The objective functions and adjoint sources behaviour for elastic full waveform inversion

Tianze Zhang, Kristopher A. Innanen

Elastic full-waveform inversion is an ill-posed data-fitting procedure that is sensitive to noise, inaccuracies of the starting model, the definition of multi-parameter classes, and inaccurate modeling of wave fields amplitudes. The objective function measures the difference between the synthetic data and observed data, which plays an essential role in the convergence property of the elastic wave equation. In this study, we investigate the behavior of the l2, l1 norm and the correlation-based objective function in FWI, and show the sensitivity of these objective functions with respect to VP, VS, and density. The objective functions show that all three objective functions show strong non-linearity with the VS parameter, which means the inversion is relatively harder to invert compared with the other two parameters since it contains a lot of local minimums. We also developed an objective function based on the multi-scale Z transform. The multi-scale Z transform objective function could release the non-linearity of the objective function since the low frequency information of the objective function is first extracted. Thus, the adjoint source contains low frequency information, which helps to build the larger-scale information of the velocity model.