This study uses the World Wide Water Resources Assessment model W3RA,. Previous studies suggest that W3RA offers a good performance at representing water storage variability when compared to other state-of-the-art models . The evaluations performed in this study also suggest a reasonable agreement between the model and satellite observations prior to DA in terms of water storage (see Sect. ), suggesting that W3RA provides a reasonable representation of the large-scale water storage variability in the Brahmaputra River basin and constitutes an appropriate baseline for the DA experiments. In addition, as a parsimonious model, W3RA presents low computational requirements and is therefore well suited to reduce the computational expense of the experiments.

The W3RA model represents natural land water and energy exchange processes. The soil profile is divided into four layers: topsoil, shallow soil, deep soil, and groundwater (Table ). In this study, topsoil and shallow soil are treated jointly as “SSM”, and deep soil and groundwater are treated jointly as “groundwater” (a detailed explanation of the reason behind this aggregation is provided in Sect. ). Water exchange processes within the soil are described through the representation of infiltration, drainage, root water uptake, and capillary rise. Soil and surface water evaporation is computed as a linear function of available energy, and groundwater discharge is estimated through a linear reservoir model . Heterogeneities in soil composition are not considered, and anthropogenic water use activities, such as groundwater pumping or agricultural irrigation, are not modeled.

Vegetation is modeled in two categories, shallow-rooted short vegetation (grass and shrubs) and deep-rooted tall vegetation (trees), which are represented as two different Hydrological Response Units (HRUs). The proportion of vegetation belonging to each category for each grid cell is determined through the satellite-based tree cover fraction map by . Vegetation transpiration is computed as the minimum of maximum root water uptake and optimum transpiration rate, where the latter is estimated using the Penman-Monteith equation. Root water uptake is modeled through a linear ramp function (as a function of soil water content) . Differences in crop types or seasonal variations of vegetation are not considered. More details on the explicit formulation of all these processes can be found in .

W3RA also has a snow module that represents snow accumulation and melt based on a degree day approach . Glaciers are not represented within the snow module. To compute surface water routing, a grid-based model was coupled with W3RA where each grid cell is represented as a linear reservoir . Lakes, reservoirs and floodplanes are not represented within the routing model, and evaporation of water routed in streams and rivers is not represented either.

The model parameterization for the Brahmaputra River basin was performed through a regionalization approach . No further calibration or correction of forcing data was implemented since possible errors resulting from these limitations are expected to be addressed through the DA process. A prior calibration of the model against streamflow would be unlikely to modify the main conclusions of this study due to the little sensitivity of streamflow estimates to water storage dynamics (later analyzed in Sect. ).

The model was run with a spatial resolution of 0.1°×0.1° (approximately 10km×10km) and a daily temporal resolution, and was forced with precipitation, maximum and minimum temperature, and downward surface solar radiation data derived from ERA5 and ERA5-Land (see more details in Sect. ).

The daily TWS was derived from the daily GRACE products of ITSG-2018 . The daily product was chosen over traditional monthly GRACE products for two reasons: (i) to avoid the temporal mismatch between TWS and SSM observations, as well as with the model; and (ii) to improve the model TWS estimates on the sub-seasonal and sub-monthly timescales . In this study, we only considered data for the period 2003–2015 to limit the temporal extent of each experiment (and hence their computational cost) and to avoid having to deal with the gap between GRACE and GRACE-FO missions (2017–2018). The post-processing applied to the data included recommended spherical harmonic corrections, geophysical inversion, and leakage error correction see more details in. The time-dependent error variance of the spherical harmonics was post-processed accordingly and rescaled at the end of the process to fit the values reported in . It should be mentioned that the Kalman Smoother approach used to generate the daily TWS product can be expected to introduce temporal autocorrelations in the daily to weekly timescale. In the present study, these temporal correlations were not accounted for because (i) the temporal autocorrelation is expected to be weaker in the seasonal to inter-annual timescale considered in this study; and (ii) the magnitude or temporal structure of these autocorrelations have not been quantified yet, and computing them would require a whole dedicated study. Accounting for temporal autocorrelations could be relevant for future studies exploring water storage estimates in weekly to sub-weekly timescales.

The daily SSM estimates originate from the European Space Agency Climate Change Initiative (ESA CCI) Combined SSM product (v08.1) . This dataset was chosen because of its daily sampling, avoiding a temporal mismatch with the model. The combined product merges SSM retrievals derived from various active and passive satellite sensors and is provided on 0.25°×0.25° tiles (Fig. ), with uncertainties computed using a triple-collocation technique. The merging process includes rescaling the retrievals to a common baseline modeled soil moisture of GLDAS Noah,. To avoid mismatches with our model, before performing DA, we re-rescaled the ESA CCI SSM estimates to the range of W3RA SSM estimates (see more details in Sect. ).

