Summary
We develop a new approach for computing Fréchet sensitivity kernels in full waveform inversion by using the discrete adjoint approach in addition to the widely used continuous adjoint approach for seismic waveform inversion. This method is particularly well suited for the forward solver AxiSEM3D, a combination of the spectral-element method (SEM) and a Fourier pseudo-spectral method, which allows for a sparse azimuthal wavefield parametrization adaptive to wavefield complexity, leading to lower computational costs and better frequency scaling than conventional 3-D solvers. We implement the continuous adjoint method to serve as a benchmark, additionally allowing for simulating off-axis sources in axisymmetric or 3-D models. The kernels generated by both methods are compared to each other, and benchmarked against theoretical predictions based on linearized Born theory, providing an excellent fit to this independent reference solution. Our verification benchmarks show that the discrete adjoint method can produce exact kernels, largely identical to continuous kernels. While using the continuous adjoint method we lose the computational advantage and fall back on a full-3-D frequency scaling, using the discrete adjoint retains the speedup offered by AxiSEM3D. We also discuss the creation of a data-coverage based mesh to run the simulations on during the inversion process, which would allow to exploit the flexibility of the Fourier parametrization and thus the speedup offered by our method.
A complexity-driven framework for waveform tomography with discrete adjoints