A numerical study on the integration radius separating convergent and divergent spherical harmonic series of topography-implied gravity

Abstract

We show that far-zone topography-implied gravitational effects may be accurately computed via external spherical harmonics not only above the limit sphere encompassing all the masses, but also inside it on planetary topographies. Although a rigorous mathematical proof is still missing, our numerical experiments indicate that this is possible, provided that near-zone masses within a certain spherical cap centred at the evaluation point are omitted from gravity forward modelling. We formulate and numerically examine a hypothesis, saying that in order to achieve convergence, the cap size needs to be larger than the highest topographical height. The hypothesis relies on the spherical arrangement of field-generating topographic masses and strictly positive topographic heights. To put our hypothesis to a test, we gravity forward model lunar degree-2160 topography using a constant mass density and expand the far-zone gravitational effects up to degree 10,800. The results are compared with respect to divergence-free reference values from spatial-domain gravity forward modelling. By systematically increasing the cap radius from 2.5 km up to 100.0 km (the maximum topographic height is ${\sim }\,20\,\mathrm {km}$), we obtained results that appear to be in line with our hypothesis. Nonetheless, a rigorous mathematical proof still needs to be found to prove whether the hypothesis is true or false. The outcomes of the paper could be beneficial for the study of convergence/divergence of spherical harmonics on planetary surfaces and for geoid computations based on spherical harmonic expansion of far-zone gravitational effects.

Appendices

Appendix A: Radius of convergence of the Taylor series from Eq. (10)—a numerical approach

After our many failed attempts to find the radius of convergence of Eq. (10) for $r > R_{\mathrm {int}}$ analytically, we were forced to opt for a numerical approach. In this “Appendix”, we provide a systematic numerical study that, at least to some extent, reveals the series behaviour for various values of $r \ge R_{\mathrm {int}}$, $\psi$ and $R_{\mathrm {int}}$. Finally, we arrive to numerical estimates of radii of convergence, based on which we formulated the hypothesis from Sect. 3.1 that anticipates the analytical solution.

Fig. 8

Coefficients $c_i$ of the Taylor series (10) for $r = R_{\mathrm {int}} + \hat{H}_{\mathrm {int}}$, $\hat{H}_{\mathrm {int}} = 15{,}000\ \mathrm {m}$, $\psi = 0.1^{\circ }$ and $R_{\mathrm {int}} = 1{,}728{,}200\ \mathrm {m}$. Since the coefficients cover thousands of orders of magnitude and change their sign, shown are as $-\mathrm {sign}(c_i)\, \log _{10}\left( |c_i| \right)$. This ensures that the depicted values share the same sign as the $c_i$ coefficients, because $\log _{10}\left( |c_i| \right) < 0$ for all studied i

Importantly, we emphasize that the numerical approach is exploited out of necessity and that it can never prove/disprove that the series converges/diverges, nor it can reveal its radius of convergence. To answer these questions rigorously, an analytical solution is necessary, no matter how advanced or sophisticated are the numerical experiments. Nevertheless, our hypothesis proved to be useful for us through the work on the paper, so is reported here, hoping that others might perhaps benefit from it, too.

To obtain meaningful information on the behaviour of the infinite Taylor series (10) for $r \ge R_{\mathrm {int}}$ and its radius of convergence, denoted as D, the series must not be truncated too soon. Our assumption is that the estimation of D improves with increasing truncation order $i_{\max }$. However, large $i_{\max }$ go hand in hand with numerical issues, since the kernels $M_i(r,\psi ,R_{\mathrm {int}})$ are singular for $r = R_{\mathrm {int}}$ and $\psi = 0$. Moreover, they involve factorials and double factorials of ${\sim }i_{\max }$ as well as raising r and $1 /l(r,\psi ,R_{\mathrm {int}})$ to high powers. As a result, a straightforward evaluation of $M_i(r,\psi ,R_{\mathrm {int}})$ via Eq. (A.5.14) of Martinec (1998) may lead to fairly inaccurate outputs in double precision, even when $i_{\max }$ is as low as 50 or so. Considering that our target $i_{\max }$ reaches at least a few hundreds, we extend the number of significant digits to 10, 240 in order to achieve accurate results. Although smaller number of digits may also be sufficient, we confirm that doubling this number does not have any meaningful effect on our experiments. To extend the number of significant digits, we use the ADVANPIX toolbox (www.advanpix.com), a multiprecision computing toolbox for MATLAB.

