The coupled physical and biological model used in this study is based on the Regional Ocean Modeling System ROMS; configured in the Gulf of Mexico (red rectangle in Fig.  shows the model domain) with a horizontal resolution of ∼ 5 km and 36 vertical sigma levels . The model used a multidimensional positive definitive advection transport algorithm MPDATA; to solve the horizontal and vertical advection of tracers, a Smagorinsky-type formula to parameterize horizontal viscosity and diffusivity, and the Mellor–Yamada 2.5-level closure scheme to calculate the vertical turbulent mixing. Atmospheric forcing is provided by the European Centre for Medium-Range Weather Forecasts ERA-Interim product ECMWF reanalysis; with a horizontal resolution of 1/8∘ (approximately 12 km × 14 km) to calculate the surface wind stress as well as the net heat fluxes and freshwater fluxes.

The biological model uses a nitrogen-based model to simulate transportation and transformation of seven pelagic variables, i.e., nitrate (NO3), ammonium (NH4), chlorophyll (Chl), phytoplankton (Phy), zooplankton (Zoo), small detritus (SDet), and large detritus (LDet). As a separate state variable, chlorophyll accounts for photo-acclimation based on . In our coupled model, the biological tracers are advected and diffused as part of the 3D circulation but provide no feedback to the physical model. Biological parameters are from the parameter optimization study of except that the half-saturation constant of nitrate was subjectively re-tuned based on the BGC-Argo float data from 0.5 mmol N m-3 to about 1.4 mmol N m-3 because the previous model underestimated the nitrate in the euphotic zone.

The coupled model receives freshwater and nutrient inputs from the Mississippi–Atchafalaya river systems, which are specified by daily measurements from the US Geological Survey river gauges and those from other major rivers that utilize climatological estimates . To ensure a dynamically consistent biological field, a 1-year spin-up is performed in 2014 wherein the physical model is initialized from the output of the 1/12∘ data-assimilative global HYCOM/NCODA and the biological model starts from a regressed 3D field of nitrate based on its climatological relationship with temperature (see Fig. S1). A semi-prognostic method is used during the spin-up period to reduce model drift by replacing model density with a linear combination of model and climatological density fields when calculating the horizontal pressure gradient . After the spin-up, experiments are performed for a year from January 2015 to December 2015.

In this study, the data assimilation scheme uses the deterministic formulation of the ensemble Kalman filter (DEnKF), which was first introduced by . The approach consists of two steps: (1) the forecast step in which an ensemble of state variables is integrated forward in time by the model and (2) the analysis step in which observations are assimilated to update the forecasted ensemble following the Kalman filter equations: 1xa=xf+Kd-Hxf,2K=PfHTHPfHT+R-1, where x represents the model state estimate, d represents the available observations, H represents the observation operator mapping the model state onto observations, and K represents the Kalman gain matrix, which is determined by the model error matrix P and observation error matrix R (Eq. ). Superscripts “a” and “f” represent analysis (i.e., updated) and forecast (i.e., prior to the update) estimates, and T represents the matrix transpose. Unlike the original stochastic EnKF, which updates each ensemble member with perturbed observations, the DEnKF updates the ensemble mean (x‾) and anomalies (A=x-x‾) separately without perturbating observations; i.e., the former is updated as in Eq. (), while the latter is updated by 3Aa=Af-12KHAf. More details on the DEnKF can be found in and .

The data assimilation framework and configurations are the same as in wherein twin experiments were performed in the same model domain. In this study, we extend the work to jointly assimilate the physical and biological observations into a coupled model. For the sake of keeping our data assimilation experiments computationally affordable, we chose an ensemble size of 20, which has been successfully used in previous studies including an idealized channel , the Middle Atlantic Bight , and the Gulf of Mexico . Spurious correlations, which can arise with relatively small ensembles, are avoided here by applying a distance-based localization with a radius of 50 km . Vertical localization is not applied. Ensemble anomalies are inflated by 1.05 in each update step to account for unrepresented sources of model uncertainty . Values of the localization radius and inflation factor were determined in .

In order to account for uncertainties in the model's initial, boundary and atmospheric forcing conditions, and biological parameters, the ensemble is initialized from 20 different daily outputs, centered on the initial date of 1 January 2015, from a previous deterministic model simulation (as described above in Sect. 2.1) and is forced by open boundary conditions, which are lagged by up to ±10 d for the different ensemble members. Furthermore, each ensemble member is forced by a perturbed version of the wind forcing. Specifically, the wind forcing from the deterministic run is decomposed into empirical orthogonal functions (EOFs) and then the first four EOFs are perturbed by multiplying random numbers with zero mean and variance of 1 as in and . In addition, four sensitive biological parameters, namely the mortality rate of phytoplankton, the maximum ratio of chlorophyll to carbon, the grazing rate of zooplankton, and the growth rate of phytoplankton at 0 ∘C, were identified in sensitivity experiments. Specifically, a 1D version of model, described in was run multiple times while incrementally perturbing one parameter at a time by factors ranging from 0.25 to 1.75 with an increment of 0.25. The four sensitive parameters were selected based on the normalized absolute differences between the perturbed and unperturbed run. In the data assimilation experiments, these parameters are subject to a Gaussian perturbation with a relative variance of 75 %, but they are not updated. The parameters are resampled from their distributions before each forecast step to prevent some extreme parameter values from being used throughout the whole data assimilation experiment.

