On the steadfastness of the least-squares reverse-time migration wavefield extrapolation via 1st-order Riemannian axis finite-difference solver

Exploring Earth's deep regions via Least-Squares Reverse-Time Migration (LSRTM) methods is of significant interest due to its exceptional structural-amplitude clarity. This cutting-edge seismic imaging technique is time-consuming and memory-intensive, thus wavefield extrapolation was proposed to be in the Pseudodepth domain (1 st -order Riemannian coordinate system’s axis) to address these issues and to prevent oversampling/aliasing when modeling deeper subsurface zones. Stabilizing the generated Riemannian wavefield involves implementing an appropriate mapping velocity and obtaining the vertical axis operator which partially converts the finite difference solver from time to frequency domains. Each Cartesian point ( x , y , z ) has a corresponding vertical-time point ( ξ 1 , ξ 2 , ξ 3 ) , allowing interpolation of the reconstructed source wavefield through a Cartesian-to-Riemannian mapping function. Our stability and convergence analysis indicates that the spatial derivatives of the 1 st -order Riemannian axis can be approximated by Fourier pseudo-spectral methods and fast-Fourier transforms using a special Gaussian-like impulse function. This function generates the source term vector-matrix within the finite-difference operator. The mapping velocity, derived as a differential form of the initial input velocity model, controls the CFL conditions of the associated Riemannian-finite difference operator. Numerical, synthetic, and seismic field data examples show that this approach is more stable and efficient in extrapolating a smooth 1 st -order Riemannian axis-based finite-difference wavefield while adhering to Claerbout’s principle for locating subsurface reflectors. Additionally, choosing the appropriate sampling rate for the new vertical axis is inversely related to the maximum frequency of the impulse wavelet and directly related to the minimum velocity value in the model.