Anderson (2003) proposed a sequential localization scheme for the EnKF that can be carried out in two steps when processing a single observation. First, for each prior ensemble, the analysis increment of the observation space is obtained. Second, a linear regression is performed for each state variable on the observation variable near the observation position in the prior ensemble. The updated increment of the state space is obtained from the increments of the observation space. The state space increments are added to the prior states to obtain the final analysis state.

LPF16 also adopts a framework of sequential assimilation. In the LPF16, the scalar weight of the particle is extended to a vector, and each model variable has a corresponding weight, which means that the influence of an observation on the weight can be localized. For a single observation, the LPF16 updates the weights of the variables only near that observation. To prevent a false reduction in posterior sample variance after resampling, the local weight is multiplied by a scalar coefficient $ \alpha $ of slightly less than 1 to make the particle weight more uniform, and this method is called $ \alpha $ inflation. Then, by a merging scheme, the variables near the observation are modified, while the prior information of the variables far from the observation remains unchanged. Finally, the higher-order corrections of particles are made by kernel density distribution mapping (KDDM). The LPF19 introduces a modified weight calculation equation based on the LPF16 and normalizes the likelihood weight before introducing the vector weight. When the weight calculation formula is changed, the updated formula of particles is also modified correspondingly. In addition, the probability mapping technology is improved in LPF19 compared with that in the LPF16. Lee and Majda (2016) show that the $ \alpha $ inflation scheme is insufficient when the observation error is much smaller than the prior error. For the LPF19, the inflation scheme has been improved by introducing the $ \beta $ inflation scheme to increase the observation error variance. The $ \beta $ inflation scheme prevents weight collapse to a single particle. Unlike $ \alpha $ inflation, the new scheme does not completely ignore observations when the likelihood is small. In the merging step, the LPF19 introduces scalar parameter $ \gamma $, which ranges between 0 and 1. Then, the updated particles are combinations of the current particles and the resampled particles.

The LWEnKF takes the perturbed EnKF as the proposal density, which is sampled to obtain the proposal particle and calculate the proposal weight, which is calculated using observations and local model variables. The proposal weight also needs to be adjusted by parameter $ \alpha $, which has the same effect as in the LPF16. The $ \beta $ inflation is used to adjust the likelihood weight, similar to the LPF19, and the local function is used to update the local likelihood weight. The particle weight is the product of the local likelihood weight and the proposal weight. In the merging step, the updated particles are combinations of the current particles and the resampled particles according to the weights of prior particles. In the merging step, the $ \gamma $ parameter is introduced, and when the number of particles is small, $ \gamma $ less than 1 can effectively alleviate filter degradation. The final model variables need to be adjusted by KDDM. The specific calculation process of the LWEnKF is described in Chen et al. (2020a).

The local radius in the assimilation system is controlled by the local parameter $ c $. In the LWEnKF, local parameters $ {c}^{B} $ and $ {c}^{D} $ are introduced to calculate the proposal weight, which correspond to the local radii of the model variables and observations, respectively. In the experiments conducted in this paper, $ {c}^{B} $ is set to 0, indicating that the model error variance rather than covariance is considered when calculating the proposal weight. The localization function adopts the fifth-order piecewise function (Gaspari and Cohn, 1999), as shown in Eq. (1).

where $ z $ represents the distance between the model variables and the positions of observations and $ l $ is the local radius corresponding to $ c $. $ {C}_{0} $ represents the incremental scale factor. The local radius in the horizontal and vertical directions can be expressed as Eq. (2).

where $ {l}_{{\rm{h}}} $ is the local radius in the horizontal direction, and its unit is km; $ R $ is the radius of the earth; $ {l}_{{\rm{v}}} $ is the local radius in the vertical direction; and $ L $ corresponds to the model layer. Figures 2a and b represent the change in the incremental scale factor with distance. In Fig. 2a, different curves correspond to different values of parameter $ c $. In Fig. 2b, different curves correspond to different values of parameter $ L $, and the dotted line in the figure is the number of ROMS model layers 24. Equation (2) is applied to the localization radius setting of sea surface satellite data assimilation.

