Scale- and parameter-adaptive model-based gradient pre-conditioner for elastic full-waveform inversion

We present a scale- and parameter-adaptive method to pre-condition the gradient of the parameters to be inverted in time-domain 2-D elastic full-waveform inversion (FWI). The proposed technique, which relies on a change of variables of the model parameters, allows to balance the value of the gradient of the Lam, parameters and density throughout the model in each step of the multiscale inversion. The main difference compared to existing gradient pre-conditioners is that the variables are automatically selected based on a least-squares minimization criteria of the gradient weight, which corresponds to the product of the gradient by a power of the parameter to be inverted. Based on numerical tests made with (1) a modified version of the Marmousi-2 model, and (2) a high-velocity and density local anomaly model, we illustrate that the value of the power helps to balance the gradient throughout the model. In addition, we show that a particular value exists for each parameter that optimizes the inversion results in terms of accuracy and efficiency. For the two models, the optimal power is similar to 2.0-2.5 and similar to 1.5 for the first and second Lam, parameters, respectively; and between 3 and 6, depending on the inverted frequency, for density. These power values provide the fastest and most accurate inversion results for the three parameters in the framework of multiscale and multishooting FWI using three different optimization schemes.