Combining EGM2008 and SRTM/DTM2006.0 residual terrain model data to improve quasigeoid computations in mountainous areas devoid of gravity data

Spherical harmonic synthesis (SHS) of gravity field functionals at the Earth's surface requires the use of heights. The present study investigates the gradient approach as an efficient yet accurate strategy to incorporate height information in SHS at densely-spaced multiple points. Taylor series expansions of commonly used functionals quasigeoid heights, gravity disturbances and vertical deflections are formulated, and expressions of their radial derivatives are presented to arbitrary order. Numerical tests show that first-order gradients, as introduced by Rapp (J Geod 71(5): 282-289, 1997) for degree-360 models, produce cm-to dm-level RMS approximation errors over rugged terrain when applied with EGM2008 to degree 2190. Instead, higher-order Taylor expansions are recommended that are capable of reducing approximation errors to insignificance for practical applications. Because the height information is separated from the actual synthesis, the gradient approach can be applied along with existing highly-efficient SHS routines to compute surface functionals at arbitrarily dense grid points. This confers considerable computational savings (above or well above one order of magnitude) over conventional point-by-point SHS. As an application example, an ultrahigh resolution model of surface gravity functionals (EurAlpGM2011) is constructed over the entire European Alps that incorporates height information in the SHS at 12,000,000 surface points. Based on EGM2008 and residual topography data, quasigeoid heights, gravity disturbances and vertical deflections are estimated at ~200 m resolution. As a conclusion, the gradient approach is efficient and accurate for high-degree SHS at multiple points at the Earth's surface.

Section: Construction Of Euralpgm2011mentioning

confidence: 99%

Efficient and accurate high-degree spherical harmonic synthesis of gravity field functionals at the Earth’s surface using the gradient approach

Hirt

2012

“…over the oceans. For a detailed description of the definition of height systems it is referred to Heiskanen and Moritz (1967) and Hofmann-Wellenhof and Moritz (2006), respectively. In order to compare with GPS-levelling geoid heights/height anomalies, one has to compute the associated quantity from the geopotential models referring to the same normal ellipsoid and the same tide system.…”

Section: Basic Relations and Gps Levelling Datamentioning

confidence: 99%

“…We are aware that also EGM2008 does not contain the complete spectral content of the gravity field. Therefore, for the German data set the size and the impact of the remaining signal by means of the residual terrain model (RTM) technique is investigated (Hirt et al 2010). For this area, RTM based geoid estimates for the remaining signal (above EGM2008 spectral content) have been computed by Hirt and were kindly made available (Hirth 2011).…”

Section: Omission Error: High Frequency Signalmentioning

confidence: 99%

Validation of GOCE gravity field models by means of orbit residuals and geoid comparisons

Gruber

Visser

Ackermann

et al. 2011

113

Three GOCE-based gravity field solutions have been computed by ESA's high-level processing facility and were released to the user community. All models are accompanied by variance-covariance information resulting either from the least squares procedure or a Monte-Carlo approach. In order to obtain independent external quality parameters and to assess the current performance of these models, a set of independent tests based on satellite orbit determination and geoid comparisons is applied. Both test methods can be regarded as complementary because they either investigate the performance in the long wavelength spectral domain (orbit determination) or in the spatial domain (geoid comparisons). The test procedure was applied to the three GOCE gravity field solutions and to a number of selected pre-launch models for comparison. Orbit determination results suggest, that a pure GOCE gravity field model does not outperform the multi-year GRACE gravity field solutions. This was expected as GOCE is designed to improve the determination of the medium to high frequencies of the Earth gravity field (in the range of degree and order 50 to 200). Nevertheless, in case of an optimal combination of GOCE and GRACE data, orbit determination results should not deteriorate. So this validation procedure can also be used for testing the optimality of the approach adopted for producing combined GOCE and GRACE models. Results from geoid comparisons indicate that with the 2 months of GOCE data a significant improvement in the determination of the spherical harmonic spectrum of the global gravity field between degree 50 and 200 can be reached. Even though the ultimate mission goal has not yet been reached, especially due to the limited time span of used GOCE data (only 2 months), it was found that existing satellite-only gravity field models, which are based on 7 years of GRACE data, can already be enhanced in terms of spatial resolution. It is expected that with the accumulation of more GOCE data the gravity field model resolution and quality can be further enhanced, and the GOCE mission goal of 1-2 cm geoid accuracy with 100 km spatial resolution can be achieved.

“…Moreover, these mass reductions can also be integrated into modern methods of gravity field modeling, such as the Remove-Compute-Restore technique (Forsberg, 1984, Chap. 4) or Residual Terrain modeling (Forsberg and Tscherning, 1997;Hirt et al, 2010). In this context, the main purpose is to smooth the signal content of gravity data in order to improve the numerical stability for interpolation or prediction tasks as well as field transformations.…”

Section: Introductionmentioning

confidence: 99%

Optimized formulas for the gravitational field of a tesseroid

Grombein

Seitz

Heck

2013

146

114

Various tasks in geodesy, geophysics, and related geosciences require precise information on the impact of mass distributions on gravity field-related quantities, such as the gravitational potential and its partial derivatives. Using forward modeling based on Newton's integral, mass distributions are generally decomposed into regular elementary bodies. In classical approaches, prisms or point mass approximations are mostly utilized. Considering the effect of the sphericity of the Earth, alternative mass modeling methods based on tesseroid bodies (spherical prisms) should be taken into account, particularly in regional and global applications. Expressions for the gravitational field of a point mass are relatively simple when formulated in Cartesian coordinates. In the case of integrating over a tesseroid volume bounded by geocentric spherical coordinates, it will be shown that it is also beneficial to represent the integral kernel in terms of Cartesian coordinates. This considerably simplifies the determination of the tesseroid's potential derivatives in comparison with previously published methodologies that make use of integral kernels expressed in spherical coordinates. Based on this idea, optimized formulas for the gravitational potential of a homogeneous tesseroid and its derivatives up to second-order are elaborated in this paper. These new formulas do not suffer from the polar singularity of the spherical coordinate system and can, therefore, be evaluated for any position on the globe. Since integrals over tesseroid volumes cannot be solved analytically, the numerical evaluation is achieved by means of expanding the integral kernel in a Taylor series with fourth-order error in the spatial coordinates of the integration point. As the structure of the Cartesian inte-T. Grombein · K. Seitz · B. Heck Geodetic Institute, Karlsruhe Institute of Technology (KIT) Englerstr. 7, Karlsruhe, D-76128 Germany Tel.: +49-721-608-42306 Fax: +49-721-608-46808 E-mail: grombein@kit.edu gral kernel is substantially simplified, Taylor coefficients can be represented in a compact and computationally attractive form. Thus, the use of the optimized tesseroid formulas particularly benefits from a significant decrease in computation time by about 45% compared to previously used algorithms. In order to show the computational efficiency and to validate the mathematical derivations, the new tesseroid formulas are applied to two realistic numerical experiments and are compared to previously published tesseroid methods and the conventional prism approach.