This chapter reviews our current knowledge of the gravity and topography of the terrestrial planets and describes the methods that are used to analyze these data. A general review of the mathematical formalism that is used in describing gravity and topography is first given. Next, the basic properties of Earth, Venus, Mars, Mercury, and the Moon are characterized. Following this, the relationship between gravity and topography is quantified, and techniques by which geophysical parameters can be constrained are detailed. Analysis methods include crustal thickness modeling, geoid/topography ratios, spectral admittance and correlation functions, and localized spectral analysis and wavelet techniques. Finally, the major results that have been obtained by modeling the gravity and topography of Earth, Venus, Mars, Mercury, and the Moon are summarized.

Gravity and Topography of the Terrestrial Planets

A MATLAB-based graphical user interface program for computing functionals of the geopotential up to ultra-high degrees and orders

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On Satellite Gravity Gradiometry

The effect of spatial truncation error on integral inversion of satellite gravity gradiometry data

Fast spherical harmonic synthesis (SHS) at multiple points based on the lumped coefficients approach is a very well-established technique. However, this method cannot be applied to SHS at irregular surfaces, as the points must be regularly spaced and refer to a regular surface such as the sphere or the ellipsoid of revolution. In this paper we present a MATLAB ? -based graphical user interface software for ultra-high degree (e.g. tens of thousands or even higher) SHS on grids at irregular surfaces, like the Earth surface. This software employs the highly efficient lumped coefficient approach for SHS at regular surfaces and the Taylor series expansions to continue the functionals to the irregular surfaces, e.g. the Earth surface. The generalized idea of continuing functionals using the Taylor series was presented by Hirt (2012) (J. Geod. 86, 729-744). We took advantage of the software GrafLab (Bucha and Janak, 2013. Comput. Geosci. 56, 186-196), which employs the lumped coefficients approach, and developed a new software isGrafLab (Irregular Surface GRAvity Field LABoratory). Compared to the commonly used "two loops" approach, the factor of increased computational speed can reach a value of several hundreds. isGrafLab allows accurate evaluation of 38 functionals of the geopotential on grids at irregular surfaces. High orders of the Taylor series can be used for the continuation. The new software offers all the other options available in GrafLab, such as the employment of three different approaches to compute the fully normalized associated Legendre functions, the graphical user interface or the possibility to depict data on a map. HighlightsA new MATLAB-based software for spherical harmonic synthesis is presented.Spherical harmonic synthesis is efficiently performed on grids at irregular surfaces.The program employs the lumped coefficients approach and Taylor series expansions.The program has been developed with a graphical user interface.

MATLAB-based graphical user interface program for computing functionals of the geopotential up to ultra-high degrees and orders

The conventional expansions of the gravity gradients in the local north-oriented reference frame have a complicated form, depending on the first- and second-order derivatives of the associated Legendre functions of the colatitude and containing factors which tend to infinity when approaching the poles. In the present paper, the general term of each of these series is transformed to a product of a geopotential coefficient $$\overline{C}_{n,m} $$
 and a sum of several adjacent Legendre functions of the colatitude multiplied by a function of the longitude. These transformations are performed on the basis of relations between the Legendre functions and their derivatives published by Ilk (1983). The second-order geopotential derivatives corresponding to the local orbital reference frame are presented as linear functions of the north-oriented gravity gradients. The new expansions for the latter are substituted into these functions. As a result, the orbital derivatives are also presented as series depending on the geopotential coefficients $$\overline{C}_{n,m} $$
 multiplied by sums of the Legendre functions whose coefficients depend on the longitude and the satellite track azimuth at an observation point. The derived expansions of the observables can be applied for constructing a geopotential model from the GOCE mission data by the time-wise and space-wise approaches. The numerical experiments demonstrate the correctness of the analytical formulas.

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Non-Singular Expressions for the Gravity Gradients in the Local North-Oriented and Orbital Reference Frames

The derivatives of the Earth gravitational potential are considered in the global Cartesian Earth-fixed reference frame. Spherical harmonic series are constructed for the potential derivatives of the first and second orders on the basis of a general expression of Cunningham (Celest Mech 2:207–216, 1970) for arbitrary order derivatives of a spherical harmonic. A common structure of the series for the potential and its first- and second-order derivatives allows to develop a general procedure for constructing similar series for the potential derivatives of arbitrary orders. The coefficients of the derivatives are defined by means of recurrence relations in which a coefficient of a certain order derivative is a linear function of two coefficients of a preceding order derivative. The coefficients of the second-order derivatives are also presented as explicit functions of three coefficients of the potential. On the basis of the geopotential model EGM2008, the spherical harmonic coefficients are calculated for the first-, second-, and some third-order derivatives of the disturbing potential T, representing the full potential V, after eliminating from it the zero- and first-degree harmonics. The coefficients of two lowest degrees in the series for the derivatives of T are presented. The corresponding degree variances are estimated. The obtained results can be applied for solving various problems of satellite geodesy and celestial mechanics.

