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Spherical Harmonic Models of Planetary Topography

This archive contains the spherical harmonic models presented in Wieczorek (2007), as well as updated versions of these, for the topography of the Earth, Moon, Venus, and Mars. These files are in ascii format, with each line containing

l, m, Clm, Slm

where l is the spherical harmonic degree, m is the spherical harmonic order, and Clm and Slm are the real spherical harmonic cosine and sine coefficients, respectively. Alternatively, using a notation where the spherical harmonic coefficients are allowed to have negative angular orders, each line corresponds to

l, m, Clm, Cl,-m.

The spherical harmonic coefficients are to be used with the "geodesy" 4π normalized spherical harmonics (excluding the Condon-Shortley phase) and are expressed in units of meters.

All models were constructed by first obtaining an N x 2N equally spaced grid (or N x N equally sampled grid for MoonUSGS359.shape) of global topography, and then using the sampling theorem of Driscoll and Healy (1994) to compute the spherical harmonic coefficients. These grids include the north pole and 0° longitude, but not the south pole and 360° longitude. When N is even, the spherical harmonic transform is exact if the function is bandlimited to degree N/2-1. These calculations were performed using the SHTOOLS routine SHExpandDH. Details specific to each model are given below and in Wieczorek (2007).

The Earth

SRTMP2160.msl	Degree 2160 model of SRTM_Plus elevations referenced to mean sea level.
SRTMP2160.shape	Degree 2160 shape model of the solid portion of the Earth derived from SRTM_Plus.
SRTMP2160.ret_shape	Degree 2160 shape model of the solid portion of the Earth derived from SRTM_Plus where the oceans have been converted to rock equivalent topography.
SRTMP2160.ellipsoid	Degree 2160 model of the solid portion of the Earth derived from SRTM_Plus referenced to the WGS-84 ellipsoid.
SRTMP2160.ret_ellipsoid	Degree 2160 model of the solid portion of the Earth derived from SRTM_Plus referenced to the WGS-84 ellipsoid, where the oceans have been converted to rock equivalent topography.

WARNING: These Earth models contain errors concerning conversions between geodetic and geocentric coordinates. These wil be updated in the near future.

SRTMP2160.msl is a spherical harmonic model of the Earth's topography, referenced to mean sea level. The data set used in constructing this model is SRTM30_Plus (v2.0), which is a combination of the 30-second Shuttle Radar Topographic Mission model for the continental land mass, and the 2-minute Smith and Sandwell bathymettry. Gaps are filled by various means as described in the above mentioned websites. Processing steps included the following:

A mosaic of the version 2.0 gridded data was created using the GMT commands xyz2grd and grdpaste.
This mosaic was (1) boxcar filtered using a Cartesian boxcar width of 0.033333°, (2) transformed to gridline registration, and (3) decimated to a 0.033333° resolution using the GMT command grdfilter. A resolution of 0.033333° corresponds to 30 pixels per degree.
The spherical harmonic algorithm of Driscoll and Heally (1994), modified to use N x 2N grids, was used to determine the spherical harmonic coefficients assuming that the function was bandlimited to degree 2699.
The spherical harmonic coefficients were truncated beyond spherical harmonic degree 2160.

SRTMP2160.shape is a spherical harmonic model for the solid portion of the Earth's shape (i.e., excluding the oceans). This model is based upon the SRTM30_Plus data set, and absolute radii were obtained by adding the WGS-84 ellipsoid, and the WGS-84 EGM 96 geoid (file ww15mgh_0.125.dat). This gridded data set was then transformed into spherical harmonics using the same method as above.

SRTMP2160.ret_shape is a spherical harmonic model for the solid portion of the Earth's shape, where the ocean mass has been converted to "rock equivalent topography." This was accomplished by adding 1030/2670 of the ocean depths to the shape model, and the spherical harmonic transform was performed as described above.

SRTMP2160.ellipsoid is a spherical harmonic model for the solid portion of the Earth's shape, referenced to the WGS-84 ellipsoid. The spherical harmonic transform was performed as described above.

SRTMP2160.ret_ellipsoid is a spherical harmonic model for the solid portion of the Earth's shape, referenced to the WGS-84 ellipsoid. The ocean mass has been converted to "rock equivalent topography" by adding 1030/2670 of the ocean depths to the ellipsoid model, and the spherical harmonic transform was performed as described above.

Venus

VenusTopo719.shape

Degree 719 shape model of Venus derived from Magellan (GTDR3.2), Pioneer Venus, and Venera 15/16 altimetry.

