## 1. Introduction

Coupled ocean–atmosphere data assimilation (DA) is an area of active research that has the potential to provide balanced analyses for coupled forecast and reanalysis, to better extract information from currently rejected observations that are sensitive to both ocean and atmosphere (such as low-peaking infrared radiances), and to provide additional observational constraints in data-sparse regions through cross-fluid covariances. While coupled DA is promising, it is a nontrivial estimation problem with unique challenges.

A key theoretical challenge for coupled DA is that an exhaustive solution to coupled DA requires specification of a forecast error covariance matrix between all model variables and all observations in both fluids. It is certain that we will not be able to specify such a complicated covariance perfectly. Using a simplified toy model, Han et al. (2013) showed that when cross-fluid covariances are specified poorly, the quality of the fully coupled DA can be degraded compared to the quality of noncoupled (multicomponent) DA. In other words, imposing additional observational constraints using a suboptimal error covariance is likely to degrade our ability to fit resident (within fluid) data and, ultimately, can degrade the accuracy of the analysis. Hence, for a coupled DA to succeed, it is critical to develop algorithms that minimize the detrimental effect of poorly specified covariances and provide greater flexibility for tuning of the error covariance models.

From a technological viewpoint, coupled DA presents both a computational and a software development challenge. Computationally, the increase in the number of state variables and measurements leads to a faster-than-linear increase in the computational cost of DA. Additionally, DA time windows over which observations are assimilated to make an analysis is much longer for the ocean (1–10 days) than the atmosphere (1–24 h). The longer time window in the ocean is justified by the fact that errors grow far less rapidly in the ocean than the atmosphere and by the greater delay in the availability of oceanic measurements for assimilation. (While we hope that this delay will be shortened in the future by technological advances, for the moment, practical real-time forecasting systems must accommodate such delays.) If one were to use an atmospheric time window as long as a typical oceanic time window, the exponential growth of errors assumed by the atmospheric tangent linear model is unlikely to be realistic at such long lead times, leading to unrealistically high error variances at the end of the assimilation window and ill-conditioned covariance matrices. Such ill-conditioning might be particularly problematic for variational schemes that either may fail to converge or only converge at a prohibitive computational cost.

From a software-development perspective, developing a new fully coupled DA solver might require discarding existing highly specialized software for single-fluid DA. Existing DA software often accumulates decades of research and careful tuning to a specific forecasting problem. Any replacement of such a highly tuned system might be challenged to initially produce analysis of equal or higher quality, which is often a requirement for transition of a research system into operations.

Several approaches to coupled DA are currently investigated by the research community. Most of the weather forecast centers have implemented “weakly” coupled DA systems where the coupled model is used as a first guess and existing (uncoupled) DA systems are used to inject observations into the system (Saha et al. 2010; MacLachlan et al. 2015; Smith et al. 2013; Shelly et al. 2014; Chen et al. 2010). The developed systems show improvements in representing coupled phenomena (such as the Madden–Julian oscillation and El Niño–Southern Oscillation, tropical monsoon, and hurricanes). However, since these studies used weakly coupled DA systems, we speculate that most of these improvements can be attributed to the improvements in the forward model rather than the DA system. Alternatively, the European Centre for Medium-Range Weather Forecasts (ECMWF) recently implemented an approximation to the strongly coupled DA system by introducing outer loops to the weakly coupled system (Laloyaux et al. 2015). Their results show that outer loop can successfully propagate information in between fluids. However, the ECMWF approach also represents only an approximation to a fully coupled system.

