Over the past two decades, the Gravity Recovery and Climate Experiment (GRACE) satellite mission and its successor, GRACE-follow on (GRACE-FO), have played a vital role in climate research. However, the absence of certain observations during and between these missions has presented a persistent challenge. Despite numerous studies attempting to address this issue with mathematical and statistical methods, no definitive optimal approach has been established. This study introduces a practical solution using Linear Regression Analysis (LRA) to overcome data gaps in both GRACE data types—mascon and spherical harmonic coefficients (SHCs). The proposed methodology is tailored to monsoon patterns and demonstrates efficacy in filling data gaps. To validate the approach, a global analysis was conducted across eight basins, monitoring changes in total water storage (TWS) using the technique. The results were compared with various geodetic products, including data from the Swarm mission, Institute of Geodesy and Geoinformation (IGG), Quantum Frontiers (QF), and Singular Spectrum Analysis (SSA) coefficients. Artificial data gaps were introduced within GRACE observations for further validation. This research highlights the effectiveness of the monsoon method in comparison to other gap-filling approaches, showing a strong similarity between gap-filling results and GRACE's SHCs, with an absolute relative error approaching zero. In the mascon approach, the coefficient of determination (R2) exceeded 91% for all months. This study offers a readily usable gap-filling product—SHCs and smoothed gridded observations—with accurate error estimates. These resources are now accessible for a wide range of applications, providing a valuable tool for the scientific community.
Keywords: GRACE; GRACE-FO; gap; linear regression; TWS; hydrology
The GRACE and GRACE-FO missions have a dedicated purpose: to provide continuous and highly detailed monitoring of the Earth's gravity field [[
GRACE successfully completed its scientific mission over a span of 15 years, concluding in June 2017. Subsequently, GRACE's successor satellite, GRACE-FO, was launched in May 2018 [[
- Utilizing other satellite data.
Some studies have attempted to fill the gaps by incorporating data from other satellite missions (e.g., Swarm, global position system (GPS), SLR) [[
- 2. Artificial Intelligence, Machine Learning, and Deep Learning.
Another approach involves leveraging artificial intelligence techniques, such as machine learning and deep learning [[
- 3. Mathematical and Statistical Methods.
The third category of solutions involves mathematical and statistical methods [[
These various approaches reflect ongoing efforts to address the challenges posed by data gaps in the GRACE and GRACE-FO missions, each with its own strengths and limitations. However, no recommendation has yet been made on the most effective method. On the contrary, despite the widespread use of mascon data by both geodesists and non-geodesist scientists, there is a notable deficiency in research efforts aimed at addressing the filling-in of gaps in mascon data.
In this study, we've taken an innovative approach to fill-in the data gaps in both the GRACE and GRACE-FO missions, including the 11-month gap in mascon and SHC data. The LRA method is employed for estimation, and the approach is novel in how the estimations are conducted. Specifically, the value for each missing month is calculated individually by utilizing the available observations for the same month from each year. For instance, to estimate the missing data for January in a particular year, observations from all the available GRACE January months across different years are drawn upon. This approach is based on the assumption that observations for the same month annually exhibit similar climate conditions, including factors like rainfall, evapotranspiration, temperature, and more.
There are some studies attempting to solve the gap issue with linear regression, and all of them use artificial intelligence to estimate the new coefficients. Zhang et al. [[
In the present studies bridging the 11-month data gap between GRACE missions, an increasing trend is observed in the adoption of artificial intelligence techniques. However, this study presents improvements in four crucial aspects: Firstly, this analysis is based on the monsoon condition, which gives accurate results in a simple manner. Secondly, this approach stands out for its simplicity and elegance. Unlike the intricate and resource-intensive experimental design often associated with artificial intelligence methods, this data-filling method is straightforward and easily understandable. This simplicity makes it accessible for implementation with minimal computational burden, making it a practical choice for a wide range of researchers and scientists. Thirdly, this approach aims to provide data products in the form of SHCs, which are versatile and applicable for various GRACE data purposes. Additionally, data products in the form of mascons are offered, catering to hydrology applications and serving the needs of both geodesists and non-geodesist scientists. Fourthly, a notable distinction from previous studies is that these previous studies were primarily designed for hydrological applications, limiting their utility in other domains, including oceanography and solid Earth studies. In contrast, this method is more versatile and adaptable across different research areas, making it valuable for a wider range of applications.
