True-amplitude versus trace-normalized full waveform inversion

Full waveform inversion (FWI) is a powerful method to estimate high-resolution physical parameters of the subsurface by iteratively minimizing the misfit between the observed and synthetic seismic data. Standard FWI algorithms measure seismic misfit between amplitude-preserved seismic data (true-amplitude FWI). However, in order to mitigate the variations in sources and recording systems acquired on complex geological structures and the physics that cannot be modelled using an approximation of the seismic wave equation, the observed and synthetic seismic data are normalized trace-by-trace and then used to perform FWI. Trace-by-trace normalization removes the amplitude effects related to offset variations and only keeps the phase information. Furthermore, trace-by-trace normalization changes the true amplitude difference because of different normalization factors used for the corresponding synthetic and observed traces. In this paper, we study the performance of true-amplitude FWI and trace-normalized-residual-based FWI in the time domain. The misfit function of trace-normalized-residual-based FWI is defined such that the adjoint source used in gradient calculation is the trace-normalized seismic residual. We compare the two inversion schemes with synthetic seismic data simulated on laterally invariant models and the more complex 2-D Marmousi model. In order to simulate realistic scenarios, we perform the elastic FWI ignoring attenuation to noisy seismic data and to the synthetic data modelled using a viscoelastic modelling scheme. Comparisons of seismic data and adjoint sources show that trace-by-trace normalization increases the magnitude of seismic data at far offsets, which are usually more cycle-skipped than those at near offsets. The inverted results on linear-gradient models demonstrate that trace-by-trace normalization increases the non-linearity of FWI, so an initial model with sufficient accuracy is required to guarantee the convergence to the global minimum. The inverted results and the final seismic residuals computed using seismic data without trace-by-trace normalization demonstrate that true-amplitude FWI provides inverted models with higher accuracy than trace-normalized-residual-based FWI, even when the unknown density is updated using density-velocity relationship in inversion or in the presence of noise and complex physics, such as attenuation.