Precipitation is the primary input to terrestrial water storage. Statistical models that reconstruct TWSA from daily precipitation and temperature can be used to quantify the influence of climatic variability on terrestrial water storage (Humphrey and Gudmundsson, 2019a). For comparison purposes, we first revisit the statistical reconstruction model proposed by Humphrey and Gudmundsson (2019a): 1TWS(t)=TWS(t-1)⋅e-1τ(t)+P(t). Here t is the daily time index, TWS(t) is the water storage on day t, P(t) is the precipitation input on day t, and τ(t) is the water residence time. A larger τ implies slower losses through evapotranspiration or runoff, while a smaller τ indicates faster depletion.

τ(t) is defined as a function of temperature to represent the seasonal variations in water residence time. 2τ(t)=a+b⋅Tz(t), with a and b are model parameters to be calibrated.

To reduce the influence of extreme temperatures on model stability, the original temperature T(t) is first detrended and standardized, then transformed using a sigmoid function: 3Tz=1-tanh⁡T0-mean(T0)SD(T0). Since subzero temperatures have negligible effects on residence time (as temperature changes exert little influence on storage losses such as evapotranspiration and runoff under freezing or insufficient thawing conditions), the original temperature series T(t) was preprocessed as follows: 4T0=0,T<0T,T≥0. The initial water storage TWS(0) is computed as the ratio of long-term mean precipitation to mean water loss rate: 5TWS(0)=mean(P)1-mean(e-1τ(t)). The daily TWS series is aggregated to monthly means TWS(tm) and calibrated against deseasonalized and detrended GRACE TWSA at the monthly scale. The model parameters a,b and β are calibrated using a Markov Chain Monte Carlo (MCMC) approach, with the objective of minimizing the sum of squared residuals between the simulated and GRACE-derived TWSA: 6anom(GRACE(tm))=β⋅anom(TWS(tm))+ϵ. The coefficient β serves as a scaling factor obtained from MCMC calibration, ϵ denotes the residual term, and anom(⋅) represents the anomaly series after removal of the seasonal cycle and linear trend.

The scaling factor β in Eq. (6) lacks clear physical meaning and serves merely as a calibration parameter, resulting in a strong dependence of the reconstruction on GRACE observations. Building on this framework, Zhong et al. (2025) proposed an improved formulation for reconstructing terrestrial water storage: 7TWS(t)=TWS(t-1)⋅e-1τ(t)+β⋅P(t). In this formulation, the parameter β has a clear physical interpretation, representing the mean fraction of daily precipitation that contributes to water storage. The residence time τ(t) is defined as in Eq. (2). It is important to note that this β is fundamentally different from the scale factor used in Eq. (6).

The daily TWS series is aggregated to monthly means TWS(tm) and calibrated against deseasonalized and detrended GRACE TWSA at the monthly scale. The model parameters a,b and β are also calibrated using the MCMC approach: 8anom(GRACE(tm))=anom(TWS(tm))+ϵ. Owing to the high nonlinearity of the estimation model, parameter optimization is prone to being trapped in local optima. To reduce the bias caused by incomplete chain convergence, the MCMC calibration was repeated 50 times for each basin. All parameter sets falling within the ±1⋅σ range were retained, and their averages were taken as the final estimates of parameters a,b and β. The standard deviation of these filtered parameters was then calculated to represent the inter-basin uncertainty of β.

The two preceding approaches describe the residence time (Eq. 2) as a temperature-dependent function. However, the parameters a and b in Eqs. (1) and (7) are obtained by calibration, and they are less directly linked to specific water-balance components. In this study, our model is derived directly from the basin-scale water balance equation and provided a systematic clarification of both the derivation and the physical interpretation of the parameters.

We start from the water balance equation: 9TWS(t)-TWS(t-1)=P(t)-ET(t)-R(t)=P(t)-ETR(t), where P(t), ET(t), and R(t) denote precipitation, evapotranspiration, and runoff at time step t, respectively, and ETR(t)=ET(t)+R(t) represents the total loss term. We assume that ETR consists of two components: (1) the contemporaneous precipitation loss ETR₁, i.e., the direct loss occurring as precipitation is converted into water storage; and (2) the storage-release loss ETR₂, i.e., the loss arising from the depletion of antecedent water storage TWS (t-1) in subsequent periods, which can be expressed as 10ETR1(t)=x⋅P(t),11ETR2(t)=y⋅TWS(t-1),1213ETR(t)=ETR1(t)+ETR2(t)=x⋅P(t)+y⋅TWS(t-1). The assumption is validated in Fig. S2, where the ETR is plotted against precipitation and TWS respectively. Both plots show significant correlations, with ETR apparently exhibiting a stronger correlation with the current precipitation than the antecedent TWS.

