Cap integration in spectral gravity forward modelling: near- and far-zone gravity effects via Molodensky’s truncation coefficients

Abstract

Spectral gravity forward modelling is a technique that converts a band-limited topography into its implied gravitational field. This conversion implicitly relies on global integration of topographic masses. In this paper, a modification of the spectral technique is presented that provides gravity effects induced only by the masses located inside or outside a spherical cap centred at the evaluation point. This is achieved by altitude-dependent Molodensky’s truncation coefficients, for which we provide infinite series expansions and recurrence relations with a fixed number of terms. Both representations are generalized for an arbitrary integer power of the topography and arbitrary radial derivative. Because of the altitude-dependency of the truncation coefficients, a straightforward synthesis of the near- and far-zone gravity effects at dense grids on irregular surfaces (e.g. the Earth’s topography) is computationally extremely demanding. However, we show that this task can be efficiently performed using an analytical continuation based on the gradient approach, provided that formulae for radial derivatives of the truncation coefficients are available. To demonstrate the new cap-modified spectral technique, we forward model the Earth’s degree-360 topography, obtaining near- and far-zone effects on gravity disturbances expanded up to degree 3600. The computation is carried out on the Earth’s surface and the results are validated against an independent spatial-domain Newtonian integration ($1\,\upmu \mathrm {Gal}$ RMS agreement). The new technique is expected to assist in mitigating the spectral filter problem of residual terrain modelling and in the efficient construction of full-scale global gravity maps of highest spatial resolution.

Appendices

Appendix A: Derivation of cap-modified spectral gravity forward modelling

In this appendix, we derive Eq. (6) that yields the gravitational effect induced by topographic masses located inside/outside a spherical cap defined by the spherical distance $\psi _0$ from the computation point. To this end, we closely follow the derivations described, for instance, in Section 7–4 of Heiskanen and Moritz (1967). At first, let us focus on the near-zone effect.

Starting from the spherical harmonic expansion as provided, for instance, by Wieczorek and Phillips (1998), the topographic potential induced by topographic masses all around the globe can be written as

$$\begin{aligned} V(r,\varphi ,\lambda )= & {} 2\pi \, G \rho R^2 \sum _{n=0}^{\infty } \left( \frac{R}{r} \right) ^{n+1}\nonumber \\&\times \,\frac{2}{2n+1} \, \sum _{p=1}^{n+3}\frac{\prod \nolimits _{i=1}^{p}(n+4-i)}{p! \, (n+3)} \, H_n^p(\varphi ,\lambda ) \, ,\nonumber \\ \end{aligned}$$

(19)

with $H_n^p(\varphi ,\lambda )$ being the Laplace surface spherical harmonic of the topographic height function (see Eq. 5),

$$\begin{aligned} H_n^p(\varphi ,\lambda ) = \sum _{m=-n}^{n} \bar{H}_{nmp} \, \bar{Y}_{nm}(\varphi ,\lambda ) \, . \end{aligned}$$

(20)

The term $H_n^p(\varphi ,\lambda )$ can equally be written as

$$\begin{aligned} H_n^p(\varphi ,\lambda )= & {} \frac{2n+1}{4\pi }\nonumber \\&\times \int \limits _{\psi =0}^{\pi } \int \limits _{\alpha =0}^{2\pi } H^p(\psi ,\alpha ) \, P_n(\cos \psi ) \, \sin \psi \, \mathrm d\alpha \, \mathrm d\psi \, ,\nonumber \\ \end{aligned}$$

(21)

where $P_n$ is unnormalized Legendre polynomial of degree n, and $\psi$ and $\alpha$ are, respectively, the spherical distance and the azimuth between the computation point and the integration element. Substituting Eq. (21) into Eq. (19) and interchanging the order of summations and integrations, we get

$$\begin{aligned} V(r,\varphi ,\lambda )= & {} G \rho R^2 \sum _{p=1}^{\infty } \int \limits _{\psi =0}^{\pi } \int \limits _{\alpha =0}^{2\pi } H^p(\psi ,\alpha ) K_p(r,\psi )\nonumber \\&\times \sin \psi \, \mathrm d\alpha \, \mathrm d\psi \, , \end{aligned}$$

