Preconditioning for Linearised Inversion of Attenuation and Velocity Perturbations

We consider the case of the linearised constant-density viscoacoustic wave equation, which involves simultaneous inversion both for velocity and attenuation contrasts or perturbations. The medium parameter can be characterised by a complex-valued velocity that includes wave speed as well as attenuation. The least-squares error measures the squared norm of the difference between modelled and observed data. Its gradient with respect to the medium parameter represents a migration image. We can use a gradient-based minimisation algorithm to invert for the model parameter. Convergence rates will improve by using a suitable preconditioner, which usually is some approximation of the Hessian. For the linearised, constant-density viscoacoustic wave-equation we derive an exact Hessian that differs from the more conventional Hessian by including the complex-valued part. For the inverse problem, we consider a single point-scatterer and investigate four different approximations of the Hessian as preconditioner. The method does not appear to improve the convergence rate. On the contrary, convergence rates are worse and the norm of the error is larger. We conclude that we have not been able to exploit the imaginary part of the Hessian with the preconditioners considered here, although there may be other options.