Hybrid domain frequency controllable envelope inversion with lowrank approximation for long-wavelength velocity model building

Full waveform inversion (FWI) needs an accurate initial model to guarantee convergence. Time domain frequency controllable envelope inversion (FCEI) is capable to build a long-wavelength velocity model that can be used as an initial model for conventional FWI. However, the implementation of time domain FCEI introduces tremendous burden in storage or computation, which hinders its efficient implementation especially on a memory-limited device. We propose a hybrid domain FCEI (HFCEI) method, in which the wave propagation is performed in the time domain while the computation of gradient is performed in the frequency domain. We demonstrate that the frequency control property of FCEI makes it suitable to be efficiently implemented in the hybrid domain. Only a few frequency domain wavefields need to be saved and no extra wave propagation is needed in calculating gradient. We also demonstrate that the frequency domain wavefields in HFCEI satisfy the lowrank assumption, which enables us to use the lowrank approximation to accurately represent the frequency domain wavefields to further reduce the storage requirement of the method. As a result, HFCEI has significant advantages than time domain FCEI in implementation especially for large-scale problems. Synthetic examples are given to verify the effectiveness of HFCEI and demonstrate its advantages over time domain FCEI.