On the computation of gravitational effects for tesseroids with constant and linearly varying density

Abstract

The accurate computation of gravitational effects from topographic and atmospheric masses is one of the core issues in gravity field modeling. Using gravity forward modeling based on Newton’s integral, mass distributions are generally decomposed into regular mass bodies, which can be represented by rectangular prisms or polyhedral bodies in a rectangular coordinate system, or tesseroids in a spherical coordinate system. In this study, we prefer the latter representation because it can directly take the Earth’s curvature into account, which is particularly beneficial for regional and global applications. Since the volume integral cannot be solved analytically in the case of tesseroids, approximation solutions are applied. However, one well-recognized issue of these solutions is that the accuracy decreases as the computation point approaches the tesseroid. To overcome this problem, we develop a method that can precisely compute the gravitational potential $\left( V\right)$ and vector $\left( V_x, V_y, V_z\right)$ on the tesseroid surface. In addition to considering a constant density for the tesseroid, we further derive formulas for a linearly varying density. In the near zone (up to a spherical distance of 15 times the horizontal tesseroid dimension from the computation point), the gravitational effects of the tesseroids are computed by Gauss–Legendre quadrature using a two-dimensional adaptive subdivision technique to ensure high accuracy. The tesseroids outside this region are evaluated by means of expanding the integral kernel in a Taylor series up to the second order. The method is validated by synthetic tests of spherical shells with constant and linearly varying density, and the resulting approximation error is less than $10^{-4}\,\hbox {m}^2\,\hbox {s}^{-2}$ for V, $10^{-5}\,\hbox {mGal}$ for $V_x$, $10^{-7}\,\hbox {mGal}$ for $V_y$, and $10^{-4}\,\hbox {mGal}$ for $V_z$. Its practical applicability is then demonstrated through the computation of topographic reductions in the White Sands test area and of global atmospheric effects on the Earth’s surface using the US Standard Atmosphere 1976.

Appendices

Appendix A

1.1 A.1  Approximation of volume integrals by Gauss–Legendre quadrature

By moving the constant term $G\rho$ into the integral kernels in Eqs. (1a)–(1d), the volume integral can be expressed as

$$\begin{aligned} \int _{\lambda _1}^{\lambda _2}\int _{\varphi _1}^{\varphi _2}\int _{r_1}^{r_2}\, f\left( r',\,\varphi ',\,\lambda '\right) \, \mathrm {d}r' \mathrm {d}\varphi ' \mathrm {d}\lambda '. \end{aligned}$$

(12)

Using the GLQ decomposition (Stroud and Secrest 1966; Asgharzadeh et al. 2007; Li et al. 2011), the least-squares numerical solution of Eq. (12) is obtained as

$$\begin{aligned} A \sum _{i=1}^{N^{r}}\sum _{j=1}^{N^{\varphi }}\sum _{k=1}^{N^{\lambda }}\, {w_i^{r}}{w_j^{\varphi }}{w_k^{\lambda }}\,f\left( {\hat{r}}_i,\,{\hat{\varphi }}_j,\,{\hat{\lambda }}_k\right) , \end{aligned}$$

(13)

where

$$\begin{aligned} A = \frac{\left( r_2 - r_1\right) \left( \varphi _2 - \varphi _1\right) \left( \lambda _2 - \lambda _1\right) }{8}, \end{aligned}$$

(14)

${w_i^{r}}$, ${w_j^{\varphi }}$, and ${w_k^{\lambda }}$ are denoted by the GLQ weights corresponding to the GLQ nodes ${x_i^{r}}$, ${x_j^{\varphi }}$, and ${x_k^{\lambda }}$ in the interval $\left[ -1,1\right]$ (Wild-Pfeiffer 2008). In practical computation using Eq. (13), the GLQ nodes must be scaled to the integration domain $\left[ r_1,r_2\right] \times \left[ \varphi _1,\varphi _2\right] \times \left[ \lambda _1,\lambda _2\right]$ using

$$\begin{aligned} {\hat{r}}_i&= \frac{x^r_i\left( r_2 - r_1\right) + \left( r_2 + r_1\right) }{2}, \end{aligned}$$

(15a)

$$\begin{aligned} {\hat{\varphi }}_j&= \frac{x^\varphi _j\left( \varphi _2 - \varphi _1\right) + \left( \varphi _2 + \varphi _1\right) }{2}, \end{aligned}$$

(15b)

$$\begin{aligned} {\hat{\lambda }}_k&= \frac{x^\lambda _k\left( \lambda _2 - \lambda _1\right) + \left( \lambda _2 + \lambda _1\right) }{2}. \end{aligned}$$

(15c)

Equation (13) shows that the effect of a tesseroid at each computation point is approximated by a weighted sum of $N^r \times N^\varphi \times N^\lambda$ equivalent point mass effects, where each point mass is located at the source coordinate $({\hat{r}}_i,\,{\hat{\varphi }}_j,\,{\hat{\lambda }}_k)$ inside the tesseroid. If $N^r = N^\varphi = N^\lambda = 1$, this method is equivalent to the point mass method.