(22)

where

$$\begin{aligned} K_p(r,\psi ) = \sum _{n=0}^{\infty } \left( \frac{R}{r} \right) ^{n+1} \, \frac{\prod \nolimits _{i=1}^{p}(n+4-i)}{p! \, (n+3)} \, P_n(\cos \psi ) \, .\nonumber \\ \end{aligned}$$

(23)

Such reordering of integration and summation is permissible as long as the series is uniformly convergent. This holds true for points with $r > \max (R+\hat{H}(\varphi ,\lambda ))$ (e.g. Balmino 1994; Wieczorek and Phillips 1998). Note that to be the series in Eq. (23) convergent, the condition $r > R$ must be satisfied (e.g. Martinec 1998).

Restricting the integration limits over $\psi$ in Eq. (22) from $\psi \in [0,\, \pi ]$ (global integration) to $\psi \in [0,\, \psi _0]$ (inside-cap integration) yields the near-zone contribution of topographic masses to the topographic potential,

$$\begin{aligned} V^\mathrm {In}(r,\varphi ,\lambda )= & {} G \rho R^2 \sum _{p=1}^{\infty } \int \limits _{\psi =0}^{\psi _0} \int \limits _{\alpha =0}^{2\pi } H^p(\psi ,\alpha ) \, K_p(r,\psi )\nonumber \\&\times \sin \psi \, \mathrm d\alpha \, \mathrm d\psi \, . \end{aligned}$$

(24)

After introducing the discontinuous function

$$\begin{aligned} K^{\mathrm {In}}_p(r,\psi )= {\left\{ \begin{array}{ll} K_p(r,\psi ) \quad &{}\mathrm {for} \quad 0\le \psi \le \psi _0 \, ,\\ 0 \quad &{}\mathrm {for} \quad \psi _0<\psi \le \pi \, , \end{array}\right. } \end{aligned}$$

(25)

Eq. (24) can be expressed as a global integration of the form

$$\begin{aligned} V^\mathrm {In}(r,\varphi ,\lambda )= & {} G \rho R^2 \sum _{p=1}^{\infty } \int \limits _{\psi =0}^{\pi } \int \limits _{\alpha =0}^{2\pi } H^p(\psi ,\alpha ) \, K^{\mathrm {In}}_p(r,\psi )\nonumber \\&\times \sin \psi \, \mathrm d\alpha \, \mathrm d\psi \, . \end{aligned}$$

(26)

Next, the kernel function $K^{\mathrm {In}}_p(r,\psi )$ can be expanded in an infinite series of Legendre polynomials

$$\begin{aligned} K^{\mathrm {In}}_p(r,\psi ) = \sum _{n=0}^{\infty } \frac{2n+1}{2} \, Q_{np}^{\mathrm {In}}(r,\psi _0) \, P_n(\cos \psi ) \end{aligned}$$

(27)

with the truncation coefficients $Q_{np}^{\mathrm {In}}(r,\psi _0)$ given by

$$\begin{aligned} Q_{np}^{\mathrm {In}}(r,\psi _0) = \int \limits _{0}^{\pi } K^{\text {In}}_p(r,\psi ) \, P_n(\cos \psi ) \, \sin \psi \, \mathrm d\psi \, . \end{aligned}$$

(28)

In Appendices B and C, respectively, we provide spectral and recurrence relations suitable to evaluate numerically the truncation coefficients $Q_{np}^{\mathrm {In}}(r,\psi _0)$ as well as $Q_{np}^{\mathrm {Out}}(r,\psi _0)$.

Substituting Eq. (27) into (26) and with the help of Eq. (21), we obtain

$$\begin{aligned} V^\mathrm {In}(r,\varphi ,\lambda ) = 2\pi \,G \rho R^2 \sum _{p=1}^{\infty } \sum _{n=0}^{\infty } Q_{np}^{\mathrm {In}}(r,\psi _0) \, H_n^p(\varphi ,\lambda ) \, .\nonumber \\ \end{aligned}$$

(29)

Finally, realizing that (cf. Lemma 4.1 of Freeden and Schneider 1998)

$$\begin{aligned} H_n^p(\varphi ,\lambda ) = 0 \quad \mathrm {for} \quad n>p\times n_{\max }, \end{aligned}$$

(30)

then utilizing Eq. (20) and, finally, after truncating the infinite series over p at some $p_{\max }$, we arrive at Eq. (6) for $z=\text {`In'}$.

