Commonly called SAM2, the current protocol for data assimilation at Mercator Ocean (Lellouche et al., 2013) computes correction over a 7-day assimilation window and is based on a modified Kalman filter named SEEK (singular evolutive ensemble Kalman filter) first introduced by Pham et al. (1998). Analysis is calculated at the middle of the assimilation window, i.e. the fourth day. The SEEK filter means, as explained by Brasseur and Verron (2006), that covariance error matrices are forced at a low rank (“Singular”) and that it computes model error covariances propagation (“Evolutive”) following the model dynamics.

The filter used in SAM2 is not evolutive in the same way as SEEK. Indeed, instead of using empirical orthogonal functions to build its error covariance matrix that will be propagated onto the model along time steps, SAM2 takes a fixed base of smoothed model anomaly fields (349 in the following experiments). This approach allows the system to get a covariance matrix that is realistic with the climatological statistics of the ocean model at the time step and saves computation time as this matrix will not be propagated in the model unlike the SEEK. Anomalies for the five control variables (sea level, U, V, T, S) are calculated from a 10-year free model, and at each date they are equal to the difference between the free run and a running mean along time over itself. At the date of an analysis, only anomalies within the past 30 days and future 30 days and from the different years are considered. The final number of anomalies that are kept for a given analysis is equal to 349. This means that the anomaly basis changes at each analysis date and follows the global model climatology. These anomalies are selected accordingly to the season of the assimilation cycle to get a basis evolving consistently with the model climatology. Our filter is not evolutive as the model error covariance is not propagated by the dynamical model. The model correction is calculated as a linear combination of the selected anomalies. Then, this correction is injected linearly over the 7 days using the incremental analysis update (IAU; Bloom at al., 1996). As explained in Lellouche et al. (2013), when in situ measurements are assimilated, a bias correction based on a 3-D Var approach is used to correct large-scale and slowly evolving errors in T, S and thus dynamical height. Bias correction uses a collection of temperature and salinity innovations from the previous 3 months and creates a correction to be added in the model's prognostic equations. Here we kept the set-up of the assimilation scheme as it is in the operational system and described in Lellouche et al. (2013) except for the following points: we did not take into account the representativity errors, we have assimilated the full sea surface height (SSH) signal instead of the sea level anomaly (SLA), and we used a uniform observing error covariance matrix (3 cm in RMSE).

In this study, both the nature run (NR) and the assimilated run (AR) are based on the NEMO model (Nucleus for European Modelling of the Ocean; Madec et al., 1998) with a global coverage and 50 vertical levels, with 22 levels within the upper 100 m and with 1 m resolution from the first level, increasing with depth up to 450 m for the last one. The system uses the OPA (Océan Parallélisé) model coupled with the LIM2 ice model (Fichefet and Morales Maqueda, 1997). The difference between the two configurations is that the NR uses a 1/12∘ tripolar grid (ORCA12) and the AR a 1/4∘ tripolar grid (ORCA025). Both models are forced using the CORE (Coordinated Ocean-Ice Reference Experiment) bulk formulation (Large and Yeager, 2009). The 1/12∘ free model is chosen for NR because it is a good estimation of the true ocean variability. The 1/12∘ NR was chosen for its capacity to better represent mesoscale variability (50–500 km) in the ocean compared to a 1/4∘ resolution simulation (Hulburt et al., 2009). Assimilating data from a higher-resolution model into the 1/4∘ configuration is a way to determine how these structures, underestimated in a free 1/4∘ model, can be forced to be closer to reality (NR).

The OSSEs were started from 7 January 2009 over an almost 1-year time period. Two different initial conditions (i.e. 7 January 2009) for the NR and for the AR are required so that we can quantify the impact of assimilating pseudo-observations of the NR in the AR. This was achieved by running the two free-run NEMO configurations initialised from climatology but at different times. The NR simulation was started in 2003 and forced with ECMWF (European Centre of Medium Weather Forecasting) operational 3 h atmospheric data and the AR was initialised from a 1/4∘ free run started from 1989 and forced by ECMWF ERA-Interim 3 h atmospheric data. The OSSEs are all forced with the ECMWF operational 3 h data. Note that as AR and NR are both forced by ECMWF operational data, our OSSEs do not address the impact of atmospheric forcing errors.