In this paper, we develop the interface solver approach—a new approximation to the strongly coupled DA that addresses the challenges for a coupled DA system outlined above. In the case of coupled ocean–atmosphere DA, the interface solver is formulated as two independent, yet coupled in the innovations vector, DA systems. Each system solves the coupled problem for one of the fluids (ocean or atmosphere). However, the coupled innovation provided to each system contains only the set of measurements that pertain to this system, such as measurements resident to this fluid and some interface measurements from nonresident fluid. Since only the minimal set of coupled observations is used, we can avoid the detrimental effect of imposing additional observational constraints with a poorly specified error covariance model. To address the difference in time scales in two fluids, each system can use differing data assimilation time window or the leading averaged coupled covariance update method of (Lu et al. 2015). Furthermore, each component of the parallel system can be developed based on the existing DA system for this fluid. The missing cross-fluid covariances can be specified using coupled ensemble covariances. The proposed interface solver approach preserves the existing investment in highly specialized single-fluid DA software, can be more computationally efficient than the exhaustive solution, and allows more flexibility in specifying the poorly known error covariances.

The proposed approach is similar to the divided state-space strategy of Luo and Hoteit (2014) in that the state space is divided into two parts that are coupled in the state vector. However, Luo and Hoteit (2014) suggested the use of all measurements from the coupled fluid, while we only intend to use a limited set of interfacial observations that we believe are related to the increments in the target fluid. Our selection should lead to important efficiencies and an implicit localization of unused observations. This paper also proposes a viable cross-fluid localization function and tests the developed ideas using a realistic mesoscale modeling system, unlike the paper by Luo and Hoteit (2014) that used an idealized one-dimensional model.

To test the proposed approach, we develop an ensemble-variational solver for a coupled, limited-area model of the Mediterranean Sea (see Fig. 1). We choose to use an ensemble-based method to specify coupled covariances because they provide a readily available way to specify covariances between fluids without the need to develop tangent linear and adjoint model approximations for a coupled system. We choose to use the ensemble-variational framework (Buehner et al. 2010a; Kuhl et al. 2013; Clayton et al. 2013) in order to reuse existing 3DVAR solvers for both the ocean and the atmosphere and to enable the use of hybrid error covariances.

This paper progresses in four parts. We first present the theory of the interface solver and coupled Gaspari–Cohn localization. Then we describe the setup for our proof-of-concept experiment followed by the presentation of the experimental results. We conclude with the interpretation of the results and limitations of our experimental framework. The details of the ensemble-variational solver are presented in the appendix.

## 2. Methods

### a. Exhaustive solver for the coupled estimation problem

*A*(atmosphere) (e.g., radiosonde observations),

*O*(ocean) (e.g., Argo floats), and

*AO*(both fluids) [e.g., satellite radiances sensitive to both sea surface temperature (SST) and lower atmosphere];

**z**above is the intermediate solution vector in the space of coupled observations. Equations (4) and (5) are the postmultiplications that project the observation-space vector into the space of the atmospheric and oceanic model grid points.

The system of Eqs. (3)–(5) specifies an exhaustive solution to the coupled DA problem. For a given ^{f}, the system of Eqs. (3)–(5) can be solved by any of the existing generic DA methods including 4D variational methods (Le Dimet and Talagrand 1986), 3D variational methods (Daley and Barker 2001), pure ensemble solvers (Hunt et al. 2007; Anderson 2001), and ensemble-variational hybrids (Buehner et al. 2010a; Kuhl et al. 2013; Clayton et al. 2013). In the following section, we derive an alternative to the exhaustive solution in Eqs. (3)–(5) that provides practical advantages that are likely to result in more accurate state estimates.

### b. Interface solver

*A*

_{bnd}stands for the subset of atmospheric states closely coupled with the ocean (such as the atmospheric boundary layer winds and temperatures) and

*O*

_{bnd}stands for the oceanic states that are closely coupled with the atmosphere (such as SSTs and temperature in the mixed layer). The implication of the sparsity pattern in Eq. (6) is that only a small subset of observations that touch the boundary layer of the coupled fluid will have a substantial impact on the target fluid. For example, only the impact of observations in the boundary layer in atmosphere might need to be considered for the impact on the ocean.

