Minimum-phase and related properties of the response of a horizontally stratified absorptive Earth to plane acoustic waves

Abstract

Abstract In a horizontally stratified absorptive earth with vertically traveling plane compressional waves, denote the Fourier transforms of the reflectivity, delayed transmissivity, and the acoustic impedance by R, T, and Z, respectively. It is shown that both Z and 1+ or -R are positive real (p-r) functions. Although R, in general, is neither p-r nor minimum phase, R<1 for real frequencies. T (the transmission function referred to a time origin t 1 --the one-way traveltime through the section) is not, in general, p-r. If the absorption law is a minimum-phase function, however, then T is minimum phase. If absorption is absent, then the inverse filter for T is not only minimum phase, but is also of a finite duration equal to 2t 1 .