the inverse of the observation error covariance is available (see the appendix). Iterative methods whose cost is on the order of the cost of applying the innovation covariance matrix are appropriate when the forecast error covariance is represented by a correlation model.

*serially*(Houtekamer and Mitchell 2001; Bishop et al. 2001). For a single observation,

*p*= 1, 𝗩

_{k}is a column vector, and the innovation 𝗗

_{k}is a scalar. In this case, a matrix square root of (𝗜 − 𝗩

_{k}

^{−1}

_{k}

^{T}

_{k}

*ansatz,*

^{ −1}

_{k}

_{k}

^{ T}

_{k}

*β*

_{k}

_{k}

^{ T}

_{k}

*β*

_{k}

_{k}

^{ T}

_{k}

^{T}

*β*

_{k}, which gives

*β*

_{k}= [𝗗

_{k}± (𝗥

_{k}𝗗

_{k})

^{1/2}]

^{−1}. The analysis ensemble update for

*p*= 1 is

^{a}

_{k}

^{f}

_{k}

*β*

_{k}

_{k}

^{ T}

_{k}

*p*×

*p*matrices. At observation locations, the analysis error ensemble is related to the forecast error ensemble by 𝗛

_{k}

^{a}

_{k}

*β*

_{k}

^{T}

_{k}

_{k})𝗛

_{k}

^{f}

_{k}

*β*

_{k}

^{T}

_{k}

_{k}) has an absolute value less than or equal to one and is positive when the plus sign is chosen in the definition of

*β*

_{k}.

^{a}

_{k}

_{k}

_{k}

^{f}

_{k}

_{k}is a solution of the nonlinear equation

_{k}

_{k}

^{f}

_{k}

_{k}

_{k}

^{T}

^{a}

_{k}

_{k}

_{k}

_{k}

^{1/2}

^{−1}

_{k}

*β*

_{k}

^{f}

_{k}

_{k}

*β*

_{k}. The corresponding analysis perturbation ensemble update,

^{−1/2}

_{k}

_{k}

^{ −1}

_{k}

^{ T}

_{k}

^{fT}

_{k}

^{ T}

_{k}

^{ −1}

_{k}

_{k}

^{f}

_{k}

^{−1}

*m*×

*m*matrix on the right-hand side of (15) is practical to compute when the inverse observation error covariance matrix

^{−1}

_{k}

*p*×

*p*matrix 𝗗

_{k}and is used in the

*ensemble transform Kalman filter*(ETKF) where the analysis update is (Bishop et al. 2001)

^{a}

_{k}

^{f}

_{k}

_{k}

**Γ**

_{k}

^{−1/2}

_{k}

**Γ**

_{k}

^{T}

_{k}

^{fT}

_{k}

^{T}

_{k}

^{−1}

_{k}

_{k}

^{f}

_{k}

_{k}of orthonormal eigenvectors is not uniquely determined.

^{1}Comparison with (15) shows that 𝗖

_{k}(

**Γ**

_{k}+ 𝗜)

^{−1}

^{T}

_{k}

_{k}

^{−1}

_{k}

^{T}

_{k}

_{k}(

**Γ**

_{k}+ 𝗜)

^{−1/2}is a square root of (𝗜 − 𝗩

_{k}

^{−1}

_{k}

^{T}

_{k}

*ensemble adjustment Kalman filter*(EAKF) the form of the analysis ensemble update is (Anderson 2001)

^{a}

_{k}

_{k}

^{f}

_{k}

_{k}is defined by

_{k}

_{k}

_{k}

_{k}

**Γ̃**

_{k}

^{−1/2}

^{ −1}

_{k}

^{ T}

_{k}

^{f}

_{k}

_{k}

^{2}

_{k}

^{T}

_{k}

^{f}

_{k}

_{k}is chosen so that

^{T}

_{k}

_{k}

^{T}

_{k}

^{T}

_{k}

^{−1}

_{k}

_{k}𝗙

_{k}𝗚

_{k}

_{k}=

**Γ̃**

_{k}is diagonal.

