On the geodetic boundary value problem for a fixed boundary surface—A satellite approach

Bjerhammar, Arne; Svensson, Leif

doi:10.1007/bf02520941

Cited by 55 publications

(12 citation statements)

References 9 publications

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“…Many geodesists have contributed to the study of BP: e.g. Koch and Pope (1972), Bjerhammar and Svensson (1983), Grafarend (1989), Heck (1989), Sacerdote and Sansò (1989), Holota (1997Holota ( , 2005, Čunderlík et al (2008) and Čunderlík and Mikula (2010). The major achievements in the present dates are local solvability results and a deep knowledge of the linearized problem which is a regular oblique derivative problem in the most general case.…”

Section: Introductionmentioning

confidence: 90%

Construction of the maximal solution of Backus’ problem in geodesy and geomagnetism

2011

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The (simplified) Backus' Problem (BP) consists in finding a harmonic function u on the domain exterior to the three dimensional unit sphere S, such that u tends to zero at infinity and the norm of the gradient of u takes prescribed values g on S. Except for a change of sign, the solution is not unique in general. However, there is uniqueness of solutions in the class of functions with the additional property that the radial component of the gradient of u on S is nonpositive. This is the geodetically relevant case. If a solutionu with this property exists, then u is the maximal solution of the problem (and −u the minimal one). In this paper we propose a method of successive approximations to get this particular solution of BP and prove the convergence for functions g close to a constant function.K e y w o r d s : harmonic functions, fully nonlinear boundary problem, geodesy, geomagnetism

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Section: Introductionmentioning

confidence: 90%

Construction of the maximal solution of Backus’ problem in geodesy and geomagnetism

2011

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“…Now the shape of the Earth can be obtained by geometric satellite triangulation and satellite altimetry over the oceans. In this way, the (linearized) fixed gravimetric boundary value problem in physical geodesy is an oblique derivative problem for the Laplace equation in the Earth's exterior, where the physical surface of the Earth is assumed to be known (see [4] and [16]). …”

Section: Formulation Of the Oblique Derivative Problemmentioning

confidence: 99%

Spectral analysis of the subelliptic oblique derivative problem

Taira

2017

Arkiv För Matematik

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Abstract.This paper is devoted to a functional analytic approach to the subelliptic oblique derivative problem for the usual Laplacian with a complex parameter λ. We solve the long-standing open problem of the asymptotic eigenvalue distribution for the homogeneous oblique derivative problem when |λ| tends to ∞. We prove the spectral properties of the closed realization of the Laplacian similar to the elliptic (non-degenerate) case. In the proof we make use of Boutet de Monvel calculus in order to study the resolvents and their adjoints in the framework of L 2 Sobolev spaces.

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“…Since the boundary condition of the original fixed GBVP is non-linear, the first step towards an analytical solution is a linearization of the problem related to an analytically given normal gravity potential U (Heck, 1989;Heck and Seitz, 1993). The linearized fixed GBVP proves to become an oblique-derivative BVP (Giraud, 1934;Mikhlin, 1965;Miranda, 1970;Bjerhammar and Svensson, 1983;Sacerdote and Sansò, 1989). Geodetic aspects of the linearized fixed GBVP have been discussed by Koch (1971), Koch and Pope (1972), Bosch (1979) and Heck (1991).…”

Section: Introductionmentioning

confidence: 96%

A Brovar-type solution of the fixed geodetic boundary-value problem

Heck

2011

Stud Geophys Geod

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For more than 150 years gravity anomalies have been used for the determination of geoidal heights, height anomalies and the external gravity field. Due to the fact that precise ellipsoidal heights could not be observed directly, traditionally a free geodetic boundary-value problem (GBVP) had to be formulated which after linearisation is related to gravity anomalies. Since nowadays the three-dimensional positions of gravity points can be determined by global navigation satellite systems very precisely, the modern formulation of the GBVP can be based on gravity disturbances which are related to a fixed GBVP using the known topographical surface of the Earth as boundary surface. The paper discusses various approaches into the solution of the fixed GBVP which after linearization corresponds to an oblique-derivative boundary-value problem for the Laplace equation. Among the analytical solution approaches a Brovar-type solution is worked out in detail, showing many similarities with respect to the classical solution of the scalar free GBVP.

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On the geodetic boundary value problem for a fixed boundary surface—A satellite approach

Cited by 55 publications

References 9 publications

Construction of the maximal solution of Backus’ problem in geodesy and geomagnetism

Construction of the maximal solution of Backus’ problem in geodesy and geomagnetism

Spectral analysis of the subelliptic oblique derivative problem

A Brovar-type solution of the fixed geodetic boundary-value problem

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