We then performed a multiple linear regression analysis to further test this assumption. In this step, temperature modulation was temporarily ignored and the coefficients x and y were treated as constants. The results show that all basins pass the F-test at the significance level of p<0.01. Moreover, 74 basins (approximately 64 % of all basins) exhibit determination coefficients R2>0.6 (Fig. S3a), indicating that the total loss term ETR can be statistically explained by the linear combination of precipitation P(t) and antecedent storage TWS(t-1).

We compared the full multiple regression model with a reduced model that includes only precipitation P(t) as the predictor to further examine whether antecedent storage TWS(t-1) provides additional explanatory power beyond precipitation alone. Figure S3b presents the spatial distribution of the R2 obtained from the precipitation-only regression, while Fig. S3c shows the spatial difference in R2 between the full model and the reduced model. Although precipitation alone explains a substantial portion of the variability in basin-scale losses, the inclusion of antecedent storage TWS(t-1) improves model performance. Across all basins, the incorporation of TWS(t-1) increases R2 by approximately 0.1. Therefore, although ETR is strongly correlated with P(t), the results demonstrate that accounting for TWS(t-1) is both statistically meaningful and physically necessary.

On this basis, we consider the modulation effect of temperature on the hydrological cycle by assuming that the proportional coefficients x and y vary with temperature and apply a Taylor expansion, which yields 14x=ϵ⋅f(TZ)=ϵ⋅(1+αTz+o(Tz2)). Retaining only the first-order approximation gives 15x=a′⋅Tz+b′, where Tz denotes standardized temperature.

Similarly, we obtain 16y=c′⋅Tz+d′. Substituting Eqs. (12), (14), and (15) into the water balance equation (Eq. 9) leads to 17TWS(t)=(1-a′⋅Tz-b′)⋅P(t)+(1-c′⋅Tz-d′)⋅TWS(t-1). After merging constant terms simplifying the signs, the recursive formulation adopted in this study is obtained as 18TWS(t)=(a⋅Tz+b)⋅P(t)+(c⋅Tz+d)⋅TWS(t-1), which is the reconstruction model adopted in this study. In this form, parameters a and c represent temperature modulation effects. The term 1-b can be interpreted as the fraction of precipitation that directly contributes to the loss component ETR1(t); therefore, parameter b represents the effective proportion of precipitation that contributes to TWS after basin losses. Similarly, 1-d represents the fraction of antecedent storage contributing to the loss term ETR2(t), and thus parameter d represents the fraction of previous storage retained in the current storage state. We will continue to discuss these parameters in subsequent sections.

When solving Eqs. (1), (7), and (17), it is necessary to specify the initial water storage value, TWS(0). Humphrey and Gudmundsson (2019a) and Zhong et al. (2025) assumed that the initial water storage corresponds to an equilibrium state, as defined in Eq. (5). Although using Eq. (5) provides a unique and reproducible starting condition, it inherently assumes that water storage is close to equilibrium at the beginning of the reconstruction period. However, if a basin experiences an extreme hydroclimatic condition at the initial time step, adopting Eq. (5) may introduce systematic biases and consequently affect the MCMC calibration of other parameters. In this study, TWS(0) is also treated as a variable and is simultaneously estimated together with other model parameters through a joint least-squares and MCMC optimization. This approach eliminates the need to assume equilibrium conditions at the start of the reconstruction period and is therefore more suitable for basins under nonequilibrium states, such as those influenced by prolonged droughts or intensive groundwater extraction (see Sect. 5.1.2 for details).

The calibration procedure in this section follows exactly the same steps as described in Sect. 3.1.2, with Eq. (8) retained throughout the process. The reconstruction logic of TWSA is illustrated in Fig. 1.

Compared with the methods of Humphrey and Gudmundsson (2019a) and Zhong et al. (2025), this study linearizes the memory term TWS(t-1)⋅e-1τ(t) as (c⋅Tz+d)⋅TWS(t-1). A key advantage of this linearization is its computational stability and rapid convergence, which substantially improves inversion efficiency. Under the same parameter calibration framework as Zhong et al. (2025), both our model and the Zhong's model were run with 50 independent MCMC chains for the Yellow River to derive the posterior parameter distributions. The parameter dispersion (indicated by the box height) in our model (Fig. 2a) is considerably narrower than that in Zhong's model (Fig. 2b), indicating that our model exhibits smaller parameter uncertainties and faster chain convergence. Consequently, a single MCMC calibration is sufficient for each basin in our method, greatly reducing computation time without a noticeable loss of fitting accuracy. Specifically, on our computing platform (Intel 13th Gen Core i9-13900H processor, 14 cores, base frequency 2.60 GHz, turbo frequency up to ∼5.4 GHz, and 16 GB RAM), the Zhong's model required 50 parameter calibrations, taking a total of 353.8 s for a single basin, whereas our method required only one calibration, completing the computation in 6.8 s.