The vertical local radius function for T/S profiles is designed as Eq. (3). The functional design is adapted to the vertical S-coordinates of the ROMS, and the local radii corresponding to different observation depths are shown in Fig. 2c.

where $ {l}'_{{\rm{v}}} $ is the vertical localization radius, and its unit is m; $ {H} $ is the adjustment parameter; $ h $ is the observation depth of meters; and $ {h}_{1},{h}_{2},a,b $ are fixed parameters, and the corresponding values are −100, −250, 220 and 30, respectively. In Fig. 2c, parameter $ c $ is set to 0.02, different curves correspond to different values of parameter $ {H} $, and the horizontal axis represents the local radius corresponding to observations at different depths.

All experiments presented in this paper use 40 particles or ensembles. The optimal localization parameter $ c $ for each method is determined through sensitivity experiments. Specifically, $ c=0.02 $ in the EnKF and $ c=0.05 $ in the LPF. In the LWEnKF, both the proposal weight and likelihood weight require local parameter $ c $. $ {c}^{D}=c=0.02 $ is used in the calculation of the proposal weight, and $ c=0.005 $ is adopted in the calculation of the likelihood weight. Regarding inflation, adaptive prior multiplicative inflation (Anderson, 2007) is adopted in the EnKF, the $ \beta $ inflation scheme is used in LPF19, and in the LWEnKF, the proposal weight is adjusted by the $ \alpha $ inflation scheme, and likelihood weight is adjusted by the $ \beta $ inflation scheme. In the merging step, both the LPF19 and LWEnKF introduce parameter $ \gamma $ to adjust the merged particles. To reduce the cost of parameter adjustment in the LWEnKF, $ \gamma $ is set equal to $ \alpha $, changing from 0.70 to 0.99.

It is important for initial state ensembles to consider the main physical quantities that govern the evolution of the state of atmospheric and oceanic systems (Hoteit et al., 2008). A similar method to Hoteit et al. (2013) is adopted to generate the initial ensemble. First, integrate the model over a long period of time (2007–2014). Second, extract the dominant variability from the long model trajectory via empirical orthogonal function (EOF) analysis. Finally, generate the initial ensemble by an exact second-order sampling scheme (Pham, 2001):

where x is the initial state of assimilation extracted from the long model trajectory; $ N $ is the size of the ensemble; ${{\boldsymbol{L}}}_{0} $ is the sample covariance matrix; and $ {{\boldsymbol{L}}}_{0} $ is a matrix whose columns consist of EOFs. $ {{\text{σ}} }_{i} $ is the ith of $ N\times (N-1) $ random matrix $ {\text{σ}} $ with orthonormal columns and zero column sums. This initial ensemble generation method has been widely used in ocean data assimilation (Li and Toumi, 2017; Chen et al. 2020a).

To increase the spread of prior ensembles during assimilation, atmospheric forcing variables are perturbed following Chen et al. (2020a):

where f are the perturbed variables, including surface wind stress and surface net heat flux. The generation of the perturbed variables is similar to the initial ensemble perturbation. EOF analysis is used to extract the main variability of the variables. The rest of the forcing variables are not perturbed.

Several questions are considered when designing the experiments. First, can the simultaneous assimilation of S-SST and AT-SSH outperform single surface observational variable assimilation? Second, is there different performance regarding different assimilation methods? If yes, what are the differences? Third, does the LWEnKF assimilation system have advantages in predicating mesoscale eddy?

The experimental design is shown in Table 1.

In Table 1, ExpA is a control experiment without assimilation; ExpB, ExpC and ExpD assimilate only satellite observations; and ExpE adds additional T/S profiles. The assimilation experiments were conducted from October 3, 2013 to February 1, 2014, for a total of 120 days, which is identical to 30 assimilation cycles (four-day assimilation window). Within a window, observations within ±2 days of the analysis time are assimilated.

