1. Introduction
Variational data assimilation (DA) (Le Dimet and Talagrand 1986; Talagrand and Courtier 1987) and the ensemble Kalman filter (EnKF; Evensen 1994) are two major advanced approaches for atmospheric DA. The variational DA approach has been successfully used at many operational numerical weather prediction (NWP) centers, first as the three-dimensional variational data assimilation (3DVar) then the four-dimensional variational data assimilation (4DVar) method (e.g., Parrish and Derber 1992; Lorenc et al. 2000; Rabier et al. 2000; Rawlins et al. 2007; Tanguay et al. 2012). Usually, 3DVar uses a static, flow-independent, climatological background error covariance (BEC) that is often spatially homogeneous and anisotropic (e.g., Parrish and Derber 1992; Purser et al. 2003a). Even though some efforts have been made to introduce spatially inhomogeneous, anisotropic BEC into the 3DVar framework (e.g., Wu et al. 2002; Purser et al. 2003b, Fisher 2003), the determination of flow-dependent BEC remains a major challenge. Compared to 3DVar, the 4DVar method allows the fitting of the model forecast trajectory to observations distributed over a period of time so as to provide more accurate model state estimations that are also more consistent with the prediction model. The use of the model as a strong constraint within the DA system also allows for the retrieval of unobserved state variables from limited observations (e.g., Sun and Crook 1997). 4DVar also implicitly evolves the BEC within the DA window so that it can be flow dependent, but the BEC itself defined at the beginning of the DA window is usually static and is not propagated from one DA window to another (Lorenc 2003).
EnKF estimates flow-dependent BEC from a set of ensemble forecasts and updates the ensemble states based on an optimal linear estimation algorithm (Evensen 1994). Because EnKF estimates and evolves flow-dependent BEC within and through the data assimilation cycles, and does not require tangent linear or adjoint model, EnKF has become increasingly popular within both research and operational communities. EnKF and its variants, including the ensemble transform Kalman filter and local ensemble transform Kalman filter, ensemble adjustment Kalman filter, and ensemble square root filter (Bishop et al. 2001; Anderson 2001; Hunt et al. 2007; Whitaker and Hamill 2002), have now been implemented at a number of operational NWP centers (e.g., Houtekamer and Mitchell, 2005; Whitaker et al. 2008; Hamill et al. 2011a).
The EnKF method can also be extended into the time dimension to assimilate observations distributed over a period of time and such algorithms usually rely on BECs across times to update the state at the analysis time. Four-dimensional EnKF algorithms include those of Hunt et al. (2004), Sakov et al. (2010), and S. Wang et al. (2013), based on variants of EnKF. Computationally, a pure ensemble-based DA system tends to be more scalable than traditional 4DVar because the ensemble forecasting portion of the system that tends to be most expensive can be easily parallelized while the time integrations of the tangent linear and adjoint models involved in the traditional 4DVar have to be performed sequentially across the minimization iterations. Further discussions on the advantages and disadvantages of 4DVar and EnKF can be found in Lorenc (2003), Kalnay et al. (2007a,b), and Gustafsson (2007).
The BEC matrix derived from an ensemble of forecasts that is typically much smaller in size compared to the number of the degrees of freedom of typical NWP models is severely rank deficient. Techniques, such as covariance localization, have been proposed to partially alleviate this problem (e.g., Burgers et al. 1998; Houtekamer and Mitchell 1998; Hamill et al. 2001; Anderson 2001; Whitaker and Hamill 2002; Evensen 2003). In comparison, the static climatological BEC typically used by 3DVar is full rank. The static BEC, being derived from typically much larger samples often contains useful balance information of less transient flows. From such considerations, the use of “hybrid”1 BEC that linearly combines static and ensemble-derived flow-dependent BECs had been proposed. The hybrid BEC can overcome limitations of ensemble BEC that spans only the space occupied by the ensemble itself.
