Divergence-free spherical harmonic gravity field modelling based on the Runge–Krarup theorem: a case study for the Moon

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–Krarup-type 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 arc-min resolution. In the context of the iterative scheme, we show that a function expressed as a truncated solid spherical harmonic expansion on a general star-shaped 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 grid-wise 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 mass-free 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 high-degree spherical harmonic gravity models.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Notes

  1. 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. 2.

    http://ddfe.curtin.edu.au/gravitymodels/2017Archive_GFM_Moon.

  3. 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. 4.

    \(\bar{V}_{nm}^{\mathrm {S}}\) in our notation.

  5. 5.

    \(\bar{V}_{nm}^{\mathrm {D}}(N_{\mathrm {D}})\) in our notation.

  6. 6.

    \(\bar{V}_{nm}^{\mathrm {D}}(N_{\mathrm {D}})\) in our notation.

  7. 7.

    Equation (3) in our paper.

  8. 8.

    Equation (2) in our paper.

References

  1. Ågren J (2004) The analytical continuation bias in geoid determination using potential coefficients and terrestrial gravity data. J Geodesy 78:314–332. https://doi.org/10.1007/s00190-004-0395-0

    Article  Google Scholar 

  2. Balmino G (1994) Gravitational potential harmonics from the shape of an homogeneous body. Celest Mech Dyn Astron 60:331–364

    Article  Google Scholar 

  3. Balmino G, Vales N, Bonvalot S, Briais A (2012) Spherical harmonic modelling to ultra-high degree of Bouguer and isostatic anomalies. J Geodesy 86:499–520. https://doi.org/10.1007/s00190-011-0533-4

    Article  Google Scholar 

  4. Barthelmes F (2013) Definition of functionals of the geopotential and their calculation from spherical harmonic models: theory and formulas used by the calculation service of the International Centre for Global Earth models (ICGEM). Scientific Technical Report STR09/02, GFZ German Research Centre for Geosciences, Potsdam, Germany, 32 pp. http://icgem.gfz-potsdam.de. Accessed 11 Oct 2017

  5. Bjerhammar A (1963) A note on gravity reduction to a spherical surface. Tellus 15:319–320. https://doi.org/10.1111/j.2153-3490.1963.tb01394.x

    Article  Google Scholar 

  6. Bucha B, Janák J (2014) A MATLAB-based graphical user interface program for computing functionals of the geopotential up to ultra-high degrees and orders: efficient computation at irregular surfaces. Comput Geosci 66:219–227. https://doi.org/10.1016/j.cageo.2014.02.005

    Article  Google Scholar 

  7. Bucha B, Janák J, Papčo J, Bezděk A (2016) High-resolution regional gravity field modelling in a mountainous area from terrestrial gravity data. Geophys J Int 207:949–966. https://doi.org/10.1093/gji/ggw311

    Article  Google Scholar 

  8. Colombo OL (1981) Numerical methods for harmonic analysis on the sphere. Report No. 310, Department of Geodetic Science and Surveying, The Ohio State University, Columbus, Ohio, 140 pp

  9. D’Urso MG (2014) Analytical computation of gravity effects for polyhedral bodies. J Geodesy 88:13–29. https://doi.org/10.1007/s00190-013-0664-x

    Article  Google Scholar 

  10. D’Urso MG, Trotta S (2015) Comparative assessment of linear and bilinear prism-based strategies for terrain correction computations. J Geodesy 89:199–215. https://doi.org/10.1007/s00190-014-0770-4

    Article  Google Scholar 

  11. Fantino E, Casotto S (2009) Methods of harmonic synthesis for global geopotential models and their first-, second- and third-order gradients. J Geodesy 83:595–619. https://doi.org/10.1007/s00190-008-0275-0

    Article  Google Scholar 

  12. Fecher T, Pail R, Gruber T, Schuh WD, Kusche J, Brockmann JM, Loth I, Müller S, Eicker A, Schall J, Mayer-Gürr T, Kvas A, Klinger B, Rieser D, Zehentner N, Baur O, Höck E, Krauss S, Jäggi A, Meyer U, Prange L, Maier A (2017) GOCO05c: a new combined gravity field model based on full normal equations and regionally varying weighting. Surv Geophys 38:571–590. https://doi.org/10.1007/s10712-016-9406-y

    Article  Google Scholar 

  13. Freeden W, Schneider F (1998a) An integrated wavelet concept of physical geodesy. J Geodesy 72:259–281

    Article  Google Scholar 

  14. Freeden W, Schneider F (1998b) Wavelet approximations on closed surfaces and their application to boundary-value problems of potential theory. Math Methods Appl Sci 21:129–163