In this study, the global scale is generally used for estimation. However, for particular parts, the focus is on eight major basins distributed across various continents, as illustrated in Figure 2. These selected basins include the Nile basin, Orange basin, Mississippi basin, Amazon basin, Volga basin, Yenisei basin, Yangtze basin, and Murray Darling basin. Importantly, all of these basins have a surface area that exceeds the spatial resolution of GRACE data, which is approximately 160,000 square kilometers. This ensures that this analysis covers basins with sufficiently large surface areas for meaningful observations and assessments [[
While the gap-filling method described below can be applied to any mascon or GSM (Gravity Recovery and Climate Experiment Science Team Mass Concentration) GRACE product (CSR, JPL, and GFZ) and is relevant to both d/o 60 and 96, the Release-06 product provided by the CSR processing center [[
Data were downloaded spanning from April 2002 to June 2017 for GRACE and from June 2018 to December 2021 for GRACE-FO. The mascon data in this context represent scaled GRACE solutions, allowing for the direct estimation of the time-series of TWS. On the other hand, the SHCs represent Earth's gravity field. CSR SHCs data with a maximum degree of 60 were specifically chosen for this analysis. To estimate the change in water mass using SHCs, the Equivalent Water Height (EWH) can be expressed as [[
(
where
(
In the analysis, the long-term value was removed using the baseline from 2004 to 2009 of the
Swarm satellite data are processed by multiple centers, but only two of these centers have maintained data continuity from December 2013 to the present day. These two centers are the Astronomical Institute of the Czech Academy of Sciences (ASU) and the International Combination Service for Time-Varying Gravity (COST-G). Both of these centers provide monthly SHC processed data, and for this study, the data were truncated at the maximum degree of 40 [[
This study employed an SLR-based recovery method that incorporates the GRACE empirical orthogonal function decomposition model developed by the Institute of Geodesy and Geoinformation at the University of Bonn, referred to as the IGG [[
The QF dataset comprises monthly gravity field models that encompass SHCs up to degree-and-order (d/o) 60. These models are constructed using tracking data obtained from two sources: HLSST (high–low satellite-to-satellite tracking) and SLR (Satellite Laser Ranging). The computation process involves the utilization of nine SLR geodetic satellites and 27 low-Earth orbiting (LEO) satellites, which include specialized gravity-focused satellites like CHAMP, GOCE, and GRACE A/B. To create these models, SLR and HLSST data are integrated using variance component estimation techniques, resulting in a spatial resolution of approximately 1000–2000 km [[
In the work conducted by Yi and Sneeuw [[
In this study, the plan is to validate the gap-filling SHCs obtained through LRA using a comparison with SHCs from dataset sources. These datasets include the IGG, QF data, Swarm-ASU, and Swarm-COST-G and data derived from SSA. This validation process will allow us to assess the accuracy and reliability of the gap-filling SHCs obtained through LRA by comparing them with SHCs from other datasets and methods.
LRA is a statistical technique used which identifies the optimal line that best fits the data by determining the regression vector parameters x and y [[
In the study, the focus was on the 11-month data gap between the GRACE and GRACE-FO missions, spanning from July 2017 to May 2018. To estimate the missing observations, Linear Regression Analysis was applied. These assumptions are deemed appropriate for the GRACE data. Regression models can be used to forecast the dependent variable's value at specific values of the independent variable. However, it is essential to note that this method is applicable only within the range of observed data, and it cannot be used to predict values for future observation months.
