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On a new approach to frequency sounding of layered media. (English) Zbl 1185.35336

Summary: Frequency sounding of layered media is modeled by a hyperbolic problem. Within the framework of this model, we formulate an inverse problem. Applying the Laplace transform and introducing the impedance function, the latter is first reduced to the inverse boundary value problem for the Riccati equation and then to the Cauchy problem for a first-order quadratic equation. The advantage of such transformations is that the quadratic equation does not contain an unknown coefficient. For a specific class of data, it is shown that the Cauchy problem is uniquely solvable. Based on the asymptotic behavior of solutions to both the Riccati and quadratic equations, a stable reconstruction algorithm is constructed. Its feasibility is demonstrated in computational experiments.

MSC:

35R30 Inverse problems for PDEs
35L10 Second-order hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
35C20 Asymptotic expansions of solutions to PDEs
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
78A45 Diffraction, scattering
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