Adaptive stabilization for $Q$-compensated reverse time migration

Abstract

Reverse time migration (RTM) for attenuating media should take amplitude compensation and phase correction into consideration. However, the attenuation compensation during seismic propagation suffers from numerical instability because of the boosted high-frequency ambient noise. We develop a novel adaptive stabilization method for $Q$-compensated RTM ($Q$-RTM), which exhibits superior properties of time-variance and $Q$-dependence over conventional low-pass filtering based method. We derive the stabilization operator by first analytically derive k-space Green’s functions for constant-$Q$ wave equation with decoupled fractional Laplacians (DFLs) and its compensated equation. The time propagator of the Green’s function for the viscoacoustic wave equation decreases exponentially, whereas that of the compensated equation is exponential divergent at high-wavenumber and is not stable after the wave is extrapolated for a long time. The Green’s functions therefore theoretically explain how the numerical instability existing in $Q$-RTM arises and shed light on how to overcome this problem pertinently. The stabilization factor required in the proposed method can be explicitly identified by the specified gain limit according to an empirical formula. The $Q$-RTM results for noise-free data using low-pass filtering and adaptive stabilization are compared over a simple five-layer model and the BP gas chimney model to verify the superiority of the proposed approach in terms of fidelity and stability. The $Q$-RTM result for noisy data from the BP gas chimney model further demonstrates that the proposed method enjoys a better anti-noise performance and helps significantly enhance the resolution of seismic images.