An exact solvable inverse problem and its application

The inverse acoustic problem for layered media may be solved by application of the well-developed inverse scattering method for the Schrodinger equation, or using the Goupillaud type inversion algorithm. The exact solvable model is useful and instructive in the sense that one can approximate a practical model by a appreciate solvable model and estimate the parameters included in your analytic representation of model. In this paper, we consider an inverse problem with potential which is a finite sum of exponential type function. The corresponding acoustic impedance profile is a local distributed inhomogenity. There are only finite numbers parameters characterizing the exponential functions. It is shown that the Marchenko equation for inverse problem is reduced to a linear algebraic equation which can be solved recursively. A new recursive algorithm of the inverse problem which is independent on Marchenko approach is proposed. As this result, we show that the inverse problem can be solved exactly. An analytic example and several numerical examples are also given.