Abstract
Recent numerical studies on external gravity field modelling show that external spherical harmonic series may diverge near or on planetary surfaces. This paper investigates an alternative solution that is still based on external spherical harmonic series, but capable of avoiding the divergence effect. The approach relies on the Runge–Krarup theorem and the iterative downward continuation. In theory, Runge–Kraruptype solutions are able to approximate the true potential within the entire space external to the masses with an arbitrary \(\varepsilon \)accuracy, \(\varepsilon >0\). Using gravity implied by the lunar topography, we show numerically that this technique avoids indeed the divergence effect, at least at the studied 5 arcmin resolution. In the context of the iterative scheme, we show that a function expressed as a truncated solid spherical harmonic expansion on a general starshaped surface possesses an infinite surface spherical harmonic spectrum after it is mapped onto a sphere. We also study the convergence of the gradient approach, which is a technique for efficient gridwise synthesis on irregular surfaces. We show that the resulting Taylor series may converge slowly when analytically upward continuing from points inside the masses. The continuation from the massfree space should therefore be preferred. As an underlying topic of the paper, spherical harmonic coefficients from spectral gravity forward modelling and their Runge–Krarup counterpart are numerically studied. Regarding their different nature, we formulate some research topics that might contribute to a deeper understanding of the current methodologies used to develop combined highdegree spherical harmonic gravity models.
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Notes
 1.
Rigorously, one should refer to the Keldysh–Lavrentiev theorem (e.g. Moritz 1980). The relation between the two theorems is discussed, for instance, in Moritz (1980). Also, whenever we speak about the \(S_{\varOmega }\) surface in theoretical parts of this paper, we assume that it fulfils the regularity condition as discussed in Sansò and Sideris (2013). Whether or not \(S_{\varOmega }\) from Fig. 1 satisfies this condition is not discussed here. After all, in Sect. 4, we end up with discretized numerical experiments and restricted computational accuracy, conditions for which such a debate appears to be meaningless.
 2.
 3.
As a note on the terminology, Hirt and Kuhn (2017) reserved the term Brillouin sphere for what we call the sphere of convergence. Note also that, for practical reasons, Hirt and Kuhn (2017) in their Section 2.2.2 (Case 3) approximated numerically the sphere of convergence by a slightly larger sphere than the actual sphere of convergence. Here, the gravity data given on this sphere are considered as lying on a Brillouin sphere (in the sense of our definition from Sect. 2, see also Fig. 1).
 4.
\(\bar{V}_{nm}^{\mathrm {S}}\) in our notation.
 5.
\(\bar{V}_{nm}^{\mathrm {D}}(N_{\mathrm {D}})\) in our notation.
 6.
\(\bar{V}_{nm}^{\mathrm {D}}(N_{\mathrm {D}})\) in our notation.
 7.
Equation (3) in our paper.
 8.
Equation (2) in our paper.
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Acknowledgements
Blažej Bucha was supported by the project VEGA 1/0954/15. The iterative downward continuation was partially performed at the HPC centre at the Slovak University of Technology in Bratislava and at the Slovak Academy of Sciences, which are parts of the Slovak Infrastructure of High Performance Computing (SIVVP project, ITMS code 26230120002, funded by the European region development funds, ERDF). Christian Hirt would like to thank the German National research foundation for funding under grant no Hi 1760/1. The Newtonian integration in the spatial domain was performed using the supercomputing resources kindly provided by Western Australia’s Pawsey Supercomputing Center. The maps were produced using the Generic Mapping Tools (Wessel and Smith 1998).
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Appendices
Harmonic analysis of a bandlimited function mapped from a bandlimited surface onto the Bjerhammar sphere
To verify numerically the conclusions drawn in Sect. 3.1, we set up a simple numerical experiment with bandlimited gravity data residing on an irregular bandlimited surface. We used the lunar degree180 topographic potential model derived by Hirt and Kuhn (2017) to synthesize the topographic potential V up to degree 180 on the lunar topography S expanded to degrees 0, 90, 180, 360 and 720. The MoonTopo2600pa.shape model by Wieczorek (2015) was used to represent the lunar shape. Note that in this appendix, the input gravity data refer to the bandlimited topographic surface S instead of the nonbandlimited \(S_{\varOmega }\) surface (see Fig. 1) as assumed throughout the paper. The case with \(S_{\varOmega }\) is covered by Sect. 4.1.
