Finite-frequency sensitivity kernels for global seismic wave propagation based upon adjoint methods

We determine adjoint equations and Frechet kernels for global seismic wave propagation based upon a Lagrange multiplier method. We start from the equations of motion for a rotating, self-gravitating earth model initially in hydrostatic equilibrium, and derive the corresponding adjoint equations that involve motions on an earth model that rotates in the opposite direction. Variations in the misfit function chi then may be expressed as delta chi = integral(V) K-m delta ln md(3)r + integral(Sigma) K-d delta ln dd(2)r + integral(Sigma FS) K-d . del(Sigma)delta ln dd(2)r, where delta ln m = delta m/m denotes relative model perturbations in the volume V, delta ln d denotes relative topographic variations on solid-solid or fluid-solid boundaries Sigma, and. del(Sigma)delta ln d denotes surface gradients in relative topographic variations on fluid-solid boundaries Sigma(FS). The 3-D Frechet kernel K-m determines the sensitivity to model perturbations delta ln m, and the 2-D kernels K-d and K-d determine the sensitivity to topographic variations delta ln d. We demonstrate also how anelasticity may be incorporated within the framework of adjoint methods. Finite-frequency sensitivity kernels are calculated by simultaneously computing the adjoint wavefield forward in time and reconstructing the regular wavefield backward in time. Both the forward and adjoint simulations are based upon a spectral-element method. We apply the adjoint technique to generate finite-frequency traveltime kernels for global seismic phases (P, P-diff, PKP, S, SKS, depth phases, surface-reflected phases, surface waves, etc.) in both 1-D and 3-D earth models. For 1-D models these adjoint-generated kernels generally agree well with results obtained from ray-based methods. However, adjoint methods do not have the same theoretical limitations as ray-based methods, and can produce sensitivity kernels for any given phase in any 3-D earth model. The Frechet kernels presented in this paper illustrate the sensitivity of seismic observations to structural parameters and topography on internal discontinuities. These kernels form the basis of future 3-D tomographic inversions.