The far-zone gravitational effect $V^\mathrm {Out}(r,\varphi ,\lambda )$ from Eq. (6), z being equal to $\text {`Out'}$, can be derived in a similar manner by integrating over $\psi \in [\psi _0,\, \pi ]$. These derivations require to introduce the discontinuous function

$$\begin{aligned} K^{\mathrm {Out}}_p(r,\psi )= {\left\{ \begin{array}{ll} 0 \quad &{}\mathrm {for} \quad 0\le \psi < \psi _0 \, ,\\ K_p(r,\psi ) \quad &{}\mathrm {for} \quad \psi _0\le \psi \le \pi \, , \end{array}\right. } \end{aligned}$$

(31)

which is then utilized in the definition of the far-zone truncation coefficients

$$\begin{aligned} Q_{np}^{\mathrm {Out}}(r,\psi _0) = \int \limits _{0}^{\pi } K^{\text {Out}}_p(r,\psi ) \, P_n(\cos \psi ) \, \sin \psi \, \mathrm d\psi \, . \end{aligned}$$

(32)

Appendix B: Spectral representation of truncation coefficients

The spectral form of the near-zone truncation coefficients can be obtained by substituting Eqs. (23) and (25) into Eq. (28). Then, after interchanging the order of summation and integration, the near-zone truncation coefficients are given by

$$\begin{aligned} Q_{np}^{\mathrm {In}}(r,\psi _0)= & {} \sum _{l=0}^{\infty } \left( \frac{R}{r} \right) ^{l+1} \, \frac{\prod \nolimits _{i=1}^{p}(l+4-i)}{p! \, (l+3)}\nonumber \\&\times \int \limits _{0}^{\psi _0} P_l(\cos \psi ) \, P_n(\cos \psi ) \, \sin \psi \, \mathrm d\psi \, . \end{aligned}$$

(33)

The integrals

$$\begin{aligned} I_{ln}(u_0,1)= & {} \int \limits _{0}^{\psi _0} P_l(\cos \psi ) \, P_n(\cos \psi ) \, \sin \psi \, \mathrm d\psi \nonumber \\= & {} \int \limits _{u_0}^{1} P_l(u) \, P_n(u) \, \mathrm du \, , \end{aligned}$$

(34)

with the substitutions $u=\cos \psi$ and $u_0=\cos \psi _0$, can be evaluated numerically via recurrence relations. For instance, Paul (1973) provides recurrence relations to compute

$$\begin{aligned} I_{ln}(-1,u_0) = \int \limits _{-1}^{u_0} P_l(u) \, P_n(u) \, \mathrm du \, , \end{aligned}$$

(35)

from which we can easily obtain $I_{ln}(u_0,1)$ by utilizing the orthogonality property of Legendre polynomials

$$\begin{aligned} I_{ln}(-1,1) = \frac{2}{2n+1} \delta _{ln} \, , \end{aligned}$$

(36)

where

$$\begin{aligned} \delta _{ln} = {\left\{ \begin{array}{ll} 1\,, \quad \quad l=n \, ,\\ 0\,, \quad \quad l\ne n \, . \end{array}\right. } \end{aligned}$$

(37)

The integrals $I_{ln}(u_0,1)$ thus read

$$\begin{aligned} I_{ln}(u_0,1)= & {} I_{ln}(-1,1) - I_{ln}(-1,u_0)\nonumber \\= & {} \frac{2}{2n+1} \delta _{ln} - I_{{ln}}(-1,u_0) \, . \end{aligned}$$

(38)

While Eq. (35) is needed for far-zone effects, Eq. (38) enters the computation of the near-zone gravity contribution. More general recurrence relations to compute $I_{ln}(u_1,u_2)$, suitable both for near- and far-zone effects, were derived by Moreaux et al. (1999).