^{2}Choosing the orthogonal matrix

_{k}to be

_{k}=

^{−1}

_{k}

^{T}

_{k}

^{f}

_{k}

_{k}gives that

_{k}=

**Γ**

_{k}and that the ensemble adjustment matrix is

_{k}

^{f}

_{k}

_{k}

_{k}

^{−1/2}

^{ −1}

_{k}

^{ T}

_{k}

^{a}

_{k}

^{f}

_{k}

_{k}

**Γ**

_{k}

^{−1/2}

^{ −1}

_{k}

^{ T}

_{k}

^{f}

_{k}

^{−1}

_{k}

^{T}

_{k}

^{f}

_{k}

^{−1}

_{k}

^{T}

_{k}

^{f}

_{k}

^{f}

_{k}

_{k}(𝗜 +

**Γ**

_{k})

^{−1/2}

^{−1}

_{k}

^{T}

_{k}

^{f}

_{k}

_{k}

^{−1}

_{k}

^{T}

_{k}

Beginning with the same forecast error covariance, the direct, serial, ETKF, and EAKF methods produce different analysis ensembles that span the same state-space subspace and have the same covariance. Higher-order statistical moments of the different models will be different, a relevant issue for nonlinear dynamics. The computation costs of the direct, ETKF, and EAKF methods are seen in Table 1 to scale comparably (see the appendix for details). There are differences in precise computational cost; for instance, the EAKF contains an additional singular value decomposition (SVD) calculation of the forecast with cost *O*(*m*^{3} + *m*^{2}). The computational cost of the serial filter is less dependent on the rank of the forecast error covariance and more sensitive to the number of observations. This difference is important when techniques to account for model error and control filter divergence, as described in the next section, result in an effective forecast error covariance dimension *m* much larger than the dynamical forecast ensemble dimension.

### b. Forecast error statistics

In the previous section we examined methods of forming the analysis ensemble given a matrix square root of the forecast error covariance. There are two fundamental problems associated with directly using the ensemble generated by (3). First, ensemble size is limited by the computational cost of applying the forecast model to each ensemble member. Small ensembles have few degrees of freedom available to represent errors and suffer from sampling error that further degrades forecast error covariance representation. Sampling error leads to loss of accuracy and underestimation of error covariances that can cause filter divergence. Techniques to deal with this problem are distance-dependent covariance filtering and covariance inflation (Whitaker and Hamill 2002). Covariance localization in the serial method consists of adding a Schur product to the definition of

_{k}is taken to have large-scale structure, a reasonable representation is an ensemble or square root decomposition, 𝗤

_{k}=

^{d}

_{k}

^{dT}

_{k}

^{d}

_{k}

*n*×

*q*matrix. Then, a square root of

^{f}

_{k}

*n*×

*m*matrix:

^{f}

_{k}

^{a}

_{k}

^{ d}

_{k}

*m*=

*m*

_{e}+

*q*and

*m*

_{e}is the number of dynamically evolved forecast perturbations. With this model error representation, ensemble size grows by

*q*with each forecast–analysis cycle. Ensemble size can be limited by computing the singular value decomposition of the ensemble and discarding components with small variance (Heemink et al. 2001). A larger ensemble with evolved analysis error and model error could be used in the analysis step, and a smaller ensemble used in the dynamical forecast stage. When the model error covariance 𝗤

_{k}is approximated as an operator, for instance using a correlation model, Lanczos methods can be used to compute the leading eigenmodes of 𝗠

_{k}

^{a}

_{k−1}

_{k}

^{a}

_{k−1}

^{T}+ 𝗤

_{k}and form

^{f}

_{k}

^{f}

_{k}

*system simulation,*explicitly accounts for some aspects of model uncertainty (Houtekamer et al. 1996).

## 4. Summary and discussion

Ensemble forecast/assimilation methods use low-rank ensemble representations of forecast and analysis error covariance matrices. These ensembles are scaled matrix square roots of the error covariance matrices, and so ensemble data assimilation methods can be viewed as square root filters (SRFs; Bierman 1977). After assimilation of observations, the analysis ensemble can be constructed stochastically or deterministically. Deterministic construction of the analysis ensemble eliminates one source of sampling error and leads to deterministic SRFs being more accurate than stochastic SRFs in some examples (Whitaker and Hamill 2002; Anderson 2001). SRFs are not unique since different ensembles can have the same covariance. This lack of uniqueness is illustrated in three recently proposed ensemble data assimilation methods that use the Kalman SRF method to update the analysis ensemble (Bishop et al. 2001; Anderson 2001; Whitaker and Hamill 2002). Identifying the methods as SRFs allows a clearer discussion and comparison of their different analysis ensemble updates.

Accounting for small ensemble size and model deficiencies remains a significant issue in ensemble data assimilation systems. Schur products can be used to filter ensemble covariances and effectively increase covariance rank (Houtekamer and Mitchell 1998, 2001; Hamill et al. 2001; Whitaker and Hamill 2002). Covariance inflation is one simple way of accounting for model error and stabilizing the filter (Hamill et al. 2001; Anderson 2001; Whitaker and Hamill 2002). Hybrid methods represent forecast error covariances with a combination of ensemble and parameterized correlation models (Hamill and Snyder 2000). Here we have shown deterministic methods of including model error into a square root or ensemble data assimilation system when the model error has large-scale representation and when the model error is represented by a correlation model. However, the primary difficulty remains obtaining estimates of model error.

