Maximum entropy spectral decomposition of a seismogram into its minimum entropy component plus noise

Abstract

We consider an N‐layered elastic medium which is perfectly insulated from its embedding space. Each layer has unit two‐way vertical traveltime, and a coincident source‐receiver pair is located just below the top interface. If the insulated medium is excited in the remote past, the power spectrum of the resulting impulsive normal‐incidence synthetic seismogram is a pure line spectrum; because no random elements exist in the system at the time of measurement, this spectrum is a minimum entropy spectrum. If we add white noise to the seismogram, the power spectrum becomes a maximum entropy spectrum. The maximum entropy spectrum can thus be decomposed into the sum of a minimum entropy spectrum plus white noise; this spectral decomposition is due to Pisarenko (1973). If the insulated medium is excited at time t = 0, the resulting synthetic seismogram differs rather remarkably from the seismogram obtained for excitation in the remote past.