Needless to say, the far-zone truncation coefficients can be derived by substituting Eqs. (23) and (31) into Eq. (32), obtaining

$$\begin{aligned} Q_{np}^{\mathrm {Out}}(r,\psi _0)= & {} \sum _{l=0}^{\infty } \left( \frac{R}{r} \right) ^{l+1} \, \frac{\prod \nolimits _{i=1}^{p}(l+4-i)}{p! \, (l+3)}\nonumber \\&\times \int \limits _{\psi _0}^{\pi } P_l(\cos \psi ) \, P_n(\cos \psi ) \, \sin \psi \, \mathrm d\psi \, . \end{aligned}$$

(39)

Appendix C: Recurrence representation of truncation coefficients

In this appendix, we derive recurrence relations with a fixed number of terms to compute the truncation coefficients for an arbitrary n and p.

From Martinec (1998) it can be seen that the integral kernels $K_p(r,\psi )$ from Eq. (23) have the following closed spatial forms

$$\begin{aligned} K_1(r,\psi )= & {} \, \frac{R}{l(r,\psi )} \, , \nonumber \\ K_2(r,\psi )= & {} \frac{1}{2} \left( K_1(r,\psi ) - r \, \frac{\partial K_1(r,\psi )}{\partial r} \right) \, ,\nonumber \\ K_p(r,\psi )= & {} \frac{1}{p!} \sum _{s=1}^{p-2} a_{ps}\, r^{p-s} \, \frac{\partial ^{p-s} K_1(r,\psi )}{\partial r^{p-s}}\, , \quad p \ge 3\, ,\nonumber \\ \end{aligned}$$

(40)

with the Euclidean distance

$$\begin{aligned} l(r,\psi )=\sqrt{r^2-2 R r \cos \psi +R^2} \end{aligned}$$

(41)

and the coefficients

$$\begin{aligned} a_{ps}=(-1)^{p-1} \frac{(p-1)! \, (p-3)!}{(p-s)! \, (p-s-2)! \, (s-1)!} \, . \end{aligned}$$

(42)

After introducing the substitution $t=R \slash r$ and the normalized Euclidean distance

$$\begin{aligned} g(t,u)=\sqrt{1-2\, tu+t^2}\, , \end{aligned}$$

(43)

we obtain from Eqs. (28) and (32) with the help of Eq. (40)

$$\begin{aligned} Q_{n1}(r,\psi _0)= & {} G_n^{(0)}(t,u_0) \, , \nonumber \\ Q_{n2}(r,\psi _0)= & {} \frac{1}{2} \left( G_n^{(0)}(t,u_0) - r \, G_n^{(1)}(t,u_0) \right) \, ,\nonumber \\ Q_{np}(r,\psi _0)= & {} \frac{1}{p!} \sum _{s=1}^{p-2} a_{ps}\, r^{p-s} G_n^{(p-s)}(t,u_0)\, , \quad p \ge 3\, .\nonumber \\ \end{aligned}$$

(44)

In Eq. (44), we introduced the substitution

$$\begin{aligned} G_n^{(j)}(t,u_0) = \frac{\partial ^j}{\partial r^j} L_n(t,u_0) \, , \quad j \ge 0\, , \end{aligned}$$

(45)

with

$$\begin{aligned} L_n(t,u_0) = t\, A_n(t,u_0) \end{aligned}$$

(46)

and

$$\begin{aligned} A_n(t,u_0) = \int \limits _{u_l}^{u_u} \frac{P_n(u)}{g(t,u)} \, \mathrm du \, , \end{aligned}$$

(47)

where $u_u$ and $u_l$ denote the upper and lower integration limits, respectively. For the computation of $Q^{\mathrm {In}}_{np}(r,\psi _0)$, we thus have $u_u=1$ and $u_l=u_0$. For $Q^{\mathrm {Out}}_{np}(r,\psi _0)$, the limits are $u_u=u_0$ and $u_l=-\,1$. For simplicity, the superscript z is omitted in Eq. (44), because these relations hold both for $Q_{np}^{\mathrm {In}}(r,\psi _0)$ and $Q_{np}^{\mathrm {Out}}(r,\psi _0)$, depending on $u_u$ and $u_l$.

