The GRACE Level 2 data consisting of monthly Stokes coefficients of Earth's geopotential was used to derive estimates of TWSC over the Indus Basin. The data from three Science Data Centers, Jet Propulsion Laboratory (JPL), University of Texas, Center for Science and Research (CSR), and GeoForschungsZentrum (GFZ) in release 6 (RL06) were utilized up to degree 90. Only the months common to all three solutions and WGHM output were used. This resulted in 147 monthly solutions from April 2002 to December 2016. The GSM (denoted as is) products were used, containing fully normalized geopotential coefficients representing the full magnitude of land hydrology, ice, and solid Earth processes. In addition, the RL06 MASCON product from CSR was also utilized to compare the overall results of level 2 processing. The ICE-6G_D model was used to account for glacial isostatic adjustment (Peltier et al., 2018). This model is available up to degree and order 256 and provides secular rate of change of gravity field in mm yr-1. The degree 1 coefficients from (Sun et al., 2016), distributed as Technical Note (TN-13) on the Tellus website, were used. The replacement C20 coefficients from (Cheng and Ries, 2017), distributed as TN-11 on the Tellus website, were used. To derive GRACE-like errors for the design of simulation experiments, the monthly normal equations provided by TU Graz for the ITSG-2018 solutions (Kvas et al., 2019) in SINEX format were used.

TWS anomalies from WGHM 2.2d, both standard and integrated versions, were used (Döll et al., 2003; Müller Schmied et al., 2016; Cáceres et al., 2020). The integrated version simulates glacier mass changes, unlike the standard version. Grids of 0.5∘ resolution from April 2002 to December 2016 were used and resampled to 1∘. For each version, there were four variants available with two climate forcings; ERA-Interim reanalysis (WFDEI) applied with WATCH Forcing Data (WFD) methodology and bias-corrected using precipitation data sets derived from rain gauge observations of either GPCC v5/v6 (Global Precipitation Climatology Centre; Schneider et al., 2015) or CRU TS3.10/TS3.21 (Climate Research Unit; Harris et al., 2014) and two irrigation variants; irrigation water use at 70 % of consumptive use (CU) and optimal CU. No single irrigation scenario can be assigned for the Indus Basin due to the heterogeneity in irrigation patterns. Hence, an average of these four variants was taken under each version (standard and integrated) to obtain the respective grids. Table 1 summarizes the variants of the model used in this study.

WGHM simulates water flows at a daily timescale on a 0.5∘ by 0.5∘ global grid, excluding Antarctica. Human intervention and its effect on water flows are also considered by simulating human water use in five sectors: irrigation, manufacturing, livestock farming, domestic use, and thermal power plants. The model requires daily meteorologic inputs of near-surface air temperature, precipitation (rainfall and snow), downward shortwave radiation, and downward longwave radiation. Reservoir data come from the Global Reservoir and Dam (GRanD) database, which includes 6862 reservoirs with a total storage capacity of 6197 km3 (Lehner et al., 2011).

Water balance is carried among three water storage compartments in the vertical direction: canopy, snow, and soil water storage. WaterGAP represents soil as a one-layer soil water storage compartment characterized by a land cover and soil-specific maximum storage capacity and soil texture. The groundwater is recharged from the soil, and storage is simulated after accounting for net abstractions from groundwater due to human use. A fraction of groundwater recharge is assumed to flow back to surface water bodies, and groundwater recharge under surface water bodies is neglected. The surface water storage comprises five sub-compartments: local lakes, local wetlands, global lakes, reservoirs, and global wetlands. At each stage, the storage is simulated by accounting for ET losses and net abstractions from lakes and reservoirs. Finally, the output from surface water bodies and groundwater flows into rivers, where the river water storage is simulated after accounting for the streamflow.

