The primary dataset for this study comes from BGC-Argo float with World Meteorological (WMO) number #5904479. Operating in the North Atlantic from April 2014 to December 2017 (Fig. 1), the float collected vertical profiles of temperature, salinity, and four biogeochemical variables: chlorophyll a concentration (Chl a), particulate backscattering coefficient (bbp), nitrate concentration (NO3-), and dissolved oxygen (O₂). The Chl a data were taken from a curated, quality-controlled archive (https://www.seanoe.org/data/00911/102324/, last access: 13 March 2025), in which raw fluorescence was corrected for dark counts, non-photochemical quenching, and physiological calibration.

Two independent BGC-Argo floats #6901485 and #6901648, are used to evaluate the accuracy and calibration of the optimized ensembles against independent observations. Both measure the same variables as the primary float (#5904479) but operate in fundamentally different environments, providing robust tests for generalization.

The first validation float #6901485, also profiled in the North Atlantic, specifically in the western part of the Subpolar Gyre. While this region also features a prominent spring bloom, it is characterized by deeper winter convection and the influence of colder, fresher Arctic waters, providing a test of parameter robustness to different physical forcing within the same ocean basin.

The second float #6901648, operated in the oligotrophic Mediterranean Sea. This basin is defined by much lower nutrient availability, being particularly limited in phosphate relative to nitrate, yet its deep winter convection still drives a distinct, albeit less intense, spring bloom. This provides a challenging test of whether biological rate parameters tuned on a high-nutrient system can generalize to a low-nutrient one.

CTD profiles, additional BGC data and trajectory information were retrieved from the Argo CORIOLIS Global Data Assembly Centre (ftp://ftp.ifremer.fr/argo, last access: February 2025). These data were quality-controlled according to the standard Argo procedures (Wong et al., 2020). Biogeochemical variables (O₂, NO3-, bbp) were processed with the established BGC-Argo procedures (Schmechtig et al., 2015, 2023; Thierry et al., 2018; Johnson et al., 2025). Particulate organic carbon (POC) concentration was estimated from bbp following Mignot et al. (2023).

To supplement the directly measured variables, we incorporated pseudo-observations of phosphate (PO43-), silicate (Si), total alkalinity (TA), and dissolved inorganic carbon (DIC). These were derived from the Copernicus Marine Service “Nutrient and Carbon Profiles Vertical Distribution” product (2025; 10.48670/moi-00048). This product supplies vertical profiles of nutrient concentrations (NO3-, PO43-, and Si) and carbonate-system variables (TA, DIC) for every Argo float equipped with an O₂ sensor. These concentrations are estimated using the CANYON-B neural-network for nutrients and the CONTENT algorithm for DIC and TA (Bittig et al., 2018); both algorithms were trained on ∼ 30 years of quality-controlled profiles from the GLODAPv2 database (Olsen et al., 2016).

First, to ensure the validity of the 1D model assumption, the time window was selected such that the float remained within a single water mass exhibiting weak horizontal gradients (Mignot et al., 2018). The prevalence of these quasi-one-dimensional dynamics is confirmed by the near-constancy of temperature and salinity below the mixed layer during the selected periods (e.g., Fig. 2a, c for float #5904479). This selection is crucial, as pronounced T-S changes at depth would otherwise imply movement across water-mass boundaries and lateral biomass variability that our 1D PISCES configuration cannot represent.

Second, to minimize bias from initial conditions, the precise start date within this window was chosen to correspond to the time of minimum Root Mean Square Error (RMSE) calculated across all available direct and pseudo-observations, between the float data and the Copernicus Marine Service Global Ocean Biogeochemistry Analysis and Forecast (CMEMS-BGC) (10.48670/moi-00015). The final, optimized start dates and resulting one-year analysis periods are: 11 January 2015–14 January 2016 for float #5904479; 26 November 2013–6 December 2014 for float #6901485; and 21 January 2015–15 February 2016 for float #6901648.

In addition to the three floats used for optimization and 1D independent ensemble evaluation, a broader set of BGC-Argo profiles from the IBI domain was used for the independent 3D validation (Fig. 10b). Profiles of O₂ with CANYON-B and CONTENT derived variables (NO3-, PO43-, Si, DIC, TA) were extracted over the 0–500 m depth range. A total of 5436 unique profiles from 2017 were retained after quality control.

We estimated the observational euphotic depth (ZEuphotic) from the Copernicus Marine Service GlobColour product (10.48670/moi-00281). This product delivers monthly 4 km maps of the diffuse-attenuation coefficient at 490 nm, Kd(490), obtained by merging multi-mission ocean-colour sensors. Following ((Morel et al., 2007), we converted Kd(490) to the photosynthetically available radiation attenuation coefficient, Kd(PAR). The euphotic depth, ZEuphotic, is defined as the depth where 1 % of surface PAR persists and was calculated as: 1ZEuphotic=-log(0.01)KdPAR

Surface chlorophyll a concentrations for the 3D validation were obtained from the CMEMS Multi-Observation Global Ocean 3D Biochemistry Reprocessed Product (MULTIOBS_GLO_BIO_BGC_3D_REP_015_010; hereafter MOB-TAC; 10.48670/moi-00046). This product combines satellite ocean-color observations with in-situ hydrological properties from BGC-Argo floats through a neural network (Sauzède et al., 2017) to provide weekly three-dimensional fields of chlorophyll a at 0.25° resolution from the surface to 1000 m. Only surface values (depth = 0 m), over the period January 2017–December 2019 were retained to match the 3D IBI simulation period.

