Harmonic Correction for Residual Terrain Modelling (RTM) Technique in Physical Geodesy Applications

<p>       In physical geodesy, the harmonic correction (HC), as one of the main problems when using residual terrain modelling (RTM), has become a research focus of high-frequency gravity field modelling. Over past decades, though various methods have been proposed to handle the HC issues for RTM technique, most of them focused on the HC for RTM gravity anomaly rather than other gravity functionals, such as RTM geoid height and gravity gradient. In practice, the HC for RTM geoid height was generally assumed to be negligible, but a quantification is yet studied. In this study, besides the highlighted HC for gravity anomaly in previous studies, the expressions of HC terms for RTM geoid height are provided in the framework of the classical condensation method under infinite Bouguer plate approximation. The errors involved by various assumption of the classical condensation method, e.g., mass inconsistency between infinite masses in the HC and limited masses in the RTM, and planar assumption of the Earth’s surface, are further studied. Based on the derived formulas, the quantification of HC for RTM geoid height when reference surface is expanded to degree and order of 2,159 is given. Our results showed the significance of HC for RTM geoid height, with values up to ~10 cm, in cm-level and mm-level geoid determination. With integration masses extending up to a sufficient distance, such as 1° from calculation point for the determination of RTM geoid height, the errors due to an infinite Bouguer plate approximation are neglectable small. The validation through comparison with terrestrial measurements proved that the HC terms provided in this study can improve the accuracy of RTM derived geoid height and are expected to be useful for applications of RTM technique in regional and global gravity field modelling.</p>

A Influência das Edificações no Cálculo do Efeito Gravitacional das Massas Topográficas – Estudo de Caso na Cidade de Porto Alegre – RS (Brasil)

Um dos objetivos da Geodésia consiste no estudo do geoide e sua determinação é obtida através do conhecimento do campo de gravidade que envolve a distribuição de massas na superfície terrestre. A abordagem clássica para a solução do problema de valor de contorno da Geodésia (PVCG) visando a determinação do geoide assume que os efeitos associados à topografia sejam levados em consideração. A técnica de Modelagem da Topografia Residual (RTM, em inglês Residual Terrain Modelling) tem como objetivo a modelagem o campo de gravidade em função da distribuição de massas associada a topografia onde, nesse tipo de estudo, o conteúdo de alta frequência do espectro relacionado a gravidade é gerado através desse método de redução associado a um modelo digital de elevação (MDE) de alta resolução. Nesse contexto, o objetivo principal desta pesquisa consiste em calcular o valor do potencial gravitacional das massas topográficas oriundas de edificações existentes na cidade de Porto Alegre – RS juntamente com a anomalia de gravidade associada à topografia. Esse estudo foi desenvolvido a partir de uma base vetorial com mais de 200 mil edificações onde o potencial gravitacional foi calculado a partir de um MDE gerado através de dados LiDAR (light detection and ranging). Para auxiliar nos cálculos, foi estimado um modelo de densidades em função das dimensões de cada edificação existentes na base de dados. Assim, foram calculados o valor do potencial gravitacional utilizando os elementos de massa tesseroide, prisma e massa pontual e também o valor da anomalia de gravidade para distâncias de 1 km, 2 km, 5 km, 10 km e 20 km usando a técnica de modelagem RTM. A influência das massas das edificações, neste estudo, representou 10,62% do valor da anomalia de gravidade em comparação com o seu correspondente em relação ao solo.

