Finite-frequency tomography using adjoint methods. Methodology and examples using membrane surface waves

We employ adjoint methods in a series of synthetic seismic tomography experiments to recover surface wave phase-speed models of southern California. Our approach involves computing the Frechet derivative for tomographic inversions via the interaction between a forward wavefield, propagating from the source to the receivers, and an 'adjoint' wavefield, propagating from the receivers back to the source. The forward wavefield is computed using a 2-D spectral-element method (SEM) and a phase-speed model for southern California. A 'target' phase-speed model is used to generate the 'data' at the receivers. We specify an objective or misfit function that defines a measure of misfit between data and synthetics. For a given receiver, the remaining differences between data and synthetics are time-reversed and used as the source of the adjoint wavefield. For each earthquake, the interaction between the regular and adjoint wavefields is used to construct finite-frequency sensitivity kernels, which we call event kernels. An event kernel may be thought of as a weighted sum of phase-specific (e.g. P) banana-doughnut kernels, with weights determined by the measurements. The overall sensitivity is simply the sum of event kernels, which defines the misfit kernel. The misfit kernel is multiplied by convenient orthonormal basis functions that are embedded in the SEM code, resulting in the gradient of the misfit function, that is, the Frechet derivative. A non-linear conjugate gradient algorithm is used to iteratively improve the model while reducing the misfit function. We illustrate the construction of the gradient and the minimization algorithm, and consider various tomographic experiments, including source inversions, structural inversions and joint source-structure inversions. Finally, we draw connections between classical Hessian-based tomography and gradient-based adjoint tomography.