A tutorial of image-domain least-squares reverse time migration through point spread functions

Least-squares reverse time migration (LSRTM) has shown great potential to improve the amplitude-fidelity and spatial resolution of reverse time migration (RTM) image. However, the main disadvantage is that it requires significant computational resources for the iterative solution. To ameliorate this problem, the image-domain least-squares reverse time migration (IDLSRTM) approach through point-spread functions (PSFs) has been proven to be a viable and alternative technique to deconvolve the standard RTM image. In this paper, we present a tutorial for the numerical implementation of IDLSRTM through PSFs. The Hessian matrix in the standard IDLSRTM approach is estimated through the spatial interpolation of precomputed PSFs on the fly, where the PSFs are computed by one-round forward modeling and migration. However, the resulting IDLSRTM scheme is highly ill-conditioned, because of the incomplete acquisition condition, irregular subsurface illumination, and band-limited data. To stabilize inversion and improve the inverted image, we suggest that the PSFs and RTM image should be deblurred by applying a deblurring filter. The deblurring PSFs can reduce the condition number of the standard Hessian matrix and make the inverse problem more well-conditioned, thus improving imaging quality and accelerating convergence. Furthermore, the regularization operators should be imposed to produce a reasonable inverted image and avoid the overfitting of the image-matching term. Based on several examples with synthetic and field data, we can conclude two significant points. The first point is that the standard IDLSRTM approach through the conventional PSFs can only recover a similar reflectivity image to the deblurring RTM and non-stationary matching filter (NMF) approaches. The second point is that the deblurring IDLSRTM approach through the deblurring PSFs can retrieve the reflectivity image with higher resolution and better amplitude-fidelity than the standard IDLSRTM approach and the standard RTM approach with the deblurring filter or NMF.