The ocean circulation and thermodynamics are simulated using the Nucleus for European Modelling of the Ocean (NEMO) platform, version 4.2 (Madec et al., 2023). The lower-trophic-level biogeochemistry is represented by PISCES version 2 (Aumont et al., 2015), which is distributed with NEMO 4.2.

PISCES is a mechanistic ecosystem model that resolves 24 prognostic state variables. Phytoplankton growth can be limited by five nutrients: nitrate, ammonium, phosphate, silicate, and iron. The plankton community is represented by four functional types defined by size: two phytoplankton groups (nanophytoplankton and diatom-dominated microphytoplankton) and two zooplankton groups (micro- and mesozooplankton). The model simulates phytoplankton biomass through its carbon, iron, and Chl a content (and silicate for diatoms), while zooplankton biomass is simulated in carbon only. Three non-living organic carbon pools – semi-labile dissolved organic matter, small particles, and large particles – are distinguished by size and reactivity. Particles carry both carbon and iron, while large particles also include calcium carbonate and biogenic silica. PISCES additionally resolves the seawater carbonate system (DIC and TA) and dissolved oxygen. In this study, we adopt the default parameter set supplied with PISCES 4.2 as our reference configuration.

To conduct our analysis, we employ three complementary configurations of the NEMO-PISCES 4.2 model. A computationally lightweight 1D column setup is used for the large optimization ensembles, while a global 3D configuration supplies the state variable estimates needed to calculate the representativeness error (as described in Sect. 3.7). Finally, a regional 3D configuration is used to assess the transferability of the optimized parameter set from 1D to 3D framework.

We define twenty observational metrics – layer-mean concentrations and vertical-structure diagnostics – derived from eight biogeochemical tracers measured or inferred from the BGC-Argo float (Mignot et al., 2023). These metrics serve as individual targets for the sensitivity analysis and optimization. Within the iIS framework, they enter the observation likelihood function, which determines the weight of each ensemble member based on how well it reproduces the observations across all 20 metrics simultaneously, accounting for the observation errors associated with each metric (Sect. 3.5). The prior parameter distributions are uniform, with bounds and inter-parameter constraints defined in consultation with the PISCES model developer (Sect. 3.4; Table S1). A concise description of all twenty metrics is provided in Table 2. Each layer-averaged metric is computed within two depth domains – the productive layer and the mesopelagic layer – using both BGC-Argo observations and the 1D model output. The following subsections detail the four steps required to calculate and compare these metrics: (1) the definition of the vertical layers used for analysis; (2) the calculation of layer-mean concentrations; (3) the diagnosis of emergent vertical features; and (4) the spatiotemporal interpolation procedure used to ensure a consistent comparison.

This section describes the parameter optimization method. The central idea is to draw a large ensemble of model simulations from broad prior parameter distributions, then iteratively reweight the ensemble so that parameter sets producing good agreement with the observations receive higher weight. After several iterations, the resulting weighted ensemble provides an approximation of the posterior distribution, from which parameter estimates, uncertainties, and correlations can be extracted. The mathematical formulation is detailed below; the Bayesian likelihood function that drives the reweighting is presented in the Supplementary Information (Eq. S14).

To quantify posterior parameter correlations, uncertainties, and the predictive spread for both assimilated and unassimilated variables, we adopt iIS, an optimization scheme built on a particle-filter assimilation framework. Particle filters, widely used in biogeochemical modelling for state and parameter estimation (Mattern et al., 2013), are gradient-free ensemble methods that assimilate observations into a population of particles – each a unique, evolving model state – thereby generating full probability distributions for states and parameters. This approach is well suited to the strong non-linearities and non-Gaussian errors characteristic of marine biogeochemistry (Ristic et al., 2004).

Iterative Importance Sampling is a technique for estimating a probability density function (PDF) using a weighted combination of samples drawn from a different, known PDF .(Raices Cruz et al., 2022). In this work, this concept is applied within the framework of Bayesian inference to estimate model parameters. More formally, the goal is to approximate the posterior PDF of the model state variables and parameters, conditioned on observational data.

The Bayesian framework provides an approach for calculating the posterior PDF using prior knowledge and observational evidence. According to Bayes' theorem, the posterior distribution can be expressed as a function of the prior distribution and the observational likelihood (Wikle et al., 2013): 2px|y=py|xpx∫py|xpxdx where x=s,θ is a random vector containing model state variables (s) and model parameters (θ), y is the observation vector, py|x is the likelihood of the observations given the model outcome and px is the probability of the model state. Here, following van Leeuwen et al. (2019) the prior probability is represented by an ensemble of N particles as: 3px=∑j=1N1Nδx-xj where xj represents the jth member of the ensemble, and  δx-xj, is the Dirac delta function that evaluates to 1 if x=xj and 0 otherwise. Equation (3) means that the probability density function is a weighted sum of the delta functions centered at each member of the ensemble, where each member has an equal weight of N-1.