In situ groundwater level observations and river water level observations of the Bangladesh Water Development Board (BWDB) see, available in the area of Bangladesh, have been used for validation. 263 groundwater observation sites were available for the study area, located mainly in the South sub-basin and covering the entire study period (Fig. ). The groundwater level data have been quality controlled by the BWDB and curated by . Specific yield information was originally obtained from around 200 pumping tests and interpolated using ground lithology maps by .

Regarding river level observation sites, a selection was carried out according to the following criteria: (i) the sites should be located on the main branch of the Brahmaputra River; and (ii) they should be as close as possible to the sub-basin outlets, to reflect the impact of sub-basin averaged TWS assimilation on river water levels. Two sites were selected for the West and South sub-basins (Fig. ), which provide data for the entire study period with few temporal gaps (Fig. ). No river observation site was available for the East sub-basin.

The Standardized Precipitation Evapotranspiration Index SPEI, is an indicator of hydrological dryness-wetness conditions, and in this study it was used to perform a consistency-check with TWS estimates. Although the SPEI reflects variability of net water fluxes rather than water storage, previous literature suggests that the aggregation of SPEI over time windows of 12 months or longer effectively works as a temporal integration, leading to higher correlations with TWS . Based on this literature, the SPEI-12 was chosen for comparisons in this study. The SPEI-12 time series for each sub-basin were derived from the Global Drought Monitor and are used to validate the inter-annual groundwater dynamics.

TWS in the East and West sub-basins is affected by trends caused by glacier retreat . These glacier trends have been quantified in previous literature using satellite stereo imagery . In this study, previously computed glacier trends were used to reduce the glacier impact from TWS in Sect. , allowing to compare the mass loss caused by glacier retreat with that caused by other hydrological processes. The Randolph Glacier Inventory 7.0 dataset was used to compute the total area covered by glaciers in each sub-basin, resulting in 7719 and 3406 km² for the East and West sub-basins, respectively.

An independent validation of SSM is not straightforward because (i) there is no in situ SSM observation openly available in the study region; and (ii) various active and passive satellite SSM retrievals are included in the assimilated ESA CCI Combined SSM, and therefore a comparison against any of these SSM products cannot be considered independent. Due to these limitations, SSM estimates are evaluated against soil water storage estimates from the WaterGAP global hydrology and water use model , which are resolved with daily and 0.5°×0.5° resolution. A 10 km resolution topographic slope dataset was used to support the interpretation of spatial SSM difference patterns .

Following previous literature on land DA , DA was performed using an Ensemble Kalman Filter (EnKF) approach . The EnKF is a Monte-Carlo approach in which the uncertainty of the model is computed through an ensemble run of Ne perturbed model realization (more details on the perturbation method can be found in Sect. ). The objective here is to update the ensemble of model states (Xk=[xk(1),xk(2),…,xk(Ne)]) for every timestep k, where the model state xk(i) represents the water storage values predicted by the model for each grid cell and each water storage component in that timestep (see a list of these components in Sect. ). The updated model state for the timestep k is computed as 1Xk+=Xk-+Kk(Yko-HXk-), where Xk- is the predicted model state ensemble matrix for the timestep k, which contains, in its column i, the model state of the ith ensemble member, and Xk+ is the updated model state ensemble matrix for the same timestep, with a similar structure. The observation matrix Yko contains each ith perturbed set of observation (here TWS, SSM or both) in its ith column. These observations are perturbed according to their uncertainty, in order to guaranty a realistic ensemble spread in the updated model states see more information on the perturbation method in Sect. . The matrix H is the forward observation operator that, when applied to the model state, generates the observation prediction. Kk is the Kalman gain matrix, which is defined as 2Kk=C(Xk-)H⊤(HC(Xk-)H⊤+Σyo,k)-1. Σyo,k refers to the error covariance matrix of the observations. C(Xk-) is the ensemble-based error covariance matrix of the model state and is computed as C(Xk)=1Ne-1ΔXkΔXk⊤, where the matrix ΔXk contains, in its ith column, the deviation of the ensemble member i from the ensemble-average of the model state, xk(i)-1Ne∑i=1Nexk(i).

