The traditional acoustic logging signal processing method is computing the slowness of each component wave by time-domain or frequency-domain methods. But both of the two methods are limited. To combine the signals’ times, frequencies, or amplitudes, we have analyzed the array acoustic logging signals by the fractional Fourier transform and the Choi-Williams distribution. First, we apply the fractional Fourier transform on an array acoustic logging waveform with proper [Formula: see text], then the Choi-Williams distribution analysis method is used to process the signal in the fractional Fourier domain, and finally the result will show in the fractional Fourier time-frequency domain. The results show the following. The array acoustic logging signal is received earlier in the mudstone and diabase formation than in the tuff and breccia formations. The basic frequencies of the compressional wave (P-wave) are not very different, but the basic frequency of the shear wave (S-wave) is highest in the tuff formation and is lowest in the diabase formation. The relative energies of each component wave in the diabase, mudstone, tuff, and breccia formation can be summarized as: for the P-wave, diabase > mudstone ≈ tuff ≈ breccia; for the S-wave, diabase ≈ mudstone > breccia > tuff; and for the Stoneley wave, diabase > mudstone > tuff > breccia. The signal processing method combining the fractional Fourier transform and the Choi-Williams distribution can comprehensively research the time, frequency, and amplitude, thereby improving the segmentation of the time and frequency domains and providing a new method for interpretation of array acoustic logging.
Gabor Analysis for Non-Rectangular Lattices and the Fractional Fourier-Transform
