Hierarchical approach to seismic full waveform inversion

Full waveform inversion (FWI) of seismic traces recorded at the free surface allows the reconstruction of the physical parameters structure on the underlying medium. For such reconstruction, an optimization problem is defined where synthetic traces, obtained through numerical techniques as finite-difference or finite-element methods in a given model of the subsurface, should match the observed traces. The number of data samples is routinely around 1 billion for 2D problems and 1 trillion for 3D problems, while the number of parameters ranges from 1 million to 10 million degrees of freedom. Moreover, if one defines the mismatch as the standard least-squares norm between values sampled in time/frequency and space, the misfit function has a significant number of secondary minima related to the ill-posedness and non-linearity of the inversion problem linked to the so-called cycle skipping.
Taking into account the size of the problem, we consider a local linearized method where the gradient is computed using the adjoint formulation of the seismic wave propagation problem. Starting for an initial model, we consider a quasi-Newton method which allows us to formulate the reconstruction of various parameters, such as P and S wave velocities, density, or attenuation factors. A hierarchical strategy is based on an incremental increase in the data complexity starting from low-frequency content to high-frequency content, from initial wavelets to later phases in the data space, from narrow azimuths to wide azimuths, and from simple observables to more complex ones. Different synthetic examples of realistic structures illustrate the efficiency of this strategy based on data manipulation.
This strategy is related to the data space, and has to be inserted into a more global framework, where we could improve significantly the probability of convergence to the global minimum. When considering the model space, we may rely on the construction of the initial model or add constraints, such as smoothness of the searched model and/or prior information collected by other means. An alternative strategy concerns building the objective function, and various possibilities must be considered which may increase the linearity of the inversion procedure.