To compute $G^{(j)}_n(t,u_0)$, representing the jth derivative of the composite function $L_n(t,u_0)$ in terms of r, we use the chain rule generalized for an arbitrarily high derivative (Faà di Bruno’s formula; e.g. Comtet 1974). This rule, expressed via the partial Bell polynomials $\tilde{B}_{jk}$, reads for $j\ge 0$

$$\begin{aligned} G^{(j)}_n(t,u_0)= & {} \sum _{k=0}^{j} J_n^{(k)}(t,u_0)\nonumber \\&\times \tilde{B}_{jk} \left( T^{(1)}(r),T^{(2)}(r), \dots , T^{(j-k+1)}(r) \right) \nonumber \\ \end{aligned}$$

(48)

with

$$\begin{aligned} J_n^{(v)}(t,u_0) = \frac{\partial ^v}{\partial t^v} L_n(t,u_0)\,, \quad v \ge 0\, , \end{aligned}$$

(49)

and

$$\begin{aligned} T^{(v)}(r) = \frac{\mathrm d^v t}{\mathrm dr^v} = (-1)^{v} \, v! \, \frac{R}{r^{v+1}}\, ,\quad v \ge 0\, . \end{aligned}$$

(50)

The partial Bell polynomials $\tilde{B}_{jk}$ can be computed recursively by the relation (e.g. Comtet 1974)

$$\begin{aligned} \tilde{B}_{jk} = \sum _{s=1}^{j-k+1} \begin{pmatrix} j-1\\ s-1 \end{pmatrix} T^{(s)} \, \tilde{B}_{j-s,k-1} \, , \quad j,k\ge 1\, , \end{aligned}$$

(51)

with

$$\begin{aligned} \tilde{B}_{00}= & {} 1 \, ,\nonumber \\ \tilde{B}_{j0}= & {} 0\,, \quad j\ge 1 \, ,\nonumber \\ \tilde{B}_{0k}= & {} 0\,, \quad k\ge 1 \, . \end{aligned}$$

(52)

Equation (49) implies that an arbitrary derivative of a product of t and $A_n(t,u_0)$ (cf. Eq. 46) has to be found. This can be done using the general Leibniz rule. Having two n-times differentiable functions $f_1(x)$ and $f_2(x)$, the rule says that the nth derivative of the product $f_1(x)\, f_2(x)$ is given as (e.g. Comtet 1974)

$$\begin{aligned} \frac{\mathrm d^n f_1(x)\, f_2(x)}{\mathrm dx^n} = \sum _{k=0}^{n} \begin{pmatrix} n\\ k \end{pmatrix} \frac{\mathrm d^{n-k} f_1(x)}{\mathrm dx^{n-k}} \, \frac{\mathrm d^{k} f_2(x)}{\mathrm dx^{k}} \, . \end{aligned}$$

(53)

Setting $f_1=t$ and $f_2=A_n(t,u_0)$ and after some simplifications, we find that the derivatives of $L_n(t,u_0)$ with respect to t read

$$\begin{aligned} J_n^{(v)}(t,u_0)= & {} \frac{\partial ^v}{\partial t^v} L_n(t,u_0)\nonumber \\= & {} t\, \frac{\partial ^v}{\partial t^v} A_n(t,u_0) + v \, \frac{\partial ^{v-1}}{\partial t^{v-1}} A_n(t,u_0) \, , \quad v\ge 1\, .\nonumber \\ \end{aligned}$$

(54)

This, however, requires to differentiate $A_n(t,u_0)$ with respect to t,

$$\begin{aligned} A_{n}^{(i)}(t,u_0) = \frac{\partial ^i}{\partial t^i} A_n(t,u_0) = \frac{\partial ^i}{\partial t^i} \int \limits _{u_l}^{u_u} \frac{P_n(u)}{g(t,u)} \, \mathrm du\, , \quad i\ge 0\, .\nonumber \\ \end{aligned}$$

(55)

Therefore, from the recurrence relations for $A^{(0)}_n(t,u_0)$ provided by Pavlis (1991), we have derived relations for $A_{n}^{(i)}(t,u_0)$, $i\ge 0$:

$$\begin{aligned} A_0^{(i)}(t,u_0)= & {} - \left[ \frac{\partial ^i}{\partial t^i} \alpha _0(t,u)\, \gamma (t,u) \right] _{u_l}^{u_u}\,, \end{aligned}$$

