# Influence of Sea Level Anomaly on Underwater Gravity Gradient Measurements
## 4 Simulation Calculation of Gravity Gradient
\begin{array}{c}\Gamma^{i}_{yy}=G\rho\int_{\mathrm{b}_{i}-\frac{\Lambda \kappa_{2}}{2}}^{\mathrm{b}_{i}+\frac{\Lambda\kappa_{2}}{2}}\int_{\mathrm{a}_{i}- \frac{\Lambda\kappa_{1}}{2}}^{\mathrm{a}_{i}+\frac{\Lambda\kappa_{1}}{2}}\frac{ 1}{\left(u^{2}+v^{2}+Z^{2}\right)^{3/2}}\bigg{(}\frac{Z\left(2u^{4}+u^{2}v^{2} -u^{4}+\left(v^{2}-u^{2}\right)Z^{2}\right)}{\left(u^{2}+v^{2}+Z^{2}\right)^{3/ 2}}\\ +\frac{\left(h_{i}-Z\right)\left(2u^{4}+u^{2}v^{2}-u^{4}+\left(v^{2}-u^{2} \right)\left(h_{i}-Z\right)^{2}\right)}{\left(u^{2}+v^{2}+\left(h_{i}-Z\right) ^{2}\right)^{3/2}}\bigg{)}dudv\\ \Gamma^{i}_{zz}=-G\rho\int_{\mathrm{b}_{i}-\frac{\Lambda\kappa_{2}}{2}}^{ \mathrm{b}_{i}+\frac{\Lambda\kappa_{2}}{2}}\int_{\mathrm{a}_{i}-\frac{\Lambda \kappa_{1}}{2}}^{\mathrm{a}_{i}+\frac{\Lambda\kappa_{1}}{2}}\frac{Z}{\left(u^ {2}+v^{2}+Z^{2}\right)^{3/2}}+\frac{h_{i}-Z}{\left(u^{2}+v^{2}+\left(h_{i}-Z \right)^{2}\right)^{3/2}}dudv\\ \Gamma^{i}_{xy}=G\rho\int_{\mathrm{b}_{i}-\frac{\Lambda\kappa_{2}}{2}}^{ \mathrm{b}_{i}+\frac{\Lambda\kappa_{2}}{2}}\int_{\mathrm{a}_{i}-\frac{\Lambda \kappa_{1}}{2}}^{\mathrm{a}_{i}+\frac{\Lambda\kappa_{1}}{2}}\frac{1}{\left(u^ {2}+v^{2}\right)^{3/2}}\bigg{(}\frac{Z\left(uv\left(3u^{2}+3v^{2}+2Z^{2} \right)\right)}{\left(u^{2}+v^{2}+Z^{2}\right)^{3/2}}\\ +\frac{\left(h_{i}-Z\right)\left(u\left(3u^{2}+3v^{2}+2\left(h_{i}-Z\right) ^{2}\right)\right)}{\left(u^{2}+v^{2}+\left(h_{i}-Z\right)^{2}\right)^{3/2}} \bigg{)}dudv\\ \Gamma^{i}_{xz}=G\rho\int_{\mathrm{b}_{i}-\frac{\Lambda\kappa_{2}}{2}}^{ \mathrm{b}_{i}+\frac{\Lambda\kappa_{2}}{2}}\int_{\mathrm{a}_{i}-\frac{\Lambda \kappa_{1}}{2}}^{\mathrm{a}_{i}+\frac{\Lambda\kappa_{1}}{2}}u\left(\frac{1}{ \left(u^{2}+v^{2}+Z^{2}\right)^{3/2}}-\frac{1}{\left(u^{2}+v^{2}+\left(h_{i}-Z \right)^{2}\right)^{3/2}}\right)dudv\\ \Gamma^{i}_{yz}=G\rho\int_{\mathrm{b}_{i}-\frac{\Lambda\kappa_{2}}{2}}^{ \mathrm{b}_{i}+\frac{\Lambda\kappa_{2}}{2}}\int_{\mathrm{a}_{i}-\frac{\Lambda \kappa_{1}}{2}}^{\mathrm{a}_{i}+\frac{\Lambda\kappa_{1}}{2}}v\left(\frac{1}{ \left(u^{2}+v^{2}+Z^{2}\right)^{3/2}}-\frac{1}{\left(u^{2}+v^{2}+\left(h_{i} -Z\right)^{2}\right)^{3/2}}\right)dudv\\ \end{array}  \tag{5}\]  
\(Z\) denotes the depth from the mean sea level, which is a negative value. Selecting a certain depth and substituting it into the formula can yield the components of the gravity gradient. When the selected depth \(Z=0\), Equations (2)-(7) can be simplified:  
\[\Gamma^{i}_{xx}=G\rho\int_{\mathrm{b}_{i}-\frac{\Lambda\kappa_{2}}{2}}^{ \mathrm{b}_{i}+\frac{\Lambda\kappa_{2}}{2}}\int_{\mathrm{a}_{i}-\frac{\Lambda \kappa_{1}}{2}}^{\mathrm{a}_{i}+\frac{\Lambda\kappa_{1}}{2}}\frac{h_{i}\left(2u ^{4}+u^{2}v^{2}-v^{4}+{h  
balance between the calculation efficiency and error. [PERSON] developed a systematic algorithm method to determine the range of terrain data [18]. [PERSON] took the relative error of 1% as the truncation distance to calculate the gravity gradient effect [7]. We adopt [PERSON]'s method and select the locations corresponding to the highest value, the lowest value, and any other value in the study area as test points, and then, gradually increase the side length of the integration area to calculate the vertical disturbing gravity gradient at different depths below the mean sea level caused by the sea level anomalies (Figure 7 and Tables 2-4). Figure 7 shows that with increasing integration area, the vertical disturbing gravity gradient tends to be stable and the change decreases. When the gradient tends to stabilize, the side length of the integral region increases as the observation depth increases. To achieve a relative error of less than 1%, the side length of the three test points at 10 m below the mean sea level in the integration area needs to reach 24 m, which is 2.4 times the observation point depth. The side length of the three test points at 30 m below the mean sea level needs to reach 68 m, which is 2.3 times the observation point depth.