Groundwater surface water use (GWSUSE) model accounts for net abstractions from surface water and groundwater. Irrigation is assumed to vary with countries (efficiency of irrigation water use) and source of irrigation water. In groundwater-depleted areas, the CU from irrigation is considered 70 % of the optimal CU since farmers have less availability to satisfy the optimal requirements. In all other areas, it is assumed to be optimal. Country-specific efficiency values are used for surface water irrigation, while in case of groundwater irrigation, water use efficiency is set to a relatively high value of 0.7 worldwide. These abstractions are accounted for in the water balance for each compartment as described above.

The glacier mass changes not included in the WGHM are obtained from global glacier model (GGM) (Marzeion et al., 2012). GGM consists of a surface mass balance model based on the temperature index approach and model that accounts for changes in glacier geometry in feedback to compute the changes in glacier mass. The model is forced by a mean ensemble of seven atmospheric datasets, reducing the uncertainty inherent in the individual forcing datasets. As an initial condition for glacier geometry, data like glacier areas and boundaries are taken from Randolph Glacier Inventory (RGI v6) (Pfeffer et al., 2014). The model is calibrated using observed surface mass balance from World Glacier Monitoring Service (WGMS).

The GRACE level 2 SH coefficients from the GSM product represent the geopotential of Earth containing the full magnitude of land hydrology, ice, and solid Earth processes during each month t as in Eq. (1), 1Vr,θ,λ,t=GMRe∑n=0NRern+1∑m=0nPnmcos⁡θCnmtcos⁡mλ+Snmtsin⁡mλ, where θλ are co-latitude and longitude respectively, n,m are degree and order of the expansion, N is the maximum degree used which was 90 in this case, Pnm are the associated Legendre's functions, Cnmt and Snmt are the spherical harmonic coefficients at month t, and Re (6378 km) is Earth's equatorial radius.

The mean static gravity field is removed to obtain gravity field anomalies. In this study, the mean period was taken as the mean of the entire study period. Monthly solutions common to all three (JPL, CSR, and GFZ) centers till December 2016 were extracted with a tolerance of 15 d in the middle of each monthly solution epoch. Running means over three consecutive solutions are taken, which reduces the noise in the solutions. Glacial isostatic adjustment (GIA) is the ongoing secular response of Earth's surface to the mass changes that occurred due to the deglaciation process on Earth and is removed using the ICE-6G model (Peltier et al., 2018). The same GIA model used in CSR MASCONS is used to maintain consistency. Assuming mass changes occur in a thin spherical shell (∼ 15 km) around Earth (Wahr et al., 1998), these coefficients are multiplied by degree-dependent factors to obtain surface mass changes coefficients in terms of equivalent water height (EWH). The resulting change in surface mass density is obtained as Eq. (2), 2Δσ(θλt)=Reρavg3∑n=0N2n+11+kn′∑m=0nPnm(cos⁡θ)ΔCnmtcos⁡mλ+ΔSnmtsin⁡mλ, where ρavg is the average density of Earth (5515 kg m-3) and kn′ are the elastic load Love numbers of Earth.

Swenson filter (Swenson and Wahr, 2006) is used for destriping, which removes the correlated errors in even and odd degrees. The filter parameters are chosen after Sasgen et al. (2018), which minimizes the signal and noise contamination between mid-latitude and polar regions during destriping. A Gaussian low pass filter of half width 300 km is employed to reduce the random errors in higher degree terms. Grids of filtered surface mass changes are obtained as Eq. (3), 3Δσ(θλt)=Reρavg3∑n=0N2n+11+kn′∑m=0nPnm(cos⁡θ)WnΔCnmtcos⁡mλ+ΔSnmtsin⁡mλ, where Wn are degree-dependent factors of Gaussian filter in spectral-domain (Sasgen et al., 2018).

From the filtered GRACE surface density coefficients, mass changes for the Indus Basin are obtained by regional integration using the region function defined in Eq. (4), 4RF(θλ)=10∀(θ,λ)∈R∀(θ,λ)∈Ω-R, where RF is the region function inside the region R of the Indus Basin and region Ω is the total surface area of Earth. This region function is transformed to spectral-domain (up to degree 90), and the corresponding coefficients are multiplied with filtered surface density coefficients to obtain mass changes for the Indus Basin, ΔM, as in Eq. (5), 5ΔM(θλt)=Reρavg3∑n=0N2n+11+kn′∑m=0nPnmcos⁡θWnRnmcΔCnmtcos⁡mλ+RnmsΔSnmtsin⁡mλ, where Rc and Rs are respectively the cosine and sine coefficients of the region function in the spectral domain.

