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Geoidbestimmung mit geopotentiellen Koten / M. Feistritzer (1998)
Titre : Geoidbestimmung mit geopotentiellen Koten Titre original : [La détermination du géoïde avec des cotes géopotentielles] Type de document : Thèse/HDR Auteurs : M. Feistritzer, Auteur Editeur : Munich : Bayerische Akademie der Wissenschaften Année de publication : 1998 Collection : DGK - C Sous-collection : Dissertationen num. 486 Importance : 90 p. Format : 21 x 30 cm ISBN/ISSN/EAN : 978-3-7696-9526-7 Note générale : Bibliographie Langues : Allemand (ger) Descripteur : [Vedettes matières IGN] Géodésie physique
[Termes IGN] cote géopotentielle
[Termes IGN] ellipsoïde (géodésie)
[Termes IGN] géoïde terrestre
[Termes IGN] isostasie
[Termes IGN] potentiel de pesanteur terrestre
[Termes IGN] problème de DirichletIndex. décimale : 30.41 Géoïde Résumé : (Auteur) Starting from geopotential numbers as data a new method of downward continuation of the disturbing potential into the interior of the earth onto an approximating surface of the geoid is introduced. In contrast to many other methods an ellipsoid of revolution is used as a reference surface instead of a sphere and furthermore the potential itself is continued downward and not derivatives of the potential (gravity anomalies, deflections of the vertical, gradiometry data). The influence of the topographic masses and their isostatic compensation is taken into account with a remove-restore technique where the kernel of the Newton-Integral is expanded into converging spherical harmonics series. The downward continuation is then performed by a generalized Tikhonov regularisation method which obtains minimal errors and it must be added that both random as well as often omitted systematic parts are considered. The downward continued disturbing potential is then used for geoid determination by using Bruns formula. To check theory test calculations have been performed for the area of Baden-Wiirttemberg. Numéro de notice : 67049 Affiliation des auteurs : non IGN Thématique : POSITIONNEMENT Nature : Thèse étrangère Permalink : https://documentation.ensg.eu/index.php?lvl=notice_display&id=61691 Réservation
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Code-barres Cote Support Localisation Section Disponibilité 67049-01 30.41 Livre Centre de documentation Géodésie Disponible 67049-02 30.41 Livre Centre de documentation Géodésie Disponible On the Unification of Indonesian Local Height Systems / Khafid (1998)
Titre : On the Unification of Indonesian Local Height Systems Type de document : Thèse/HDR Auteurs : Khafid, Auteur Editeur : Munich : Bayerische Akademie der Wissenschaften Année de publication : 1998 Collection : DGK - C Sous-collection : Dissertationen num. 488 Importance : 110 p. Format : 21 x 30 cm ISBN/ISSN/EAN : 978-3-7696-9528-1 Langues : Anglais (eng) Descripteur : [Vedettes matières IGN] Systèmes de référence et réseaux
[Termes IGN] archipel
[Termes IGN] données altimétriques
[Termes IGN] données GPS
[Termes IGN] données ITGB
[Termes IGN] Earth Gravity Model 1996
[Termes IGN] harmonique sphérique
[Termes IGN] Indonésie
[Termes IGN] Java (île de)
[Termes IGN] niveau moyen des mers
[Termes IGN] problème des valeurs limites
[Termes IGN] réseau altimétrique local
[Termes IGN] Sumatra
[Termes IGN] système de référence altimétriqueIndex. décimale : 30.10 Systèmes de référence et réseaux géodésiques Résumé : (Auteur) Height systems are usually referred to the mean sea level at tide gauges. Due to the deviation of mean sea level from one equipotential surface of the Earth's gravity field, the geoid, off-sets exist between the various height systems. These off-sets also affect derived quantities such as gravity anomalies. As long as the direct determination of the potential difference between two datum points, by geodetic levelling and gravimetry , is possible the connection, i.e. unification, of height systems is straightforward. It is a serious problem, however, whenever height datum zones are disconnected by sea. With the advent of advanced space positioning techniques and satellite altimetry new solution strategies for the worldwide unification of height systems came into discussion. This investigation deals with the regional unification of height systems for the area of the Indonesian archipelago.