Next, we set \(V_0(P_0)=V(P)\), where \(P_0(R_0,\varphi ,\lambda ) \in \varOmega _0\) and \(P(r,\varphi ,\lambda ) \in S\). Then, we harmonically analysed \(V_0(\varOmega _0)\) on \(\varOmega _0\) via the Gauss–Legendre quadrature (e.g. Sneeuw 1994) up to degree 1800. In this way, we obtained a set of spherical harmonic coefficients \(\bar{V}_{nm}\) for each of the five data sets.
In Fig. 8, we show dimensionless degree variances
for the five sets of recovered coefficients and also for the rescaled original coefficients. It can be seen that even though the potential V(S) is evaluated from truncated expansions to degree 180, after mapping from the topography onto the Bjerhammar sphere, \(V_0(\varOmega _0)\) may contain a signal even well beyond this degree, depending on the spectral content of the topography. The smoother the topography, the steeper is the decrease beyond degree 180. We also observe a sort of steplike decline of the spectra beyond degree 180 with sudden drops, which are apparently related to the maximum harmonic degrees of the topography and of the original gravitational potential (see Eq. 15). Also, we see that if the topography S is approximated by a sphere, that is, \(\bar{h}_{00}\ne 0\) m and \(\bar{h}_{nm}=0\) m for all \(n>0\), then there are no additional harmonic degrees beyond 180, resulting in a sudden drop of the spectrum at degree 181. This is in agreement with Eq. (19) if p runs to infinity.
Finally, we performed a closedloop test to show numerically that Eqs. (3) and (14) are equivalent. At the grid points given by the Gauss–Legendre quadrature (degree 1800) and lying on the Bjerhammar sphere, we synthesized the potential from each set of the recovered coefficients via surface spherical harmonic synthesis up to degree 1800 (Eq. 14). These values were then compared with the original input potential on the topography (Eq. 3). The maximum differences in absolute value were found to be \(1.6 \times 10^{7}\ \text {m}^2 \, \text {s}^{2}\) for the topographies expanded up to degrees 0, 90, 180 and 360. We have also verified that a variation of the experiment with the maximum degree 3600 does not yield further improvement. For the degree720 topography, we observed an inferior accuracy of \(2.5 \times 10^{4}\ \text {m}^2 \, \text {s}^{2}\) (maximum difference in absolute value) which reflects that some relevant parts of the spectrum were apparently not recovered by the spherical harmonic analysis up to degree 1800. After the same experiment but with the maximum degree 3600, the maximum difference in absolute value dropped to \(6.3 \times 10^{7}\ \text {m}^2 \, \text {s}^{2}\) which supports our interpretation.
Gradient approach
In this appendix, we discuss the gradient approach, which we deploy to evaluate \(f_i\) on the irregular \(S_{\varOmega }\) surface (the fourth step in the flowchart depicted in Fig. 2) and to compute the summation in Eq. (7) (the sixth step in the same flowchart). We start by introducing the gradient approach in terms of the negative firstorder radial derivatives of the topographic potential, here called gravity disturbances,
Here, the gravity disturbances play the role of the function \(f_i\). When working with some other gravity field quantity, one can proceed analogously.
The goal of this appendix is to obtain efficiently \(\delta g^{\mathrm {D}}\) at grids residing on \(S_{\varOmega }\). Since the direct pointwise evaluation of Eq. (26) can be extremely timeconsuming (e.g. Hirt 2012), we employ the gradient approach, which offers a substantial acceleration. The gradient approach combines a fast FFTbased synthesis on a regular surface (e.g. a sphere or an ellipsoid of revolution) and an analytical continuation via the truncated Taylor expansion. In our case, we get
where \(\varOmega _{\mathrm {a}}\) is an auxiliary sphere from which the gravity disturbances are analytically continued to the boundary surface \(S_{\varOmega }\) and \(h_\mathrm{a}\) denotes the radial distance between \(S_{\varOmega }\) and \(\varOmega _{\mathrm {a}}\). In Eq. (27), the zerothorder derivative is given by Eq. (26) and the radial derivatives for \(k \ge 1\) read (Hirt 2012)
Here, both Eqs. (26) and (28) are evaluated on the auxiliary sphere via the efficient FFTbased synthesis and then are substituted into Eq. (27), thus obtaining the sought \(\delta g^{\mathrm {D}}\) on \(S_{\varOmega }\).