(56)

$$\begin{aligned} A_1^{(i)}(t,u_0)= & {} -\frac{1}{3} \left[ \frac{\partial ^i}{\partial t^i} \alpha _1(t,u)\, \gamma (t,u) \right] _{u_l}^{u_u}\,, \end{aligned}$$

(57)

$$\begin{aligned} A_n^{(i)}(t,u_0)= & {} \frac{1}{2n+1} \Bigg [ -(t^2+1) B_n^{(i)}(t,u_0)\nonumber \\&+\, 2t\,B_{n-1}^{(i)}(t,u_0) - 2i\, t \, B_{n}^{(i-1)}(t,u_0)\nonumber \\&+\,2i\,B_{n-1}^{(i-1)}(t,u_0) - i(i-1) B_{n}^{(i-2)}(t,u_0)\nonumber \\&+\,2(n+1)\, M_n(u_0) \, \gamma ^{(i)}(t,u_0) \Bigg ]\, , \quad n\ge 2\, ,\nonumber \\ \end{aligned}$$

(58)

where the ith derivatives of $\alpha _0$, $\alpha _1$ and $\gamma$ with respect to t read

$$\begin{aligned} \alpha _0^{(i)}(t,u)= & {} {\left\{ \begin{array}{ll} \begin{aligned} &{}t+\frac{1}{t}-2u\,, \quad &{}&{}i=0\,,\\ &{}(-1)^i \, \frac{i!}{t^{i+1}} + \delta _{\alpha _0}^{(i)}\,, \quad &{}&{}i\ge 1\,, \end{aligned} \end{array}\right. } \end{aligned}$$

(59)

$$\begin{aligned} \alpha _1^{(i)}(t,u)= & {} {\left\{ \begin{array}{ll} \begin{aligned} &{}t^2-tu-\frac{u}{t}+\frac{1}{t^2}-2u^2+2\,, \quad &{}&{} i=0\,,\\ &{}(-1)^{i+1} \, \frac{i!\, u}{t^{i+1}}\\ &{}\quad + (-1)^{i} \, \frac{(i+1)!}{t^{i+2}} + \delta _{\alpha _1}^{(i)}\, , \quad &{}&{}i\ge 1\,, \end{aligned} \end{array}\right. } \end{aligned}$$

(60)

$$\begin{aligned} \gamma ^{(i)}(t,u)= & {} {\left\{ \begin{array}{ll} \begin{aligned} &{}\frac{1}{g(t,u)}\,, &{}&{} i=0\,,\\ &{} \frac{\partial ^i}{\partial t^i} \frac{1}{g(t,u)} = \sum _{ \begin{matrix} (s+i)\ \text {is even}\\ s=0 \end{matrix} }^{i} (-1)^{\frac{i+s}{2}}\\ &{} \times \frac{(i-s+1)!!\, (i+s-1)!!}{(i-s+1)!}&{}&{}\\ &{}\times \frac{i!}{s!} \, \frac{(t-u)^s}{g^{i+s+1}(t,u)} \, ,\quad &{}&{}i\ge 1\,, \end{aligned} \end{array}\right. } \end{aligned}$$

(61)

with

$$\begin{aligned} \delta _{\alpha _0}^{(i)}= {\left\{ \begin{array}{ll} \begin{aligned} &{}1\,, \quad &{}&{} i=1\,, \\ &{}0\,, \quad &{}&{} i\ge 2\,, \end{aligned} \end{array}\right. } \end{aligned}$$

(62)

and

$$\begin{aligned} \delta _{\alpha _1}^{(i)}= {\left\{ \begin{array}{ll} \begin{aligned} &{}2t-u\,, \quad &{}&{} i=1\,, \\ &{}2\,, \quad &{}&{} i=2\,,\\ &{}0\,, \quad &{}&{} i\ge 3\,.\\ \end{aligned} \end{array}\right. } \end{aligned}$$

(63)

After simple manipulations, it can be shown that Eqs. (56) and (57) for $i=0$ are equal to Eq. (A.30b) of Pavlis (1991). By differentiating with respect to t, we obtained Eqs. (56) and (57) for $i\ge 1$. For their numerical evaluation, we use the general Leibniz rule (Eq. 53). Similarly, Eq. (58) for $i\ge 1$ can be derived by differentiating the relation for $i=0$, which is provided by Pavlis (1991) in Eq. (A.30a). Note that in Eq. (58), the negative derivatives that occur for $i=0,1$ need to be set to zero. Finally, Eq. (61) for $i\ge 1$ follows from Eq. (A.5.12) of Martinec (1998).

