Least-squares RTM in viscoacoustic media

LSRTM aims to minimize the misfit between the observed data and predicted data measured by the quadratic function:

$$J(\mathbf{m}) = \frac{1}{2}\Vert \widehat{\mathbf{d}} - \mat... ...c{1}{2}\Vert \mathbf{A} \mathbf{m} - \mathbf{d} \Vert^2_2 \; .$$ (18)

Since $\mathbf{A}$ is a linear operator, a gradient-based local optimization method, such as the Conjugate Gradient method (CG), is usually applied to iteratively update the image (Dai and Schuster, 2013; Xue et al., 2014). $J(\mathbf{m})$ is minimized when $\mathbf{m}$ satisfies (Tarantola, 2005)

$$\mathbf{m} = (\mathbf{A}^* \mathbf{A})^{-1}\mathbf{A}^* \mathbf{d} \; .$$ (19)

The square matrix $\mathbf{A}^* \mathbf{A}$ is known as the wave-equation Hessian, and its condition number affects the convergence rate of LSRTM implemented as an iterative inversion (Plessix and Mulder, 2004). In acoustic media, RTM usually provides a good approximation to the inverse of RTDM, and the Hessian matrix is well-conditioned (Symes, 2008). However, in viscoacoustic media, because both RTM and RTDM operators attenuate seismic waves, the image obtained by the aforementioned algorithm suffers from twice the amplitude loss accumulated along the reflection wavepath. Therefore, differently from the pure acoustic case, viscoacoustic RTM provides a poor approximation to the inverse of viscoacoustic RTDM, which makes the Hessian matrix $\mathbf{A}^* \mathbf{A}$ ill-conditioned. In the presence of strong attenuation and without proper preconditioning, this could slow down the convergence rate of an iterative method like CG and, in practice, may require a prohibitively large number of iterations to achieve a satisfactory result.