To assess the impact of the number of altimeter data, three satellites have been considered: Jason-1, Jason-2 and Envisat (Fig. 1a). Jason-1 and Jason-2 have a 10-day repeat cycle and Envisat a 35-day repeat cycle. Jason-1 was in its interleaved orbit with its ground tracks just in between Jason-2 tracks and with a time shift of 5 days. This orbit was chosen to optimise mesoscale variability sampling by Jason-1 and Jason-2. The OSSEs were carried out over the year 2009. Jason-1, Jason-2 and Envisat simulated observations were derived from the NR with a resolution of 7 km between two points along the tracks. An observation white noise of 3 cm rms was simulated and added to these pseudo-observations.

Mercator Ocean operational systems assimilate SLA observations. The absolute sea level (i.e. sea level relative to the geoid) is obtained by using an external mean dynamic topography (MDT) based on the CNES-CLS MDT. In our case, the nature and assimilated runs have different MDTs because of the grid resolution, the model parametrisations and different initialisation procedures. We thus chose to assimilate the absolute sea level (which include the MDT and the SLA) from the NR at 1/12∘.

Argo in situ temperature and salinity observations from the surface down to 2000 m were simulated using the 2009 Argo profile positions in the Coriolis CORA3.2 database (Fig. 1b).

The four different OSSEs that have been carried out are summarised in Table 1. The first three simulations address the question of the number of altimeters required to constrain ocean analyses and forecasts. There are three experiments with one (Jason-2), two (Envisat and Jason-2) and three (Jason-1, Envisat and Jason-2) assimilated satellite data sets. They are respectively called Sat1, Sat2 and Sat3 experiments. The other OSSE addresses the impact of Argo profiling floats together with the three satellite data sets.

All the assimilated experiments start on 7 January 2009 and end 30 December 2009. The difference between a given simulation and the NR are used to derive statistics on errors on analyses and forecasts over the last 7 months (June–December 2009). For each assimilation experiment, time series of errors on analyses and forecasts (up to 7 days) are obtained. Seven-day forecast errors will be used in this study.

Figure 2 shows the mean square error (MSE) for the free run (FR) and for the analyses and forecasts of the three different assimilation runs (Sat1, Sat2 and Sat3) estimated as the difference with the NR. As expected, the FR shows large differences with the NR as they provide two uncorrelated mesoscale variability fields. Assimilation of one satellite leads to a significant reduction of both analysis and 7-day forecast errors due to a strong correction of the mean sea level. Adding a second altimeter significantly reduces the errors. The impact of assimilating a third altimeter remains positive but not as large as the addition of a second altimeter. Moreover, errors are largely reduced between the 7-day forecast and the analysis for each of the three assimilation runs.

The evolution in time of the global MSE of sea level for both the analysis and 7-day forecast fields is shown in Fig. 3. The system constrained by the 1/12∘ simulated SSH observations converges toward a stable state in 2 to 3 months. The free-run MSE is about 97 cm2 (not shown on the plot) over the time period of the experiment; it is reduced to 20 cm2 in Sat1. The analysis MSE in Sat2 is lower than in Sat1 and approximatively equal to 15 cm2. Sat3 provides a slight improvement of a few square centimetres compared to Sat2. In fact, the first altimeter brings the biggest error reduction compared to the free run but the second and third altimeters keep reducing this error.

