## 3 Methodology  
The inverse Vening Meinesz method is a commonly used method for gravity anomalies inversion using vertical deflections. Therefore, prior to calculating the gravity anomalies, the gridded vertical deflection components must first be derived from the simulated spaceborne GNSS-R SSH altimetry results in Section 2. Additional filtering was applied to the simulated data, removing points with errors greater than 3 times the value of the altimetry accuracy \(\sigma\). Since the gravity anomaly inversion relies on the geoid height \(N\), where \(N=MSS-DTU\), the MDT_CNES_CLS_18_global model obtained from AVISO was subtracted from the simulated SSH to eliminate the influence of mean dynamic topography (MDT) and obtain the simulated geoid height. The gridded geoid surface is then obtained by gridding the geoid heights of all simulated altimetry points in a month. Finally, the vertical deflection on the grid (_i,j_) can be calculated using Equations (2) and (3) for the north-south \(\xi\)(_i,j_) and east-west directions \(\eta\)(_i,j_), respectively. This algorithm not only improves efficiency but has also been verified for accuracy and applicability [63,64]. \[\xi(i,j)=-\frac{1}{R}\frac{(N_{i+1}-N_{i})}{(\varphi_{i+1}-\varphi_{i})} \tag{2}\]  
\[\eta(i,j)=-\frac{1}{Rcos\varphi_{i}}\frac{\left(N_{j+1}-N_{j}\right)}{\left( \lambda_{j+1}-\lambda_{j}\right)} \tag{3}\]  
where \(R\) is the Earth's mean radius, \(N\) is the mean geoid height at the grid point, and \(\varphi\) and \(\lambda\) represent the latitude and longitude of the grid point, respectively. In 1997, [PERSON] derived the Inverse Vening Meinesz formula, which converts vertical deflections into gravity anomalies by utilizing the gradient of the H function, as Equations (4) and (5). \[\Delta\text{g}(p)=\frac{\gamma_{0}}{4\pi}\iint\limits_{\sigma}H^{\prime}( \xi_{q}\cos\alpha_{qp}+\eta_{q}\sin\alpha_{qp})d\sigma_{q} \tag{4}\]  
\[H^{\prime}=\frac{\text{d}H}{d\psi_{pq}}=-\frac{\cos\frac{\psi_{pq}}{2}}{2\sin^ {2}\frac{\psi_{pq}}{2}}+\frac{\cos\frac{\psi_{pq}}{2}\left(3+2\sin\frac{\psi_ {pq}}{2}\right)}{2\sin\frac{\psi_{pq}}{2}\left(1+\sin\frac{\psi_{pq}}{2} \right)} \tag{5}\]  
Figure 5: Example of simulated SSH altimetry results for an Operational track: (**a**) distribution and selection of GNSS-R specular point trajectories, (**b**) SSH comparison between simulated and true MSS, and (**c**) altimetry errors along the selected track. where the average gravity of the Earth \(\gamma_{0}=\frac{GM}{R^{2}}\), \(H^{\prime}\) is the gradient of the H function, and \(p\) and \(q\) are two points on the sphere. \(\alpha_{qp}\) and \(\psi_{pq}\) represent the azimuth and spherical distance between the two points and can be calculated using Equations (6) and (7). \[\tan\alpha_{qp}=\frac{-\cos\phi_{p}\sin\Delta\lambda_{qp}}{-\sin(\phi_{q}-\phi_ {p})+2\sin\phi_{q}\cos\phi_{p}\sin^{2}\frac{\Delta\lambda_{qp}}{2}} \tag{6}\]  
\[\sin^{2}\biggl{(}\frac{\psi_{qp}}{2}\biggr{)}=\sin^{2}\biggl{(}\frac{\Delta \phi_{qp}}{2}\biggr{)}+\sin^{2}\biggl{(}\frac{\Delta\lambda_{qp}}{2}\biggr{)} \cos\phi_{q}\cos\phi_{p} \tag{7}\]  
where \(\Delta\phi_{qp}=\phi_{q}-\phi_{p}\), \(\Delta\lambda_{qp}=\lambda_{q}-\lambda_{p}\), \(\phi_{p}\) and \(\phi_{q}\) represent the latitude of points \(p\) and \(q\), while \(\lambda_{p}\) and \(\lambda_{q}\) represent the longitude of point \(p\) and \(q\).