New Relations Among Associated Legendre Functions and Spherical Harmonics

Several new relations among associated Legendre functions (ALFs) are derived, most of which relate a product of an ALF with trigonometric functions to a weighted summation over ALFs, where the weights only depend on the degree and order of the ALF. These relations are, for example, useful in applications such as the computation of geopotential coefficients and computation of ellipsoidal corrections in geoid modelling. The main relations are presented in both their unnormalised and fully normalised ($4\pi$-normalised) form. Several approaches to compute the weights involved are discussed, and it is shown that the relations can also be applied in the case of first- and second-order derivatives of ALFs, which may be of use in analysis of satellite gradiometry data. Finally, the derived relations are combined to provide new identities among ALFs, which contain no dependency on the colatitudinal coordinate other than that in the ALFs themselves

Experiences with Point-Mass Gravity Field Modelling in the Perth Region, Western Australia

Gravimetric geoid modelling in the Perth region of Western Australia has attracted much attention in recent years because of the numerous restrictions in this area, such as the presence of the Darling Fault system and variable gravity data coverage and quality. This paper presents the results of experiments to determine the effectiveness of free-positioned point-mass modelling in relation to the previous attempts based on Stokes's integral. It is shown that the point-mass modelling technique does not yield improved fits to 99 local GPS-levelling data with a standard deviation of the fit of +/- 17.5cm. However, the experiments do show that the technique is a useful tool for conveniently identifying areas where there are large topographic mass density contrasts and mismatches among the gravity data used. Therefore, the technique, though apparently not optimal for geoid modelling in this region at this point in time, may provide a useful indicator of problems that will affect geoid omputations using other techniques

A surface spherical harmonic expansion of gravity anomalies on the ellipsoid

A surface spherical harmonic expansion of gravity anomalies with respect to a geodetic reference ellipsoid can be used to model the global gravity field and reveal its spectral properties. In this paper, a direct and rigorous transformation between solid spherical harmonic coefficients of the Earth’s disturbing potential and surface spherical harmonic coefficients of gravity anomalies in ellipsoidal approximation with respect to a reference ellipsoid is derived. This transformation cannot rigorously be achieved by the Hotine–Jekeli transformation between spherical and ellipsoidal harmonic coefficients. The method derived here is used to create a surface spherical harmonic model of gravity anomalies with respect to the GRS80 ellipsoid from the EGM2008 global gravity model. Internal validation of the model shows a global RMS precision of <1 nGal. This is significantly more precise than previous solutions based on spherical approximation or approximations to order e2 or e3, which are shown to be insufficient for the generation of surface spherical harmonic coefficients with respect to a geodetic reference ellipsoid. Numerical results of two applications of the new method (the computation of ellipsoidal corrections to gravimetric geoid computation, and area means of gravity anomalies in ellipsoidal approximation) are provided

A new functional model for determining minimum and maximum detectable deformation gradient resolved by satellite radar interferometry

In this paper, a functional model for determining the minimum and maximum detectable deformation gradient in terms of coherence for synthetic aperture radar (SAR) sensors is presented. The model is developed based on a new methodology that incorporates both real and simulated data. Sets of representative surface deformation models have been simulated, and the associated phase from these models introduced into real SAR data acquired by European Remote Sensing 1 and 2 satellites. Subsequently,interferograms were derived, and surface deformation was estimated. A number of cases of surface deformation with varying magnitudes and spatial extent have been simulated. In each case, the resultant surface deformation has been compared with the "true" surface deformation as defined by the deformation model. Based on these comparisons, a set of observations that lead to a new functional model has been established.Finally, the proposed model has been validated against external datasets and proven viable. Although the major weakness of the model is its reliance on visual interpretation of interferograms, this model can serve as a decision-support tool to determine whether or not to apply satellite radar interferometry to study a given surface deformation

On the accurate numerical evaluation of geodetic convolution integrals

In the numerical evaluation of geodetic convolution integrals, whether by quadrature or discrete/fast Fourier transform (D/FFT) techniques, the integration kernel is sometimes computed at the centre of the discretised grid cells. For singular kernels - a common case in physical geodesy - this approximation produces significant errors near the computation point, where the kernel changes rapidly across the cell. Rigorously, mean kernels across each whole cell are required. We present one numerical and one analytical method capable of providing estimates of mean kernels for convolution integrals. The numerical method is based on Gauss-Legendre quadrature (GLQ) as efficient integration technique. The analytical approach is based on kernel weighting factors, computed in planar approximation close to the computation point, and used to convert non-planar kernels from point to mean representation. A numerical study exemplifies the benefits of using mean kernels in Stokes?s integral. The method is validated using closed-loop tests based on the EGM2008 global gravity model, revealing that using mean kernels instead of point kernels reduces numerical integration errors by a factor of ~5 (at a grid-resolution of 10 arc minutes). Analytical mean kernel solutions are then derived for 14 other commonly used geodetic convolution integrals: Hotine, Etvs, Green-Molodensky, tidal displacement, ocean tide loading, deflection-geoid, Vening-Meinesz, inverse Vening-Meinesz, inverse Stokes, inverse Hotine, terrain correction, primary indirect effect, Molodensky's G1 term and the Poisson integral. We recommend that mean kernels be used to accurately evaluate geodetic convolution integrals, and the two methods presented here are effective and easy to implement

Is Australian data really validating EGM2008, or is EGM2008 just in/validating Australia data?

