## 1. Introduction

In recent years, development of hybrid background error covariance (BEC) models has been an area of active research in atmospheric data assimilation (Hamill and Snyder 2000; Etherton and Bishop 2004; Wang et al. 2007). It has been shown in particular that hybrid models tend to be more robust than conventional ensemble-based data assimilation schemes, especially when the model errors are larger than observational ones (Wang et al. 2007, 2008, 2009). This feature is attractive for the regional assimilation problems in oceanography, where information on the background state is often scant and incomplete.

Sequential data assimilation schemes developed so far for regional oceanographic studies can be classified in two categories. The first one is the Kalman filter (KF)-type algorithms with low-rank BEC matrices *T*/*S*) soundings by Argo drifters and ships. The second type of assimilation algorithms employ steady-state covariances

A typical oceanographic setting of such kind is a near-coastal survey by autonomous gliders, which have recently become a fast-developing operational technology in oceanography (Rudnick et al. 2004). Gliders are capable of making remotely controllable surveys of limited areas at high spatiotemporal resolution. Such a dense 4D coverage is usually accompanied by a relatively poor knowledge of the background ocean state: near-coastal regions are often affected by poorly known peculiarities of the bottom topography and the associated tidal/inertial motions that cannot be resolved by global OGCMs. Considerable model error covariances also persist at scales comparable with the size of the domain due to inconsistencies in the boundary conditions and/or local atmospheric forcing.

Because of the relative novelty of glider technology, examples of glider assimilation are rare in literature (Heaney et al. 2007; Shulman et al. 2009). Recently, Dobricic et al. (2010) have shown that three-dimensional variational data assimilation (3DVar) assimilation of glider data significantly improves the forecast skill of a regional model. Most importantly, glider data were able to capture basin-scale BE correlations, which improved the model’s forecast skill several weeks after termination of glider observations. Dobricic et al. utilized the second category 3DVar algorithm based on stationary Gaussian-shaped BECs in the horizontal combined with EOF decomposition in the vertical (Dobricic and Pinardi 2008) and did not explicitly include adaptive error covariances inferred from model statistics.

In this study we propose a hybrid 3DVar assimilation system specifically targeted on preserving survey-scale correlations that could be resolved by gliders in coastal areas. Similar to the existing atmospheric hybrid models, the “flow dependent” part of the covariance *Ocean Modell.*). For that reason the proposed BEC model is formulated in terms of the inverse covariances and the assimilation problem is solved in the state space

Another distinctive feature of the BEC model is an explicit separation of the covariance components in *m* and the magnitudes (scaling coefficients) for both

The rest of the paper is organized as follows. We start with the description of the hybrid BEC model (section 2), then briefly review the Navy Coastal Ocean Model (NCOM) forecast model and the experimental design for the Monterey Bay area (section 3). We continue with an examination of the forecast skills of the assimilation system for the twin-data experimental setting and subsequent real-data experiment (section 4). Section 5 concludes the paper.

## 2. A hybrid 3DVar assimilation scheme

### a. The BEC model

*δ*

**x**of the sequential data assimilation scheme considered here is obtained by minimizing the cost function:

*K*×

*K*observation error covariance matrix, T denotes transposition, and

*K*components are the model-data misfits of the background solution.

*δ*

**x**, we introduce a diagonal matrix

*δ*

**x**and spatial coordinates. Farther below we will assume that all the quantities in (1) are normalized by the respective error variances and introduce new variables:

*J*invariant:

*J*/∂

*δ*

**x**= 0 now take the following form:

*α*

^{−1}and

*β*

^{−1}are empirically defined positive scalar parameters often constrained by the requirement

*α*

^{−1}+

*β*

^{−1}= 1 (e.g., Wang et al. 2008). We adopt the traditional representation of

**e**

*} derived from the eigenvector analysis of the ensemble covariance:*

_{k}*m*×

*M*matrix with the columns

**e**

*,*

_{k}*k*= 1, … ,

*m*and

**Λ**

*is the diagonal*

_{m}*m*×

*m*matrix whose nonzero elements represent the variances of

**e**

*.*

_{k}*ν*depends on spatial coordinates to simulate inhomogeneity and anisotropy of the background flow and

*τ*is the scalar parameter, specifying in 3D the local correlation radii

*ρ*via the eigenvalues

^{i}*τ*can be also interpreted as “integration time” of the corresponding finite-difference diffusion equation.

