Plane‐wave Q deconvolution

Abstract

Often the information content of measured signals from distance sources is hidden, because the signal distorts, weakens, and loses resolution as it propagates. For seismic energy traveling in the earth, these propagation effects can be approximated by the constant (frequency‐independent) Q model for attenuation and dispersion. For a propagating plane wave, this model leads to a spatial attenuation factor that is an unbounded function of frequency. Consequently, the broadband inverse of the constant-Q filter does not exist. For a fixed distance between the source and receiver the effects of the propagation path can be deconvolved (removed) within the seismic band by reversing the propagation of the plane wave. This propagation reversal is done by a time reversal with Q replaced by —Q, thereby changing absorption to gain in the complex wavenumber. Normally, measured seismic traces contain returns from a variety of depths. The interference of waves with different amounts of attenuation complicates the inversion process. From a superposition of plane waves with reversed propagation, a general inverse to an attenuation earth filter is proposed. To account for the increased attenuation with depth, the plane‐wave inverse filter is now time‐varying. This time‐varying inverse filter has a simple Fourier integral representation where the wavenumber is complex, and the direction of propagation is chosen such that the wave is growing rather than attenuating with distance. To control the wavelet side lobes a frequency‐domain window function (Hanning window) is applied to the trace. This two‐step plane‐wave deconvolution scheme was demonstrated to be superior to conventional deconvolution procedures. Tests with field data indicate the method is effective in removing attenuation effects from both VSP (Vertical Seismic Profile) and surface measurements. Phase distortions are eliminated and interference between events is reduced within the seismic band. This inverse is nearly exact for events where the time‐bandwidth (propagation time‐signal bandwidth) product is less than the effective Q. For depths where the time‐bandwidth product is greater than [Formula: see text] large wavelet side lobes appear. The wavelet side lobes can be partially suppressed by tapering the edges of the spectrum. However, the large side lobes of wavelets from shallow reflectors limit the bandwidth that can be recovered from the deeper events to aproximately [Formula: see text], where t is the propagation time to the event. Advances in the inversion algorithm (e.g., a Wiener filter could be used in place of the Hanning window to control side lobes) could probably improve upon our results, but in most cases even a small amount of measurement noise limits the reflection sequences to time‐bandwidth products that are less than twice the effective Q.