In the uni-variate (i.e., single-observation) TWS DA experiment Yk,TWS0 and ΣyTWSo,k are used within the EnKF equations, which represent the perturbed TWS observations and their uncertainty, respectively. The forward TWS computation operator, HTWS, vertically aggregates all water storage components and horizontally averages gridded model estimates on three sub-basins (Fig. ). For the uni-variate SSM DA experiment, Yk,SSM0 and ΣySSMo,k represent the perturbed SSM observations and their uncertainty, respectively. The forward SSM computation operator (HSSM) aggregates the water storage components corresponding to SSM and averages gridded model estimates on the 0.25° × 0.25° ESA CCI tiles (Fig. ).

Multi-variate DA can be implemented in a similar way to uni-variate DA by extending the observation matrix to include all observations in the following way 3Yk,multi-variateo=Yk,TWSoYk,SSMo.

The observation error covariance matrix and the forward operator matrix also need to be updated accordingly, that is, Σymulti-variateo,k=ΣyTWSo,k00ΣySSMo,k and Hmulti-variate=HTWSHSSM. Note that the matrix Σymulti-variateo,k represents the empirical observation errors of the GRACE TWS observations and satellite-derived SSM observations. Such empirical errors arise from factors including, but not limited to, instrumental noise, GRACE orbit determination uncertainties, and post-processing errors. Given the fundamentally different measurement principles and processing chains underlying TWS and SSM products, it is theoretically unlikely that correlated empirical observation errors would exist between the two, and it is therefore realistic to set the extra-block-diagonal elements in Σymulti-variateo,k to zero.

Previous studies by , , include two-step update schemes or temporal smoothers to address the temporal mismatch issue between the model and observations. These approaches are not necessary in this study, as both observations are assimilated in the same daily timescale as the model. This implies that the forward observation operator matrices HTWS/SSM/multi-variate here perform a vertical aggregation and horizontal averaging, but do not apply any temporal averaging.

Most of the previous TWS and SSM DA experiments share a similar EnKF update framework, but tune the DA implementation differently (see Table ). In this paper, we study the impact of two of these tuning approaches: uncertainty adjustment and covariance localization.

The uncertainty adjustment see, e.g., is implemented to achieve a better balance between the TWS and SSM observations, and to avoid that one observation type dominates over the other. It consists of multiplying the TWS and SSM observation uncertainties with scaling factors (αTWS and αSSM, respectively), which must be set heuristically (see more details in Sect. ), 4Σymulti-variateo,kadjusted=(αTWS)2⋅ΣTWS,k00(αSSM)2⋅ΣSSM,k. The optimal value of the factors might differ depending on the value of the uncertainties and the amount of each type of observation that is assimilated .

The covariance localization approach is typically used in the field of DA to mitigate the impact of spurious long-range correlations in model ensemble statistics, which can negatively affect the DA update, especially when small ensemble sizes are used . In short, the approach consists of damping correlations that are above a defined radius that represents the expected correlation range.

Previous studies by implement observation space covariance localization, where correlations between model variables and observations are damped based on the distance between them. This type of localization ensures that each model variable is influenced only by nearby observations that could realistically affect them (Fig. b).

The localization can be discrete i.e., all observations further than a cutoff radius should be ignored, or smooth i.e., the observations are weighted depending on their distance to each model grid cell,. Domain localization or local analysis can also be seen as a type of discrete observation space covariance localization. Smooth observation space localization matrices are typically built based on compactly supported distant-dependent functions, such as Gaspari-Cohn functions . The localization matrix (ρ1/2) is then applied to the predicted model-observation covariance matrix (C(Xk-)H⊤) through a Schur (elementwise) multiplication, resulting in the following Kalman gain 5Kk=ρ1∘(C(Xk-)H⊤)(ρ2∘(HC(Xk-)H⊤)+Σyo,k)-1.

The localization radius typically reflects the expected radius of influence of each observation, and thus larger localization radii are typically assigned to TWS observations (3–10° in previous studies) and smaller radii to SSM observations (0.25–3° in previous studies, Table , Fig. b). In this study, we call this “observation space mixed localization”.

In addition to observation space covariance localization, it is also possible to implement a model space covariance localization (Fig. c). In this approach, localization is applied to the matrix C(Xk-), which represents the ensemble-based error correlation between the model state variables. In other words, correlations between model grid cells with other model grid cells are damped depending on their distance, no matter the kind of assimilated observation or location of the observations.

The localization is typically performed through smooth distance-dependent functions such as the Gaspari-Cohn function, and applied to the covariance matrix through a Schur multiplication, that is, ρ∘C(Xk-) , so that the Kalman gain is expressed as 6Kk=(ρ∘C(Xk-))H⊤(H(ρ∘C(Xk-))H⊤+Σyo,k)-1.