    Article  Google Scholar 

  15. Garmier R, Barriot JP (2001) Ellipsoidal harmonic expansions of the gravitational potential: theory and applications. Celest Mech Dyn Astron 79:235–275

    Article  Google Scholar 

  16. Grafarend EW (2001) The spherical horizontal and spherical vertical boundary value problem—vertical deflections and geoidal undulations—the completed Meissl diagram. J Geodesy 75:363–390

    Article  Google Scholar 

  17. Grafarend EW, Engels J (1993) The gravitational field of topographic-isostatic masses and the hypothesis of mass condensation. Surv Geohpys 140:495–524

    Article  Google Scholar 

  18. Grombein T, Seitz K, Heck B (2013) Optimized formulas for the gravitational field of a tesseroid. J Geodesy 87:645–660. https://doi.org/10.1007/s00190-013-0636-1

    Article  Google Scholar 

  19. Heck B, Seitz K (2007) A comparison of tesseroid, prism and point-mass approaches for mass reductions in gravity field modelling. J Geodesy 81:121–136. https://doi.org/10.1007/s00190-006-0094-0

    Article  Google Scholar 

  20. Heiskanen WA, Moritz H (1967) Physical geodesy. W. H. Freeman and Company, San Francisco, p 364

    Google Scholar 

  21. Hirt C (2012) Efficient and accurate high-degree spherical harmonic synthesis of gravity field functionals at the Earth’s surface using the gradient approach. J Geodesy 86:729–744. https://doi.org/10.1007/s00190-012-0550-y

    Article  Google Scholar 

  22. Hirt C, Kuhn M (2012) Evaluation of high-degree series expansions of the topographic potential to higher-order powers. J Geophys Res. https://doi.org/10.1029/2012JB009492

  23. Hirt C, Kuhn M (2014) Band-limited topographic mass distribution generates full-spectrum gravity field: gravity forward modeling in the spectral and spatial domains revisited. J Geophys Res Solid Earth 119:3646–3661. https://doi.org/10.1002/2013JB010900

    Article  Google Scholar 

  24. Hirt C, Kuhn M (2017) Convergence and divergence in spherical harmonic series of the gravitational field generated by high-resolution planetary topography—a case study for the Moon. J Geophys Res Planets 122:1727–1746. https://doi.org/10.1002/2017JE005298

    Article  Google Scholar 

  25. Hirt C, Reußner E, Rexer M, Kuhn M (2016) Topographic gravity modeling for global Bouguer maps to degree 2160: validation of spectral and spatial domain forward modeling techniques at the 10 microgal level. J Geophys Res Solid Earth 121:6846–6862. https://doi.org/10.1002/2016JB013249

    Article  Google Scholar 

  26. Hoffmann-Wellenhof B, Moritz H (2005) Physical geodesy. Springer, Wien, p 403

    Google Scholar 

  27. Hotine M (1969) Mathematical geodesy. U.S. Department of Commerce, Washington, p 416

    Google Scholar 

  28. Hu X, Jekeli C (2015) A numerical comparison of spherical, spheroidal and ellipsoidal harmonic gravitational field models for small non-spherical bodies: examples for the Martian moons. J Geodesy 89:159–177. https://doi.org/10.1007/s00190-014-0769-x

    Article  Google Scholar 

  29. Ivanov KG, Pavlis NK, Petrushev P (2018) Precise and efficient evaluation of gravimetric quantities at arbitrarily scattered points in space. J Geodesy. https://doi.org/10.1007/s00190-012-0558-3

  30. Jekeli C (1981) The downward continuation to the Earth’s surface of truncated spherical and ellipsoidal harmonic series of the gravity and height anomalies. Report No. 11, Department of Geodetic Science and Surveying, The Ohio State University, Columbus, Ohio, 140 pp

  31. Jekeli C (1983) A numerical study of the divergence of spherical harmonic series of the gravity and height anomalies at the Earth’s surface. Bull Géod 57:10–28

    Article  Google Scholar 

  32. Klees R, Tenzer R, Prutkin I, Wittwer T (2008) A data-driven approach to local gravity field modelling using spherical radial basis functions. J Geodesy 82:457–471. https://doi.org/10.1007/s00190-007-0196-3

    Article  Google Scholar 

  33. Krarup T (1969) A contribution to the mathematical foundation of physical geodesy. Meddelelse No. 44, Geodætisk Institut, København