In summary, this study improves upon previous models in three aspects:

Linearization of the memory term. The linearization not only accelerates computational performance but also enhances the physical interpretability of parameters.

To demonstrate the compatibility of the proposed model with existing approaches, we consider two extreme cases:

In the model of Zhong et al. (2025), the memory factor is expressed as e-1τ(t). When applying a first-order Taylor expansion to the exponential term in Eq. (7), it can be approximated as: 19TWS(t)=β⋅P(t)+1-1τ(t)⋅TWS(t-1). Our model (Eq. 17) becomes mathematically similar to Eq. (18) when the temperature and precipitation modulation is neglected (i.e., a=0). This indicates that our model degenerates to the first-order linear form of the Zhong model when temperature effects are not considered.

To quantify the uncertainty associated with meteorological forcing data, this study estimates observational errors based on the differences between multiple data sources. Specifically, the standard deviation of daily precipitation differences between ERA5-Land and MSWEP is used as the uncertainty estimate for ERA5-Land precipitation, while the standard deviation of daily temperature differences between ERA5-Land and GLDAS-GLSM is used to estimate the uncertainty in ERA5-Land temperature.

Temperature uncertainty: 22σT=std(TERA5-Land-TGLDAS-GLSM). Precipitation uncertainty: 23σP=std(PERA5-Land-PMSWEP). Building on this, we adopt a Monte Carlo simulation framework in which Gaussian noise with zero mean and standard deviations of σT and σP is added to the temperature and precipitation inputs, respectively. For each perturbed dataset, the model is recalibrated to obtain an optimal set of parameters under the given realization. After completing 1000 simulations, all sets of estimated parameters are collected to derive the uncertainty distributions of both the model parameters and the reconstructed TWS.

At the grid scale, we first compared the reconstruction results from this study (JPL-REC and GSFC-REC) with those of Humphrey and Gudmundsson (2019a), which were trained using the JPLM and GSFCM products, respectively (hereafter referred to as Humphrey-JPL-REC and Humphrey-GSFC-REC). Figure 4 presents the spatial distribution of NSE values between the reconstructed and GRACE-derived TWSA after deseasonalization and detrending over the period from 2002 to 2019. Figure 4a and d show the NSE performance of JPL-REC and GSFC-REC relative to their corresponding mascon products (JPLM and GSFCM), respectively. Both panels indicate high model performance (NSE >0.7) across large parts of North America, Europe, Australia, and southern Africa, whereas low NSE values (NSE <0.4) are evident in parts of North Africa, Central Asia, and the Arabian Peninsula. Figure 4b and 4e show the NSE performance of Humphrey-JPL-REC and Humphrey-GSFC-REC relative to their corresponding mascon products (JPLM and GSFCM), respectively. While the overall spatial patterns are broadly similar to those in Fig. 4a and d, the extent of low-performance areas in Africa and Central Asia is noticeably larger in the Humphrey reconstructions. Figure 4c and f present the differences in NSE (ΔNSE), calculated as Fig. 4a minus Fig. 4b and Fig. 4d minus Fig. 4e, respectively. Blue shading (ΔNSE > 0) indicates regions where the present method outperforms the approach by Humphrey and Gudmundsson (2019a). Notably, in Fig. 4c, approximately 63 % of the grid cells show ΔNSE >0, with substantial improvements in regions such as the Arabian Peninsula and sub-Saharan Africa, along with scattered enhancements across other continents. Figure 4f demonstrates that under GSFCM constraints, the present method substantially outperforms that of Humphrey and Gudmundsson (2019a). One potential explanation lies in the difference in training datasets: while (Humphrey and Gudmundsson, 2019a) utilized the GSFC mascon v2.4 (an older version that is no longer publicly available), the present study employs the updated GSFC mascon RL06v2.0. This difference may account for the discrepancies between our results in Fig. 4e and those reported in the original publication by Humphrey and Gudmundsson (2019a).

We compare our reconstruction (JPL-REC) with two representative types of products to systematically evaluate model performance at the basin scale: (1) a previous GRACE-based reconstruction product (Zhong-REC from (Zhong et al., 2025)), and (2) physically based land surface model simulations forced by meteorological data (GLDAS CLSM).