Here, the localization radius should reflect the expected spatial variability of water storage. Although it is possible to define a unique localization radius for all water storage components see, e.g.,working with TWS DA, it might be more realistic to consider different localization radii for different vertical water storage components (e.g., a sharper localization might be applied to SSM compared to groundwater, Fig. c). This second approach is the one we propose in this study and is from now on called “model space mixed localization”. In a previous work performing daily TWS DA, it was found that the groundwater localization half-radius should be larger than 3° to avoid too sharp groundwater storage updates during DA .

Model space covariance localization is especially useful when working with non-local observations , where computing a distance between observations and model variables is not straightforward (e.g., when TWS is assimilated as a sub-basin average, it is not clear which is the “position” of the observation, and taking the geometrical center of the sub-basin as a reference might decrease the influence of the observations around the sub-basin edges). One of its main drawbacks is the need to explicitly calculate the matrix C(Xk-) (of dimension ∼34000×34000 in our experiment) and its Schur multiplication with the matrix ρ, of the same dimension, which can be very memory intensive .

In this study, four different tuning approaches are compared: Approach #1 is a plain multi-variate DA without tuning; Approach #2 applies observation uncertainty adjustment; Approach #3 applies observation space mixed localization; and Approach #4 applies model space mixed localization (see more details of the implementations in Sect. ). It is worth noting that the Approaches #3 and #4 do not include any observation uncertainty adjustment (Approach #2), and Approach #4 does not require any additional observation space localization such as that in the Approach #3.

One of the interesting aspects of this work is the evaluation of the Kalman gain (K) structure in different DA settings (Sect. ). In this section, we delve deeper into the meaning of the Kalman gain and the simplified view displayed in the manuscript.

K (or Kk, where k indicates a specific timestep) is the element in the EnKF equations that determines the extent to which each observation will impact each model variable (see the EnKF equations in Sect. ). The structure of K exclusively depends on the model error structure and observation error structure (Eq. ), and can be modified through DA tuning approaches such as observation uncertainty adjustment and localization (see Eqs. , and ). Note that the final impact of each observation on the model will also depend on the difference between observations and model predictions (Yko-HXk-), see Eq. (), but for the sake of simplicity this factor is omitted from our investigations.

Each column of K corresponds to one observation (i.e., in this study, 3 columns for sub-basin averaged TWS observations, and up to 368 columns for SSM observations). Each row of K corresponds to one model variable (that is, a water storage component in a model grid cell, with 8 water storage components and 4226 model grid cells, resulting in 8×4226=33808 model variables). The matrix element in the row i and column j represents the impact that the observation number j will have on the model variable number i.

Analyzing the structure of K allows us to investigate the impact of each observation on each of the model variables. However, K typically has very large dimensions (33808×371 in this study), making it difficult to visualize and analyze. To ease this task, in Sect.  we represent a “simplified view” of K by selecting some representative observations and model variables. In this simplified view, columns 1–3 represent the three sub-basin averaged TWS observations, and columns 4–12 represent 9 selected SSM observations. Likewise, rows 1–3 represent the model sub-basin averaged TWS; rows 4–8 represent model SSM in grid cells for which SSM observations are available and rows 9–12 represent model SSM in grid cells for which no SSM observations are available. The value of each of the matrix elements is represented by a color scale, and the matrix is separated into blocks to facilitate its interpretation. Additional details on the computations of the simplified K displayed in the manuscript are available in Sect. .

The W3RA model was used to perform the experiments. The model run was carried out for the Ganges-Brahmaputra-Meghna river system area, in a daily and 10 km resolution, with forcing data derived from ERA5 and ERA5-Land (see Sect. ). An ensemble size of Ne=30 members was chosen to perform the experiments, following previous TWS and SSM DA experiments , together with covariance localization techniques to avoid possible spurious correlations between distant grid cells. The 30 model realizations were generated by perturbing the forcing fields and the parameters of the model according to their own uncertainties, following the approach of . The model was warmed up for the period 2000–2003 to reach an appropriate ensemble spread in the water storage estimates. The run was extended from 2003–2015 to generate the Open Loop (OL) solution.

An observation pre-processing was performed before DA. As the modeled and satellite-retrieved SSM typically refer to different soil depths, SSM observations were rescaled so that their statistics match those of the model via cdf-matching , and the SSM uncertainties provided by ESA CCI were also rescaled accordingly. The ESA CCI tiles for which less than 50 % observations were available were excluded from the DA process (the remaining tiles are colored red in Fig. ). In the case of TWS, the temporal mean of the sub-basin averaged TWS time series was adjusted to that of the model, but the time series were not rescaled to avoid modifying the amplitude of seasonal, inter-annual and decadal variability, which might be overestimated or underestimated within the model. Both observations were perturbed using a Cholesky decomposition approach and according to their respective uncertainties .