To get a robust idea on the series behaviour, we set the parameters of the Taylor series as follows:

• $r=R_{\mathrm {int}}+\hat{H}_{\mathrm {int}}$ with $\hat{H}_{\mathrm {int}} = 0\ \mathrm {m}$, $5000\ \mathrm {m}$, $10{,}000\ \mathrm {m}$, $\dots$, $30{,}000\ \mathrm {m}$;

• $\psi = 0^{\circ }$, $0.001^{\circ }$, $0.01^{\circ }$, $0.1^{\circ }$, $0.5^{\circ }$, $1^{\circ }$, $5^{\circ }$, $10^{\circ }$, $30^{\circ }$, $90^{\circ }$;

• $R_{\mathrm {int}} = 1{,}728{,}200\ \mathrm {m}$, $6{,}378{,}137\ \mathrm {m}$;

• $i_{\max }=1500$.

Clearly, we focus mostly on small integration radii, for instance, $\psi = 0.001^{\circ }$ (${\sim }\,30\ \mathrm {m}$ for Moon, ${\sim }\,111\ \mathrm {m}$ for Earth), but included are also large radii up to $\psi = 90^{\circ }$ (${\sim }\,2700\ \mathrm {km}$ for Moon, ${\sim }\,10{,}000\ \mathrm {km}$ for Earth). The values of $R_{\mathrm {int}}$ are chosen to correspond to the internal sphere $\Omega _{\mathrm {int}}$ from Sect. 4 and to the equatorial radius of the Earth, respectively. Having in mind near-surface applications, the height of evaluation points $\hat{H}_{\mathrm {int}}$ gradually grows from 0 up to $30{,}000\ \mathrm {m}$. Note that the case with negative heights is not studied here. Finally, the truncation order $i_{\max }=1500$ was empirically found to be a reasonable compromise between the accuracy and computational costs.

Considering all combinations of the input parameters, this set-up yields 138 Taylor series (cases $\hat{H}_{\mathrm {int}} = 0$ and $\psi =0$ are not investigated due to the singularity issue). As an example, we show in Fig. 8 the coefficients $c_i$ (see Eq. 10),

$$\begin{aligned} c_i = \frac{1}{i!} \, \frac{M_i(R_{\mathrm {int}} + \hat{H}_{\mathrm {int}},\psi ,R_{\mathrm {int}})}{R_{\mathrm {int}}^i}{,} \quad i = 0,1,2,\dots \end{aligned}$$

(20)

that were obtained for one of the series. The figure demonstrates that the $c_i$ coefficients decrease exponentially and change their sign with i. While for $r=R_{\mathrm {int}}$ the sign follows almost exclusively the pattern $+$, $+$, −, − (or vice versa) as already known from Martinec (1998), the number of successive positive/negative signs grows with increasing r. For instance, in Fig. 8, this number equals ${\sim }\,15$ for $\hat{H}_{\mathrm {int}} = 15{,}000\ \mathrm {m}$, while it is ${\sim }\,30$ for the same radii, but with $\hat{H}_{\mathrm {int}} = 30{,}000\ \mathrm {m}$ (not shown here). The sign pattern is affected also by $\psi$. For instance, for $\psi = 0$, all studied $c_i$ are positive.

We employ two independent methods to numerically estimate the radii of convergence of these series.

• The root test Using the root test, the radius of convergence D can be found as $D = 1 / C$, where

  $$\begin{aligned} C = \limsup \limits _{i \rightarrow \infty } \root i \of {| c_i|}. \end{aligned}$$

  (21)

  In our case, we find a numerical approximation of C simply as

  $$\begin{aligned} C \approx \root i_{\max } \of {| c_{i_{\max }} |}{,} \quad i_{\max } > 0{.} \end{aligned}$$

  (22)

