Our model is based on version 3.7 of the Regional Ocean Modeling System (ROMS; ), a primitive equation, hydrostatic, free surface regional ocean physical circulation model. The model domain is designed to simulate physical and biogeochemical dynamics in the CCS. It extends from 30 to 48° N, and longitudinally from the U.S. west coast to 134° W, with a horizontal resolution of 0.1° × 0.1° (resulting in a ≈ 10 km resolution) The model is vertically divided into 42 terrain-following layers with a higher resolution near the ocean surface. Our implementation uses the Coupled Ocean Atmosphere Mesoscale Prediction System COAMPS; for physical forcing at the surface (wind, longwave solar radiation, air temperature, pressure, and humidity). Boundary conditions for the physical model are provided by Copernicus Global 1/12° reanalysis GLORYS;. The GLORYS product is further used to nudge model temperature and salinity toward their reanalysis values in a 1° wide “sponge” region along the open model boundaries. This physical model setup closely follows that in previous studies, such as and .

As a biogeochemical model, we use the NEMUCSC model, an extension of the North Pacific Ecosystem Model for Understanding Regional Oceanography NEMURO;. Our NEMUCSC configuration has been extended from the setup presented in with carbon and oxygen cycling and phytoplankton photoadaptation (i.e., dynamic nitrogen to chlorophyll ratios for phytoplankton, which we detail below).

The base NEMUCSC model accounts for a range of biogeochemical constituents and interactions in the ecosystem, including: (1) three limiting macronutrients (nitrate, ammonium, and silicate); (2) two phytoplankton functional types (nanophytoplankton and diatoms); (3) three zooplankton size classes (microzooplankton, mesozooplankton, and predatory zooplankton); and (4) three detritus pools (dissolved and particulate organic nitrogen and particulate silica). The NEMUCSC carbon extension, presented in with refinements to alkalinity and carbon dynamics described in , adds dissolved inorganic carbon (DIC), total alkalinity, and calcium carbonate variables. It computes ocean pH and pCO₂ based on the OCMIP carbonate chemistry (Fig. ).

The second extension to the NEMUCSC model adds a chlorophyll variable for each of the two nitrogen-based phytoplankton variables. The photoadaptation model presented in is used to dynamically adjust the stoichiometric ratio of nitrogen to chlorophyll for the two phytoplankton functional types represented in the model, assuming a constant nitrogen-to-carbon ratio based on the Redfield ratio.

We further expanded the code required by the DA (NEMURO's tangent-linear and adjoint code; see ) to fully support the extended model and permit the assimilation of chlorophyll, oxygen and carbonate system variables. For DA, we added a “total chlorophyll” variable which is the sum of diatom chlorophyll and nanophytoplankton chlorophyll, so that chlorophyll observations can be directly assimilated, informing both phytoplankton functional types. Following and , chlorophyll is log-transformed in the DA procedure. A log-transformation is not applied to pH, alkalinity, DIC, oxygen or the physical variables for which we also assimilate data in this study.

For the carbonate system variables DIC and alkalinity, as well as oxygen, nitrate and silicate, boundary conditions are created using statistical relationships from ESPER, based on temperature and salinity boundary values. Following a comparison with climatological nutrient values from the World Ocean Atlas , nitrate boundary values in the mixed layer were reduced to match World Ocean Atlas values, and to prevent unrealistically high nitrate input through the boundaries. To further prevent unrealistic values for other variables, boundary conditions for the remaining BGC variables are set to low values that roughly match model estimates near the boundaries. For the year of our experiments, 2019, atmospheric pCO₂ is set to a constant value of 400 ppm.

In this study, we use a variety of satellite, glider, and float observations, complemented by estimated data derived from statistical relationships using an average of three algorithms (ESPER-LIR, ESPER-NN, and CANYON-B). We conducted a series of experiments in which various data subsets were excluded from the assimilation process to examine their influence on model state estimates. Furthermore, we used model state estimates with and without DA and ESPER-LIR to derive hybrid (dynamical model + statistical model) state estimates.

The datasets are classified into three broad categories:

Reference dataset: this represents what is typically assimilated in a coupled physical-BGC model, encompassing:

Temperature data from satellites

In this study, we conducted several DA experiments to systematically investigate the impact of assimilating different observation subsets. Table  provides an overview of all experiments and the data assimilated in each.

