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Ellipsoidische und topographische Effekte im geodätischen Randwertproblem / Kurt Seitz (1997)
Titre : Ellipsoidische und topographische Effekte im geodätischen Randwertproblem Titre original : [Les effets ellipsoïdiques et topographiques dans le problème géodésique de valeurs limites] Type de document : Thèse/HDR Auteurs : Kurt Seitz, Auteur Editeur : Munich : Bayerische Akademie der Wissenschaften Année de publication : 1997 Collection : DGK - C Sous-collection : Dissertationen num. 483 Importance : 139 p. Format : 21 x 30 cm ISBN/ISSN/EAN : 978-3-7696-9523-6 Note générale : Bibliographie Langues : Allemand (ger) Descripteur : [Vedettes matières IGN] Géodésie physique
[Termes IGN] analyse harmonique
[Termes IGN] approximation
[Termes IGN] champ de pesanteur terrestre
[Termes IGN] linéarisation
[Termes IGN] potentiel de pesanteur terrestre
[Termes IGN] problème des valeurs limitesIndex. décimale : 30.40 Géodésie physique Résumé : (Editeur) To determine both the external gravity potential W and the geometry of the earth's surface 5, various boundary value problems (bvp) can be formulated. They depend on the utilised observaWes L and whether the boundary is supposed to be known or unknown. If the geometry of 5 is already determined by the classical terrestrial techniques or by methods of satellite geodesy, then the fixed boundary value problem is under consideration. Otherwise the resulting bvp is of free type. The relation between the unknowns W, S and the observables L is given by boundary conditions. Generally, they are of non-linear structure.
This thesis focuses on the scalar free bvp and the fixed gravimetric bvp.
Analytical and numerical aspects require the linearisation of the boundary conditions. Therefore, suitable approximations have to be introduced for the unknowns. For the gravity potential, the normal gravity potential w is introduced. In the case of the scalar free boundary value problem, the boundary surface is approximated by the telluroid s, resulting from some telluroid mapping. In this context, the gravity potential W is substituted by the disturbing potential 6w := W w.
As shown in previous studies (Heck and Seitz, 1993, 1995; Seitz et al., 1994), the non-linear terms in the boundary condition can amount to maximum values of ±200 10~8ma~2 in case of the scalar free bvp, and ±40 10~8ms~2 for the fixed gravimetric problem. As a consequence of omitting the non-linear terms, the equipotential surfaces of the solution for the disturbing potential are shifted in vertical direction. Utilising a normal gravity field of Somigiiana-Pizzetti-type (e.g. GRS80), this shift takes values up to 4mm. If the gravitational portion of the normal gravity field is represented by a truncated series of spherical harmonics (e.g. Nv = 20), this effect can be reduced to 2mm. In the vicinity of the earth's physical surface, the accuracy of present day global solutions of the earth's gravity potential varies in the scope of dm-m. Therefore, the non-linear effects in the boundary condition can be neglected for the purpose of this thesis.
f
The linear boundary condition is now based on a linear operator D acting on the disturbing potential 6w. Applying the evaluation operator E,, the resulting linear functional is restricted to the telluroid. In case of the fixed bvp, the evaluation operator es is applied to D{Sw} and restricts the boundary condition to the physical earth's surface. The coefficients of the differential operator D are functional of the normal gravity potential w. Introducing an approximation wa of the normal gravity potential tu, the coefficients of D can be represented in terms of a Taylor series expansion.
For the scalar free boundary value problem as well as for the fixed gravimetric bvp the second order approximation of the differential operator D is derived. Thereby, the approximation wa of the normal gravity potential is represented by the potential of a Helmert's normal spheroid. This potential is symmetrical with respect to the mean earth's equatorial plane and independent of the geocentric longitude. In addition to the centrifugal part, this normal potential contains the isotropic term fj?/r and, furthermore, two gravitational terms, which are proportional to the zonal harmonic coefficients «/2 and «/4.
The analytical representation of the differential operator is carried out on the basis of wa. The extension of wa by the coefficient J± leads to the second order approximation of D. Further simplifications of wa result in the first order approximation and the isotropic approximation of D{6w} (fundamental equation of physical geodesy). This evaluation is done for a general normal potential and a Somigliana-Pizzetti reference field respectively.
The numerical studies clearly show that the error in the boundary condition, caused by isotropic approximation, exceeds the accuracy of relative gravimetric measurements by the multiple. It is also larger, by a factor of 5-10, than the non-linear terms in the corresponding boundary condition. According to this result, the ellipsoidal terms must not be neglected, as it is done in the isotropic approximation. If a Somigiiana-Pizzetti-field is used as normal field w, the first order approximation of the linear boundary condition produces a maximum error of ±0.5 10~8ms~2. The analytical
approximation of the differential operator finds its qualitative end, if the second order ellipsoidal terms, which are caused by the zonal coefficient J^, are taken into account. The errors due to the resulting second order approximation of the ellipsoidal terms in the linear boundary condition can be neglected in practice. The residuals are less than ±0.01 10~8ms~2.
