Enhancing low-wavenumber components of full-waveform inversion using an improved wavefield decomposition method in the time-space domain

Full-waveform inversion (FWI) is a successful technique that attempts to build high-resolution velocity models by minimizing the residuals between observed and calculated data after a series of iterations. However, its successful application still relies on the quality of the estimated initial velocity model, as well as on the design of a suitable hierarchical workflow. If the initial model is far away from the true model, the widely used gradient-based FWI will inevitably get trapped into meaningless local minima. Given that the properties of the low-wavenumber components can help construct the initial model and overcome cycle skipping, we propose a low-wavenumber update FWI method based on the Hilbert transform called the hybrid method. This method provides a new form of FWI in the time-space domain that appears to be capable of enhancing the low-wavenumber updates. In this method, the updates of low wavenumbers and high wavenumbers are divided into two parts. In the first step, a gradient formula based on the Hilbert transform plays a key role in updating the low-wavenumber components. The gradient is formed by cross-correlation of upgoing and downgoing waves of source and residual wavefields. Instead of using the wavefield decomposition method in the frequency-wavenumber domain with high computational complexity, we apply a novel method to extract the upgoing and downgoing wavefields in the time-space domain. In addition to the acceptable computational burden, the memory is reduced greatly. The second step is the standard FWI, and its task is to update the high-wavenumber components. Numerical tests on an anomaly model and the Marmousi model demonstrate the effectiveness of the hybrid method for extracting low-wavenumber components even for noisy data or when low-frequency data are missing.