Forward modelling of global gravity fields with 3D density structures and an application to the high-resolution (~ 2 km) gravity fields of the Moon

Abstract

Rigorous modelling of the spherical gravitational potential spectra from the volumetric density and geometry of an attracting body is discussed. Firstly, we derive mathematical formulas for the spatial analysis of spherical harmonic coefficients. Secondly, we present a numerically efficient algorithm for rigorous forward modelling. We consider the finite-amplitude topographic modelling methods as special cases, with additional postulates on the volumetric density and geometry. Thirdly, we implement our algorithm in the form of computer programs and test their correctness with respect to the finite-amplitude topography routines. For this purpose, synthetic and realistic numerical experiments, applied to the gravitational field and geometry of the Moon, are performed. We also investigate the optimal choice of input parameters for the finite-amplitude modelling methods. Fourth, we exploit the rigorous forward modelling for the determination of the spherical gravitational potential spectra inferred by lunar crustal models with uniform, laterally variable, radially variable, and spatially (3D) variable bulk density. Also, we analyse these four different crustal models in terms of their spectral characteristics and band-limited radial gravitation. We demonstrate applicability of the rigorous forward modelling using currently available computational resources up to degree and order 2519 of the spherical harmonic expansion, which corresponds to a resolution of ~ 2.2 km on the surface of the Moon. Computer codes, a user manual and scripts developed for the purposes of this study are publicly available to potential users.

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Acknowledgements

This research was supported partially by the Australian Government through the Australian Research Council’s Discovery Projects funding scheme (project DP160104095). We thank Mark Wieczorek for sharing his own results of topography gravitational potential fields and the surface grain density, and Greg Neumann for his advice on the LOLA topography grid data. Thoughtful and constructive comments of the three anonymous reviewers are gratefully acknowledged. Thanks are also extended to the editor-in-chief Prof. Jürgen Kusche and to the responsible editor Prof. Ilias N. Tziavos for handling our manuscript.

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Correspondence to M. Šprlák.

Appendix A: Computer codes, bash scripts, and manual

Appendix A: Computer codes, bash scripts, and manual

The computer codes, LINUX bash scripts, and a user manual can be downloaded from the public-domain folder available at https://drive.google.com/drive/folders/0By2RsmhxzXIyLU96SzZPclVNUnM. It contains the following files:

  1. 1.

    manual.pdf Short description of the programs, input parameters, and functionality,

  2. 2.

    gravtess_uniform.f FORTRAN source code for calculation of the SHCs inferred by the uniform lunar crust,

  3. 3.

    gravtess_lateral.f FORTRAN source code for calculation of the SHCs inferred by the lunar crust with laterally variable density,

  4. 4.

    gravtess_radial.f FORTRAN source code for calculation of the SHCs inferred by the lunar crust with radially variable density,

  5. 5.

    gravtess_spatial.f FORTRAN source code for calculation of the SHCs inferred by the lunar crust with spatially variable density,

  6. 6.

    alf_sr_v121305.f FORTRAN source code for calculation of the integrals of the associated Legendre functions of the first kind,

  7. 7.

    FFTCC.f FORTRAN source code necessary for the 1D discrete Fourier transform,

  8. 8.

    script_uniform.sh Script for compiling the FORTRAN code gravtess_uniform.f90 and executing the program,

  9. 9.

    script_lateral.sh Script for compiling the FORTRAN code gravtess_lateral.f90 and executing the program,

  10. 10.

    script_radial.sh Script for compiling the FORTRAN code gravtess_radial.f90 and executing the program,

  11. 11.

    script_spatial.sh Script for compiling the FORTRAN code gravtess_spatial.f90 and executing the program,

  12. 12.

    depth_density_0p175_350_2850.dat Sample density file required by the programs gravtess_radial and gravtess_spatial,

  13. 13.

    pressure_0p175_350_2850.dat Sample pressure file required by the program gravtess_spatial.

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Šprlák, M., Han, S. & Featherstone, W.E. Forward modelling of global gravity fields with 3D density structures and an application to the high-resolution (~ 2 km) gravity fields of the Moon. J Geod 92, 847–862 (2018). https://doi.org/10.1007/s00190-017-1098-7

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Keywords

  • Bulk density
  • Newton’s integral
  • Spherical harmonic expansion
  • LOLA
  • GRAIL