It is important to note that Eqs. (7) and (8) are just approximations to the exhaustive solution in Eqs. (3)–(5). In fact, the intermediate solution vectors

### c. Coupled forecast error covariance model

^{climo}is the 3DVAR error covariance from one of the existing 3DVAR systems (Cummings and Smedstad 2013; Daley and Barker 2001);

^{loc}is the ensemble localization matrix described in section 2d;

^{ens_raw}is the raw ensemble matrix;

*α*is the weighting factor. Our preliminary experiments (not shown here) showed that air–sea coupling had a strong episodic nature and, hence, we set the climatological cross covariances (

### d. Localization of the coupled ensemble

^{loc}in Eq. (9), we specify localization function

^{loc}matrix in Eq. (9); and

^{ens_raw}and

^{ens}matrices, respectively). The location of each variable

*x*,

*y*, and

*z*for Cartesian coordinates in space,

*t*for time, and an integer index

*υ*that identifies the type of a variable (e.g., ocean salinity or atmospheric temperature).

*υ*along one of the coordinate directions. We define

*x*,

*y*, and

*t*directions as the average of the scales in each fluid, similar to intervariable correlations specified in the 3DVAR code developed by Daley and Barker (2001) and Cummings and Smedstad (2013). We choose to normalize the vertical distance separately in ocean and atmosphere to capture the information about how far the variable locations are from the fluid interface. Notice that when points

*υ*

_{A}=

*υ*

_{O}), Eq. (13) reduces to the univariate Eq. (12). Our computational experiments with a wide variety of observing network configurations showed that the localization function in Eqs. (10)–(13) remained positive definite for a wide range of localization scales

Figure 2 shows four examples of the cross-fluid localization functions based on Eq. (13): two for observations locations in pseudo-atmosphere (Figs. 2a,b) and two for pseudo-ocean (Figs. 2c,d). For example, Fig. 2b shows localization of a scatterometer-like observation and Fig. 2c shows that of an SST-like observation. Figure 2 shows that both the horizontal and vertical extent of the localization function change as the function crosses the fluid interface (*z* = 0).

## 3. Experimental framework

### a. Perfect-twin model experiments

To test the effectiveness of the proposed interface solver, we designed a perfect-twin model experiment based on the coupled model of the Mediterranean Sea (described in section 3b below). Using the ensemble transform method (described in section 3c below) and the coupled model we generated 20 cycling ensemble members that were centered on the control run that was constrained using real observations over the modeling domain that were assimilated every 6 h (Fig. 3a). Observations for the control run were assimilated using the weakly coupled DA system.

Figure 4 shows that during the period of our experiment, the Mediterranean domain experienced a strong burst of mistral winds (middle row) followed by the relaxation of the circulation (bottom row). As a result of this wind event a basin-wide cooling of the sea surface and mixed layer temperature was documented (cf. the trend of ocean temperatures shown in the central and right columns). In addition to the cooling and weakening of the mixed layer, a strong vertical mixing event occurred in the Gulf of Lions (well seen in the ocean transects around 42°N latitude).

After the time series of 20 cycling members was generated, we conducted a series of noncycling perfect-twin experiments for 20 independent daytime groups (Fig. 3b). Every 24 h from 0600 UTC 2 January to 0600 UTC 22 January 2012 we assimilated observational data extracted from one of the ensemble members that was selected as a “true” state. We excluded the true member of the ensemble from the computation of the ensemble covariance matrix. Because we assumed that our forward model was perfect, we did not cycle the perfect-twin DA system. Instead, we evaluated the ability of perfect-twin DA to reproduce the 3D fields of the “truth” ensemble member. In other words, we never produced a forecast based on the new state estimate. Instead, we treated each daytime group as a new experiment.

To exclude variability of the actual observational system as a factor influencing our results, we fixed the configuration of the observational system for the coupled experiments to the types and locations of observations active between 0300 and 0900 UTC 3 January 2012 (see Table 1 for the number and types of assimilated observations). To avoid biasing our results by the overwhelming number of atmospheric measurements over the continental Europe, we restricted atmospheric observations in the perfect-twin experiments to the locations of observations over the sea surface. We set observation errors for each observation types to be equal to half the average error variance at observation locations (on average, observations were twice as accurate as the model). At the locations of observations, the twin ensemble showed good spread–skill relationship (as seen in Fig. 5).