Here, assume that
(
where
Using the existing observations, the monthly
(
(
where the design matrix is:
(
the vector of regression coefficients is expressed as
(
and the vector of error is expressed as:
(
Expressing the vector of regression observations as
(
then
(
solving Equation (
(
W is the weight matrix that adjusts the contribution of each observation to the parameter estimation. The goal of W is to give more weight to observations with smaller variances. W is an m × m matrix with (w
(
Then, to solve Equation (
(
(
(
(
where
(
(
Then, the values of
(
(
After estimating the
(
where
To validate the accuracy of the model, the gravity models discussed in Section 2.2 are employed. To provide an overview of the SHCs using these gravity models, it is important to note that they can be broadly categorized into four distinct groups for the purpose of gap-filling datasets: (
To rigorously validate the LRA-based gap-filling approach in the spectral domain, an artificial data gap from January 2009 to December 2009 was intentionally created. SHCs were estimated within this period and compared directly with the original GRACE values for robust validation.
In these following sections, the five datasets mentioned above (three models + two Swarm) will be thoroughly examined in terms of their spectral characteristics. Their consistency will be evaluated, and their suitability for specific scenarios will be explored using an LRA model.
As depicted in Figure 3, the illustration presents a detailed comparison between the original GRACE-CSR SHCs and the coefficients generated by the LRA model for each month of the year 2009, denoted from (a) to (l). The figure is segmented into twelve boxes, one for each month, where the original month's SHCs are on the upper-left side and the corresponding LRA-generated coefficients are on the upper-right side. The subtraction of these values results in the data shown in the lower-left side of each box, illustrating the discrepancies between the GRACE and LRA SHCs. The observed disparities between the two datasets are minimal, demonstrating a high level of consistency between the GRACE and LRA model. Additionally, the absolute relative error for each month was calculated, and the results are displayed in the lower-right side of each sub-plot. Remarkably, the absolute relative error is nearly negligible across the board. This outcome underscores the remarkable accuracy of this model in effectively filling the data gaps.
We evaluated the root mean square (RMS) for spherical harmonic coefficients (SHCs) up to degree/order 60 to validate correlations between GRACE and LRA, IGG, and QF. As depicted in Figure 4, during an artificial gap in 2009, we calculated the RMS. Notably, GRACE and LRA exhibited the smallest RMS (1.79 × 10
In this comprehensive analysis, a time-series examination from January 2009 to December 2009 was undertaken. The objective was a meticulous comparison of datasets, including GRACE CSR, QF, IGG, and LRA-derived SHCs. This comparative assessment encompassed the scrutiny of a diverse set of SHCs, ranging from low to high degrees, such as C
As delineated in Figure 5, the time-series data visually present the outcomes of this rigorous comparison, revealing patterns of consistency or inconsistency. The correlation coefficient values between each dataset and the LRA-derived coefficients are prominently displayed at the top of each subfigure, representing GRACE CSR, QF, IGG, and LRA SHCs, respectively. The overarching findings indicate a noteworthy level of overall consistency among the datasets. Particularly striking is the high degree of coherence observed between the IGG and LRA SHCs with the GRACE CSR coefficients, in contrast to the QF model. This discrepancy with the QF model may arise from the significant reliance of the IGG data on GRACE/GRACE-FO sources. At lower degrees (up to degree 4), all datasets exhibit similar behavior. However, beyond this degree, they generally exhibit comparable patterns, except for the QF coefficients. Notably, in the case of C
In contrast, the LRA gap-filling approach exhibits higher consistency with the GRACE coefficients across various SHCs, as evident in the correlation coefficients. For instance, in S
We computed these correlation coefficients for the entire spherical harmonic coefficients (SHCs) up to degree/order 60 in order to validate the correlations between GRACE and LRA, IGG, and QF. In 2009, we calculated the correlation coefficients during an artificial gap.