It is a well-known fact that height datum unification is a global problem. However under certain circumstances it is possible to restrict the unification to a certain part of the Earth. Solution strategies make use of a land, a sea and an integrated component. The sea approach is based on satellite altimetry and oceanic levelling, the land approach on space positioning and spirit levelling, including tide gauge registrations. The link between both components is the solution of the geodetic boundary value problem applied regionally, with the aim to provide potential differences between the various vertical datum systems.
The five fundamental elements related to unification of height systems are investigated individually. Mean sea surface heights are determined by satellite radar altimetry, precise orbit ephemerides and corrections for tides and mesoscale ocean variations. By applying collinear pass analysis, crossover adjustment and a least-squares fit to the TOPEX/POSEIDON (T/P) mean sea surface, the mean sea surface in Indonesian waters is determined using the combined altimetry data from the missions Geosat/ERM, ERS-1/35 days, and T/P with an estimated accuracy of 10 cm. The high resolution data set of the ERS-1/168 days repeat mission has been processed with a simplified procedure. The result is consistent with that of the combination solution. As second element sea surface topography is derived from ocean levelling employing hydrographic data. The sea surface topography for Indonesian waters, especially in the eastern Indonesian region, is computed using several oceanographic data sets, the spherical harmonic model of the old Levitus data set up to degree and order 36, data from the Snellius-I and II expeditions, and the recent data set of the World Ocean Atlas 1994. The accuracy of ocean levelling depends on the measurement errors, data sampling and the reference surface. The assumption of a level of no motion at a certain depth, in this investigation 2000 dB is required to eliminate a fundamental indeterminancy. It is a problematic assumption. Further improvement will be possible once better data coverage and higher accuracy measurements are available. To increase the accuracy of local height datum connection, as third element the relative geoid computation is investigated. The relative geoid computation with inclusion of the height off-sets as unknowns should be the appropriate strategy for the solution of height datum connection over small distances between computation points. Finally as fourth and fifth element, the levelling network and the possibility of geodetic space techniques (GPS and VLBI) for height datum connection in Indonesia are reviewed. The levelling networks of Sumatra, Java, Kalimantan, Bali, Lombok, Sulawesi East Timor, Maluku, Seram have been extended by BAKOSURTANAL in recent years. A geodynamic GPS network is under development in the context of the GEODYSSEA project. They offer a promising prospect for use in a precise regional vertical datum connection. From the comparison of sea surface topography as derived from steric levelling and from altimetry and a spherical harmonic geoid model (EGM96) up to degree and order 360, it can be concluded that sea surface topography as computed from the World Ocean Atlas 1994 can be used for preliminary Indonesian height datum connections. The sea surface topography in the South-West coast of Sumatra is about -16 cm, in the South coast of Java it increases from -14 cm (west) to -8 cm (east).Numéro de notice : 28005 Affiliation des auteurs : non IGN Thématique : POSITIONNEMENT Nature : Thèse étrangère Permalink : https://documentation.ensg.eu/index.php?lvl=notice_display&id=63352 Réservation
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Code-barres Cote Support Localisation Section Disponibilité 28005-01 30.10 Livre Centre de documentation Géodésie Disponible 28005-02 30.10 Livre Centre de documentation Géodésie Disponible 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 Figure de la Terre / F. Chambat (1996)
Titre : Figure de la Terre : gravimétrie, régime de contraintes et vibrations propres Type de document : Thèse/HDR Auteurs : F. Chambat, Auteur Editeur : Paris : Université de Paris 7 Denis Diderot Année de publication : 1996 Importance : 316 p. Format : 21 x 30 cm Note générale : Bibliographie
Thèse de doctorat spécialité géophysique interneLangues : Français (fre) Descripteur : [Vedettes matières IGN] Géodésie physique
[Termes IGN] altitude
[Termes IGN] calcul tensoriel
[Termes IGN] densité
[Termes IGN] ellipsoïde de référence
[Termes IGN] figure de la Terre
[Termes IGN] géoïde
[Termes IGN] gravimétrie
[Termes IGN] harmonique sphérique
[Termes IGN] perturbation
[Termes IGN] potentiel de pesanteur terrestre
[Termes IGN] surface de référence
[Termes IGN] tectonique
[Termes IGN] tenseur
[Termes IGN] vibrationIndex. décimale : THESE Thèses et HDR Résumé : (Auteur) Le lien entre forme de la Terre et contraintes est étudié sans hypothèse rhéologique a priori. Le potentiel externe de gravité est exprimé harmonique par harmonique au deuxième ordre en fonction des variations latérales de densité sur les équipotentielles et de l'altitude des interfaces. Ces variations latérales sont reliées aux composantes du tenseur des contraintes grâce à l'équation d'équilibre. On en déduit l'expression générale du potentiel en fonction des contraintes. L'importance du deuxième ordre pour le calcul de ce potentiel est mise en évidence.