Hamill and Snyder (2000) were the first to test this idea within a 3DVar framework, with an algorithm that they called hybrid EnKF–3DVar scheme. They demonstrated this method with a low-resolution quasigeostrophic model and simulated data in a perfect model setting and found that the static BEC is helpful to the analysis when flow-dependent BEC is derived from a small ensemble. For their implementation, the ensemble-derived BEC was explicitly calculated and stored, and combined with the static BEC in their 3DVar framework; while this implementation is attractive for low-dimension problems because it is simple and straightforward, it would become prohibitively expensive for a real NWP model when the ensemble BEC matrix is huge.
Lorenc (2003) proposed a more elegant alternative hybrid algorithm that utilizes a set of extended control variables preconditioned by the ensemble perturbations in the variational cost function. Also, a correlation function is used to localize the ensemble covariance. Wang et al. (2007) demonstrated that this extended control variable formulation is mathematically equivalent to that of Hamill and Snyder (2000) in the 3D framework.
There are some advantages when utilizing the ensemble-derived covariance within a variational framework, as in the aforementioned hybrid algorithms, when compared to pure EnKF. The variational formulation allows for the application of BEC localization in the state space rather than observation space; the latter has problems with observations whose forward operators are nonlocal (Campbell et al. 2010). Additional benefits include easier implementation of equation constraints (Gauthier and Thépaut 2001) and bias correction in the variational framework (Dee and Uppala 2009). In general, variational algorithms that incorporate ensemble-derived BECs are called ensemble–variational or EnVar algorithms, and the hybrid algorithms of Hamill and Snyder (2000) and Lorenc (2003) are such examples. Ensemble-derived BECs have been shown to improve DA in 3DVar frameworks for operational global (Buehner 2005; Hamill et al. 2011b; X. Wang et al. 2013; Kleist and Ide 2015a) and regional (Pan et al. 2014) models.
More recently, EnVar algorithms have been extended into the fourth time dimension to form 4D EnVar algorithms. There are two basic types of such algorithms. One is a direct extension of the 3D extended control variable algorithm of Lorenc (2003) into four dimensions, which can also be considered as introducing the ensemble BEC into a standard 4DVar framework using the extended control variable approach. Being based on the standard 4DVar that involves the integration of tangent linear and adjoint models, we call this algorithm En4DVar. Buehner et al. (2010a,b) applied the extended control variable approach to a 4DVar framework (denoted 4D-Var-Benkf in their papers) and tested with real observations in the Canadian operational global model, and found positive impact of the ensemble-derived BEC. Very recently, a hybrid ensemble–4DVar system is implemented for the Met Office operational global model (Clayton et al. 2013). Zhang and Zhang (2012) also applied the extended control variable approach to a 4DVar framework of a regional research model. Kuhl et al. (2013) implemented En4DVar within the Naval Research Laboratory Atmospheric Variational Data Assimilation System-Accelerated Representer (NAVDAS-AR) data assimilation framework. It was found that the forecast error was significantly reduced by their En4DVar system. These systems were all built on existing 4DVar capabilities that already have an adjoint model, and correspond to the algorithm that we call En4DVar in this paper.
