Resolution analysis in full waveform inversion

We propose a new method for the quantitative resolution analysis in full seismic waveform inversion that overcomes the limitations of classical synthetic inversions while being computationally more efficient and applicable to any misfit measure. The method rests on (1) the local quadratic approximation of the misfit functional in the vicinity of an optimal earth model, (2) the parametrization of the Hessian in terms of a parent function and its successive derivatives and (3) the computation of the space-dependent parameters via Fourier transforms of the Hessian, calculated with the help of adjoint techniques. In the simplest case of a Gaussian approximation, we can infer rigorously defined 3-D distributions of direction-dependent resolution lengths and the image distortion introduced by the tomographic method. We illustrate these concepts with a realistic full waveform inversion for upper-mantle structure beneath Europe. As a corollary to the method for resolution analysis, we propose several improvements to full waveform inversion techniques. These include a pre-conditioner for optimization schemes of the conjugate-gradient type, a new family of Newton-like methods, an approach to adaptive parametrization independent from ray theory and a strategy for objective functional design that aims at maximizing resolution. The computational requirements of our approach are less than for a typical synthetic inversion, but yield a much more complete picture of resolution and trade-offs. While the examples presented in this paper are rather specific, the underlying idea is very general. It allows for problem-dependent variations of the theme and for adaptations to exploration scenarios and other wave-equation-based tomography techniques that employ, for instance, georadar or microwave data.