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  • View in gallery

    The assimilation process for the original variant of the scheme V0. The dashed lines show the model forecast and the solid line shows the analysis. The analysis is obtained by adding the observational update at time tk in the model operator between times tk−1 and tk. The forecast is obtained using the nominal values of the parameters p0.

  • View in gallery

    The assimilation process for the variant V1. The dashed lines show the model forecast, and the solid line shows the analysis. The analysis is obtained by modifying the model operator between times tk−1 and tk using the parameters pka estimated from the observations at time tk. The forecast between times tk and tk+1 is obtained using the updated values of the parameters pka.

  • View in gallery

    The assimilation process for the variant V2. The dashed lines show the model forecast, and the solid line shows the analysis. The analysis is obtained by adding the observational update of time tk, in the model operator between times tk−1 and tk, without modifying the model parameters pk − 1a. The forecast between times tk and tk+1 is obtained using the updated values of the parameters pka.

  • View in gallery

    (top) Average values and (bottom) standard deviations of the turbulent exchange coefficients (left) CE and (right) CH in the reference model (original ORCA2 bulk formulation). The values are scaled by a factor of 103.

  • View in gallery

    (top) Evolution of error variances in (left) SST (°C) and (right) SSS (psu) for the World Ocean (except the northern polar zones). The black solid line shows the error corresponding to the free simulation (without assimilation). The green, yellow, blue, and red lines correspond to V0, V0*, V1, and V2, respectively. The solid lines represent the continuous analysis, while the bullets represent the 10-day model forecasts. (bottom) A duplication of the (top) but with a finer y-axis scale.

  • View in gallery

    Evolution of error variances in (left) SST (°C) and (right) SSS (psu) for the Atlantic Ocean only. The green, yellow, blue, and red lines correspond to V0, V0*, V1, and V2, respectively. The solid lines represent the continuous analysis and the bullets represent the 10-day model forecasts.

  • View in gallery

    Spatial distributions of the standard deviation errors in (left) SST and (right) SSS 10-day forecasts for (top) V0, (middle) V1, and (bottom) V2.

  • View in gallery

    Standard deviation error values in (left) temperature (°C) and (right) salinity (psu) 10-day forecast profiles for (top) V0, (middle) V1, and (bottom) V2.

  • View in gallery

    Standard deviation errors in (left) the latent heat flux exchange coefficient CE and (right) the sensible heat flux exchange coefficient CH, for (top) V0, (middle) V1, and (bottom) V2.

  • View in gallery

    Detailed statistics for two specific locations of the World Ocean: (top) 16°S, 2°W and (bottom) 36°N, 60°W: (left) CE parameter evolution in time, (middle) temperature innovations, and (right) CE parameter corrections. The black dashed line stands for the true ocean, and the green, yellow, blue, and red lines correspond to V0, V0*, V1, and V2, respectively. [The green curve in (right) displays the correction to the parameters that is computed in V0 but never applied.]

  • View in gallery

    Spatial distribution of standard deviation error values on the (left) latent heat flux and (right) sensible heat flux for (top) V0, (middle) V1, and (bottom) V2. Unit: W m−2.

  • View in gallery

    Evolution of the (left) SST (°C) and (right) error variance for a 3-month forecast corresponding to V0, V1, and V2 (green, blue, and red curves, respectively). The forecasts start on 4 Apr 1993 (4 months after the beginning of the experiments).

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Improved Turbulent Air–Sea Flux Bulk Parameters for Controlling the Response of the Ocean Mixed Layer: A Sequential Data Assimilation Approach

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  • 1 LEGI/CNRS, UMR 5519, Grenoble, France
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Abstract

Bulk formulations parameterizing turbulent air–sea fluxes remain among the main sources of error in present-day ocean models. The objective of this study is to investigate the possibility of estimating the turbulent bulk exchange coefficients using sequential data assimilation. It is expected that existing ocean assimilation systems can use this method to improve the air–sea fluxes and produce more realistic forecasts of the thermohaline characteristics of the mixed layer. The method involves augmenting the control vector of the assimilation scheme using the model parameters that are to be controlled. The focus of this research is on estimating two bulk coefficients that drive the sensible heat flux, the latent heat flux, and the evaporation flux of a global ocean model, by assimilating temperature and salinity profiles using horizontal and temporal samplings similar to those to be provided by the Argo float system. The results of twin experiments show that the method is able to correctly estimate the large-scale variations in the bulk parameters, leading to a significant improvement in the atmospheric forcing applied to the ocean model.

Corresponding author address: Jean-Michel Brankart, LEGI/CNRS, BP 53X, 38041 Grenoble CEDEX, France. Email: jean-michel.brankart@hmg.inpg.fr

Abstract

Bulk formulations parameterizing turbulent air–sea fluxes remain among the main sources of error in present-day ocean models. The objective of this study is to investigate the possibility of estimating the turbulent bulk exchange coefficients using sequential data assimilation. It is expected that existing ocean assimilation systems can use this method to improve the air–sea fluxes and produce more realistic forecasts of the thermohaline characteristics of the mixed layer. The method involves augmenting the control vector of the assimilation scheme using the model parameters that are to be controlled. The focus of this research is on estimating two bulk coefficients that drive the sensible heat flux, the latent heat flux, and the evaporation flux of a global ocean model, by assimilating temperature and salinity profiles using horizontal and temporal samplings similar to those to be provided by the Argo float system. The results of twin experiments show that the method is able to correctly estimate the large-scale variations in the bulk parameters, leading to a significant improvement in the atmospheric forcing applied to the ocean model.

Corresponding author address: Jean-Michel Brankart, LEGI/CNRS, BP 53X, 38041 Grenoble CEDEX, France. Email: jean-michel.brankart@hmg.inpg.fr

1. Introduction

Turbulent momentum, heat, and freshwater fluxes at the air–sea interface remain among of the main sources of error in present-day ocean models (Large 2006). They strongly limit the capacity of such models to provide a realistic forecast of the thermohaline characteristics of the mixed layer and of the surface ocean currents. This is the reason why we decided to investigate ways of improving the knowledge of these fluxes through assimilation of oceanic observations.

