Regularized Solutions of the Two Layers Inverse Gravimetric Problem in the Space of Bounded Variation Functions
The two layers inverse gravimetric problem is to determine the shape of the two layers in a body B, generating a given gravitational potential in the exterior of B. If the constant density of the two layers is given, the problem is reduced to the determin
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Abstract
The two layers inverse gravimetric problem is to determine the shape of the two layers in a body B, generating a given gravitational potential in the exterior of B. If the constant density of the two layers is given, the problem is reduced to the determination of the geometry of the interface between the two. The problem is known to be ill posed and therefore it needs a regularization, that for instance could have the form of the optimization of a Tikhonov functional. In this paper it is discussed why neither L2 nor H 1;2 are acceptable choices, the former giving too rough solutions, the latter too smooth. The intermediate Banach space of functions of Bounded Variation is proposed as a good solution space to allow for discontinuities, but not too wild oscillations of the interface. The problem is analyzed by standard variational techniques and existence of the optimal solution is proved. Keywords
Bounded variation functions Inverse gravimetric problem Regularization methods
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Introduction to the Problem
The inverse gravimetric problem is to determine the mass distribution in a body B from the known gravity potential C in the complementary region ˝ D B . As it is well known the problem is undetermined (Sansò 2014; Ballani and Stromeyer 1982) and it is generally solved by restricting the space where a solution can be sought. One typical model that, under suitable hypotheses on the regularity of the surfaces involved (Barzaghi and Sansò 1986; Isakov 1990), implies the uniqueness of the solution, is that of a body B consisting by two layers, each of known constant density. This can be considered as a perturbation of a larger known gravity field that acts as reference and imposes the main direction of the vector g. Here M. Capponi · F. Sansò Politecnico di Milano, DICA, Milano, Italy e-mail: [email protected]; [email protected] D. Sampietro () Geomatics Research & Development srl, Lomazzo, Italy e-mail: [email protected]
we will work on the problem when the reference field is considered as parallel and opposed to the verse of the z axis of the Cartesian system in which we frame the problem. The two layers (see Fig. 1) are the upper layer, .H C ıH / Z HC and the lower layer H Z .H C ıH / with densities respectively C and ; we will call ı D C the density contrast and D Gı, with G the universal gravitational constant. Always referring to Fig. 1, we will assume that the interface S between the two layers, fz D .H C ıH .x//g, is agreeing with the plane fz D H g outside the support C , that is a bounded set on the horizontal plane. Moreover, we will set the hypothesis that S has to be constrained, to stay in depth within the layer H z HC , for geophysical/geological reasons. We will simplify this constraint by assuming that jıH ./j L ;
2C;
L D max.HC CH ; H CH /
without any loss of generality. Finally we assume that the modulus of gravity go .x/, is observed on the z D 0 plane. Such a hypothesis is just comfortable for the
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