VenusTopo719.shape is a spherical harmonic model of the shape of Venus expanded to spherical harmonic degree 719. This model is based primarily on the sinusoidally projected GTDR3.2 data set, with gaps filled by Pioneer Venus (file pven001a.img) and Venera 15/16 (file mpi-venus-alt.dat) data. The following steps were followed in constructing the spherical harmonic model:

Gaps in the sinusoidally projected GTDR3.2 data set were first filled by Pioneer Venus, and then Venera 15/16 data.
The sinusoidally projected data were transformed to a cylindrical projection. After mirroring along all four boundaries, the remaining gaps were filled by interpolation using the GMT command surface with a tension parameter of T0.35.
The spherical harmonic algorithm of Driscoll and Heally (1994), modified to use N x 2N grids, was used to determine the spherical harmonic coefficients assuming that the function was bandlimited to degree 719.

Mars

MarsTopo719.shape

Degree 719 shape model of Mars derived from the PDS 1/32° resolution gridded data set.

MarsTopo719.shape is a spherical harmonic model of the shape of Mars expanded to spherical harmonic degree 719. The data source is the 1/32° gridded data product at the PDS (file megr90n000fb.img), and the following steps were followed in constructing the spherical harmonic model:

The 0.03125° resolution data were translated from pixel to gridline registration using the GMT command grdsample.
These data were decimated to a 0.125° resolution using the boxcar filter of the GMT command grdfilter.
The spherical harmonic algorithm of Driscoll and Heally (1994), modified to use N x 2N grids, was used to determine the spherical harmonic coefficients assuming that the function was bandlimited to degree 719.

The Moon

ULCN359_lpo.shape	Degree 359 shape model of the Moon derived from the USGS Unified Lunar Control Network 2005 (smoothed using a local polynomial).
ULCN359_grid.shape	Unified Lunar Control Network 2005 (gridded using a triangulated irregular network).

ULCN359_lpo.shape and ULCN359_grid.shape are spherical harmonic models of the Moon's shape, expanded to spherical harmonic degree 359, derived from the USGS Unified Lunar Control Network (ULCN 2005; Archinal, 2006). The data used in the spherical harmonic expansion are the lunar radii given in the files ULCN2005_lpo.txt.gz and ULCN2005_grid.txt.gz (available in this directory). The model "lpo" smoothed the original data points using a local polynomial fit, whereas the model "grid" was obtain by constructing a grid using a triangulated irregular network.

These model were constructed using the following steps:

The 0.0625° resolution data were translated from pixel to gridline registration using the GMT command grdsample, and then mirrored longitudinally.
The transformed data were boxcar filtered (using a filter width of 0.25° with the GMT command grdfilter) and then decimated to a 0.25° resolution using the GMT command grdsample.
The spherical harmonic algorithm of Driscoll and Heally (1994), modified to use N x 2N grids, was used to determine the spherical harmonic coefficients assuming that the function was bandlimited to degree 359. The spherical harmonic coefficients were then rotated by 180 degrees by multiplying each coefficient by (-1)^m.

MoonUSGS359.shape

Degree 359 shape model of the Moon derived from the USGS (2002) model.

MoonUSGS359.shape is a spherical harmonic model of the Moon's shape, expanded to spherical harmonic degree 359. The data used in the spherical harmonic expansion are the lunar radii given in the file lunarDEM_ASCII_interp_dd_geocentric.asc (provided by Mark Rosiek) which was used in generating the USGS topographic map I-2769. This data set is an interpolated version of the Clementine altimetry combined with polar DEMs derived from Clementine stereo image data.

This model was constructed using the following steps:

The 0.0625° resolution data from lunarDEM_ASCII_interp_dd_geocentric.asc were translated from pixel to gridline registration using the GMT command grdsample.
The transformed data were decimated to a 0.25° resolution using the GMT command grdsample.
After discarding every other longitude, the spherical harmonic algorithm of Driscoll and Heally (1994) was used to determine the spherical harmonic coefficients assuming that the function was bandlimited to degree 359.

References

Archinal, B. A., M. R. Rosiek, R. L. Kirk, B. L. Redding, The unified lunar control network 2005, Open-File Report 2006-1367 (v. 1.0), 2006.

Driscoll, J.R. and D.M. Healy, Computing Fourier transforms and convolutions on the 2-sphere, Adv. Appl. Math., 15, 202-250, 1994.

U. S. Geological Survey, Color-coded topography and shaded relief map of the lunar near side and far side hemispheres, I-2769, 2002.

Wessel, P., and W. H. F. Smith, Free software helps map and display data, EOS, 72(441), 1991.

Wieczorek, M. A., Gravity and topography of the terrestrial planets, Treatise on Geophysics, 10, 165-205, doi:10.1016/B978-044452748-6/00156-5, 2007.

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Institut de Physique du Globe de Paris