A way to address this problem would be to use a four-dimensional variational data assimilation (4DVAR) scheme, by including the fluxes in the control parameters of the model in addition to the initial conditions (Roquet et al. 1993; Stammer et al. 2004). However, many existing ocean assimilation systems are based on sequential assimilation schemes that would greatly benefit from more precise estimates of the fluxes. Hence, it is useful to look for alternative schemes that are numerically less expensive and easy to implement in a wide range of existing systems. Our starting point is a reduced-order Kalman filter in which the background error covariance matrix is represented by a set of 3D error modes in the state space of the ocean model (Pham et al. 1998). In this paper, we will use an optimal interpolation scheme derived from the singular evolution of the extended Kalman (SEEK) filter (Brasseur and Verron 2006). However, the method that we propose using to correct the fluxes is directly applicable in any similar assimilation scheme, such as ensemble optimal interpolation schemes, ensemble Kalman filters (Evensen 2003), or any variant of reduced-order Kalman filters: the reduced-rank square root (RRSQRT) scheme (Heemink et al. 2001), the error subspace statistical estimation (ESSE) system (Lermusiaux and Robinson 1999), the SEEK filter (Pham et al. 1998), etc.

A general method used to identify model parameters (like the air–sea flux bulk parameters) by means of a Kalman filter is to augment the filter control space by including these parameters in addition to the state variables. This method, although common in the engineering literature (Cox 1964; Ho and Whalen 1963; Nelson and Stear 1976; Ljung 1979), has not often been used in atmospheric or oceanographic applications of the Kalman filter. A first demonstration of the applicability of the method to atmospheric problems was performed by Anderson (2001), who used it for estimating one forcing parameter of the 40-variable Lorenz model, using an ensemble-based data assimilation scheme. Second, a work by Losa et al. (2003) deals with the joint state and parameter estimation, in a zero-dimensional ecosystem model, using a sequential-importance resampling filter. In a third study, by Annan et al. (2005), the method is applied to the tuning of the surface temperature climatology in a spectral primitive equation atmospheric GCM, using the ensemble Kalman filter. Similarly, Aksoy et al. (2006) also use an ensemble Kalman filter to estimate up to six (spatially constant) model parameters in a two-dimensional sea-breeze model. These four studies show the relevance of sequential data assimilation methods for estimating model parameters, but also point up the difficulty in proving the effectiveness of the method in all applications. Within that context, the objective of this paper is to demonstrate the possibility of estimating parameters driving the turbulent air–sea fluxes of a realistic ocean GCM, using a sequential assimilation scheme to invert oceanic observations.

If the purpose of the Kalman filter is to control the air–sea fluxes, the control parameters could be the fluxes themselves or the atmospheric fields from which they are computed. But it seems preferable to include a few key parameters of the bulk formula in the control vector (i) because they are more likely to persist over time (the aim is to improve the forecast), (ii) because they are likely to be as easy to control by ocean observations (provided that we only include parameters that are linearly linked to the value of the flux), and (iii) because they can be assumed (Large 2006) to be the real source of error (even if this will probably result in compensating errors on the atmospheric parameters by correcting the bulk coefficients).

In this paper, we present an example in which only the latent heat flux coefficient (CE) and the sensible heat flux coefficient (CH) are included in the control vector because they can be assumed to be one of the main sources of error (although in realistic experiments, other sources of error, like the wind stress drag coefficient, precipitation, or cloud cover, need to be considered). The procedure is tested using twin assimilation experiments with a low-resolution (2° × 2°) global ocean configuration. Within this context, assimilation experiments, with sequential corrections of the bulk coefficients, can be performed if we provide a background error covariance matrix in the augmented control space. This will be done using ensemble simulations characterized by various values of CE and CH.

The synthetic observations that will be assimilated to correct the fluxes are temperature and salinity profiles with horizontal and temporal samplings comparable to those of the Argo floats system (about 3000 free-drifting profiling floats measuring the temperature and salinity of the upper 2000 m of the ocean; information online at http://www.argo.ucsd.edu/). In particular, we will study the ability to reconstitute the bulk coefficients by assimilating these kinds of oceanic observations, and examine how correcting the parameters can help improve the quality of the air–sea fluxes and thereby the quality of the temperature and salinity forecasts. Investigations will also be conducted into the behavior of the method in the presence of systematic errors arising from inaccurate parameters.

The structure of the paper is as follows: sections 2 and 3 present the ocean model and the assimilation methodology, respectively; section 4 describes the experimental protocol; and section 5 presents the results of the experiments.

2. The ocean model

The ocean model used in this study is the ORCA2 configuration of the Océan Parallélisé (OPA) model (Madec et al. 1998). This is a global ocean configuration using a 2° × 2° ORCA-type horizontal grid, with the meridional grid spacing reduced to 1/2° in the tropical regions in order to improve the representation of the equatorial dynamics. This is a free-surface configuration based on the resolution of the ocean dynamic primitive equations, with a z-coordinate vertical discretization. There are 31 levels along the vertical, and the vertical resolution varies from 10 m in the first 120 m to 500 m at the bottom. The lateral mixing for active tracers (temperature and salinity) is parameterized along isopycnal surfaces, and the model uses a turbulent kinetic energy (TKE) closure scheme to evaluate the vertical mixing of the momentum and tracers. The vertical eddy viscosity and diffusivity coefficients are computed from a 1.5 turbulent closure model based on a prognostic equation for the turbulent kinetic energy, and a closure assumption for the turbulent length scales (see Blanke and Delecluse 1993 for more details).

The model is forced at the surface boundary with heat, freshwater, and momentum fluxes. The momentum flux is derived from the European Remote Sensing Satellite (ERS) scatterometer wind stresses complemented by Tropical Atmosphere–Ocean (TAO) derived stresses (Menkes et al. 1998). The heat and freshwater turbulent fluxes are computed using bulk formulation from National Centers for Environmental Prediction (NCEP) atmospheric interannual data provided by the National Oceanic and Atmospheric AdministrationCooperative Institute for Research in Environmental Sciences (NOAA–CIRES) Earth System Research Laboratory/Physical Sciences Division/Climate Diagnostics Center (ESRL/PSD/CDC), Boulder, Colorado. In this model, a common way of dealing with errors in the atmospheric forcing is to add a local restoring term to relax the model solution toward the sea surface temperature (SST) and sea surface salinity (SSS) climatological data. Since the objective of this work is to develop a method for correcting for errors in the air–sea fluxes, none of these restoring terms have been applied in the experiments described below.

In this paper, the focus will be on the latent heat flux coefficient (CE) and on the sensible heat flux coefficient (CH), both of which are involved in the computation of the following quantities:

  • the latent heat flux,
    i1520-0426-26-3-538-e1
    where L is the vaporization latent heat, W the wind speed, and qs and qa are the saturation and atmospheric specific humidities, respectively;
  • the evaporation freshwater flux,
    i1520-0426-26-3-538-e2
  • the sensible heat flux,
    i1520-0426-26-3-538-e3
    where ρa is the air density, C the air specific heat, and Tw and Ta are the air and sea surface temperature, respectively.