The algorithm described in this article makes use of a self-normalized importance sampling techniques to estimate the posterior px|y using weighted samples of px. Importance sampling generally involves the use of an auxiliary PDF, called proposal density function, whose main objective is to restrict sampling to regions of the state space with high probability (Owen and Zhou, 2000). In our setting, we used the prior px as the proposal density since there is no reasonable estimate of the posterior distribution of the parameters. In this case the weights of each particle j is simply calculated using the observation likelihood (van Leeuwen et al., 2019) : 4ωj=py|xj∑j=1Npy|xj The posterior is then written as a weighted combination of the prior states: 5px|y=∑j=1Nωjδx-xj Estimating model parameters from observations of the state is challenging because we often lack reliable prior information about the parameter distributions, and the relationship between the state and the parameters is typically highly non-linear. Moreover, models generally exhibit biases, partly due to uncertainty in the parameters and partly because of structural deficiencies, which include necessary simplifications, incomplete knowledge of key processes, and uncertainties in their mathematical representation. In the context of importance sampling, the small magnitude of observational noise significantly reduces the probability of obtaining model trajectories with high likelihood, thereby amplifying the impact of model imperfections.

Consequently, plain Monte Carlo sampling would produce many samples with negligible weights, making the estimation inefficient. To address this, the proposed algorithm evaluates Eqs. (4) and (5) iteratively, modifying the likelihood py|xto py|xα, where α is an inflation parameter. This parameter is dynamically adjusted based on the effective sample size (ESS) defined as Martino et al. (2017): 6ESS=1∑j=1Nωj2 At each iteration, both the prior and the likelihood function are updated. The goal of each step is to construct a more informative prior, which allows for a gradual reduction of the inflation parameter α. Importantly, α plays a critical role in preserving parameter diversity throughout the iterations. It can also be interpreted as an inflation factor accounting for unresolved dynamics, analogous to error inflation techniques commonly used in the data assimilation community (see, e.g., Minamide and Zhang, 2017; Ohishi et al., 2022).

The sampling procedure is designed to efficiently explore the high-dimensional parameter space while keeping computational costs manageable. Model parameter samples are first drawn using Sobol's sequence, a low-discrepancy Quasi-Monte Carlo method (Sobol' et al., 2011), implemented through the scipy.stats.qmc.Sobol function in the SciPy library (Virtanen et al., 2020). Low-discrepancy sampling ensures more uniform coverage of the parameter space and reduces the risk of missing key regions of variability (Renardy et al., 2021). This method requires the sample size (N) to be a power of two.

To balance computational expense with sampling density, a dynamic ensemble size is used. For the first iteration, which must sample the entire broad prior parameter space, a larger ensemble of N=8192 is used. In subsequent iterations, where the sampling is focused on a narrower region of interest, the ensemble size is reduced to N=2048 to lower the computational cost.

The model is then integrated with these parameter sets to generate an ensemble of trajectories. Self-normalized importance weights are computed using the adaptive inflation factor (α), which is adjusted to maintain an effective sample size (ESS) of at least 25 % of the current ensemble size. In the final iteration of the iIS algorithm, the resampling process retains the top-ranked particles corresponding to the ESS, which resulted in a final optimized ensemble of 512 members.

For emergent metrics related to depth (HO2min, Hnitracline, and HDCM), we apply a uniform error distribution around the observational values. In practice, if a model realization produces a value outside the specified error range for any of these metrics, it is assigned a weight of zero. Additionally, some observed variations in these depth metrics are not captured by any simulation in the ensemble. This is particularly true for variations in the depth of the oxygen minimum, which appear to be driven more by physical than biogeochemical processes. To account for this, an additional condition is introduced: If more than 25 % of the simulations fall outside the error range for a given observation, that observation is excluded from the analysis. This approach assumes that such observations are influenced by physical processes not well represented in the model. Finally, DCM are not present in all vertical profiles, resulting in discontinuities in the corresponding time series. By applying this method, simulations that produce an unobserved DCM, as well as those that fail to reproduce an observed one, are filtered out.

To ensure biologically realistic growth rates of mesozooplankton in the model ensemble, an additional constraint based on temperature-dependent physiological dynamics has been implemented. Specifically, building on the work of Huntley (1992) and using the temperature measured by float #5904479, the maximum generation time of mesozooplankton should not exceed 30 d. Accordingly, the minimum plausible accumulation rate was set to 0.01 d⁻¹. To prevent the regeneration of ensemble members yielding unrealistically low mesozooplankton accumulation rates, any member with a maximum accumulation rate below 0.01 d⁻¹was assigned a weight of zero and therefore excluded from subsequent resampling.

To approximate the posterior distribution px|y, a kernel density function (KDF) is fitted using the KernelDensity class from the sklearn.neighbors module of the scikit-learn library (Pedregosa et al., 2011). The top-ranked particles corresponding to the ESS are retained, while the remaining particles are resampled from the fitted KDF. All weights are reset to N-1 for the next iteration, and the PISCES model is re-run for the newly resampled particles (Fig. 3).

This procedure is repeated iteratively until the inflation factor α converges or a predefined maximum number of iterations is reached. The proposed algorithm can also be interpreted as an adaptative IS method, following the classification of Elvira and Martino (2021), due to the use of a new proposal distribution, here, the prior p(x), at each iteration.