To analyse further the structure of errors in areas of high mesoscale variability, MSEs for analyses and 7-day forecasts are shown for the GS (Figs. 4 and 5), AC (Figs. 6 and 7) and KU (Figs. 8 and 9) regions. Diamond-like structures can be seen on the analysis error maps for all regions when only one altimeter is assimilated revealing the repetitive spatial sampling of Jason-2. Adding Envisat observations suppresses this effect. In those energetic regions, the MSE for the free run is very high in the core of the main current. The increase in the number of assimilated altimeter data sets allows for a clear reduction of both 7-day forecast and analysis errors.

To summarise results shown on the different maps, the following score is defined as the MSE for a given AR in percentage of the free-run MSE:

σ=100×Meansquare(errorAR)Meansquare(errorFR).

These statistics are presented in Table 2.

The greatest impact is made with the assimilation of the first altimeter which strongly reduces the large-scale biases existing between the NR and FR. Sat1 sea level global analysis MSE reaches 21 % of the free-run MSE. Adding a second satellite (Sat2) reduces the analysis errors by 6 %. The third satellite (Sat3) reduces further the errors by about 2 %.

Compared to Sat1 global analysis MSE, Sat2 analysis MSE is reduced by 28 % and for Sat3 compared to Sat2 error is reduced by 11 %. In high-eddy-energy regions this ratio can reach respectively 42 and 22 %.

For the same assimilation experiment, the analysis error is always lower than the 7-day forecast error. The error level of the analysis with one altimeter is close to the 7-day forecast error level when two or three altimeter data sets are assimilated. This is true for all of the considered regions and globally (Table 2). The largest error reduction due to data assimilation occurs in the Agulhas and Kuroshio regions.

The error increase between the analysis and 7-day forecast for each experiment highlights the “model predictability” in the different regions. The relative MSE in percent between analysis and forecast increase is 28 % globally for Sat1, 35 % for Sat2 and 37 % for Sat3. In western boundary currents (WBCs), values are around 34 % for Sat1, around 49 % for Sat2 and 54 % for Sat3. The error increase is thus the largest when more altimeter data are assimilated. Analyses are thus better constrained, but this does not fully translate into improved forecasts.

Note that as the NR and the AR use the same atmospheric forcing, 7-day forecast errors are only related to internal mesoscale dynamics and initialisation issues.

Estimation of the sea level wavenumber spectrum from altimetry data (e.g. Le Traon et al., 1990; Stammer, 1997; Le Traon and Dibarboure, 2008) has allowed major progresses in the characterisation of ocean mesoscale dynamics. Wavenumber spectra are used here to characterise sea level analysis and 7-day forecast errors in the Gulf Stream, the Agulhas Current and the Kuroshio regions.

Wavenumber spectra were calculated from the sea level model error fields using fast Fourier transform (FFT). The FFT was applied in 10 × 20∘ boxes within the previously defined WBCs regions but did not fit exactly to the areas shown on the maps. Longitudinal spectra were estimated from daily error fields and meridionally averaged. Figure 10 shows the mean sea level error spectrum calculated in the GS (Fig. 11a), AC (Fig. 11b) and KU (Fig. 11c) regions. The computation is made from June to December 2009 both for the analysis and for each 7-day forecast of the assimilation cycle.

The error reduction due to altimeter data assimilation is visible for all of the three selected regions: the free model run error spectrum is higher at all wavelengths larger than 100 km. The assimilation corrects the 1/4∘ model sea level below its own capacity to represent small scales. Below this limit of 100 km, all the simulations are gathered in one curve. This curve follows the same slope as the full sea level spectrum of the nature run (not shown in the plot).

As seen before, the error is reduced each time an additional altimeter is assimilated, for all wavelengths larger than 100 km and up to 1000 km. It is also the case for the analysis compared to the 7-day forecast. Analysis of spectra in a variance preserving form (Fig. 11) shows that, compared to analysis errors, 7-day forecast errors occur at larger wavelengths; they have a maximum variance at wavelengths of 300–500 km, while it is about 200–300 km for analysis errors.

