Estimation of parameters in lossless layered media systems

In this paper we develop a procedure for estimating the parameters (reflection coefficients and travel times) of lossless layered media systems. The estimation criterion is maximum likelihood. It is shown that maximum-likelihood estimates of the parameters of a layer may be obtained by first determining the maximum-likelihood estimate of the upgoing waveform (one of the states) in that layer, and then estimating the parameters from the estimated upgoing waveform as if the estimated waveform is the output of the layer under consideration. A maximum-likelihood state estimator is developed. It is shown that the state waveforms are composed of overlapping wavelets, and estimation of the parameters of a layer is essentially equivalent to estimation of the amplitude and time-delay of the first wavelet in the upgoing waveform of the layer under consideration. A suboptimal maximum-likelihood parameter estimator is developed. It consists of two filters connected in series: a matched filter to treat noise; and a transversal equalizing-filter to correct for wavelet overlapping effects. Numerical results which illustrate the performance of this maximum-likelihood procedure for different noise levels are presented.