Toward Seismic Tomography based upon Adjoint Methods

We demonstrate that Fréchet derivatives for tomographic inversions may be obtained based upon just two calculations for each earthquake: one calculation for the current 3D model and a second, `adjoint', calculation that uses time-reversed signals at the receivers as simultaneous, fictitious sources. For a given model~m, we consider objective functions χ(m) that minimize differences between frequency-dependent traveltime and amplitude anomalies. We show that the Fréchet derivatives of such objective functions may be written in the generic form δχ=∫VKm(x) δln m(x) d3x, where δ ln m=δ m/m denotes the relative model perturbation. The volumetric kernel Km is defined throughout the model volume V and is determined by time-integrated products between spatial and temporal derivatives of the regular displacement field s and the adjoint displacement field s? obtained by using time-reversed signals at the receivers as simultaneous sources. For each event, the construction of the kernel Km requires one forward calculation for the regular field s and one adjoint calculation involving the fields s and s?. For multiple events the kernels are simply summed. The final summed kernel is controlled by the distribution of events and stations and thus determines image resolution. The summed kernel is a weighted combination of finite-frequency `banana-donut' kernels. We illustrate the characteristics of these 3D finite-frequency kernels based upon adjoint simulations for a variety of arrivals, e.g., P, S, PS, Pdiff, PKIKP, and SKS. One particularly nice feature of an adjoint calculation of a finite-frequency kernel is that one need not be able to identify the arrival; the adjoint calculation will automatically reveal how this arrival `sees' the compressional- and shear-wave speed of the Earth. For this reason, any arrival in the data that is reasonably well fit by the current synthetics becomes a set of measurements, which rapidly leads to a large data base with a wide variety of arrivals and related kernels. To facilitate the iterative non-linear inverse problem, we demonstrate how to find basis functions for the model parameters that are optimally concentrated where the summed kernel is large. Specifically, we determine the smallest-dimensional set of band-limited orthonormal model parameter basis functions that result in the largest gradient of a misfit function. These optimal basis functions are used in a standard conjugate gradient algorithm to determine the (local) minimum of the misfit function.