Extraction of reflected events from sonic-log waveforms using the Karhunen-Loève transform

Abstract

Sonic-reflection logging, a recently developed borehole geophysical scheme, is in principle capable of providing a clear view of outside the well bore. In this type of acoustic well logging, a key technical obstacle is that the reflected wave signal is almost entirely obscured by the directly arriving P-, S-, and Stoneley wave modes. Effective extraction of these reflection signals from the full acoustic waveforms is therefore a critical data-processing step. We have examined the use of the Karhunen-Loève (KL) transform, combined with a band-limiting filter, as a technique for the extraction of reflections of interest from a mixture with directly arriving wave modes of much higher amplitude. Under the assumption that large energy (squared-amplitude) differences exist between each wave component, the direct Stoneley wave, S-wave, and the P-wave are eliminated sequentially by subtracting the most significant principal components, after which the remaining signal is seen to be dominated by reflected events. Thereafter, the extracted reflections can be used in migration to provide interpretable images of the structures outside the borehole. Synthetic data are used to develop and justify our procedure for subtraction of appropriate KL principal components. Laboratory data are used to demonstrate in detail the suppression of unwanted modes. For comparison, the multiscale slowness-time-coherence method is applied to extract reflections from the same data set. The procedure is exemplified on a field data case with attention paid in particular to the consequences to imaging of near-borehole structures.