Analogously, the recurrence relations for $B_n^{(i)}(t,u_0)$, $i\ge 0$, (see Eq. 58) were derived from the formulae for $B_n^{(0)}(t,u_0)$ that are provided by Pavlis (1991):

$$\begin{aligned} B_0^{(i)}(t,u_0)= & {} \left[ \frac{\partial ^i}{\partial t^i} \beta _0(t,u)\, \gamma (t,u) \right] _{u_l}^{u_u}\,, \end{aligned}$$

(64)

$$\begin{aligned} B_1^{(i)}(t,u_0)= & {} \left[ \frac{\partial ^i}{\partial t^i} \beta _1(t,u)\, \gamma (t,u) \right] _{u_l}^{u_u}\,, \end{aligned}$$

(65)

$$\begin{aligned} B_n^{(i)}(t,u_0)= & {} \frac{t^2+1}{t} B_{n-1}^{(i)}(t,u_0) - B_{n-2}^{(i)}(t,u_0)\nonumber \\&-\,\frac{i}{t} B_{n}^{(i-1)}(t,u_0) + 2i B_{n-1}^{(i-1)}(t,u_0)\nonumber \\&-\,\frac{i}{t} B_{n-2}^{(i-1)}(t,u_0)+\frac{i(i-1)}{t} B_{n-1}^{(i-2)}(t,u_0)\nonumber \\&-\,\frac{1}{t}\, M_{n-1}(u_0)\, \gamma ^{(i)}(t,u_0)\, , \quad n\ge 2\, , \end{aligned}$$

(66)

where the ith derivatives of $\beta _0$ and $\beta _1$ with respect to t are given as

$$\begin{aligned} \beta _0^{(i)}= & {} (-1)^i\, i! \frac{1}{t^{i+1}}\, , \quad i\ge 0\,, \end{aligned}$$

(67)

$$\begin{aligned} \beta _1^{(i)}= & {} {\left\{ \begin{array}{ll} \begin{aligned} &{}\frac{1}{t^2}-\frac{u}{t}+1\, , \quad &{}&{}i=0\,,\\ &{}(-1)^i\, (i+1)! \frac{1}{t^{i+2}} + (-1)^{i+1}\, i! \, \frac{u}{t^{i+1}}\, , \quad &{}&{}i\ge 1\,, \end{aligned} \end{array}\right. } \end{aligned}$$

(68)

Starting from Eqs. (64) and (65) for $i=0$ (Eq. A.29b of Pavlis 1991), the relations for $i\ge 1$ are derived by differentiation with respect to t. Again, we evaluate these equations by the general Leibniz rule (Eq. 53). The recurrence relation in Eq. (66) for $i\ge 1$ was derived by differentiating Eq. (66) for $i=0$ (Eq. A.29a of Pavlis 1991) with respect to t. Setting the negative derivatives to zero, Eq. (66) holds for an arbitrary $i\ge 0$.

For the term $M_n(t,u_0)$, Pavlis (1991) provides recurrence relations with the initial values $u_l=-1$ and $u_u=u_0$ (the far-zone effect in terms of the truncation coefficients). After simple manipulations, these initial values can be rewritten in a unified manner both for near- and far-zone effects, obtaining

$$\begin{aligned} M_0(u_0)= & {} [P_{1}(u)]_{u_l}^{u_u}\,, \end{aligned}$$

(69)

$$\begin{aligned} M_1(u_0)= & {} \frac{1}{3} \left[ P_2(u) - P_0(u)\right] _{u_l}^{u_u}\,, \end{aligned}$$

(70)

$$\begin{aligned} M_n(u_0)= & {} \frac{1}{n+1} [(2n-1) \, u_0 \, M_{n-1}(u_0)\nonumber \\&- (n-2) \, M_{n-2}(u_0)] \, , \quad n\ge 2\, , \end{aligned}$$