Number of observations in a twin experiment.

The twin model experiments were conducted in three configurations: uncoupled, interface, and exhaustive solver. Localization scales were optimized for each of the experiments and are listed in Table 2. The exhaustive solver was implemented as a special case of the interface solver, where all available measurements were assimilated into both Eqs. (7) and (8). We illustrate implementation of these experiments for the oceanic subsystem [Eq. (8)] in Fig. 6. The uncoupled ocean solver assimilated only the oceanic measurements (the light blue vertical line shows the extent of the assimilated measurements). The series of interface solvers assimilated oceanic measurements and the interface measurements in the atmosphere that were progressively higher from the ocean surface (two red vertical arrows). The number of assimilated interfacial (atmospheric) measurements was quantified by the distance from the surface of the ocean to the highest assimilated atmospheric observation. This distance was normalized by the atmospheric localization scale *L*_{z|A}. The green vertical line in Fig. 6 extends from the bottom of the ocean to the top of the atmosphere and shows that all ocean and atmospheric measurements were assimilated by the exhaustive solver.

Tuned localization values. Tuning was performed to minimize the mismatch between the true twin state and the analyzed state; *L*_{z} values were not tuned for each experiment and they were fixed to *L*_{z} = 200 m in the ocean and *L*_{z} = 1500 m for the atmosphere, and *L*_{t} was set to infinity for both atmosphere and ocean.

### b. Forward model

Our forward modeling system was based on the mesoscale ocean–atmosphere ensemble forecasting system described in Holt et al. (2011). The dynamical core of the coupled system consisted of the COAMPS atmospheric model (Hodur 1997) and the Navy Coastal Ocean Model (NCOM) (Rhodes et al. 2002). The atmospheric model was run at a 42-km resolution and the oceanic model was run at a 12-km resolution. Both models were coupled dynamically using the Earth System Modeling Framework (Hill et al. 2004) at a 6-min time step. Boundary conditions to the atmospheric COAMPS ensemble were provided by the Navy Operational Global Atmospheric Prediction System (NOGAPS) (Hogan and Rosmond 1991) ensemble (Kuhl et al. 2013). Because of the enclosed nature of the Mediterranean basin and the relatively coarse resolution of our model, we treated it as “lake” and did not impose flow through the strengths of the Gibraltar and Dardanelles.

### c. Ensemble generation

The ensemble generation scheme used in this system is the computationally inexpensive ensemble transform (ET) technique originally devised for adaptive sampling problems by Bishop and Toth (1999) but later reconfigured for global ensemble generation by McLay et al. (2008) and Wei et al. (2008), and for regional ensemble generation by Bishop et al. (2009). Like Toth and Kalnay’s (1997) breeding method for ensemble generation, the ET method selects relatively fast growing dynamical modes by a process similar to natural selection. Unlike the breeding method, the ET approach also ensures 1) that all of the ensemble perturbations are quasi orthogonal, and 2) that their amplitudes are consistent with prescribed estimates of analysis error covariances.

*n*×

*K*matrix

*K*forecast ensemble deviations about the ensemble mean via a transformation matrix

*n*is the number of variables. The details of the

We cycled the ET method every 6 h to scale and rotate the current ensemble forecast

### d. Hybrid covariance

The ensemble-variational approach presented in this paper supports hybrid modeling of the error covariance between ocean variables in Eq. (9). The static covariance for ocean variables is described in Cummings and Smedstad (2013). In most of our experiments (sections 4a and 4b), we did not use the hybrid formulation of the covariance and instead used only the flow-dependent part of the covariance [parameter *α* was set to 0 in Eq. (9)]. We chose not to use the hybrid covariance in these experiments because we wanted to focus on the effect of cross-fluid correlations that are only captured by the ensemble covariance. In section 4c, where we focused on the impact of the hybrid covariance on the ocean analysis, we did use the hybrid formulation by setting parameter *α* in Eq. (9) to 0.5.