The properties are different, when the normal field w contains tessera! and sectorial terms. In the present thesis, the normal gravity potential is given in terms of spherical harmonics, with a maximum degree Nv = 20. The apposition of the zonal coefficient J^ in the approximated normal field wa gives no better results for the approximated ellipsoidal terms. The maximum residuals are ±7 10~8m.s~2 for both levels of analytical approximation, the first order and the second order approximation. Even if second order tesseral and sectorial terms are included in w0, the accuracy of the approximation cannot be improved. According to the magnitude of the low degree spherical harmonic coefficients, all terms (e.g. up to degree and order 4) have to be included into the analytical approximation of the ellipsoidal terms in the boundary condition. The necessary analytical and numerical expenditure cannot be justified. Because of the slowly decreasing spectrum of this normal gravity potential, it must be concluded that the maximum degree Nv of the spherical harmonic representation has to be in accordance with the order of analytical approximation.
To solve the fixed or scalar free boundary value problem by the aid of a harmonic analysis, the respec--tive boundary condition must be related to a geometrically defined surface of revolution, symmetrical to the mean earth's rotation axis. In this context, a sphere K 9 k with radius r = a and the surface E 9 e of an ellipsoid of revolution is considered. The explained algorithm used for the harmonic ana-lysis results in the spherical harmonic spectrum, whether the (reduced) boundary data are given on K 3 k or E 9 e.
The analytical continuation of the boundary condition is realised by a formal Taylor series expansion of the evaluation operator E, or es respectively. Additionally, the optimal choice of the Taylor point has to be discussed. The Taylor point can be chosen at the auxiliary surface on which the data are analyticaly continued (sphere or surface of an ellipsoid). The Taylor point can also be situated at the original boundary surface. In case of the fixed bvp it can be situated at the physical earth's surface S 9 P, and on the telluroid s 3 p in case of the scalar free boundary value problem. The order of the Taylor series expansion of the evaluation operator is set up in accordance with the error level of the second order analytical approximation of the boundary condition. By this consideration the absolute error level of 10~8ms~2, which is aspired for the analytical approximation of the differential operator, is assigned to the analytically continued boundary condition. For each boundary condition we end up with a representation, where the analytical side contains an isotropic term and so-called ellipsoidal and topographical constituents. The isotropic term refers to the surface K or E. The ellipsoidal and topographical terms in the boundary condition are caused by the anisotropy of the normal potential and by the difference between the boundary surface and the auxiliary surface (K or E), on which the boundary condition is continued. They have to be evaluated either on the original boundary surface (S 3 P - fixed bvp; s 9 p - scalar free bvp) or on the auxiliary surface (K 9 k or E B e) according to the choice of the Taylor point. The ellipsoidal and topographical terms can be applied as a reduction to the original boundary data. The result is a spherical boundary value problem on the sphere K (Neumann-problem in the case of the fixed bvp), or rather a quasi-isotropic bvp with an ellipsoidal boundary surface. If the complete ellipsoidal and topographical terms are neglected, the constant radius approximation will be obtained. The reduction terms are functionals of the disturbing potential Sw we solve for. Therefore, an iterative procedure is required.
To guarantee an absolute accuracy of 1 lQ~8ms~2 for the continuation terms (ellipsoidal and topogra--phical terms), the partial derivatives of the disturbing potential are considered up to the 8th order. The partial derivatives of
j%- are considered up to the 4th order, with respect to the geocentric distance r. The numerical studies confirmed that this high degree of evaluation is necessary in the framework of the analytical continuation to the sphere. Otherwise, the level of accuracy obtained for the ellipsoidal terms cannot be reached for the continuation terms. If the analytical continuation is performed to the surface of an ellipsoid, the Taylor series can be truncated after the 5th order terms without anyloss of accuracy. The accuracy of approximation is not influenced by the choice of the Taylor point. The extremal values of the whole ellipsoidal and topographical part of the linear boundary condition, that has to be modelled by the analytical representation, decreases from ±100 10~5ms~2 in the case of a sphere to ±20 10~5ms~2, if the boundary operator is continued to the surface of an ellipsoid. These terms are identical with the neglected terms in case of the spherical approximation on K or the isotropic approximation on E, respectively. They have the same order of magnitude as the boundary data itself. Undoubtedly, this kind of approximation, where all ellipsoidal and topographical terms are neglected, leads to no high-precision solution of the fixed or scalar free boundary value problem.
First of all, the error of approximation depends on the order of the Taylor series, which are setup for the analytical continuation of the boundary condition. Therefore, the surface of an ellipsoid should be favoured above the sphere. The modelling of the normal field has a minor effect. With regard to the analytical handling, a normal field of SomigJiana-Pizzetti-type should be preferred to a truncated sphe--rical harmonic axpansion. The error in the developed analytical representation of the ellipsoidal and topographical terms reaches maximum values of ±1 10~8ms~2, if the GRS80 (Somigliana-Pizzetti-field) is used as a normal field. If the normal gravity potential is modelled by a spherical harmonic expansion, (OSU91alf) truncated at degree Nv = 20, the approximation errors slightly increase up to ±7 10-8ms-2.
When no ellipsoidal and topographical reductions are applied to the scalar gravity anomalies and when this data is analysed on a sphere, the resulting effect in the solution of the disturbing potential increases to ±40m2s~2. This corresponds to a vertical shift up to ±4m of the equipotential surfaces in the vicinity of the earth's surface. If the unreduced scalar gravity anomalies are analysed on the surface of an ellipsoid, the corresponding effect will be ±2m.
