Stability Investigations and Insights on the Pseudodepth Domain- Based (1 st -order Riemannian Solver) Seismic Wavefield Modeling

Abstract

Abstract The enlightenment of Earth's nether regions presents itself as a matter of profound intrigue. A novel and avant-garde method, known as the Least-squares reverse-time migration is considered as the cutting-edge seismic imaging technique due to its outstanding structural irradiating. Its procedure consumes a lot of time and occupies huge memory amounts thus wavefield extrapolation has been juxtaposed in a Riemannian coordinate system to transform the vertical axis to overcome the above-mentioned issues and oversampling/aliasing for modelling deeper parts of the seismic section. The problem of stabilizing this Riemannian wavefield arises from the introduction of the mapping velocity and the vertical axis operator that converts the finite difference solution partially from time into frequency domains. For every point in the Cartesian space, there is a corresponding vertical-time point with the coordinates hence we can interpolate the reconstructed source wavefield by drawing a Cartesian-to-Riemannian mapping function. Our stability and convergence analysis suggests that the spatial derivatives of 1st order Riemannian axis can be approximated by a Fourier pseudo-spectral methods as well as fast-Fourier transform using a special Gaussian-like impulse function to generate the vector-matrix of the source term within the finite-difference operator, in addition to the mapping velocity, which is a differential form of the initial input velocity model, manifestly controls the CFL conditions of the associated Riemannian-finite difference operator. Numerical and synthetic examples indicate that this approach is more stable and efficient in extrapolating a smooth Riemannian wavefield while maintaining Claerbout’s principle for locating subsurface reflectors also choosing an appropriate sampling rate for the new vertical axis is related inversely by the maximum frequency of the impulse wavelet and directly with minimum velocity value in the given model.