On the spherical and spheroidal harmonic expansion of the gravitational potential of the topographic masses

Abstract

This paper is devoted to the spherical and spheroidal harmonic expansion of the gravitational potential of the topographic masses in the most rigorous way. Such an expansion can be used to compute gravimetric topographic effects for geodetic and geophysical applications. It can also be used to augment a global gravity model to a much higher resolution of the gravitational potential of the topography. A formulation for a spherical harmonic expansion is developed without the spherical approximation. Then, formulas for the spheroidal harmonic expansion are derived. For the latter, Legendre’s functions of the first and second kinds with imaginary variable are expanded in Laurent series. They are then scaled into two real power series of the second eccentricity of the reference ellipsoid. Using these series, formulas for computing the spheroidal harmonic coefficients are reduced to surface harmonic analysis. Two numerical examples are presented. The first is a spherical harmonic expansion to degree and order 2700 by taking advantage of existing software. It demonstrates that rigorous spherical harmonic expansion is possible, but the computed potential on the geoid shows noticeable error pattern at Polar Regions due to the downward continuation from the bounding sphere to the geoid. The second numerical example is the spheroidal expansion to degree and order 180 for the exterior space. The power series of the second eccentricity of the reference ellipsoid is truncated at the eighth order leading to omission errors of 25 nm (RMS) for land areas, with extreme values around 0.5 mm to geoid height. The results show that the ellipsoidal correction is 1.65 m (RMS) over land areas, with maximum value of 13.19 m in the Andes. It shows also that the correction resembles the topography closely, implying that the ellipsoidal correction is rich in all frequencies of the gravity field and not only long wavelength as it is commonly assumed.

Appendix

Integrating Legendre’s functions of the first and second kinds, $P_{nm}$ and $Q_{nm}$, over variable $u$ can reduce the spheroidal harmonic expansion into two-dimensional surface harmonic analysis. To accomplish this, Legendre’s functions are expanded in Laurent series and then scaled into two real functions as follows.

The Legendre function $P_{nm}$ of the complex variable $z=i\frac{u}{E}$ is defined as (Hobson 1931, p. 91)

$$\begin{aligned} P_{nm} (z)=\frac{1}{2^{n}n!}(z^{2}-1)^{m/2}\frac{\mathrm{d}^{n+m}}{\mathrm{d}z^{n+m}}(z^{2}-1)^{n} \end{aligned}$$

(43)

Expanding $(z^{2}-1)^{n}$ into a binominal series, then differentiating it $n +m$ times, we obtain

$$\begin{aligned} P_{nm} (z)&= \frac{1}{2^{n}n!}(z^{2}\!-\!1)^{m/2}\frac{\mathrm{d}^{n+m}}{\mathrm{d}z^{n+m}}(z^{2}-1)^{n}\nonumber \\&= \frac{1}{2^{n}n!}(z^{2}\!-\!1)^{m/2}\sum _{l=0}^r {(-1)^{l}C_n^l \frac{(2n-2l)!}{(n-m-2l)!}} z^{n-m-2l}\nonumber \\ \end{aligned}$$

(44)

where $r=(n-m)/2$ or $r=(n-m-1)/2$ whichever is an integer, and $C_n^l$ is the binominal coefficient given by

$$\begin{aligned} C_n^l =\frac{n!}{l!(n-l)!} \end{aligned}$$

(45)

Using the binominal expansion

$$\begin{aligned} (z^{2}-1)^{m/2}=z^{m}\left( 1-\frac{1}{z^{2}}\right) ^{m/2}=z^{m}\sum _{k=0}^\infty {(-1)^{k}C_{m/2}^k } z^{-2k}\nonumber \\ \end{aligned}$$

(46)

where

$$\begin{aligned} C_{m/2}^k=\frac{\frac{m}{2}\left( \frac{m}{2}-1\right) \left( \frac{m}{2}-2\right) \cdots \left( \frac{m}{2}-k+1\right) }{k!}, \end{aligned}$$

(47)

gives

$$\begin{aligned} P_{nm} (z)&= \frac{1}{2^{n}n!}\sum _{k=0}^\infty {(-1)^{k}C_{m/2}^k } z^{m-2k}\sum _{l=0}^r {(-1)^{l}C_n^l \frac{(2n-2l)!}{(n-m-2l)!}} z^{n-m-2l}\nonumber \\&= \frac{1}{2^{n}n!}\sum _{k=0}^\infty {\sum _{l=0}^r {(-1)^{l+k}C_{m/2}^k C_n^l \frac{(2n-2l)!}{(n-m-2l)!}} } \frac{(n-m)!}{(2n)!}z^{n-2k-2l}\nonumber \\ \end{aligned}$$

(48)

Note that the power of $z$ is independent of $m$. This reflects the fact that the reference ellipsoid is a spheroid and the function $P_{nm} (z)$ is independent of the longitude. The summation over $k$ is infinite when $m$ is odd. For even values of $m$, the binominal coefficient $C_{m/2}^k$ becomes zero after $k=m/2$, thus the summation over $k$ is only to $k=m/2$.