To establish a baseline, we use two reference simulations: a non-assimilative “free simulation” and a reference DA experiment. The free simulation does not assimilate any observations and is solely driven by the coupled physical-biogeochemical model. Our reference DA experiment represents a typical joint physical-biogeochemical DA approach. It assimilates remotely sensed and in situ temperature and salinity data, satellite sea level anomaly (SLA), and satellite-derived chlorophyll. This experiment serves as a benchmark for current physical-biogeochemical DA without any carbonate system observations.

Building upon the reference DA, we conducted three main experiments that progressively incorporate additional observation types: DA exp 1 assimilates all reference data plus CUGN pH and alkalinity estimates. This experiment examines the impact of jointly assimilating carbonate system variables with the standard physical and chlorophyll data.

DA exp 2 builds on DA exp 1 by additionally assimilating pH-sensor glider line temperature, salinity, pH, and alkalinity data. This experiment aim is to examine the model estimates achievable through direct assimilation of the pH-sensor glider data , compared to the assimilation of neighboring CUGN pH and alkalinity in DA exp 1.

DA exp 3 assimilates all reference data plus CUGN pH, alkalinity, and oxygen data. This experiment investigates the impact of including oxygen observations in the assimilation framework.

To assess the spatial influence of the pH DA and validate our approach, we conducted three cross-validation experiments: we focused on three CUGN glider transects out of Monterey Bay along Line 67 (see highlighted line in Fig. g) in different seasons (starting in late March, July, and October 2019). During each transect, the glider moved westward out of Monterey Bay, covered the full extent of the line, and returned nearshore. We denote these three lines as CUGN L67Mar, CUGN L67Jul, and CUGN L67Oct (started in March, July, and October 2019, respectively) and performed an experiment excluding the data for each line. DA exp 1.1, 1.2, and 1.3 assimilate all data from DA exp 1 except CUGN L67Mar, L67Jul, or L67Oct data, respectively. These cross-validation experiments allow us to evaluate the model's ability to estimate pH and other variables at locations where observations are withheld from the assimilation.

In addition to these direct assimilation experiments, we create hybrid estimates by using dynamical ocean model outputs as inputs to the ESPER statistical model. ESPER can provide estimates of various biogeochemical seawater properties, including pH, alkalinity, DIC, and oxygen. It generates these estimates based on location information, temperature and salinity, and optionally, additional inputs such as oxygen. We can use the model estimates of temperature and salinity as input to ESPER to obtain pH, alkalinity, DIC, and oxygen estimates. These hybrid estimates, combining the dynamical model with the statistical model, may benefit from physical DA as it improves the temperature and salinity estimates that enter ESPER. We created hybrid estimates using the published version of ESPER (ESPER_LIR V1.01) without any additional training and based on model estimates from

the free simulation (ESPER based on temperature and salinity only),

The 4dVar DA system requires the specification of observation error and background error values, the latter to quantify the model uncertainty associated with the background model state, prior to performing DA. Previous experience working with joint 4dVar DA for physical and chlorophyll a observations, and differences between the observed variables in the carbonate system submodule and the rest of the model suggest different approaches for specifying error values for these two parts of the model.

outlines an iterative technique to adjust observation error and background error values to improve the consistency of prior and posterior error values. The adjustments were shown to have a positive effect on 4dVar performance diagnostics and model performance metrics, such as the forecasting skill. This technique, which we hereafter refer to as a fixed point iteration, or “FPI”, can only be applied to the background error values of directly observed variables or variables with a simple relationship to assimilated observations. In the physical model, the temperature, salinity, and SSH variables are directly observed, which means that the observations of temperature, salinity, and SSH correspond directly to these model variables. Chlorophyll a is also directly observed, but the value of the chlorophyll a variable is a function of, and mostly determined by the values of the model's two phytoplankton variables, diatom and nanophytoplankton. By assuming a constant phytoplankton nitrogen:chlorophyll a ratio, we can create a simple (linear) relationship between the phytoplankton variables and the chlorophyll a observations. This approach permits us, starting from initial estimates, to utilize the FPI to adjust temperature, salinity, SSH, and chlorophyll a observation error values and the background error values for the model variables associated with these observations (see for details).

Because the relationship between observations of the carbonate system and model variables is often more complex, the FPI cannot easily be applied to variables of the carbonate system. For example, model pH is a complex, non-linear function of alkalinity, DIC, temperature, and salinity. Thus, we rely on our initial estimates of observation errors and background error values for the carbonate system variables.