• The Domb–Sykes plot (e.g. Hinch 1991) with coefficients evaluated after Mercer and Roberts (1990). Using the $c_i$ coefficients, we compute new coefficients $b_i^2$ as (Mercer and Roberts 1990)

  $$\begin{aligned} b_i^2 = \frac{c_{i+1}\, c_{i-1} - c_i^2}{c_i\, c_{i-2} - c_{i-1}^2}, \quad i = 100,101,\dots ,i_{\max }-1. \end{aligned}$$

  (23)

  The $b_i$ coefficients are then plotted against $1 /i$ and extrapolated for $1 /i = 0$. The vertical intercept finally yields an approximation of C. In practice, we find the vertical intercept by fitting a linear function $f(x) = \alpha \,x + \beta$, $x = 1 /i$, through $b_i$. The least-squares estimate of $\beta$ then approximates C, from which D is found similarly as in the previous case. Note that we excluded the $b_i$ coefficients for $i=2,3,\dots ,99$ from the fitting process. This ensures that we use only those $b_i$, for which the behaviour of the plotted curve is already settled down, at least to some extent. The threshold value of $i = 100$ was found empirically and is not necessarily the best possible choice.

Fig. 9

Radius of convergence of the series from Fig. 8 estimated via the root test and the Domb–Sykes approach (solid lines). Note that each estimation method is depicted using its own vertical axis, given that the Domb–Sykes method converges significantly faster. For each axis, we also show the (same) prediction for the radius of convergence being equal to $l(r,\psi ,R_{\mathrm {int}})$ (dashed lines). While the root test is used for $i_{\max }=1,2,3,\dots ,1500$, the Domb–Sykes approach is applied only to $i_{\max } = 110, 120, 330, \dots ,1500$ and then is extrapolated for the other values of $i_{\max }$ between 110 and 1500

As an indirect check of the accuracy of the two methods, we assume at first that $r=R_{\mathrm {int}} = 1{,}728{,}200\ \mathrm {m}$ and $\psi$ is some small but nonzero spherical distance, say, $\psi =0.01^{\circ }$. Substituting these values in Eq. (10), we obtain a Taylor series, for which Martinec (1998) has found the radius of convergence to be $l(R_{\mathrm {int}},\psi ,R_{\mathrm {int}})$ (see Eq. 14). This analytical solution can therefore play the role of the reference value that can be used to check the two proposed methods. Using $i_{\max }=1500$, we have found that the root test and the Domb–Sykes approach give radii of convergence of $303.5645\ \mathrm {m}$ and $301.6281\ \mathrm {m}$, respectively. Considering that the reference value equals $301.6278\ \mathrm {m}$, the Domb–Sykes approach yields a sub-millimetre accuracy and the root test is accurate at the level of a few metres. Combining these results with other experiments—still relying on $r=R_{\mathrm {int}} = 1{,}728{,}200\ \mathrm {m}$ but different $\psi$ (see the $\hat{H}_{\mathrm {int}} = 0$ row in Table 3)—we conclude that the two numerical methods can indeed provide meaningful estimates of the radius of convergence.

Next, we apply the two methods to estimate radii of convergence D for all studied Taylor series with $r \ge R_{\mathrm {int}}$, that is, including the case $r > R_{\mathrm {int}}$, for which an analytical solution is not known to us. In Fig. 9, we show estimates of D as a function of truncation order $i_{\max }$ for one such series. The figure demonstrates that our numerical estimates of D appear to converge to some unknown value as $i_{\max }$ grows. It is seen that the Domb–Sykes curve has generally a significantly smaller slope which may indicate that this method converges to the true (but unknown) value faster than the root test. This is in agreement with the experiment from the previous paragraph.