This section focuses on the discussion of the experimental results of ExpA−ExpD and explores whether simultaneous assimilation of S-SST and AT-SSH can achieve better results than single variable assimilation.

In Fig. 3, the dotted lines show the root mean square errors (RMSEs) of the three methods with only S-SST assimilation, solid lines denote the results with both AT-SSH and S-SST assimilation, and the black solid line shows the result of the control experiment. In the control experiment, the RMSE increased gradually with time without constraints from observations. Based on the RMSE trends, the three assimilation methods improved the observation variable SST, but the assimilation effect of the LPF19 is not ideal. The RMSE of the LPF19 shows the same trend as that of the control experiment, while the assimilation effect of the EnKF and LWEnKF is obvious. Compared to the control experiment, the EnKF and LWEnKF reduce the RMSE by 41.95% and 47.99% according to the forecast results (Fig. 3a) for assimilating S-SST alone, respectively. The four-day forecast in figures means the forecast results at the fourth forecast day. For the analysis results (Fig. 3b), the EnKF and LWEnKF reduce the average RMSE by 55.61% and 59.82%, respectively. When both S-SST and AT-SSH are assimilated, the forecast results (Fig. 3a) show that EnKF and LWEnKF reduce the average RMSE by 51.97% and 52.33%, respectively. For the analysis results (Fig. 3b), the EnKF and LWEnKF reduce the average RMSE by 61.15% and 61.73%, respectively. The readers should note that the percentages are calculated using the data before the two significant digits are retained, so the average RMSEs in the figure are equal and the calculated percentages are slightly different. For both AT-SSH and S-SST assimilation, the EnKF and LWEnKF show better assimilation effects than for S-SST assimilation alone based on the RMSE time series and the statistical average RMSE results. This result indicates that AT-SSH assimilation has a positive effect on the simulation of SST. The LWEnKF is comparable with the EnKF for S-SST assimilation alone. In the case of both AT-SSH and S-SST assimilation, the LWEnKF is no worse than the EnKF.

In Fig. 4, the dotted lines show the RMSEs of the three methods with only AT-SSH assimilation, the solid lines are the results with both AT-SSH and S-SST assimilation, and the black solid line shows the result of the control experiment. The trends of the RMSEs of the three methods are basically the same, and RMSEs fluctuate greatly at the beginning and tend to be stable in the later stage. Compared to the control experiment, the EnKF and LWEnKF reduce the RMSE by 36.60% and 30.58% according to the forecast results (Fig. 4a) for assimilating AT-SSH alone, respectively. For the analysis results (Fig. 4b), the EnKF and LWEnKF reduce the average RMSE by 50.47% and 44.29%, respectively. When both S-SST and AT-SSH are assimilated, the forecast results (Fig. 4a) show that the EnKF and LWEnKF reduce the average RMSE by 36.66% and 38.89%, respectively. For the analysis results (Fig. 4b), the EnKF and LWEnKF reduce the average RMSE by 49.53% and 50.81%, respectively.

The assimilation of both AT-SSH and S-SST does not show a better effect than AT-SSH assimilation alone based on the RMSE time series and the statistical average RMSE results. This result indicates that S-SST assimilation has a limited influence on the SSH simulation, as opposed to the positive effect of AT-SSH assimilation on SST simulation. For AT-SSH assimilation alone, the LWEnKF does not have the same effect as the EnKF, but the LWEnKF can be comparable to the EnKF for both AT-SSH and S-SST assimilation.

Whether considering single variable assimilation or joint variable assimilation, the LPF19 is not ideal for the simulation of SST and SSH. The assimilation effects of the EnKF and LWEnKF are much better than those of the LPF19. For SSH and SST prediction and analysis, the LWEnKF performs no worse than the EnKF.