(71)

where the Legendre polynomials $P_n(u)$ for $n=0,1,2$ read

$$\begin{aligned} P_0(u)= & {} 1\, ,\nonumber \\ P_1(u)= & {} u\, ,\nonumber \\ P_2(u)= & {} -\,\frac{1}{2} + \frac{3}{2} \, u^2\, . \end{aligned}$$

(72)

As for the proofs of the newly derived Eqs. (58) and (66) for $i\ge 1$, they are omitted here for the sake of brevity, but the validity of these recurrence relations can be proved using mathematical induction.

Our last remark concerns Eq. (47). Following the derivations by Pavlis (1991), recurrence relations to compute directly $A_n(r,\psi _0)$ could possibly be obtained. As an advantage, Faà di Bruno’s formula could be avoided (cf. Eqs. 45 and 48), thus slightly simplifying the computations. However, though not confirmed by numerical experiments, an increased numerical instability can be expected when working with the Euclidean distance $l(r,\psi )$ (Eq. 41) instead of its normalized counterpart g(t, u) (Eq. 43). This is the reason why we prefer to work with $A_n(t,u_0)$, even at the cost of somewhat increased complexity of the computation.

Appendix D: Spectral and recurrence relations for an arbitrary radial derivative of truncation coefficients

Differentiating Eq. (33) k times with respect to r, $k\ge 1$, leads to spectral relations

$$\begin{aligned} \frac{\partial ^k}{\partial r^k}Q_{np}^{\mathrm {In}}(r,\psi _0)= & {} \frac{(-1)^k}{R^k} \sum _{l=0}^{\infty } \prod \limits _{j=1}^{k}(l+j)\nonumber \\&\times \left( \frac{R}{r} \right) ^{l+k+1} \, \frac{\prod \limits _{i=1}^{p}(l+4-i)}{p! \, (l+3)}\nonumber \\&\times \int \limits _{0}^{\psi _0} P_l(\cos \psi ) \, P_n(\cos \psi ) \, \sin \psi \, \mathrm d\psi \, .\nonumber \\ \end{aligned}$$

(73)

A similar equation for $\partial ^k (Q_{np}^{\mathrm {Out}}(r,\psi _0)) \slash \partial r^k$, $k\ge 1$, can analogously be obtained from Eq. (39). Let us recall that the integrals can be computed analytically (cf. Appendix B).

The recurrence relations, holding both for near- and far-zone effects, were derived by differentiating Eq. (44) with respect to r. After considering Eq. (45), we get for $k\ge 0$

$$\begin{aligned} \frac{\partial ^k}{\partial r^k}Q_{n1}(r,\psi _0)= & {} G_n^{(k)}(t,u_0) \, , \nonumber \\ \frac{\partial ^k}{\partial r^k}Q_{n2}(r,\psi _0)= & {} \frac{1}{2} \left( -(k-1) G_n^{(k)}(t,u_0) - r \, G_n^{(k+1)}(t,u_0) \right) \, ,\nonumber \\ \frac{\partial ^k}{\partial r^k}Q_{np}(r,\psi _0)= & {} \frac{1}{p!} \sum _{s=1}^{p-2} a_{ps} \sum _{i=0}^{k} \begin{pmatrix} k\\ i \end{pmatrix}\nonumber \\&\times R_{p-s}^{(k-i)}(r)\,G_n^{(p-s+i)}(t,u_0)\, , \quad p \ge 3\, , \end{aligned}$$

(74)

with

$$\begin{aligned} R_w^{(v)}(r) = \frac{\mathrm d^v}{\mathrm dr^v} r^w = {\left\{ \begin{array}{ll} \begin{aligned} &{}r^w\, , \ \ &{}&{}v=0 \, ,\, w\ge 1\,,\\ &{}\prod _{j=1}^{v} (w-j+1)\, r^{w-v}\, , \ \ &{}&{}v\ge 1 \,,\, w\ge 1. \end{aligned} \end{array}\right. } \end{aligned}$$

(75)

In Eq. (74), we applied the general Leibniz rule (Eq. 53) in the third case for $p\ge 3$.