The second type of algorithm formulates a 4D variational cost function that projects the ensemble perturbations to the observation space so that tangent linear model and adjoint model can be avoided in the absence of the static BEC term (Liu et al. 2008). In Liu et al. (2009), a localized matrix is introduced into the algorithm for ensemble covariance localization and Liu and Xiao (2013) further applied it to real data problems. This algorithm was originally called En4DVar but is renamed 4DEnVar in Liu and Xiao (2013) to better distinguish it from algorithms that are more closely linked to the traditional 4DVar and include the integration of an adjoint model (Lorenc 2013). Following Liu and Xiao (2013), 4DEnVar is also used to refer to this type of algorithms in this paper while En4DVar is used to refer to algorithms involving the integration of an adjoint model. In Buehner et al. (2010a,b), a version of the 4DEnVar algorithm of Liu et al. (2008, 2009) was implemented within a global spectral model (called En-4D-Var in their papers) and compared with traditional 4DVar, EnKF, and 4D-Var-Benkf methods. These data assimilation schemes had a similar performance in the northern extratropics and tropics. In the southern extratropics, En-4D-Var was slightly better than EnKF but slightly worse than 4D-Var-Benkf. Here we point out that the En-4D-Var implementation of Buehner et al. (2010a,b) corresponds to the algorithm that we will call 4DEnVar-NPC in this paper. Buehner et al. (2013) further compared En-4D-Var (corresponding to our 4DEnVar-NPC), 3DVar, and 4DVar for global weather prediction. They found that En-4D-Var is always better than 3DVar and is either similar or better than 4DVar in the tropical troposphere and the winter extratropical regions. Kleist and Ide (2015b) evaluated hybrid 4DEnVar with various initialization techniques within the National Centers for Environmental Prediction Global Data Assimilation System (GDAS) using simulated data. They found that the hybrid 4DEnVar can reduce analysis error for most variables at most levels, especially in the extratropics, compared to hybrid 3DEnVar.
Despite the successful applications of the various 4D ensemble–variational approaches, their relationships, as well as the approximations involved in their implementations, are still unclear. Desroziers et al. (2014) used a generalized variational formulation in terms of a 4D state vector to discuss different possible implementations of 4DEnVar. They proposed two new preconditioned algorithms and compared 4DEnVar and 4DVar for a Burgers equation model. However, approximations related to covariance localization in a 4D space was not discussed in detail in their paper. Fairbairn et al. (2014) pointed out that no explicit localization of the correlations in time was included in their experiments, or in most other implementations of 4DEnVar.
Most recently, Lorenc et al. (2015) compared hybrid-4DVar (which is our hybrid En4DVar) and hybrid-4DEnVar (which is actually our hybrid 4DEnVar-NPC). Hybrid-4DVar was found to perform better than hybrid-4DEnVar in the Met Office global operational system. They suggested the fact that the hybrid-4DVar evolves the static background error covariance within the assimilation window while the hybrid-4DEnVar does not was the main cause of the differences. It is the purpose of this paper to clarify the relationships and understand the approximations involved in the derivations and implementations of various 4D ensemble–variational algorithms. We present various derivative algorithms based on the two approaches in a common framework that can incorporate both static and ensemble-derived BECs. An understanding is also sought on the effects of various approximations made in the algorithms. Furthermore, in order to introduce static BEC to 4DEnVar while still avoiding an adjoint model, we propose the use of the first guess at the appropriate time (FGAT) approximation within the 4D hybrid schemes.
The organization of the paper is as follows. Four 4D ensemble–variational algorithms are first derived and discussed in a common framework in section 2. FGAT formulations for three hybrid algorithms that avoid adjoint model are also introduced. Section 3 provides mathematical proofs of the relationships among En4DVar and 4DEnVar algorithms. Section 4 presents results from single-observation experiments for a one-dimensional linear advection model to demonstrate the behaviors and relationships of the algorithms and illustrate the effects of covariance localization. A summary and some further discussion are provided in the conclusions.
2. Four-dimensional ensemble–variational algorithms
a. Hybrid En4DVar with extended control variable
b. 4DEnVar-NPC
In Eqs. (14) and (15), adjoint model and tangent linear model are avoided.2 With the NPC approximation mentioned earlier, we name the algorithm more specifically as 4DEnVar-NPC, which is consistent with 4DEnVar in Lorenc et al. (2015). We label this algorithm 4DEnVar instead of En4DVar because it does not involve the integration of the tangent linear model or adjoint, and is closer to EnVar than to 4DVar. When all observations are taken at the analysis time, no time integration of the prediction model is involved and the algorithm becomes 3D and is called 3DEnVar.3 The issue of propagating the control variable also disappears.
