Analysis of filtering characteristics of density waveform inversion and full model wavenumber construction

Based on the born weak scattering approximation, the essence of full waveform inversion is a process to construct the full wavenumber spectrum of the perturbation model. The low-medium-high wavenumber (as we known as tomography+migration) reconstruction successful is the key to the full waveform inversion. Density inversion is a big challenge for full waveform inversion, the high coupling of density and velocity make a big problem to density inversion that has not be solved well. However, density inversion has another major problem: under the velocity-density parameterization, the density inversion results has the migration effect, and the low wavenumber information in the model cannot be constructed. In this paper, under the condition of ignoring the coupling of velocity-density parameterization, we focus on solving the migration characteristics. Firstly, the chain-rules is used to derive the sensitive kernel functions under the velocity-density parameterization mode, and with the far-field plane wave approximation, the relationship between amplitude, wavenumber and scattering angle in the new velocity-density gradient is derived to explain the migration effect of density inversion. Then the new density gradient with Laplace filtering form is derived, and we establish a new objective function based on integration for the density. So that a new density gradient is derived to suppress the filtering effect and implement a full wavenumber model inversion by releasing the low wavenumber information. Finally, the contribution of the new and old method to tomography+migration parts in density gradient is analyzed by the two-layer model. The accuracy and effectiveness of the density full-wavenumber residual model spectrum reconstruction is verified by the Marmousi model with density single-parameter and velocity-density two-parameter simultaneous inversion test.