Compared to the free-run errors, adding one satellite (Sat1) reduces analysis errors for all wavelengths larger than 250 km. Addition of a second (Sat2) and third (Sat3) altimeter allows for the reduction of analysis errors down to 150 km wavelength. In the KU and the GS regions, the Sat2 and Sat3 analysis errors are similar for most of the length scale. In the AG region, the assimilation of the third satellite still allows for significant analysis error reduction.

In most cases, the 7-day forecast error spectrum for the Sat3 experiment is lower than the analysis error for the Sat1 experiment for wavelengths smaller than 300 km.

To assess the system ability to reproduce the nature run, it is necessary to analyse how non-assimilated model variables are improved when assimilating sea level altimeter data. The unobserved variables are impacted by assimilating only sea level observation through two mechanisms. The first one is the multivariate characteristic of the analysis corrections computed by SAM2. The model error covariance matrix is defined with a collection of model anomalies used to calculate increment for all the model prognostic variables, SL, U, V, T and S. The second one is the non-linear model dynamics that implies changes on temperature, salinity and velocities when the SSH analysis correction on sea level is added to the model 7-day forecast.

Because of geostrophy, we expect, in particular, that assimilating more altimetry data will better constrain surface velocity fields. Figure 12 presents the MSE of analysis and 7-day forecast for the surface zonal velocity U. The free run shows everywhere higher values for the velocity MSEs both for U and V (not shown).

Table 3 shows the same score as the one used for the sea level but for the MSE of the analysis and 7-day forecast errors of the zonal and meridional velocity components in cm2 s-2. Globally and in the Gulf Stream region, the meridional and zonal velocity MSEs are similar; meridional velocity MSEs are slightly higher (∼ 10 %) than zonal errors in the Agulhas Current and slightly lower (10 % again) in the Kuroshio.

The absolute MSEs decrease from Sat1 to Sat3, and are much lower than the free run. For each experiment, the analysis error is again reduced compared to the 7-day forecast error. The level of error for the 7-day forecast of Sat3 is, in most regions, comparable to the level of the analysis error of Sat1. The assimilation of a second satellite leads to a higher error reduction than the third one, for both analysis and 7-day forecast and in all regions.

Sat1 global analysis velocity MSEs represent 55 % of the free-run MSEs. Additional error reductions of 10 and 4 % occur for Sat2 and Sat3. In high-eddy-energy regions (GS, AC, KU), the analysis MSEs are smaller and can reach 35 % of the free-run MSE for Sat1; they continue to be reduced by 13 and 4 % for Sat2 and Sat3 (on average in the WBCs).

Seven-day forecast surface velocity errors are less reduced when an additional altimeter data set is assimilated. They globally represent 64, 56 and 53 % of the free-run MSE for respectively Sat1, Sat2 and Sat3.

Assimilation of multiple altimeter data does not only improve the surface velocity; it also improves velocity fields at depth. Figure 13 shows global RMSE profiles for U and V. These plots are similar for the two velocity components and show decreasing error profile with depth. There is a clear positive impact of the assimilation of additional altimetry observations up to 2000 m depth. The improvement brought by each additional satellite is almost uniform on the vertical and even the third altimeter improves the 3-D velocity field estimation.

Assimilating sea level altimeter data also improves the temperature and salinity at depths as shown on RMSE profiles for temperature (T) and salinity (S) of Fig. 14. Temperature error profiles show a maximum at the thermocline depth as the salinity error decreases with depth. Globally, Sat1 gives a good improvement for T at depth, compared to free run with 0.2 ∘C of RMSE in temperature at 200 m depth. Sat2 and Sat3 are not distinguishable and only improve the RMSE score by less than 0.05 ∘C. The experiments with altimeter data assimilation only slightly improved salinity fields. Sea level as measured by altimetry is to a large extent the signature of baroclinic processes and represents an integral of the density anomaly. As density variations are mainly correlated with temperature variations and less to salinity variations in most of the ocean regions, this explains why assimilating altimeter data improves the representation of the upper temperature fields (e.g. Guinehut et al., 2012).