The purpose of the sensitivity analysis is to identify which of the 95 PISCES parameters have a measurable influence on the assimilated metrics, and to quantify whether this influence operates directly (first-order effect) or through interactions with other parameters (higher-order effects). This information is used to define two reduced parameter subsets for comparison with the full 95-parameter optimization. The following subsections describe the theoretical framework (Sect. 3.3.1), the quantities of interest used to measure parameter influence (Sect. 3.3.2), and the selection criterion applied to define the parameter subsets (Sect. 3.3.3)

The aim is to include in the iIS procedure only those parameters that have an impact on the assimilated metrics, and therefore only those parameters that can be significantly constrained by the metrics defined in Table 2. Parameters that do not significantly influence these metrics, either in terms of phenology or amplitude, are not perturbed. For such non-influential parameters, we assume that their reference values should be retained, as there is insufficient information to constrain their uncertainties.

To identify influential parameters, we select all parameters with a Sobol' sensitivity index (first-order or total-order; see Sect. 3.3 for details) greater than 0.02 for at least one quantity of interest (QoI), following the approach of Prieur et al. (2019). This filtering step yields a reduced parameter set, thereby lowering the dimensionality of the parameter uncertainty space.

We have shown in Sect. 3.2 and 3.3 that both iIS and Sobol' sensitivity analysis require a prior estimate of the parameters' probability density functions. In practice, however, limited information about these parameters is available in the literature; most often, only plausible value ranges are reported (Denman, 2003).

Given this lack of detailed prior knowledge, we assume uniform distributions. This choice reflects a non-informative prior assumption, treating all values within the specified interval as equally likely. It provides a conservative starting point for inference, minimizing subjective bias.

To account for the possibility that fitting the ensemble of assimilated metrics may require the model to reach a substantially different equilibrium state, we define broad perturbation intervals, ranging from one-hundredth to twice the reference values. To ensure physical consistency, certain parameters are subject to additional constraints to prevent unrealistic phenomena such as the artificial generation or loss of matter for example (Table S1 in the Supplement).

In the Sect. 3.2 each particle is weighted using the observation likelihood. As noted by van Leeuwen et al. (2019), the likelihood function p(y|x) quantifies the probability of obtaining the observation y assuming that x represents the true state. Given that observations are modeled as y=Hx+ϵ, with H a possibly non-linear observation operator mapping from the model state to the observation space and ϵ a random noise following a known distribution, the likelihood becomes a function of the discrepancy between the observed and model-predicted values, shifted by the noise distribution pϵ : 9py|x=pε(y-H(x))

Therefore, accurately estimating observational errors is crucial, as these errors directly influence the weighting of particles in the importance sampling procedure. We consider two main sources of observational error: measurement error and representativity error. Measurement error accounts for uncertainties in mapping from the model state to the observation space. This includes uncertainties associated with the observation operator (H) itself, the instrumental precision of direct sensor measurements, and the inferred uncertainty of variables derived from neural networks (e.g., CANYON-B). Representativity error, in contrast, arises from discrepancies between what the 1D model can simulate and the finer-scale or unresolved 3D features present in the observations. The explicit form of the likelihood function is given in the Supplementary Information (Eq. S14).

The observation error ϵ is assumed to follow a Gaussian distribution for most metrics. However, for metrics related to the depth of the nitracline, the DCM, and the depth of the oxygen minimum, a uniform distribution is used. Within the iIS procedure, this choice effectively excludes particles that produce unrealistic values for these depth-related variables.

The observation likelihood in iIS requires an estimate of the measurement uncertainty for each metric. We used four distinct methods to compute these errors, each tailored to a specific group of metrics. The resulting observation errors, expressed in percentages, are shown for each metric in Table 2.

Using a 1D model to represent a 3D ocean introduces a “representativity error” due to neglected 3D physical processes. To account for this, the cost function used in our assimilation combines observational errors with this representativeness term. Including this error limits the extent to which the optimization compensates for missing physics, yielding a more reliable calibration. The magnitude of this error was quantified at each time step as the absolute difference between the value from the full “3D-Free” simulation (sampled along the float's trajectory) and the value from the corresponding 1D simulation (“1D-Free”, initialized and forced with the same 3D fields).

This time-dependent representativity error is then added in quadrature to the measurement error at each corresponding time step, creating the total observational error used in the likelihood calculation. This approach allows for greater uncertainty during periods when the 1D assumption is weakest, prevents overconfidence in observational constraints, and reduces the risk that the parameter optimization compensates for missing physics with unrealistic parameter values.

In addition to measurement and representativity errors, we also evaluated two other potential sources of uncertainty: (i) error covariances among the observed and derived variables, and (ii) grid discretization errors.

First, to assess the impact of error covariances, we estimated the full error covariance matrix for the observation vector. We propagated a 3 % perturbation in the input oxygen observations through the CANYON-B and CONTENT algorithms to quantify the covariance terms both among the network-derived variables (e.g., DIC, TA) and between those variables and the uncertain input (O₂). A comparative analysis demonstrated that including these off-diagonal covariance terms in the likelihood calculation had a negligible effect on the final results; the RMSE between the optimized solution and the observations changed by less than 2 %.

