Wave arrival detection in acoustic well logs via the wigner distribution and generalized Allan Variance

Some of the first acoustic well logs were done in the 1950s by Vogel [l]. The tool (commonly called a sonde) used for these logs consisted of one transmitter and one or two receivers. At that time only a rudimentary knowledge of the acoustics of the borehole was known; however, it was known that the compressional wave in the formation would be the first to effect the received signal. Using this fact, the logs were developed to measure the compressional wave travel time through the formation. The present-day sonde employs two transmitters and two receivers, in order to obtain a more accurate measure of compressional travel times. It is apparent that compressional travel time is only a part of the information that is available from the acoustic log. Observing the time trace produced at a receiver, several different occurrences can be seen. These occurrences are the results of various wave types, including the compressional wave, the shear wave, and others. The characteristics of the various waves and how they relate to the surrounding formation is the subject of a great deal of research. A relationship between acoustic travel times and the porosity of the rock formation has been found. Also the ratio of the shear and compressional wave velocities can be related to the lithology of the surrounding formation. This paper explores techniques for the detection and measurement of the compressional and shear waves. First the Generalized Allan Variance is used to detect the compressional wave. The detection of the shear wave is complicated by the trailing end of the compressional wave, which covers the shear wave arrival. The arrival of the shear wave can be detected by applying nonstationary spectral analysis to the wavetrain and identifying the spectral character associated with the wave arrival. A second technique for measuring the shear wave velocity involving the processing of wavetrains from multiple receivers is described.