In this study, physical and biological observations are jointly assimilated to constrain the coupled model. The observations assimilated include sea surface height (SSH), sea surface temperature (SST), Argo T–S profiles, and satellite estimates of surface chlorophyll.

The SSH observations for assimilation are obtained by adding the 1/4∘ mapped sea level anomaly (SLA) from Archiving Validation and Interpretation of Satellite Oceanographic Data (AVISO) to a mean dynamic topography (MDT) from , and they are adjusted by removing the spatially averaged mismatches between assimilated and forecasted SSH to account for differences in reference time between the SLA data (1993–2012) and our coupled model (2015) . This is equivalent to assimilating the SSH gradient into the model, as it is the only dynamically meaningful quantity for driving the geostrophic component of ocean currents and adjusting subsurface thermohaline structures. The SST observations are the Advanced Very High Resolution Radiometer AVHRR; product with a horizontal resolution of 0.01∘. Observation errors are specified as 0.02 m for SSH and 0.3 ∘C for SST .

The surface chlorophyll is provided by the Ocean-Colour Climate Change Initiative project OC-CCI; at a daily frequency with a spatial resolution of 1/24∘. However, for the daily chlorophyll field, a large portion of data can be missing due to cloud cover and inter-orbit gaps. In 2015 for the Gulf of Mexico, the spatial coverage of surface chlorophyll varies from 0 % to 63 % with a mean coverage of 9.5±9.0 %. Hence, to increase the availability of observations, an asynchronous data assimilation method is applied so that not only the daily records of surface chlorophyll at the date of update but also the daily records within the preceding 7 d are assimilated. Errors associated with the surface chlorophyll are set to be 35 % of the measured concentrations, which has been commonly used in previous applications e.g.,. In this study, the update is performed on actual chlorophyll concentrations because our prior tests showed that it outperforms assimilating log-chlorophyll in the open gulf (with depth >1000 m). There are previous examples in which the actual chlorophyll values have been assimilated successfully e.g.,, although we note that assimilating the actual chlorophyll values is theoretically suboptimal because of their non-Gaussian distribution.

Profiling observations are from the International Argo project (hereafter referred to as Argo floats) and five BGC-Argo floats, which were funded by the Bureau of Ocean Energy Management (hereafter referred to as BOEM floats). In 2015, the Argo floats provided nearly 800 T–S profiles extending from the surface to a depth of 2000 m in the Gulf of Mexico. These are treated either as independent observations for model skill assessment or, in the DAargo experiment (see Sect. 2.4), assimilated with uncertainties of 0.3∘C for temperature and 0.01 for salinity. The BOEM floats collected more than 500 profiles of temperature, salinity, chlorophyll, and backscatter at a biweekly frequency from 2011 to 2015, 114 of which were collected in 2015 (see Fig.  for their locations) and are used as independent observations. Backscatter is converted into phytoplankton and particulate organic carbon (POC) concentrations following . In the absence of direct measurements for nitrate, we estimate it along the BOEM float trajectories based on their climatological relationship with temperature (Fig. S1).

We performed five 1-year simulations in 2015. The first one is a deterministic model simulation without data assimilation (henceforth referred to as the Free simulation). The second one is an ensemble run assimilating satellite data (SSH, SST, and satellite surface chlorophyll) only (henceforth DAsat), and the third one is an ensemble run assimilating Argo T–S profiles in addition to satellite data (henceforth DAargo). The calculations (Att=0.04+0.025×Chl) used in these three simulations are from the literature e.g.,; the light attenuation coefficient, Att, is strongly determined by water depth and not very sensitive to chlorophyll concentrations. The Free run and DAsat run are repeated by using an alternative light parameterization (henceforth referred to as Free-alt and DAsat-alt simulations, respectively) to evaluate its effect on the data assimilation impact on subsurface biological properties. This alternative light parameterization (Att=0.027+0.075×Chl1.2) is subjectively tuned based on the BGC-Argo observations and emphasizes the self-shading effect of chlorophyll on light attenuation.

