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 = ∫ K_m(x) δ ln m(x) d^3x, where δ ln m = δm/m denotes the relative model perturbation and K_m 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 K_m 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.