The ellipsoidal correction to the Stokes kernel for precise geoid determination

Najafi-Alamdari, Mehdi; Emadi, Seyed-Rohallah; Moghtased-Azar, Khosro

doi:10.1007/s00190-006-0050-z

Cited by 5 publications

(4 citation statements)

References 13 publications

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“…The boundary values of T or N can be determined by: spirit levelling (HOFMANN-WELLENHOF et al, 2006;SANSO and SIDERIS, 2013), a global geopotential model (PAVLIS et al, 2012;PAIL et al, 2011PAIL et al, , 2010, satellite altimetry if the region is over the seas (HOFMANN-WELLENHOF et al, 2006;SANSO and SIDERIS, 2013) or the remove-compute-restore technique (HOFMANN-WELLENHOF et al, 2006;SANSO and SIDERIS, 2013;ODERA et al, 2012) which can be based on ellipsoidal corrections (MARTINEC and GRAFAREND, 1997;ARDESTANI and MARTINEC, 2003;HIPKIN, 2004;NAJAFI-ALAMDARI et al, 2006). This way, T is determined from the linear system derived from Eq.…”

Section: Numerical Solution Of the Dirichlet Problem By Fdm And Fdm2mentioning

confidence: 99%

“…Present day needs for highly accurate geoid models have driven many attempts to modify Stokes's Integral and to compute ellipsoidal corrections for it (MARTINEC and GRAFAREND, 1997;ARDESTANI and MARTINEC, 2003;HIPKIN, 2004;NAJAFI-ALAMDARI et al, 2006) which can be as big as 1m in some places (MARTINEC and GRAFAREND, 1997;NAJAFI-ALAMDARI et al, 2006). Current high resolution geoid models do not consider these corrections and show decimeter big uncertainties, e.g.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On accurate geoid modeling: derivation of dirichlet problems that govern geoidal undulations and geoid modeling by means of the finite difference method and a hybrid method

Rio

2014

Bol. Ciênc. Geod.

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The geoid is the reference surface used to measure heights (orthometric). These are used to study any mass variability in the Earth system. As the Earth is represented by an oblate spheroid (Ellipsoid), the geoid is determined by geoidal undulations (N) which are the separation between these surfaces. N is determined from gravity data by Stokes's Integral. However, this approach takes a Spherical rather than an Ellipsoidal Earth. Here it is derived a Partial Differential Equation (PDE) that governs N over the Earth by means of a Dirichlet problem and show a method to solve it which precludes the need for a Spherical Earth. Moreover, Stokes's Integral solves a boundary value problem defined over the whole Earth. It was found that the Dirichlet problem derived here is defined only over the region where a geoid model is to be computed, which is advantageous for local geoid modeling. Moreover, the method eliminates several of the sources of uncertainty in Stokes's Integral. However, estimates indicate that the errors due to discretization are very large in this new method which calls for its modification. So, here it is also proposed an RESUMO O geóide é a superfície de referência utilizada para se medire altitudes (ortométrica). Estas são utilizadas para estudar qualquer variação de massa no sistema terrestre. Como a Terra é representada por um esferóide oblatado (elipsóide), o geóide é determinado por meio de ondulações geoidais (N) que são a separação entre essas superfícies. N é determinado a partir de dados gravitacionais pela integral de Stokes. Todavia, esta abordagem considera uma Terra esférica ao invés de elipsóidica. Neste artigo é deduzida uma equação diferencial parcial (sigla em inglês PDE) que governa N ao redor da Terra por vias de um Problema de Dirichlet. Também mostra-se aqui um método para resolver esta PDE que dispensa a necessidade de uma Terra esférica. Além do mais, a Integral de Stokes resolve um problema de valor de contorno definido por toda a Terra. Descobriu-se que o Problema de Dirichlet aqui proposto está definido apenas ao longo da região de cálculo o que é vantajoso para modelamento local do geóide. Além do mais, o método elimina diversas das fontes de incertezas presentes na Integral de Stokes. Todavia, estimativas indicam que o erro devido à discretização é muito grande neste novo método o que pede por modificações. Sendo assim, aqui também propõe-se uma combinação ótima de técnicas por meio de um método Híbrido. Mostra-se que que este método híbrido atenua as incertezas do método das Diferenças Finitas. Além do mais, uma rigorosa análise de erros indica que o método Híbrido aqui proposto pode bem desempenhar melhor do que a Integral de Stokes. Palavras-chave: Modelagem do Geóide; Problema de Dirichlet no Elipsóide; Método das Diferenças Finitas; Integral de Stokes.

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Section: Numerical Solution Of the Dirichlet Problem By Fdm And Fdm2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On accurate geoid modeling: derivation of dirichlet problems that govern geoidal undulations and geoid modeling by means of the finite difference method and a hybrid method

Rio

2014

Bol. Ciênc. Geod.

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“…In this section, we shall validate our direct LS approach with the standard ellipsoidal correction h that is usually applied to the gravity data, for example in Jekeli (1981), Cruz (1986), and recently discussed in Najafi-Alamdari et al (2006). In our approach, the value h will also be obtained directly from the LS SHA with two Jacobian matrices and this value will be compared with that from the standard analytic expression.…”

Section: Ellipsoidal Correctionmentioning

confidence: 99%

“…The first is a closed-loop test with simulated data and as such it confirms the concept and the codes. In the second test, we discuss the ellipsoidal correction that usually compensates the problem of the vertical when the gravity anomaly/disturbance is to be processed with standard formula in the spherical approximation (Jekeli 1981;Cruz 1986;Pavlis 1988;Najafi-Alamdari et al 2006). The accuracy of this correction is verified through the least squares spherical harmonic analysis (LS SHA) of data on the reference ellipsoid.…”

Section: Introductionmentioning

confidence: 99%

Short guide to direct gravitational field modelling with Hotine’s equations

Sebera

Wagner

Bezděk

et al. 2012

J Geod

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This paper presents a unified approach to the least squares spherical harmonic analysis of the acceleration vector and Eötvös tensor (gravitational gradients) in an arbitrary orientation. The Jacobian matrices are based on Hotine's equations that hold in the Earth-fixed Cartesian frame and do not need any derivatives of the associated Legendre functions. The implementation was confirmed through closedloop tests in which the simulated input is inverted in the least square sense using the rotated Hotine's equations. The precision achieved is at the level of rounding error with RMS about 10 −12 −10 −14 m in terms of the height anomaly. The second validation of the linear model is done with help from the standard ellipsoidal correction for the gravity disturbance that can be computed with an analytic expression as well as with the rotated equations. Although the analytic expression for this correction is only of a limited accuracy at the submillimeter level, it was used for an independent validation. Finally, the equivalent of the ellipsoidal correction, called the effect of the normal, has been numerically obtained also for other gravitational functionals and some of their combinations. Most of the numerical investigations are provided up to spherical harmonic degree 70, with degree 80 for the computation time comparison using real GRACE data. The relevant Matlab source codes for the design matrices are provided.

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IAG Newsletter

Tóth

2007

J Geod

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The ellipsoidal correction to the Stokes kernel for precise geoid determination

Cited by 5 publications

References 13 publications

On accurate geoid modeling: derivation of dirichlet problems that govern geoidal undulations and geoid modeling by means of the finite difference method and a hybrid method

On accurate geoid modeling: derivation of dirichlet problems that govern geoidal undulations and geoid modeling by means of the finite difference method and a hybrid method

Short guide to direct gravitational field modelling with Hotine’s equations

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