Sten Claessens

Ellipsoidal topographic potential: New solutions for spectral forward gravity modeling of topography with respect to a reference ellipsoid

Forward gravity modeling in the spectral domain traditionally relies on spherical approximation. However, this level of approximation is insufficient for some present day high-accuracy applications. Here we present two solutions that avoid the traditional spherical approximation in spectral forward gravity modeling. The first solution (the extended integration method) applies integration over masses from a reference sphere to the topography and applies a correction for the masses between ellipsoid and sphere. The second solution (the harmonic combination method) computes topographic potential coefficients from a combination of surface spherical harmonic coefficients of topographic heights above the ellipsoid, based on a relation among spherical harmonic functions introduced by Claessens (2005). Using a degree 2160 spherical harmonic model of the topographic masses, both methods are applied to derive the Earths ellipsoidal topographic potential in spherical harmonics. The harmonic combination method converges fastest and—akin to the EGM2008 geopotential model—generates additional spherical harmonic coefficients in spectral band 2161 to 2190 which are found crucial for accurate evaluation of the ellipsoidal topographic potential at high degrees. Therefore, we recommend use of the harmonic combination method to model ellipticity in spectral-domain forward modeling. The method yields ellipsoidal topographic potential coefficients which are “compatible” with global Earth geopotential models constructed in ellipsoidal approximation, such as EGM2008. It shows that the spherical approximation significantly underestimates degree correlation coefficients among geopotential and topographic potential. The topographic potential model is, for example, of immediate value for the calculation of Bouguer gravity anomalies in fully ellipsoidal approximation.

Is Australian Data Really Validating EGM2008, or Is EGM2008 Just in/Validating Australian Data?

The tide-free release of the EGM2008 combined global geopotential model and its pre-release PGM2007A are compared with Australian land and marine 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. The results show that we cannot legitimately claim to truly validate EGM2008. Instead, EGM2008 confirms 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 because it can confirm the errors in our regional data. Simply, EGM2008 is a good model over Australia.

Study of the Earth's short-scale gravity field using the ERTM2160 gravity model

This paper describes the computation and analysis of the Earths short-scale gravity field through high-resolution gravity forward modelling using the Shuttle Radar Topography Mission (SRTM) global topography model. We use the established residual terrain modelling technique along with advanced computational resources and massive parallelisation to convert the high-pass filtered SRTM topography - complemented with bathymetric information in coastal zones - to implied short-scale gravity effects. The result is the ERTM2160 model (Earth Residual Terrain Modelled-gravity field with the spatial scales equivalent to spherical-harmonic coefficients up to degree 2160 removed). ERTM2160, used successfully for the construction of the GGMplus gravity maps, approximates the short-scale (i.e., ~10km down to ~250m) gravity field in terms of gravity disturbances, quasi/geoid heights and vertical deflections at ~3 billion gridded points within ?60? latitude. ERTM2160 reaches maximum values for the quasi/geoid height of ~30cm, gravity disturbance in excess of 100mGal, and vertical deflections of ~30? over the Himalaya mountains. Analysis of the ERTM2160 field as a function of terrain roughness shows in good approximation a linear relationship between terrain roughness and gravity effects, with values of ~1.7cm (quasi/geoid heights), ~11mGal (gravity disturbances) and 1.5? (vertical deflections) signal strength per 100m standard deviation of the terrain. These statistics can be used to assess the magnitude of omitted gravity signals over various types of terrain when using degree-2160 gravity models such as EGM2008. Applications for ERTM2160 are outlined including its use in gravity smoothing procedures, augmentation of EGM2008, fill-in for future ultra-high resolution gravity models in spherical harmonics, or calculation of localised or global power spectra of Earths short-scale gravity field. ERTM2160 is freely available via http://ddfe.curtin.edu.au/gravitymodels/ERTM2160. Display Omitted Residual gravity model ERTM2160 computed from the SRTM topography at 250m resolution.Supercomputing resources used for forward gravity modelling at ~3 billion points.Global short-scale RMS signal magnitudes are 1.6cm for geoid, 11mGal for gravity.Linear relation between terrain roughness and RMS gravity signal magnitudes found.

