Time-reverse modelling of acoustic wave propagation in attenuating media

Time-reverse modelling (TRM) of acoustic wave propagation has been widely implemented in seismic migration and time-reversal source imaging. The basic assumption of this modelling is that the wave equation is time-invariant in non-attenuating media. In the Earth, attenuation often invalidates this assumption of time-invariance. To overcome this problem, I propose a TRM approach that compensates for attenuation and dispersion effects during the wave propagation in attenuating media. This approach is based on a viscoacoustic wave equation which explicitly separates attenuation and dispersion following a constant-Q model. Compensating for attenuation and dispersion during TRM is achieved by reversing the sign of the attenuation operator coefficient while leaving the counterpart dispersion parameter unchanged in this viscoacoustic wave equation. A low-pass filter is included to avoid amplifying high-frequency noise during TRM. I demonstrate the effects of the filter on the attenuation and the phase velocity by comparing with theoretical solutions in a 1-D Pierre shale homogeneous medium. Three synthetic examples are used to demonstrate the feasibility of attenuation compensation during TRM. The first example uses a 1-D homogeneous model to demonstrate the accuracy of the numerical implementation of the methodology. The second example shows the applicability of source location using a 2-D layering model. The last example uses a 2-D cross-well synthetic experiment to show that the methodology can also be implemented in conjunction with reverse-time migration to image subsurface reflectors. When attenuation compensation is included, I find improved estimation of the source location, the excitation timing of the point source, the magnitude of the focused source wavelet and the reflectivity image of reflectors, particularly for deep structures underneath strongly attenuating zones.