## 4. Results

### a. Convergence of exhaustive and interface solutions

*L*

_{obs}) normalized by the localization scale in the coupled fluid (

*L*

_{z}). For example the ratio of

*L*

_{obs}/

*L*

_{z}= 1 means that all observations within the native fluid were assimilated and all coupled observations within the first localization scale (

*L*

_{z}) from the fluid interface were assimilated as well (see Fig. 6). We used coupled Gaspari–Cohn localization as described in section 2d. The localization scales used for this experiment are listed in Table 2 (first row).

Figure 7 shows that when no coupled observations are assimilated (*L*_{obs}/*L*_{loc} = 0), interface and exhaustive solvers can differ by as much as 30% in case of ocean velocities (shown off the scale in Fig. 7). However, the normalized differences decrease below 0.5% when we include cross-fluid observations within half of the vertical localization distance from the ocean–air interface (*L*_{obs}/*L*_{loc} = 0.5). The solutions of the two solvers continue to converge as more and more observations are included.

It is notable, that the two solvers do not converge perfectly when we include all of the observations within one vertical localization distance (*L*_{obs}/*L*_{loc} = 1). This is especially apparent for ocean variables. We attribute this to the fact that

To further illustrate the ability of each solver to represent forecast error, we plotted both the forecast error and the analysis increments for one daytime group (0600 UTC 6 January 2012). Atmospheric temperatures are shown in Fig. 8, and oceanic, in Fig. 9. In our twin experiment, the map of the forecast error showed that the true ensemble member had cooler surface atmospheric temperatures than the first guess (top row, Fig. 8). Both uncoupled (second row in Fig. 8) and exhaustive coupled solutions (third row in Fig. 8) reproduce this atmospheric cooling well. Notice that because we limited our observational data over the surface of the Mediterranean Sea, analysis increments relax to zero over the continent. In our analysis, we ensured that all of our aggregate statics account only for the reduction of error over the sea surface.

When we compared coupled (third row) and uncoupled (second row in Fig. 8) solutions, we noticed warmer atmospheric temperatures next the strait of Gibraltar, Adriatic Sea, and along the Egyptian coast. These areas of warming are also apparent in the maps of the true error (top row of Fig. 8) and correspond to the areas of warm SST forecast errors (top row of Fig. 9).

In case of the ocean temperature (Fig. 9), all of the DA schemes were able to reproduce SST forecast errors almost perfectly. We attribute this to an extensive SST data coverage. However, these corrections did not propagate through the water column as deep as the transect of the forecast errors suggests (right column of Fig. 9). We attribute this deficiency to ocean vertical localization scales that were too short.

### b. Benefits of interface solver: Asymmetric localization

Our experiments show that the flexibility of the interface solver design can lead to more accurate analyzed states than it is possible with an exhaustive solution. Specifically, in the interface solver it is possible to tune covariance localizations separately for ocean + interface and atmosphere + interface analysis (recall that the two are run separately). Such separate tuning breaks the symmetry of the cross-fluid covariance matrix and makes it impossible to be implemented in an exhaustive solver, which is run using a single analysis. Figure 10 shows that when we used tuned asymmetric localization, we achieved lower RMSE than solution using a tuned symmetric localization (see Table 2 for tuned localization values).

Tuned values in Table 2 suggest that when, for example, we run ocean + interface analysis it is beneficial to decrease localization scale in the atmosphere as compared to the symmetrically tuned localization (from 500 to 400 km). Compare rows 2 and 1 in Table 2. This decrease essentially suggests that ocean variables are influenced by atmospheric variables that are closer in range than atmospheric decorrelation scales alone would suggest. The reverse is opposite when we tuned localization scales for the atmospheric + interface analysis. When we are interested in producing the best coupled analysis in the atmosphere, the tuned interface solver for atmosphere suggests to use the atmospheric localization radius of 900 km [which is more consistent with the decorrelation scales in atmosphere of about 1000–1500 km; Baker and Daley (2000)] than the localization radius tuned for either exhaustive solver (500 km) or the ocean + interface solver (400 km).