For observation error values, we used the uncertainty values provided with the datasets when available and constant errors otherwise. Specifically, we started with 0.1 °C, 0.05, 10 cm, and 30 % (relative error) as the observation error estimates for in situ temperature, in situ salinity, SSH, and chlorophyll, respectively, when no uncertainty values were provided by the datasets. These values were adjusted using the FPI described above; the resulting error values used in our experiments are listed in Table  alongside the error values used for the other observations that were not modified. For oxygen, the reported uncertainty is 0.5 %, which is about 1–1.5 µmol kg⁻¹ at the surface in the CCS . However, to account for biases due to the response time of the sensor, we use 3 µmol kg⁻¹ in this study. For the Line 67 pH glider, we use 0.013 for pH and the error values reported in .

The error values for statistically estimated alkalinity, DIC, and pH using T, S, and oxygen as inputs were calculated based on comparisons with discrete samples. We assessed the performance of three algorithms (ESPER-LIR, ESPER-NN, and CANYON-B) using data from three cruises (WCOA 2021 (33RO20210613), 32WF20190511, and 32WF20190723) that span a large part of the model domain (Fig. ). These cruises were not included in the training datasets for the algorithms and thus represent an independent assessment for the algorithms. We focus on the upper 500 m, as the errors are higher in the upper water column, they provide an upper bound for the error estimate. In general, all three algorithms produced reliable estimates of alkalinity, DIC, and pH across the whole model domain. The median difference was -4.5 µmol kg⁻¹ for alkalinity, -0.1 µmol kg⁻¹ for DIC, and -0.001 for pH, demonstrating small or insignificant systematic bias. Given that there was no clear “best” algorithm in this domain, we chose to average the output of the three algorithms. The error values for these parameters were estimated as the range of the lowest and highest values from the error bars in Fig.  and were ±15 µmol kg⁻¹ for alkalinity, ±20 µmol kg⁻¹ for DIC, and ±0.05 for pH. These represent conservative estimates of the errors.

The prescribed background error values (specifying the diagonal entries of the background error covariance matrix) for all model variables are assumed to be constant at the surface and decrease with depth following an S-shaped logistic function (for all variables except SLA, which has no depth-dependence). Below 200 m, where oceanic variability is weaker, background error values are set to approximately 10 % of their magnitude at the surface (following the approach in ).

The background error values for temperature, salinity, SSH, diatom, and nanophytoplankton are set using the FPI. For the carbonate system variables and oxygen, we use a long, non-assimilative model simulation to set the surface value: for each variable, we assume that all daily average surface values are contained within a window of ±3 standard deviations from its mean. Thus, we take the difference between the maximum daily average surface value in the simulation and its minimum and divide it by 6 to obtain the standard deviation (this is referred to as “3σ-approach” in Table ). The background error values for the other model variables are set to low values to prevent large increments to unobserved variables. All surface background error values used in our experiments are listed in Table .

In a state estimation setting, assimilating pH data (observed or estimated) alone can deteriorate carbon state estimates. This issue arises because, in the model, pH is primarily a function of, and calculated from model estimates of DIC and alkalinity. Different combinations of alkalinity and DIC can yield the same pH value, resulting in model alkalinity and DIC estimates being underdetermined by pH data, analogous to temperature and salinity estimates being underdetermined by density observations in a physical assimilation setting.

In our initial experiments, where we assimilated pH observations alone, state estimates were moved to points in alkalinity-DIC space with the correct pH value. However, the resulting estimates of alkalinity and DIC were largely based on their initial estimates and the prescribed background model error values, rather than reflecting the true state of the system (see also Fig. 5 in ). To move state estimates to the correct point in alkalinity-DIC space, we found it necessary to assimilate two out of the three properties: pH, alkalinity, and DIC.

For this study, we elected to assimilate pH, as it was directly measured on Line 67, and estimated alkalinity because its uncertainty is lower than that of DIC. This choice means that we are using and relying on statistical estimates of alkalinity in the DA procedure, even when pH is directly observed. While this approach introduces the reliance on statistically estimated alkalinity, alkalinity can be estimated with an accuracy of < 7 µmol kg⁻¹ along line 67, even near the coast .

Our DA experiments differ only in the subsets of data being assimilated (see Sect. ). To establish a baseline, we use two reference simulations: a reference DA experiment and a free simulation that assimilates no data.

We first examine the root mean squared error (RMSE) between model and data for each dataset introduced in Sect. . Comparing the two experiments reveals that, similar to previous studies jointly assimilating chlorophyll and physical data e.g., the DA substantially improves the fit to the assimilated data compared to the free simulation (compare gray and blue bars in Fig. ). However, this reduction in error for physical and chlorophyll data (first row in Fig. ) does not translate to improved fit for glider observations of carbonate system variables (pH, alkalinity, DIC) or oxygen (second row in Fig. ). Notably, there is no improvement in the reference DA experiment even though it assimilates temperature and salinity data from the same CUGN glider lines associated with the carbonate system variables.