The proposed iterative procedure for solving the linear boundary value problem is investigated with respect to its convergence behaviour. When the analytical continuation to the surface of an ellipsoid is applied, a strong convergence can be observed. Already after 6 iteration steps, the spectrum of the solved disturbing potential does not change any more. The spectrum of the residuals represents the error of the solution in the frequency domain. It is the difference between the solved and the given spectrum, which was used to calculate the boundary data. The residuals illustrate the error in the space domain. The resulting vertical shift of the equipotential surfaces in the vicinity of the earth's physical surface reaches a maximum of ±4mm.
In case of the analytical continuation to a sphere, the iterative approach diverges. At first, the residual spectrum is decreasing within the scope of low and medium frequencies. Simultaneously, the error in the high frequency coefficients is increasing. The spectrum deteriorates in the whole frequency domain after about 5 iteration steps.
Only if the reduction terms (ellipsoidal and topographical terms) are generated in the first iteration step by the use of the given (exact) disturbing potential, convergence will be obtained. This was done in order to check the algorithms. For that reason it is supposed that the inaccurate initial solution is responsible for the divergent behaviour of the iterative process. It can be presumed that the analytical continuation to a sphere reacts more sensibly upon errors in the initial solution, than the analytical continuation to the surface of an ellipsoid. In this context, the problem of downward continuation is crucial, requiring further investigations.
(Editeur) To determine both the external gravity potential W and the geometry of the earth's surface 5, various boundary value problems (bvp) can be formulated. They depend on the utilised observaWes L and whether the boundary is supposed to be known or unknown. If the geometry of 5 is already determined by the classical terrestrial techniques or by methods of satellite geodesy, then the fixed boundary value problem is under consideration. Otherwise the resulting bvp is of free type. The relation between the unknowns W, S and the observables L is given by boundary conditions. Generally, they are of non-linear structure.
This thesis focuses on the scalar free bvp and the fixed gravimetric bvp.
Analytical and numerical aspects require the linearisation of the boundary conditions. Therefore, suitable approximations have to be introduced for the unknowns. For the gravity potential, the normal gravity potential w is introduced. In the case of the scalar free boundary value problem, the boundary surface is approximated by the telluroid s, resulting from some telluroid mapping. In this context, the gravity potential W is substituted by the disturbing potential 6w := W ? w.
As shown in previous studies (Heck and Seitz, 1993, 1995; Seitz et al., 1994), the non-linear terms in the boundary condition can amount to maximum values of ±200 10~8ma~2 in case of the scalar free bvp, and ±40 10~8ms~2 for the fixed gravimetric problem. As a consequence of omitting the non-linear terms, the equipotential surfaces of the solution for the disturbing potential are shifted in vertical direction. Utilising a normal gravity field of Somigiiana-Pizzetti-type (e.g. GRS80), this shift takes values up to 4mm. If the gravitational portion of the normal gravity field is represented by a truncated series of spherical harmonics (e.g. Nv = 20), this effect can be reduced to 2mm. In the vicinity of the earth's physical surface, the accuracy of present day global solutions of the earth's gravity potential varies in the scope of dm-m. Therefore, the non-linear effects in the boundary condition can be neglected for the purpose of this thesis.
f
The linear boundary condition is now based on a linear operator D acting on the disturbing potential 6w. Applying the evaluation operator E,, the resulting linear functional is restricted to the telluroid. In case of the fixed bvp, the evaluation operator es is applied to D{Sw} and restricts the boundary condition to the physical earth's surface. The coefficients of the differential operator D are functional of the normal gravity potential w. Introducing an approximation wa of the normal gravity potential tu, the coefficients of D can be represented in terms of a Taylor series expansion.
For the scalar free boundary value problem as well as for the fixed gravimetric bvp the second order approximation of the differential operator D is derived. Thereby, the approximation wa of the normal gravity potential is represented by the potential of a Helmert's normal spheroid. This potential is symmetrical with respect to the mean earth's equatorial plane and independent of the geocentric longitude. In addition to the centrifugal part, this normal potential contains the isotropic term fj?/r and, furthermore, two gravitational terms, which are proportional to the zonal harmonic coefficients «/2 and «/4.
The analytical representation of the differential operator is carried out on the basis of wa. The extension of wa by the coefficient J± leads to the second order approximation of D. Further simplifications of wa result in the first order approximation and the isotropic approximation of D{6w} (fundamental equation of physical geodesy). This evaluation is done for a general normal potential and a Somigliana-Pizzetti reference field respectively.
The numerical studies clearly show that the error in the boundary condition, caused by isotropic approximation, exceeds the accuracy of relative gravimetric measurements by the multiple. It is also larger, by a factor of 5-10, than the non-linear terms in the corresponding boundary condition. According to this result, the ellipsoidal terms must not be neglected, as it is done in the isotropic approximation. If a Somigiiana-Pizzetti-field is used as normal field w, the first order approximation of the linear boundary condition produces a maximum error of ±0.5 10~8ms~2. The analytical
approximation of the differential operator finds its qualitative end, if the second order ellipsoidal terms, which are caused by the zonal coefficient J^, are taken into account. The errors due to the resulting second order approximation of the ellipsoidal terms in the linear boundary condition can be neglected in practice. The residuals are less than ±0.01 10~8ms~2.