1.2 A.2  Analytical integration over radial direction and surface integrals by Gauss–Legendre quadrature

In this method, Newton’s integral in Eqs. (1a)–(1d) is decomposed into a one-dimensional integral over the radial direction $r'$ and a two-dimensional integral over $\varphi '$ and $\lambda '$, yielding

$$\begin{aligned} V\left( r,\varphi ,\lambda \right)&= G\rho \,\int _{\lambda _1}^{\lambda _2}\int _{\varphi _1}^{\varphi _2} \underbrace{\left[ \int _{r_1}^{r_2}\,\frac{1}{\ell } r'^2\,\mathrm {d}r'\right] }_ {H^{V}\left( \varphi ',\,\lambda '\right) } \cos \varphi ' \mathrm {d}\varphi ' \mathrm {d}\lambda ', \end{aligned}$$

(16a)

$$\begin{aligned} V_x\left( r,\varphi ,\lambda \right)&= G\rho \,\int _{\lambda _1}^{\lambda _2}\int _{\varphi _1}^{\varphi _2} \underbrace{\left[ \int _{r_1}^{r_2}\,\frac{r'C_\varphi }{\ell ^3} r'^2\,\mathrm {d}r'\right] }_ {H^{V_x}\left( \varphi ',\,\lambda '\right) }\nonumber \\&\cos \varphi ' \mathrm {d}\varphi ' \mathrm {d}\lambda ', \end{aligned}$$

(16b)

$$\begin{aligned} V_y\left( r,\varphi ,\lambda \right)&= G\rho \,\int _{\lambda _1}^{\lambda _2}\int _{\varphi _1}^{\varphi _2} \underbrace{\left[ \int _{r_1}^{r_2}\,\frac{r'\cos \varphi '\sin \delta {\lambda }}{\ell ^3} r'^2\,\mathrm {d}r'\right] }_ {H^{V_y}\left( \varphi ',\,\lambda '\right) } \times \nonumber \\&\cos \varphi ' \mathrm {d}\varphi ' \mathrm {d}\lambda ', \end{aligned}$$

(16c)

$$\begin{aligned} V_z\left( r,\varphi ,\lambda \right)&= G\rho \,\int _{\lambda _1}^{\lambda _2}\int _{\varphi _1}^{\varphi _2} \underbrace{\left[ \int _{r_1}^{r_2}\,\frac{r'\cos \psi - r}{\ell ^3} r'^2\,\mathrm {d}r'\right] }_ {H^{V_z}\left( \varphi ',\,\lambda '\right) } \nonumber \\&\times \cos \varphi ' \mathrm {d}\varphi ' \mathrm {d}\lambda ', \end{aligned}$$

(16d)

where $H^{V}$, $H^{V_x}$, $H^{V_y}$, and $H^{V_z}$ are the integral kernels for the surface integral, whose analytical formulas are available in Wild-Pfeiffer (2008).

By moving the term $G \rho \times \cos \varphi '$ into the integral kernels in Eqs. (16a)–(16d), the general expression can be written in the form of

$$\begin{aligned} \int _{\lambda _1}^{\lambda _2}\int _{\varphi _1}^{\varphi _2}\, g\left( \varphi ',\,\lambda '\right) \, \mathrm {d}\varphi ' \mathrm {d}\lambda ', \end{aligned}$$

(17)

which can be approximated by GLQ as

$$\begin{aligned} B \sum _{j=1}^{N^{\varphi }}\sum _{k=1}^{N^{\lambda }}\, {w_j^{\varphi }}{w_k^{\lambda }}\,g\left( {\hat{\varphi }}_j,\,{\hat{\lambda }}_k\right) , \end{aligned}$$

(18)

with

$$\begin{aligned} B = \frac{\left( \varphi _2 - \varphi _1\right) \left( \lambda _2 - \lambda _1\right) }{4}. \end{aligned}$$

(19)

1.3 A.3  Taylor series expansion

Another method to solve the elliptical integral is achieved by means of expanding the integral kernels of Eqs. (1a)–(1d) in a Taylor series and solving a subsequent volume integral. Similar to Eqs. (10a) and (10b), the gravitational potential and vector of a homogeneous tesseroid read

$$\begin{aligned} V\left( r,\varphi ,\lambda \right) =&\,G\rho \,\varDelta r \varDelta \varphi \varDelta \lambda \left[ K_{000}^{V} + \frac{1}{24}\left( K_{200}^{V}\varDelta r^2 \right. \right. \nonumber \\&\, + \left. \left. K_{020}^{V}\varDelta \varphi ^2 + K_{002}^{V}\varDelta \lambda ^2\right) + O\left( \varDelta ^4\right) \frac{}{}\right] , \end{aligned}$$

