The use of FFT techniques in physical geodesy

Schwarz, K. P.; Sideris, M. G.; Forsberg, R.

doi:10.1111/j.1365-246x.1990.tb00701.x

Cited by 298 publications

(127 citation statements)

References 21 publications

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“…A significant reduction of computation time can be achieved by applying fast Fourier transform techniques (cf. Forsberg, 1985;Schwarz et al, 1990;Klose and Ilk, 1993). The price to be paid is a decreasing accuracy when the bounding surface is too rough.…”

Section: Introductionmentioning

confidence: 99%

Optimized formulas for the gravitational field of a tesseroid

Grombein

Seitz

Heck

2013

J Geod

146

114

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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.

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Section: Introductionmentioning

confidence: 99%

Optimized formulas for the gravitational field of a tesseroid

Grombein

Seitz

Heck

2013

J Geod

146

114

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“…A thorough description of the computational process of power spectra of gravity related quantities is given in Flury (2006) and Schwarz et al (1990), therefore only some basic relations are repeated.…”

Section: Spectral Analysis Methodsmentioning

confidence: 99%

“…In this sense geoid determined from gravity data could not be considered as a fully EGM2008 independent quantity since gravity from EGM2008 was utilized outside the border of the country, however our investigations regarding the covariance analysis rely exclusively to the previously selected rectangular test area in the central part of the country. The ACF of gravimetric geoid height was computed by radially averaging the isotrop 2D planar ACF obtained by spectral method (Schwarz et al 1990). For comparison EGM2008 geoid height covariances was also derived with spectral content of l,m = 600 2160.…”

Section: Covariance Analysismentioning

confidence: 99%

A study of different wavelength spectral components of the gravity field derived from various terrestrial data sets

2014

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In the general scheme of gravity field modelling long , medium and short wavelength constituents of the gravity field derived from e.g. geopotential model, terrestrial data and digital terrain model respectively, are routinely combined. In this study, spectral characteristics of terrestrial data sets are investigated. The estimation of spectral sensitivity of gravity related quantities such as gravity anomaly, vertical deflections and gravity gradients was accomplished through Fourier PSD and covariance analysis depending on the spatial distribution of data points. The information content of the estimated spectra were validated on global and local levels to access their further utilization. The spectra were compared to the 1D spectrum of the gravitational field derived from spherical harmonic coefficients using a high resolution global gravitational model as well as to an analytical approximation. Besides the frequency domain investigations the information content regarding the different wavelength structure comprised in terrestrial and EGM2008 model is investigated in the space domain based on covariance analysis. As a combined validation process the gravity degree variances were transformed to the necessary auto and cross covariance functions to predict geoid height from gravity anomaly, which ensures an independent validation process of the computed spectrum. Based on the spectral characteristics of terrestrial measurement spectral weights for spectral combination were derived involving global gravity field model, gravity and gravity gradient data in gravity field modelling. To determine the geoid in the whole spectral band the specific integral kernels in the spectral domain should be modified using the suggested spectral weights.

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“…Dentre esses modelos podemos citar a aplicação da integral de Stokes (Heiskanen & Moritz, 1967), com o método de colocação por mínimos quadrados (Moritz, 1980) ou com a transformada de Fourier (Schwarz et al, 1989). Essas técnicas baseiam-se em relações funcionais e/ou estocásticas entre anomalias ar livre e ondulações geoidais.…”

Section: Introductionunclassified

Estimativa de alturas geoidais para o estado de São Paulo baseada em redes neurais artificiais

Veronez¹,

Souza²,

Matsuoka³

et al. 2009

Rev. Bras. Geof.

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ABSTRACT. The information of height provided by the GNSS (Global Navigation Satellite System) is purely geometrical, and in most engineering papers, the height must be referenced to the geoid. Provided we have a sufficient number of Bench Marks (BMs) with known horizontal and vertical coordinates, it is nearly always possible to adjust mathematical expressions that allow for the interpolation of geoidal heights. The aim of this paper is to evaluate the efficiency of Artificial Neural Network (ANN) in the process of predicting geoidal heights, having the State of São Paulo as the area of study. The information used is based on a set of 157 BMs, evenly distributed all across the State. The horizontal coordinates (latitude and longitude) and the vertical coordinates (geometrical, orthometrical and geoidal heights) of these BMs are known. From the 157 BMs, 115 were used for the training of RNA and 42 in the process of simulation to assess the efficiency of the model proposed. Efficiency is based in determining the discrepancies (error) between known geoidal heights and those which were obtained by the neural model. As a contribution to this research, we have compared the values simulated with the Earth Gravitational Model 2008 (EGM2008) and with the MAPGEO2004 as well. In terms of results, the RNA produced a mean absolute error of 0.19 m ± 0.14 m and a strong correlation (R 2 = 0.9871) with the values taken as true. Statistically, the tests showed that there was no difference between known geoidal heights and those which were provided by the neural model for a level of significance of 5%. When we compare these results with the EGM2008 and MAPGEO2004, the RNA has an error reduction of 0.07 and 0.44 m, respectively. Keywords: GPS, MAPGEO2004, EGM2008, artificial neural networks, geoidal height. RESUMO.A informação da altitude fornecida pelo sistema GNSS (Global Navigation Satellite System )é puramente geométrica, e na maioria dos trabalhos de engenharia a altitude deve estar referenciada ao geóide. Com um número suficiente de Referências de nível (Rn's) com coordenadas horizontais e verticais conhecidas, quase sempre,é possível ajustar-se, pelo Método dos Mínimos Quadrados, expressões matemáticas que permitem interpolar as alturas geoidais. O objetivo deste trabalho foi avaliar a eficiência das Redes Neurais Artificiais (RNAs) no processo de predição de alturas geoidais tendo comoárea de estudo o Estado de São Paulo. As informações utilizadas basearam-se em um conjunto de 157 Referências de nível (Rn's) distribuídas uniformemente em todo Estado. Para estas Rn's são conhecidas suas coordenadas horizontais (latitude e longitude) e verticais (altitudes geométrica e ortométrica e altura geoidal). Das 157 Rn's, 115 foram utilizadas para o treinamento da RNA e 42 no processo de simulação para avaliar a eficiência do modelo proposto. A eficiência baseou-se em determinar as discrepâncias (erro) entres as alturas geoidais conhecidas e as obtidas pelo modelo neural. Como contribuição da pesquisa comparou-se também os valores...

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The use of FFT techniques in physical geodesy

Cited by 298 publications

References 21 publications

Optimized formulas for the gravitational field of a tesseroid

Optimized formulas for the gravitational field of a tesseroid

A study of different wavelength spectral components of the gravity field derived from various terrestrial data sets

Estimativa de alturas geoidais para o estado de São Paulo baseada em redes neurais artificiais

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