The properties are different, when the normal field w contains tessera! and sectorial terms. In the present thesis, the normal gravity potential is given in terms of spherical harmonics, with a maximum degree Nv = 20. The apposition of the zonal coefficient J^ in the approximated normal field wa gives no better results for the approximated ellipsoidal terms. The maximum residuals are ±7 10~8m.s~2 for both levels of analytical approximation, the first order and the second order approximation. Even if second order tesseral and sectorial terms are included in w0, the accuracy of the approximation cannot be improved. According to the magnitude of the low degree spherical harmonic coefficients, all terms (e.g. up to degree and order 4) have to be included into the analytical approximation of the ellipsoidal terms in the boundary condition. The necessary analytical and numerical expenditure cannot be justified. Because of the slowly decreasing spectrum of this normal gravity potential, it must be concluded that the maximum degree Nv of the spherical harmonic representation has to be in accordance with the order of analytical approximation.
To solve the fixed or scalar free boundary value problem by the aid of a harmonic analysis, the respec--tive boundary condition must be related to a geometrically defined surface of revolution, symmetrical to the mean earth's rotation axis. In this context, a sphere K 9 k with radius r = a and the surface E 9 e of an ellipsoid of revolution is considered. The explained algorithm used for the harmonic ana-lysis results in the spherical harmonic spectrum, whether the (reduced) boundary data are given on K 3 k or E 9 e.
The analytical continuation of the boundary condition is realised by a formal Taylor series expansion of the evaluation operator E, or es respectively. Additionally, the optimal choice of the Taylor point has to be discussed. The Taylor point can be chosen at the auxiliary surface on which the data are analyticaly continued (sphere or surface of an ellipsoid). The Taylor point can also be situated at the original boundary surface. In case of the fixed bvp it can be situated at the physical earth's surface S 9 P, and on the telluroid s 3 p in case of the scalar free boundary value problem. The order of the Taylor series expansion of the evaluation operator is set up in accordance with the error level of the second order analytical approximation of the boundary condition. By this consideration the absolute error level of 10~8ms~2, which is aspired for the analytical approximation of the differential operator, is assigned to the analytically continued boundary condition. For each boundary condition we end up with a representation, where the analytical side contains an isotropic term and so-called ellipsoidal and topographical constituents. The isotropic term refers to the surface K or E. The ellipsoidal and topographical terms in the boundary condition are caused by the anisotropy of the normal potential and by the difference between the boundary surface and the auxiliary surface (K or E), on which the boundary condition is continued. They have to be evaluated either on the original boundary surface (S 3 P - fixed bvp; s 9 p - scalar free bvp) or on the auxiliary surface (K 9 k or E B e) according to the choice of the Taylor point. The ellipsoidal and topographical terms can be applied as a reduction to the original boundary data. The result is a spherical boundary value problem on the sphere K (Neumann-problem in the case of the fixed bvp), or rather a quasi-isotropic bvp with an ellipsoidal boundary surface. If the complete ellipsoidal and topographical terms are neglected, the constant radius approximation will be obtained. The reduction terms are functionals of the disturbing potential Sw we solve for. Therefore, an iterative procedure is required.
To guarantee an absolute accuracy of 1 lQ~8ms~2 for the continuation terms (ellipsoidal and topogra--phical terms), the partial derivatives of the disturbing potential are considered up to the 8th order. The partial derivatives of
j%- are considered up to the 4th order, with respect to the geocentric distance r. The numerical studies confirmed that this high degree of evaluation is necessary in the framework of the analytical continuation to the sphere. Otherwise, the level of accuracy obtained for the ellipsoidal terms cannot be reached for the continuation terms. If the analytical continuation is performed to the surface of an ellipsoid, the Taylor series can be truncated after the 5th order terms without anyloss of accuracy. The accuracy of approximation is not influenced by the choice of the Taylor point. The extremal values of the whole ellipsoidal and topographical part of the linear boundary condition, that has to be modelled by the analytical representation, decreases from ±100 10~5ms~2 in the case of a sphere to ±20 10~5ms~2, if the boundary operator is continued to the surface of an ellipsoid. These terms are identical with the neglected terms in case of the spherical approximation on K or the isotropic approximation on E, respectively. They have the same order of magnitude as the boundary data itself. Undoubtedly, this kind of approximation, where all ellipsoidal and topographical terms are neglected, leads to no high-precision solution of the fixed or scalar free boundary value problem.
First of all, the error of approximation depends on the order of the Taylor series, which are setup for the analytical continuation of the boundary condition. Therefore, the surface of an ellipsoid should be favoured above the sphere. The modelling of the normal field has a minor effect. With regard to the analytical handling, a normal field of SomigJiana-Pizzetti-type should be preferred to a truncated sphe--rical harmonic axpansion. The error in the developed analytical representation of the ellipsoidal and topographical terms reaches maximum values of ±1 10~8ms~2, if the GRS80 (Somigliana-Pizzetti-field) is used as a normal field. If the normal gravity potential is modelled by a spherical harmonic expansion, (OSU91alf) truncated at degree Nv = 20, the approximation errors slightly increase up to ±7 10-8ms-2.