The trends of GRACE TWS in the East and West sub-basins include, among others, ice water loss due to glacier retreat in the Himalayan mountains , but W3RA does not model glaciers. Glacier retreat rates are available from previous work based on satellite stereo imagery (see Sect. ), and therefore a possibility is to exclude these glacier declines from TWS before DA. In the main experiments presented in this study, we do not apply this reduction to equate the experimental conditions to regions where no robust glacier retreat estimates are available. As part of the evaluation, we assess how the W3RA accommodates these trends. Additionally, a side-experiment involving an a-priori glacier trend removal is described and evaluated in Appendix .

All DA experiments were performed for the period September 2003 to December 2015, in the three sub-basins in Fig. . The DA update occurs at the end of each daily timestep following the method described in Sects.  and . In addition to the multi-variate DA, uni-variate (i.e., single-observation) TWS DA and SSM DA experiments were also run for comparison.

The TWS observations were assimilated as a summation of all the W3RA land water storage components. SSM was assimilated into the SSM component (that is, in this study, the summation of topsoil and shallow soil components, see Table ). Although the ESA CCI SSM corresponds to the first few centimeters of the soil, and therefore could be equated to the model topsoil, due to the small field capacity of shallow soil we decided to update these two components jointly. The motivation behind this decision was to simplify the design of the mixed localization for the different localization implementations, as well as the evaluation of the results. Equating a deeper model soil layer with satellite-derived soil moisture could lead to a systematic phase difference between them, where satellite soil moisture variations precede model variations. However, this was not the case in this study, with satellite soil moisture peaking and decaying later in the season compared to model SSM estimates (Sect. ). Note that, to further simplify localization and validation, deep soil water and groundwater are also treated jointly (the summation of both is here referred to as groundwater).

In all DA experiments, river water and snow components were excluded from the DA update step as it was found that including them led to degradations in these components (see Appendix ), and the vegetation water storage component was also excluded due to its marginal contribution to the variability of TWS in the basin. The water storage estimates of the river, snow and vegetation of the model were subtracted from the TWS observations prior to assimilation. As a result, only SSM, surface runoff and groundwater components were updated during the DA process.

Spatially, TWS was assimilated as a sub-basin averaged field, and SSM was assimilated as an average over the 0.25°×0.25° ESA CCI tiles (Fig. ). The observations were downscaled to the model resolution within the DA process through ensemble statistics.

The novel approach suggested and analyzed in this study (Approach #4) implements a model space mixed covariance localization. To the best of our knowledge, this is the first time such an approach has been implemented in a multi-variate land DA experiment. The localization was defined through a Gaspari-Cohn function (see Sect. ).

Localization radii in a model-space localization approach should in principle follow the spatial wavelength of water storage variability represented by the model for each component. When analyzing the spatial SSM variability in W3RA, a very smooth variability was found (variations correlated within distances of a few hundreds kilometers), probably due to the coarse parameterization of the model. Since satellite-based SSM observations are likely capable to reflect spatial heterogeneities that are not represented in the model parameters, a localization half-radius 0.5° was applied to SSM (as well as to the covariance between SSM and groundwater) based on the spatial variability that can be expected from SSM observations see, e.g.,, to allow introducing the observed spatial patterns. Regarding groundwater, the model displayed a decorrelation length of around 300 km, so a localization half-radius of 3° was initially deemed appropriate based on. However, this was found to produce very sharp anomalies in groundwater estimates, and finally, a half-radius of 10° was adopted to enhance smoother groundwater variability patterns. In both cases, the values are in the range of TWS localization degrees reported in the literature (see Table ).

For comparison, three other approaches were implemented based on the previous literature. Approach #1 was a plain multi-variate DA run without any tuning. In Approach #2, (uncertainty adjustment), SSM uncertainty multiplier factors of αSM=1,2,5,10,15,30,60,240 were considered, while αTWS was kept to 1. Approach #3 (observation space mixed localization) was implemented by applying a smooth localization based on a Gaspari-Cohn function. For consistency with the model space localization, an influence half-radius of 10° was used for the TWS observations and 0.5° for the SSM observations. For the implementation of observation space localization, the TWS observations were assumed to correspond to the center of each sub-basin, and the SSM observations were assumed to be located in the center of each 0.25°×0.25° tile.

The Kalman gain here displayed was computed based on the model covariance structure of the 1st of July of 2004. This date was chosen because of the high correlation between SSM and TWS that appear on this date, which leads to notable cross-variable influences and therefore helps to explain the results observed in the DA experiments. Regarding the observation covariance structure, average errors of 3.31 and 13.38 mm were taken for all SSM and TWS observations, respectively.

