Source Wavefield Reconstruction in Fractional Laplacian Viscoacoustic Wave Equation-Based Full Waveform Inversion

Abstract

We develop a fractional Laplacian viscoacoustic wave equation-based full waveform inversion (FWI) method. The main novelty is efficient reconstruction of the source wavefields in gradient computation by the adjoint state method. Our FWI is based on Fourier pseudo-spectral time-domain (PSTD) numerical solutions of the fractional Laplacian viscoacoustic wave equation, which can describe the frequency independent <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${Q}$ </tex-math></inline-formula> (quality factor) behaviors of seismic waves accurately. The presented wavefield reconstruction strategy utilizes reverse time-marching formulae to compute the source wavefields backward in time, including an implicit formula in the interior domain and an explicit one in the perfectly matched layer (PML) absorbing domains. Since the wavefields in the entire domain are recovered, our method does not require storing massive boundary values like conventional reconstruction methods. To avoid numerical instability in reverse time reconstruction, we design a global-local checkpointing technique. When reverse time reconstruction encounters numerical instability, the forward propagation restarts from the nearest global checkpoint and proceeds to the unstable time step. Then, the reverse time reconstruction continues. The local checkpoints help to delay the numerical instability in the PML domains. Numerical examples verify the feasibility of our reconstruction strategy and the efficiency gain achieved by using this strategy in FWI.