March 1987 LIDS - P - 1659 Inversion of Generalized Parabolic Projections ' All Ozbek and Bernard

scattering problem for the corresponding wave field at the receiver locations. Depending on the difference beThe multidimensional inverse scattering problem for an tween the observed and the computed scattered waves, acoustic medium is considered within the homogeneous a new estimate of the medium model is obtained and a new estimate of the medium model is obtained and background Born approximation. A constant density the next iteration is performed. These methods are acoustic medium is probed by a wide-band plane wave very time consuming, and the convergence of the cursource, and the scattered field is observed along a rerent methods depends on the accuracy of the initial ceiver array located outside the medium. The inversion estimate. problem is formulated as a generalized tomographic The objective of exact direct inversion methods is problem. It is shown that the observed scattered field to reconstruct the medium properties exactly, with no can be appropriately filtered so as to obtain generalized iterations involved. These methods require large numprojections of the scattering potential. For a 2-D experbers of sources and receivers with particular observaimental geometry, these projections are weighted intetion geometries which limit their applicability to pracgrals of the scattering potential over regions of parabolic tical problems. From a theoretical point of view, onesupport. The inversion problem is therefor similar to dimensional exact direct inverse scattering methods that of x-ray tomography, except that instead of behave reached an advanced level of development (see [1] ing given projections of the object to be reconstructed for an overview), whereas their extension to higher difor an overview), whereas their extension to higher dialong straight lines, projections along parabolas are mensions has proved to be difficult. given. The inversion procedure that we propose is simConsequently, the logical approach to the practiiHar to the x-ray solution, in the sense that it consists cl solution of the multidimensional inverse scatterof a backprojection operation followed by 2-D space ining problems is the use of approximate direct invers variant filtering. A "Projection-Slice Theorem" is demethods. The differential equation for wave propagamethods. The differential equation for wave propagarived relating the generalized projections and the scattion in a medium can be transformed into the so-called tering potential in the Fourier transform domain. Lippmann-Schwinger integral equation [121. For example, for an acoustic medium with constant density, this equation expresses in integral form the scattered