(20a)

$$\begin{aligned} V_\alpha \left( r,\varphi ,\lambda \right) =&\,G\rho \,\varDelta r \varDelta \varphi \varDelta \lambda \left[ K_{000}^{V_\alpha } + \frac{1}{24}\left( K_{200}^{V_\alpha }\varDelta r^2 \right. \right. \nonumber \\&\, + \left. \left. K_{020}^{V_\alpha }\varDelta \varphi ^2 + K_{002}^{V_\alpha }\varDelta \lambda ^2\right) + O\left( \varDelta ^4\right) \frac{}{}\right] . \end{aligned}$$

(20b)

The zeroth and second order coefficients $K^{V}_{ijk}$ and $K^{V_\alpha }_{ijk}$ are given in Heck and Seitz (2007) and Wild-Pfeiffer (2008). Notice that the zeroth-order approximation is equivalent to the point mass method.

1.4 A.4  Approximation of tesseroids by rectangular prisms

If a tesseroid is replaced by a rectangular prism in the spherical system, the gravitational effects are computed in the local edge system of the prism first, and then these effects are transformed to the local topocentric coordinate system of the computation point. Since the horizontal dimension of the used tesseroid is very small $\left( \sin \varDelta \varphi \approx \varDelta \varphi \right)$ and its vertical dimension is much smaller than the distance to the origin of the geocentric coordinate system $\left( \varDelta {r} \ll r_1\right)$ in this study, the conversion of a tesseroid into a rectangular prism is achieved by (Wild-Pfeiffer 2008)

$$\begin{aligned} \varDelta {x} = r_0 \varDelta \varphi , \quad \varDelta {y} = r_0 \cos \varphi _0 \varDelta \lambda , \quad \varDelta {z} = \varDelta {r}, \end{aligned}$$

(21)

where $r_0$, $\varphi _0$, $\lambda _0$ refer to Eq. (5d) and $\varDelta {r}$, $\varDelta \varphi$, $\varDelta \lambda$ refer to Eq. (11). The condition assuming that both mass bodies have the same mass is also taken into account.

The formulas of the gravitational effects of a rectangular prism in a rectangular coordinate system can be found in, e.g., Mader (1951), Forsberg (1984), Nagy et al. (2000, 2002), and Wild-Pfeiffer (2008). The transformation of the prism effect from the local edge system into the system of the computation point follows the formulas given in Heck and Seitz (2007) and Wild-Pfeiffer (2008).

1.5 A.5  Approximation of tesseroids by point masses

A tesseroid can also simply be replaced by a point mass of the same mass located at the mass center $\left( r_c,\varphi _c,\lambda _c\right)$ of the tesseroid. According to Anderson (1976), the mass center of a small-size homogeneous tesseroid can be approximated as

$$\begin{aligned} r_c = \frac{3}{4}\frac{r_2^4 - r_1^4}{r_2^3 - r_1^3}, \quad \varphi _c = \frac{\varphi _1 + \varphi _2}{2}, \quad \lambda _c = \frac{\lambda _1 + \lambda _2}{2}, \end{aligned}$$

(22)

and the mass of tesseroid is computed by

$$\begin{aligned} m_\text {tess} = m_\text {pm} = \frac{\rho }{3}\left( r_2^3 - r_1^3\right) \left( \sin \varphi _2 - \sin \varphi _1\right) \left( \lambda _2 - \lambda _1\right) . \end{aligned}$$

(23)

Replacing $\rho \,\int _{\lambda _1}^{\lambda _2}\int _{\varphi _1}^{\varphi _2}\int _{r_1}^{r_2}\kappa \,\mathrm {d}r' \mathrm {d}\varphi ' \mathrm {d}\lambda '$ by $m_\text {pm}$ and $\left( r',\varphi ',\lambda '\right)$ by $\left( r_c,\varphi _c,\lambda _c\right)$ in the integral kernels of Eqs. (1a)–(1d) yields the point mass formulas for the gravitational potential and vector:

$$\begin{aligned} V\left( r,\varphi ,\lambda \right)&= G\,m_\text {pm}\frac{1}{\ell _c}, \end{aligned}$$

(24a)

$$\begin{aligned} V_x\left( r,\varphi ,\lambda \right)&= G\,m_\text {pm} \frac{r_c\,{C_\varphi }_c}{\ell _c^3}, \end{aligned}$$

(24b)

$$\begin{aligned} V_y\left( r,\varphi ,\lambda \right)&= G\,m_\text {pm} \frac{r_c \cos {\varphi _c} \sin \delta {\lambda _c}}{\ell _c^3}, \end{aligned}$$