Second, we assessed the impact of grid discretization error. This error arises because the 1D model is forced by a single grid point from the 1/4° physical model, while the true float position varies within that grid cell. To test the sensitivity to this choice, we ran an ensemble of 1D simulations where each member was forced by a different, randomly selected 1/12° sub-grid point from within the same 1/4° grid cell. By comparing simulations with identical parameters but different physical forcing, we could quantify the influence of this sub-grid variability. The analysis indicated that the resulting grid-induced differences were minor compared to measurement and representativity errors; including this physical uncertainty in the optimization process changed the final RMSE between the optimized solution and the observations by less than 2 %.

Given the significant computational cost required to quantify these two error sources and their minor impact on the results, both were excluded from our final error budget.

We evaluate the effectiveness of the three optimization strategies using statistical criteria that assess three key aspects of performance:

Goodness of fit: The improvement in skill against the assimilated data is quantified using the Normalized Root Mean Square Error (NRMSE).

The Normalised Root Mean Square Error (NRMSE) is our primary metric for quantifying the misfit between simulated and observed metrics. We calculate two versions for each metric M: one for the single best ensemble member and one for the weighted-mean ensemble. The NRMSE normalises the RMSE by the total observation error εεtM (measurement and representativity errors combined in quadrature) so that a value of ∼ 1 indicates model-data misfit comparable to the expected observation error, while values less than 1 indicate a good fit (Eqs. S5–8 in the Supplement). The performance of the optimized simulations is then evaluated by comparing their NRMSE values to that of the reference simulation with default PISCES parameters (hereafter REF). The optimized simulations are referred to as OPTI. For both the best-member and ensemble NRMSE, the relative change in performance is quantified as Δ%NRMSE = (NRMSEOPTI- NRMSE_REF) / NRMSEREF×100 %, so that negative values indicate improvement

To assess the performance of the optimized ensembles against independent observations, we use the Reduced Centred Random Variable (RCRV). At each time step, the RCRV normalises the residual between the weighted ensemble mean and the observation by the total uncertainty, combining observation error and ensemble spread in quadrature (Eqs. S9–12 in the Supplement). The time-mean RCRV (bias) measures systematic error: values near zero indicate an unbiased ensemble. The standard deviation of the RCRV (dispersion) measures calibration: a value of 1.0 indicates a perfectly calibrated ensemble, values below 1.0 indicate a conservative (over-dispersive) ensemble, and values above 1.0 indicate an overconfident one.

We quantify the reduction in parameter uncertainty by the percentage change in the 67 % Highest Density Interval (HDI) between the prior and posterior distributions (Eq. S13 in the Supplementary Information). The HDI represents the narrowest credible interval containing 67 % of the probability mass. Negative values indicate that the posterior is narrower than the prior, i.e., the observations have reduced parameter uncertainty.

Finally, we calculate the pairwise correlation coefficients between all optimized parameters using the 512 best-weighted ensemble members from each strategy. This analysis tests whether the assimilation framework successfully found an independent solution for each parameter, a key indicator of a well-constrained system.

For the surface chlorophyll a validation against MOBTAC, we adopt the five regional boxes defined by Gutknecht et al. (2019) in their formal skill assessment of the CMEMS IBI biogeochemical system (Fig. 10b): Box 1 (Celtic Sea/Rockall Trough), Box 2 (Canary/Madeira region), Box 3 (Bay of Biscay), Box 4 (Gulf of Lion), and Box 5 (Balearic Sea). Using the same geographical framework ensures direct comparability with the established IBI validation protocol.

These boxes are, however, too small to yield robust statistics for the subsurface BGC-Argo validation, where float coverage is much sparser – particularly in the North Atlantic, which concentrates only 10 %–15 % of the matched profiles. The IBI domain was therefore divided into three broader sub-regions for the BGC-Argo assessment (Fig. 10b): (i) Northern basin (20° W–17° E, 48–65° N), (ii) Iberian Atlantic (20–5° W, 26–48° N), and (iii) Mediterranean and Bay of Biscay (5° W–17° E, 26–48° N). For the aggregated analysis presented in Sect. 5, the Northern and Iberian Atlantic basins are further combined into a single “North Atlantic” region.

The technical details of the model–observation colocation procedure and the adaptation of the NRMSE metric for the 3D assessment are provided in the Supplementary Information (Sect. S2). In brief, subsurface BGC-Argo profiles are matched to the IBI model in space, depth, and time over the 0–500 m range, with model values interpolated horizontally, vertically, and temporally to each float observation. Surface chlorophyll a is compared on the MOBTAC 0.25° grid using monthly means in log₁₀-transformed space, displayed on Taylor diagrams. Model performance is assessed using the same NRMSE framework as in Sect. 3.9, with two differences: the representativity error term is omitted (since the 3D model resolves the processes it was designed to account for), and the observation uncertainty uses the point-by-point error field associated with each BGC-Argo measurement rather than the propagated Monte Carlo estimates. The Δ%NRMSE is computed as defined in Sect. 3.9. When both NRMSE values fall below unity, a positive Δ%NRMSE does not constitute a meaningful degradation, since both simulations already reproduce observations within their associated uncertainties.