TGF: A New MATLAB-based Software for Terrain-related Gravity Field Calculations

With knowledge of geometry and density-distribution of topography, the residual terrain modelling (RTM) technique has been broadly applied in geodesy and geophysics for the determination of the high-frequency gravity field signals. Depending on the size of investigation areas, challenges in computational efficiency are encountered when using an ultra-high-resolution digital elevation model (DEM) in the Newtonian integration. For efficient and accurate gravity forward modelling in the spatial domain, we developed a new MATLAB-based program called, terrain gravity field (TGF). Our new software is capable of calculating the gravity field generated by an arbitrary topographic mass-density distribution. Depending on the attenuation character of gravity field with distance, the adaptive algorithm divides the integration masses into four zones, and adaptively combines four types of geometries (i.e., polyhedron, prism, tesseroid and point-mass) and DEMs with different spatial resolutions. Compared to some publicly available algorithms depending on one type of geometric approximation, this enables accurate modelling of gravity field and greatly reduces the computation time. Besides, the TGF software allows to calculate ten independent gravity field functionals, supports two types of density inputs (constant density value and digital density map), and considers the curvature of the Earth by involving spherical approximation and ellipsoidal approximation. Further to this, the TGF software is also capable of delivering the gravity field of full-scale topographic gravity field implied by masses between the Earth’s surface and mean sea level. In this contribution, the TGF software is introduced to the geoscience community and its capabilities are explained. Results from internal and external numerical validation experiments of TGF confirmed its accuracy at the sub-mGal level. Based on TGF, the trade-off between accuracy and efficiency, values for the spatial resolution and extension of topography models are recommended. The TGF software has been extensively tested and recently been applied in the SRTM2gravity project to convert the global 3” SRTM topography to implied gravity effects at 28 billion computation points. This confirms the capability of TGF for dealing with large datasets. Together with this paper, the TGF software will be released in the public domain for free use in geodetic and geophysical forward modelling computations.

TGF: A New MATLAB-based Software for Terrain-related Gravity Field Calculations

<p>With knowledge of geometry and density-distribution of topography, the residual terrain modelling (RTM) technique has been broadly applied in geodesy and geophysics for the determination of the high-frequency gravity field signals. Depending on the size of investigation areas, challenges in computational efficiency are encountered when using an ultra-high-resolution digital elevation models (DEM) in the evaluation of Newtonian integration. This paper presents a new MATLAB-based program, terrain gravity field (TGF), for the accurate and efficient determination of the terrain-related gravity field based on an adaptive algorithm. Depending on the attenuation character of gravity field with distance, the adaptive algorithm divides the integration masses into four zones, and adaptively combines four types of geometries and DEMs with different spatial resolutions. The most accurate but least efficient polyhedron together with the finest DEM are only considered for the innermost zone, while prism approximation for the second zone, the third zone with the more efficient tesseroid and a coarse DEM, and the most efficient but least accurate point-mass with the coarsest DEM for distant masses. Compared to some publicly available algorithms depending on one type of geometric approximation, the TGF achieves accurate modelling of gravity field and greatly reduces the computation time. Besides, the TGF software allows to calculate ten independent elements of gravity field, supports two types of density inputs (constant density value and digital density map), and considers the sphericity of the Earth by involving spherical approximation and ellipsoidal approximation. Further to this, the TGF software is also capable of delivering the gravity field of full-scale topographic gravity field implied by masses between the Earth’s surface and mean sea level. Results from internal and external numerical validation experiments of TGF confirmed its accuracy of sub-mGal level. Based on TGF, the trade-off between accuracy and efficiency, values for the spatial resolution and extension of topography models are recommended. The TGF software has been extensively tested and recently been applied in the SRTM2gravity project to convert the global 3” SRTM topography to implied gravity effects at 28 billion computation points. This confirms TGF the capability of dealing with large datasets.</p>

Comparison of remove-compute-restore and least squares modification of Stokes' formula techniques to quasi-geoid determination over the Auvergne test area

Comparison of remove-compute-restore and least squares modification of Stokes' formula techniques to quasi-geoid determination over the Auvergne test areaThe remove-compute-restore (RCR) technique for regional geoid determination implies that both topography and low-degree global geopotential model signals are removed before computation and restored after Stokes' integration or Least Squares Collocation (LSC) solution. The Least Squares Modification of Stokes' Formula (LSMS) technique not requiring gravity reductions is implemented here with a Residual Terrain Modelling based interpolation of gravity data. The 2-D Spherical Fast Fourier Transform (FFT) and the LSC methods applying the RCR technique and the LSMS method are tested over the Auvergne test area. All methods showed a reasonable agreement with GPS-levelling data, in the order of a 3-3.5 cm in the central region having relatively smooth topography, which is consistent with the accuracies of GPS and levelling. When a 1-parameter fit is used, the FFT method using kernel modification performs best with 3.0 cm r.m.s difference with GPS-levelling while the LSMS method gives the best agreement with GPS-levelling with 2.4 cm r.m.s after a 4-parameter fit is used. However, the quasi-geoid models derived using two techniques differed from each other up to 33 cm in the high mountains near the Alps. Comparison of quasi-geoid models with EGM2008 showed that the LSMS method agreed best in term of r.m.s.