(24c)

$$\begin{aligned} V_z\left( r,\varphi ,\lambda \right)&= G\,m_\text {pm} \frac{r_c\cos \psi _c - r}{\ell _c^3}, \end{aligned}$$

(24d)

where

$$\begin{aligned}&\ell _c = \sqrt{r^2 + r_c^2 - 2rr_c\cos {\psi _c}}, \end{aligned}$$

(25a)

$$\begin{aligned}&\cos {\psi _c} = \sin {\varphi }\sin {{\varphi _c}} + \cos {\varphi }\cos {{\varphi _c}} \cos \delta {\lambda _c},\end{aligned}$$

(25b)

$$\begin{aligned}&{C_\varphi }_c = \cos \varphi \sin \varphi _c - \sin \varphi \cos \varphi _c \cos \delta {\lambda _c},\end{aligned}$$

(25c)

$$\begin{aligned}&\delta {\lambda _c} = \lambda _c - \lambda . \end{aligned}$$

(25d)

Appendix B

The analytical gravitational potential $V^\text {s}$ and attraction $V_z^\text {s}$ of a homogeneous spherical shell at an arbitrary point with a radial distance r above, below, on, or inside the shell mass can be computed by (Tsoulis 1999; Heck and Seitz 2007; Makhloof and Ilk 2008; Grombein et al. 2013)

$$\begin{aligned} V^\text {s} = \left\{ \begin{array}{lll} \frac{4 \pi G \rho }{3r} \left( R_2^3 - R_1^3\right) &{} \quad r \ge R_2\\ 2 \pi G \rho \left( R_2^2 - \frac{r^2}{3} - \frac{2 R_1^3}{3 r}\right) &{} \quad R_1< r < R_2\\ 2 \pi G \rho \left( R_2^2 - R_1^2\right) &{} \quad r \le R_1 \end{array}\right. , \end{aligned}$$

(26)

and

$$\begin{aligned} V_z^\text {s} = \left\{ \begin{array}{ll} -\frac{4 \pi G \rho }{3 r^2}\left( R_2^3 - R_1^3\right) &{} \quad r \ge R_2 \\ [1.0ex] -\frac{4 \pi G \rho }{3 r^2}\left( r^3 - R_1^3\right) &{} \quad R_1< r < R_2 \\ [1.0ex] 0 &{} \quad r \le R_1 \end{array}\right. . \end{aligned}$$

(27)

Due to the isotropy of the homogeneous spherical shell, $V_x^\text {s}$ and $V_y^\text {s}$ are zero at an arbitrary point.

Appendix C

1.1 C.1  Zeroth and second order coefficients ${\widetilde{K}}^V_{000}$, ${\widetilde{K}}^V_{200}$, ${\widetilde{K}}^V_{020}$, and ${\widetilde{K}}^V_{002}$ in Eq. (10a)

$$\begin{aligned} {\widetilde{K}}^V_{000}= & {} \frac{r_0^3 \,\cos \varphi _0}{\ell _0}, \end{aligned}$$

(28)

$$\begin{aligned} {\widetilde{K}}^V_{200}= & {} \,\frac{3 \,r_0^3 \,\cos \varphi _0 \,{\widetilde{B}}_0^2}{\ell _0^5} - \frac{6 \,r_0^2 \,\cos \varphi _0 \,{\widetilde{B}}_0 + r_0^3 \,\cos \varphi _0}{\ell _0^3}\nonumber \\&\, + \frac{6 \,r_0 \,\cos \varphi _0}{\ell _0},\end{aligned}$$

(29)

$$\begin{aligned} {\widetilde{K}}^V_{020}= & {} \,\frac{3 \,r_0^5 \,r^2 \,\cos \varphi _0 \,{\widetilde{A}}_0^2}{\ell _0^5}\nonumber \\&\,- \frac{r \,r_0^4\left( 2 \,\sin \varphi _0 \,{\widetilde{A}}_0 + \cos \varphi _0 \,\cos \psi _0\right) }{\ell _0^3}\nonumber \\&\, - \frac{r_0^3 \,\cos \varphi _0}{\ell _0}, \end{aligned}$$

(30)

$$\begin{aligned} {\widetilde{K}}^V_{002}= & {} \,\frac{3 \,r^2 \,r_0^5 \,\sin ^2\delta \lambda _0 \,\cos ^2\varphi \,\cos ^3\varphi _0}{\ell _0^5}\nonumber \\&\,- \frac{r \,r_0^4 \,\cos \delta \lambda _0 \,\cos \varphi \,\cos ^2\varphi _0}{\ell _0^3}, \end{aligned}$$

(31)

where $\ell _0$, $\cos \psi _0$, $\delta \lambda _0$, $r_0$, $\varphi _0$, and $\lambda _0$ are the same as in Eq. (5), and

$$\begin{aligned} {\widetilde{A}}_0&= \sin \varphi \,\cos \varphi _0 - \cos \varphi \,\sin \varphi _0 \,\cos \delta \lambda _0, \end{aligned}$$