Let us study the convergence of the Taylor series in Eq. (27) for three different auxiliary spheres \(\varOmega _{\mathrm {a}}\) of the Moon:

(A)
\(\varOmega _{\mathrm {a}}\) coincides with the Bjerhammar sphere \(\varOmega _0\) with the radius \(R_{\mathrm {a}}=R_0=R1\ \text {m}\), where \(R=1738\ \text {km}\), that is, \(\varOmega _{\mathrm {a}}\) is completely below the boundary surface,

(B)
\(\varOmega _{\mathrm {a}}\) is taken as the Brillouin sphere \(\varOmega _1\) with the radius \(R_{\mathrm {a}}=R_1=R+10\ \text {km}\), that is, the whole sphere \(\varOmega _{\mathrm {a}}\) passes above the fieldgenerating masses,

(C)
\(\varOmega _{\mathrm {a}}\) passes somewhere in the middle between the highest and the lowest point of \(S_{\varOmega }\) (\(R_{\mathrm {a}}=R+5\ \text {km}\)), that is, \(\varOmega _{\mathrm {a}}\) is located either within or outside the masses, depending on the position.
The first case leads to an upward continuation from \(\varOmega _0\), the second variant to a downward continuation from \(\varOmega _1\), and the last variant relies on a combined upward/downward continuation. The main advantage of the first case, where \(\varOmega _{\mathrm {a}}=\varOmega _0\), is that the radial derivatives for \(k=1,\ldots ,K\) can later be utilized—as precomputed quantities—in the iterative scheme (see Eq. 7). From a practical computational point of view, this merges the two synthesis steps in the iterative scheme (the fourth and sixth step in the flowchart in Fig. 2) into one. This is not possible for the other two variants, at least not in a straightforward manner.
To set up a realistic numerical experiment, we took from Hirt and Kuhn (2017) those grids of gravity disturbances that reside on \(S_{\varOmega }\) and are implied by the degree720 and degree2160 lunar topographic mass distributions. Next, we mapped the grids onto the Bjerhammar sphere \(\varOmega _0\) and, finally, harmonically analysed them via the Gauss–Legendre quadrature. In this way, we obtained two sets of \(\bar{V}^\mathrm {D}_{nm}(N_{\mathrm {D}})\) up to degree 2160. Since the data are given originally at global 5 arcmin equiangular grids, we applied a 2D spline interpolation to compute the values at the grid points required by the quadrature. These two steps (mapping onto \(\varOmega _0\) and spherical harmonic analysis) are, in fact, the initial steps 2 and 3 from the iterative scheme depicted in Fig. 2, so this is a realistic scenario.
From the two sets of coefficients, we computed gravity disturbances on the respective \(S_{\varOmega }\) surfaces via the exact pointwise spherical harmonic synthesis. These values serve as the reference (or exact) ones for the validation. To investigate the convergence of the three variants A–C, we used then the same coefficients to synthesize gravity disturbances on \(S_{\varOmega }\) via the Taylor series in Eq. (27) with \(K=0,1,\ldots ,10,15,\ldots ,45\). To this end, we employed the isGrafLab software by Bucha and Janák (2014), which relies on the gradient approach.