In the standard OPA parameterization, CE and CH are the products of very complex parameterization depending, in particular, on the stability of the air column close to the sea surface. This complex parameterization will not be described in this paper since it is only used to produce the reference model simulation (see section 4a). It is only important to note that the values of the CE and CH parameters depend on the current state of the system (SST, SSS, and the state of the atmosphere), so that the CE and CH parameters in the reference simulation are not constant in space and time.

3. The assimilation scheme

The assimilation method used in this study is derived from the SEEK filter, which is a reduced-order Kalman filter introduced by Pham et al. (1998). This sequential method has already been used in a number of studies (Verron et al. 1999; Brasseur et al. 1999; Testut et al. 2003; Birol et al. 2004). In the present implementation, the background error covariance matrix is kept unchanged from one assimilation cycle to the next, transforming the Kalman filter into an optimal interpolation filter. The main feature of the scheme is the structure of the background error covariance matrix, which is parameterized in a low-order subspace of the state space. This means that error is assumed to occur only in a few known directions within the state space [see Brasseur and Verron (2006) or Rozier et al. (2007) for more details]. We also make use of the local variant of the SEEK filter described in Brankart et al. (2003) and Testut et al. (2003). The scheme considers that the long-range correlation coefficients in the reduced-rank covariance matrix are irrelevant and sets them to zero, so that the analysis for each water column of the model will depend only on the observations within a specified influence bubble. In this study, the size of each influence bubble is parameterized inside a box of 10 × 10 grid points. Additional details about the parameterization of the background error covariance matrix are given in section 4b.

In addition to this standard procedure, we also apply the incremental scheme implemented by Ourmières et al. (2006). Indeed, a significant drawback of sequential methods is the time discontinuity of the solution resulting from intermittent corrections of the model state. To tackle this problem, Bloom et al. (1996) have proposed an algorithm called the incremental analysis update (IAU). The principle of this algorithm is to incorporate the sequential observational update (δx) in a more gradual way. At each assimilation cycle, the model is restarted from the initial conditions of the current assimilation cycle with a forcing term added in the model equations:
i1520-0426-26-3-538-e4
where M is the model operator and Δt is the duration of the assimilation cycle. The state vector x of our model contains four variable fields: temperature (T), salinity (S), and zonal and meridional velocities (U and V). The model operator M also depends on a vector of parameters p, which are going to be included in the control vector. Here, p0 denotes the nominal values of the parameters, without any correction through data assimilation. The procedure is illustrated in Fig. 1, where xkf and xka denote, respectively, the forecast state vector and the analysis state vector at time tk. Here, xka is computed from xk − 1a by running the model M as in Eq. (4) with the observational increment computed at time tk. This is the original algorithm that we use to start with, without any correction of the fluxes. It is the reference assimilation scheme, denoted as variant V0, with which new schemes will be compared. (It is, however, important to notice here that using the IAU scheme is not required to apply the parameter correction scheme that will shortly be presented; an intermittent correction algorithm could be used as well.)

To reach our objective of using such an assimilation scheme to correct the bulk parameters, the control vector is augmented by the current values of the bulk parameters p = [CE,CH], in addition to the four prognostic variables of the model. The control vector is then written as = [T,S,U,V,CE,CH] instead of x = [T,S,U,V]. Thus, from observations of temperature and salinity we will be able to make a correction for the bulk parameters. Obviously, the estimation will depend very much on the background error covariances between these six variables; it is the pattern of the correlation between [T,S] and [CECH] that will govern the computation of a correction to [CE,CH] from the diagnostics of the error in the [T,S] fields. A method of building such a background error covariance matrix will be discussed in section 4b. Before that, however, we need to determine the best way to incorporate a correction on the model parameters into the model simulation. Three different approaches will be compared: in variant V0 (described above), we compute corrections of the model parameters, but they are never used in the model. In variant 1, denoted as V1, we only correct the model parameters p, and in variant 2, denoted as V2, we correct both the model parameters p and the model state x. The remainder of this section gives the details of variants V1 and V2.

In V1, we assume that the system can be controlled by adjusting only the bulk parameters, that is, by correcting only the vector p. The state vector is therefore no longer corrected directly [the IAU increment δx in Eq. (4) is set to zero], and the correction is applied by restarting the model from the initial condition of the current assimilation cycle using only the corrected values of the parameters p. This is illustrated in Fig. 2, where pka denotes the estimated value of the parameter after the observational update of time tk. This variant of the scheme differs from V0 only in that the application of the IAU increment (δx) is replaced by a correction of the bulk coefficients and that the upgraded values of the coefficients are used in the next assimilation cycle.

In V2, the state vector of the model x is corrected as in the original scheme V0, but the new bulk parameters (p) are taken into account in the next forecast. Again the observational update is incorporated gradually as in the original IAU scheme (Fig. 3). Since the corrected bulk parameters are used as background values for the next analysis, this variant of the scheme differs from V0 only in that upgraded values of the coefficients are used in the next assimilation cycle.

Variant V2 can be considered to be an assimilation scheme written for the augmented state vector, . Up to now, we have tacitly admitted (as in most parameter estimation studies using a Kalman filter) that the model equations are augmented by the equations = 0; that is, the parameter are assumed to be constant over time. However, in order to avoid unstable estimations of the bulk coefficients due to repeated observational updates without any feedback loop, we introduce a complementary model that governs the bulk parameters. This model computes the “forecast values” of the parameters pkf from the analyzed values pka. Such a model proved necessary to forecast the bulk parameters just because, in our experiments, the estimate is not sufficiently constrained by the observations: some noise is introduced for each new estimate, which causes the model to “blows up” if prior nonstatistical knowledge about the parameters (like a nominal value, or smoothness) is not supplied to the system.

In our case study, we found that sufficient information is provided if this complementary model for the parameters involves applying (i) a window mean filter to p (with a window of 10 model grid points) and (ii) a relaxation toward the nominal value p0 of the parameters:
i1520-0426-26-3-538-e5
where pkf represents the “forecast values” of the parameters, F(·) is the filter applied to the parameters, and K is a relaxation constant. The values pkf are used instead of pka in the next assimilation cycle as model parameters and as background values for the analysis (see Figs. 2 and 3, where pka must be replaced by pkf everywhere, if the complementary model is used). This procedure stabilizes the sequential estimation of the coefficients by preventing them from drifting away from the reference value and by removing small-scale noise in the estimation. In all experiments, the constant K is set to 0.4 for CH (without that relaxation, the assimilation process gives unstable estimations for CH) and to 0 for CE (for which this proved unnecessary).

