Inversion of sonic waveforms with unknown source and receiver functions

Abstract

SUMMARY A seismogram that is several times the length of the source-receiver wavelet is windowed into two parts—these may overlap—to obtain two seismograms with approximately the same source function but different Green's functions. A similarly windowed synthetic seismogram gives two corresponding synthetic seismograms. The spectral product of the window 1 data with the window 2 synthetic is equal to the spectral product of the window 1 synthetic with the window 2 data only if the correct earth model is used to compute the synthetic. This partition principle is applied to well-log sonic waveform data from Ocean Drilling Project hole 806B, a slow formation, and used there to estimate Poisson's ratio from a single seismogram whose transmitter and receiver functions are unknown. A multichannel extension of the algorithm gives even better results. The effective borehole radius Rb, was included in the inversion procedure, because of waveform sensitivity to Rb. Inversion results for Rb agreed with the sonic caliper, but not the mechanical caliper; thus if Rb is not included in the inversion its value should be taken from the sonic caliper.