Ali Reza Azmoude Ardalan; Abdorreza Safari; Yahya Tavakkoli
Abstract
The methods applied to regularization of the ill-posed problems can be classified under “direct” and “indirect” methods. Practice has shown that the effects of different regularization techniques on an ill-posed problem are not the same, and as such each ill-posed problem requires its own investigation ...
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The methods applied to regularization of the ill-posed problems can be classified under “direct” and “indirect” methods. Practice has shown that the effects of different regularization techniques on an ill-posed problem are not the same, and as such each ill-posed problem requires its own investigation in order to identify its most suitable regularization method. In the geoid computations without applying Stokes formula, the downward continuation based on Abel-Poisson integral is an inverse problem, which requires regularization. Since so far the regularization of this ill-posed problem has been thoroughly studied, in this paper the regularization of the downward continuation problem based on Abel-Poisson integral, is investigated and various techniques falling into the aforementioned classes of regularizations are applied and their efficiency is compared. From the first class Truncated Singular Value Decomposition (TSVD) and Truncated Generalized Singular Value Decomposition (TGSVD) methods and from the second class Generalized Tikhonov (GT) with the norms and semi-norms in Sobolev subspaces , are applied and their capabilities for the regularization of the problem is compared. Our numerical results derived from simulated studies reveal that the GT method with discretized norm of Sobolev subspace gives the best results among the studied methods for the regularization of the downward continuation problem based on the Abel-Poisson integral. On the contrary, the TGSVD method with the discretized second order derivatives has less consistency with the ill-posed problem and yields less accuracy. Finally, the GT method with discretized norm of Sobolev subspace is applied to the downward continuation of real gravity data of the type modulus of gravity acceleration within the geographical region of Iran to derive a geoid model for this region.
Abdorreza Safari; Yahyallah Tavakkoli
Abstract
The problem of downward continuation of the gravity field from the Earth’s surface to the reference ellipsoid arises from the fact that the solution to the boundary value problem for geoid determination without applying Stokes formula is sought in terms of the disturbing potential on the ellipsoid ...
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The problem of downward continuation of the gravity field from the Earth’s surface to the reference ellipsoid arises from the fact that the solution to the boundary value problem for geoid determination without applying Stokes formula is sought in terms of the disturbing potential on the ellipsoid but the gravity observations are only available on the Earth’s surface. Downward continuation is achieved via Abel-Poisson integral and its derivatives. Before solving downward continuation problem it should be checked the solvability of the problem. The solvability of the problem is guarantied if Picard condition is satisfied. The topic of this paper is the study of solvability of downward continuation problem via Picard condition.