The conclusions of the paper are based on a comparison of the three variants of the assimilation scheme described in Figs. 1 –3.

4. Design of the assimilation experiments

a. Experimental setup

The three variants of the assimilation scheme will be tested using twin experiments designed as follows. The reference simulation (the true ocean) is a standard ORCA2 simulation for the year 1993, with the original ORCA2 bulk formula. Figure 4 illustrates the average values and standard deviations of CE and CH in this experiment. Figure 4 shows that the average value of the coefficients is rather non homogeneous over the World Ocean, varying typically between 10−3 (in the equatorial band) and 2 × 10−3 (in the subpolar regions). The time variability of the coefficients is generally quite low, with standard deviations lower than 0.5 × 10−4 except in a few regions where they can be 4–10 times larger: south of the Antarctic Circumpolar Current, along the western coasts of the northern subpolar gyres, along the path of the Gulf Stream and of the Kuroshio, in the eastern equatorial Pacific, etc.

Synthetic observations of temperature and salinity profiles are then sampled from this reference simulation to be assimilated into a perturbed simulation (the false ocean) in which the nominal values of the bulk coefficients p0 = [CE0,CH0] are defined as being constant in space and time (CE0 = 1.18 × 10−3,CH0 = 1.14 × 10−3). These values agree with the range of bulk coefficient values determined in the most popular bulk parameterizations (e.g., Friehe and Schmitt 1976; Blanc 1985; Smith 1988; DeCosmo et al. 1996). The initial condition of the false ocean is kept the same as the initial condition of the reference simulation, so that the only source of difference between the true ocean and the false ocean is model error due to CE and CH. Model error is indeed defined as the forecast error occurring if the model is started from perfect initial conditions.

From this, we can also anticipate that the model error in the false ocean due to inaccurate bulk parameters CE and CH will not usually be centered: the expected value of the model error (the model bias) is not zero (see examples in section 5d). For instance, overestimating CE or CH will systematically lead to underestimated SSTs in the forecast. Such a situation is likely to occur often in practice with real atmospheric forcings, so our demonstration experiments must take this into account. In principle, Kalman filters are not designed to deal optimally with model errors that are not centered: in such a situation, a specific procedure must be added to estimate the model bias separately (Dee and da Silva 1998). On the contrary, the method presented in this paper deals optimally with a systematic model error due to errors in the parameters. This will be discussed in section 5d.

In the assimilation experiments, we assume that we know the false ocean, the initial conditions of the true ocean, and that we have temperature and salinity profiles observed from the true ocean every 10 days, down to a depth of 216 m (i.e., for the first 17 levels of the model), and every two nodes of the model horizontal grid. This sampling of observations is meant to roughly simulate the horizontal and temporal distributions of Argo floats. We do not assimilate observations that are deeper than 216 m because flux errors corresponding to the last 10 days have only a small impact on the deeper layers. With respect to shallow observations, the deep observations would therefore have negligible weight on the correction of the bulk coefficients. Note also that in our experiments no observational update will be applied north of 75°N.

Our aim is to control the false ocean by shifting it toward the true ocean through the assimilation of these observations. We will therefore attempt to obtain an effective control, not only of the state of the system, but also of the bulk coefficients or the air–sea heat and freshwater fluxes. The evaluation will be performed by computing errors on the model state vector and the parameters. In twin experiments, errors can indeed be directly computed as differences with respect to the reference simulation (i.e., the true ocean).

Since the experimental setup is designed in such a way that the only source of error in the system is model error due to the bulk parameters CE and CH, this provides us with an ideal framework for checking the procedure we have designed (given that we have a priori knowledge that the origin of the error in the system can only be in the bulk parameters). Thus, we simply have to demonstrate the possibility of tracing back the errors in the bulk parameters from observations of the state of the ocean. However, we can already anticipate that this framework will only be that perfect at the beginning of the experiments. With time, imperfections in our corrections will result in initial condition errors being present in the subsequent assimilation cycles. Thus, this kind of experimental setup will also allow us to evaluate the robustness of the scheme with regard to the presence of initial condition errors in the system.

Another important feature of this system is that the parameters included in the augmented state vector are not constant over time (see the standard deviation in Fig. 4). Hence, this is more like a forcing function that will be estimated by the assimilation scheme. However, as stated in the introduction, we expect the bulk coefficients to be easier to estimate than the fluxes themselves, because their variations over time are, relatively, smaller and more gradual. For the same reasons, estimating the parameters is also expected to be more useful for improving forecasts. Nevertheless, the simple fact that they are not constant means that the model = 0 for the parameters is inaccurate; there is still model error in the augmented model. If the parameters were constant, the augmented model would be free of model error because model error due to incorrect parameters (which is the only source of model error in our system) in the original system becomes the initial error in the parameters in the augmented system. In such a case, the parameters would be much easier to estimate from the accumulation of observational information over time. However, parameters that vary over time are much more difficult to estimate. This difficulty partly explains why we modified the model = 0 by using the more robust model in Eq. (5). Based on these arguments, we may also anticipate that it will be more difficult to control the augmented system in those regions where the standard deviation of the parameters is greatest (see Fig. 4).

b. Parameterization of background error covariance

As explained in section 3, the key element of the algorithm governing the inversion of the temperature and salinity innovations used to compute the correction for the bulk coefficients is the background error covariance matrix for the augmented state vector . To generate this matrix, we built an ensemble of ocean models, characterized by different values of CE and CH (which are constant in space and time for each model). For every member of the ensemble, CE and CH values are obtained as random numbers sampled from a normal distribution with mean values CE0 = 1.18 × 10−3,CH0 = 1.14 × 10−3 equal to the nominal values of the parameters (corresponding to the false ocean) and a variance of σ = 0.15 × 10−3. The dispersion of the parameters in this ensemble is chosen so as to be consistent with a priori information on parameter error covariance (see Fig. 4).