First, we adopted the JPLM as a common reference to evaluate the relative performance of different statistical reconstruction methods. We compared basin-averaged monthly TWSA from our JPL-REC and Zhong-REC against the JPLM and computed the NSE for each (Fig. 5). By comparing the NSE values for the two approaches, we assessed their relative reconstruction accuracy. In the comparison across 116 major global river basins, our proposed model achieved a median NSE of 0.76, outperforming Zhong-REC, which had a median NSE of 0.70. Among these basins, 84 (73 %) achieved NSE >0.7, compared to only 59 (51 %) for Zhong-REC. Model performance was closely linked to the quality of the forcing data and the dominant hydrological processes (Yi et al., 2023). Poor performance (NSE <0.5) was observed in basins such as the Congo, Niger, and Sanaga in Africa, and the Tarim River in Central Asia, where sparse ground-based meteorological observations result in large uncertainties in climate forcing data (Xu et al., 2020; Chen et al., 2008). Cold-region basins such as the Yukon, Brahmaputra, Lena, Indigirka, Kolyma, Olenek, Yana, and Khatanga are strongly influenced by glacial and permafrost-driven seasonal accumulation and ablation processes (Riegger and Tourian, 2014; Yi et al., 2023). The linear reservoir assumption fails to adequately capture the mass variations in these basins (Liu et al., 2022), resulting in NSE values ranging from 0.45 to 0.70. In contrast, river basins where hydrological signals are dominant, such as the Amazon, Mississippi, Nelson, and Amur, showed good reconstruction performance with NSE values greater than 0.80. The histogram of ΔNSE (this model minus Zhong-REC) shown in Fig. 5c reveals a positively skewed distribution, indicating performance improvements in the majority of basins. Specifically, 37 basins (approximately one-quarter of the sample) exhibited ΔNSE >0.1. The most pronounced improvements were found in arid and semi-arid regions (Fig. S4), including the Tarim River, Lake Chad, and Yellow River basins. The spatial distribution of regions with the most notable improvements in Fig. 5c closely aligns with the ΔNSE pattern between the proposed model and Humphrey-JPL-REC (Fig. S5), further supporting the role of the coupling between temperature and precipitation and the joint inversion of initial conditions in enhancing reconstruction accuracy in areas with low NSE.

Second, we compared basin-averaged monthly TWSA from JPL-REC and GLDAS CLSM to highlight the improvement of our reconstruction method relative to the land surface model (Fig. S6). Across 116 major river basins worldwide, NSE values for both products are computed against the JPLM, and the spatial distribution of the resulting ΔNSE is presented (Fig. S6c). The results show that NSE values from our reconstruction are higher in most basins, with ΔNSE predominantly positive, indicating that the proposed method more effectively captures temporal TWSA variations at the basin scale.

It should be noted that TWSA estimates from GLDAS CLSM are derived from explicit simulations of individual storage components and are therefore influenced by the completeness and accuracy of the represented physical processes. In contrast, our approach statistically characterizes climate-driven storage responses under direct constraints from GRACE observations, implicitly integrating multiple storage processes. This leads to improved representation of total water storage variations and better agreement across many river basins.

This section presents a comparison of deseasonalized and detrended monthly TWSA time series across eight representative river basins. Figure 6 illustrates the differences among GRACE/GRACE-FO observations (black line), the present reconstruction (JPL-REC, red line), Zhong-REC (blue line), and Humphrey-JPL-REC (green line) in terms of basin-averaged TWSA time series.

Interannual variations in terrestrial water storage are primarily driven by atmospheric circulation anomalies induced by ENSO events, which often trigger hydrological extremes such as floods and droughts, thereby significantly altering regional water balance (Chen et al., 2022). In the Amazon Basin, the La Niña event of 2009 markedly increased regional precipitation (Chen et al., 2010), while the strong El Niño of 2015–2016 led to severe drought conditions (Tian et al., 2021). The proposed reconstruction model successfully captured both the water storage peak in 2009 and the pronounced depletion in 2016. In the Yangtze River Basin, the early 2010 La Niña event was one of the primary drivers of drought, as it weakened the East Asian summer monsoon and caused an eastward retreat of the western Pacific subtropical high, thereby reducing moisture transport and triggering drought in the middle and lower reaches (Li et al., 2023; Zhang et al., 2015). The sharp decline in TWSA observed by GRACE during this period was accurately reproduced by the present model. In the Nile Basin, due to the slower response of TWS compared to precipitation, the reduction in net precipitation caused by the 2008 La Niña event did not immediately lead to a decline in TWSA. Instead, TWSA exhibited a gradual downward trend from 2008 to 2010 (Forootan et al., 2019), and this phase lag was also captured by the model. During the strong La Niña of the period from 2010 to 2011, northern and inland Australia experienced abnormally wet conditions. In the Lake Eyre Basin, intense rainfall and widespread flooding caused a significant rise in TWSA (Chen et al., 2022). The reconstructed TWSA time series in this basin is highly consistent with GRACE observations.