*Ocean Modell.*).

**e**

*, the respective background error covariance matrix can be readily written as*

_{k}*α*,

*β*of its components and the number of eigenvectors

*m*spanning the

*α*,

*β*, and

*m*are determined from the model states and the data.

### b. Definition of m and α

Accurate determination of the first term in the BEC model in (8) is important because this term is responsible for capturing error correlations on scales comparable with the size of the domain. In oceanographic applications these errors are generated by poorly known open boundary conditions and errors in atmospheric forcing, which tend to have larger scales than those of the internal oceanic variability. In addition,

**e**

*of the first-guess BEC estimate provide poor approximation to the true eigenvectors of the background error covariance. To assess reliability of*

_{k}**e**

*we employ the Bayesian information criterion (Schwarz 1978) and define the optimal number of “trusted” eigenvectors as the minimum of*

_{k}*N*data samples by

*m*modes. Here

*k*th observation location at the

*n*th analysis time, the coefficients

*n*for a given number of modes

*m*≪

*K*, and the overbar denotes averaging over

_{n}*N*time layers.

The relationship in (10) gives an asymptotic (*N* ≫ 1) approximation to the Bayesian posterior probability for a model with *m* parameters (linear regression on *m* eigenfunctions) given *N* observations (*T*/*S* fields sampled by gliders at the analysis times) under the assumption that model-data misfits are normally distributed. A similar, but less restrictive *m* criteria could be also used (Akaike 1974; Hannah and Quinn 1979).

*m*-dimensional subspace

**e**

_{k}}. Because the Gaussian part of

*δ*

**x**on

**e**

*can be obtained (see the appendix):*

_{k}Here *δ***e** is the *m*-dimensional vector of the expansion coefficients such that

*α*in several ways. The matrices

**Λ**are known and the matrix on the left-hand side can be estimated by approximating

*δ*

**y**by the linear combinations of

**e**

*at*

_{k}*N*analysis times and computing the time-averaged covariances between the vectors

*δ*

**e**of the optimal fit coefficients. Optimal

*α*can be then computed by minimizing a norm of the difference between the left- and right-hand sides of (12). Since all the matrices in (12) are positive definite, a convenient option is to set the difference between their traces to zero. In the application considered below, the background model errors are much larger than observational errors

*α*: its value is found by minimizing the mean squared difference between the diagonal elements of 〈

*δ*

**e**

*δ*

**e**

^{T}〉 and

**Λ**

*/*

_{m}*α*.

In principle, one can generalize the covariance model in *δ***e***δ***e**^{T}〉, by adjusting the diagonal elements of **Λ*** _{m}*. In oceanographic applications, however, there is no reason to refine the covariance model by finetuning the eigenvalues because even the leading eigenvectors of

*α*.

### c. Definition of β

*β*of the Gaussian part by equating the trace of the sample forecast error covariance Tr〈

*δ*

**y**

*δ*

**y**

^{T}〉 derived form the innovation statistics to the trace of

*M*×

*M*system of equations with random right-hand sides.

### d. Numerical implementation

*ν*= diag

*ν*, assuming that local decorrelation radii

^{i}*ρ*are directly proportional to the model grid steps Δ

^{i}*x*,

^{i}*i*= 1, 2, 3 spatially varying in 3D:

Since gliders directly measure only the temperature and salinity fields, the operator *T*/*S* components of the state vector under the prior assumption of zero correlations between them. The temperature/salinity background error correlations were taken into the account by the **e*** _{k}* were only estimated for the temperature and salinity fields).