According to Figure 6 and Table 2, GRACE and LRA showed the greatest correlation coefficient (0.8258), indicating significant agreement. With GRACE, the IGG dataset showed a modest correlation coefficient of 0.6763. The QF data, on the other hand, had the lowest correlation coefficient, at 0.5428. The high correlation between GRACE and LRA suggests strong agreement between these two datasets.
In December 2013, the Swarm mission released products, which was a significant development. Coefficients for that month were estimated using the LRA approach. A comparative assessment included GRACE-CSR, Swarm (ASU and COST-G), IGG, QF, and the LRA gap-filling model. The evaluation focused on PDV. For a fair comparison, all coefficients for all datasets were standardized by truncating them up to degree 40, aligning with Swarm products. Figure 7a illustrates the results of this comparison, revealing that the PDVs for all products, including the LRA model, exhibit a high degree of consistency with one another.
We opted to analyze a specific month, July 2014, which featured a data gap in GRACE but also included data from the Swarm mission. In this analysis, the LRA gap-filling approach was employed to estimate missing observations for this month. Subsequently, a comparative assessment was conducted, contrasting this estimated data with datasets obtained from Swarm (ASU and COST-G), IGG, QF, and SSA data, as documented in the work by Yi and Sneeuw [[
As portrayed in Figure 7b, this comparison was focused on assessing the potential degree variance (PDV) among these datasets, with all coefficients truncated up to degree 40 to align with Swarm data standards. The outcomes of this evaluation underscore a remarkable level of consistency among all products, including the LRA model, in terms of PDV. In December 2013 (as shown in Figure 7a), the aim was to determine which product closely aligns with GRACE values. To achieve this, the variance for the PDV differences between GRACE-CSR and the other products was calculated. The resulting variances were as follows: Swarm ASU (4.47 × 10
In geodesy, gravity anomalies are used to define the figure of the Earth, notably the geoid (the equipotential surface of the Earth's gravity field that corresponds most closely to the mean sea level). They are used in geodesy in three main areas: (
(
where
Following the validation process, involving comparing the effectiveness of the LRA gap-filling approach with both the GRACE data and other gap-filling models during the GRACE era, the LRA gap-filling model has been employed to bridge the data gap that occurred between GRACE and GRACE-FO from July 2017 to May 2018.
In this phase of the study, a thorough and extensive comparison was conducted, encompassing the LRA-derived SHCs alongside datasets from IGG, QF, and Swarm mission data (ASU and COST-G) and SSA for data filling purposes. Figure 8 illustrates a time-series comparison that involves the examination of random SHCs, including C
Despite some higher volatility observed in the Swarm SHCs' time-series compared to other datasets; Swarm generally demonstrates a noteworthy degree of consistency on average. It is noteworthy that a higher level of consistency is observed between SSA, QF, and IGG with LRA SHCs when compared to the Swarm data. This could potentially be attributed to the SSA and IGG data having a substantial derivation from GRACE/GRACE-FO sources.
The correlation coefficient values consistently exceed 0.89 for all coefficients between SSA and LRA, indicating a maximum level of compatibility since both datasets are extracted from GRACE. Conversely, in the case of IGG and QF, the correlation coefficient consistently exceeds 0.86 for all SHCs except C
In this analysis, we estimate the RMS values for several gravity field models during the GRACE/GRACE-FO gap from July 2017 to May 2018 (Figure 10). The following models are considered: LRA, SSA, IGG, QF, and ASU. LRA exhibits the smallest RMS values across all degrees, particularly in the lowest degrees (up to degree-10). SSA shows a more dispersed pattern of RMS values. IGG and QF exhibit higher RMS values compared to LRA. ASU demonstrates a spread similar to SSA but denser. The RMS values for ASU are generally higher than those for LRA.
The LRA gap-filling model is versatile and can also be applied to the estimation of Equivalent Water Height (EWH) as a mascon data type. To evaluate the performance of the gap-filling approach, assessments were conducted in eight well-distributed geographical regions, as illustrated in Figure 11. These eight basins span across all continents.