A two-step update is used on a weekly data assimilation cycle in the assimilative experiments, in which the physical observations are first assimilated to update both physical and biological state variables through the multivariate covariance, and chlorophyll observations are assimilated next to update only biological state variables. Although the DEnKF can update all state variables based on their cross-covariance, we limit updates to two physical variables (temperature and salinity) and four biological variables (nitrate, chlorophyll, phytoplankton, and zooplankton) that are key to the coupled physical–biogeochemical system. As the circulation features in the open gulf (the Loop Current and its associated mesoscale eddies) are primarily in geostrophic balance, an update of temperature and salinity can improve three-dimensional circulation features in large scales effectively, as shown in the twin experiments in . All these state variables are updated throughout the whole water column, while other variables are adjusted by internal model dynamics.

To evaluate the prediction skill, we calculate the root mean square errors (RMSEs), the bias, and the correlation coefficient (Corr) of the model forecast (M) with respect to assimilated and independent observations (O): 4RMSE=1N∑(M-O)2,5bias=1N∑(M-O), where N represents the number of model–data pairs available. To account for the overestimation of nitrate in warm waters, which typically occurs in the euphotic zone (Fig. S1), an unbiased root mean square error (unbiased RMSE) is used to quantify the model–data misfit of nitrate. 6unbiased RMSE=1N∑(M-O-bias)2

As the biological model provides no feedback to the physical model, the alternative light parameterization does not affect physical properties. The physical results from Free-alt and DAsat-alt runs are thus not displayed in this section.

The dominant circulation features in the Gulf of Mexico, the Loop Current and Loop Current eddies, are assessed by comparing their fronts, defined here as the 10 cm SSH contour, from satellite data, the Free run, and two data-assimilative runs (i.e., the DAsat and DAargo runs). In the first 2 months, all model estimates of the Loop Current are different from satellite observations due to the influence of initial conditions (Fig. ). After March, the SSH field shows a similar northward and westward extension of the Loop Current intrusion between two assimilative runs and satellite observations, but large deviations from observations remain in the Free run. In addition, all estimates except for the Free run reproduce the satellite-observed timing of eddy shedding well, as well as the size, shape, and position of Loop Current eddies.

For a more quantitative assessment, the daily output of SSH and SST fields from the three runs is compared with the satellite estimates. The spatial distribution of RMSE from the Free run and the RMSE changes in two data-assimilative runs are shown in Fig. . In the regions influenced by the Loop Current and Loop Current eddies, this figure shows high RMSE for SSH in the Free run (Fig. a) and large RMSE reductions in two data-assimilative runs (Fig. b–c). In contrast, the reductions in SST RMSEs are more spatially homogeneous. A summary of the overall RMSE, the bias, and the correlation coefficient (Coef) for physical variables from the Free run and two data-assimilative runs are shown in Table 1. In general, the two data-assimilative runs both significantly improved SSH and SST with reduced RMSEs and increased correlation coefficients. Although the two data-assimilative runs tend to underestimate the satellite observations of SST, the bias (-0.06 ∘C) is relatively small.

The correction of mesoscale features by data assimilation was not limited to the surface but extended to the subsurface and even deep waters. Specifically, the two assimilative runs corrected the position, the amplitude, and the polarity of mesoscale eddies and hence better represented the elevated and depressed thermoclines within these eddies (Fig. ). The most noticeable improvement (by 60 %–61 %) was witnessed by float 287, which captured a newly detached Loop Current eddy with features of high SSH and depressed thermoclines during July and October. In addition, assimilation of Argo T–S profiles in the DAargo run led to slight further improvements in the subsurface temperature distributions when compared with the DAsat run. For instance, although the DAsat run greatly improved subsurface temperature distributions along the trajectory of float 285, an underestimation of temperature at a depth of about 200 m remains within the peak of the anticyclonic eddy. Corrections imposed by assimilating Argo profiles increased temperature here and decreased the bias from observations. These small but localized further improvements can also be observed by other floats, e.g., in July–October for float 289 and February for float 290.

In general, assimilating the satellite data in the DAsat run resulted in large reductions in RMSEs of 3D temperature (by 46 %–48 %; Table 1) and salinity (by 36 %–39 %; Table ) with respect to Argo floats and BOEM floats (Fig. ). The reductions extend to over 1000 m and a depth of about 800 m for temperature and salinity, respectively. It should be noted again that data from both Argo and BOEM floats are independent in the DAsat run. Although assimilating the Argo profiles in the DAargo run only yields marginal further improvements in RMSEs of temperature (∼ 3 %) and salinity (∼ 5 %), it notably reduces the overestimation of temperature that occurs below the surface in the DAsat run (Table ).