In contrast to the basins discussed above, interannual anomalies in river basins such as the Colorado and Shatt al-Arab are closely coupled with human water extraction. In the Colorado Basin, groundwater levels experienced a brief recovery from 2009 to 2010 due to increased precipitation. However, an anomalous drought event in 2012, combined with record-low snowfall in the Rocky Mountains, led to a rapid decline in water storage (Castle et al., 2014). Similarly, since the onset of drought in 2007, the Shatt al-Arab Basin has experienced a sharp decline in precipitation, compounded by a rapid drop in upstream reservoir levels along the Euphrates River. As surface water supplies became insufficient to meet demand, groundwater abstraction surged and became the primary source for agricultural and domestic use, leading to a rapid depletion of groundwater storage (Voss et al., 2013). In the high-latitude Indigirka Basin, winter precipitation is primarily stored in solid form, with snow water equivalent (SWE) gradually accumulating throughout the frozen season and then melting intensively in the following summer. GRACE effectively captured this “snow accumulation–melt” process, showing a spring peak of approximately 50 mm from 2006 to 2007. By contrast, the four-parameter recursive model developed in this study treats all precipitation P(t) as an immediate input flux, which is attenuated through a memory term (c⋅Tz+d)⋅TWS(t-1). This assumption leads to premature dissipation of the snow accumulation signal during winter, resulting in significant underestimation of the seasonal peak. Meanwhile, the relatively large constant memory coefficient d caused part of the residual water to be released with a delay, resulting in overall higher reconstructed TWSA compared to GRACE observations from 2008 to 2010. Among all basins examined, this basin exhibited the poorest performance of our method relative to previous models.

Although our model is primarily constrained by monthly GRACE observations, the method can also reconstruct daily characteristics in TWS changes based on the daily climate drivers. Here, we evaluated our reconstructed daily TWSA by comparing it with two different reference datasets, namely the daily ITSG-Grace2018 solution derived using a Kalman smoothing approach and the GLDAS-2.2 Daily product. The comparison with ITSG-Grace2018 evaluates whether the daily reconstruction is consistent with GRACE-based sub-monthly variability, whereas the comparison with GLDAS-2.2 assesses whether the daily reconstruction is consistent with physically based land-surface simulations.

We first assessed reconstruction performance at the basin scale by computing basin-averaged daily TWSA over major river basins worldwide. The difference in the time series was quantified using NSE (Fig. 7a and b). The NSE values are positive across most basins, with over half exceeding 0.5, indicating that our reconstruction results align well with the previous models. Although the reconstruction performs well overall relative to both reference datasets, its agreement with ITSG-Grace2018 is stronger in many basins. Specifically, 63 basins show NSE values greater than 0.6 in comparison with ITSG-Grace2018, whereas the count slightly reduces to 51 when compared with GLDAS-2.2. This is likely because our reconstruction is calibrated under constraints from GRACE/GRACE-FO TWSA, while ITSG-Grace2018 is also a daily product derived from GRACE observations.

To better illustrate sub-monthly variability, we showed daily TWSA time series in the Mississippi River basin in Fig. 7c. The results show similar sub-monthly variabilities among the reconstructed GRACE-REC, ITSG-Grace2018, and GLDAS-2.2 time series. One exception is found in the Indus basin between our GRACE-REC and GLDAS-2.2. The main reason is that GLDAS-2.2 fails to capture a three-year drought spanning from 2016 and 2018 (see Fig. S7 for details). The reconstructed daily TWSA is consistent with the corresponding reduction in the 6-month moving average of precipitation. By contrast, the GLDAS-2.2 daily TWSA remains comparatively flat during this period, suggesting that it may underestimate the sensitivity of basin-scale storage to interannual hydroclimatic forcing in this basin. Figure S7b further shows that our monthly reconstruction remains close to the GRACE/GRACE-FO observations, particularly around the resumption of GRACE-FO observations in June 2018, whereas GLDAS-2.2 tends to overestimate TWSA at that time. This discrepancy highlights the indispensable role of observation-based reconstruction approaches.

We conducted a self-validation experiment to further evaluate the robustness of the proposed method by dividing the JPLM product into separate training and validation periods. Specifically, the period from April 2002 to December 2012 was designated as the training period and the model parameters were estimated, while the period from January 2013 to December 2023 was treated as an independent validation period.