L'inversion des données de géopotentiel et de topographies est effectuée jusqu'au degré 360 en minimisant le déviateur des contraintes, ou plus exactement, la différence entre contraintes quasi verticale et quasi horizontale. L'épaisseur de croûte, quantité la mieux déterminée, est en général plus importante sous les reliefs, où l'état tectonique est une compression verticale. Les compilations de profondeur sismique du "moho" corroborent nos résultats. Cette méthodologie est appliquée à la Lune jusqu'au degré 70. Les anomalies de masses positives sous les grands bassins sont clairement visibles jusqu'à grande profondeur.
Le calcul des fréquences propres d'une Terre latéralement hétérogène est étendu afin de tenir compte du déviateur des contraintes. Sous l'hypothèse transversalement isotrope les observables sont exprimés en fonction des variations latérales des coefficients élastiques sur les équipotentielles, du déviateur de contraintes et de l'altitude des surfaces de contraintes. L'inversion des données correspondantes permet d'avoir accès aux premiers degrés pairs des variations latérales des paramètres mécaniques de la Terre.Note de contenu : Introduction générale
1 Préliminaires géophysiques
1.1 Géodésie et gravimétrie
1.2 Observations et théories géophysiques
2 Préliminaires mathématiques
2.1 Opérateurs de dérivation - Calcul tensoriel
2.2 Déformations et calcul de perturbations
2.3 Harmoniques sphériques et potentiels
3 Gravimétrie, tectonique et forme de la Terre
3.1 Équation de Poisson
3.2 Équilibre hydrostatique
3.3 Équilibre non hydrostatique
4 Gravimétrie et tectonique: inversion des données
4.1 Les données terrestres
4.2 Modèle le plus hydrostatique
4.3 Détermination des paramètres moyens
4.4 Détermination de l'altitude
4.5 Détermination du déviateur
5 Gravimétrie et tectonique locales
5.1 Résolution théorique
6 Gravimétrie et topographie lunaires
6.1 Données lunaires
6.2 Encadrement de la densité moyenne
6.3 Construction de modèles moyens
6.4 Inversion gravimétrique au degré 1=70
7 Modes propres et figure non hydrostatique de la Terre
7.1 Caractérisation variationnelle des modes propres
7.2 Perturbation et splittings
Conclusions et perspectivesNuméro de notice : 46396 Affiliation des auteurs : non IGN Thématique : POSITIONNEMENT Nature : Thèse française Note de thèse : Thèse de doctorat : Géophysique interne : Paris 7 : 1996 nature-HAL : Thèse DOI : sans Permalink : https://documentation.ensg.eu/index.php?lvl=notice_display&id=45611 Réservation
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Code-barres Cote Support Localisation Section Disponibilité 46396-01 THESE Livre Centre de documentation Thèses 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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