Application of the maximum‐likelihood method (MLM) for sonic velocity logging

Modern digital sonic tools can record full waveforms using an array of receivers. The recorded waveforms are extremely complicated because wave components overlap in time. Conventional beamforming approaches, such as semblance processing, while robust, sometimes do not resolve the interfering wave components propagating at similar speeds, such as multiple compressional arrivals due to the formation alteration surrounding the borehole. Here the maximum‐likelihood method (MLM), a high‐resolution array processing algorithm, is modified and applied to process borehole array sonic data. Extensive modifications of the original MLM algorithm were necessary because of the transient character of the sonic data and its effect upon the spectral covariance matrix. We applied MLM to several array sonic data sets, including laboratory data, synthetic waveforms, and field data taken by a Schlumberger array sonic tool. MLM’s slowness resolution is demonstrated in resolving a secondary compressional arrival from the primary compressional arrival in an altered formation, and the formation compressional arrival in the presence of a stronger casing arrival in an unbonded cased hole. In comparison with the semblance processing results, the MLM results clearly show a better slowness resolution. However, in the case of a weak formation arrival, the semblance processing tends to enhance and resolve the weak arrival by the semblance normalization procedure, while the MLM, designed to estimate the signal strength, does not. The heavy computational requirement (mainly, many matrix inversions) in the MLM makes it much slower than semblance processing, which may prohibit implementation of the MLM algorithm in a real‐time environment.