When no ellipsoidal and topographical reductions are applied to the scalar gravity anomalies and when this data is analysed on a sphere, the resulting effect in the solution of the disturbing potential increases to ±40m2s~2. This corresponds to a vertical shift up to ±4m of the equipotential surfaces in the vicinity of the earth's surface. If the unreduced scalar gravity anomalies are analysed on the surface of an ellipsoid, the corresponding effect will be ±2m.
The proposed iterative procedure for solving the linear boundary value problem is investigated with respect to its convergence behaviour. When the analytical continuation to the surface of an ellipsoid is applied, a strong convergence can be observed. Already after 6 iteration steps, the spectrum of the solved disturbing potential does not change any more. The spectrum of the residuals represents the error of the solution in the frequency domain. It is the difference between the solved and the given spectrum, which was used to calculate the boundary data. The residuals illustrate the error in the space domain. The resulting vertical shift of the equipotential surfaces in the vicinity of the earth's physical surface reaches a maximum of ±4mm.
In case of the analytical continuation to a sphere, the iterative approach diverges. At first, the residual spectrum is decreasing within the scope of low and medium frequencies. Simultaneously, the error in the high frequency coefficients is increasing. The spectrum deteriorates in the whole frequency domain after about 5 iteration steps.
Only if the reduction terms (ellipsoidal and topographical terms) are generated in the first iteration step by the use of the given (exact) disturbing potential, convergence will be obtained. This was done in order to check the algorithms. For that reason it is supposed that the inaccurate initial solution is responsible for the divergent behaviour of the iterative process. It can be presumed that the analytical continuation to a sphere reacts more sensibly upon errors in the initial solution, than the analytical continuation to the surface of an ellipsoid. In this context, the problem of downward continuation is crucial, requiring further investigations.Numéro de notice : 28009 Affiliation des auteurs : non IGN Thématique : POSITIONNEMENT Nature : Thèse étrangère Permalink : https://documentation.ensg.eu/index.php?lvl=notice_display&id=63356 Réservation
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Code-barres Cote Support Localisation Section Disponibilité 28009-01 30.40 Livre Centre de documentation Géodésie Disponible 28009-02 30.40 Livre Centre de documentation Géodésie Disponible Studies on the use of the boundary element method in physical geodesy / Rüdiger Lehmann (1997)
Titre : Studies on the use of the boundary element method in physical geodesy Type de document : Monographie Auteurs : Rüdiger Lehmann, Auteur Editeur : Munich : Bayerische Akademie der Wissenschaften Année de publication : 1997 Collection : DGK - A Sous-collection : Theoretische Geodäsie num. 113 Importance : 103 p. Format : 21 x 30 cm ISBN/ISSN/EAN : 978-3-7696-8194-9 Note générale : Bibliographie Langues : Anglais (eng) Descripteur : [Vedettes matières IGN] Géodésie physique
[Termes IGN] discrétisation
[Termes IGN] équation intégrale
[Termes IGN] problème des valeurs limites
[Termes IGN] surface de référence
[Termes IGN] Terre (planète)Index. décimale : 30.40 Géodésie physique Résumé : (Auteur) This report investigates various aspects of the application of the boundary element method in physical geodesy. Mainly the increasing accuracy requirements for the high resolution gravity field and geoid deter-mination have induced the further development of classical methods for the solution of geodetic boundary value problems, but also new techniques were established. The boundary element method permits the nu-merical solution of linearized geodetic boundary value problems, formulated as geodetic boundary integral equations. Previous geodetic investigations have focused on the strongly singular boundary integral equations in a local scale, which were solved successfully by means of Galerkin discretization methods with piecewise constant trial functions.
We extend these results in various directions : We consider the closed surface of the Earth as the bound-ary surface. Additionally, we solve a hypersingular boundary integral equation, and also piecewise linear trial/test functions are applied. Modern numerical cubature methods for the different types of integrals are tested and implemented. For the solution of the resulting linear system of equations, we apply highly efficient generalized CG methods.
A very important aspect of this report is the application of modern parallel computers. Problems of imple-mentation of the boundary element method on such types of computers are in the focus of current research in engineering sciences. Some new problems arise, concerning the parallelization of data structures and algo-rithms, and their solutions are discussed comprehensively. The performance of the parallelization is tested on a MIMD computer with distributed memory of type IBM SP.
Final numerical investigations make the pros and cons of the applied solution methods clearer.Numéro de notice : 28249 Affiliation des auteurs : non IGN Thématique : POSITIONNEMENT Nature : Monographie Permalink : https://documentation.ensg.eu/index.php?lvl=notice_display&id=63595 Réservation
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Code-barres Cote Support Localisation Section Disponibilité 28249-01 30.40 Livre Centre de documentation Géodésie Disponible 28249-02 30.40 Livre Centre de documentation Géodésie Disponible Analyse und Numerik überbestimmter Randwertprobleme in der Physikalischen Geodäsie / M. Hirsch (1996)
Titre : Analyse und Numerik überbestimmter Randwertprobleme in der Physikalischen Geodäsie Titre original : [Analyse et problème de valeur aux limites numériques surdéterminées en géodésie physique] Type de document : Thèse/HDR Auteurs : M. Hirsch, Auteur Editeur : Munich : Bayerische Akademie der Wissenschaften Année de publication : 1996 Collection : DGK - C Sous-collection : Dissertationen num. 453 Importance : 154 p. Format : 21 x 30 cm ISBN/ISSN/EAN : 3-7696-9596-1 Note générale : Bibliographie Langues : Allemand (ger) Descripteur : [Vedettes matières IGN] Géodésie physique
[Termes IGN] Aristoteles
[Termes IGN] modèle mathématique
[Termes IGN] problème des valeurs limitesIndex. décimale : 30.40 Géodésie physique Résumé : (Auteur)The determination of the Earth gravity field is a primary objective of geodesy. In order to solve this task, gravity values must be available in global covering and high density; till now this condition is only insufficiently fulfilled. Therefore, great expectations are focused on new developed observation technologies realizing high precision measurements of gravity or gravitational field signals on moved platforms (airplanes, satellites). These processes promise strongly improved qualitative and quantitative information about the gravity field. Moreover, they are much more effective than traditional methods.