The tide-free release of the EGM2008 combined global geopotential model and its tide-free pre-release PGM2007A are compared with Australian land, marine and airborne gravity observations, co-located GPS-levelling on the [admittedly problematic] Australian Height Datum, astrogeodetic deflections of the vertical, and the AUSGeoid98 regional gravimetric quasigeoid model. In all comparisons, EGM2008 performs better than any previous global gravity model. The standard deviation of the differences between free-air gravity anomalies from EGM2008 and free-air gravity anomalies from Australian land gravity observations is 5.5 mGal, compared to, e.g., 11.7 mGal for EGM96. Furthermore, the standard deviation of the differences between height anomalies from EGM2008 and anation-wide set of 254 GPS-levelling points is 17.3 cm, compared to, e.g., 33.4 cmfor EGM96. In the comparisons with GPS-levelling, EGM2008 also outperforms AUSGeoid98 (standard deviation of 19.1 cm in the differences with the nation-wide set of 254 GPS-levelling points), and the same holds for the comparison to astrogeodetic deflections of the vertical. However, due to the poor quality of some of the Australian data, we cannot legitimately claim to truly validate EGM2008. Instead, EGM2008 confirms the already-known problems with the Australian data, as well as revealing some previously unknown problems. If one wants to claim validation, then EGM2008 is validated implicitly because it can confirm the errors in our regional data. Simply, EGM2008 is a good model over Australia

Kilometer-resolution gravity field of Mars: MGM2011

We present a model that resolves the gravity field of Mars down to km-scales: Mars Gravity Model 2011 (MGM2011). MGM2011 uses Newtonian forward-modelling and the MOLA (Mars Orbiter Laser Altimeter) topography model to estimate the short-scale gravity field (scales of ~3 km to ~125 km). Combined with a reference gravity field and the satellite-tracking model MRO110B2, MGM2011 provides surface gravity accelerations and vertical deflections over the entire Martian surface at 3 arc-min resolution. MGM2011 is beneficial for gravity field simulation, inversion and statistics, as well as engineering-driven applications such as topographic mapping and inertial navigation

An experimental Indian gravimetric geoid model using Curtin University’s approach

Over the past decade, numerous advantages of a gravimetric geoid model and its possible suitability for the Indian national vertical datum have been discussed and advocated by the Indian scientific community and national geodetic agencies. However, despite several regional efforts, a state-of-the-art gravimetric geoid model for the whole of India remains elusive due to a multitude of reasons. India encompasses one of the most diverse topographies on the planet, which includes the Gangetic plains, the Himalayas, the Thar desert, and a long peninsular coastline, among other topographic features. In the present study, we have developed the first national geoid and quasigeoid models for India using Curtin University’s approach. Terrain corrections were found to reach an extreme of 187 mGal, Faye gravity anomalies 617 mGal, and the geoid-quasigeoid separation 4.002 m. We have computed both geoid and quasigeoid models to analyse their representativeness of the Indian normal-orthometric heights from the 119 GNSS-levelling points that are available to us. A geoid model for India has been computed with an overall standard deviation of ±0.396 m but varying from ±0.03 m to ±0.158 m in four test regions with GNSS-levelling data. The greatest challenge in developing a precise gravimetric geoid for the whole of India is data availability and its preparation. More densely surveyed precise gravity data and a larger number of GNSS/levelling data are required to further improve the models and their testing

Using AUSGeoid2020 and its error grids in surveying computations

We present summarised formulas and worked examples for the propagation of geoid and vertical deflection errors through some common geodetic surveying computations, as well as a demonstration of their effects on least squares adjustments of small simulated geodetic networks. We also present location-specific uncertainties for the vertical deflections derived from the horizontal gradients of the AGQG2017 gravimetric-only quasigeoid model, upon which AUSGeoid2020 is based

Towards the New AusGeoid Model

Since November 1998, all high-precision GPS users in Australia have adopted the AUSGeoid98 gravimetric geoid model to transform GPS-derived ellipsoidal heights to the Australian Height Datum (AHD) and vice versa. Since AUSGeoid98 was released by Geoscience Australia (http://www.ga.gov.au/nmd/geodesy/ausgeoid/) based on recommendations by the first-named author, several new theories have been formulated and refined datasets have been released, all of which can improve the Australian geoid model. This paper reports our latest implementations of these theories and datasets, which comprise a global geopotential model derived from the GRACE (Gravity Recovery And Climate Experiment) dedicated satellite gravimetry mission, gravimetric terrain corrections from the version-2 DEM-9S 9"x9" digital elevation model, approximately 200,000 additional land gravity observations in Geoscience Australia's database, improved gravity data cleaning methods, refined marine gravity data from multi-mission satellite radar altimetry, a least-squares crossover adjustment of ship-track gravity observations, and new GPS data collected by State and Territory geodetic survey agencies at key tide-gauges, some junction points and other benchmarks of the AHD. The refined gravimetric geoid solution will be fitted to the GPS-AHD data using least-squares collocation so as to deliberately provide a more direct transformation to the AHD that obviates the need to occupy nearby AHD benchmarks during a GPS survey. This pragmatic solution, while not producing a classical equipotential geoid model, does provide a very useful product for GPS users in Australia until the AHD is rigorously redefined. Our results show that the new model will deliver GPS-derived AHD heights with an RMS of less than ~12 cm in an absolute sense over most parts of Australia, which reduces when used in relative mode over shorter GPS baselines. In short, the new model will deliver height results that are commensurate with or better than Australian class LC (third order) geodetic levelling methods. As with AUSGeoid98, this new product will be released and administered by Geoscience Australia based on our recommendations, hopefully towards the end of 2005