The GSA revealed that the parameter controlling organic matter recycling (e.g., the half-saturation constant for DOC remineralization, KDOC) exerted the strongest first-order influence, primarily through its impact on mesopelagic concentrations of DIC, nitrate, phosphate, and dissolved oxygen (Fig. 4a). This parameter, however affected only a limited subset of metrics. In contrast, the phytoplankton light response (e.g., the P-I slope for nanophytoplankton, αNano) showed significant effects on almost all state variables, including those in the mesopelagic layer. Zooplankton parameters also emerged as important drivers, particularly those related to grazing and prey preference (e.g., the microzooplankton preference for nanophytoplankton, PNanoMicroZoo, and the half-saturation constant for microzooplankton grazing, KGMicroZoo).

The total-order analysis demonstrated that many of the most influential parameters, especially for zooplankton, acted primarily through interactions rather than direct effects (Fig. 4b). For instance, the non-assimilated fractions of nanophytoplankton consumed by microzooplankton (σMicroZoo) and mesozooplankton (σMesoZoo) were ranked as the first and second most sensitive parameters overall, despite having only weak first-order effects (Fig. 4a). In contrast, KDOC displayed similar sensitivity in both the first-order and total-order analyses, suggesting its influence is largely independent of parameter interactions (Fig. 4a). When accounting for interactions, seven parameters had significant impacts on all model metrics (αNano, PNanoMicroZoo, KGMicroZoo, KGMesoZoo, θMaxFeNano, GmMicroZoo, GmMesoZoo). These correspond to parameters of microzooplankton, mesozooplankton and nanophytoplankton, highlighting their central role in biogeochemical dynamics (Fig. 4b).

By contrast, nine parameters related to processes irrelevant to the float's open-ocean location – such as the half-saturation constant for anoxia, coastal iron release, and iron concentration in sea ice – showed no measurable influence.

All three parameter optimization strategies improved the simulation of the productive-layer seasonal cycle, correcting biases present in REF. The three optimized ensembles – All-parameters (Fig. 5), Total effects (Fig. S2), and Main effects (Fig. S5) – produce nearly identical corrections. In all cases, the ensembles capture the magnitude and timing of the spring phytoplankton bloom, as indicated by Chl a concentrations (Fig. 5a). This improvement is reflected across related variables: the model reproduces the observed seasonal drawdown of NO3-, PO43- and DIC, and the concentration of Chl a and POC aligns with float data (Fig. 5a–d, f).

These visual improvements are confirmed by a large and statistically robust reduction in the model-data misfit (Table 3a). For the assimilated float, the three strategies achieved a comparable median NRMSE reduction across the eight productive-layer metrics: -55.6 % (± 36.1 %) for Main effects, -54.2 % (± 24.2 %) for Total effects, and -53.6 % (± 29.9 %) for All-parameters. In all cases, the median improvement is larger than the associated interquartile uncertainty, indicating an enhancement of the goodness of fit. Given the small sample size (n=8 metrics), we used a non-parametric Kruskal-Wallis H-test to formally compare the distributions of NRMSE reductions. For the weighted-mean ensembles, the test confirmed that there is no statistically significant difference among the three strategies (p=0.99). This similarity in performance demonstrates that while a small subset of highly sensitive parameters drives most of the improvement, perturbing all 95 parameters achieves the same skill.

The median NRMSE of the weighted-mean ensemble for each strategy was comparable to that of its single best-performing member (Table 3a). However, the ensemble weighted mean (solid lines, Fig. 5) consistently provided a smoother and more physically plausible representation of the seasonal cycle than any individual “best” member simulation (dashed lines), which is a known advantage of the ensemble approach (Germineaud et al., 2019). Regarding the parameters themselves, while the overall posterior distributions were largely similar across the three strategies, the specific values for the single best-performing member of each strategy show notable differences. These parameter sets are provided in Table S1 as a concrete and reusable outcome for future studies.

In contrast to the productive layer, the optimized ensembles closely resemble REF for the mesopelagic layer and for emergent vertical properties (Figs. 6, 7 and S3, S4, S6, S7). For most deeper metrics, such as nutrients, oxygen and carbonate chemistry, the optimized ensembles remain nearly indistinguishable from REF. This outcome can be attributed to several factors: REF already simulated these variables with reasonable skill, leaving little mean bias for the optimization to correct (e.g., Fig. 7a, c); mesopelagic properties have long adjustment timescales, making a one-year assimilation period insufficient to induce substantial changes; and the 1D model configuration neglects the deep advective processes that drive much of the variability in the ocean interior.

Mesopelagic POC is an exception. Its dynamics are forced by particle export from the surface, giving it a shorter response timescale than the other deep tracers. For this variable, the optimization mitigated a bias present in the reference run, improving the simulation skill (Fig. 6a).

For these deeper properties, the optimization therefore did not correct large mean-state biases, but penalized parameter sets that drifted away from the observations. For instance, the assimilation penalized any parameter set that produced an unrealistic nitracline depth or oxygen minimum, ensuring the optimized ensembles remained consistent with the float data throughout the simulation period.