*Ocean Modell.*):

*τ*/

*n*is the length of the implicit “time step” and

*n*is the number of explicit time steps. With the definition (16), the square root of the integration time

*τ*has the meaning of the proportionality coefficient between the local decorrelation scales

*ρ*and the grid steps in the corresponding directions:

^{i}*τ*= 20 was determined through preliminary experiments described in section 3. The second parameter

*n*in (17) was chosen as a compromise between the numerical complexity and accuracy in approximation of

*n*= 2 the approximation error is close to 15% (Fig. 1), which is quite reasonable given the overall uncertainty in the definition of the heuristic covariance operator

*δ*

**x**was obtained by solving (9) with a generalized minimum residual solver (Saad 2003). In correspondence with the approximation (17), the inverse of

*K*, the solution of (9) required, as a rule, 150–300 iterations, keeping the computational cost of the analysis well below the cost of a 12-h model run between the assimilations. In the reported experiments, the state space dimension

*M*was 515 102, which is the number of observations

*K*varied between 1500 and 3000, and

*m*never exceeded 2. Within these ranges of

*K*and

*m*the CPU time required for the estimation of

*α*and

*m*was negligible compared to the time

*τ*

_{cpu}of solving the normal equation. Conversely, estimation of

*β*required several (usually 4–7) iterative inversions of

*τ*

_{cpu}, and was the most expensive part of the analysis.

The only type of data used in the present study were temperature and salinity profiles from gliders. Therefore, balance constraints were introduced by applying the linearized equation of state and the geostrophic–hydrostatic relationships directly to the temperature and salinity increments (e.g., Li et al. 2008) obtained from minimization of the cost function (1).

## 3. Experiment design

The BEC model was verified by 3DVar assimilation experiments with the Navy Coastal Ocean Model (NCOM) configured in the Monterey Bay (Fig. 2) for processing of the data acquired during the Autonomous Ocean Sampling Network (AOSN II) experiment (Ramp et al. 2008). The experiment was conducted in the summer of 2003 with the ultimate goal of developing an adaptive sampling technique that combines numerical forecasts with the data flows from controllable observation platforms. Observations were performed by several types of autonomous underwater vehicles (AUVs) including gliders, high-frequency radars, two moorings, bottom-mounted ADCPs, surface drifters, and CTD casts. In the present study, we focus the analysis on the temperature/salinity data from gliders only: space–time coordinates of the gliders are used to define observation operators

### a. Numerical model, observations, and validation technique

To simulate oceanic variability during the experiment we used a version of NCOM forced by the Coupled Ocean–Atmosphere Mesoscale Prediction System (COAMPS; Hodur et al. 2002) winds in the time period between 1 August (*t* = 0) and 27 August (*t* = 27) of 2003. The model was configured on a curvilinear orthogonal grid (Fig. 2) with horizontal resolution ranging from 1 to 4 km, and a hybrid *σ*/*z* vertical coordinate system with 9 *σ* levels in the upper ocean and 32 *z* levels below. At the open boundaries, the model was one-way coupled to the global NCOM model (Shulman et al. 2009).

Glider observations during the experiment covered the central part of the model domain (Fig. 2). With a typical dive cycle of about 1 h, a glider would travel approximately 0.5 km between surfacings, which is well below the grid resolution. For that reason we prescribed observational operators

**g**was used. The diagonal elements of

**g**(

*g*,

_{T}*g*,

_{S}*g*,

_{u}*g*, and

_{υ}*g*) were depth dependent and were obtained as horizontally averaged time variances of temperature

_{ζ}*T*, salinity

*S*, horizontal velocity {

*u*,

*υ*}, and SSH

*ζ*, respectively, at a grid point

**r**:

*ξ*stands for either

*T*,

*S*,

*u*,

*υ*, or

*ζ*and angular brackets denote the horizontal average at level

*z*.

