Far-Zone Effects for Spherical Integral Transformations I: Formulas for the Radial Boundary Value Problem and its Derivatives

Abstract

Integral transformations represent an important mathematical tool for gravitational field modelling. A basic assumption of integral transformations is the global data coverage, but availability of high-resolution and accurate gravitational data may be restricted. Therefore, we decompose the global integration into two parts: (1) the effect of the near zone calculated by the numerical integration of data within a spherical cap and (2) the effect of the far zone due to data beyond the spherical cap synthesised by harmonic expansions. Theoretical and numerical aspects of this decomposition have frequently been studied for isotropic integral transformations on the sphere, such as Hotine’s, Poisson’s, and Stokes’s integral formulas. In this article, we systematically review the mathematical theory of the far-zone effects for the spherical integral formulas, which transform the disturbing gravitational potential or its purely radial derivatives into observable quantities of the gravitational field, i.e. the disturbing gravitational potential and its radial, horizontal, or mixed derivatives of the first, second, or third order. These formulas are implemented in a MATLAB software and validated in a closed-loop simulation. Selected properties of the harmonic expansions are investigated by examining the behaviour of the truncation error coefficients. The mathematical formulations presented here are indispensable for practical solutions of direct or inverse problems in an accurate gravitational field modelling or when studying statistical properties of integral transformations.

Appendix A Global integral identities

Global integrals of $T_{[k]}$ are required to get the spectral formulas for the far-zone effects in Sect. 3. These global integral identities are listed in Table 13. Equation (A3) can be found in (e.g. Heiskanen and Moritz 1967, p. 30). Hagiwara (1972) proved validity of Eqs. (A4) and (A5). Laborious mathematical derivations of Eqs. (A6) and (A7) can be found in (Thalhammer 1994, 1995; Eshagh 2009).

Equations (A8) and (A9) are, to our knowledge, original. We opt to illustrate a proof of Eq. (A8) for the interested reader. Firstly, we take the difference $\mathcal{{D}}^{xxx} - 3 \mathcal{{D}}^{xyy}$ of the differential operators for the components of the third-order gravitational tensor in LNORF, see Eqs. (28) and (31). We set $r = R$ $(t = 1)$ and get:

$$\begin{aligned}&\cos 3 \alpha \, (1 - u^2)^{3/2} \frac{\partial ^3}{\partial u^3} \end{aligned}$$

$$\begin{aligned}&= \bigg (\frac{3}{\cos ^2 \varphi } - 2\bigg ) \frac{\partial }{\partial \varphi } + 3 \tan \varphi \frac{\partial ^2}{\partial \varphi ^2} - \frac{6 \tan \varphi }{\cos ^2 \varphi } \frac{\partial ^2}{\partial \lambda ^2} + \frac{\partial ^3}{\partial \varphi ^3} - \frac{3}{\cos ^2 \varphi } \frac{\partial ^3}{\partial \varphi \partial \lambda ^2}. \end{aligned}$$

(A1)

Secondly, we apply the left-hand side of the last expression to the left-hand side of the global integral identity (A3) and use the fact that $(1 - u^2)^{3/2} \frac{\partial ^3}{\partial u^3} P_{n,0}(u) = P_{n,3}(u)$. Correspondingly, the right-hand side of Eq. (A1) acts on $T_{[k],n}$ on the right-hand side of Eq. (A3). Finally, we exploit the Legendre differential equation (e.g. Heiskanen and Moritz 1967, p. 21) to further simplify the derivatives of $T_{[k],n}$ with respect to $\varphi$ and $\lambda$.

The global integral identity (A9) can be proved equivalently. In this case, however, the initial point is the difference of the differential operators $\mathcal{{D}}^{yyy} - 3 \mathcal{{D}}^{xxy}$, see Eqs. (29) and (34).

Table 13 Global integral identities