WGHM anomalies were centered around the mean of the entire observation period (April 2002–December 2016) by including only GRACE months. Unfiltered mass change time series for the Indus Basin from both versions were obtained using the region function in Eq. (4), by integrating the global grids over the surface of Earth as in Eq. (6), 6ΔMregiont=∫Δσθ,λ,tRFθ,λdΩ, where dΩ=Re2sin⁡θdθdλ is an element of the area on the region surface. The global grids are then transformed to the spherical harmonic domain up to degree 90 and applied with Swenson destriping and Gaussian low pass filter in the same manner as described in Sect. 2.3.1. The filtered coefficients are transformed back to the spatial domain to obtain filtered model grids. Regional integration in the spectral domain is performed with filtered model coefficients to get filtered model mass change time series.

A single scaling factor (k) for the entire basin is derived by a least squares regression between unfiltered model time series and filtered model time series, that is, by minimizing L in Eq. (7), 7L=∑t=1TΔStu-kΔStf2, over the entire period of study. In Eq. (7), ΔStu is the unfiltered and ΔStf is the filtered model mass for the tth month. A scaling factor for each grid cell in a 10×10 grid is derived using the unfiltered model grid and filtered model grid and minimizing L in Eq. (7) but at every grid cell in the basin. This is done for both versions of WGHM to obtain a 1∘ resolution map of scaling factors.

To derive frequency-dependent scaling factors, the total mass changes from unfiltered and filtered model versions are decomposed into long-term linear, seasonal, and residual components as Eq. (8), 8ΔS=ΔSlongterm+ΔSsesonal+ΔSresidual. For this purpose, a time series model is fit to the data using least squares as in Eq. (9), 9yi=β0+β1t+∑k=1nβkccos⁡2πTkt+βkssin⁡2πTkt+νt, where β1 is the linear trend, βkc and βks are the amplitudes of the cosine and sine terms of the periodic component with a period Tk, and t represents the month with respect to the mean of the observation period. However, the periods (Tk) of the seasonal terms are unknown. For this study, the unknown periods are found from the data itself using a Lomb–Scargle (LS) periodogram analysis (Scargle, 1982) which allows detection of weak periodic signals in otherwise random, unevenly sampled data.

From this analysis, the peaks were found at annual and semi-annual periods. The false alarm probabilities associated with each peak were minimal (∼10-5), which shows that these periods are significant. These periods were used in Eq. (9) for time series decomposition. The trend, annual, and semi-annual components are separated in both model versions' unfiltered and filtered time series. Then a least squares fit is carried out for all three components separately to obtain three scaling factors that minimize the expression in Eq. (10), 10L=∑j=1NΔSju-kjΔSjf, where j= trend  annual  semi-annual , where kj is the scaling factor for jth component.

Figure 3 presents the schematic of deriving scaling factors (in green boxes) from filtered and unfiltered model time series (in yellow boxes) and applying them to corresponding GRACE time series (in blue boxes) to obtain scaled GRACE estimates (in purple boxes).

The DDC approach or the method of deviation (Vishwakarma et al., 2017) uses twice-filtered GRACE fields to determine two correction terms for the regional averages of filtered GRACE fields, namely the leakage and deviation integrals for the entire basin. We used the method codes provided by the authors (and through personal communication with the lead author) to compute basin-averaged DDC corrected time series for the Indus Basin to compare with our scaled basin averaged time series.