*r*and

^{s}*r*between the model states were computed in both observational and state spaces:

^{g}**y**

*and salinity*

_{T}**y**

*were extracted from the “true” fields*

_{S}*T*,

^{t}*S*(Fig. 3, left panel) at glider locations

^{t}*ε*with zero mean and 0.1 rms variation:

To simulate model errors and assess the impact of assimilation in the twin-data experiments, the “first guess” model solution **x**^{fg}(*t*) was generated by integrating the model for 27 days starting from the initial condition specified by **x*** ^{t}*(

*t*= 8.5) (Fig. 3, right panel).

**x**

^{fg}(0) as the background state at

*t*= 0, a series of 3DVar assimilation experiments were performed: on every 12-h assimilation cycle, model forecasts

**x**

*were updated with the analyses increment*

^{f}**x**

*=*

^{a}**x**

*+*

^{f}*δ*

**x**and the next 12-h integration was started from

**x**

*. The skill of assimilation*

^{a}*q*(

*t*) was assessed in both observational and state spaces by calculating the normalized distances between the 12-h model forecasts and the true states:

Experiments with real data were conducted and the results were validated in a similar manner, except that **x*** ^{t}* was taken as the first guess and

### b. Parameters of the hybrid covariance model

In contrast to atmospheric applications, regional oceanographic problems have more difficulties with the BEC estimation from ensembles. The reason is that realistic ensembles simulating ocean variability on regional scales are rarely available. In the present study, the first-guess background error statistics was obtained from the ensemble of the differences *l* and the respective 12-h forecasts (background states) *α* = 0. The latter were treated as a rough approximation to the ensemble of the true ocean states. Since the number of ensemble members (*l* = 1, … , 55) was limited by the duration of the glider survey, the expected number of statistically sensible eigenvectors of the ensemble covariance matrix was rather small and never exceeded 2 (see section 4).

In the hybrid assimilation runs (with *α* ≠ 0) this first-guess ensemble *l* preceding the current analysis time were replaced by the members derived from the forecasts already made with the hybrid scheme. The quality of updated ensembles was monitored by the number of eigenvectors accepted by the information criterion (10) and by the percentage of *δ***y** variance explained by these eigenvectors.

To increase the robustness in estimating *α* and *β*, we utilized the method of Wang et al. (2007) and performed additional time averaging while computing the sample variances in (13) and (15). This averaging was done over the ensemble of 30 states (15 days) preceding the analysis time. In the initial 15 days of the assimilation run, the missing background states were taken from the respective forecasts **x**^{fg}(*t*) generated by the first-guess solution. Similar averaging over *N* = 30 samples following the analysis time was done when estimating *σ _{m}* in (11).

There are two parameters in the definition of *τ* controlling decorrelation length scales, and the other is the order *n* of approximation of the exponent in (17). These parameters were tuned by numerical experimentation.

Comparison of the model solutions (Fig. 3) with the grid (Fig. 2) gives an indication that horizontal correlations are likely to decay at 3–6 grid steps. We checked this hypothesis by twin experiments with *α* = 0 and computed the forecast skill of the assimilated solutions with various values of *n* was also varied in the range between 1 and 4. The best overall result was obtained with *τ* = 20 (*ρ _{i}* ~ 4.5Δ

*x*) and

_{i}*n*= 2. Although assimilation quality (with

*τ*= 15,

*n*= 3) was similar and in some periods slightly better, the computational cost appeared to be much larger. We therefore selected

*τ*= 20, and

*n*= 2 as basic parameters for the assimilation experiments.

## 4. Results

### a. Twin-data experiments

Figure 4 compares the skill of 3DVar assimilation runs performed with the Gaussian and hybrid BEC models. During the first 8 days of assimilation, the hybrid scheme was unable to detect any statistically reliable modes. Between days 8 and 11 the first mode was detected, accounting for 8% of the forecast error on day 8, 14% on day 11 (Fig. 4), and 17% on day 17. On day 12 the second mode was detected, accounting for 4% of the forecast error variance. Contribution of the second mode increased to almost 10% on day 18. Later, the modes appear to lose their predictive skill with the contributions dropping to 12% and 7%, respectively, on day 25.