THe NZGEOID09 model of New Zealand

Abstract The NZGeoid09 gravimetric quasigeoid model of New Zealand was computed through FFT-based Stokesian integration with a deterministically modified kernel and an iterative computation approach that accounts for offsets among New Zealands 13 different local vertical datums (LVDs). NZGeoid09 is an improvement over the previous NZGeoid05 due to use of the EGM2008 and DNSC08GRA models, and due to improvements to the data processing strategy. The integration parameters of degree of kernel modification L=40 and cap radius ψ0=2.5° were determined empirically through a comparison with 1422 GPS/levelling observations, after the LVD offsets had been removed. The precision of NZGeoid09 was assessed using the same GPS/levelling dataset, yielding an overall standard deviation of 6.2 cm. NZGeoid09 performs better than NZGeoid05 and marginally better than EGM2008, but few data are available in the Southern Alps of New Zealand to give a better evaluation.

Layer-Based Modelling of the Earth’s Gravitational Potential up to 10-km Scale in Spherical Harmonics in Spherical and Ellipsoidal Approximation

Global forward modelling of the Earth’s gravitational potential, a classical problem in geophysics and geodesy, is relevant for a range of applications such as gravity interpretation, isostatic hypothesis testing or combined gravity field modelling with high and ultra-high resolution. This study presents spectral forward modelling with volumetric mass layers to degree 2190 for the first time based on two different levels of approximation. In spherical approximation, the mass layers are referred to a sphere, yielding the spherical topographic potential. In ellipsoidal approximation where an ellipsoid of revolution provides the reference, the ellipsoidal topographic potential (ETP) is obtained. For both types of approximation, we derive a mass layer concept and study it with layered data from the Earth2014 topography model at 5-arc-min resolution. We show that the layer concept can be applied with either actual layer density or density contrasts w.r.t. a reference density, without discernible differences in the computed gravity functionals. To avoid aliasing and truncation errors, we carefully account for increased sampling requirements due to the exponentiation of the boundary functions and consider all numerically relevant terms of the involved binominal series expansions. The main outcome of our work is a set of new spectral models of the Earth’s topographic potential relying on mass layer modelling in spherical and in ellipsoidal approximation. We compare both levels of approximations geometrically, spectrally and numerically and quantify the benefits over the frequently used rock-equivalent topography (RET) method. We show that by using the ETP it is possible to avoid any displacement of masses and quantify also the benefit of mapping-free modelling. The layer-based forward modelling is corroborated by GOCE satellite gradiometry, by in-situ gravity observations from recently released Antarctic gravity anomaly grids and degree correlations with spectral models of the Earth’s observed geopotential. As the main conclusion of this work, the mass layer approach allows more accurate modelling of the topographic potential because it avoids 10–20-mGal approximation errors associated with RET techniques. The spherical approximation is suited for a range of geophysical applications, while the ellipsoidal approximation is preferable for applications requiring high accuracy or high resolution.

Procrustean Solution of the 9-Parameter Transformation Problem

The Procrustean “matching bed” is employed here to provide direct solution to the 9-parameter transformation problem inherent in geodesy, navigation, computer vision and medicine. By computing the centre of mass coordinates of two given systems; scale, translation and rotation parameters are optimised using the Frobenius norm. To demonstrate the Procrustean approach, three simulated and one real geodetic network are tested. In the first case, a minimum three point network is simulated. The second and third cases consider the over-determined eight- and 1 million-point networks, respectively. The 1 million point simulated network mimics the case of an air-borne laser scanner, which does not require an isotropic scale since scale varies in the X, Y, Z directions. A real network is then finally considered by computing both the 7 and 9 transformation parameters, which transform the Australian Geodetic Datum (AGD 84) to Geocentric Datum Australia (GDA 94). The results indicate the effectiveness of the Procrustean method in solving the 9-parameter transformation problem; with case 1 giving the square root of the trace of the error matrix and the mean square root of the trace of the error matrix as 0.039 m and 0.013 m, respectively. Case 2 gives 1.13×10−12 m and 2.31×10−13 m, while case 3 gives 2.00×10−4 m and 1.20 × 10−5 m, which is acceptable from a laser scanning point of view since the acceptable error limit is below 1 m. For the real network, the values 6.789 m and 0.432 m were obtained for the 9-parameter transformation problem and 6.867 m and 0.438 m for the 7-parameter transformation problem, a marginal improvement by 1.14%.