We evaluated the model performance using the NSE by comparing JPL-REC with the JPLM in both the training and validation periods (Fig. 8). During the training period (Fig. 8a), most basins exhibit high reconstruction skill, with NSE values exceeding 0.6 over large parts of North and South America as well as Eurasia. A very similar spatial pattern is obtained for the validation period (Fig. 8b), with only a modest reduction in overall skill. Approximately 60 % of the basins retain NSE values above 0.5 in the validation period. This indicates that the model retains stable predictive capability beyond the calibration window and does not exhibit significant overfitting behavior.

The spatial pattern of the results during validation period reveals distinct regional characteristics. High NSE values are mainly found in large, humid basins with relatively dense observations, such as the Amazon, Mississippi, Yangtze River basins. In these regions, variability in TWS is strongly controlled by climate forcing. Consequently, our model can reliably reproduce the TWSA signals observed by GRACE/GRACE-FO.

We further selected eight representative basins spanning high-latitude cold regions, tropical humid regions and typical monsoon-affected areas (Fig. 9). For each basin, the validation period TWSA (blue line) was compared with the reconstructed values during the training period (red line) and GRACE observations (black line). The results show that in certain basins, such as the Amazon, Yangtze, Mississippi and Murray, model performance during the extrapolation period remained comparable to or even exceeded that during the training period. This indicates that under these hydroclimatic conditions, the fixed-parameter model can reliably capture interannual TWSA variability driven by precipitation and temperature. In contrast, basins such as Chad and Khatanga show lower NSE during the validation period and larger discrepancies in certain years. This decline may be attributed to the limitations imposed by parameter nonstationarity and errors in the forcing data. These results suggest that when only precipitation and temperature forecasts are available, the proposed recursive model can be directly applied for estimating TWS changes during missing periods or into the future. Nevertheless, its applicability varies with hydroclimatic conditions, and for regions requiring high prediction accuracy, periodic parameter updates or the inclusion of additional forcing information may still be necessary.

We first evaluated the impact of the temporal resolution of meteorological forcing data on reconstruction performance, specifically whether daily forcing provides additional benefit compared to monthly forcing. To this end, all 116 basins were recalibrated using both daily and monthly ERA5-Land precipitation and temperature. The monthly forcing datasets were aggregated from daily values: precipitation was summed to monthly totals, and temperature was averaged over each month. The deseasonalized and detrended TWSA reconstructions were then compared on a monthly basis, and NSE was computed for each case. The difference between the two forcing schemes was quantified as ΔNSE=NSEDaily-NSEMonthly, with its spatial distribution shown in Fig. 10a. Positive values indicate better performance under daily forcing.

The results show that 108 out of 116 basins (approximately 95 %) exhibit positive ΔNSE values, suggesting that daily forcing generally outperforms monthly forcing. The most significant improvements were observed in high-latitude permafrost–glacier transition zones such as Siberia and Alaska. In contrast, tropical humid basins (e.g., the Paraná Basin) exhibited minimal differences (ΔNSE ≤±0.05). Only seven basins (approximately 5 %), such as the Huai River Basin, showed slightly negative differences (-0.1<ΔNSE<0), mostly located in regions with strong human regulation or sparse observational data.

From a time series perspective, the reconstructed TWSA driven by monthly climate forcing exhibits significantly lower amplitude compared to that driven by daily forcing. This is mainly due to two factors: daily forcing preserves extreme precipitation events and high-frequency temperature fluctuations on a day-to-day basis, allowing short-term water flux pulses to enter the recursive system as sharp peaks. In contrast, monthly forcing involves pre-integrating or averaging over 30 d precipitation and temperature records before inputting them into the model, which smooths out extreme events and leads to underestimated amplitude in the simulations (Humphrey et al., 2016). In addition, the memory coefficient calibrated at the daily time step typically ranges from dday≈ 0.97 to 0.995, indicating very limited attenuation on a daily basis. When directly scaled to a monthly time step, this corresponds to dday30≈0.74; alternatively, recalibration at the monthly scale yields an equivalent memory coefficient of approximately dday30≈0.7 to 0.8. Both cases imply a stronger low-pass filtering effect, which further dampens peak values and elevates troughs. Therefore, we argue that daily forcing is essential for improving model fidelity across nearly all basins and should be considered necessary for accurate reconstruction.