In order to apply the Cauchy theorem for multiplication of two infinite series, we can define the coefficients above $r$ as zeros and extend the upper limit of the summation over $l$ to infinity. Then Eq. (48) can be abbreviated as

$$\begin{aligned} P_{nm} (z)=k_{nm} \sum _{k=0}^\infty {(-1)^{k}c_k^{nm}\,z^{n-2k}} \end{aligned}$$

(49)

where

$$\begin{aligned} k_{nm} =\frac{(2n)!}{2^{n}n!(n-m)!}, \end{aligned}$$

(50)

and $c_k^{nm}$ is the Cauchy product defined as

$$\begin{aligned} c_k^{nm}&= \sum _{l=t}^k {c_{k,l}^{nm}}\end{aligned}$$

(51)

$$\begin{aligned} c_{k,l}^{nm}&= C_{m/2}^l C_n^{k-l} \frac{(n-m)!}{(2n)!}\frac{[2n-2(k-l)]!}{[n-m-2(k-l)]!} \end{aligned}$$

(52)

The summation of the Cauchy product generally starts from zero while the summation in (51) starts from $t$ defined as

$$\begin{aligned} t=k-r\ge 0 \end{aligned}$$

(53)

since the coefficients below $t$ in (51) are zeros.

As mentioned above, the binominal coefficient $C_{m/2}^l$ is zero for $l\ge t>m/2$ and even values of $m$. In this situation the coefficient $c_{k,l}^{nm}$ becomes zero and the infinite series in (49) is reduced to finite.

The coefficient $c_{k,l}^{nm}$ satisfies the following recurrence relationship:

$$\begin{aligned}&c_{k,l+1}^{nm} \nonumber \\&=\frac{(m-2l)(k-l)(2n-2k+2l+1)}{(l+1)(n-m-2k+2l+1) (n-m-2k+2l+2)}c_{k,l}^{nm}\nonumber \\ \end{aligned}$$

(54)

$$\begin{aligned}&c_{k+1,l}^{nm} =\frac{(n-m-2k+2l-1)(n-m-2k+2l)}{2(k-l+1)(2n-2k+2l-1)}c_{k,l}^{nm}\nonumber \\ \end{aligned}$$

(55)

The initial value (54) can be computed using the following recurrence relation:

$$\begin{aligned} \begin{aligned} c_{0,0}^{nm}&=1 \\ c_{k,0}^{nm}&=\frac{n!(n-m)!}{(2n)!}\frac{(2n-2k)!}{k!(n-k)!(n-m-2k)!}\\ c_{k+1,0}^{nm}&=\frac{(n-m-2k-1)(n-m-2k)}{2(k+1)(2n-2k-1)}c_{k,0}^{nm} \end{aligned} \end{aligned}$$

(56)

The Legendre function $Q_{nm}$ can be expressed in a hypergeometric series as (Hobson 1931, p. 108):

$$\begin{aligned} Q_{nm} (z)&= j_{nm} (z^{2}-1)^{m/2}\frac{1}{z^{n+m+1}}\nonumber \\&\times F\left( \frac{n+m+2}{2}, \frac{n+m+1}{2};n+\frac{3}{2};\frac{1}{z^{2}}\right) \nonumber \\&= j_{nm} (z^{2}-1)^{m/2}\frac{1}{z^{n+m+1}}\sum _{l=0}^\infty {f_l \frac{1}{z^{2l}}}\nonumber \\&= j_{nm} \frac{1}{z^{n+1}}\sum _{k=0}^\infty {\sum _{l=0}^\infty {(-1)^{k}} } C_{m/2}^k f_l \frac{1}{z^{2k+2l}} \end{aligned}$$

(57)

where

$$\begin{aligned} j_{nm}&= (-1)^{m}\frac{2^{n}n!(n+m)!}{(2n+1)!}\end{aligned}$$

(58)

$$\begin{aligned} f_l&= \frac{\left( \frac{n+m+2}{2}\right) _l \left( \frac{n+m+3}{2}\right) _l }{(\frac{2n+3}{2})_l l!} \end{aligned}$$

(59)

The Pochhammer symbol $(n)_l$ in (59) is defined as

$$\begin{aligned} (n)_l =\left\{ \begin{array}{ll} 1&{}\quad {l=0} \\ n(n+1)\cdots (n+l-1)&{}\quad l>0 \\ \end{array}\right. \end{aligned}$$

(60)

It is easy to show that the coefficient $f_l$ satisfies the following recurrence relation:

$$\begin{aligned} f_0&= 1 \nonumber \\ f_{l+1}&= \frac{(n+m+2l+2)(n+m+2l+1)}{2(l+1)(2n+2l+3)}f_l \end{aligned}$$

(61)

Equation (61) is useful for computing the coefficient $f_l$ to $l$th term of the hypergeometric series.

