Using Adjoint Methods to Construct 3-D Banana-Doughnut Kernels

We use adjoint methods popular in climate and ocean dynamics to calculate Fréchet derivatives for tomographic inversions. The Fréchet derivative of an objective function χ(m), where m denotes the Earth model, may be written in the generic form δ χ=∫ K_{m( {x}) δ ln} m( {x}) d^{3 {x}}, where δ ln m=δ m/m denotes the relative model perturbation. We construct the 3-D finite-frequency `banana-donut' kernel K<sub>m</sub> by simultaneously computing the so-called `adjoint' wave field {s} forward in time and reconstructing the regular wave field {s} 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 upon the spectral-element method, and only two simulations are needed to produce density, shear-wave speed, and compressional-wave speed sensitivity kernels. This method is applied to a complicated 3-D southern California model. Various density, shear-wave speed, and compressional-wave speed sensitivity kernels are presented for different phases in the seismograms. These kernels illustrate the sensitivity of the observations to the structural parameters, and form the basis for fully 3-D tomographic inversions to update the structural model.