ExpE assimilates EN4.2.1 T/S profiles, including T/S profiles from both Argo and CTD and temperature profiles from XBT. There are 577 T/S profiles in total, of which 446 are from Argo. The locations of all the profiles are marked in Fig.1 with red points. The Argo data are uniformly distributed in the whole assimilation period, and the data from XBT and CTD are less. The T/S profiles from Argo are used to validate the performance of the assimilation system. The RMSEs of T/S profiles are calculated to evaluate the performance of different assimilation methods.

Figure 5 shows the RMSEs of the temperature profiles. It should be noted that no matter what data are assimilated, there is no significant difference between the results of LPF (blue lines) and those of the control test (black lines), and assimilation does not affect. Hence the focus is only put on EnKF and LWEnKF hereafter.

In the experiment of AT-SSH assimilation only (Fig. 5b), the prior and posterior results are not as good as the control experiment above 200 m, and AT-SSH does not bring positive adjustment to the internal ocean temperature. When S-SST is only assimilated (Fig. 5a), the temperature field in the upper layer of the ocean is well constrained, and there is no similar situation with AT-SSH only assimilated, and the temperature field in the internal ocean is well adjusted. Compared to the control experiment, the RMSEs decrease above 200 m when assimilating AT-SSH and S-SST together (Fig. 5c). When all the data including T/S profiles are assimilated (Fig. 5d), the RMSE curves of the EnKF and LWEnKF have roughly the same shapes. The RMSE values increase with depth and reach the maximum value near 150 m depth. Below 150 m, RMSE values decrease with increasing depth. When comparing the three methods, the EnKF and LWEnKF are shown to be significantly better than the LPF19, and the LWEnKF has the smallest mean RMSE. The LWEnKF is significantly better than the EnKF at depths below 100 m and is equivalent to the EnKF at depths of above 100 m. In terms of posterior RMSE, LWEnKF and EnKF assimilation remarkably adjust the prior field, while the LPF19 assimilation effect is not visible. From the average RMSEs, for the prior, the LWEnKF reduces by 22.63% compared to the EnKF, and for the posterior, it reduces by 55.11%.

Figure 6 shows the RMSEs of the salinity profiles. The significant feature of the RMSE curves is that they fluctuate greatly in the upper layer. The experiments conducted in this study do not assimilate sea surface salinity (SSS) data. Compared to temperature data assimilation, the results of the model deviate from the actual upper ocean salinity without the constraint of SSS observations. The model is unstable after the assimilation of the salinity profiles, and the RMSE fluctuates at depths of shallower than 100 m. However, under the constraint of S-SST data, the RMSE of the temperature profiles does not have similar instability.

No matter what data are assimilated, there is no improvement in the salinity field for LPF compared with the control experiment. The results of EnKF and LWEnKF are not much better than those of the control experiment when AT-SSH (Fig. 6a) or S-SST (Fig. 6b) are assimilated alone. The situation improves slightly when S-SST and AT-SSH are assimilated together (Fig. 6c). EnKF and LWEnKF show a slightly better prediction effect than the control experiment above 200 m and LWEnKF is obviously better than EnKF.

When all the data including T/S profiles are assimilated (Fig. 6d), the LWEnKF has the smallest mean RMSE, and its effect is better than that of the EnKF at depths of below 100 m. From the posterior RMSE, the assimilations of the LWEnKF and EnKF make visible adjustments to the forecast, and the effect of the LPF19 assimilation seems null (also see in Fig. 7) and therefore the results from LPF19 will not be discussed hereafter. From the average RMSEs, for the prior, the LWEnKF reduces by 11.58% compared to the EnKF, and for the posterior, it reduces by 35.22%.

To explore the changes brought by the assimilation of T/S profiles to SSH, the RMSEs of SSH are shown in Fig. 7. There is no significant difference in the RMSEs of SSH before and after T/S profiles assimilation. From the RMSEs values, the effect of the assimilation of the T/S profiles on SSH is limited. It may be because the T/S profiles are too sparse for the model domain, and it is difficult to make obvious adjustments to SSH.