To validate the effectiveness of the method in the context of EWH estimation, a virtual data gap was simulated during the GRACE era, specifically in January 2016. During this simulation, the LRA-derived EWH was employed and compared with GRACE-EWH obtained from mascon data. The results, depicted in Figure 11, showcase the original GRACE data alongside the EWH values generated using the LRA gap-filling approach, along with the differences between them. It is notable that while there may be slightly larger discrepancies in the Amazon basin compared to other regions, the differences in a global context tend to approach minimum values.
Additionally, Table 3 presents the standard deviation values for all eight basins, facilitating a comparative analysis. These standard deviation values demonstrate minimal differences between the results obtained using GRACE data and the LRA gap-filling approach for all basins.
Furthermore, as part of the analysis, an artificial data gap was deliberately introduced into the GRACE dataset, specifically during the year 2009. We analyzed the TWS time-series for eight river basins and computed the root mean square (RMS) difference between GRACE and the LRA (Figure 12). The RMS values, which reflect variability, remained small across all basins relative to the TWS range. Specifically, the Amazon basin exhibits significant seasonal fluctuations, with an RMS of 3.69 mm, indicating high variability. The Mississippi basin shows fluctuations that are less pronounced than the Amazon, with an RMS of 3.13 mm. In the Murray Darling, Orange, and Yangtze basins, we observe moderate fluctuations, with RMS values below 2 mm. The Volga and Nile basins exhibit significant fluctuations over time, with RMS values of 2.78 mm and 1.22 mm, respectively. Finally, the Yenisei basin shows present but less pronounced fluctuations compared to the Amazon or Volga basin, with an RMS of 2.24 mm.
The LRA gap-filling approach was then utilized to estimate the EWH for each of the 12 months within that year. Subsequently, the mean (M), standard deviation (σ), and variance for the EWH values during this period were calculated. Remarkably, when comparing the mean EWH values obtained from both the GRACE dataset and the LRA approach for all months in 2009 (Table 4), a high degree of similarity was observed. This consistency also extended to the standard deviation and variance values, which exhibited matching characteristics. This convergence of numerical results strongly indicates that the LRA approach is a reliable method for filling gaps in GRACE data during the GRACE era.
Then, the weighted mean
(
(
where w is the variance (1/σ
(
The weighted mean provides a way to calculate the average when different data points have varying levels of importance or significance, while the standard deviation represents the measure of the variability or spread of the weighted mean [[
The standard deviation of the weighted mean offers insights into the expected variation in the weighted mean across diverse datasets. The convergence of these values between GRACE and the LRA approach underscores the convenience and suitability of this approach as a replacement for GRACE during data gaps.
To assess the model, various error metrics were calculated here to quantify the differences between the LRA gap-filling model and the GRACE observations. The estimated error metrics include the Mean Percentage Error (MPE) and R-squared (coefficient of determination), which measures the proportion of variance in the observed data that is explained by the model's predictions. The Mean Percentage Error (MPE) is used to calculate the percentage difference between each observed value and its corresponding predicted value, while R-squared measures the proportion of variance in the observed values that can be explained by the predicted values. It ranges from 0 to 1, where higher values indicate a better fit. The Mean Percentage Error (MPE) can be estimated as follows:
(
Choosing March 2016 as a random new virtual gap, the EWH observations were estimated using the LRA approach during this month and compared with GRACE-EWH observations. Figure 13 illustrates the MPE globally, indicating that the MPE is close to zero except for small dots in the Atlantic and Indian oceans.
In estimating the Mean Percentage Error (MPE), the estimation can be made using data from a single month. However, for estimating the coefficient of determination (R
Figure 14 visually represents the R
The consistently high levels of agreement observed between GRACE and the LRA gap-filling approach provide us with confidence in the reliability of this method for various applications involving GRACE data.