Groundwater level changes (Δh) were converted to groundwater storage changes (ΔSGW) using specific yield data (Sy), through the relation ΔSGW=Δh×Sy. The groundwater validation is performed on an inter-annual timescale, which is derived by averaging the time series on a sliding window of 12 months. The in situ groundwater time series presented many small gaps that, if left unaddressed, generated larger 12-month gaps after the smoothing process. Therefore, a gap-filling algorithm was applied before smoothing (see more details in Appendix ). For spatial validation of groundwater variability, a clustering based on Principal Component Analysis (PCA) was performed on the observed groundwater level time series more details on PCA can be found in. Groundwater sites for which the first principal component contributed to more than 65 % of the variability were included in cluster 1, and all other sites were included in cluster 2.

The river water storage validation was performed against in situ water level time series. Comparing river level to river storage requires assumptions on the river cross-section. Due to the absence of this information, we here perform a simple transformation by rescaling the modeled water storage estimates to match the variance of the observed river level data. This is equivalent to assuming a rectangular cross-section and we set the width of the rectangle to obtain a maximum agreement between observed and modeled water levels. Although this approach has many limitations, it does not affect the final conclusion regarding little sensitivity of the model estimates to water storage updates in the seasonal to sub-seasonal timescale (Sect. ). The agreement between model estimates and observations was computed in terms of Nash-Sutcliffe efficiency.

For the interpretation of the results in Sect. , the impact of glacier retreat was reduced from the sub-basin averaged TWS time series. suggest that glacier retreat does not show strong inter-annual changes between 2004 and 2016 in this region, and therefore we approximated the impact as a linear trend. The trend was calculated as ΔSsubb=ΔSglac⋅AglacAsubb, where Asubb and Aglac are the area of each sub-basin and of the glaciers of each sub-basin, respectively, and ΔSglac is the decrease in water storage produced by the glacier retreat. Glacier trend estimates of previous studies present slight discrepancies, and therefore in this study the largest trend estimate was considered (Table ).

We evaluate the realism of the assimilated TWS dynamics using the independent SPEI-12 drought index. Because SPEI is derived from climatic water balance rather than direct observations of storage, the comparison is interpreted as a consistency check rather than a strict validation.

Although SPEI is standardized and is not suitable for validating the magnitude of TWS variability, it reveals clear negative trends for the three sub-basins in the period 2004–2014 (Fig. ). These trends are absent in the OL simulations. The disagreement could partly originate from model limitations, such as a limited representation of evaporation in surface water bodies, or vegetation and transpiration. However, it likely also reflects a more general difficulty of hydrological models to represent decadal TWS variability and trend related to climate variations and human water use, as reported in a multi-model experiment by .

The decadal trends emerge distinctly through the multi-variate DA. Regarding the sub-decadal variability, multi-variate DA preserves the inter-annual variability of OL, although it also excessively smooths certain anomalies (e.g., 2007 and 2010), possibly due to GRACE TWS resolution or interference with the decadal trends. In general, the correlation with SPEI is significantly improved through DA from 0.13/-0.25/0.55 (OL) to 0.74/0.79/0.82 (DA) for the East, West, and South sub-basins, respectively (Table ). An especially large improvement is observed in the West sub-basin, where the OL estimates suggest a positive TWS trend as opposed to the DA TWS and SPEI data.

We compare groundwater estimates in the South sub-basin against monthly groundwater storage changes derived from in situ observations, to assess the realism of their temporal and spatial patterns.

To convert groundwater level observations to groundwater storage data, spatially distributed specific yield data from was first used (see Sect.  for more information). The results show a significant scale difference between the modeled and observed groundwater variability, both in the seasonal timescale (STD of 182 and 140 mm for OL and DA respectively, and 65 mm for observations, Fig. a) and the inter-annual timescale (STD of 35/50/14 mm for OL, DA and observations respectively, Fig. c).

Various reasons could be behind this discrepancy in magnitude, which concern both the groundwater level observations and the model. On the observation side, the groundwater wells used for validation in this study measure water levels in shallow aquifers with a mean depth of 30 m , while the model and GRACE water storage changes reflect water mass variations at all ground levels including deep aquifers, and can therefore be expected to present a larger variability see, e.g.,. The specific yield values used to convert observed groundwater levels to groundwater storage values can also be affected by biases and uncertainties : factors in its measurement, such as an insufficient duration of the pumping test or the assumption of partial penetration, can lead to an underestimation of the specific yield, consequently leading to an underestimation of groundwater storage variability. The spatial interpolation of specific yield data, which was performed based on ground lithology maps , can additionally contribute to their uncertainty.