This ensemble of models is used to perform a set of 10-day forecasts from a series of initial conditions distributed at 10-day intervals over the year 1993 (the period of our assimilation experiments). The 10-day interval is chosen to correspond to the frequencies of observational updates in the assimilation experiments (which are set to the characteristic time scale of the ARGO data collection), and that series of initial conditions is used so that the covariance of the ensemble will be representative of the full period of the experiments. More specifically, a set of five model forecasts is performed for each of the 37 initial conditions, using different random CE and CH values for each of these 185 simulations. An ensemble of 185 forecast anomalies (augmented with CE and CH anomalies) is then obtained by subtracting from each forecast the forecast obtained from the same initial conditions, using the nominal values of the parameters.

The background error covariance matrix is then parameterized using the first 40 empirical orthogonal functions (EOFs) of this ensemble. This covariance matrix is a consistent estimator of the 10-day model error covariance and is adequate for parameterizing the filter background error covariance, assuming that the initial error covariance remains small, cycle after cycle. From this assumption, it can already be anticipated that, over such a short period of time (10 days), the correlation between deep state variables (below the mixed layer) and the bulk parameters is weak, meaning that the observations below the mixed layer are not likely to be useful in correcting the forcing error that occurred in the last 10 days. This is, however, consistent with the target application of this study, which is to produce more realistic short-term forecasts of the thermohaline characteristics of the mixed layer by improving the quality of the air–sea fluxes.

However, it should be pointed out that this ensemble parameterization is inadequate for representing the horizontal error correlation patterns. Indeed, using horizontally constant CE and CH coefficients for every member of the ensemble means that we assume that the error in the coefficients is always uniform worldwide or, similarly, that the horizontal autocorrelation function of the coefficient error is equal to 1 for any pair of geographical locations. To overcome this difficulty, we decided to rely on the local SEEK parameterization to constrain the horizontal correlation pattern. Indeed, reducing the weight of an observation inside each influence bubble as a function of the distance from the center of the bubble is similar to imposing a spatial correlation pattern on the background error covariance matrix [see Brankart et al. (2003) and Testut et al. (2003) for more details]. Hence, the role of the local gain algorithm is essentially to drive the horizontal correlation patterns included in the background error covariance matrix. And the role of the ensemble parameterization described above is to drive (i) the vertical correlation patterns, (ii) the multivariate correlations between the ocean-state variables, and (iii) the multivariate correlations between the ocean variables and the flux parameters. In this way, we will be able to obtain local corrections for the CE and CH coefficients even from an ensemble of simulations with constant coefficients worldwide.

5. Comparison of the three variants in the assimilation scheme

According to the experimental setup described in the previous section, three assimilation experiments were performed using the three different variants of the scheme presented in section 3 and summarized in Figs. 1 –3. All three experiments use the same background error covariance matrix (same EOFs) described in section 4b. In this section, the results of V0 will be compared to those obtained using the new algorithms (V1 and V2), including the surface flux correction.

a. Time evolution of temperature and salinity errors

Figure 5 shows the evolution of the error variances in SST and SSS over the World Ocean (except the northern polar zones). The black solid line in Fig. 5 shows the error corresponding to the free simulation, without data assimilation (i.e., the false ocean). The green, yellow, blue, and red lines in Fig. 5 correspond to V0, V0*, V1, and V2, respectively. (The yellow curve in Fig. 5 shows the results for the variant V0*, which will be discussed in section 5d.) The solid lines in Fig. 5 represent the continuous analysis (see Figs. 1 –3) and the bullets represent the 10-day model forecasts. Figure 6 presents the same information for the Atlantic Ocean only. (The error in the free simulation is not shown in Fig. 6 because it exhibits the same kind of behavior as in Fig. 5.)

First, we observe that V0 is already able to control the model error due to inaccurate bulk parameters: the large SST and SSS drifts of the false ocean with respect to the true ocean are easily canceled by the assimilation of the temperature and salinity profiles. The analysis is stable over time for both variables, and the error variances are generally 3–6 times lower than the corresponding values in the free run. This means that the reduced space, in which the error covariance matrix is parameterized, has been correctly defined. Most of the temperature and salinity innovations have been captured by the reduced-order analysis and interpreted as a correction on the full-model state vector.

Second, it may be observed that the forecasts obtained from V1 and V2 are always better than those of V0. The poor quality of the V0 forecast is easy to understand because it is always performed using the false model operator M (x,p0) (i.e., without correction of the bulk parameters). The error increase in the forecast is smaller using V1 and V2, meaning that the correction of the bulk parameters has a positive impact on the subsequent model forecasts of SST and SSS. The model operators modified using the bulk parameters estimated by the assimilation scheme are thus more accurate than the false model operator used to forecast SST and SSS. A large part of the 10-day model error standard deviation for SST and SSS has been directly eliminated by correcting the model parameters.

Third, the analyses and forecasts obtained using V2 rapidly become superior to those obtained using V0. This is an indirect consequence of the improved forecasts, as they lead to smaller innovations that can be interpreted more easily by statistical analysis. Smaller model errors for SST and SSS mean smaller errors accumulating from one cycle to the next and better analysis of the evolution of the system. However, V1 behaves differently. Correcting the bulk parameters only is sufficient to stabilize the error standard deviation at values that are much smaller than those of the free simulation, although the results are not as accurate as those obtained with V2. Moreover, in Fig. 5, the SST and SSS error variances of V1 exhibit large increases over the last four months of the experiments. This increase is localized in the western tropical Pacific Ocean (see Fig. 7, described in section 5b), which is why it is omitted from Fig. 6. The reason for these differences between V1 and V2 is that correcting only the parameters quickly becomes insufficient as soon as there are initial errors present at the beginning of the assimilation cycles. These initial errors accumulate from the slightly inaccurate corrections of the parameters in the previous assimilation cycles. To deal with these errors, state corrections in V2 are necessary.

b. Distribution of temperature and salinity errors

Figure 7 shows maps of the standard deviation errors for SST and SSS 10-day forecasts for the three variants of the scheme. The maps show the spatial distribution of the errors instead of their evolution over time as in Fig. 5. Again, the standard deviation error computed for V0 is larger everywhere than the corresponding errors in V1 and V2. As expected, the largest remaining errors are found where the time variance of the parameters is the largest (cf. Fig. 4): in the Gulf Stream or Kuroshio regions, in the confluence region, etc. Generally speaking, however, the standard deviation errors in the SST and SSS 10-day forecast are significantly reduced by the correction of the parameters especially in regions where the error was large in V0, making the resulting error maps more homogeneous over the World Ocean.