Assimilating satellite observations in the DAsat run reduced RMSEs of surface chlorophyll almost everywhere, with only 3 % of the model domain experiencing degradation (Fig. b). Although large reductions in RMSE were achieved in the coastal regions, e.g., in the northern Gulf of Mexico, on Campeche Bank, and in Campeche Bay, the simulated chlorophyll concentrations remained much lower than the satellite estimates because of high observational uncertainties and a large background misfit in the Free run (Fig. a). This was expected because the biological model was optimized for the open gulf . Table  shows the RMSE, the bias, and the correlation coefficient for biological variables from the Free run and the data-assimilative runs. A relative reduction in RMSE equal to or exceeding 10 % is considered a significant improvement. In the open gulf, encompassed by the 1000 m isobath, the overall RMSE of surface chlorophyll was reduced by 19 % from 0.13 mg m-3 in the Free run to 0.11 mg m-3 in the DAsat run (Table ). In addition, the correlation coefficient increased from 0.52 to 0.68. Assimilating Argo T–S profiles in the DAargo run led to lower reductions in the overall RMSEs of surface chlorophyll (Table ) and even more deteriorations (Fig. c).

To evaluate the impacts of data assimilation on subsurface biological properties, the temporal evolution of nitrate in different model experiments is shown in Fig.  in comparison to nitrate estimated based on its climatological relationship with temperature. The temperature-based nitrate tends to be overestimated in the upper layers (Fig. S1). Because of its high correlation with temperature, the nitrate distribution was modulated in the two assimilative runs along with the improvement in temperature fields. For instance, the two assimilative runs reproduce the Loop Current eddy observed by float 287 and hence capture the depressed thermoclines that are not present in the Free run (Fig. ). At the same time, the nitraclines are also depressed and the nitrate concentrations become lower within this Loop Current eddy (Fig. ). As a result, the unbiased RMSE of nitrate following this float is reduced by 40 % in the DAsat run and 38 % in the DAargo run. These depressed (upwelled) nitraclines due to the increase (decrease) in SSH by data assimilation can also be observed elsewhere, e.g., in August for float 285, in April–July for float 286, January–April for float 287, and in August–October for float 290, although the amplitude of these mesoscale eddies is smaller. In general, data assimilation improved the overall agreement of subsurface nitrate with correlation coefficients and decreased RMSEs by 28 % and 30 % in the DAsat and DAargo runs relative to the Free run (Table  ).

The impacts of assimilation on subsurface chlorophyll are more complicated because of the high nonlinearity of the model with regard to chlorophyll. Although the mean vertical profiles of chlorophyll are well reproduced in all three experiments (Fig. S2), all failed to resolve the high spatiotemporal variability in subsurface chlorophyll, which is at least partly due to the presence of mesoscale eddies (Fig. ). As a result, assimilation improved subsurface chlorophyll RMSEs marginally, even in the Loop Current eddy for float 287 for which the most noticeable improvements of temperature (∼ 60 %) and nitrate (∼ 40 %) RMSEs were obtained. Results for phytoplankton and POC are similar to chlorophyll, although the reductions in their RMSEs are larger because assimilating the satellite data reduces their biases, especially in the upper layer (Fig. S2, Table ).

The model's inability to reproduce the spatiotemporal variability of subsurface chlorophyll is also reflected by the positions of the deep chlorophyll maximum (DCM, denoted by red lines in Fig. ). As a ubiquitous phenomenon in the oligotrophic regions, a distinct DCM is observed throughout the whole year in the open Gulf of Mexico, and its depth is inversely correlated with SSH (correlation coefficient -0.6). Although the mean position and magnitude of the DCM are well reproduced by the model with and without data assimilation (Fig. S2), the simulated DCM depth is much more stable and less sensitive to SSH variations. As a result, the reduction in the RMSE of DCM depth is limited to 18 % in DAsat run but is significant (Table ).

Both with and without data assimilation, the alternative parameterization led to higher correlations between simulated SSH and DCM depth with correlation coefficients of -0.60 in Free-alt run and -0.67 in DAsat-alt run. As a result, the alternative parameterization produces slightly lower RMSEs and a higher correlation coefficient for DCM depth (Table ) and yields larger improvements in chlorophyll within the Loop Current eddy for float 287 (Fig. ). To illustrate the underlying reasons, the mean vertical profiles of nitrate, the intensity of photosynthetically active radiation (PAR), the chlorophyll, and the phytoplankton within the center of this Loop Current eddy are shown in Fig. . When using the original parameterization, assimilating the satellite data depresses the DCM depth from 70 m in the Free run to 90 m in the DAsat run but with a considerable bias of 20 m when compared to the observations. However, the chlorophyll is underestimated in the DAsat run, and as a result its RMSEs are barely improved. In contrast, in the DAsat-alt run the DCM depth is corrected to 120 m, in agreement with the observations, and represents the vertical chlorophyll distribution more accurately, although the nitrate profile is almost the same as in DAsat run. This was because the alternative parameterization accounted for the elevated PAR intensity as a response to reduced chlorophyll concentrations in the upper layer, which in turn facilitated the synthesis of chlorophyll and hence corrected their concentrations toward the observations.