Abstract

In their paper, Bhattacharryya and Chan address the problem of reduction of magnetic and gravity data on an arbitrary surface acquired in a region of high topographic relief. In their work, the authors are kind enough to mention our contribution to the solution of the problem of finding the sources responsible for an observed magnetic or gravity anomaly, using the general formalism of inverse problems (Courtillot et al, 1974). Unfortunately, however, the authors seem to be unaware of our other publications which are far more relevant to their subject. Courtillot et al (1973) solved the problem of continuation of a potential field measured on an uneven profile, using the Backus and Gilbert approach. Another reference relevant to this problem (solved by Bhattacharryya and Chan on p. 1424) is Parker and Klitgord (1972), who used the Schwartz-Christoffel transformation. The work was extended to the case of three-dimensional potential fields measured on an uneven surface by Ducruix et al (1974). Indeed, the development of our paper is strikingly similar to that of Bhattacharryya and Chan, although the method is quite different. In our paper, we give many illustrations of both theoretical and real cases, in which our method is seen to perform very well. We leave it to the reader to compare the results provided by both methods and to compare the methods themselves. In a third paper (Le Mouel et al, 1975), we generalized the method and showed how one could obtain excellent approximate analytic solutions of the Dirichlet and Neumann problems in the two-dimensional case for a contour with any arbitrary shape. Finally, let us take the opportunity of this discussion to mention a review of the subject which appears in French in Courtillot (1977) and in English, much expanded, in Courtillot et al (1978). In this last paper, which should be of interest in solving a variety of geophysical problems, we show how our method allows one to continue a potential field measured on an entirely arbitrary set of data points in any number of dimensions for the various coordinate systems in which the Laplace and Helmholtz equations are separable. We also establish the relationship between our method and a generalization of the theory of generalized inverse matrices. One other relevant reference on that subject is Parker (1977). In the case of spherical coordinates, an application can be the continuation of satellite data, a problem studied by Bhattacharryya (1977).

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