After successfully estimating the Equivalent Water Height using the LRA gap-filling approach to represent TWS, the method was applied to eight geographically diverse basins worldwide. The aim was to estimate TWS trends and observe TWS fluctuations over the period from 2002 to 2022 (Figure 15; Table 6). It is worth noting that only 14% of the months had no available observations, suggesting that the impact of data gap filling on the analysis is expected to be minimal.
Figure 15 illustrates the seasonal time-series of TWS for these basins, along with the spatial trend derived from the mascon data. Table 6 presents the TWS trends with and without the inclusion of LRA gap-filling data.
The time-series include both the original TWS values based on the original GRACE grids and the estimated values obtained through this new gap-filling method. The orange rectangle in the figure highlights the period of the GRACE/GRACE-FO data gap. The estimated values align closely with the existing data, reinforcing the confidence in the effectiveness of this gap-filling method between GRACE and GRACE-FO. However, differences in trends were observed, which could be as large as 0.22 mm/year in certain areas (Figure 15i; Table 6). Importantly, these variations in trends are not a result of extreme values introduced by the method but are primarily attributable to the uneven sampling of the original time-series.
By comparing trends before and after gap filling, differences of several millimeters per year could be noted. For example, in the Amazon basin, the TWS trend is estimated at 2.17 ± 0.15 mm/year using LRA, whereas it is 2.21 ± 0.16 mm/year without it. Across all basins, positive trends are observed, except for the Volga and Yenisei basins, where negative trends are attributed to ice melting in high-latitude basins in the northern hemisphere.
Nonetheless, it is important to highlight that the LRA gap-filling approach has proven to be highly effective in bridging the data gap between GRACE and GRACE-FO. Considering its capability in filling gaps, it is natural to question whether this approach could be adapted for predictive purposes, whether looking back retrospectively or forward prospectively. From a technical standpoint, the process involves introducing data gaps at the start or end of a sequence, mimicking prediction scenarios. However, it is essential to acknowledge the substantial uncertainty linked to such purely mathematical extrapolation. Given these uncertainties, this study refrains from pursuing this application at its current stage.
In this study, a novel strategy based on the LRA gap-filling method is proposed to address missing data in the GRACE missions. The LRA gap-filling method is applied to fill gaps within the GRACE missions and the 11-month gap between the missions, along with a two-month gap in GRACE-FO. The LRA gap-filling method is implemented in two types of GRACE data, SHCs and mascon. In the SHC data type, five comparisons were conducted, including (
Graph: Figure 1 GRACE observation months. The blue cells indicate the months that correspond to the existing GRACE observations, while the green cells indicate the months that correspond to the existing GRACE-FO observations. The red cells represent the months for which both GRACE and GRACE-FO observations are missing. Finally, the yellow cells show the gap that exists between the GRACE and GRACE-FO observations.
Graph: Figure 2 Eight major basins, including the Nile, Orange, Mississippi, Amazon, Volga, Yenisei, Yangtze, and Murray Darling.
Graph: Figure 3 The SHCs of GRACE-CSR and the LRA model for the period from January 2009 to December 2009 are presented in this figure. The figure is divided into 12 subfigures, labeled from (a–l), with each subfigure representing a specific month. Within each subfigure, four figures are shown in a clockwise arrangement. The upper-left figure displays the original GRACE-CSR SHCs, followed by the LRA SHCs in the virtual gap. The next figure shows the absolute relative error, and the final figure depicts the differences between the two sets of coefficients.
Graph: Figure 4 The RMS between GRACE and LRA, IGG, and QF during an artificial gap in 2009.
Graph: Figure 5 A time-series comparison is conducted among four types of data: GRACE-CSR, QF, IGG, and LRA. The figure consists of six subfigures, each focusing on a specific SHC comparison. Subfigure (a) compares C3,1, subfigure (b) C4,0, subfigure (c) C6,5, subfigure (d) C41,20, subfigure (e) S3,1, and subfigure (f) compares S16,6. The correlation coefficient values between GRACE-CSR, QF, and IGG with LRA SHCs are provided above each corresponding subfigure.
