Introduction

Seismic attenuation is caused by the effective anelastic properties of the Earth (Aki and Richards, 2002; Carcione, 2007), and may lead to poor illumination and misplacement of reflectors in a migration image. To directly compensate for seismic attenuation during reverse time migration (RTM) (Whitmore, 1983; McMechan, 1983; Baysal et al., 1983), Zhang et al. (2010) proposed a viscoacoustic wave equation involving a pseudo-differential operator based on the constant-$Q$ model (Kjartansson, 1979) with decoupled effects of amplitude loss and velocity dispersion. Suh et al. (2012) extended the operator to vertically transversely isotropic (VTI) media. Bai et al. (2013) adopted a similar approach for attenuation compensation in RTM, but used a viscoacoustic wave equation without memory variables. Using fractional Laplacians, Zhu and Harris (2014) proposed a constant-$Q$ viscoacoustic wave equation with separate terms accounting for amplitude loss and velocity dispersion, which was further applied for $Q$-compensated RTM using both synthetic and field data (Zhu et al., 2014; Zhu and Harris, 2015). Fletcher et al. (2012); Sun and Zhu (2015) investigated stable approaches for Q-compensation in RTM.

The imaging problem can also be cast as an inverse problem, with the objective of minimizing the $L_2$ norm of the difference between recorded data and predicted data (Ronen and Liner, 2000). Such approaches are known as least-squares migration (Nemeth et al., 1999; Tang, 2009; Dai et al., 2011), and more specifically least-squares RTM (LSRTM) in the context of RTM (Xue et al., 2014; Liu et al., 2013; Dai et al., 2012; Sun et al., 2014a; Dai and Schuster, 2013; Hou and Symes, 2015; Zhang et al., 2013; Wong et al., 2011). LSRTM is capable of mitigating imaging artifacts and enhancing subsurface illumination, and may have a correlative objective function to relax the amplitude matching requirement (Zhang et al., 2014). Pioneering works of linearized inversion in viscoacoustic and viscoelastic media have been done by Ribodetti et al. (2000,1995) using an asymptotic theory and by Blanch and Symes (1995,1994) using the wave equation. Recently, Dutta and Schuster (2014) and Sun et al. (2014b) have shown that LSRTM can be applied for attenuation compensation in viscoacoustic media. Dutta and Schuster (2014) used the standard linear solid (SLS) model (Robertsson et al., 1994; Blanch et al., 1995), with a simplified stress-strain relation and incorporated a single relaxation mechanism (Blanch and Symes, 1995). Sun et al. (2014b) employed the lowrank one-step method to solve the constant-$Q$ wave equation, which allows for an efficient formulation involving fractional Laplacians (Zhu and Harris, 2014; Carcione, 2010). The computational cost of LSRTM depends on the number of iterations, which hinges on the conditioning of the wave-equation Hessian that it tries to invert. In acoustic media, RTM is an efficient approximation to the inverse of reverse time de-migration (RTDM), the forward modeling operator, and provides accurate kinematic information of subsurface structures (Symes, 2008). In viscoacoustic media, however, RTM is a poor approximation to the inverse of RTDM, because the wave amplitude suffers from attenuation during both forward and backward propagation (Sun et al., 2015; Zhu and Harris, 2014). As a result, the wave-equation Hessian becomes ill-conditioned, and iterative LSRTM suffers from a slow convergence rate. To improve the convergence rate of LSRTM in viscoacoustic media, we propose to seek for a proper preconditioner that mitigates the effect of attenuation in the inversion.

In this paper, to construct LSRTM in viscoacoustic media, we use the lowrank one-step wave extrapolation (Sun et al., 2016) and derive its adjoint operator based on non-stationary linear filtering theory (Margrave, 1998). Sun et al. (2015) have successfully applied the lowrank one-step wave extrapolation operator to solve the constant-$Q$ wave equation with fractional Laplacians. To solve the problem of slow convergence of LSRTM in viscoacoustic media, we propose to construct a preconditioned formulation by replacing the viscoacoustic RTM operator, i.e. RTM based on the solution of the viscoacoustic wave equation forward and backward in time, with a better approximate inverse of the RTDM operator, i.e. the $Q$-compensated RTM or $Q$-RTM (Bai et al., 2013; Zhu et al., 2014; Zhang et al., 2010; Suh et al., 2012). $Q$-RTM involves a modeling operator with separate control over amplitude and phase, and is designed to compensate for the amplitude loss along the attenuated wavepaths. As a result, the preconditioned wave-equation Hessian is well-conditioned, helping the new framework to quickly converge to the true amplitude solution within only a few iterations. Since the inverted matrix is numerically non-Hermitian, we adopt the Generalized Minimum Residual (GMRES) algorithm, a Krylov subspace method (Saad and Schultz, 1986), for iterative inversion. Using a synthetic model, we test the ability of the proposed $Q$-LSRTM to dramatically enhancing image quality at a reasonable cost.