On the model side, an inadequate partitioning of the soil profile in its different layers could result in attributing part of the soil water storage variability to the groundwater component . In theory, general model limitations to represent water storage variability could also contribute to this discrepancy, although this possibility is here considered less likely due to the good agreement found between W3RA and GRACE TWS variability (Sect. ).

To avoid the impact of possible uncertainties in the specific yield data, a second comparison was performed by considering a uniform specific yield of 15 % for all the groundwater wells, which is close to values of 10 % used by for Bangladesh or average values of 14 % reported by for the US in previous studies. This significantly increases the magnitude of the observed groundwater variability (STD of 171 mm in the seasonal timescale and 40 mm in the inter-annual timescale, Fig. b and d). The new comparison suggests that the agreement between modeled and observed groundwater is increased, and RMSD is reduced from 95 to 88 mm after DA. Especially, the DA results display a high agreement with observed data during the wet periods, as also reported by , although the DA results seem to slightly underestimate the seasonal amplitude when compared to the OL. In the inter-annual timescale, DA reduces the RMSD from 40 to 36 mm (Fig. d).

To ensure that the results obtained are not an artifact of the chosen specific yield, the comparison is repeated in terms of Peason correlation coefficient, a metric that is scale invariant. Improvements are especially observed in the inter-annual timescale (Fig. d), in which the correlation with observed groundwater storage is increased from 0.43 to 0.71 after DA. Especially, the groundwater storage derived from DA presents a negative trend for the period 2004–2014, with higher levels in 2005–2008 than in 2013–2014, which is clearly featured in the observed data and not reflected in the OL model estimates. DA also displays realistic sub-decadal variability, capturing key anomalies such as the 2007 peak and the wet period of 2010–2011, which remain distinguishable despite the overarching negative trend.

To evaluate the realism of spatial groundwater patterns of the DA results, a comparison against distributed groundwater time series provided by individual wells is shown in Fig. . Biases in the specific yield estimates are likely to be more severe when the assessment is performed for specific wells, and therefore this comparison is exclusively performed in terms of correlation coefficient.

The results show that DA improved groundwater estimates in many parts of the basins, but also suggest degradations for some specific sites and regions (Fig. ). To better interpret the results, a PCA-based clustering of in situ data is applied (see details in Sect. ). For the sites belonging to the main cluster (Fig. a, b, and c), overall improvements are observed as a consequence of the negative trends introduced by the multi-variate DA process, increasing the average correlation coefficient from 0.47 (OL) to 0.54 (multi-variate DA). Certain short-term anomalies, such as the positive anomaly in 2010–2012, may appear better represented in the OL solution, posing a need for further investigation in future studies, particularly to assess potential links to sub-decadal glacier- or snow-related anomalies . Conversely, other anomalies, such as the 2007–2009 event, are more accurately reproduced by the multi-variate DA.

For the remaining sites (Figs. d and e), of which most are located close to the main rivers branches, notable degradations can be observed. Multi-variate DA introduces a negative trend in these regions too, but the observed groundwater level of this cluster is characterized by a less strong trend, which reduces the average correlation from 0.33 (OL) to 0.28 (multi-variate DA). The absence of trend in the observations might be explained by the fact that large river branches in these regions allows greater groundwater recharge, either through natural percolation processes or enhanced by anthropogenic actions . W3RA does not account for the feedback from the surface to the sub-surface components and therefore the DA is not informed about this mechanism.

In conclusion, the multi-variate DA appears to improve the overall groundwater storage dynamics in the South sub-basin, especially through the modification of inter-annual groundwater variability and the introduction of a negative trend for the period 2004–2014 that is also reflected in groundwater level observations. It is important to note that the comparisons performed here validate the groundwater dynamics but not magnitude of the trend, due to uncertainties in the scale between the modeled and observed groundwater data. Namely, the magnitude of the DA trend and inter-annual variability could be overestimated due to the over-allocation of surface water or shallow soil variability to the groundwater component in the DA process.

Here we compare the OL and DA outputs with the soil simulations of the WaterGAP global hydrology and water use model . Although model outputs might be affected by limitations such as errors in forcing data, process simplification, or missing representation of certain processes, they can provide an additional perspective to perform a consistency check and assess the realism of the DA results.

The basin averaged time series show that, after multi-variate DA, SSM becomes saturated later in the season, and the decay of water at the end of the monsoon season is also slower (Fig. a). In other words, DA results mainly in a light phase shift in the SSM time series. The phase shift brings the estimates closer to those of the WaterGAP estimates.

