Gibbs Phenomenon Suppression and Optimal Windowing for Attenuation and Q Measurements*

There are basically four known techniques, spectral ratios (Hauge 1981, Johnston and Toksoz 1981, Moos 1984, Goldberg et al 1985, Patten 1988, Sams and Goldberg 1990), forward modeling (Chuen and Toksoz 1981), inversion (Cheng et al 1986), and first pulse rise time (Gladwin and Stacey 1974, Moos 1984), that have been used in the past to measure the attenuation coefficient or quality factor from seismic and acoustic data. Problems have been encountered in using the spectral techniques, which included: (1) the correction for geometrical divergence of the acoustic wave front; (2) the suppression of the Gibbs phenomenon or the ringing effect in the spectra; and (3) the elimination of contamination from interfering wave modes. Geometrical corrections have been presented by Patten (1988) and Sams and Goldberg (1990) for the borehole acoustics case. The remaining difficulties in the application of the spectral ratios technique come mainly from the suppression of the Gibbs phenomenon and optimal windowing of wave modes. This report deals with these two problems.