In the same way, we analysed all the studied Taylor series. One exception is that for $\psi = 0$ and all r, we did not apply the Domb–Sykes approach. This is because we encountered the issues discussed by Mercer and Roberts (1990) in their “Appendix” which, for this case, prevent the use of the method. By inspecting all the other results, we have noticed that all the estimations of D appear to approach the value of the spatial distance $l(r,\psi ,R_{\mathrm {int}})$ (see Eq. 9). Using once again Fig. 9 as an example, the root test and the Domb–Sykes method (both with $i_{\max }=1500$) yield $D \approx 15{,}440.72\ \mathrm {m}$ and $15{,}302.87\ \mathrm {m}$, respectively, while the spatial distance $l(r,\psi ,R_{\mathrm {int}})$ for this series equals to ${\sim }\,15{,}302.84\ \mathrm {m}$. Some further sample outputs from the experiment are reported in Table 3. For instance, for $\hat{H}_{\mathrm {int}} = 20{,}000\ \mathrm {m}$, $\psi = 0.1^{\circ }$ and $R_{\mathrm {int}} = 1{,}728{,}200\ \mathrm {m}$, the absolute value of the difference between the Domb–Sykes approach and $l(r,\psi ,R_{\mathrm {int}})$ is only ${\sim }\,0.006\ \mathrm {m}$, which is the smallest absolute error that we achieved for $r > R_{\mathrm {int}}$ (relative error $3.2 \times 10^{-7}$).

Table 3 Differences between numerical estimates of D and $l(r,\psi ,R_{\mathrm {int}})$ for $R_{\mathrm {int}} = 1{,}728{,}200\ \mathrm {m}$ and various $r = R_{\mathrm {int}} + \hat{H}_{\mathrm {int}}$ and $\psi$

Finally, we acknowledge that the same experiments but for $R_{\mathrm {int}} = 6{,}378{,}137\ \mathrm {m}$ (the Earth’s equatorial radius) led to the same observations as already discussed, so are not reported here for the sake of brevity.

Combining all the results and observations from this “Appendix”, we are finally able to formulate the hypothesis from Sect. 3.1. The hypothesis leads to the following convergence condition for the series (10),

$$\begin{aligned} \psi > \psi _{\mathrm {conv}} = \arccos \left( \frac{r^2 + R_{\mathrm {int}}^2 - (\max (\hat{H}_{\mathrm {int}}))^2}{2\, r\, R_{\mathrm {int}}} \right) {.} \end{aligned}$$

(24)

Here, it is assumed that r satisfies $R_{\mathrm {int}} \le r \le R_{\mathrm {int}} + \max (\hat{H}_{\mathrm {int}})$. For $r>R_{\mathrm {int}}+\max (\hat{H}_{\mathrm {int}})$, it follows directly from the hypothesis that the series (10) converges for all $\psi$. Note that if $r = R_{\mathrm {int}}$, then $\psi _{\mathrm {conv}}$ from Eq. (24) equals $\psi _{\mathrm {conv}}$ from Eq. (15). Similarly as in Sect. 3.1, the hypothesis and Eq. (24) rely on the assumption of positive topographic heights $\hat{H}_{\mathrm {int}}(\varphi ^{\prime },\lambda ^{\prime }) > 0$ for all $(\varphi ^{\prime },\lambda ^{\prime })$.

Fig. 10

The same as Fig. 3, but with input heights $\hat{H}$ referenced to the mean sphere $\Omega$ (Eq. 25). Beyond the $10^8$ mGal upper limit of the vertical axis, the curves for $l_0=2.5$, 5.0, 7.5 and 10.0 km continue in raising and reach maxima of $4.9 \times 10^{26}$, $1.4 \times 10^{22}$, $1.6 \times 10^{17}$ and $1.9 \times 10^{10}$ mGal for $p_{\max }=65$, respectively

Once again, we emphasize that the validity of the hypothesis is not known to us and that it needs to undergo a rigorous mathematical investigation, confirming analytically whether it is true or false.

Assuming now the hypothesis is true, the following observations can be made in relation to cap-modified spectral gravity forward modelling.

• With increasing r (but satisfying $R_{\mathrm {int}} < r \le R_{\mathrm {int}} + \max (\hat{H}_{\mathrm {int}})$), $\psi _{\mathrm {conv}}$ from Eq. (24) decreases. Therefore, as the computation point moves radially upwards from the internal sphere, smaller and smaller amount of near-zone masses needs to be omitted from gravity forward modelling in order to achieve convergence of far-zone gravitational effects.

• If $\psi _0 = 0$, Eq. (5) converges for all $r > \max (R_{\mathrm {int}} + \hat{H}_{\mathrm {int}})$. This is the same convergence condition as that for global spectral gravity forward modelling (e.g. Wieczorek and Phillips 1998) which could be expected, given that Eq. (5) with $\psi _0 = 0$ is identical to global spectral gravity forward modelling (for further details, see Eq. 7 of Bucha et al. 2019a).