Graph: Figure 6 Correlation coefficients between GRACE and geodetic data sources (LRA, IGG, and QF) during an artificial gap in 2009.
Graph: Figure 7 Potential degree variance (PDV). (a) PDV in December 2013 using GRACE CSR, Swarm ASU and Swarm COST-G, IGG, LRA, and QF, (b) PDV in July 2014 using SSA, Swarm ASU and Swarm COST-G, IGG, LRA, and QF.
Graph: Figure 8 Gravity anomaly in December 2013. The gravity anomalies for GRACE-CSR- and LRA-derived SHCs are shown in (a,b), respectively, while the difference between them is shown in (c). In addition, the RMS is shown in (d).
Graph: Figure 9 A time-series comparison is conducted among four types of data: SSA, QF, IGG, Swarm (ASU and COST-G), and LRA. The figure consists of four subfigures, each focusing on a specific SHC comparison. Subfigure (a) compares C3,1, subfigure (b) C5,2, subfigure (c) C6,3, and subfigure (d) S3,1. The correlation coefficient values between SSA, QF, IGG, Swarm ASU, and Swarm-COST-G with LRA SHCs are provided above each corresponding subfigure.
Graph: Figure 10 The RMS for 5 SHC products, SSA, LRA, IGG, QF, and ASU, as triangles during the GRACE/GRACE-FO gap from July 2017 to May 2018.
Graph: Figure 11 Predictions and observations. (a) The GRACE-derived TWS in January 2016, (b) the predicted TWS in the same month using LRA, (c) the difference between both of them.
Graph: Figure 12 Time-series of total water storage (TWS) for each river basin, including the root mean square (RMS). The green rectangle denotes an artificial gap. (a) Amazon, (b) Mississippi, (c) Murray-Darling, (d) Nile, (e) Orange, (f) Volga, (g) Yangtze, and (h) Yenisei River basins.
Graph: Figure 13 The Mean Percentage Error (MPE) in the estimation of EWH in March 2016, comparing results obtained from GRACE CSR mascon data with those from the LRA gap-filling approach.
Graph: Figure 14 The coefficient of determination (R2) in the estimation of EWH during 2016, comparing results obtained from GRACE CSR mascon data with those from the LRA gap-filling approach.
Graph: Figure 15 TWS time-series for GRACE from April 2002 to February 2022. The figure is divided into nine images, with eight TWS time-series of the Amazon, Mississippi, Murray Darling, Nile, Orange, Volga, Yangtze, and Yenisei River basins (a–h). The GRACE TWS is represented by the blue line, while the orange rectangle represents the GRACE/GRACE-FO 11-month gap. Subfigure (i) represents the spatial TWS trend.
Table 1 The river basins include their area and their locations.
Basin Approximate Area (km2) Location Nile basin 3,038,100 East and North Africa Orange basin 850,000 Southern Africa Mississippi basin 2,980,000 North America Amazon basin 5,500,000 South America Volga basin 1,360,000 Europe Yenisei basin 2,580,000 Asia Yangtze basin 1,800,000 Asia Murray Darling basin 1,061,469 Australia
Table 2 Correlation coefficients between GRACE and LRA, IGG, and QF.
Source Correlation Coefficients LRA 0.8258 IGG 0.6763 QF 0.5428
Table 3 The TWS standard deviation for the 8 basins in January 2016 using GRACE, LRA, and the subtraction results of GRACE-LRA.