The 12-h forecast errors measured in terms of the normalized distances *α* = 0 (thin lines in Fig. 4). A similar level of error reduction was observed by Wang et al. (2008) in twin-data experiments with a hybrid assimilation into the Weather Research and Forecasting (WRF) model. In terms of the 12-h forecasts of temperature and salinity in glider observation points the error reduction is somewhat smaller (11% for temperature and 13% for salinity), but can still be considered as a satisfactory improvement (Fig. 5).

Assimilation experiments with different noise in observations have shown that the patterns in Figs. 4–5 are robust up to the noise levels of 0.5. At higher noise levels, the approximation (13) becomes less accurate and it is necessary to use the relationship (12) for estimating *α*. Larger errors in estimating *α* result in the loss of accuracy in estimating the number of modes *m* and the magnitude *β* of the Gaussian part of the covariance. We therefore assume that the proposed algorithm is valid when observation errors are considerably smaller than the background errors. This is not a severe restriction for regional assimilation problems in oceanography where the first-guess/background model solutions are rarely preconditioned by data and often appear to be rather far from reality.

### b. Real-data experiments

Figure 6 shows a typical situation we encountered in the experiments with real data in the Monterey Bay: The first-guess model solution does not have much in common with the mooring record at 40 m (left panel). Moreover, the mean profiles in the right panel demonstrate considerable salinity biases above 30 m and in the depth range between 50 and 200 m. The rms variations of salinity measured by gliders and moorings are generally consistent with each other in magnitude (cf. horizontal bars and the width of light shading around the thick profile in the right panel). A noticeable bias between the mean salinity measured by moorings (solid dots) and gliders (thick line) could be attributed to differences in averaging: the glider profile is obtained by averaging over all the glider positions (Fig. 2), whereas the mooring profile (solid dots) is obtained as the mean of only two moorings. Similar biases between the first-guess solution and observations were obtained for the temperature field (not shown).

To estimate observation errors, we compiled the glider *T*/*S* records at times when gliders passed closer than 200 m of either of the moorings and compared these data with the corresponding observations at moorings. In total, 168 of such “pass-by events” were found. Comparison of these observations has shown that the rms discrepancies in temperature and salinity were fairly stable with depth and varied within 0.26–0.35 after normalization by the rms variances *σ _{m}*(

*z*) recorded at the moorings (horizontal bars in the right panel of Fig. 6). Based on these computations, the observation error variances were estimated as

*R*

^{1/2}(

*z*) = 0.3

*σ*(

_{m}*z*) and assumed not to vary in the horizontal.

*ξ*and temperature, salinity, or velocity

^{f}*ξ*measured at the points of moored observations:

^{m}*T*/

*S*observations or 18 levels of ADCP data) and over two moorings shown in Fig. 2.

Figure 7 demonstrates the differences *δq* between the salinity forecast skills *α* = 0 and similar forecast skills obtained with the hybrid BEC model. Although the skill improvement does not look as good as in twin-data experiments (Fig. 4), it appears to be robust: the differences in skill

Compared with the time-averaged assimilation skill in twin-data experiments (e.g., *n _{e}* (20–55 members) have shown that with

*n*= 20, the number of detection events dropped to 3, whereas with

_{e}*n*= 50, it increased from 30 (Fig. 7) to 35 without any substantial improvement of the forecast skill. Further increase of the ensemble size was limited by the duration of the assimilation experiments.

_{e}In principle, the ensemble could be expanded, for example, by the breeding technique, but the problem with the poor quality of the first-guess solution (Fig. 6) may still persist, because the bred vectors would still show unstable modes of the background state that is rather far from reality. In the present study, we used a simple approximation to the error fields by considering an ensemble of differences between a free model run and an assimilation run with the Gaussian covariance model. This ensemble was able to generate just a few members in the 27-day period. One may hope, however, that for longer observation periods the BEC model will gain enough skill to show better performance.