Appendix B: Cap-modified spectral technique applied directly to the $\hat{H}$ masses referenced to $\Omega$

In practical applications of global and cap-modified spectral gravity forward modelling, one frequently forward models directly the $\hat{H}$ masses instead of combining $\hat{H}_{\mathrm {int}}$ with $\hat{H}_{\mathrm {B}}$ as discussed in Eqs. (4), (5) and (6). This leads to the following counterpart of the three equations,

$$\begin{aligned} V^{\mathrm {Out}}(r,\varphi ,\lambda )= & {} 2\pi \,G\,\rho \,R^2 \sum _{p=1}^{p_{\max }} \sum _{n=0}^{p\times n_{\max }} Q^{\mathrm {Out}}_{np}(r,\psi _0,R)\nonumber \\&\times \sum _{m=-n}^{n} \bar{H}_{nmp} \, \bar{Y}_{nm}(\varphi ,\lambda ) {,} \end{aligned}$$

(25)

where $\bar{H}_{nmp}$ are the spherical harmonic coefficients of $\left( \hat{H} /R \right) ^p$.

One of the reasons to use Eq. (25) is that $\max (|\hat{H}|)$ is generally about half of $\max (|\hat{H}_\mathrm {int}|)$, provided that $R_{\mathrm {int}}$ is close to but smaller than $\min (r_{\mathrm {S}})$ and R is close to the mean topographic sphere (see Fig. 1). This causes the binomial series involved in the spectral techniques¹ to converge more rapidly for the former case (e.g. Sun and Sjöberg 2001). However, we did not employ Eq. (25) in Sects. 2–4, because $\hat{H}$ measured from the reference sphere $\Omega$ may take either a positive or a negative value for $(\varphi ^{\prime }, \lambda ^{\prime })$ (see Fig. 1), thus violates the condition of strictly positive topographic heights (cf. Sects. 2 and 3).

In this “Appendix”, we repeat the experiments from Sect. 4.4, but this time we apply directly Eq. (25) instead of Eqs. (4), (5) and (6). This is done despite the fact that the theoretical discussion from Sect. 3 on the convergence/divergence may not be applicable here due to the negative heights $\hat{H}$ covering roughly half of the sphere. Nevertheless, the experiments may reveal how much the selection of the sphere to which the topographic heights refer (mean vs. internal) affects the cap-modified spectral technique in practical applications. A similar comparison study for global spectral gravity forward modelling (Eq. 25 with $\psi _0 = 0$) was done by Šprlák et al. (2018), but no such tests have yet been presented for the cap-modified spectral technique. Naturally, this experiment required to compute an entirely new set of spherical harmonic coefficients for powers of $\hat{H}/R$ as well as of truncation coefficients $Q^{\mathrm {Out}}_{np}(r,\psi _0,R)$, both of which employ now the reference sphere with the radius R instead of $R_{\mathrm {int}}$.

In the same manner as for Fig. 3, results of the experiment are shown in Fig. 10. When compared with Fig. 3, this figure shows that Eq. (25) appears to converge/diverge faster when using the mean sphere to reference the topographic heights. More specifically, while $p_{\max }$ had to reach ${\sim }\,30$ to observe the divergence effect in Fig. 3 for $l_0 \le 12.5\ \mathrm {km}$, here the respective series diverge already with $p_{\max }=15$ or even with $p_{\max }=5$. For $l_0 = 15.0\ \mathrm {km}$, the RMS error starts to slightly oscillate and grow when $p_{\max }\ge 30$. If Eq. (25) indeed converges/diverges faster than Eqs. (4), (5) and (6), this might indicate the onset of a divergence behaviour. In that case, this could support our statement made in relation to Fig. 3 that the RMS errors for $l_0=15.0\ \mathrm {km}$ and $20.0\ \mathrm {km}$ in Fig. 3 might start to grow once $p_{\max }$ exceeds 65 in that figure.

Similarly as in Sect. 4, the computations were repeated also for $l_0=18$, 19, 21, 22 and 25 km, including the experiments in meridian planes from Sect. 4.5. The obtained results are in line with the here presented conclusions.