If the observation operator is linear and when spatial localization is configured to be the same, the pure 3DEnVar is the same as the ensemble mean analysis from an EnKF using the same set of ensemble perturbations. Note that even though Eq. (13) is considered an approximation to 
c. 4DEnVar
d. 4DEnVar-NPL
In the original 4DEnVar cost function in Eq. (20), tangent linear model 
We note there that even though the NPL approximation was used in the implementation of Liu et al. (2009), and Liu and Xiao (2013), who proposed the original 4DEnVar algorithm, the NPL approximation and its implications were not explicitly described there.
e. 4D hybrid schemes with FGAT
Although the 4DEnVar and 4DEnVar-NPC algorithms with ensemble covariance only do not need an adjoint model, when the static BEC is included, Eq. (10) has to be used, which does involve adjoint model 
It is also equivalent to assuming 
We note there that the use of the FGAT strategy to avoid adjoint model in a hybrid 4DEnVar system is also used in a very recent paper of Lorenc et al. (2015). Such a choice is perhaps not surprising given that a key benefit of 4DEnVar compared to 4DVar or En4DVar is the elimination of the need to develop and maintain an adjoint of a full NWP model. Such an idea was also presented in Liu and Xue (2013). Related to this issue, in the generalized 4D hybrid framework of Desroziers et al. (2014), it was mentioned in passing that the climatological background error could be static in time (within the assimilation window). This is equivalent to making the FGAT approximation but their paper did not directly refer to the FGAT terminology. In Kleist and Ide (2015b), it was also pointed that FGAT-like approximation can be made in an En4DVar system.
f. Temporal localization in 4D ensemble-based algorithms
One aspect that has not been discussed so far is the need for ensemble BEC localization in time, which arises because of the existence of noise in the covariances calculated between ensemble perturbation states of different times due to the limited ensemble size. Temporal localization should in general be applied in all 4D ensemble-based algorithms, and has been used in, for example, the 4D ensemble square root filter algorithm of S. Wang et al. (2013). Placing the analysis time at the middle of assimilation window (when the algorithm does not involve adjoint model integration) can help somewhat, as was done in Liu et al. (2009).
Temporal localization in 4DEnVar and En4DVar algorithms can be achieved by multiplying the localization matrix 
3. Equivalence among the ensemble–variational algorithms
We show in this section that some of the ensemble–variational algorithms presented above are actually mathematically equivalent.
a. Equivalence of En4DVar and 4DEnVar
Both schemes apply the full covariance localization without approximation, and the tangent linear model is required for their full implementation. If the static background error covariance is not involved, an adjoint model is not required by 4DEnVar but is still required by En4DVar. The need for an adjoint model due to the static BEC terms can be removed by applying the FGAT approximation, as discussed earlier.
b. Equivalence of 4DEnVar-NPC and 4DEnVar-NPL
We have therefore proven that 4DEnVar-NPC and 4DEnVar-NPL are mathematically equivalent, and the approximations made in these algorithms to avoid tangent linear and adjoint models are also effectively the same; they both sacrifice the time-evolving or flow-following aspect of the covariance localization.
All algorithms and equivalent proof can be rewritten in a nonpreconditioning variational format, just like the formula for the Gridpoint Statistical Interpolation analysis system (GSI; Wu et al. 2002; Kleist et al. 2009). The equivalent proofs are still the same, which we will not address repeatedly here.
4. Single-observation tests with a one-dimensional linear advection model
The previous section demonstrated through mathematical derivation the equivalence of two groups of ensemble–variational algorithms with and without covariance localization approximations. In this section, we further demonstrate through simple numerical experiments other such equivalences. For all the experiments, the static BEC is excluded, and, therefore, we focus on the treatment of the ensemble covariance and the associated covariance localization effects. For this reason, 4DEnVar-FGAT is not considered here.