Construction of spherical harmonic series for the potential derivatives of arbitrary orders in the geocentric Earth-fixed reference frame

Due to the complicated structure of their expressions, the ellipsoidal harmonic series for the derivatives of the Earth’s gravitational potential are commonly applied only on a reference ellipsoid. They depend on the first- and second-order derivatives of the associated Legendre functions of both kinds and contain a few singular terms. We construct ellipsoidal harmonic expansions in the exterior space for the first and second potential derivatives, which are similar to the series on the reference ellipsoid enveloping the Earth. We take a point P at an arbitrary altitude above the reference ellipsoid and construct the ellipsoid of revolution confocal to it, which passes through this point. The conventional complicated singular expressions for the first and second potential derivatives in the local north-oriented ellipsoidal reference frame, with the origin at the point P, are transformed into non-singular ellipsoidal harmonic series, which do not contain the first- and second-order derivatives of the associated Legendre functions. The resulting series have an accuracy of the squared eccentricity. These series can be applied for constructing a geopotential model, which is based, simultaneously, on the surface gravity data and the data of satellite missions, which provide measurements of the accelerations and/or the gravitational gradients. When the eccentricity of the considered external ellipsoid is equated to zero, the ellipsoid becomes an external sphere passing through the point P and the constructed ellipsoidal harmonic expansions are converted into non-singular spherical harmonic series for the first and second potential derivatives in the local north-oriented spherical reference frame.

Compact non-singular expansions in the exterior space of a reference ellipsoid for the gravitational gradients in the local north-oriented ellipsoidal and spherical reference frames

The second-order derivatives of the Earth’s potential in the local north-oriented reference frame are expanded in series of modified spherical harmonics. Linear relations are derived between the spectral coefficients of these series and the spectrum of the geopotential. On the basis of these relations, recurrence procedures are developed for evaluating the geopotential coefficients from the spectrum of each derivative and, inversely, for simulating the latter from a known geopotential model. Very simple structure of the derived expressions for the derivatives is convenient for estimating the geopotential coefficients by the least-squares procedure, at a certain step of processing satellite gradiometry data. Due to the orthogonality of the new series, the quadrature formula approach can be also applied, which allows avoidance of aliasing errors caused by the series truncation. The spectral coefficients of the derivatives are evaluated on the basis of the derived relations from the geopotential models EGM96 and EIGEN-CG01C at a mean orbital sphere of the GOCE satellite. Various characteristics of the spectra are studied corresponding to the EGM96 model.

Development of the second-order derivatives of the Earth’s potential in the local north-oriented reference frame in orthogonal series of modified spherical harmonics

The standard analytical approach which is applied for constructing geopotential models OSU86 and earlier ones, is based on reducing the boundary value equation to a sphere enveloping the Earth and then solving it directly with respect to the potential coefficients n,m. In an alternative procedure, developed by Jekeli and used for constructing the models OSU91 and EGM96, at first an ellipsoidal harmonic series is developed for the geopotential and then its coefficients n,me are transformed to the unknown n,m. The second solution is more exact, but much more complicated. The standard procedure is modified and a new simple integral formula is derived for evaluating the potential coefficients. The efficiency of the standard and new procedures is studied numerically. In these solutions the same input data are used as for constructing high-degree parts of the EGM96 models. From two sets of n,m (n≤360,|m|≤n), derived by the standard and new approaches, different spectral characteristics of the gravity anomaly and the geoid undulation are estimated and then compared with similar characteristics evaluated by Jekeli's approach (`etalon' solution). The new solution appears to be very close to Jekeli's, as opposed to the standard solution. The discrepancies between all the characteristics of the new and `etalon' solutions are smaller than the corresponding discrepancies between two versions of the final geopotential model EGM96, one of them (HDM190) constructed by the block-diagonal least squares (LS) adjustment and the other one (V068) by using Jekeli's approach. On the basis of the derived analytical solution a new simple mathematical model is developed to apply the LS technique for evaluating geopotential coefficients.

M. S. Petrovskaya

Papers

Non-Singular Expressions for the Gravity Gradients in the Local North-Oriented and Orbital Reference Frames

Construction of spherical harmonic series for the potential derivatives of arbitrary orders in the geocentric Earth-fixed reference frame

Compact non-singular expansions in the exterior space of a reference ellipsoid for the gravitational gradients in the local north-oriented ellipsoidal and spherical reference frames

Development of the second-order derivatives of the Earth’s potential in the local north-oriented reference frame in orthogonal series of modified spherical harmonics

New analytical and numerical approaches for geopotential modeling