Using the Cauchy product, (57) can be written as

$$\begin{aligned} Q_{nm} (z)=j_{nm} \frac{1}{z^{n+1}}\sum _{k=0}^\infty {d_k^{nm} } \frac{1}{z^{2k}} \end{aligned}$$

(62)

where

$$\begin{aligned} d_k^{nm} =\sum _{l=0}^k {(-1)^{l}C_{m/2}^l\,f_{k-l} } =\sum _{l=0}^k {d_{k,l}^{nm}} \end{aligned}$$

(63)

The coefficient $d_{k,l}^{nm}$ satisfies the following recurrence relation:

$$\begin{aligned}&d_{k,0}^{nm} = f_k \nonumber \\&d_{k,l+1}^{nm}\nonumber \\&= -\frac{(m\!-\!2l)(k-l)(2n\!+\!2k\!-\!2l+1)}{(l\!+\!1) (n\!+\!m\!+\!2k-2l)(n+m+2k-2l-1)}d_{k,l}^{nm}\nonumber \\ \end{aligned}$$

(64)

The coefficient $f_k$ can be computed using the recurrence relation (61). Since $d_{k,0}^{nm}$ never becomes zero for every $k$, the series in (63) is an infinite series.

Now we substitute $z=i\frac{u}{E}$ in (49) and (62), and introduce two real functions by scaling the Legendre functions $P_{nm}$ and $Q_{nm}$ in such a way that the series are power series of $\varepsilon ^{2}$:

$$\begin{aligned} p_{nm} (u)&= \frac{1}{i^{n}k_{nm} }\varepsilon ^{n}P_{nm} \left( i\frac{u}{E}\right) =\sum _{k=0}^\infty {\varepsilon ^{2k}c_k^{nm} } \left( \frac{u}{b}\right) ^{n-2k}\end{aligned}$$

(65)

$$\begin{aligned} q_{nm} (u)&= \frac{i^{n+1}}{j_{nm} \varepsilon ^{n+1}}Q_{nm} \left( i\frac{u}{E}\right) \nonumber \\&= \sum _{k=0}^\infty {(-1)^{k}\varepsilon ^{2k}d_k^{nm}}\times \left( \frac{b}{u}\right) ^{n+2k+1} \end{aligned}$$

(66)

where $\varepsilon =E/b$, the second eccentricity of the reference ellipsoid.

On the ellipsoid, the radial functions are reduced to

$$\begin{aligned} p_{nm} (b)&= \sum _{k=0}^\infty {\varepsilon ^{2k}c_k^{nm} } =1+O(\varepsilon ^{2})\end{aligned}$$

(67)

$$\begin{aligned} q_{nm} (b)&= \sum _{k=0}^\infty {(-1)^{k}\varepsilon ^{2k}d_k^{nm} } =1+O(\varepsilon ^{2}) \end{aligned}$$

(68)

where $O(\varepsilon ^{2})$ denotes terms with powers of $\varepsilon ^{2}$.

The radial functions are scaled Legendre’s functions, thus they should satisfy the recurrence relations of Legendre’s functions. For different $n$, the recurrence relations of Legendre’s functions are (Hobson 1931, p. 290)

$$\begin{aligned} \left\{ \begin{array}{l} (2n+1)zP_{nm} (z)-(n-m+1)P_{n+1,m}(z)\\ \,\,\,-(n+m)P_{n-1,m} (z)=0 \\ (2n+1)zQ_{nm} (u)-(n-m+1)Q_{n+1,m} (u)\\ \,\,\,-(n+m)Q_{n-1,m} (u)=0 \\ \end{array}\right. \end{aligned}$$

(69)

For different $m$:

$$\begin{aligned} \left\{ \begin{array}{l} P_{n,m+2} (z)+2(m+1)\frac{z}{\sqrt{z^{2}-1}}P_{n,m+1} (z)\\ \,\,\,-(n-m)(n+m+1)P_{nm} (z)=0 \\ Q_{n,m+2} (z)+2(m+1)\frac{z}{\sqrt{z^{2}-1}}Q_{n,m+1} (z)\\ \,\,\,-(n-m)(n+m+1)Q_{nm} (z)=0 \\ \end{array}\right. \end{aligned}$$

