Discrete model for solving one-dimensional inversion on finite media

The authors develop a one-dimensional discrete-time model for wave propagation in an inhomogeneous layered medium of finite length. The problem of acoustic inversion requires the computation of the acoustic impedance as a function of travel time from knowledge only of boundary excitation and measurement. A discrete state-space ARMA model is developed whose parameters are an explicit function of the boundary conditions of the finite-length medium and sampled values of the inhomogeneous acoustic layered impedance profile. The inversion problem is formulated in the framework of a recursive-least-squares predictive error method in which good initial parameter estimates are derived from transient data. Examples are included showing identification from transient reflected data subject to measurement noise and undersampling