Due to the unavailability of distributed in situ data, we designed a simulation experiment using WGHM and monthly GRACE TWS errors to assess the extent to which the scaling schemes can recover the signal lost to filtering. We use WGHM fields as a proxy for true TWS signals and corrupt them with GRACE-like errors (Willen et al., 2022). For this, we first derived the error covariance matrices from the normal equations provided by TU Graz for the ITSG solutions (Kvas et al., 2019) from degree 2 onwards. For the degree 1 coefficients, the corresponding error covariances were computed from an ensemble of 12 different estimates derived following Swenson et al. (2008), arising from four GRACE products (JPL, CSR, and GFZ release 6, ITSG-2018) and three GIA models (A et al., 2013; Caron et al., 2018; Peltier et al., 2018). Assuming the errors to follow multivariate random distribution, we derived their random realizations in the SH domain from the covariance matrices using Cholesky decomposition. We then added these errors to the WGHM fields in SH domain and filtered using the same filter used in the study for GRACE (Swenson destriping +300 km Gaussian). These filtered–corrupted WGHM fields now represent GRACE-like observations. Finally, using the scaling factors derived in the study, we rescaled these filtered–corrupted WGHM fields as per the respective schemes (Fig. 3) to recover the lost signals as rescaled WGHM fields.

We use GRACE errors from the ITSG solutions for assessing scaled estimates of GRACE solutions in our study since the normal equations from three centers – JPL, CSR, and GFZ – are not publicly available. Moreover, using ITSG-based errors can be justified by the following: our study uses a monthly running average of the three solutions, which means the errors are already reduced. The ITSG solution which has been shown to have less noise than the JPL, CSR, and GFZ solutions (Ditmar, 2022) thus provides a reasonable estimate of the average error content.

Two quantities are computed using WGHM as a proxy to the true signal to assess the uncertainty specific to leakage and scaling. Equation (11) provides the initial leakage due to filtering, and Eq. (12) provides the amount of leakage not covered by the scaling factors, termed residual leakage. 11Initial Leakage Error=stdΔStu-ΔStfc×RMSΔGtRMSΔStf12Residual Leakage Error=stdΔStu-kΔStfc×RMSΔGtRMSΔStf, where std represents the standard deviation, RMS represents the root mean square of the time series indexed by t, k represents any one of the scaling factors schemes, ΔGt represents GRACE mass time series, and ΔStfc represents the filtered–corrupted model time series. If the scaling factors work, the residual leakage should be less than the initial leakage. This quantification assumes all the variation in the difference between filtered–corrupted and unfiltered model time series to represent leakage error (making no distinction between individual effects on underlying signal and noise). It is thus meaningful for intercomparison of different schemes but not as a true estimate of leakage error. This standard deviation is multiplied by the ratio of RMS of GRACE and filtered–corrupted model time series to account for the amplitude difference in GRACE and model (Landerer and Swenson, 2012).

The uncertainty estimation of GRACE mass estimates is done following Groh et al. (2019), which uses only the time series of derived GRACE masses to determine noise. The unscaled GRACE time series contains errors from measurement, leakage, GIA, and C20. Since this approach provides the temporally uncorrelated component of total GRACE errors, the effect of scaling on the random component of time series (thus a random component of leakage) can be compared. The uncertainties from GIA models (<1 mm yr-1 equivalent sea level) and C20 (<0.1 mm yr-1 equivalent sea level) coefficients are almost negligible for the Indus Basin and, hence, are not considered (Caron et al., 2018; Blazquez et al., 2018). A high pass filter is applied to the residuals of the GRACE time series (after removing trend and seasonal components) to remove the un-modeled inter-annual components in the residuals. The filter width is chosen to be 18 months (6 sigma width of Gaussian hat =18 months), which means all the signal with a larger than 18-month period is removed, leaving temporally uncorrelated errors referred to as noise. This noise contains the GRACE measurement error and the random component of the initial leakage error. A scaling factor generated from high pass filtering random white noise signals is multiplied to account for the damping of any white noise component during high pass filtering. Similarly, this process is repeated for scaled GRACE time series. The corresponding scaled noise contains the scaled measurement and residual random leakage errors.