This section investigates the influence of the initial terrestrial water storage, TWS(0), on reconstruction performance. Humphrey and Gudmundsson (2019a) demonstrated through experiments that if TWS (0) is simply set to zero, the model requires a spin-up period before reaching equilibrium. To avoid discarding multiple years of data, they derived an analytical equilibrium value to serve as a uniform initialization for TWS(0). However, when a basin is not in climatic equilibrium, this equilibrium-based initialization can introduce systematic bias. Since the model's memory decay coefficient (c⋅Tz+d) is approximately equal to 1, such bias can persist for several years in basins with long water residence times, making the initial condition a critical factor influencing reconstruction accuracy in long-memory basins. To mitigate this bias, TWS(0) in this study is jointly estimated along with the model parameters (a,b,c,d). Three initialization strategies were compared (Fig. 11): (1) the optimal solution where TWS(0) is treated as an unknown and solved through least squares estimation; (2) a suboptimal solution, obtained by increasing the optimal value by 10 %; and (3) the equilibrium solution, derived from Eq. (5). Using the Chad Basin as an example, the NSE of the equilibrium initialization was 0.48, which increased to 0.57 under the suboptimal case and further improved to 0.73 under the globally optimal initialization. Therefore, in this study, TWS(0) is retrieved through parameter inversion for all basins, substantially reducing systematic bias and enhancing reconstruction accuracy.

Figure 12 illustrates the global distribution of parameter a and its associated uncertainty. That parameter a arises from the first-order Taylor expansion of the temperature modulation term in the model formulation (Eq. 13). Therefore, it represents a mathematical sensitivity coefficient describing how temperature perturbs the relationship between precipitation input and TWS response. As shown in Fig. 12a, the sign of parameter a varies among basins, with positive values in some basins and negative values in others. In many high-latitude regions, the calibrated values tend to be negative, whereas some humid basins exhibit positive values. These spatial differences likely reflect variations in regional hydroclimatic conditions and the ways in which temperature influences hydrological processes such as snowmelt, evapotranspiration, and soil moisture dynamics.

Figure 13a compares the full four-parameter model with a reduced three-parameter version in which the temperature and precipitation coupling term (i.e., parameter a) is excluded. The global distribution of ΔNSE indicates that, on average, the contribution of parameter a is limited. However, under specific hydroclimatic conditions, incorporating parameter a can lead to improvements in model performance with ΔNSE >0.1. Notable improvements are concentrated in high-latitude snow-dominated basins, such as the Olenek River, and in certain arid regions with high evapotranspiration, such as the Volta Basin. For the remaining 112 basins, ΔNSE values are below 0.1, suggesting no statistically significant difference. From a time series perspective, excluding parameter a in the Volta Basin leads to an overestimation of summer peaks during 2005–2006 and an underestimation during 2009–2012. In regions with high temperatures, removing parameter a impairs the model's ability to capture interannual peak and trough variability.

From Eq. (17) it follows that parameter b represents the fraction of precipitation that is ultimately converted into TWS: a larger b value indicates that a higher fraction of precipitation contributes to storage, whereas a smaller b suggests that a greater proportion of precipitation is dissipated through evapotranspiration or rapid runoff before becoming part of the basin water storage. In this sense, parameter b characterizes whether precipitation tends to be directly lost from the system or effectively retained as storage.

Figure 14 shows the global distribution of parameter b and its associated uncertainty. Overall, parameter b exhibits spatial heterogeneity across different regions. As shown in a Fig. 14, relatively low parameter b values are found in the Congo, Niger, and Orange River basins in Africa; the Lake Eyre and Murray basins in Australia; and the Rio Grande and Colorado basins in North America. These regions are characterized by relatively high long-term mean ET / P (Fig. S8); a large fraction of precipitation is exhausted by evaporation and runoff before contributing to TWS, which yields a low conversion efficiency. Consequently, the efficiency of precipitation-to-storage conversion is low, resulting in smaller calibrated b values.

In contrast, higher b values are observed in high-latitude basins such as the Lena and Yenisey. In these regions, winter precipitation is predominantly stored as snow or ice, delaying immediate losses to the atmosphere or river discharge. During the snowmelt season, the accumulated solid water is released and contributes to TWS. This storage–delay–release mechanism reduces the fraction of direct precipitation loss and enhances the effective conversion of precipitation into TWS, thereby leading to higher b values.

To further substantiate this physical interpretation, we examined the statistical relationship between parameter b and the multi-year mean loss ratio (ET+R)/P at the basin scale (Fig. S9). The results reveal a negative correlation between parameter b and (ET+R)/P. This statistical relationship provides independent observational support for interpreting parameter b. Furthermore, the global spatial pattern of b is consistent with the findings of (Zhong et al., 2025).