Assimilation increments from the interface solver (optimized for each fluid separately, bottom row in Figs. 8 and 9) and the exhaustive solver (optimized for both fluids, third row in Figs. 8 and 9) show that the interface solver had smoother corrections in the atmosphere. These smoother corrections appeared to be more consistent (visually) with the smoothness of the forecast error (Fig. 8, top row). We suggest that the larger localization scales in the ocean for the atmosphere + interface solver were associated with the atmosphere acting as a low-pass filter that averages SST perturbations in the ocean over a larger area than oceanic Rossby radius would suggest. The differences between the exhaustive and ocean + interface solvers were less apparent for the ocean temperature corrections (Fig. 9), which was consistent with the very small change in the localization scales between the exhaustive solver and the ocean + interface solver. We suggest that these small changes were due to the overwhelming influence of SST and SSH observations on the ocean analysis in our test case.

### c. Benefits of including uncoupled static error covariance

Our experimental results show that including hybrid covariances lead to decreased root-mean-square error (RMSE) scores for some of the assimilated variables (cf. rows 1 and 2 in Table 3). For example, RMSE were decreased for salinities and components of vector velocities (higher values in Table 3 means bigger decrease in RMSE). However, the errors were increased for temperature fields (lower values in Table 3 means smaller decrease in RMSE). We explain these changes in the RMSE by better modeling of the salinity errors in the static covariance as compared to the flow-dependent covariance (and the reverse for the temperature). For example, if we compare coupled ensemble results in Table 3 (first row) with uncoupled static (third row), we can see that pure static covariance was significantly better at representing salinity errors (16.4% vs 5.9% RMSE reduction) than for the temperature errors (54.9% vs 74.6% reduction). Because hybrid covariances weighted ensemble and static covariances equally, the RMSE reductions were moved toward the average of the two results. These results suggest that the representation of the salinity errors by the model ensemble can be further improved.

RMSE reduction by DA (larger numbers are better). RMS errors are evaluated against 3D fields of the “true” ensemble member. We only list error reductions for oceanic variables because we do not yet have an implementation of a static error covariance for atmospheric variables in our coupled ensemble-variational solver. Surface velocities were averaged over the top 3 m.

## 5. Summary and conclusions

The interface solver approach is developed as an alternative to the exhaustive solution of the coupled ocean–atmosphere DA problem. The new solution method solves an approximation to the strongly coupled DA problem. Computational results show that interface and exhaustive solutions converge as more cross-fluid (interface) observations are assimilated by the interface solver. Results of our proof-of-concept study suggest that for a 6-h assimilation cycle it might be sufficient to assimilate as few cross-fluid (interface) observations as within a half of the vertical localization distance from the interfluid interface. We expect that in other applications where there is more connectedness between the ocean surface and the upper atmosphere (e.g., in a tropical cyclone) it might be important to include more interfacial observations to achieve convergence. However, the benefit of the interface solver is in providing flexibility to design an appropriate DA solution for each particular problem of interest.

Our limited experimental results support earlier findings of Han et al. (2013) that an exhaustive solution to the coupled DA problem can result in degraded state estimation results in presence of poorly known cross-fluid covariances. Specifically, we show that RMSEs were higher for the exhaustive solver than for the interface solver solution, where coupled localization was tuned separately for the ocean and the atmosphere. This result suggests that the proposed cross-fluid localization scheme is likely suboptimal and further improvements to cross-fluid location can be developed [see, e.g., Gaspari et al. (2006) for possible formulations of positive-definite cross-fluid covariance functions]. However, because of the limited number of ensemble members, localization alone may not be sufficient to completely remedy the detrimental effects of poorly specified cross-fluid covariances.

