Letters to the Editor Comments on the Paper; "phase Distortion and Hilbert Transformation in Multiply Reflected and Refracted Body Waves", by G. L. Choy And

In a recent paper Choy and Richards (1975) have demonstrated that phase-shifted arrivals can introduce systematic travel-time errors due to relative changes in the peak and trough positions of the arrival wavelets. The method they suggest to correct the problem is to cross-correlate an estimated zero-phase-shifted wavelet with the seismogram after all the arrivals have been restored to a zero phase shift. The peaks of the cross-correlation are then taken as the reference times for the arrivals. This method is awkward to use in practice because the arrivals will in general be phase-shifted by various angles that are not usually known a priori. This necessitates that each arrival be treated separately by searching for the correct restoring phase shift. The purpose of this note is to point out the existence of a much easier method which is simply to compute the envelope of the cross-correlation of the seismogram with an estimated zero-phase-shifted wavelet (Helmberger and Wiggins, 1971). The envelope of a signal x(t) is the modulus of the analytic signal 2(0 which is defined to be (Bracewell, 1965)