Inversion strategies for visco-acoustic waveform inversion

We analyse the cross-talk issue by partitioning the inversion parameters into two classes; the velocity parameter class, and the attenuation parameter class. Both parameters are defined at a reference frequency, and a dispersion relation is assumed that describes these parameters at any other frequency. We formulate the model gradients at a forward modelling frequency, and convert them to the reference frequency by employing the Jacobian of the coordinate change represented by the dispersion relation. We show that at a given modelling frequency, the Frechet derivatives corresponding to these two parameter classes differ only by a 90 degrees phase shift, meaning that the magnitudes of resulting model updates will be unscaled, and will not reflect the expected magnitudes in realistic (Q(-1) < 1) media. Due to the lack of scaling, cross-talk will be enhanced by poor subsurface illumination, by errors in kinematics, and by data noise. To solve these issues, we introduce an attenuation scaling term (the inverse of a penalty term) that is used to pre-condition the gradient by controlling the magnitudes of the updates to the attenuation parameters. Initial results from a suite of synthetic cross-hole tests using a three-layer randomly heterogenous model with both intrinsic and extrinsic (scattering) attenuation demonstrate that cross-talk is a significant problem in attenuation inversion. Using the same model, we further show that cross-talk can be suppressed by varying the attenuation scaling term in our pre-conditioning operator. This strategy is effective for simultaneous inversion of velocity and attenuation, and for sequential inversion (a two-stage approach in which only the velocity models are recovered in the first stage). Further regularization using a smoothing term applied to the attenuation parameters is also effective in reducing cross-talk, which is often highly oscillatory. The sequential inversion approach restricts the search space for attenuation parameters, and appears to be important in retrieving a reliable attenuation model when strong time-damping is applied. In a final test with our synthetic model, we successfully carry out visco-acoustic inversions of noise-contaminated data.