This study deals with suitable mathematical modelling of two of these new measurement methods: airborne gravimetry and satellite gradiometry. In particular, the combination between already available gravity information and new observations in a consistent model is investigated. Overdetermined boundary value problems are used for a mathematical description of this task. In contrast to the classical geodetic boundary value problem, the solution of these problems is not uniquely determinable. The sought quantities rather have to be estimated in function spaces. For this reason, the well known BLUE principle was expanded in order to apply it in infinite dimensional spaces. The direct parameter estimation in the overdetermined boundary value problem is not possible, since the equation types are different while the BLUE principle requires an identical equation type. Therefore, a transformation into a homogenous system of integral equations using the theory of pseudodifferential operators (PDO) has to be performed.
Starting from a general formulation of the overdetermined boundary value problem, two special problems are studied; a linear fixed problem to model the local determination of the gravity field by means of airborne gravimetry, and a nonlinear free boundary value problem, describing the global determination of the gravitational field by means of satellite gradiometry. The solution of the nonlinear problem is based upon an imbedding technique by Hormander. Using this imbedding technique the problem can be decomposed into a sequence of linear boundary value problems with the same structure.
In order to be able to solve the problems with an uniform procedure, the problems are transformed in systems of PDO-equations and interpreted as an analogy to the Gauss-Markov-Model. Inversion-free solution formulae are derived for optimal estimation of the sought potential in the space and frequency domains. Using assumptions about stochastic properties of measurement noise, error formulae, describing expected accuracy of the solution, can be obtained.
In order to verify derived solutions, numerical studies are carried out, which can be divided into the following two parts:
In the first part, an overdetermined boundary value problem in local formulation is investigated. This problem is applied for modelling the stabilized downward continuation of airborne gravimeter data. Using three numerical experiments, the possibility of achieving the stabilization of continuation process without a smoothing of the measurements can be proved. This means that the overdetermined boundary value problem is an alternative to the usually applied Tikhonov's regularization, also in numerical case.
The second part discusses the numerical studies of an overdetermined boundary value problem, which has been formulated to determine the global gravitational field in high resolution. First, the numerical experiments are described. This description explains the simulation of the satellite gradiometry mission ARISTOTELES, the data reduction to given boundary surfaces and the error modelling. In the sequel, the successful numerical verification of the derived estimation formulae is covered. A detailed graphical representation illustrates the accuracy potential of the satellite gradiometry data. Further on, analyses of the influence of the polar data gaps and of the aliasing effect are carried out. The obtained results are compared with the results of other authors.Numéro de notice : 28036 Affiliation des auteurs : non IGN Thématique : POSITIONNEMENT Nature : Thèse étrangère Permalink : https://documentation.ensg.eu/index.php?lvl=notice_display&id=63383 Réservation
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Code-barres Cote Support Localisation Section Disponibilité 28036-01 30.40 Livre Centre de documentation Géodésie Disponible 28036-02 30.40 Livre Centre de documentation Géodésie Disponible Tailored numerical solution strategies for the global determination of the Earth's gravity field / W.D. Schuh (1996)
Titre : Tailored numerical solution strategies for the global determination of the Earth's gravity field : study of the complementary use of the gradiometry and Global Positioning System (GPS) for the Earth's gravity field Type de document : Rapport Auteurs : W.D. Schuh, Auteur Editeur : Graz : Technische Universität Graz Année de publication : 1996 Collection : Mitteilungen der geodätischen Institute num. 81 Importance : 156 p. Format : 21 x 30 cm Note générale : Bibliographie Langues : Anglais (eng) Descripteur : [Vedettes matières IGN] Géodésie physique
[Termes IGN] champ de pesanteur terrestre
[Termes IGN] compensation par moindres carrés
[Termes IGN] équation linéaire