The nonuniqueness of SRFs has been exploited in estimation theory to design filters with desirable computational and numerical properties. An open question is whether there are ensemble properties that would make a particular SRF implementation better than another, or if the only issue is computational cost. For instance, it may be possible to choose an analysis update scheme that preserves higher-order, non-Gaussian statistics of the forecast ensemble. This question can only be answered by detailed comparisons of different methods in a realistic setting where other details of the assimilation system such as modeling of systematic errors or data quality control may prove to be as important.

## Acknowledgments

Comments and suggestions of two anonymous reviews improved the presentation of this work. The authors thank Ricardo Todling (NASA DAO) for his comments, suggestions, and corrections. IRI is supported by its sponsors and NOAA Office of Global Programs Grant NA07GP0213. Craig Bishop received support under ONR Project Element 0601153N, Project Number BE-033-0345, and also ONR Grant N00014-00-1-0106.

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## APPENDIX

### Computational Costs

Here we detail the computational cost scalings summarized in Table 1. All the methods require applying the observation operator to the ensemble members to form 𝗛_{k}^{f}_{k}_{k}^{f}_{k}*O*(*mnp*), but may be significantly less when 𝗛_{k} is sparse or can be applied efficiently. We also assume that the inverse observation error covariance ^{−1}_{k}

#### Direct method

- Solve (𝗛
_{k}𝗣 ^{f}_{k} + 𝗥𝗛 ^{T}_{k}_{k})𝗬_{k}= 𝗛_{k} for 𝗬𝗭 ^{f}_{k}_{k}. If 𝗥^{−1}is available, the solution can be obtained using the Sherman–Morrison–Woodbury identity (Golub and Van Loan 1996),and only inverting*m*×*m*matrices. Cost:*O*(*m*^{3}+*m*^{2}*p*). Form 𝗜 − (𝗛

_{k} )𝗭 ^{f}_{k}^{T}𝗬_{k}. Cost:*O*(*pm*^{2}).Compute matrix square root of the

*m*×*m*matrix 𝗜 − (𝗛_{k} )𝗭 ^{f}_{k}^{T}𝗬_{k}. Cost:*O*(*m*^{3}).Apply matrix square root to 𝗭

_{f}. Cost:*O*(*m*^{2}*n*).

Total cost: *O*(*m*^{3} + *m*^{2}*p* + *m*^{2}*n*).

#### Serial method

For each observation:

Form 𝗗. Cost:

*O*(*m*).Form 𝗜 −

*β*𝗩𝗩^{T}and apply to . Cost:𝗭 ^{f}_{k}*O*(*nm*).

Total cost: *O*(*mp* + *mnp*).

#### ETKF

Form 𝗭

^{fT}𝗛^{T}𝗥^{−1}𝗛𝗭^{f}. Assume 𝗥^{−1}inexpensive to apply. Cost:*O*(*m*^{2}*p*).Compute eigenvalue decomposition of

*m*×*m*matrix. Cost:*O*(*m*^{3}).Apply to 𝗭

^{f}. Cost:*O*(*m*^{2}*n*).

Total cost: *O*(*m*^{2}*p* + *m*^{3} + *m*^{2}*n*).

#### EAKF

Cost in addition to ETKF:

Eigenvalue decomposition of 𝗣

^{f}(low rank). Cost:*O*(*m*^{2}*n*+*m*^{3}).Form 𝗙

^{T}𝗭^{f}. Cost:*O*(*m*^{2}*p*).

Total cost: *O*(*m*^{2}*p* + *m*^{3} + *m*^{2}*n*).

Summary of analysis ensemble calculation computational cost as a function of forecast ensemble size *m,* number of observations *p,* and state dimension *n*

^{1}

For instance, the columns of 𝗖_{k} that span the (*m* − *p*)-dimensional null space of ^{fT}_{k}^{T}_{k}^{−1}_{k}_{k}^{f}_{k}*p* is less than the ensemble size *m.*

^{2}

The appearance of ^{−1}_{k}^{f}_{k}_{k} is *m*′ × *m*′ and 𝗙_{k} is *n* × *m*′ where *m*′ is the number of nonzero eigenvalues of ^{f}_{k}

^{*}

International Research Institute for Climate Prediction Contribution Number IRI-PP/02/03.