Spatial and Spectral Representations of the Geoid-to-Quasigeoid Correction

In geodesy, the geoid and the quasigeoid are used as a reference surface for heights. Despite some similarities between these two concepts, the differences between the geoid and the quasigeoid (i.e. the geoid-to-quasigeoid correction) have to be taken into consideration in some specific applications which require a high accuracy. Over the world’s oceans and marginal seas, the quasigeoid and the geoid are identical. Over the continents, however, the geoid-to-quasigeoid correction could reach up to several metres especially in the mountainous, polar and geologically complex regions. Various methods have been developed and applied to compute this correction regionally in the spatial domain using detailed gravity, terrain and crustal density data. These methods utilize the gravimetric forward modelling of the topographic density structure and the direct/inverse solutions to the boundary-value problems in physical geodesy. In this article, we provide a brief summary of existing theoretical and numerical studies on the geoid-to-quasigeoid correction. We then compare these methods with the newly developed procedure and discuss some numerical and practical aspects of computing this correction. In global applications, the geoid-to-quasigeoid correction can conveniently be computed in the spectral domain. For this purpose, we derive and present also the spectral expressions for computing this correction based on applying methods for a spherical harmonic analysis and synthesis of global gravity, terrain and crustal structure models. We argue that the newly developed procedure for the regional gravity-to-potential conversion, applied for computing the geoid-to-quasigeoid correction in the spatial domain, is numerically more stable than the existing inverse models which utilize the gravity downward continuation. Moreover, compared to existing spectral expressions, our definition in the spectral domain takes not only the terrain geometry but also the mass density heterogeneities within the whole Earth into consideration. In this way, the geoid-to-quasigeoid correction and the respective geoid model could be determined more accurately.

Variance component estimation uncertainty for unbalanced data: application to a continent-wide vertical datum

Variance component estimation (VCE) is used to update the stochastic model in least-squares adjustments, but the uncertainty associated with the VCE-derived weights is rarely considered. Unbalanced data is where there is an unequal number of observations in each heterogeneous data set comprising the variance component groups. As a case study using highly unbalanced data, we redefine a continent-wide vertical datum from a combined least-squares adjustment using iterative VCE and its uncertainties to update weights for each data set. These are: (1) a continent-wide levelling network, (2) a model of the ocean’s mean dynamic topography and mean sea level observations, and (3) GPS-derived ellipsoidal heights minus a gravimetric quasigeoid model. VCE uncertainty differs for each observation group in the highly unbalanced data, being dependent on the number of observations in each group. It also changes within each group after each VCE iteration, depending on the magnitude of change for each observation group’s variances. It is recommended that VCE uncertainty is computed for VCE updates to the weight matrix for unbalanced data so that the quality of the updates for each group can be properly assessed. This is particularly important if some groups contain relatively small numbers of observations. VCE uncertainty can also be used as a threshold for ceasing iterations, as it is shown—for this data set at least—that it is not necessary to continue time-consuming iterations to fully converge to unity.

Computation of geopotential coefficients from gravity anomalies on the ellipsoid

One of the most important stages in the computation of a global geopotential model is the computation of the spherical harmonic coefficients from gravity anomalies on the normal ellipsoid. None of the existing methods provides an exact solution, and all show severe shortcomings in the high degrees of the spectrum. In this paper, a new, theoretically exact method is proposed, which is moreover easily applicable up to very high degree and order (2160 and beyond). The solution of the geopotential coefficients is presented as a weighted sum over “spherically approximated” coefficients of equal order, where the gravity anomalies are presumed to reside on a sphere. The weights depend solely upon the degree and order of the coefficient and the definition of the normal ellipsoid and its gravity field. Numerical comparisons with existing methods show substantial differences, especially in the high degrees, which can be explained by the fact that all previous methods are of limited accuracy

Evaluation of Gravity and Altimetry Data in Australian Coastal Regions

Satellite altimetry in near-coastal marine areas is notoriously problematic, and gravity anomalies derived from various satellite-altimetry-derived gravity models differ significantly near the coast. In this paper, gravity anomalies from the DNSC08 and Sandwell & Smith v18.1 (SS18) models are compared and ‘validated’ against shipborne and airborne gravity anomalies around the Australian coast. Due to the scarcity of high-quality gravity observations just off the coast, a true validation of the models cannot be achieved. However, DNSC08 conforms slightly better to both shipborne and airborne gravity observations closest to the coast in selected test areas, although the standard deviation of differences between the models and the test data barely exceeds the estimated test data accuracy.