Maximum entropy and the relationship of the partial autocorrelation to the reflection coefficients of a layered system

A horizontally stratified half space bounded by a perfect reflector at the top gives rise to a seismogram which, when completed by the direct downgoing pulse at zero time and by symmetry about the origin for negative time, produces an autocorrelation function. If this autocorrelation is convolved with the corresponding prediction error operators of increasing length, we obtain a "gapped function," which deviates more and more from the perfect symmetry exhibited by the autocorrelation. This gapped function consists of the downgoing and upgoing waveforms at the top of each layer. The gap separates the two waveforms, and the gapwidth increases as deeper and deeper layers are reached. In particular, the width of the gap is a measure of the entropy of the seismogram at a given depth level-the deeper we go into the sub-surface, the higher the entropy of the corresponding gapped function. We explore the nature of the gapped function as it relates to the Toeplitz recursion generating the prediction error operators, and we re-derive the synthetic seismogram in terms of wave motion measured in units proportional to the square root of energy. We obtain an explicit relationship between the partial autocorrelation function on the one hand, and the reflection coefficient sequence on the other. This formulation allows us to generalize earlier results, so that we can treat the case for which the surface reflection coefficient is less than unity in magnitude. We investigate both the physical as well as the mathematical foundations of the stratified model, and the relation that this model bears to maximum entropy spectral analysis [1], [6]. In particular, we discuss Burg's fundamental result on the relationship of the maximum entropy spectrum to the reflection coefficient sequence characterizing the given subsurface model.