  34. Kuhn M, Hirt C (2016) Topographic gravitational potential up to second-order derivatives: an examination of approximation errors caused by rock-equivalent topography (RET). J Geodesy 90:883–902. https://doi.org/10.1007/s00190-016-0917-6

    Article  Google Scholar 

  35. Landweber L (1951) An iteration formula for Fredholm integral equations of the first kind. Am J Math 73:615–624

    Article  Google Scholar 

  36. Martinec Z (1989) Program to calculate the spectral harmonic expansion coefficients of the two scalar fields product. Comput Phys Commun 54:177–182

    Article  Google Scholar 

  37. Martinec Z (1998) Boundary-value problems for gravimetric determination of a precise geoid. Springer, Berlin, p 223

    Google Scholar 

  38. Martinec Z, Pěč K (1989) The Phobos gravitational field modeled on the basis of its topography. Earth Moon Planet 45:219–235

    Article  Google Scholar 

  39. Meissl P (1971) A study of covariance functions related to the Earth’s disturbing potential. Report No. 151, Department of Geodetic Science, The Ohio State University, Columbus, Ohio, 86 pp

  40. Moazezi S, Zomorrodian H, Siahkoohi HR, Azmoudeh-Ardalan A, Gholami A (2016) Fast ultrahigh-degree global spherical harmonic synthesis on nonequispaced grid points at irregular surfaces. J Geodesy 90:853–870. https://doi.org/10.1007/s00190-016-0915-8

    Article  Google Scholar 

  41. Moritz H (1980) Advanced physical geodesy. Herbert Wichmann Verlag, Karlsruhe, p 500

    Google Scholar 

  42. Nagy D, Papp G, Benedek J (2000) The gravitational potential and its derivatives for the prism. J Geodesy 74:552–560

    Article  Google Scholar 

  43. Olver FWJ, Lozier DW, Boisvert RF, Clark CW (2010) NIST handbook of mathematical functions. Cambridge University Press, Cambridge, p 968

    Google Scholar 

  44. Pavlis NK (1988) Modeling and estimation of a low degree geopotential model from terrestrial gravity data. Report No. 386, Department of Geodetic Science, The Ohio State University, Ohio, USA, 173 pp

  45. Pavlis NK, Holmes SA, Kenyon SC, Factor JK (2012) The development and evaluation of the Earth Gravitational Model 2008 (EGM2008). J Geophys Res 117(B04406):1–38. https://doi.org/10.1029/2011JB008916

    Google Scholar 

  46. Pohánka V (1988) Optimum expression for computation of the gravity field of a homogeneous polyhedral body. Geophys Prospect 35:733–751

    Article  Google Scholar 

  47. Reimond S, Baur O (2016) Spheroidal and ellipsoidal harmonic expansions of the gravitational potential of small Solar system bodies. Case study: comet 67P/Churyumov–Gerasimenko. J Geophys Res Planets 121:497–515. https://doi.org/10.1002/2015JE004965

    Article  Google Scholar 

  48. Rexer M (2017) Spectral solutions to the topographic potential in the context of high-resolution global gravity field modelling. PhD thesis, Technische Universität München, München, Germany, 212 pp

  49. Rexer M, Hirt C (2015) Ultra-high-degree surface spherical harmonic analysis using the Gauss–Legendre and the Driscoll/Healy quadrature theorem and application to planetary topography models of Earth, Mars and Moon. Surv Geophys 36:803–830. https://doi.org/10.1007/s10712-015-9345-z

    Article  Google Scholar 

  50. Rummel R, van Gelderen M (1992) Spectral analysis of the full gravity tensor. Geophys J Int 111:159–169

    Article  Google Scholar 

  51. Rummel R, van Gelderen M (1995) Meissl scheme–spectral characteristics of physical geodesy. Manuscr Geod 20:379–385

    Google Scholar 

  52. Sacerdote F, Sansò F (2010) Least squares, Galerkin and BVPs applied to the determination of global gravity field models. In: Mertikas SP (ed) Gravity, geoid and earth observation: proceedings of the IAG commission 2: gravity field, Chania, Crete, Greece, 23–27 June 2008, vol 135. Springer, pp 511–517. ISBN: 978-3-642-10633-0, e-ISSN 978-3-642-10634-7

  53. Sansò F (1993) Theory of geodetic B.V.P.s applied to the analysis of altimetric data. In: Rummel R, Sansò F (eds) Satellite altimetry in geodesy and oceanography. Springer, Berlin, pp 318–371. https://doi.org/10.1007/BFb0117924