[Termes IGN] harmonique sphérique
[Termes IGN] matrice de covariance
[Termes IGN] modèle stochastique
[Termes IGN] potentiel de pesanteur terrestre
[Termes IGN] problème des valeurs limitesIndex. décimale : 30.40 Géodésie physique Numéro de notice : 18259 Affiliation des auteurs : non IGN Thématique : POSITIONNEMENT Nature : Rapport d'étude technique Permalink : https://documentation.ensg.eu/index.php?lvl=notice_display&id=41459 Réservation
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Code-barres Cote Support Localisation Section Disponibilité 18259-01 30.40 Livre Centre de documentation Géodésie Disponible Geodetic theory today / Fernando Sanso (1995)
Titre : Geodetic theory today : third Hotine-Marussi symposium on mathematical geodesy Type de document : Actes de congrès Auteurs : Fernando Sanso, Éditeur scientifique ; International association of geodesy, Auteur ; International union of geodesy and geophysics, Auteur Editeur : Berlin, Heidelberg, Vienne, New York, ... : Springer Année de publication : 1995 Collection : International Association of Geodesy Symposia, ISSN 0939-9585 num. 114 Conférence : IAG 1994, 3rd Hotine-Marussi symposium on mathematical geodesy, Geodetic theory today 30/05/1994 03/06/1994 L'Aquila Italie Importance : 446 p. Format : 21 x 30 cm ISBN/ISSN/EAN : 978-3-540-59421-5 Langues : Anglais (eng) Descripteur : [Vedettes matières IGN] Géodésie
[Termes IGN] approximation
[Termes IGN] cartographie
[Termes IGN] estimation statistique
[Termes IGN] géodésie mathématique
[Termes IGN] géopositionnement
[Termes IGN] Global Positioning System
[Termes IGN] gravimétrie
[Termes IGN] interférométrie
[Termes IGN] mécanique céleste
[Termes IGN] modélisation
[Termes IGN] ondelette
[Termes IGN] probabilité
[Termes IGN] problème des valeurs limites
[Termes IGN] problème inverse
[Termes IGN] théorie de la relativitéNote de contenu : Foreword : Report on the III Hotine-Marussi Symposium
(B. Benciolini)
Monday, May 30, 1994
The Newton Form of the Geodesic Flow on SR2 and EA,B2 in Maupertuis Gauge
(R.J. You, E. Grafarend)
The Lie Series Describing the Geodesic Flow on SR2 and EA,B2
(E.W. Grafarend, R. Syffus)
Conformal Structures and Reference Frames in the Post-Newtonian Approximation to General Relativity
(D.S. Theiss)
3D Linear Similarity Coordinate Transformation Between a Global Geodetic System and a Local Geodetic System Without Local Ellipsoidal Heights
(E. Grafarend, F.I. Okeke)
Application of Moebius Barycentric Coordinates (Natural Coordinates) for Geodetic Positioning
(W. Pachelski, E.W. Grafarend)
DEM Generation with ERS-1 Interferometry
(C. Prati, F. Rocca)
Fundamental GPS Network in Lithuania
(K. Borre, P. Petroâkevi~ius)
The Rotation of the Celestial Equatorial System, With the So-called "Non-rotating Origin"
(B. Richter)
On the GPS Double-difference Ambiguities and Their Partial Search Spaces
(P.J.G. Teunissen)
The Exact Solution of the Nonlinear Equations of the 7-parameter Global Datum Transformation C7 (3)
(E. Grafarend; F. Krumm)
GPS - Spacetime Observables
(V.S. Schwarze)
Tuesday, May 31, 1994
The Optimal Universal Transverse Mercator Projection
(E. Grafarend)
The Generalized Mollweide Projection of the Biaxial Ellipsoid
(E. Grafarend, A. Heidenreich)
The Hotine Oblique Mercator Projection of Ea,b
(J. Engels, E. Grafarend)
The Embedding of the Plumbline Manifold : Orthometric Heights
(E. Grafarend, R.J. You)
Inverse Cartographic Problems
(B. Marana, F. Sansà)
Accuracy of Orbit Computation for Geodetic Satellites : the Ordered and the Chaotic Case
(A. Milani)
The Short-arc Approach to Laser Ranging Analysis
(S. Usai, M. Carpino, A. Milani, G. Catastini, A. Rossi)
Orbital Injection Analysis for Twin LAGEOS Satellites in Supplementary Orbits and the Measurement of the Lense-Thirring Gravitomagnetic Effect
(F. Vespe, L. Anselmo)
Determination of the Gravity Field from Satellite Gradiometry - A Simulation Study -
(M. Belikov, M. van Gelderen, R. Koop)
New Wavelet Methods for Approximating Harmonic Functions
(W. Freeden, M. Schreiner)
Satellite Gradiometry - A New Approach
(W. Freeden, M. Schreiner)
Satellite Orbit Geometry Under Nonconservative Forces
(E. Grafarend, R.J. You)
Spherical Cap Models of Laplacian Potentials and General Fields
(A. De Santis, C. Falcone)
Solving the Inverse Gravimetric Problem: On the Benefit of Wavelets
(L. Ballani)
Integrated Inverse Gravimetric Problems
(R. Barzaghi)
Wednesday, June 1, 1994
Mathematical Statistics for Spatial Data ; the Use of Geostatistics for Geodetic Purposes
(A. Stein)
On the Reconstruction of Regular Grids from Incomplete, Filtered or Unevenly Sampled Band-limited Data
(M.G. Sideris)
Non-parametric Statistics and Bootstrap Methods for Testing the Data Quality of a Geographic Information System
