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. |