# GRACE/GFO and Swarm Observation Analysis of the 2023-2024 Extreme Drought in the Amazon River Basin
## 3 Data and Methods
### Method  
#### 3.2.1 Improvement in TWSC Uncertainty  
The TWSC results based on different GRACE/GFO solutions exhibit some inconsistencies due to variations in data processing models and parameters [46]. These inconsistencies increased the uncertainty of our results, making it crucial to evaluate the uncertainties of the different GRACE/GFO solutions and improve the reliability of the GRACE/GFO TWSC data. Since there is no true value for TWSC, we applied the generalized three-cornered hat approach to assess the uncertainties of the five GRACE/GFO TWSC results. This approach is particularly useful, as it does not require any prior information [47]. Following this, the five GRACE/GFO TWSC results were integrated using a least-squares method weighted  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Variable Name** & **Data Source** & **Time Span** & **Spatial Resolution** & **Temporal** & **Data Source** \\ \hline GRACE/GFO & CSR & & 1\({}^{\circ}\times 1^{\circ}\) & & [[https://icgem.giz-potsdam.de/sl/temporal](https://icgem.giz-potsdam.de/sl/temporal)]([https://icgem.giz-potsdam.de/sl/temporal](https://icgem.giz-potsdam.de/sl/temporal)), accessed on 25 July 2025 \\ TWSC & CSR Mascon & 200301–202402 & 0.25\({}^{\circ}\)\(\times\) 0.25\({}^{\circ}\) & Monthly & [https://www2 csr.utexas.edu/grace/RL05_m](https://www2 csr.utexas.edu/grace/RL05_m) & \\  & JPL Mascon & & 0.5\({}^{\circ}\)\(\times\) 0.5\({}^{\circ}\) & & [[https://gazep1m.goz.gov/data/get-data/jpla-jpla0_mascons/](https://gazep1m.goz.gov/data/get-data/jpla-jpla0_mascons/)]([https://gazep1m.goz.gov/data/get-data/jpla-jpla0_mascons/](https://gazep1m.goz.gov/data/get-data/jpla-jpla0_mascons/)), accessed on 25 July 2025 \\ \hline PRE & GPM & 200301–202402 & 0.1\({}^{\circ}\)\(\times\) 0.1\({}^{\circ}\) & Monthly & ERGM\({}_  
according to the uncertainty estimates. The specific methodological principles and steps were as follows [48]. 1. Uncertainty assessment of TWSC  
Suppose there are \(N\) different observation series. If any one set of observation series is selected as the reference series, the difference series between the remaining observation series and the reference series can be expressed in matrix form as:  
\[Y= \begin{bmatrix}y_{11}&y_{12}&\dots&y_{1(N-1)}\\ y_{21}&y_{22}&\dots&y_{2(N-1)}\\ \vdots&\vdots&\vdots&\vdots\\ y_{M1}&y_{M2}&\dots&y_{M(N-1)}\end{bmatrix}  \tag{1}\]  
where \(y_{i,j}(i=1,2,\dots,M;j=1,2,\dots,N-1)\) indicates the difference between the elements of the reference series and the corresponding elements of the observation series and \(M\) indicates the number of elements in the observation series. The covariance matrix corresponding to the difference matrix is:  
\[S= \begin{bmatrix}s_{11}&s_{12}&\dots&s_{1(N-1)}\\ s_{21}&s_{22}&\dots&s_{2(N-1)}\\ \vdots&\vdots&\vdots&\vdots\\ S_{(N-1)1}&s_{(N-1)2}&\dots&s_{(N-1)(N-1)}\end{bmatrix}  \tag{2}\]  
where \(s_{i,k}\) indicates the covariance or variance in the corresponding difference elements. Introducing the \(N\times N\) noise covariance matrix R, its relationship with \(S\) is:  
\[S=J\times R\times J^{T} \tag{3}\]  
where the matrix \(J\) and \(R\) is:  
\[J_{N-1,N}= \begin{bmatrix}1&0&\dots&0&-1\\ 0&1&\dots&0&-1\\ \vdots&\vdots&\ddots&\vdots&\dots\\ 0&0&0&\dots&-1\end{bmatrix}  \tag{4}\]  
\[J_{N-1,N}= \begin{bmatrix}1&0&\dots&0&-1\\ 0&1&\dots&0&-1\\ \vdots&\vdots&\ddots&\vdots&\dots\\ 0&0&0&\dots&-1\end{bmatrix}  \tag{5}\]  
It can be obtained from Equation (3) that the following relationship holds:  
\[r_{ij}=s_{ij}-r_{NN}+r_{iN}+r_{jN} \tag{6}\]  
Since there are \(N\times(N+1)/2\) unknown parameters, but only \(N\times(N-1)/2\) equations, Formula (5) cannot be solved. Therefore, the remaining \(N\) free parameters need to be obtained with a reasonable approach to derive a unique solution. The free parameters were estimated by minimizing the \"global correlation\" of all observation series while ensuring the positive definiteness of \(R\).