(F. Crosilla, G. Pillirone)
The Effects of Fuzzy Weight Matrices on the Results of Least Squares Adjustments
(H. Kutterer)
On Some Alternatives to Kalman Filtering
(B. Schaffrin)
Chaotic Behaviour in Geodetic Sensors and Fractal Characteristics of Sensor Noise
(Z. Li, K.-P. Schwarz)
Discriminant Analysis to Test Non-Nested Hypotheses
(B. Betti, M. Crespi, F. Sansà, D. Sguerso)
Thursday, June 2, 1994
On Stochastic Boundary Conditions for Laplace Equation
(Yu.A. Rozanov)
A Series Solution for Zagrebin's Problem
(J. Otero, J. Capdevila)
Solution of the Geodetic Boundary Value Problem in Spectral Form
(M.S. Petrovskaya)
Gravity Reductions Versus Approximate B.V.P.s
(F. Sansà, G. Sona)
Classical Methods for Non-spherical Boundary Problems in Physical Geodesy
(P. Holota)
On a Scalar Fixed Altimetry - Gravimetry Boundary Value Problem
(W. Keller)
Non-linear Effects in the Geodetic Version of the Free Geodetic Boundary Value Problem Based on Higher Order Reference Fields
(B. Heck, K. Seitz)
Perturbation Expansion for Solving the Fixed Gravimetric Boundary Value Problem
(R. Klees)
Application of the Concept of Biorthogonal Series to a Simulation of a Gradiometric Mission
(A. Albertella, F. Migliaccio, F. Sansà)
Local Geoid Accuracies from Different Kinds of Data
(A. Albertella, F. Sacerdote)
On Calculating the Attraction of the Topographic-Isostatic Masses
(H.A. Abd-Elmotaal)
The Fast Hartley Transform Applied to the Iberian Geoid Calculation
(M.J. Sevilla, G. Rodriguez-Velasco)
The Impact of the Gravitational Potential of Topographic and Isostatically Adjusted Masses on the Geoid
(J. Engels, E. Grafarend, P. Sorcik)
The Topographic Potential with Respect to the Reference Ellipsoid
(M. Feistritzer)
Friday, June 3, 1994
On the Application of Wavelets in Geodesy
(F. Barthelmes, L. Ballani, R. Klees)
Geodetic Applications of Wavelets : Proposals and Simple Numerical Experiments
(L. Battha, B. Benciolini, P. Zatelli)
How Accurately Do We Know the Marine Geoid in Shallow Water Regions ?
(M. Metzner, S. Dick, E.W. Grafarend, M. Stawarz)
Rank Deficiency of Altimetry-Gravimetry SST Determination
(R. Barzaghi, M.A. Brovelli, F. Sansà)Numéro de notice : 64248 Affiliation des auteurs : non IGN Thématique : POSITIONNEMENT Nature : Actes Permalink : https://documentation.ensg.eu/index.php?lvl=notice_display&id=36364 Réservation
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Code-barres Cote Support Localisation Section Disponibilité 64248-02 CG.94 Livre Centre de documentation Congrès Disponible 64248-01 DEP-PMT Livre SGM Dépôt en unité Exclu du prêt Bestimmung von Modellparametern der Erde durch Analyse ihrer Drehbewegung / H. Fröhlich (1994)PermalinkEffects of Non-Linearity in the Geodetic Boundary Value Problems / B. Heck (1993)PermalinkUntersuchungen zur Lösung der fixen gravimetrischen Randwertprobleme mittels sphäroidaler und Greenscher Funktionen / C.T. Nguyen (1993)PermalinkUntersuchungen zur Lösung der fixen gravimetrischen Randwertprobleme mittels sphäroidaler und Greenscher Funktionen / N. Thong (1993)PermalinkLösung des fixen geodätischen Randwertproblems mit Hilfe der Randelementmethode / R. Klees (1992)PermalinkEine approximative Lösung der fixen gravimetrischen Randwertaufgabe im Innen- und Außenraum der Erde / J. Engels (1991)PermalinkGeodetic work in the Netherlands 1987-1990 / W. Baarda (1991)PermalinkOn the linearized boundary value problems of physical geodesy / B. Heck (1991)PermalinkThe geodetic boundary value problem in two dimensions and its iterative solution / M. Van Gelderen (1991)PermalinkGeodetic boundary value problems 4 / K. Arnold (1990)PermalinkThe role of the topography in gravity gradiometer reductions and in the solution of the geodetic boundary value problem using analytical downward continuation / Y.M. Wang (1990)PermalinkAdvanced physical geodesy / Helmut Moritz (1989)PermalinkGeodetic boundary value problems 3 / K. Arnold (1989)PermalinkA simulation study of the overdetermined geodetic boundary value problem using collocation / L. Tsaoussi (1989)PermalinkNumerical recipes / William H. Press (1988)PermalinkProceedings, Tome 1. Theory of Integrated Geodesy and Boundary value problems, Gravity Field within the Integrated Geodesy Experiences, applications and numerical results / GGRI of Hungary academy (1988)PermalinkSphärische Integralformeln zur Lösung des freien geodätischen Randwertproblems / W.R. Barth (1988)PermalinkTreatment of geodetic levelling in the integrated geodesy approach / Dennis G. Milbert (1988)PermalinkFigure and dynamics of the Earth, Moon and planets: proceedings of the international symposium, Prague, 1986, 1. Tome 1 / P. Holota (1987)PermalinkGeodätische Netze im Kontinuum / Friedrich W. Krumm (1987)Permalink