The observation at the initial or ending time is assimilated by each of the 4D ensemble–variational methods discussed above. The tangent linear and adjoint models of the advection equation needed by the En4DVar are developed based on the discrete model. For this linear system, the tangent linear model is the same as the finite-difference prediction model. Because all algorithms without localization have the same cost function as in Eq. (16), a reference analysis without localization (Fig. 1) is obtained by minimizing Eq. (16). In all cases, analysis at the initial time (i.e., at the beginning of the DA window) is sought, just like in a standard 4DVar. When the single observation is located at the initial time, there is no need for time integration. Therefore, all algorithms degenerate to 3D algorithms. In this case, it is easy to find that all cost functions are the same assuming that the same spatial localization is applied at the initial time. Figure 1 shows that there is significant noise at places far from the observation when no localization is performed because of the covariance sampling noise. When using a spatial localization, a clean Guassian-shaped analysis increment symmetric around the observation point is obtained (black line in Fig. 1), and the increments obtained by En4DVar and 4DEnVar are the same; they are indistinguishable in Fig. 1.
Analysis increment at the initial time when assimilating a single observation located at gridpoint 50 at the initial time. The thin dotted line denotes the observation location, the red line indicates the analysis increment without localization, and the black line is for the analysis increment with localization.
Citation: Monthly Weather Review 144, 2; 10.1175/MWR-D-15-0203.1
When the single observation is located at the end of the DA window, the largest analysis increment at the initial time should be found at about (160 × 0.001 × 2
Analysis increments at the initial time assimilating a single observation located at the 50th grid point at the ending time, when no localization is used (red line), when using flow-following localization (black line), and when using non-flow-following localization (blue line).
Citation: Monthly Weather Review 144, 2; 10.1175/MWR-D-15-0203.1
As can be seen, consistent with the mathematical proofs, the analyses of En4DVar and 4DEnVar are the same while the analyses of 4DEnVar-NPC and 4DEnVar-NPL are the same. When the advection speed is lower or the required spatial localization scale is larger, the effects of localization approximations would have smaller effects and vice versa. When the advection speed is doubled in the next set of experiments, the upstream shift of the correct increment peak is also doubled, as given by the flow-following algorithms (black line in Fig. 3). Limited by the non-flow-following localization centered at the 50th grid point, the analysis increment upstream of grid point 40 is severely damped. When the spatial localization scale (as well as the spatial decorrelation scale used by the initial perturbations in our tests) is halved, the analysis increment becomes smaller and the negative effects of the non-flow-following localization become larger (blue line in Fig. 4). In practice, when no better solution is available (i.e., when a flow-following localization algorithm is not available or too expensive to implement), the optimal radius of non-flow-following localization used in the approximated 4D algorithms is expected to be larger than that of the corresponding 3D algorithm in order to better accommodate the propagation effects. Flow-following localization in ensemble data assimilation remains an unsolved problem that requires more research.
As in Fig. 2, but for doubling the advection speed to 4
Citation: Monthly Weather Review 144, 2; 10.1175/MWR-D-15-0203.1
As in Fig. 2, but for half of the localized radius and the error spatial decorrelation scale of the initial ensemble perturbations.
Citation: Monthly Weather Review 144, 2; 10.1175/MWR-D-15-0203.1
5. Summary and discussion
Flow-dependent BECs have been shown to be valuable for improving the quality of state estimation for atmospheric and oceanic as well as other geophysical flows in the past decade. Introducing ensemble-derived flow-dependent BEC into 3D and 4D variational DA frameworks has advantages. Instead of directly calculating the ensemble covariance matrix and using it inside a variational framework, which is computationally impractical for full atmospheric models, Lorenc (2003) proposed an extended control variable approach that introduces the ensemble BEC through an additional extended control variable “background” term in a 3DVar cost function (named En3DVar or 3DEnVar here). The formulation can be easily extended into a 4DVar framework (named En4DVar). The ensemble covariance localization can be achieved by introducing a correlation matrix that “preconditions” the extended control variable and its effect can be achieved by applying a recursive filter. The need to apply tangent linear and adjoint models in En4DVar also carries high computational costs.
An alternative approach for utilizing ensemble BEC within a 4D variational framework was proposed by Liu et al. (2008; 2009), which projects the ensemble perturbations to the observation space so that the tangent linear and adjoint models can be avoided. Their original formulations (called 4DEnVar here) did not include the static BEC part in the 4D cost function. Liu et al. (2009) introduced a large localization matrix to modify the ensemble perturbations before they are used so as to achieve ensemble covariance localization, a procedure that is also computationally very expensive.