    Google Scholar 

  54. Sansò F, Sideris MG (2013) Geoid determination: theory and methods. Springer, Berlin, p 734

    Google Scholar 

  55. Sansò F, Sideris MG (2017) Geodetic boundary value problem: the equivalence between Molodensky’s and Helmert’s solutions. Springer, New York

    Google Scholar 

  56. Sansò F, Sona G (1995) Gravity reductions versus approximate B.V.P.s. In: Sansò F (ed) Geodetic theory today. Springer, Berlin, pp 304–314. https://doi.org/10.1007/978-3-642-79824-5

    Google Scholar 

  57. Sebera J, Šprlák M, Novák P, Bezděk A, Val’ko M (2014) Iterative spherical downward continuation applied to magnetic and gravitational data from satellite. Surv Geophys 35:941–958. https://doi.org/10.1007/s10712-014-9285-z

    Article  Google Scholar 

  58. Sebera J, Bezděk A, Pešek I, Henych T (2016) Spheroidal models of the exterior gravitational field of asteroids Bennu and Castalia. Icarus 272:70–79. https://doi.org/10.1016/j.icarus.2016.02.038

    Article  Google Scholar 

  59. Sjöberg L (1977) On the errors of spherical harmonic developments of gravity at the surface of the Earth. Report No. 12, Department of Geodetic Science, The Ohio State University, Columbus, Ohio, 74 pp

  60. Sjöberg L (1980) On the convergence problem for the spherical harmonic expansion of the geopotential at the surface of the Earth. Boll Geod Sci Affini 39:261–270

    Google Scholar 

  61. Sneeuw N (1994) Global spherical harmonic analysis by least-squares and numerical quadrature methods in historical perspective. Geophys J Int 118:707–716

    Article  Google Scholar 

  62. Takahashi Y, Scheeres DJ (2014) Small body surface gravity fields via spheical harmonic expansions. Celest Mech Dyn Astron 119:169–206. https://doi.org/10.1007/s10569-014-9552-9

    Article  Google Scholar 

  63. Tsoulis D (2012) Analytical computation of the full gravity tensor of a homogeneous arbitrarily shaped polyhedral source using line integrals. Geophysics 77:F1–F11. https://doi.org/10.1190/GEO2010-0334.1

    Article  Google Scholar 

  64. Wang YM (1997) On the error of analytical downward continuation of the earth’s external gravitational potential on and inside the earth’s surface. J Geodesy 71:70–82

    Article  Google Scholar 

  65. Wessel P, Smith WHF (1998) New, improved version of generic mapping tools released. EOS, Trans Am Geophy Union 79:579. https://doi.org/10.1029/98EO00426

    Article  Google Scholar 

  66. Wieczorek MA (2015) Gravity and topography of the terrestrial planets. In: Schubert G (ed) Treatise on geophysics, 2nd edn. Elsevier, New York, pp 153–193. https://doi.org/10.1016/B978-0-444-53802-4.00169- chap 10.5

    Google Scholar 

  67. Wieczorek MA, Phillips RJ (1998) Potential anomalies on a sphere: applications to the thickness of the lunar crust. J Geophys Res 103:1715–1724

    Article  Google Scholar 

  68. Wieczorek MA, Simons FJ (2005) Localized spectral analysis on the sphere. Geophys J Int 162:655–675. https://doi.org/10.1111/j.1365-246X.2005.02687.x

    Article  Google Scholar 

  69. Zeng X, Li X, Su J, Liu D, Zou H (2013) An adaptive iterative method for downward continuation of potential-field data from a horizontal plane. Geophysics 78:J43–J52. https://doi.org/10.1190/GEO2012-0404.1

    Article  Google Scholar 

Download references

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).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Blažej Bucha.

Appendices

Harmonic analysis of a band-limited function mapped from a band-limited surface onto the Bjerhammar sphere

To verify numerically the conclusions drawn in Sect. 3.1, we set up a simple numerical experiment with band-limited gravity data residing on an irregular band-limited surface. We used the lunar degree-180 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 band-limited topographic surface S instead of the non-band-limited \(S_{\varOmega }\) 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.