Finally, in order to provide an idea of the vertical structure of the standard deviation errors, in the temperature and salinity 10-day forecasts, Fig. 8 shows a meridional section across the Atlantic at 30°W. The top plots Fig. 8 (for V0) show the error on the 10-day forecast without corrections on the parameters. At this time scale (10 days), errors in the atmospheric forcing can only have a significant impact inside the ocean surface mixed layer. In this simulation, the largest errors occur in the North Atlantic and South Atlantic subpolar gyres. As Fig. 8 indicates, both parameter correction algorithms (V1 and V2) are able to significantly reduce the error with respect to V0. This error reduction is effective down to the bottom of the mixed layer (where most of the error occurs), meaning that a correct history of the heat and salt vertical transfers is restored in the water column when the bulk parameters are controlled. This is also consistent with the statement that the error correlations between the bulk parameters and deep state variables are weak, so that deep observations should have little impact on the correction. Figure 8 also reveals that V2 produces better results than V1, although this does not prevent patches of residual errors from remaining, especially in the subpolar gyres.

c. Quality of the parameter estimates

In the previous section, the three variants of the assimilation scheme were compared according to their effects on temperature and salinity errors. However, good results can sometimes be obtained for the wrong reasons. Therefore, we have to analyze the method’s capacity to produce reliable estimates of the model parameters. It is only if the parameters are improved by assimilation that the real source of error has been properly identified, and that the reduction in the error on temperature and salinity values has occurred for the right reasons.

Figure 9 shows maps of standard deviation errors on the CE and CH coefficients (i.e., the root-mean-square difference with respect to the CE and CH values in the true ocean) for the three variants of the scheme. Remember that in V0, the coefficients used to produce the forecasts are the constant values defining the false ocean. It is thus the standard deviation error for these constant values that is presented in Fig. 9. As can be seen in Fig. 9, there are large discrepancies between the constant parameters of the false model and the parameters of the reference simulation almost everywhere in the World Ocean, with the error values being especially large in the subpolar regions. This type of error is undoubtedly the reason behind the poor quality of the forecasts of SST and SSS described in the previous section.

Clearly, the standard deviation error of the CE coefficient calculated from V0 is generally much larger than the corresponding error values for V1 and V2. This is also true for the standard deviation error of the CH coefficient except in the western tropical Pacific (for V1 and V2) and along the path of the Gulf Stream and Kuroshio (for V1). In these regions, the error in CH can become much larger than in the V0 experiment especially if V1 is used. Although it is difficult to speculate about the origin of such behavior, the most likely explanation (partly confirmed in section 5e) is that, in these regions, the sensible heat flux is very small [because TwTa; see Eq. (3)], so that errors in the CH parameter have virtually no impact on the ocean state. Consequently, it is very difficult to infer a correction to CH from temperature and salinity differences: the observability of CH through temperature and salinity becomes quite bad as soon as the linear relation between flux and CH becomes close to singular (i.e., with a coefficient TwTa close to zero).

On the other hand, as a general rule, the largest remaining errors occur where the time variance of the parameters is the largest (cf. Figs. 4 and 9). However, variants V1 and V2 of our system demonstrate that they are able to correctly reproduce the spatial distribution of the true coefficients from their constant values in most regions of the World Ocean for the entire period of our experiments. The overall quality of the bulk coefficients for these two methods is quite similar, though V2 provides more reliable results for the CH coefficient.

Note that the possibility of controlling these two coefficients through temperature and salinity observations could be anticipated. We chose two coefficients that linearly govern the value of two important contributions to the heat flux and one important contribution to the freshwater flux. These two fluxes also have a roughly linear impact (assuming that the mixed layer depth remains fairly constant during the 10-day forecast) on the surface value of two ocean variables: the temperature and the salinity. It may therefore be expected that, by observing these two variables, it would be possible to trace back a correction to the bulk coefficients.

d. Correction of systematic errors

The improvements to the parameters were presented in the previous section, but we need to analyze in more detail how the system actually works. How do the parameters evolve over time compared to the true ocean? What are the statistics of the temperature and salinity innovations, and the corresponding statistics of the parameter corrections? Figure 10 (first column) shows the evolution of the CE parameter at two specific locations in the Atlantic (16°S, 2°W and 36°N, 60°W) for the true ocean, the false ocean, and for V1 and V2. (Parameter values in V0 are constant and equal to those of the false ocean.) The first thing that we can see in Fig. 10 is that the parameter value in the false ocean is very different from the averaged parameter value in the true ocean. This implies that, at this ocean location, the model error in the false ocean model = M(x,p0) is biased. This is also obvious in Fig. 10 (second column), which shows the surface temperature innovations at the same location. The average innovation in V0, using the model = M(x,p0) is not close to zero, but is systematically smaller given that CE is systematically smaller than in the true ocean. In V0, the model is biased due to a systematic error on the parameters p0.

On the other hand, we can see that in V1 and V2 the average coefficient value is similar to that of the true ocean, the temperature innovations are centered, and the parameter corrections are also centered (see Fig. 10). How can this new behavior be interpreted? The reason for it is that variants V1 and V2 of the assimilation scheme work in the augmented state space, where the model = M(x,p), = 0 is unbiased. We stated in section 4a that model error remains in the model for the parameters = 0 since the parameters are not constant, but it is easy to see that this model is unbiased. Indeed, unless there is a drift in the parameter value (which is unlikely for CE and CH), the distribution of the model error for the model = 0 is obviously centered. Thus, the augmented state vector methodology has transformed the biased model error in the model state space into unbiased model error in the augmented state space [as explained in Dee (2005)], together with a (possibly large) initial error on the parameter value: p(t = 0) = p0.

Thus, we have been able to show that the augmented state vector method is appropriate for dealing with model biases that are due to inaccurate model parameters. However, we have to admit that, since the model = M(x,p0) is biased, V0 cannot be considered to be an optimal data assimilation scheme. One could argue that a classical method for correcting the bias [e.g., the one proposed by Dee and da Silva (1998)] could have provided a better solution than V0, and that V1 and V2 should have been compared with such a solution. There are two possible answers to this question. First, most ocean assimilation systems drive models with biased atmospheric forcing, using a scheme of the V0 kind. It is this kind of assimilation system that our method is intended to improve. Second, correcting the model parameters that are known to be the source of the bias is certainly more effective than using a general bias identification technique that would be much more difficult to implement and to parameterize. (Identifying the error on a few parameters is undoubtedly much easier than identifying a bias on the full state vector.)