(70)

Using the relationship between the Legendre functions $P_{nm}$ and $Q_{nm}$, we obtain the recurrence relation of the radial functions for different $n$:

$$\begin{aligned} \left\{ \begin{array}{l} \frac{u}{b}p_{nm} (u)-p_{n+1,m} (u)+\varepsilon ^{2}\frac{n^{2}-m^{2}}{4n^{2}-1}p_{n-1,m} (u)=0 \\ \frac{u}{b}q_{nm} (u)+\frac{(n+m+1)(n-m+1)}{(2n+1)(2n+3)}\varepsilon ^{2}q_{n+1,m} (u)\\ \,\,\,-q_{n-1,m} (u)=0 \\ \end{array}\right. \end{aligned}$$

(71)

For different $m$, the recurrence relation is given by

$$\begin{aligned} \left\{ \begin{array}{l} {p_{n,m+2}(u)+\frac{2(m+1)}{n-m-1}\frac{u}{\sqrt{u^{2}+E^{2}}}p_{n,m+1}(u)}\\ \,\,\,{-\frac{n+m+1}{n-m-1}p_{nm} (u)=0} \\ {q_{n,m+2}(u)-\frac{2(m+1)}{n+m+2}\frac{u}{\sqrt{u^{2}+E^{2}}}q_{n,m+1}(u)}\\ \,\,\,{-\frac{n-m}{n+m+2}q_{nm} (u)=0} \\ \end{array}\right. \end{aligned}$$

(72)

Based on (71), one can see that the function $p_{nm}$ can be computed in forward recurrence. However, it is difficult to do the same for $q_{n+1,m}$ since the computation of the latter involves the difference of similar quantities ($q_{n-1,m} -(u/b)q_{nm} )$ divided by the small number $\varepsilon ^{2}$, which is unstable numerically. This agrees with Gil and Segura’s (1998) assessment, and they use backward recurrence relation to compute $Q_{nm}$. The quantity $q_{n+1,m}$ can be computed in the same way.

The initial values of $p_{nm}$ and $q_{nm}$ to degree and order 1 are needed for the above recurrence computation and are shown in the rest of this Appendix. Degree and order 2 are included also and can be used for the verification of the recurrence relations.

Using the definition of $P_{nm}$ (44) and $p_{nm}$ (67), we get

$$\begin{aligned} \begin{aligned} P_{00} (u)&=1, \quad \qquad \qquad \qquad P_{10} (u)=\frac{u}{b},\\ P_{11} (u)&=\frac{1}{b}\sqrt{u^{2}+E^{2}}, \quad P_{20} (u)=\left( \frac{u}{b}\right) ^{2}+\frac{\varepsilon ^{2}}{3},\\ P_{21} (u)&=\frac{u}{b^{2}}\sqrt{u^{2}+E^{2}}, \quad P_{22} (u)=\frac{1}{b^{2}}(u^{2}+E^{2}). \end{aligned} \end{aligned}$$

(73)

The close form of $Q_{nm}$ to degree and order 4 can be found in (Suschowk 1959). Using the definition of $q_{nm}$ in (66), we list the function to degree and order 2 as follows:

$$\begin{aligned} q_{00} (u)&= \frac{1}{\varepsilon }\tau _R , \quad \qquad \qquad \qquad \qquad q_{10} (u)=-\frac{3}{\varepsilon ^{2}}\left( \frac{u}{E}\tau _R -1\right) ,\nonumber \\ q_{11} (u)&= \frac{3}{2\kappa \varepsilon ^{2}}\left( \kappa ^{2}\tau _R -\frac{u}{E}\right) , \quad q_{20} (u)\!=\!\frac{15}{4\varepsilon ^{3}}\left[ \left( 1\!+\!3\frac{u^{2}}{E^{2}}\right) \tau _R \!-\!3\frac{u}{E}\right] ,\nonumber \\ q_{21} (u)&= -\frac{5}{2\kappa \varepsilon ^{3}}\left( 3\frac{u}{E}\kappa ^{2}\tau _R -3\frac{u^{2}}{E^{2}}-2\right) ,\nonumber \\ q_{22} (u)&= \frac{5}{8\kappa ^{2}\varepsilon ^{3}}\left( 3\kappa ^{4}\tau _R -3\frac{u^{3}}{E^{3}}-5\frac{u}{E}\right) . \end{aligned}$$

(74)

where

$$\begin{aligned} \kappa =\frac{\sqrt{u^{2}+E^{2}}}{E}, \tau _R =\tan ^{-1}\frac{E}{u} \end{aligned}$$

(75)