(32a)

$$\begin{aligned} {\widetilde{B}}_0&= r_0 - r \,\cos \psi _0,\end{aligned}$$

(32b)

$$\begin{aligned} {\widetilde{C}}_0&= r_0 \,\cos \psi _0 - r,\end{aligned}$$

(32c)

$$\begin{aligned} {\widetilde{D}}_0&= \cos \varphi \,\sin \varphi _0 - \sin \varphi \,\cos \varphi _0 \,\cos \delta \lambda _0,\end{aligned}$$

(32d)

$$\begin{aligned} {\widetilde{E}}_0&= \cos \varphi \,\cos \varphi _0 + \sin \varphi \,\sin \varphi _0 \,\cos \delta \lambda _0. \end{aligned}$$

(32e)

1.2 C.2  Zeroth and second order coefficients ${\widetilde{K}}^{V_x}_{000}$, ${\widetilde{K}}^{V_x}_{200}$, ${\widetilde{K}}^{V_x}_{020}$, and ${\widetilde{K}}^{V_x}_{002}$ in Eq. (10b)

$$\begin{aligned} {\widetilde{K}}^{V_x}_{000}= & {} \frac{r_0^4 \,\cos \varphi _0 \,{\widetilde{D}}_0}{\ell _0^3}, \end{aligned}$$

(33)

$$\begin{aligned} {\widetilde{K}}^{V_x}_{200}= & {} \,\frac{3 \,r_0^2 \,\cos \varphi _0 \,{\widetilde{D}}_0}{\ell _0^3} \left[ 4 - \frac{r_0}{\ell _0^2} \left( 9 \,r_0 - 8 \,r \,\cos \psi _0\right) \right. \nonumber \\&\,+ \left. \frac{5 \,r_0^2 \,{\widetilde{B}}_0^2}{\ell _0^4}\right] , \end{aligned}$$

(34)

$$\begin{aligned} {\widetilde{K}}^{V_x}_{020}= & {} \,-\frac{r_0^4}{\ell _0^3} \left\{ \frac{3 \,r \,r_0}{\ell _0^2} \left[ \cos \varphi _0 \left( \cos \psi _0 \,{\widetilde{D}}_0 - 2 \,{\widetilde{A}}_0 \,{\widetilde{E}}_0\right) \right. \right. \nonumber \\&\,+ \left. 2 \,\sin \varphi _0 \,{\widetilde{D}}_0 \,{\widetilde{A}}_0\right] + 2 \left( \cos \varphi _0 \,{\widetilde{D}}_0 + \sin \varphi _0 \,{\widetilde{E}}_0\right) \nonumber \\&\,- \left. \frac{15 \,r^2 \,r_0^2 \,\cos \varphi _0 \,{\widetilde{D}}_0 \,{\widetilde{A}}_0^2}{\ell _0^4}\right\} , \end{aligned}$$

(35)

$$\begin{aligned} {\widetilde{K}}^{V_x}_{002}= & {} \,\frac{r_0^4 \,\cos ^2\varphi _0}{\ell _0^3} \left[ \sin \varphi \,\cos \delta \lambda _0\right. \nonumber \\&\,- \frac{3 \,r \,r_0 \,\cos \varphi }{\ell _0^2}\nonumber \\&\,\times \left( 2 \,\sin \varphi \,\cos \varphi _0 \,\sin ^2\delta \lambda _0 + \cos \delta \lambda _0 \,{\widetilde{D}}_0\right) \nonumber \\&\,+ \left. \frac{15 \,r^2 \,r_0^2 \,\cos ^2\varphi \,\cos \varphi _0 \,\sin ^2\delta \lambda _0 \,{\widetilde{D}}_0}{\ell _0^4}\right] . \end{aligned}$$

(36)

1.3 C.3  Zeroth and second order coefficients ${\widetilde{K}}^{V_y}_{000}$, ${\widetilde{K}}^{V_y}_{200}$, ${\widetilde{K}}^{V_y}_{020}$, and ${\widetilde{K}}^{V_y}_{002}$ in Eq. (10b)

$$\begin{aligned} {\widetilde{K}}^{V_y}_{000}= & {} \frac{r_0^4 \,\cos ^2\varphi _0 \,\sin \delta \lambda _0}{\ell _0^3}, \end{aligned}$$

(37)

$$\begin{aligned} {\widetilde{K}}^{V_y}_{200}= & {} \,\frac{3 \,r_0^2 \,\cos ^2\varphi _0 \,\sin \delta \lambda _0}{\ell _0^3} \left[ 4 - \frac{r_0}{\ell _0^2} \left( 9 \,r_0 - 8 \,r \,\cos \psi _0\right) \right. \nonumber \\&\,+ \left. \frac{5 \,r_0^2 \,{\widetilde{B}}_0^2}{\ell _0^4}\right] , \end{aligned}$$