In this paper, the Liu et al. (2009) formulation is extended to include the static BEC to form a hybrid system. This observation-space-perturbation 4DEnVar formulation is compared with the extended control variable En4DVar formulation. It is shown that before any approximation is made with the localization treatment, the two formulations are mathematically equivalent. The control variable w introduced by Liu et al. (2009) is the same as the transformed extended control variable 
Approximations are then introduced into the En4DVar algorithm based on the extended control variable so that tangent linear model integrations on the extended control variable are avoided, and the time evolution of the ensemble perturbations are provided by the ensemble forecasts. This approximate formulation is called 4DEnVar-NPC, because there is no propagation of the extended control variable and the algorithm does not inherently require an adjoint model (except when static BEC is included in the hybrid formulation). Approximations which avoid separate integrations on localized ensemble perturbations are introduced into the original 4DEnVar algorithm, resulting in the approximate 4DEnVar-NPL formulation, where label NPL indicates no propagation of the localization matrix in time. This paper proves that 4DEnVar-NPC and 4DEnVar-NPL are also mathematically equivalent.
All algorithms can include static BEC to form hybrid algorithms but this inclusion would make adjoint model integration necessary. To address this issue, the FGAT approximation is introduced to the static BEC portion of the hybrid En4DVar and 4DEnVar formulations, including the 4DEnVar-NPC, 4DEnVar, and 4DEnVar-NPL algorithms. This approximation avoids the adjoint model while still allowing for the use of observations distributed over a time window in the static portion of the cost function (they can already be used in the ensemble BEC portion), although the formulation is no longer truly 4DVar in the traditional sense. With the FGAT approximation, the static BEC, unlike in the traditional 4DVAR, is no longer implicitly evolved in time or flow following within the assimilation window. A comparison of the pure 4D ensemble–variational algorithms discussed in this paper is given in Table 1.
A comparison of four-dimensional ensemble–variational algorithms without a static background error term.
Single-observation tests for a one-dimensional linear advection system are performed to confirm the mathematical equivalence of the algorithms, and to examine the effects of the localization approximations. When the flow speed is low or the desired BEC localization scale is large, the effects of the non-flow-following localization approximations are smaller and vice versa. Attempts have been made in the literature (e.g., Bishop and Hodyss 2007, 2009, 2011) to realize flow-dependent covariance localization in an ensemble framework, but all schemes proposed so far are computationally very expensive. Ota et al. (2013) estimated observation impact by a flow-following localization within EnKF framework. However, how to realize effective and efficient flow-following covariance localization in a 4D variational DA framework is still a major and important area for future research, and neglecting the following aspect of covariance localization will remain an important source of approximations before effective solutions are found.
Acknowledgments
This research was primarily supported by NSF Grants AGS-0802888 and OCI-0905040, and the NOAA Warn-on-Forecast Grant NA080AR4320904. The second author was also supported by NSF Grants AGS-0941491 and AGS-1046171. Computations were carried out on the CAPS Linux Cluster machines.
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In this study, we use the word “hybrid” to mainly refer to the use of a combination of the static and ensemble-derived flow-dependent covariances (i.e., the hybrid covariance). We do not intend to use “hybrid” to refer to any algorithm, although in the literature it had sometimes been used to refer to ensemble–variational algorithms.
Note that when the static BEC term is included, as in Eqs. (9) and (10), the tangent linear model and adjoint model will again be needed even with the approximations.
Unlike the 4D case, for a 3D formulation, the inclusion of the static BEC term in the cost function does not necessitate the use of a tangent linear model or adjoint model, a 3DEnVar can easily include the static BEC term to use hybrid covariance. For this reason, the algorithm can also be called En3DVar.
In principle, one can implement iterative outer loops as with the regular variational algorithms; in that case, additional nonlinear ensemble integrations will be needed for each outer loop iteration.