Approximate Inverse Geophysical Scattering on a Small Body

• Published in: SIAM journal on applied mathematics Vol. 56; no. 1; pp. 192 - 218
• Main Authors: Katsevich, Alexander I., Ramm, Alexander G.
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01-02-1996
• Subjects: Accuracy; Approximation; Ellipsoids; Geophysics; Greens function; Information retrieval noise; Inhomogeneity; Integral equations; Inverse problems; Inverse scattering; Laboratories; Localization; Mathematical functions; Mathematical moments; Tensors
• ISSN: 0036-1399; 1095-712X
• DOI: 10.1137/S0036139994263446

A rigorous theoretical investigation of an inverse geophysical scattering problem for a small body D characterized by a real-valued function$\nu(z), z \in D \subset \mathbb{R}^3$, is given. Using this investigation, a two-step method for an approximate solution of the inverse problem is developed. First, the zeroth moment (total intensity) ν̃D≈ ∫Dν (z)dz and the first moment (center of gravity) z̃(0)≈ ∫Dzv (z) dz/∫Dv (z) dz of the unknown function v(z) are approximately found. Second, the above moments are refined and the tensor of the second central moments of ν(z) is found. Using this information, an ellipsoid D̃ and a real constant ν̃ are found, such that the inhomogeneity ν̃(z) = ν̃, z ∈ D̃, and$\tilde{\nu}(z) = 0, z \not\in \tilde{D}$, best fits the surface data and has the same zeroth, first, and second moments. The accuracy of such procedure is established. Both low-frequency and fixed-frequency cases are considered. The proposed method is very simple numerically and is relatively stable with respect to small perturbations of the data. Model numerical experiments showed effectiveness of the method.