Figure 8 shows the time evolution of the ratio between the weighting parameters *α* and *β* in the twin- and real-data experiments. By an order of magnitude, the ratio *γ* is consistent with the results of Wang et al. (2008) who set *γ* = *β*/*α* = const in time and found the optimal *γ* to vary between 1 and 4 in a series of twin-data experiments with the WRF model. In our case, the relative weight *β* of the Gaussian term in the cost function appeared to be approximately 2 times smaller in the twin-data experiment (thin curve in Fig. 7). This is consistent with a better skill in explaining model-data misfits by the modes retrieved in the twin-data experiment (cf. Figs. 4 and 7). Larger relative values of *α* on days 10–13 (before the mode rejection) can be explained by the tendency of the algorithm to keep the deteriorating mode “alive.”

We also investigated the impact of the algorithms for definition of *m* and *α* on the forecast skill. In the twin-data experiments with fixed *m* the 27-day-averaged skill was always worse than that in Fig. 4 for 3 tested values of *γ* = 0.5, 1, and 2. When *m* was computed through (10) and *γ* was kept constant at 0.95, the forecast skill was virtually the same as in Fig. 4, but somewhat below using other values of *γ*. Similar results were obtained with real data: keeping *m* = const degraded the forecast skill, often below the one obtained with Gaussian BEC model. Several runs with an adjustable *m* and *γ* = const were difficult to interpret as the skill improvements were small, highly variable, and did not show any deterministic dependence on the value of *γ* ∈ [0.5, 2.5].

## 5. Summary and discussion

In this study we proposed a hybrid BEC model specifically designed for 3DVar analysis of regional circulations supported by glider surveys. The model is supplied by an algorithm for weighting the ensemble-generated error covariance

The proposed BEC model is formulated in terms of the inverse covariances with the restriction of

The hybrid BEC model was validated by numerical experiments with simulated and real data. In the twin-data setting, the hybrid formulation was capable of improving the model’s forecast skill by 15%–20%, which is comparable with the improvement reported by Wang et al. (2008) for a hybrid scheme with the atmospheric WRF model. Results of the experiments with real data showed a few percent improvement with sporadic detection of only one mode. We attribute this to a poor quality of the background solution, which was heavily biased and demonstrated considerably lower time variation in the temperature and salinity fields (left panel in Fig. 6). Thus, finding a better background solution appears to be the first priority in upcoming studies of the algorithm.

Other developments may include elaboration of the structure of the diffusion tensor in *f*/*H* contours and/or isopycnals of the geostrophically balanced modes if the latter are detected. The ensemble could be enriched by the vectors bred from the eigenvectors of

One of the drawbacks of the proposed model is the computational cost of estimating the weight *β* of the static covariance [(15)]. Our experience shows, however, that the numerator in (15) weakly depends on the structure of *α* and *β* when the background errors are comparable with the observation errors.

The benefit of the proposed hybrid model may also be diminished for global assimilation problems where some sort of localization is needed and the impact of the ensemble-generated covariances may be smaller with higher observation density (e.g., Whitaker et al. 2008). Nevertheless, we assume that the proposed approach may have a prospect for further development for regional data assimilation problems with poorly known background states.

## Acknowledgments

This study was supported by the Office of Naval Research (Program Element 0602435N) and NSF Grant 0629400. Helpful discussions with Dr. Shulman are acknowledged.

## APPENDIX

### Derivation of Equation (12)

*δ*

**e**such that

*δ*

**y**covariance, covariances of

*δ*

**e**could be written as follows:

*δ*

**y**between the background state and the data have the following covariance:

**e**

*, is*

_{k}^{−2}, allowing the relationship in (A6) to be reduced to

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