In acoustic sensing of subterranean formations, a transmitting source in the borehole can be used to create pressure waves that travel through the formation and are recorded at a number of receivers. The pressure wave-train can be very complex. For example, in a fast formation the recorded wave-train from a monopole source may consist of a compressional wave, a shear wave, a Stoneley wave, and a pseudo-Rayleigh wave. These waves are referred to as primary or “direct” waves. In conventional acoustic processing, receiver waveforms are processed using semblance techniques to extract formation properties from the direct waves such as compressional and shear wave velocities.
In a borehole environment, the wave-train also contains coherent reflection events from bedding planes, planar fractures, and clean borehole breakouts. These events are referred to as ‘secondary’ waves because they are usually much smaller than the direct waves. In addition to the secondary waves, incoherent scattering events occur due to borehole rugosity, road noise or drilling noise, and scattering from complex geological structures such as non-planar fracture networks. In conventional acoustic processing, the reflections and scattering are treated as noise and the semblance techniques are applied to reduce their effect on the measurement of the formation properties from the direct waves. Known techniques can include the use of one-dimensional (1D) wave separation in the frequency domain to estimate the direct waves. The wave separation treats the direct waves as the signal and the reflected waves as incoherent noise, which is averaged out during the wave separation. The estimated direct waves are subtracted from the original waveforms to estimate the reflected waves. In such techniques, uniform depth spacing is also assumed and aliasing errors are ignored. This subtraction method can suffer from large residual errors in the reflected wave estimate because the reflections are typically much smaller than the direct waves. The residual errors are reduced after wave separation by stacking the waveforms in the time domain using an expected time of arrival estimate from ray tracing theory. However, the usefulness of such measurements and analysis may be related to the precision or quality of the information derived from such measurements.