Preconditioned least-squares reverse-time migration using random phase encoding

Introduction

Reverse-time migration (RTM) uses the full wave equation to image the subsurface with high accuracy. However, RTM images suffer from several operator artifacts such as low-frequency noise, decreased resolution due to squaring the wavelet, and imbalanced amplitudes. These artifacts appear because, by migrating the data, we apply only the adjoint of the linear modeling operator, as opposed to its inverse. The imaging operator can be inverted in several ways, including iterative least-squares inversion. Although inversion can remove these artifacts, it is not widely used because the computational cost of applying the forward and adjoint operators in each iteration is extremely high.

Several methods have been proposed to reduce the computational cost of LSRTM. One of these methods is to reduce the data size by encoding the sources to create a super source (Sun et al., 2002; Jing et al., 2000; Morton and Ober, 1998; Romero et al., 2000). This technique has also been applied to full waveform inversion (Boonyasiriwat and Schuster, 2010; Ben-hadj ali et al., 2011; Gao et al., 2010; Krebs et al., 2009). The cost of applying the forward or adjoint of the modeling operator becomes independent of the number of sources, which greatly reduces the computational cost of the inversion. On the other hand, combining the sources causes crosstalk artifacts in the estimated image. These artifacts can be suppressed by changing the encoding function over iterations. Romero et al. (2000) and Krebs et al. (2009) showed that one-sample random phase encoding gives the best convergence rate.

In this paper, I compare the convergence rate of conventional LSRTM to phase-encoded LSRTM to test whether the computational reduction justifies the additional realizations of the encoding function. I first measure the norm of the model error of the two models. Then, I measure the norm of the model error after processing the estimated model at each iteration. The processing steps, which are a low-cut filter and an automatic gain control (AGC), are used to reduce the low-frequency noise and amplitude imbalance in order to measure the error in the image resolution.

The convergence rate of LSRTM inversion can be accelerated by preconditioning the gradient with the diagonal of the Hessian matrix. The Hessian matrix can be estimated by applying the encoding function either to the receiver side only or to both the receiver and source sides (Tang, 2009). A cheaper approximation to the Hessian matrix is the source intensity function, which ignores the receiver side of the Hessian matrix (Tang and Lee, 2010). The source intensity function can also be encoded along the source axis. I compare the convergence rates of these methods and also introduce a new approximation to the Hessian matrix that is based on the blended source-wavefield only. Finally, I test each of these preconditioners in the LSRTM and compare the results as a function of their cost.

Preconditioned least-squares reverse-time migration using random phase encoding