Regarding the spatially distributed estimates, original W3RA (OL) estimates present an especially strong discrepancy with WaterGAP in areas with a sharp topography (slope > 1 %, correlation coefficients mostly between 0.4–0.7, see triangles in Fig. b). After DA, notable increases in the correlation with WaterGAP arise in these mountain regions, reaching increases of up to +0.3 correlation points in areas where SSM observations are available (Fig. c).

There could be various reasons why the two models disagree in mountainous areas. WaterGAP is calibrated for the Brahmaputra River basin and is able to reflect a broader range of land cover conditions and soil compositions when compared to the W3RA model . This could make the WaterGAP model more suited to represent land surface hydrological processes in areas with a sharp topography such as the Himalayan Mountains. Taking this into account, the results could indicate that the assimilation of ESA CCI SSM helps to better account for SSM variations in topographically complex areas. However, these results should be interpreted with caution, as the WaterGAP model might also be affected by its own limitations in other aspects.

The observed differences could also be related to the depth considered for the SSM components, which is larger in WaterGAP when compared to W3RA. To exclude this possible cause, we have repeated a comparison equating WaterGAP SSM with the summation of all soil water storage components of W3RA (that is, topsoil, shallow soil, and deep soil, see Table ). The results, reported in Appendix , lead to conclusions similar to those presented here, thus excluding the depth difference as the origin of the disagreement.

An important consequence of these results is that the assimilation of SSM could be accounting for the limited performance of the W3RA model at estimating SSM variability in mountain areas, but it is not clear if improvements could be expected from the assimilation of SSM products if a better adapted base model was used (e.g., WaterGAP itself). Evaluating the added value of the satellite-based SSM product with respect to different hydrological models is not within the scope of this study, but should be considered in future satellite SSM DA experiments, especially regarding its operational implementation.

In the more flat regions located along the Brahmaputra River (slope < 1 %, circles in Figs. b and c), for which the W3RA OL and WaterGAP estimates are more similar (correlation coefficients between 0.7–1.0), DA is found to slightly reduce the agreement between the two models by up to 0.1 correlation points. For the Himalayan Mountain area, where no SSM observation was assimilated, the impact of DA is driven by the TWS observations and leads to slight improvements and degradations.

River water storage estimates are here validated against in situ river water level observations. We select the two sites that are closest to the outlet of the West and South sub-basins (water level data is not available for the East sub-basin, see site locations in Fig. ). Comparing river level to river storage requires assumptions on the river cross-section. Due to the absence of this information, the modeled water storage estimates were rescaled to match the variance of the observed river level data before validation (Sect. ). Although this approach has important limitations, it does not affect the main conclusions described in what follows.

The OL time series show dynamics similar to the observed water levels (Fig. ), with a high Nash-Sutcliffe efficiency (NSE) of 0.84 and 0.83 for the West and South sub-basins, respectively. The multi-variate DA introduces a decadal negative trend and inter-annual variability in both sub-basins, as can be noted by the higher river water storage values in 2004–2005 and notably lower values after 2010. As a consequence, the NSE is reduced to 0.75 and 0.74 for the West and South sub-basin, respectively, after DA.

Although river water storage was not directly updated during our DA experiments, the observed changes in river water storage likely stem from trends and inter-annual variability introduced in groundwater by multi-variate DA, which then have an impact in streamflow through baseflow generation. Re-visiting runoff parameterization or applying joint Calibration/Data Assimilation CDA; could help refine this process. In addition, groundwater also includes negative trends related to glacier retreat (see Sect. ), which likely increases negative trends in river storage. Incorporating a glacier component into the model representation and guaranteeing an accurate TWS trend partitioning between glacier and groundwater could help mitigate this issue.

Validation using detrended time series yields NSE values of 0.84/0.84 (OL) and 0.86/0.82 (DA) for the West and South sub-basins, respectively, suggesting that multi-variate DA has limited influence on sub-annual river storage dynamics. This aligns with the findings of , where SMAP SSM assimilation into Noah-MP showed minimal impact on streamflow in high-flow tributaries such as the Brahmaputra River. The broader literature also points to methodological choices, catchment characteristics, and land–river coupling limitations as key factors that restrict the impact of DA on river discharge . As a consequence, this comparison cannot be further utilized to validate the hydrological estimates obtained through the DA experiment.

It is worth mentioning that we also attempted a multi-variate DA experiment in which we updated the river water storage component together with the land water storage states. The results are reported in detail in Appendix , which show significant degradations in the sub-seasonal timescale. This suggests that to improve the surface water component in the Brahmaputra River basin, a direct DA of water level measurements, such as lake and river altimetry, might be more beneficial than a SSM and TWS DA see, e.g.,.