(38)

$$\begin{aligned} {\widetilde{K}}^{V_y}_{020}= & {} \,-\frac{r_0^4}{\ell _0^3} \big [2 \,\cos 2\varphi _0 \,\sin \delta \lambda _0 \nonumber \\&\,- \frac{15 \,r^2 \,r_0^2 \,\cos ^2\varphi _0 \,\sin \delta \lambda _0 \,{\widetilde{A}}_0^2}{\ell _0^4}\nonumber \\&\,+ \frac{3 \,r \,r_0 \,\cos \varphi _0 \,\sin \delta \lambda _0}{\ell _0^2}\nonumber \\&\,\times \left. \left( 4 \,\sin \varphi _0 \,{\widetilde{A}}_0 + \cos \varphi _0 \,\cos \psi _0\right) \right] ,\end{aligned}$$

(39)

$$\begin{aligned} {\widetilde{K}}^{V_y}_{002}= & {} \,-\frac{r_0^4 \,\cos ^2\varphi _0 \,\sin \delta \lambda _0}{\ell _0^3}\nonumber \\&\,\times \left( 1 + \frac{9 \,r \,r_0 \,\cos \varphi \,\cos \varphi _0 \,\cos \delta \lambda _0}{\ell _0^2}\right. \nonumber \\&\,- \left. \frac{15 \,r^2 \,r^2_0 \,\cos ^2\varphi _0 \,\cos ^2\varphi \,\sin ^2\delta \lambda _0}{\ell _0^4}\right) . \end{aligned}$$

(40)

1.4 C.4  Zeroth and second order coefficients ${\widetilde{K}}^{V_z}_{000}$, ${\widetilde{K}}^{V_z}_{200}$, ${\widetilde{K}}^{V_z}_{020}$, and ${\widetilde{K}}^{V_z}_{002}$ in Eq. (10b)

$$\begin{aligned} {\widetilde{K}}^{V_z}_{000}= & {} \frac{r_0^3 \,\cos \varphi _0 \,{\widetilde{C}}_0}{\ell _0^3}, \end{aligned}$$

(41)

$$\begin{aligned} {\widetilde{K}}^{V_z}_{200}= & {} \,\frac{3 \,r_0 \,\cos \varphi _0}{\ell _0^3} \left[ \frac{r_0^2}{\ell _0^2} \left( \frac{5 \,{\widetilde{B}}_0^2 \,{\widetilde{C}}_0}{\ell _0^2} - 2 \,\cos \psi _0 \,{\widetilde{B}}_0 - {\widetilde{C}}_0\right) \right. \nonumber \\&\,- \left. 2 \,r_0 \left( \frac{3 \,{\widetilde{B}}_0 \,{\widetilde{C}}_0}{\ell _0^2} - \cos \psi _0\right) + 2 {\widetilde{C}}_0\right] , \end{aligned}$$

(42)

$$\begin{aligned} {\widetilde{K}}^{V_z}_{020}= & {} \,\frac{r_0^3}{\ell ^3_0} \left[ \frac{3 \,r_0^2 \,r \,\cos \varphi _0 \,{\widetilde{A}}_0^2}{\ell ^2_0} \left( 2 + \frac{5 \,{\widetilde{C}}_0 \,r}{\ell ^2_0}\right) \right. \nonumber \\&\,- r_0 \left( 2 \,\sin \varphi _0 \,{\widetilde{A}}_0 + \cos \varphi _0 \,\cos \psi _0\right) \left( 1 + \frac{3 \,{\widetilde{C}}_0 r}{\ell ^2_0}\right) \nonumber \\&\,- \left. \cos \varphi _0 \,{\widetilde{C}}_0 \right] , \end{aligned}$$

(43)

$$\begin{aligned} {\widetilde{K}}^{V_z}_{002}= & {} \,\frac{r_0^4 \,\cos \varphi \,\cos ^2\varphi _0}{\ell ^3_0}\nonumber \\&\,\times \left[ \frac{3 \,r_0 \,r \,\sin ^2\delta \lambda _0 \,\cos \varphi \,\cos \varphi _0}{\ell ^2_0} \left( 2 + \frac{5 \,{\widetilde{C}}_0 \,r}{\ell ^2_0}\right) \right. \nonumber \\&\,- \left. \cos \delta \lambda _0 \,\left( 1 + \frac{3 \,{\widetilde{C}}_0 \,r}{\ell ^2_0}\right) \right] . \end{aligned}$$

(44)

