[1] EGM2008 is a spherical harmonic model of the Earth's gravitational potential, developed by a least squares combination of the ITG-GRACE03S gravitational model and its associated error covariance matrix, with the gravitational information obtained from a global set of area-mean free-air gravity anomalies defined on a 5 arc-minute equiangular grid This grid was formed by merging terrestrial, altimetry-derived, and airborne gravity data Over areas where only lower resolution gravity data were available, their spectral content was supplemented with gravitational information implied by the topography EGM2008 is complete to degree and order 2159, and contains additional coefficients up to degree 2190 and order 2159 Over areas covered with high quality gravity data, the discrepancies between EGM2008 geoid undulations and independent GPS/Leveling values are on the order of ±5 to ±10 cm EGM2008 vertical deflections over USA and Australia are within ±11 to ±13 arc-seconds of independent astrogeodetic values These results indicate that EGM2008 performs comparably with contemporary detailed regional geoid models EGM2008 performs equally well with other GRACE-based gravitational models in orbit computations Over EGM96, EGM2008 represents improvement by a factor of six in resolution, and by factors of three to six in accuracy, depending on gravitational quantity and geographic area EGM2008 represents a milestone and a new paradigm in global gravity field modeling, by demonstrating for the first time ever, that given accurate and detailed gravimetric data, asingle global model may satisfy the requirements of a very wide range of applications

https://agupubs.onlinelibrary.wiley.com/doi/epdf/10.1029/2011JB008916

The development and evaluation of the Earth Gravitational Model 2008 (EGM2008)

Three gravity field models, parameterized in terms of spherical harmonic coefficients, have been computed from 71 days of GOCE (Gravity field and steady-state Ocean Circulation Explorer) orbit and gradiometer data by applying independent gravity field processing methods. These gravity models are one major output of the European Space Agency (ESA) project GOCE High-level Processing Facility (HPF). The processing philosophies and architectures of these three complementary methods are presented and discussed, emphasizing the specific features of the three approaches. The resulting GOCE gravity field models, representing the first models containing the novel measurement type of gravity gradiometry ever computed, are analysed and assessed in detail. Together with the coefficient estimates, full variance-covariance matrices provide error information about the coefficient solutions. A comparison with state-of-the-art GRACE and combined gravity field models reveals the additional contribution of GOCE based on only 71 days of data. Compared with combined gravity field models, large deviations appear in regions where the terrestrial gravity data are known to be of low accuracy. The GOCE performance, assessed against the GRACE-only model ITG-Grace2010s, becomes superior at degree 150, and beyond. GOCE provides significant additional information of the global Earth gravity field, with an accuracy of the 2-month GOCE gravity field models of 10 cm in terms of geoid heights, and 3 mGal in terms of gravity anomalies, globally at a resolution of 100 km (degree/order 200).

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First GOCE gravity field models derived by three different approaches

SUMMARY The fast Fourier transform (FFT) technique is a very powerful tool for the efficient evaluation of gravity field convolution integrals It can handle heterogeneous and noisy data, and thus presents a very attractive alternative to the classical, time consuming approaches, provided gridded data are available This paper reviews the mathematics of the FFT methods as well as their practical problems, and presents examples from physical geodesy where the application of these methods is especially advantageous The spectral evaluation of Stokes’, Vening Meinesz’ and Molodensky’s integrals, least-squares collocation in the frequency domain, integrals for terrain reductions and for airborne gravity gradiometry , and the computation of covariance and power spectral density functions are treated in detail Numerical examples illustrate the efficiency and accuracy of the FFT methods Key words: FFT, physical geodesy, spectral methods 1 INTRODUCTION Physical geodesy is the branch of geodesy which uses measured gradients of the anomalou6gravity potential T to determine a unique and coherent representation of the terrestrial gravity field at the Earth’s surface and in outer space The anomalous potential T is the difference between the actual gravity potential of the Earth and the reference potential of an ellipsoid with the same mass, flattening, and angular rotation rate as the Earth An approximation of T is needed to model geodetic measurements, to predict perturbations of satellite orbits, to determine global ocean circulation patterns, to assist global geophysics, and to support oil and mineral exploration In recent years, the amount of data available for the solution of this problem has increased dramatically, both in quantity and in type This has made the data processing problems more severe and has created a demand for efficient numerical solutions Since much of the data is available in gridded form, the use of fast spectral techniques was clearly appropriate Progress in the application of these methods to geodetic problems has been rapid during the last three years and it is almost certain that, because of their efficiency and accuracy, they will become standard procedures for a number of applications However, it has also become clear that geodetic and, more generally, geophysical data often present specific problems not usually encountered in typical electrical engineering applications The problems are with the heterogeneity of the data, the complicated surface on which they are given, the uneven spatial distribution, and the non-uniformity of the data noise This paper will discuss the use of

The use of FFT techniques in physical geodesy

Two new geopotential coefficient models to spherical harmonic degree 360 are developed using recent advances made in theoretical modeling methods, satellite gravitational models, and expanded and improved terrestrial data. The new models are based on the combination of a satellite potential coefficient model (GEM-T2) to degree 50 with 30-arc mean gravity anomalies, yielding an adjusted set of coefficients and gravity anomalies, which were then harmonically analyzed to yield a set of potential coefficients to degree 360. The models were verified in several ways including satellite orbit residual analysis, demonstrating a substantial improvement over previous high-degree expansions.

The development and analysis of geopotential coefficient models to spherical harmonic degree 360

Three presentations on geographical analysis and modeling

: This report presents some numerical methods for estimating spherical harmonic coefficients from data sampled on the sphere. The data may be given in the form of area means or of point values, and it may be free from errors or affected by measurement 'noise'. The case discussed to greatest length is that of complete, global data sets on regular grids (i.e., lines of latitude and longitude, the latter, at least, separated by constant interval); the case where data are sparsely and irregularly distributed is also considered in some detail. The first section presents some basic properties of spherical harmonics, stressing their relationship to two-dimensional Fourier series. Algorithms for the evaluation of the harmonic coefficients by numerical quadratures are given here, and it is shown that the number of operations is the order of N cubed for equal angular grids, where N is the number of lines of latitude, or 'Nyquist frequency', of the grid. The second section introduces a quadratic measure for the error in the estimation of the coefficients by linear techniques. This is the error measure of least squares collocation, which is a method that can be used for harmonic analysis. Efficient algorithms for implementing collocation on the whole sphere are described. a formal relationship between collocation and least squares adjustment is used to obtain an alternative form of the collocation algorithm that is likely to be stable with dense data sets and, with a minor modification, can be used to implement least squares adjustment as well. The basic principle is that for regular grids the variance-convariance matrix of the data consists of Toeplitz-circulant blocks, so it can be both set up and inverted very efficiently.

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Oscar L Colombo

Papers

Numerical Methods for Harmonic Analysis on the Sphere