Basin Terrestrial Water Storage (mm) GRACE LRA GRACE-LRA Amazon 5.53 6.48 5.14 Mississippi 1.18 1.53 0.78 Murry-Darling 0.42 0.64 0.45 Nile 1.82 1.34 0.64 Orange 0.41 0.54 0.58 Volga 1.69 1.05 0.96 Yangtze 1.62 0.97 1.05 Yenisei 1.31 1.08 0.65
Table 4 The global mean, standard deviation, and variance values for EWH computed for each month throughout the year 2009. The values are presented for both the GRACE CSR mascon data and the EWH derived through the LRA gap-filling approach.
Month ) Standard Deviation (σ) Variance (1/σ2) GRACE LRA GRACE LRA GRACE LRA January −0.653 −0.701 6.415 6.869 0.0243 0.0212 February −0.447 −0.526 6.817 7.071 0.0215 0.0200 March −0.580 −0.483 7.618 7.920 0.0172 0.0159 April −0.396 −0.348 7.819 7.593 0.0164 0.0173 May 0.018 0.164 8.098 8.117 0.0152 0.0152 June 0.315 0.316 7.819 8.070 0.0164 0.0154 July 0.126 −0.089 7.902 8.042 0.0160 0.0155 August −0.134 −0.184 9.149 9.281 0.0119 0.0116 September −0.290 −0.217 9.698 9.545 0.0106 0.0110 October −0.225 −0.506 9.666 9.441 0.0107 0.0112 November −0.556 −0.564 9.542 9.158 0.0110 0.0119 December −0.707 −0.734 8.949 9.218 0.0125 0.0118
Table 5 The global weighted mean and standard deviation of the weighted mean during 2009 comparing results obtained from GRACE CSR mascon data with those from the LRA gap-filling approach.
Source Weighted Standard Deviation of the Weighted Mean GRACE/GRACE-FO −0.308 1.047 LRA −0.329 1.149
Table 6 TWS trend in 8 basins with and without using the LRA gap-filling approach.
Basin TWS Trend (mm/year) Using LRA Without LRA Amazon 2.17 ±0.15 2.21 ±0.16 Mississippi 3.43 ± 0.22 3.65 ± 0.21 Murry Darling 1.74 ± 0.13 1.75 ± 0.13 Nile 4.94 ± 0.25 5.06 ± 0.23 Orange 0.94 ± 0.36 0.94 ± 0.37 Volga −4.47 ± 0.30 −4.53 ± 0.28 Yangtze 3.49 ± 0.32 3.52 ± 0.30 Yenisei −0.56 ± 0.05 −0.53 ± 0.05
Conceptualization, H.A.M. and J.J.; methodology, H.A.M.; software, H.A.M.; validation, H.A.M., J.J. and W.S.; writing—original draft preparation, H.A.M.; writing—review and editing, J.J. and W.S.; visualization, W.S.; supervision, W.S. All authors have read and agreed to the published version of the manuscript.
The data presented in this study are available on request from the corresponding author and/or on the corresponding websites in the Acknowledgment section. CSR RL06 GRACE/GRACE-FO Mascon Solutions can be downloaded from https://www..csr.utexas.edu/grace/RL06%5fmascons.html (accessed on 15 November 2023), GRACE CSR SHCs can be found at https://www..csr.utexas.edu/grace/RL061LRI.html/ (accessed on 16 November 2023), and GRACE-FO CSR SHCs data can be downloaded from "ftp:/isdcftp.gfz-potsdam.de/grace-fo/Level-2/CSR/RL06/" (accessed on 16 November 2023). The Swarm ASU and COST-G data are available for download at the following links, respectively, "
The authors declare no conflicts of interest.
The authors extend their gratitude to the Supercomputing Center of Wuhan University for generously providing access to their supercomputing system, which was instrumental in conducting the numerical calculations for this research. Special thanks are also due to Löcher and Kusche for granting access to the IGG data, as well as to Shuang Yi and Nico Sneeuw for sharing their valuable SSA data. Furthermore, the authors express their appreciation to Matthias Weigelt for providing access to the QF data utilized in this study.
By Hussein A. Mohasseb; Wenbin Shen and Jiashuang Jiao
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