In order to illustrate this argument, the two new techniques, V1 and V2, will now be compared to a modified V0 scheme, denoted as V0*, in which the model bias is explicitly corrected. Since we are using twin experiments, a “perfect” value of the bias can be estimated directly in the following way. We compute a series of 10-day forecast using the false (biased) model from a series of the true ocean states as initial conditions (distributed in time every 10 days in 1993). The 10-day forecast bias is then estimated as the time average of the differences between these 10-day forecasts and the true ocean. It is this “perfect” value of the 10-day forecast bias that is subtracted from each 10-day forecast in V0*, so that V0* is just V0 with a perfect forecast bias correction in the model state space (better than any possible bias identification scheme). Despite this, the results show that V2 is most often preferable to V0*. This is illustrated in Figs. 5 and 6, showing that the error variances in the SST and SSS 10-day forecasts (with bias correction in V0*) are always twice as large in V0* as in V2. Figure 10 (middle column) shows the temperature innovations corresponding to V0*. Unlike V0, the innovations are centered in V0* because of the bias correction. The innovation values of V0*, however, usually remain larger than in V1 and V2, which is consistent with Figs. 5 and 6.

In addition, as we can gather from Fig. 10, the method is not only able to identify a correction to the average value of the parameters; the low-frequency variations of the parameters are also correctly captured by the assimilation scheme. For instance, at the first location (16°S, 2°W), the CE coefficient increases in the reference simulation from 1.25 × 10−3 in January to 1.4 × 10−3 in July and then decreases to 1.35 × 10−3 between July and December. At the second location (36°N, 60°W), on the other hand, the CE coefficient decreases in the reference simulation from 1.45 × 10−3 in January to 1.25 × 10−3 in August and then increases to 1.45 × 10−3 between August and December. We can see that these seasonal variations of the parameters are correctly reproduced by the assimilation of the temperature and salinity data.

e. Quality of the flux estimates

Since both SST and SSS are improved in V1 and V2 with respect to V0, and since the bulk parameters are also improved, we can almost be certain that the fluxes themselves computed using Eqs. (1) and (3) will also be improved. This is illustrated in Fig. 11, which shows the spatial distributions of the standard deviation error values on the latent (top plots) and sensible (bottom plots) heat fluxes for V0 (left-hand column), V1 (center column), and V2 (right-hand column). The evaporation flux is linked to the latent heat flux by Eq. (2), so that the results for evaporation are obvious from the latent heat flux results. The fluxes errors corresponding to the false ocean and to V0* are not represented in Fig. 11 because they are similar to those of V0.

Figure 11 also reveals that, except in the western equatorial Pacific for the sensible heat flux, the standard deviation error corresponding to V0 is everywhere larger than the corresponding errors in V1 and V2. As expected, the largest errors occur where the parameter errors are the largest, and the greatest improvements in the fluxes also occur in these regions. In particular, the V1 scheme is slightly more favorable than the V2 scheme on the latent heat flux, because of the slightly more accurate latent heat flux coefficients. Concerning the regions where the estimation of the sensible heat flux coefficient (CH) is inaccurate (especially for V1; see section 5.c), we observe that the sensible heat flux remains correct (except for a slight error increase in the equatorial–subtropical Pacific), confirming that, in these regions, the sensitivity of sensible heat flux errors to CH errors is small (due to a small differences between the air and sea temperatures). We can see that both new methods provide results of similar quality overall so that, using this criterion only, it is not easy to say which of the two variants is the best. The conclusion would not be the same in every region of the ocean.

f. Quality of the model forecasts

As stated previously, one objective of the correction of the model parameters is to improve the model forecast. The evaluation of the forecast that has been presented up to now has dealt with the 10-day forecast, that is, the model forecast that provides the background state for the next observational update. The purpose of this section is to analyze the improvements in forecast quality (for SST) over longer time scales. Figure 12 shows the evolution in the SST error variance for a 3-month forecast. The forecast is performed for each of the three variants of the scheme. The initial conditions for each forecast are the analysis from the corresponding simulation on 4 April 1993, 4 months after the beginning of the assimilation experiments. The parameters are the ones that would have been used to produce the 10-day forecast in each variant. For V0, the parameters are set to their false values, while for V1 and V2 they are set to the improved values produced by the complementary model [Eq. (5)]. These experiments amount to extending the 10-day forecast of that assimilation cycle (the one beginning on 4 April 1993) to a 3-month forecast.

The results show that the error variances for V0, V1, and V2 increase monotonically during the entire period of the 3-month forecast. Both V1 and V2 provide significantly better quality model forecasts, with error values that are one order of magnitude smaller than in V0. As expected, better parameters also improve the long-term forecasts, as a consequence of their relatively small and slow time variations. The same 3-month error evolution cannot be shown for V0* because the forecast bias has only been computed for a 10-day forecast and not for other forecast periods. This illustrates another advantage of our technique over bias estimation methods working in the state space, which make it more cumbersome to estimate the forecast bias for several forecast periods.

6. Conclusions

The objective of this paper was to investigate ways of improving the turbulent air–sea fluxes by a sequential data assimilation approach, with the aim of better controlling and forecasting the ocean mixed layer characteristics. For that purpose, the control vector of the assimilation scheme has been augmented with the parameters that must be estimated, and the covariance matrices in the augmented control space have been computed using ensemble experiments. Two specific procedures have been proposed. The first procedure (V1) focuses on the correction of the model parameters, without any possibility of correcting other sources of errors. This should be useful for ocean modelers who want to improve simulations that have been corrupted by systematic errors in the air–sea fluxes; and can be viewed as an optimal flux correction using ocean observations (like SST and SSS). It would be advantageous to use it to replace the classical Newtonian relaxation to surface temperature and salinity data, without introducing any nonphysical term into the model equations. The second procedure (V2) combines a correction of the model parameters with a correction of the full-model state vector. This was designed as a complement to an existing assimilation scheme, thus making it possible to deal with other sources of errors, controlled by other kinds of observations.

In addition, errors in the model parameters most often lead to a model bias. It was thus natural to study a problem in which a model bias is present. The conclusion is that the method is able to deal optimally with such model bias, which comes from model parameters included in the augmented control vector, transforming the biased model error in the model state space into unbiased errors in the augmented state space. The procedure even gives better results than an ideal bias identification procedure working in the state space.