Figure 15 shows the global distribution of parameter c and its associated uncertainty. The parameter c arises from the first-order Taylor expansion of the temperature modulation term in the model formulation (Eq. 15). Therefore, it represents a mathematical sensitivity coefficient describing how temperature perturbs the storage memory term. The parameter c exhibit clear spatial variability. In many high-latitude basins, the magnitude of c is relatively larger (mostly negative), whereas in most mid- and low-latitude basins the values are close to zero. This pattern indicates that the influence of temperature on the model's water storage memory term is more pronounced in colder regions, while in many other basins the recursive water storage component is largely temperature-independent within our reconstruction model. The spatial distribution of parameter uncertainty (Fig. 15b) further shows that estimation errors are relatively small in most regions, suggesting that the adopted calibration approach is stable and robust across the majority of basins.

Figure 16 compares the full four-parameter model with a reduced three-parameter version excluding parameter c. Among the 116 basins analyzed, most exhibit limited sensitivity to parameter c; however, incorporating c still yields measurable performance improvements in some cases, with 11 basins showing ΔNSE >0.1. Notable improvements are concentrated in high-latitude cold regions around the Arctic (e.g., Yana, Yukon, and Lena), while limited improvements are found in parts of North America (e.g., Columbia, Colorado, and Rio Grande) and in certain Chinese basins (e.g., Yangtze and Yellow River).

From a time series perspective, in the Yukon Basin, the average temperatures during the snowmelt seasons of the period from 2006 to 2007 and in 2009 were markedly below normal (Fig. S10). Under these conditions, the decay factor (c⋅Tz+d) remained close to the baseline value d, resulting in a slower dissipation rate. Meanwhile, SWE during these years was also below the multi-year average, limiting the contribution from snowmelt. GRACE observations captured a slight increase in TWS during spring, followed by a continuous decline. The four-parameter model successfully reproduced this slow depletion process (Fig. 16c). In contrast, the three-parameter model, due to the absence of a temperature modulation term, maintains a higher retention rate under the same precipitation input and consequently produces an overall overestimation in the reconstructed TWS curve. Conversely, 2019 was an exceptionally warm year, with precipitation anomalies remaining positive from 2017 to 2021. Rising temperatures substantially reduce(c⋅Tz+d), thereby accelerating the depletion of antecedent water storage. The abundant snowpack was rapidly flushed during the melt season, resulting in a deeper TWS trough recorded by GRACE compared to that of 2006 to 2007. The four-parameter model captured this deep trough due to the dynamic weakening effect of the parameter c, whereas the three-parameter model produced a shallower minimum. A similar pattern is observed in the Yana Basin: exceptionally high SWE in 2007 led to elevated TWS levels, while reduced snowmelt and elevated temperatures in the following year (Fig. S11) triggered rapid snowmelt and a sharp decline in TWS during 2008. The four-parameter model exhibits better agreement with GRACE observations, whereas the three-parameter model shows excessive smoothing (Fig. 16b). These comparisons highlight the critical role of the temperature modulation term c in accurately capturing TWS extremes during warm years or periods of rapid snowmelt.

In the linear storage model, when no additional precipitation input is introduced, water depletion follows an exponential decay process. As shown in Fig. 17, when parameter d=0.97, and in the absence of both additional precipitation input and temperature effects, only 40 % of the stored water remains after one month. In contrast, when d=0.99, more than 70 % of the stored water can still be retained after one month under the same condition. This indicates that the interannual variability of TWS is highly sensitive to the value of parameter d.

To examine spatial variability, Fig. 18 presents the global distribution of the 30 d retention factor (d30) and its associated uncertainty (Fig. 18). According to the model formulation, parameter d in Eq. (17) represents the fraction of antecedent storage that remains in the system after a period of decay. From Eq. (10) it follows that, when d approaches 1, the storage at the previous time step is almost entirely preserved, implying that the current total loss (ET+R) is primarily attributable to contemporaneous precipitation losses. Conversely, when d deviates from 1, current losses are also driven by the release of antecedent storage. This mechanism is consistent with the spatial pattern shown in Fig. 18a. In high-latitude regions of the Northern Hemisphere, the widespread presence of permafrost limits the infiltration of meltwater during the spring thaw, causing most of the water to be rapidly discharged as surface runoff. As a result, the calibrated parameter d is relatively low.

Notably, the Ob and Mackenzie River basins exhibit considerably higher water retention after 30 d of decay compared to other high-latitude basins (Fig. 18a). In the Ob basin, approximately 15 % of the upstream and midstream region between the Irtysh River and the upper Ob is occupied by endorheic depressions, where inflow is retained and does not contribute to the main channel (Yi et al., 2023). Additionally, springtime ice jams, channel backwater, and overbank flooding further prolong the average residence time of water, resulting in a higher calibrated 30 d retention coefficient (d30). In the Mackenzie basin, the presence of multiple large lakes leads to a highly fragmented river network (Yi et al., 2023), impeding the continuous drainage of liquid water and thereby extending the residence time of water within the basin.