A closer examination of a representative glider line reveals the spatial structure of the model-observation differences for pH. We focus on a CUGN Line 67 glider transect out of Monterey Bay (see highlighted line in Fig. g) starting in late March 2019, which we refer to as CUGN L67Mar (Fig. ). The discrepancies between pH observations and free simulation model estimates along this line are highest between 50 and 200 m depth (Fig. d, e, f), corresponding to density surfaces of 1025.5–1027 kg m⁻³. Near the surface and at depths below 300 m, the free simulation shows a positive bias, overestimating pH by ≈ 0.1. Assimilation of the reference data, including temperature and salinity data from this line, does not remove the surface or deep pH bias, and results in minor shifts in the largest model-observations differences at intermediate depths between 50 and 200 m (compare Fig. f and h). The improvement in pH estimates, measured by the reduction in the absolute model-observation difference of the reference DA in comparison to the free simulation (Fig. i) confirms that the largest change in model pH through the assimilation of the reference data occurs in the same depth range (50 to 200 m), where the reference DA leads to positive and negative changes in pH estimates.

In our state estimation setup, the joint assimilation of temperature, salinity, SLA, and chlorophyll data has negligible impact on estimates of pH, oxygen, alkalinity, or DIC. This finding aligns with previous state estimation studies reporting little impact of physical DA on chlorophyll estimates, which we will discuss further in Sect. .

Because the joint physical and chlorophyll DA showed no consistent improvements in pH estimates, we next examine the simulation that assimilates CUGN-based pH and alkalinity estimates in addition to the reference data, in the experiment referred to as “DA exp 1”. The joint assimilation of pH and alkalinity estimates with physical and chlorophyll data leads to significant improvement in the model carbonate chemistry parameters. DA exp 1 more than halves the average RMSE of the CUGN pH data compared to the free simulation or the reference DA, while achieving a similar fit to the reference data temperature, salinity, SLA and chlorophyll as the reference DA (Fig. ). The improvement in pH is accompanied by improved fit to CUGN alkalinity and DIC data (Fig. i, j), with little change in oxygen estimates (Fig. h).

Examining the CUGN L67Mar transect (Fig. j–l), the assimilation of pH data leads to overall improvement in pH estimates: surface and deep biases shown in the reference DA are removed, and larger pH discrepancies at intermediate depths disappear.

In the results above, we have only examined the fit to data that has been assimilated. To better understand the spatial influence of pH DA, we examine the increments in DA exp 1 to the carbonate system variables. When only assimilating physical and chlorophyll data, the DA system does not create corrective increments to the carbonate system variables (as these have no direct influence on the physical or chlorophyll state estimates; compare Fig. ). That is, in the reference DA, the carbonate system variables are only affected indirectly by the DA through increments in the physical and nitrogen-based variables and subsequent changes in physical and biogeochemical model dynamics. In DA exp 1, however, the assimilation creates significant increments to alkalinity and DIC to improve the fit to the CUGN pH and alkalinity observations.

The average depth-integrated absolute alkalinity increment (Fig. a) illustrates the horizontal extent of influence of the pH assimilation, clearly outlining the 2019 CUGN glider lines that were assimilated (compare Fig. g, i). Each increment represents the difference between the model state before and after DA at the start of each cycle, which is influenced by the 50 km horizontal length scale prescribed in the DA setup. Thus, it does not capture the subsequent effect of DA due to the model dynamics (the cumulative effect of the increments on the physical and biogeochemical model dynamics through each cycle).

To assess the downstream effect of pH assimilation, we examine the difference in pH estimates at 100 m depth between DA exp 1 and the reference DA at the end of the experiments (31 December 2019). After a full year of assimilating data, the pH difference at 100 m depth has spread from the pH observation locations along the coast and offshore (Fig. b). The snapshot of pH difference reveals complex structures, with areas of positive pH change bordering water masses with negative change. In Sect.  and , we examine if the pH DA improves the fit to pH data that has not been assimilated.

Hybrid estimates use the output of the dynamical model as input to the statistical ESPER model (see Sect. ). As DA can lead to substantial improvements in dynamical model estimates, it is likely to produce better hybrid estimates.