Basin scaling factors of ks=1.14 from the standard version and kg=1.22 from the integrated version are obtained. These values are greater than one, indicating that filtering causes the overall mass changes to leak out; hence, a factor greater than 1 is required to restore the signal. However, the values are small, indicating that the Indus Basin has a small leakage amount overall. Basin scale factors from two studies have reported similar values (Table 3). The addition of glacier mass changes in the model leads only to a minor effect on the basin-scale factor. Glacier mass changes are located at the edge of the basin and suffer from more leakage out. This explains the slightly larger scaling factor from the integrated version. The similarity of our results to results from other studies that used different models indicates that basin-scale factors are not very sensitive to the model used for the Indus Basin.

Figure 10 shows the gridded scaling factors in 1∘ resolution maps from the standard (left) and the integrated version (right). The maps include 108 scaling factors, one for each pixel inside the Indus Basin. From the standard version, the scaling factors ranged from -0.4 to 10.1. From the integrated version, the scaling factors ranged from -0.5 to 8.5. Significant differences between the two occur in the upper Indus Basin, where the glaciers are located. Table 4 lists the standard interpretation of gridded scaling factor values used in most studies (Long et al., 2015). Figure 11 shows the histogram of scaling factors from both model versions.

From the standard version, scaling factors for 13 grid cells are negative. These grid cells were excluded for scaling, considering out-of-phase behavior with respect to their neighboring grid cells. Most scaling factors are less than 1 (67 grid cells), out of which most are less than 0.5 (55 grid cells), indicating that a large part of the basin suffers from moderate to significant leakage-in. These small scaling factors occur mainly in the lower reaches of the basin (Indus plains) and upper Indus Basin, where non-glacier mass changes are small in magnitude. Few grid cells stand out with larger scaling factors (>3), which depict large local mass changes in the pixel. These occur in the southeast Indus Basin region, which has a larger magnitude of TWS changes due to ISM and human intervention in the form of irrigation and groundwater depletion. These more considerable changes leak out to nearby dry regions of the upper Indus Basin (as represented in the standard version), requiring greater than 1 scaling factor.

From the integrated version, scaling factors at 11 grid cells are found negative and excluded. The number of grid cells with significant leakage in (k<0.5) is larger (62) than for the standard version. The grid cells with large scaling factors (k>3) are more distributed than the standard version. This is because the glacier mass changes cause large local mass variations in the upper Indus Basin, requiring larger scaling factors to account for leakage out to nearby dry regions (north-east of the Indus Basin with arid Tibetan plateau). The spatial variability of scaling factors obtained is higher (CV=1.69) from the integrated version than from the standard version (CV=1.48) due to a more heterogeneous representation of mass changes in the integrated version.

Figure 12 shows the histograms of grid scaling factors only for the non-glaciated region of the Indus Basin (i.e., the non-glaciated pixels in Fig. 1). The increase in the frequency of small scaling factors (<0.5) and decrease in large ones (>3) indicates that the distribution of values from the integrated version is shifted towards zero as compared to the standard version. This shows that the addition of a localized water storage compartment in the model (glacier in this case) not only affects the scaling factors in that region but in the entire basin.

The frequency-dependent scaling factors for the Indus Basin are also shown in Table 3. For the trend component, the scaling factors from both versions (1.14 from standard and 1.23 from integrated) are almost identical to the frequency-independent basin-scale factors. This reinforces that the Indus Basin is dominated by long-term trends rather than seasonal signals, which causes the basin-scale factors to be driven by filtering the trend component. The scaling factors for the annual component from both versions (1.64 from standard and 1.59 from integrated) are significantly higher than 1 to account for leakage out of the annual mass changes described earlier. This also shows that the basin-scale factor alone would under-scale the annual component. The semi-annual scaling factors are nearly identical and close to 1 in the standard and integrated version, showing that filtering has a negligible effect on this component. These deviations from a single basin-scale factor reinforce the need for frequency-dependent scaling factors in basins where mass changes occur at different frequencies and the filtering has a significantly different effect on these individual frequencies.