Figure 9 clearly shows that the Taylor series converges most rapidly in case C, where the auxiliary sphere is in the middle between the Bjerhammar and Brillouin sphere. This is probably caused by the approximately halved continuation distances when compared to A and B (see, e.g. Hirt 2012). It is interesting to observe how slowly the Taylor series converges in case A. Based on Fig. 9, it can be expected that the lunar topography expanded beyond degree 2160 may require even larger values of K to achieve an accurate approximation of \(f_i\) on \(S_{\varOmega }\) with A. We attribute this to the fact that the scenario A relies on an upward continuation from the Bjerhammar sphere, which is completely within the body. As a result, the downward continued external potential may oscillate widely on that sphere. In other words, we need radial derivatives of a strongly varying function \(f_i(\varOmega _0)\) which slows down the convergence of the Taylor series.
Unfortunately, this also implies that we may need a very large number of the radial derivatives of \(f_i\) on \(\varOmega _0\) in Eq. (7) to achieve the desired accuracy. The main problem with very high values of K, say around 100 or larger, is that the term \(\prod _{i=0}^{k}(n+i+1)\) in Eq. (28) overflows in double precision when, for instance, \(k=92\) and \(n=2160\). So sooner or later, we will face a situation where an accurate computation of the summation in Eq. (7) will become difficult in double precision.
Fortunately, there is an easy way how to evaluate accurately both Eq. (7) and (27). From the Taylor expansion, it follows that
where h is the radial distance between \(P \in S_{\varOmega }\) and \(P_0 \in \varOmega _0\). This means that in order to get the sum in Eq. (7), we do not have to compute the individual derivatives of \(f_i\) on \(\varOmega _0\) separately. Instead, we only need the values of \(f_i\) at both P and \(P_0\). From these values, we can then easily compute the sum via Eq. (29). As a benefit of Eq. (29), it gives us the sum for \(k=1,\ldots ,\infty \), while the approach with the individual computation of radial derivatives forces us to truncate the summation to some finite K. The Taylor series in Eq. (29) can be interpreted as an upward continuation of \(f_i\) from the Bjerhammar sphere \(\varOmega _0\) to \(S_{\varOmega }\). However, note that the difference \(f_i(P)  f_i(P_0)\) can also be interpreted via the downward continuation from \(S_{\varOmega }\) to \(\varOmega _0\),
When the Taylor series is infinite, the lefthand sides of Eqs. (29) and (30) become equal. Note that this holds true only for an infinite Taylor expansion.
Concluding, the sum in Eq. (7) can be obtained (even for \(K=\infty \)) by Eq. (29), provided that we have the values of \(f_i\) on the boundary surface \(S_{\varOmega }\) and on the Bjerhammar sphere \(\varOmega _0\). Since the latter can easily be done via the FFTbased spherical harmonic synthesis, the problem of both computationally demanding spherical harmonic synthesis steps (Eqs. 7 and 27) is reduced to a single one, that is, the evaluation of \(f_i\) on \(S_{\varOmega }\). According to Fig. 9, this task can most efficiently be completed with a combined upward/downward continuation (case C). However, it has to be emphasized that, in case C, approximately one half of the gradients are computed in those areas of \(\varOmega _{\mathrm {a}}\) that are buried within the topographic masses. From A we know that this might not be the best choice when the input signal is complex and the continuation distances are large. So sooner or later, we may expect similar difficulties which might then again result in a slow convergence of the Taylor series.
As a general recommendation, the upward continuation from points located within the body should be avoided (cases A and C), as this may slow down the speed of convergence of the Taylor series. Obviously, this problem becomes more serious with increasing harmonic degree of the expansion. In general, the analytical continuation should therefore be performed from a Brillouin sphere (case B), which is completely outside the body. This ensures that the behaviour of the function under consideration will be much more reasonable, resulting in a potentially faster convergence of the Taylor series. For our numerical experiments in Sect. 4, the combined upward/downward continuation seems to be a reasonable compromise and was therefore used in all computations.
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Bucha, B., Hirt, C. & Kuhn, M. Divergencefree spherical harmonic gravity field modelling based on the Runge–Krarup theorem: a case study for the Moon. J Geod 93, 489–513 (2019). https://doi.org/10.1007/s0019001811774
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Keywords
 Gravity field modelling
 Spherical harmonics
 Runge–Krarup theorem
 Iterative downward continuation
 Divergence effect
 Gradient approach