Because the temperature and salinity model estimates do not appear to benefit from the assimilation of carbonate system data (Fig. e, f), we only produce ESPER estimates based on the output of the free simulation and the reference DA. We do not use model oxygen as inputs for the ESPER estimates because model initial conditions for oxygen are based on ESPER estimates and DA experiments have shown little improvement in oxygen estimates compared to the free simulation (compare Fig. h). Instead, here we use ESPER based on temperature and salinity only, and examine the assimilation of oxygen data and ESPER estimates based on assimilated temperature, salinity, and oxygen in Sect. .

Using model temperature and salinity, we generate ESPER-based estimates throughout the entire model domain. Here, we focus on ESPER estimates of pH, alkalinity, and DIC at the observation locations, that is, at the time and place glider observations were recorded. Comparing these values to the CUGN data reveals that hybrid estimates have a better fit than model-based estimates for pH, alkalinity, and DIC (Fig. g, i, j), except for DA exp 1, 2, 3 which all directly assimilate CUGN pH and alkalinity data. For oxygen, hybrid estimates create the closest fit to CUGN observations among all estimates considered so far. In all cases, ESPER estimates based on the reference DA outperform those based on the free simulation, benefiting from improved temperature and salinity estimates obtained through physical DA.

The hybrid pH estimates for ESPER based on the reference DA along CUGN L67Mar (Fig. m–o), show a reduced misfit to data compared to the free simulation and reference DA. The largest misfit occurs at depths around 100 m, corresponding to the 1026 kg m⁻³ density surface.

In summary, hybrid estimates based on model temperature and salinity, combined with the ESPER statistical model, produce some of the best carbonate system estimates in our study, unless the carbonate system data itself is assimilated. Notably, these hybrid estimates do not require a biogeochemical ocean model and can be obtained from a physical ocean model and physical DA, an aspect we examine in more detail in Sect. .

In previous sections, we examined model-based and hybrid estimates for data that was assimilated in some experiments and where pH and alkalinity were not directly measured but estimated using a statistical model. Here, we evaluate the same estimates against a new glider-based dataset that is not part of the CUGN data, contains direct pH measurements along with temperature, salinity, and oxygen, and was not assimilated in the experiments introduced so far. This dataset, referred to as the pH-sensor glider line data, was recorded during several MBARI glider deployments. These deployments spatially overlap with the CUGN Line 67 (see Fig. ) and are thus in the region strongly influenced by the CUGN pH DA (see Fig. ).

For comparison with the pH-sensor glider line data, we introduce an additional experiment, DA exp 2, which assimilates pH-sensor glider line temperature, salinity, pH, and alkalinity data, as well as reference data and CUGN pH and alkalinity data. DA exp 2 serves two main functions: to determine the estimates for the pH-sensor glider line achievable through direct assimilation and to demonstrate that CUGN pH data and measured pH-sensor glider line pH data can be jointly assimilated without deteriorating estimates for either dataset. To run DA exp 2, we rely on statistical estimates of alkalinity for the pH-sensor glider line, jointly assimilated with measured pH data, and include estimates of DIC by combining measured pH and estimated alkalinity (which has a lower error relative to DIC estimated directly from algorithms) in the pH-sensor glider line data for comparison.

We first examine the 4 d RMSE values of model and hybrid estimates with respect to the pH-sensor glider line data (Fig. k–p). For temperature and salinity, DA experiments perform similarly to the free simulation, with only DA exp 2 showing improvement. Assimilation of large temperature and salinity datasets, including CUGN data, does little to improve overall estimates at the pH-sensor glider line.

These results from physical data also apply to the carbonate system variables. For pH, DA exp 1 performs slightly worse on average than the reference DA and free simulation, with only hybrid estimates and direct assimilation (DA exp 2) performing better. Results for alkalinity and DIC are similar to pH; for oxygen, all estimates perform comparably, and DA-based estimates only slightly outperform the free simulation.

For pH, the largest misfit between model estimates and pH-sensor glider line observations occurs near the surface and at intermediate depths down to 150 m, with no deep pH bias for simulations assimilating CUGN pH data (not shown). The structure of pH differences between estimates from the reference DA and DA exp 1 (Fig. b) provides context for why the DA cannot substantially improve intermediate depth estimates along glider Line 67 extending out of Monterey Bay. The region is highly active in terms of physical circulation and DA impact, with large pH changes occurring on relatively small spatial scales. We conclude that our DA requires spatially or temporally denser observations to inform the model about the evolving pH field, and only the direct assimilation of pH-sensor glider line pH observations significantly reduces pH misfit in the top 150 m.

