# Influence of Sea Level Anomaly on Underwater Gravity Gradient Measurements
## 4 Simulation Calculation of Gravity Gradient  
Considering the large amplitudes of the annual mean and annual range in the Kuroshio extension area in Figures 2 and 3, the right rectangular prism method is used to calculate the disturbing gravity gradient from sea level anomalies in the range of \(5^{\circ}\times 5^{\circ}\) (34\({}^{\circ}\) N-39\({}^{\circ}\) N, 152\({}^{\circ}\) E-157\({}^{\circ}\) E) in the Kuroshio extension area with large sea level anomalies. Moreover, the data of the day with the highest sea level anomaly in the region (8 October 2020) is selected from the 366-day data for subsequent calculation (Figure 4 and Table 1). directions (Figure 5). The projection of the computation point on the mean sea level is set as the origin, which will change the horizontal coordinates of the computation point P to 0, subsequently simplifying the calculation. Ignoring the influence of the centrifugal force generated by the earth's rotation, the sea level anomaly data are divided into rectangular prisms based on the longitude and latitude grid, whose length and width are determined as the average length and width of all grids and the upper and lower surfaces of the prism are determined as the sea level anomaly surface and mean sea level, respectively. Then, the horizontal position of the computation point is at the center of each grid (Figure 6). To simplify the formula, using \(x_{1}\), \(x_{2}\), and \(x_{3}\) to represent \(x\), \(y\), and \(z\), respectively, we obtain the gravity gradient generated by the \(i\)th prism at the computation point P:  
\[\Gamma_{jk}^{i}=G\int_{0}^{h_{i}}\int_{0}^{h_{j}}\int_{0_{i}-\frac{Ax_{2}}{2}} ^{h_{i}+\frac{Ax_{1}}{2}}\int_{a_{i}-\frac{Ax_{1}}{2}}^{a_{i}+\frac{Ax_{1}}{2 }}\frac{\partial^{2}}{\partial x_{j}\partial x_{k}}\frac{1}{r}\rho_{i}dx_{1}^ {\prime}dx_{2}^{\prime}dx_{3}^{\prime} \tag{1}\]  
Here, \(G\) denotes [PERSON]'s gravitation constant, which is taken as 6.67 \(\times\) 10\({}^{-11}\) m\({}^{3}\).kg\({}^{-1}\)-\(\,\)s\({}^{-2}\); \(\rho_{i}\) is the seawater density of the \(i\)th prism, which is assumed to be a constant value of 1025 kg\(\cdot\)m\({}^{-3}\); \(h_{i}\) is the sea level anomaly value of the \(i\)th prism; and (a\({}_{i}\), b\({}_{i}\)) is the horizontal coordinate of the center point of the \(i\)th prism. Additionally, \(\Delta x_{1}\) and \(\Delta x_{2}\) are the average length and width, respectively, of each grid in the study area, which are calculated to be 22,371.490 and 27,742.137 m. Furthermore, \(r=\sqrt{\left(x_{1}-x_{1}^{\prime}\right)^{2}+\left(x_{2}-x_{2}^{\prime}\right) ^{2}+\left(x_{3}-x_{3}^{\prime}\right)^{2}}\) is the distance between the computation point P (\(x_{1}\), \(x_{2}\), \(x_{3}\)) and element point P\({}^{\prime}\) (\(x_{1}^{\prime}\), \(x_{2}^{\prime}\), \(x_{3}^{\prime}\)), \(j\) and \(k\) are the subscripts of \(x\), which can be taken as 1, 2, and 3, representing the \(x\), \(y\), and \(z\) directions, respectively. Substitute \(x_{1}=0\), \(x_{2}=0\), and \(x_{3}=Z\) into Equation (1), place \(\rho\) before the integral sign, and replace \(x_{1}^{\prime}\), \(x_{2}^{\prime}\) with \(u\), \(v\) to represent the horizontal coordinates of the element point. Next, integrate Equation (1) in the vertical direction to obtain six components:  
\[\begin{array}{c}\Gamma_{xx}^{i}=G\rho\int_{ \begin{subarray}{c}b_{i}-\frac{Ax_ {2}}{2}\\ b_{i}-\frac{Ax_{2}}{2}\end{subarray} }^{b_{i}+\frac{Ax_{1}}{2}}\int_{a_{i}- \frac{Ax_{1}}{2}}^{a_{i}+\frac{Ax_{1}}{2}}\frac{1}{\left(u^{2}+v^{2}\right)^{2 }}\bigg{(}\frac{Z\left(2u^{4}+u^{2}v^{2}-v^{4}+\left(u^{2}-v^{2}\right)Z^{2} \right)}{\left(u^{2}+v^{2}+Z^{2}\right)^{3/2}} \tag{2}\]  
Figure 5: Right rectangular prism. Figure 6: Block representation of the sea level anomalies (unit: m). \[