Vertical and horizontal spheroidal boundary-value problems

Abstract

Vertical and horizontal spheroidal boundary-value problems (BVPs), i.e., determination of the external gravitational potential from the components of the gravitational gradient on the spheroid, are discussed in this article. The gravitational gradient is decomposed into the series of the vertical and horizontal vector spheroidal harmonics, before being orthogonalized in a weighted sense by two different approaches. The vertical and horizontal spheroidal BVPs are then formulated and solved in the spectral and spatial domains. Both orthogonalization methods provide the same analytical solutions for the vertical spheroidal BVP, and give distinct, but equivalent, analytical solutions for the horizontal spheroidal BVP. A closed-loop simulation is performed to test the correctness of the analytical solutions, and we investigate analytical properties of the sub-integral kernels. The systematic treatment of the spheroidal BVPs and the resulting mathematical equations extend the theoretical apparatus of geodesy and of the potential theory.

Equivalence of the analytical solutions for the horizontal spheroidal BVP

In this Appendix, we show that the two distinct analytical solutions of the horizontal spheroidal BVP, see Eqs. (43), (44), and (46), are mathematically equivalent. We restrict the proof only to the spectral domain. Implications and conclusions are identical also in the spatial domain.

The starting point of the proof is Eq. (44), which can be written as follows:

$$\begin{aligned} \bar{C}_{n,m}= & {} \frac{a_0}{4 \pi GM} \frac{1}{n(n + 1)} \int _{{\varOmega }'} \bigg [L(b_0,\beta ')\ V^{\beta }(b_0,{\varOmega }')\ \frac{\partial }{\partial \beta '}\ \bar{Y}_{n,m}({\varOmega }') \nonumber \\&\quad -\, \frac{a_0}{\cos \beta '}\ V^{\lambda }(b_0,{\varOmega }')\ \frac{\partial }{\partial \lambda '}\ \bar{Y}_{n,m}({\varOmega }')\bigg ]\ {\mathrm {d}}{\varOmega }' \nonumber \\&\quad +\, \frac{1}{4 \pi GM} \frac{\varepsilon ^2}{n(n + 1)} \int _{{\varOmega }'} \cos \beta '\ V^{\lambda }(b_0,{\varOmega }')\ \frac{\partial }{\partial \lambda '}\ \bar{Y}_{n,m}({\varOmega }')\ {\mathrm {d}}{\varOmega }' \nonumber \\&\quad +\, \frac{a_0}{4 \pi GM} \frac{\varepsilon ^2 m^2}{n(n + 1)\big [a_0^2\ n(n + 1) - \varepsilon ^2 m^2\big ]} \nonumber \\&\quad \times \int _{{\varOmega }'} \bigg [L(b_0,\beta ')\ V^{\beta }(b_0,{\varOmega }')\ \frac{\partial }{\partial \beta '}\ \bar{Y}_{n,m}({\varOmega }') \nonumber \\&\quad -\, \frac{L^2\left( b_0,\beta '\right) }{a_0 \cos \beta '}\ V^{\lambda }(b_0,{\varOmega }')\ \frac{\partial }{\partial \lambda '}\ \bar{Y}_{n,m}({\varOmega }')\bigg ]\ {\mathrm {d}}{\varOmega }'. \end{aligned}$$

(53)

The last equation was obtained with the help of the formula:

$$\begin{aligned} \frac{L^2\left( b_0,\beta '\right) }{a_0^2 \cos ^2 \beta '} = \frac{1}{\cos ^2 \beta '} - \frac{\varepsilon ^2}{a_0^2}\ , \end{aligned}$$

(54)

which can be deduced from Eq. (32) by considering $u' = b_0$ and $v(b_0) = a_0$. In addition, we exploited the partial fraction expansion of the term in front of the integral of Eq. (44), i.e.:

$$\begin{aligned}&\frac{a_0^2}{a_0^2\ n(n + 1) - \varepsilon ^2\ m^2} \nonumber \\&\quad = \frac{1}{n + 1} + \frac{\varepsilon ^2\ m^2}{n(n+1)\big [a_0^2\ n (n + 1) - \varepsilon ^2\ m^2\big ]}. \end{aligned}$$

(55)

The addition of the second integral with the third one in Eq. (53) reads:

$$\begin{aligned}&\frac{1}{4 \pi GM} \frac{\varepsilon ^2}{n(n + 1)} \int _{{\varOmega }'} \cos \beta ' \nonumber \\&\quad \times \bigg [- \frac{GM}{a_0} \sum _{n'=0}^{\infty } \sum _{m'=-n'}^{+n'} \frac{1}{a_0 \cos \beta '}\ \bar{C}_{n',m'}\ \frac{\partial }{\partial \lambda '}\ \bar{Y}_{n',m'}({\varOmega }') \bigg ] \nonumber \\&\quad \times \frac{\partial }{\partial \lambda '}\ \bar{Y}_{n,m}({\varOmega }')\ {\mathrm {d}}{\varOmega }' \nonumber \\&\quad +\, \frac{a_0}{4 \pi GM} \frac{\varepsilon ^2 m^2}{n(n + 1)\big [a_0^2\ n(n + 1) - \varepsilon ^2 m^2\big ]} \int _{{\varOmega }'} \Bigg \{L(b_0,\beta ')\ \nonumber \\&\quad \times \bigg [\frac{GM}{a_0} \sum _{n'=0}^{\infty } \sum _{m'=-n'}^{+n'} \frac{1}{L(b_0, \beta ')}\ \bar{C}_{n',m'}\ \frac{\partial }{\partial \beta '}\ \bar{Y}_{n',m'}({\varOmega }')\bigg ] \nonumber \\&\quad \times \frac{\partial }{\partial \beta '} \bar{Y}_{n,m}({\varOmega }') \nonumber \\&\quad -\, \frac{L^2(b_0,\beta ')}{a_0 \cos \beta '}\ \bigg [- \frac{GM}{a_0} \sum _{n'=0}^{\infty } \sum _{m'=-n'}^{+n'} \frac{1}{a_0 \cos \beta '}\nonumber \\&\quad \times \bar{C}_{n',m'}\ \frac{\partial }{\partial \lambda '}\ \bar{Y}_{n',m'}({\varOmega }') \bigg ] \frac{\partial }{\partial \lambda '}\ \bar{Y}_{n,m}({\varOmega }')\Bigg \} {\mathrm {d}}{\varOmega }' \nonumber \\&\quad = - \frac{\varepsilon ^2}{a_0^2\ n(n + 1)} \sum _{n'=0}^{\infty } \sum _{m'=-n'}^{+n'} \bar{C}_{n',m'}\ \frac{1}{4 \pi } \nonumber \\&\qquad \times \int _{{\varOmega }'} \frac{\partial }{\partial \lambda '}\ \bar{Y}_{n',m'}({\varOmega }')\ \frac{\partial }{\partial \lambda '}\ \bar{Y}_{n,m}({\varOmega }')\ {\mathrm {d}}{\varOmega }' \nonumber \\&\qquad +\, \frac{\varepsilon ^2 m^2}{n(n + 1)\big [a_0^2\ n(n + 1) - \varepsilon ^2 m^2\big ]} \sum _{n'=0}^{\infty } \sum _{m'=-n'}^{+n'} \bar{C}_{n',m'} \nonumber \\&\qquad \times \frac{1}{4 \pi } \int _{{\varOmega }'} \bigg [\frac{\partial }{\partial \beta '}\ \bar{Y}_{n',m'}({\varOmega }')\ \frac{\partial }{\partial \beta '}\ \bar{Y}_{n,m}({\varOmega }') \nonumber \\&\qquad +\, \frac{1}{\cos ^2 \beta '} \frac{\partial }{\partial \lambda '}\ \bar{Y}_{n',m'}({\varOmega }')\ \frac{\partial }{\partial \lambda '}\ \bar{Y}_{n,m}({\varOmega }')\bigg ]\, {\mathrm {d}}{\varOmega }' \nonumber \\&\qquad -\, \frac{\varepsilon ^4 m^2}{a_0^2\ n(n + 1)\big [a_0^2\ n(n + 1) - \varepsilon ^2 m^2\big ]} \sum _{n'=0}^{\infty } \sum _{m'=-n'}^{+n'} \bar{C}_{n',m'} \nonumber \\&\qquad \times \frac{1}{4 \pi } \int _{{\varOmega }'} \frac{\partial }{\partial \lambda '}\ \bar{Y}_{n',m'}({\varOmega }')\ \frac{\partial }{\partial \lambda '}\ \bar{Y}_{n,m}({\varOmega }')\ {\mathrm {d}}{\varOmega }' \nonumber \\&\quad = \bar{C}_{n,m} \Bigg \{\frac{\varepsilon ^2 m^2}{a_0^2\ n(n + 1) - \varepsilon ^2 m^2} - \frac{\varepsilon ^2 m^2\ }{a_0^2\ n(n + 1)} \nonumber \\&\qquad \times \,\bigg [1 + \frac{\varepsilon ^2 m^2}{a_0^2\ n(n + 1) - \varepsilon ^2 m^2} \bigg ] \Bigg \} \nonumber \\&\quad = - \bar{C}_{n,m} \Bigg \{\frac{\varepsilon ^2 m^2\ }{a_0^2\ n(n + 1)} \bigg [1 - \frac{a_0^2\ n(n + 1) - \varepsilon ^2 m^2}{a_0^2\ n(n + 1) - \varepsilon ^2 m^2} \bigg ] \Bigg \} \nonumber \\&\quad = 0. \end{aligned}$$

(56)

The left-hand side of Eq. (56) results from inserting Eqs. (18) and (19), i.e., the spheroidal harmonic expansions of the horizontal gravitational gradient components $V^{\beta }$ and $V^{\lambda }$, into Eq. (53). The interchange of the order of summation and integration (assuming the uniform convergence of the series expansion) and the application of Eq. (54) lead to the expression after the first equality. The expression after the second equality is obtained by making use of the orthogonality relationship of Eq. (25) and another one defined as:

$$\begin{aligned}&\frac{1}{4 \pi } \int _{{\varOmega }'} \frac{\partial }{\partial \lambda '} \bar{Y}_{n,m}({\varOmega }')\ \frac{\partial }{\partial \lambda '} \bar{Y}_{n',m'}({\varOmega }')\ {\mathrm {d}}{\varOmega }' \nonumber \\&\quad = m\ m'\ \delta _{n,n'}\ \delta _{m,m'}. \end{aligned}$$

(57)

Simple algebraic operations gave the last two equalities of Eq. (56).

Taking into account the result of Eq. (56), the only nonzero term of Eq. (53) is the first integral. This is identical with the spheroidal harmonic analysis by Eq. (43). Thus, we have proved that Eqs. (43) and (44) are mathematically equivalent.