To better understand model pH estimates for non-assimilated data, we performed three additional cross-validation experiments. In these experiments, different subsets of CUGN pH and alkalinity data were withheld from assimilation. We focus on the Line 67, and three glider transects in different seasons: CUGN L67Mar, CUGN L67Jul, and CUGN L67Oct (CUGN L67Mar is used in Fig. ). For each line, we created an experiment that assimilates all data from DA exp 1 (reference data and CUGN pH, alkalinity), except for the CUGN temperature, salinity, pH, and alkalinity data associated with that specific transect. These three new experiments are denoted DA exp 1.1, 1.2, and 1.3 (see Table ).

We examine the RMSE of DA exp 1.1, 1.2, 1.3 in comparison to DA exp 1, the reference DA, and free simulation for the three datasets (Fig. ). In some cases, such as CUGN L67Mar salinity (Fig. b), the error of DA exp 1.1 is highest and even exceeds that of the free simulation. This suggests that in some cases, the advection of increments may cause model-data discrepancies downstream, likely due to the filamentous structure of the change in pH after less than a year of carbonate system DA (Fig. b). However, for most datasets, the cross-validation experiments perform better than no assimilation of CUGN data. A clear example is the pH RMSE for CUGN L67Mar (Fig. c), where the free simulation and reference DA show the highest error (both not assimilating CUGN pH data), while the experiments assimilating CUGN L67Mar data (DA exp 1, 1.2, and 1.3) have the lowest error values. The cross-validation experiment DA exp 1.1 is in between the two results, and benefits from the assimilation of the surrounding CUGN data.

A closer look at pH estimates from cross-validation experiment DA exp 1.1 along CUGN L67Mar reveals that assimilation of neighboring CUGN data positively affects deep pH, largely removing the positive bias seen in free simulation and reference DA (Fig. p–r). However, small patches of high model-data misfit at intermediate depths (≈ 100 m) indicate that advection of pH increments from other parts of the CUGN line can cause local increases in error.

In summary, the cross-validation experiments demonstrate that glider DA of pH data typically positively affects pH estimates near the observation network, but advection of assimilative increments can negatively affect downstream estimates. We examine this interesting point further in Sect. .

Changes in pH and dissolved oxygen are closely related because both are affected by important biogeochemical processes such as primary production, plankton respiration, and the decomposition of organic matter. These pathways are included in the NEMUCSC model; however, despite this connection, the assimilation of pH data leads only to minor changes in oxygen in our previous experiments (Fig. h, n). Two main factors in the setup of our assimilation system contribute to this limited impact: (1) The ROMS DA system sets covariance terms between different variables to zero. (2) Our DA configuration favors corrections to observed variables (such as alkalinity) over corrections to unobserved variables (such as the nutrient variables). Thus, the assimilation of pH data mainly modifies model alkalinity and DIC and because changes in these two variables do not directly affect oxygen, the effect of pH DA on oxygen is low. We discuss these factors in more detail in Sect. .

To address this limitation, our final set of experiments includes oxygen observations in the assimilation. DA exp 3 assimilates the reference data along with CUGN pH, alkalinity, and oxygen (Table ). This additional assimilation of oxygen leads to marked improvement in oxygen estimates for the CUGN data compared to DA exp 1 (Fig. h). It also results in a minor increase in error for most other assimilated variables (Fig. a–g, i). This increase is consistent with the inclusion of more observations in the DA, which downweights other observations in the 4D-Var state estimation procedure. Importantly, this result does not imply that a better fit to oxygen observations necessarily creates worse estimates for temperature or other variables.

Notably, DA exp 3 creates better oxygen estimates than the hybrid ESPER-based estimates. This effect mirrors the results for pH estimates, where direct assimilation in DA exp 1 outperformed the hybrid estimates based on temperature and salinity. Building on the improved oxygen estimates from DA exp 3, we create an additional hybrid estimate. This new estimate is based on DA exp 3 and ESPER, including oxygen as an input to ESPER alongside temperature and salinity. The resulting hybrid model produces the best pH estimates and generally outperforms the previously introduced hybrid models, which are based on temperature and salinity alone.

In summary, the addition of dissolved oxygen observations in the assimilation yields improved oxygen estimates without a substantial negative effect on other observed variables. Furthermore, the inclusion of assimilated oxygen in hybrid estimates improves their performance compared to those based on temperature and salinity alone. However, it is important to note that this approach requires a DA system that includes a biogeochemical model with oxygen.