This limited overall impact is confirmed by the error statistics. For the seven non-productive-layer metrics, the median NRMSE improvements for the weighted-mean ensembles were negligible: -0.31 % (± 1.3 %) for Main effects, -0.38 % (± 1.2 %) for Total effects, and -0.36 % (± 1.4 %) for All-parameters (Table 3b). The associated uncertainty (IQR) for each strategy is several times larger than the median improvement, and a Kruskal-Wallis H-test confirmed that the differences among the strategies are not statistically significant (p=0.99). These results suggest that while the framework can constrain specific, well-observed features at any depth, the one-year assimilation period is insufficient to correct for potential systemic biases in slower, deeper biogeochemical processes.

As with the productive layer, the median NRMSE for the weighted-mean ensemble of each strategy was statistically indistinguishable from that of its corresponding single “best” member (Table 3b). For these deeper properties, where model-data misfits are often smaller and less dynamic, the visual difference between the smoother ensemble mean and the more variable best-member simulation is less pronounced than in the surface layer, but the principle remains that the ensemble mean provides a more robust estimate.

While all three optimization strategies showed comparable skill against assimilated data, their performance diverged when estimating the predictive uncertainty for unassimilated variables (Fig. 8). The Main effects ensemble, which perturbed only 29 parameters, produced the smallest uncertainty spread, with a median standard deviation of 0.17 (± 0.16) relative to the seasonal cycle of the unobserved metrics. In contrast, the Total effects (66 parameters) and All-parameters (95 parameters) ensembles produced larger spreads, with median values of 0.29 (± 0.21) and 0.27 (± 0.26), respectively.

Formally, a Kruskal-Wallis H-test on these distributions of uncertainty spreads does not indicate a statistically significant difference between the three strategies at the α=0.05 level (p = 0.26). The difference in median spread is nonetheless a direct consequence of where each variable's uncertainty comes from. For the assimilated variables, the observations constrain the predictions: parameter sets that fit the data poorly are given little weight, so all three strategies converge on a similar, narrow spread regardless of how many parameters were free to vary – hence their comparable skill. The unassimilated variables carry no such constraint, so their predictive spread reflects only the parametric uncertainty that each strategy propagates. The Main effects strategy holds most parameters – including many that the GSA showed exert influence mainly through interactions – fixed at their default values, and so removes part of that uncertainty by assumption rather than because the data exclude it. The resulting narrow spread is therefore an underestimate of the model's parametric uncertainty.

Conversely, the Total effects and All-parameters strategies allow a larger set of influential parameters to vary, propagating their uncertainty into the ensemble spread. The wider predictive spreads they produce reflect the propagated parametric uncertainty rather than a degradation in model skill. Therefore, we conclude that while any of the strategies can produce a skillful “best-guess” simulation, approaches that perturb more parameters provide a fuller quantification of predictive uncertainty, which is required for forecasting and climate-projection applications.

Assimilating the BGC-Argo data reduced parameter uncertainty unevenly across the 95 parameters, as measured by the percentage reduction in the 67 % HDI of each parameter distribution (Table 4). The reduction was concentrated in the parameters with the highest Sobol indices. This targeted nature is most evident in the All-parameters experiment. In this run, the median HDI reduction showed a clear cascade based on parameter sensitivity: the reduction was greatest for the Main effects subset (-27.4 %), smaller for the broader Total effects subset (-20.4 %), and smallest when averaged over all 95 parameters (-16.3 %). A Kruskal-Wallis test confirms that this cascade is significant at α=0.10 (p=0.08). The cascade indicates that the algorithm allocates its constraining power to the parameters that most influence the assimilated observations. The strongest overall constraint was achieved, as expected, in the most focused experiment: when only the 29 Main effects parameters were optimized, their median HDI shrank by 41.3 %.

Furthermore, the optimization produced posterior parameter ensembles that were effectively decorrelated (Table 4). Strong posterior correlations are a common outcome in data assimilation, often indicating that the available data are insufficient to constrain parameters independently. In contrast, our analysis reveals an absence of significant linear dependencies. Across all three optimization strategies, the maximum correlation coefficient observed between any two parameters was low (peaking at 0.34), while the median correlation was statistically indistinguishable from zero (0.032–0.04). These low correlations indicate that the comprehensive, multi-variable BGC-Argo dataset provided sufficient orthogonal constraints to allow the framework to find a solution in which the parameters are uncorrelated with one another.

To evaluate the accuracy and robustness of the optimized ensembles against independent observations, the three ensembles were used to simulate conditions along the trajectories of two BGC-Argo floats that were not involved in the optimization: one in the North Atlantic Subpolar Gyre (#6901485) and one in the oligotrophic Mediterranean Sea (#6901648) (Fig. 1). A visual comparison of the model performance for these floats is provided in the Supplement (Figs. S8–25). Ensemble accuracy and reliability were assessed using the RCRV bias and dispersion metrics (Fig. 9).

The RCRV bias analysis shows that all three optimization strategies produce ensembles with low systematic error against these out-of-sample observations. For both validation floats, the median ensemble bias is small across most metrics, confirming that the optimized simulations accurately represent independent data (Fig. 9a, b). A Kruskal-Wallis H-test confirms that the bias distributions are statistically indistinguishable among the three strategies (p=0.99). All ensembles do, however, produce two large biases for the Mediterranean float. The first concerns the nitracline depth: the optimization, constrained by a North Atlantic float with a shallow nitracline, yields biological rate parameters that maintain this shallow structure in the Mediterranean, where the observed nitracline is much deeper. The second concerns the productive-layer nitrate concentration, and is a statistical artifact of the RCRV metric: in the oligotrophic Mediterranean, observed nitrate concentrations and their associated errors are extremely small, causing the normalisation to amplify a physically negligible model–data mismatch into a large bias score.