Appendix D

1.1 D.1  The gravitational potential $V^{*\text {c}}$ of a spherical cap with a linear density $d\rho \times r'$

Analogous to the derivations in Appendix A3 of Heck and Seitz (2007), the gravitational potential $V^{*\text {c}}$ of a spherical cap with a linear density $d\rho \times r'$ at the computation point located on the axis passing through the north pole is given by Newton’s integral

$$\begin{aligned} V^{*\text {c}}\left( r; \, r_1, r_2, \theta _c\right) = G d\rho \int _0^{2\pi } \int _0^{\theta _c} \int _{r_1}^{r_2} \frac{r'^3 \sin \theta ' \mathrm {d}r' \mathrm {d}\theta ' \mathrm {d}\alpha '}{\ell '}, \end{aligned}$$

(45)

where $\ell ' = \sqrt{r^2 + r'^2 - 2rr'\cos \theta '}$ is the Euclidean distance between the computation point $P\left( r, \theta = 0^{\circ }\right)$ and the running integration point $Q\left( r', \theta ', \alpha '\right)$. Integration with respect to the azimuth $\alpha '$ for the rotational symmetric cap and the spherical distance $\theta '$ results in

$$\begin{aligned} \begin{aligned} V^{*\text {c}}\left( r; \, r_1, r_2, \theta _c\right) = \,&\frac{2 \pi G d\rho }{r} \int _{r_1}^{r_2} \left\{ \sqrt{r^2 + r'^2 - 2rr'\cos {\theta _c}}\right. \\ \,&- \left. \left| r - r'\right| \right\} r'^2\mathrm {d}r'. \end{aligned} \end{aligned}$$

(46)

Finally, after the integration with respect to the radial distance (Gradshteyn and Ryzhik 2007), the formula for $V^{*\text {c}}$ is given by

$$\begin{aligned} \begin{aligned}&V^{*\text {c}}\left( r; \, r_1, r_2, \theta _c\right) = 2 \pi G d\rho \\&\,\times \left\{ \frac{3 r' + 5 r \cos \theta _c}{12 r} {\ell '_c}^3 + \frac{5 r \cos ^2\theta _c - r}{8} \left( r' - r \cos \theta _c\right) \ell '_c \right. \\&\,+ \left. \left. \frac{5 r \cos ^2\theta _c - r}{8} r^2 \sin ^2\theta _c \,\mathrm {ln}\left( \ell '_c + r' - r \cos \theta _c\right) \right\} \right| ^{r' = r_2}_{r' = r_1}\\&\,+ \left. 2 \pi G d\rho \left( \frac{r'^4}{4 r} - \frac{r'^3}{3}\right) \right| ^{r' = r_2}_{r' = r_1} \times \left\{ \begin{array}{ll} +1 &{} \quad r \ge r_2\\ -1 &{} \quad r \le r_1 \end{array}\right. , \end{aligned} \end{aligned}$$

(47)

where $\ell '_c = \sqrt{r^2 + r'^2 - 2rr'\cos \theta _c}$. If the extension of the cap tends to $\theta _c = \pi$, the gravitational potential of a spherical shell with a linear density $d\rho \times r'$ simplifies to

$$\begin{aligned} V^{*\text {s}}\left( r; \, r_1, r_2, \pi \right) = \left\{ \begin{array}{ll} \frac{\pi G d\rho }{r} \left( r_2^4 - r_1^4\right) &{} \quad r \ge r_2\\ \frac{4 \pi G d\rho }{3} \left( r_2^3 - r_1^3\right) &{} \quad r \le r_1 \end{array}\right. . \end{aligned}$$

(48)

When the computation point is inside the shell mass (i.e., $r_1< r < r_2$), the analytical expression for $V^{*\text {s}}$ is given by $\pi G d\rho \left( \frac{4 r_2^3}{3} - \frac{r^3}{3} - \frac{r_1^4}{r}\right)$.

1.2 D.2  The gravitational vector ${V_x}^{*\text {c}}$, ${V_y}^{*\text {c}}$, and ${V_z}^{*\text {c}}$ of a spherical cap with a linear density $d\rho \times r'$

Since the density of the spherical cap only varies linearly along the radial direction, the horizontal components of gravitational vector ${V_x}^{*\text {c}}$ and ${V_y}^{*\text {c}}$ are zero at the computation point P. In contrast, the radial component ${V_z}^{*\text {c}}$ can be derived from Eq. (47) by differentiation with respect to r:

$$\begin{aligned} \begin{aligned}&{V_z}^{*\text {c}}\left( r; \, r_1, r_2, \theta _c\right) \\&\quad =\frac{\partial V^{*\text {c}}\left( r; \, r_1, r_2, \theta _c\right) }{\partial r} \\&\quad =2 \pi G d\rho \left\{ \frac{\ell '_c}{4 r} \left[ \left( 3 r' + 5 r \cos \theta _c\right) \left( r - r' \cos \theta _c\right) - \frac{r'}{r}{\ell '_c}^2\right] \right. \\&\qquad \,+ \frac{5 \cos ^2\theta _c - 1}{8} \left[ \frac{}{}\left( r' - 2 r \cos \theta _c\right) \ell '_c\right. \\&\qquad \,+ \left. \frac{\left( r' r - r^2 \cos \theta _c\right) \left( r - r' \cos \theta _c\right) }{\ell '_c}\right] \\&\qquad \,+ \frac{5 \cos ^2\theta _c - 1}{8} \left[ 3 r^2 \sin ^2\theta _c \,\mathrm {ln}\left( \ell '_c + r' - r \cos \theta _c\right) \right. \\&\qquad \,+ \left. \left. \left. r^3 \sin ^2\theta _c \frac{r - \left( r' + \ell '_c\right) \cos \theta _c}{\ell '_c \left( \ell '_c + r' - r \cos \theta _c\right) }\right] \right\} \right| ^{r' = r_2}_{r' = r_1}\\&\qquad \,+ \left. 2 \pi G d\rho \left( -\frac{1}{4 r^2} r'^4\right) \right| ^{r' = r_2}_{r' = r_1} \times \left\{ \begin{array}{ll} +1 &{} \quad r \ge r_2\\ -1 &{} \quad r \le r_1 \end{array}\right. . \end{aligned} \end{aligned}$$

(49)

The gravitational attraction ${V_z}^{*\text {s}}$ of a spherical shell with a linear density $d\rho \times r'$ results from Eq. (49) with $\theta _c = \pi$:

$$\begin{aligned} {V_z}^{*\text {s}}\left( r; \, r_1, r_2, \pi \right) = \left\{ \begin{array}{ll} -\frac{\pi G d\rho }{r^2} \left( r_2^4 - r_1^4\right) &{} \quad r \ge r_2 \\ [1.0ex] 0 &{} \quad r \le r_1 \end{array}\right. . \end{aligned}$$

(50)

According to the above equation, the analytical formula for ${V_z}^{*\text {s}}$ at the point inside the shell mass (i.e., $r_1< r < r_2$) is expressed as $-\,\frac{\pi G d\rho }{r^2}\left( r^4 - r_1^4\right)$. Notice that the vector components ${V_x}^{*\text {s}}$ and ${V_y}^{*\text {s}}$ of the spherical shell are zero at an arbitrary point.

1.3 D.3  The gravitational potential ${V}^{*\text {b}}$ and attraction ${V_z}^{*\text {b}}$ of a spherical zonal band with a linear density $d\rho \times r'$

From the analytical solutions of the gravitational potential ${V}^{*\text {c}}$ (Eq. 47) and attraction ${V_z}^{*\text {c}}$ (Eq. 49), the potential ${V}^{*\text {b}}$ and attraction ${V_z}^{*\text {b}}$ of a spherical zonal band between the spherical distances $\theta _i$ and $\theta _{i+1}$ at the computation point P can be derived as follows:

$$\begin{aligned} {V}^{*\text {b}}\left( r; \, r_1, r_2, \theta _i, \theta _{i+1}\right)&= \,\,{V}^{*\text {c}}\left( r; \, r_1, r_2, \theta _{i+1}\right) \nonumber \\&\,\,-{{V}^{*\text {c}}\left( r; \, r_1, r_2, \theta _{i}\right) ,} \end{aligned}$$

(51a)

$$\begin{aligned} {V_z}^{*\text {b}}\left( r; \, r_1, r_2, \theta _i, \theta _{i+1}\right)&= \,\,{V_z}^{*\text {c}}\left( r; \, r_1, r_2, \theta _{i+1}\right) \nonumber \\&\,\,-{{V_z}^{*\text {c}}\left( r; \, r_1, r_2, \theta _{i}\right) .} \end{aligned}$$

(51b)

Assuming that the spherical zonal band can be uniformly decomposed into K sectors (i.e., tesseroids), the gravitational potential and attraction of each sector at the computation point P are obtained by means of dividing ${V}^{*\text {b}}$ and ${V_z}^{*\text {b}}$ by K, respectively.

Appendix E

List of abbreviations:

2DGLQ	Analytical integration over radial
	direction and surface integrals by
	Gauss–Legendre quadrature
3DGLQ	Volume integrals by
	Gauss–Legendre quadrature
AD2D	Two-dimensional adaptive
	subdivision of the tesseroids
AD3D	Three-dimensional adaptive
	subdivision of the tesseroids
GLQ	Gauss–Legendre quadrature
NS	No subdivision of the tesseroids
PM	Point mass
PR	Prism
RE	Regular subdivision of the tesseroids
RTM	Residual terrain model
STD	Standard deviation
TC	Topographic correction
TE	Topographic effect
TSE2	Taylor series expansion up to the second order
USSA1976	US Standard Atmosphere 1976

1. As an example, the abbreviation such as 3DGLQ_AD2D in the text means a combination of the 3DGLQ method and the AD2D technique