Theoretical background for continental- and global-scale full-waveform inversion in the time-frequency domain

We propose a new approach to full seismic waveform inversion on continental and global scales. This is based on the time-frequency transform of both data and synthetic seismograms with the use of time- and frequency-dependent phase and envelope misfits. These misfits allow us to provide a complete quantification of the differences between data and synthetics while separating phase and amplitude information. The result is an efficient exploitation of waveform information that is robust and quasi-linearly related to Earth's structure. Thus, the phase and envelope misfits are usable for continental- and global-scale tomography, that is, in a scenario where the seismic wavefield is spatially undersampled and where a 3-D reference model is usually unavailable. Body waves, surface waves and interfering phases are naturally included in the analysis. We discuss and illustrate technical details of phase measurements such as the treatment of phase jumps and instability in the case of small amplitudes. The Fréchet kernels for phase and envelope misfits can be expressed in terms of their corresponding adjoint wavefields and the forward wavefield. The adjoint wavefields are uniquely determined by their respective adjoint-source time functions. We derive the adjoint-source time functions for phase and envelope misfits. The adjoint sources can be expressed as inverse time-frequency transforms of a weighted phase difference or a weighted envelope difference. In a comparative study, we establish connections between the phase and envelope misfits and the following widely used measures of seismic waveform differences: (1) cross-correlation time-shifts; (2) relative rms amplitude differences; (3) generalized seismological data functionals and (4) the L2 distance between data and synthetics used in time-domain full-waveform inversion. We illustrate the computation of Fréchet kernels for phase and envelope misfits with data from an event in the West Irian region of Indonesia, recorded on the Australian continent. The synthetic seismograms are computed for a heterogeneous 3-D velocity model of the Australian upper mantle, with a spectral-element method. The examples include P body waves, Rayleigh waves and S waves, interfering with higher-mode surface waves. All the kernels differ from the more familar kernels for cross-correlation time-shifts or relative rms amplitude differences. The differences arise from interference effects, 3-D Earth's structure and waveform dissimilarities that are due to waveform dispersion in the heterogeneous Earth.