Fig. 8
figure8

Dimensionless degree variances (Eq. 25) of the degree-180 lunar topographic potential mapped from various band-limited surfaces onto the Bjerhammar sphere. The spectra refer to the Bjerhammar sphere. In order to ensure a mutual consistency, the original spherical harmonic coefficients (SHCs) were rescaled properly to the Bjerhammar sphere (Eq. 6)

In Fig. 8, we show dimensionless degree variances

$$\begin{aligned} \sigma {V}_{n}^2=\sum _{m=-n}^{n} \bar{V}^2_{nm} \end{aligned}$$
(25)

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 step-like 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 closed-loop 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 degree-720 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 first-order radial derivatives of the topographic potential, here called gravity disturbances,

$$\begin{aligned} \delta g^{\mathrm {D}}(r,\varphi ,\lambda )= & {} - \frac{\partial V^{\mathrm {D}}(r,\varphi ,\lambda )}{\partial r}\nonumber \\= & {} \frac{\mathrm{GM}}{R_0^2} \sum _{n=0}^{N_{\mathrm {D}}} (n+1) \left( \frac{R_0}{r} \right) ^{n+2} \nonumber \\&\quad \times \,\sum _{m=-n}^{n} \bar{V}^\mathrm {D}_{nm}(N_{\mathrm {D}}) \, \bar{Y}_{nm}(\varphi ,\lambda ). \end{aligned}$$
(26)

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 point-wise evaluation of Eq. (26) can be extremely time-consuming (e.g. Hirt 2012), we employ the gradient approach, which offers a substantial acceleration. The gradient approach combines a fast FFT-based 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

$$\begin{aligned} \delta g^{\mathrm {D}}(r,\varphi ,\lambda ) \approx \sum _{k=0}^{K} \frac{1}{k!} \frac{\partial ^k \delta g^{\mathrm {D}}}{\partial r^k} \Biggr |_{P_\mathrm{a} \in \varOmega _{\mathrm {a}}} \, h_\mathrm{a}^k(\varphi ,\lambda ), \end{aligned}$$
(27)

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 zeroth-order derivative is given by Eq. (26) and the radial derivatives for \(k \ge 1\) read (Hirt 2012)

$$\begin{aligned} \frac{\partial ^k \delta g^{\mathrm {D}}}{\partial r^k}= & {} (-1)^{k} \frac{\mathrm{GM}}{R_0^{k+2}} \sum _{n=0}^{N_{\mathrm {D}}} \left( \frac{R_0}{r} \right) ^{n+2+k} \prod _{i=0}^{k} (n+i+1)\nonumber \\&\quad \times \,\sum _{m=-n}^{n} \bar{V}^\mathrm {D}_{nm}(N_{\mathrm {D}}) \, \bar{Y}_{nm}(\varphi ,\lambda ). \end{aligned}$$
(28)

Here, both Eqs. (26) and (28) are evaluated on the auxiliary sphere via the efficient FFT-based 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:

  1. (A)

    \(\varOmega _{\mathrm {a}}\) coincides with the Bjerhammar sphere \(\varOmega _0\) with the radius \(R_{\mathrm {a}}=R_0=R-1\ \text {m}\), where \(R=1738\ \text {km}\), that is, \(\varOmega _{\mathrm {a}}\) is completely below the boundary surface,

  2. (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 field-generating masses,

  3. (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 degree-720 and degree-2160 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 arc-min 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 point-wise 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.

Fig. 9
figure9

Differences between gravity disturbances from the gradient approach and the exact point-wise synthesis as a function of the Taylor order \(K=0,1,\ldots ,10,15,\ldots ,45\). The gravity disturbances are implied by the lunar degree-720 and degree-2160 topographies and are evaluated at the Gauss–Legendre grid for degree 2160 (\(2161 \times 4322\) nodes) residing on \(S_{\varOmega }\). Shown are the RMS of differences and their maxima in absolute values

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

$$\begin{aligned} \sum _{k=1}^{\infty }\frac{1}{k!} \frac{\partial ^k f_i}{\partial r^k} \Biggr |_{P_0 \in \varOmega _0} h^k = f_i(P) - f_i(P_0), \end{aligned}$$
(29)

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 \(\varOmega _0\),

$$\begin{aligned} -\sum _{k=1}^{\infty }\frac{1}{k!} \frac{\partial ^k f_i}{\partial r^k} \Biggr |_{P \in S_{\varOmega }} (-h)^k = f_i(P) - f_i(P_0). \end{aligned}$$
(30)

When the Taylor series is infinite, the left-hand 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 FFT-based 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.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bucha, B., Hirt, C. & Kuhn, M. Divergence-free 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/s00190-018-1177-4

Download citation

Keywords

  • Gravity field modelling
  • Spherical harmonics
  • Runge–Krarup theorem
  • Iterative downward continuation
  • Divergence effect
  • Gradient approach