Finite-Frequency Kernels Based on Adjoint Methods

We derive the adjoint equations associated with the calculation of Fréchet derivatives for tomographic inversions based upon a Lagrange multiplier method. The Fréchet derivative of an objective function χ (m), where m denotes the Earth model, may be written in the generic form δχ = ∫ Kₘ (x) δ ln m (x) d³x, where δ ln m = δm/m denotes the relative model perturbation and Kₘ the associated 3D sensitivity or Fréchet kernel. Complications due to artificial absorbing boundaries for regional simulations as well as finite sources are accommodated. We construct the 3D finite-frequency “banana-doughnut” kernel Kₘ by simultaneously computing the so-called “adjoint” wave field forward in time and reconstructing the regular wave field backward in time. The adjoint wave field is produced by using time- reversed signals at the receivers as fictitious, simultaneous sources, while the regular wave field is reconstructed on the fly by propagating the last frame of the wave field, saved by a previous forward simulation, backward in time. The approach is based on the spectral-element method, and only two simulations are needed to produce the 3D finite-frequency sensitivity kernels. The method is applied to 1D and 3D regional models. Various 3D shear- and compressional-wave sensitivity kernels are presented for different regional body- and surface-wave arrivals in the seismograms. These kernels illustrate the sensitivity of the observations to the structural parameters and form the basis of fully 3D tomographic inversions.