The RCRV dispersion metric assesses whether the ensemble uncertainty is realistically calibrated. A dispersion of 1.0 indicates a perfectly calibrated ensemble, where the predicted spread matches the actual model–data error; values below 1.0 indicate an over-dispersive (conservative) ensemble, while values above 1.0 indicate an under-dispersive (overconfident) one. For both validation floats, the median dispersion scores are consistent across all three strategies, approximately 0.3 for the North Atlantic float and 0.4 for the Mediterranean float (Fig. 9c, d), with no significant difference among strategies (Kruskal-Wallis p=0.99). These values, being well below unity, indicate that the optimized ensembles provide a conservative estimate of the model's true uncertainty: the predicted spread is wider than the observed errors. This rules out the risk of overconfident calibration with artificially tight posteriors described by Hermans et al. (2022) and Yang and Zhu (2018).

Although not perfectly calibrated, these results confirm that the assimilation has constrained the model while maintaining a cautious uncertainty envelope. A conservative estimate of uncertainty is generally preferable to an overconfident one in operational forecasting applications, and the consistently low bias across both floats and all strategies demonstrates that the optimized parameter set produces accurate simulations in bioregions distinct from the training site.

The 3D validation dataset comprises approximately 1430 unique BGC-Argo profiles matched to the IBI model over the period 2017–2019 (Fig. 10b). After quality control and outlier filtering, this yields between 430 000 and 490 000 observation–model pairs per variable across the 0–500 m depth range. The observation distribution is strongly asymmetric: the Mediterranean basin concentrates 85 %–90 % of the matched pairs, while the North Atlantic (Northern + Iberian Atlantic) contributes the remaining 10 %–15 %, reflecting the denser BGC-Argo float deployment in the Mediterranean during this period.

The OPTI simulation generally outperforms REF across the IBI domain (Table 6). Of the 18 variable–basin combinations, 12 show a negative Δ%NRMSE, indicating that the optimized parameters improve model skill. The largest improvements occur for total alkalinity in the Northern basin (Δ%NRMSE =-59.3 %), dissolved oxygen in the Mediterranean (-16.4 %), phosphate in the Northern basin (-14.3 %), and DIC in the Northern basin (-13.5 %). Phosphate is the only variable for which OPTI outperforms REF in every basin, with Δ%NRMSE ranging from -4.1 % to -14.3 %.

Across basins, the Northern and Iberian Atlantic show the largest improvements. Nitrate, phosphate, and total alkalinity show consistent improvements in both basins, with Δ%NRMSE ranging from -4.1 % (PO43-, Iberian Atlantic) to -59.3 % (TA, Northern basin). DIC improves markedly in the Northern basin (-13.5 %) but remains essentially unchanged in the Iberian Atlantic (+1.1 %). These improvements are genuine because, for these variables, NRMSE values exceed unity in at least one simulation, confirming that the optimization measurably reduces model errors that are detectable relative to observation uncertainty. Dissolved oxygen is the only variable where OPTI slightly underperforms REF in the Northern basin (+1.8 %), though both simulations exceed unity only marginally. Silicate stands apart: both simulations achieve NRMSE well below unity (0.27–0.40) across both basins, indicating that the model reproduces CANYON-B-derived silicate observations within less than half the associated uncertainty regardless of the parameter set.

The Mediterranean presents a more contrasted picture. All NRMSE values exceed unity – by factors of 2 to 10 – indicating that model errors surpass observation uncertainty in this basin. Dissolved oxygen and phosphate show genuine improvements (Δ%NRMSE =-16.4 % and -8.5 %, respectively), confirming that the optimized parameters partially correct the free-running model even in a biogeochemical regime different from the training site. Nitrate and silicate, however, degrade under OPTI (+5.2 % and +19.8 %), suggesting that the biological rate parameters tuned on a single North Atlantic float do not fully transfer to the oligotrophic Mediterranean, consistent with the nitracline bias identified in the 1D independent validation (Sect. 4.6). The carbonate system variables exhibit very high NRMSE values (8–10 for both DIC and TA), reflecting the fact that the CONTENT-derived error estimates are much smaller than the actual model–observation discrepancies; within this context, OPTI provides only marginal improvements (-0.8 % and -6.6 %, respectively).

Surface chlorophyll a provides a complementary, independent assessment of the optimized parameters (Fig. 10). The Taylor diagram confirms that both simulations reproduce the MOBTAC-observed seasonal cycle across all five boxes with correlations typically exceeding 0.8–0.9. The OPTI simulation outperforms REF: for every box, the green markers sit closer to the MOBTAC reference point than the corresponding red markers, reflecting both a reduced centred RMSE and a normalised standard deviation brought closer to unity. The improvement is most visible in the Atlantic box 1 where REF tends to underestimate the amplitude of the seasonal cycle – a bias that the optimized parameters correct.