Furthermore, we speculate that any further improvements in localization functions (in a context of a variation-based solver) will have to be accompanied by the development of better preconditioners for the coupled variational minimization. In fact, our experience with the developed ensemble-variational system showed that, coupled data assimilation schemes present special challenges to the preconditioners that speed the rate at which variational schemes find the most likely state. In our system, the preconditioner was based on observation volumes that are a few correlation length scales wide in the horizontal and are not a function of the vertical coordinate (see the appendix for details). When the localization scales varied substantially between the two fluids, our simple preconditioners would fail. (However, these regimes were away from the optimal parameters discussed in this paper). Hence, if one would develop localization functions that would remain positive definite in cases where horizontal localization scales differ substantially in the two fluids, one would also need to develop more sophisticated preconditioners to ensure that the covariances matrices remain well-conditioned. At the same time, our experiments with the interface solver suggest that when asymmetry in localization is allowed, it is actually beneficial to specify localization scales in ocean and atmosphere that are more similar to each other than uncoupled experiments would suggest.

Our results also show that the developed interface solver framework and the ensemble-variational implementation that we used in this study can accommodate and benefit from the inclusion of the hybrid error covariances. These benefits were apparent even when static covariance had zero correlations for cross-fluid variables. Specifically, we found that hybrid estimates had some of the errors lower than either the pure coupled ensemble solution or the pure ocean-only static solution. Inclusion of the hybrid covariances also highlighted the deficiencies in representation of salinity errors in the ensemble-based covariance that we used. Our results with the coupled covariance are encouraging because they suggest that coupled DA would be able to retain benefits that hybrid DA achieved for uncoupled DA.

Results presented in this study were obtained using a complex, realistic, coupled model of the Mediterranean. However, our experiment configuration was simplistic (idealized twin experiment), short (20 daytime groups), and constituted a proof-of-concept study. Longer-duration, cycling data assimilation studies over multiple regions of the globe will be needed to collect sufficient error statics and understand specific benefits of the coupled DA. Also the tuned localization scales presented in this study are specific to the region of the Mediterranean and to the specific (short) period of our study. We expect that future studies will need to develop their own (regionally dependent) localization parameters.

Finally, the interface solver methodology presented in this paper is for a case of two 3DVAR-like hybrids. Similar coupled hybrids can also be developed for 4DVAR solvers. In fact, using the interface solver methodology, it might be beneficial to develop separate hybrids for ocean and the atmosphere: a 3DVAR hybrid in the ocean and a 4DVAR hybrid in the atmosphere. Furthermore, if updates to the forecast error variance are properly accounted for, it should be possible to formulate an interface solver that uses different length of assimilation windows in the ocean and atmosphere. Technical details of 4DVAR coupled hybrids are beyond the scope of this paper. We plan to present these details in a separate publication.

## Acknowledgments

This work was supported by the Office of Naval Research Award N0001412WX20323. We are grateful to Keith Sashegyi formally of NRL-MRY for his support with the atmospheric 3DVAR software.

## APPENDIX

### Implementation of the Hybrid 3DVAR Solver

To solve Eqs. (7) and (8) of the interface solver, we incorporated hybrid ensemble error covariances [Eq. (9)] into the existing 3DVAR algorithm (Cummings and Smedstad 2013). We chose this approach for several reasons. First, we wanted to reuse existing highly specialized software for ocean DA, including the static error covariance for the ocean. Second, the 3DVAR algorithms splits the large algebraic problem in Eqs. (7) and (8) into a large number of small problems for each of which a full-rank error covariance can be used; hence, allowing a straightforward implementation of the hybrid system.

*ij*indexing of the computational grid. The quilt cells are sized to optimize the convergence of the preconditioned conjugate gradient solver. Because of the efficient preconditioner, the conjugate gradient solver converges in as few as 6–10 iterations.

**y**

_{i}and

**z**

_{i}as the parts of the observational-space vectors corresponding to the

*i*th quilt cell, then we can decompose Eq. (A1) as a sequence of following independent operations:

**z**in Eq. (A2) is found, a postmultiplication step is performed that interpolates vector

**z**from the observational space into the space of grid points:

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