The results of twin experiments presented in the paper show that both procedures can lead to accurate estimations of the two parameters included in the control vector. They also show that, at least in this ideal situation, significant improvement in the forecasting of the mixed layer thermohaline characteristics can be obtained. Better results are obtained if both the state vector and the parameters are corrected; correcting the parameters alone quickly becomes insufficient as soon as initial errors, resulting from slightly inaccurate corrections in the previous assimilation cycles, begin appearing in the system. These experiments were conducted with synthetic observations of temperature and salinity profiles that roughly simulate the horizontal and temporal distributions of Argo floats expected in the near future. The positive results gives us a background for future work in which this method will be applied to the more realistic problem of assimilating real data profiles, or satellite observations of surface temperature and salinity.

Our experiments, however, are ideal in the sense that the only source of error is due to two bulk coefficients, CE and CH: the atmosphere and the ocean initial conditions were both perfectly known. In realistic applications, the sources of error are more diverse, requiring the control of additional forcing parameters, such as cloud cover or precipitation. It can also be expected that increasing the number of parameters to be identified may require more careful Monte Carlo simulations, with refined prior assumptions about the structure of the parameter error covariance. For instance, a low-order representation of the horizontal error covariance in parameter space (e.g., using their time variability in a free model simulation) should facilitate the identification of a larger number of parameters.

In summary, to go further, several questions still need to be answered. For instance, what is the impact of the initial errors in the ocean state? How many bulk parameters is it possible to control using only oceanic observations? What perspectives may be provided by the complementarity of space and in situ data? In other words, is it possible to control the full system, that is, the ocean state and the flux parameters, with the available ocean observation system?

Acknowledgments

This work was conducted as part of the MERSEA project funded by the European Union (Contract AIP3-CT-2003-502885), with additional support from CNES. The calculations were performed with the support of IDRIS/CNRS.

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Fig. 1.
Fig. 1.

The assimilation process for the original variant of the scheme V0. The dashed lines show the model forecast and the solid line shows the analysis. The analysis is obtained by adding the observational update at time tk in the model operator between times tk−1 and tk. The forecast is obtained using the nominal values of the parameters p0.

Citation: Journal of Atmospheric and Oceanic Technology 26, 3; 10.1175/2008JTECHO603.1

Fig. 2.
Fig. 2.

The assimilation process for the variant V1. The dashed lines show the model forecast, and the solid line shows the analysis. The analysis is obtained by modifying the model operator between times tk−1 and tk using the parameters pka estimated from the observations at time tk. The forecast between times tk and tk+1 is obtained using the updated values of the parameters pka.

Citation: Journal of Atmospheric and Oceanic Technology 26, 3; 10.1175/2008JTECHO603.1

Fig. 3.
Fig. 3.

The assimilation process for the variant V2. The dashed lines show the model forecast, and the solid line shows the analysis. The analysis is obtained by adding the observational update of time tk, in the model operator between times tk−1 and tk, without modifying the model parameters pk − 1a. The forecast between times tk and tk+1 is obtained using the updated values of the parameters pka.

Citation: Journal of Atmospheric and Oceanic Technology 26, 3; 10.1175/2008JTECHO603.1

Fig. 4.
Fig. 4.

(top) Average values and (bottom) standard deviations of the turbulent exchange coefficients (left) CE and (right) CH in the reference model (original ORCA2 bulk formulation). The values are scaled by a factor of 103.

Citation: Journal of Atmospheric and Oceanic Technology 26, 3; 10.1175/2008JTECHO603.1

Fig. 5.
Fig. 5.

(top) Evolution of error variances in (left) SST (°C) and (right) SSS (psu) for the World Ocean (except the northern polar zones). The black solid line shows the error corresponding to the free simulation (without assimilation). The green, yellow, blue, and red lines correspond to V0, V0*, V1, and V2, respectively. The solid lines represent the continuous analysis, while the bullets represent the 10-day model forecasts. (bottom) A duplication of the (top) but with a finer y-axis scale.

Citation: Journal of Atmospheric and Oceanic Technology 26, 3; 10.1175/2008JTECHO603.1

Fig. 6.
Fig. 6.

Evolution of error variances in (left) SST (°C) and (right) SSS (psu) for the Atlantic Ocean only. The green, yellow, blue, and red lines correspond to V0, V0*, V1, and V2, respectively. The solid lines represent the continuous analysis and the bullets represent the 10-day model forecasts.

Citation: Journal of Atmospheric and Oceanic Technology 26, 3; 10.1175/2008JTECHO603.1

Fig. 7.
Fig. 7.

Spatial distributions of the standard deviation errors in (left) SST and (right) SSS 10-day forecasts for (top) V0, (middle) V1, and (bottom) V2.

Citation: Journal of Atmospheric and Oceanic Technology 26, 3; 10.1175/2008JTECHO603.1

Fig. 8.
Fig. 8.

Standard deviation error values in (left) temperature (°C) and (right) salinity (psu) 10-day forecast profiles for (top) V0, (middle) V1, and (bottom) V2.

Citation: Journal of Atmospheric and Oceanic Technology 26, 3; 10.1175/2008JTECHO603.1

Fig. 9.
Fig. 9.

Standard deviation errors in (left) the latent heat flux exchange coefficient CE and (right) the sensible heat flux exchange coefficient CH, for (top) V0, (middle) V1, and (bottom) V2.

Citation: Journal of Atmospheric and Oceanic Technology 26, 3; 10.1175/2008JTECHO603.1

Fig. 10.
Fig. 10.

Detailed statistics for two specific locations of the World Ocean: (top) 16°S, 2°W and (bottom) 36°N, 60°W: (left) CE parameter evolution in time, (middle) temperature innovations, and (right) CE parameter corrections. The black dashed line stands for the true ocean, and the green, yellow, blue, and red lines correspond to V0, V0*, V1, and V2, respectively. [The green curve in (right) displays the correction to the parameters that is computed in V0 but never applied.]

Citation: Journal of Atmospheric and Oceanic Technology 26, 3; 10.1175/2008JTECHO603.1

Fig. 11.
Fig. 11.

Spatial distribution of standard deviation error values on the (left) latent heat flux and (right) sensible heat flux for (top) V0, (middle) V1, and (bottom) V2. Unit: W m−2.

Citation: Journal of Atmospheric and Oceanic Technology 26, 3; 10.1175/2008JTECHO603.1

Fig. 12.
Fig. 12.

Evolution of the (left) SST (°C) and (right) error variance for a 3-month forecast corresponding to V0, V1, and V2 (green, blue, and red curves, respectively). The forecasts start on 4 Apr 1993 (4 months after the beginning of the experiments).

Citation: Journal of Atmospheric and Oceanic Technology 26, 3; 10.1175/2008JTECHO603.1

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