Hessian kernels of seismic data functionals based upon adjoint techniques

We present an extension of the adjoint method that allows us to compute the second derivatives of seismic data functionals with respect to Earth model parameters. This work is intended to serve as a technical prelude to the implementation of Newton-like optimization schemes and the development of quantitative resolution analyses in time-domain full seismic waveform inversion. The Hessian operator H applied to a model perturbation δm can be expressed in terms of four different wavefields. The forward field u excited by the regular source, the adjoint field u† excited by the adjoint source located at the receiver position, the scattered forward field δu and the scattered adjoint field δu†. The formalism naturally leads to the notion of Hessian kernels, which are the volumetric densities of Hδm. The Hessian kernels appear as the superposition of (1) a first-order influence zone that represents the approximate Hessian, and (2) second-order influence zones that represent second-order scattering. To aid in the development of physical intuition we provide several examples of Hessian kernels for finite-frequency traveltime measurements on both surface and body waves. As expected, second-order scattering is efficient only when at least one of the model perturbations is located within the first Fresnel zone of the Fréchet kernel. Second-order effects from density heterogeneities are generally negligible in transmission tomography, provided that the Earth model is parameterized in terms of density and seismic wave speeds. With a realistic full waveform inversion for European upper-mantle structure, we demonstrate that significant differences can exist between the approximate Hessian and the full Hessian—despite the near-optimality of the tomographic model. These differences are largest for the off-diagonal elements, meaning that the approximate Hessian can lead to erroneous inferences concerning parameter trade-offs. The full Hessian, in contrast, allows us to correctly account for the effect of non-linearity on model resolution.