S U M M A R YWe present a full seismic waveform tomography for upper-mantle structure in the Australasian region. Our method is based on spectral-element simulations of seismic wave propagation in 3-D heterogeneous earth models. The accurate solution of the forward problem ensures that waveform misfits are solely due to as yet undiscovered Earth structure and imprecise source descriptions, thus leading to more realistic tomographic images and source parameter estimates. To reduce the computational costs, we implement a long-wavelength equivalent crustal model. We quantify differences between the observed and the synthetic waveforms using time-frequency (TF) misfits. Their principal advantages are the separation of phase and amplitude misfits, the exploitation of complete waveform information and a quasi-linear relation to 3-D Earth structure. Fréchet kernels for the TF misfits are computed via the adjoint method. We propose a simple data compression scheme and an accuracy-adaptive time integration of the wavefields that allows us to reduce the storage requirements of the adjoint method by almost two orders of magnitude.To minimize the waveform phase misfit, we implement a pre-conditioned conjugate gradient algorithm. Amplitude information is incorporated indirectly by a restricted line search. This ensures that the cumulative envelope misfit does not increase during the inversion. An efficient pre-conditioner is found empirically through numerical experiments. It prevents the concentration of structural heterogeneity near the sources and receivers.We apply our waveform tomographic method to ≈1000 high-quality vertical-component seismograms, recorded in the Australasian region between 1993 and 2008. The waveforms comprise fundamental-and higher-mode surface and long-period S body waves in the period range from 50 to 200 s. To improve the convergence of the algorithm, we implement a 3-D initial model that contains the long-wavelength features of the Australasian region. Resolution tests indicate that our algorithm converges after around 10 iterations and that both long-and short-wavelength features in the uppermost mantle are well resolved. There is evidence for effects related to the non-linearity in the inversion procedure.After 11 iterations we fit the data waveforms acceptably well; with no significant further improvements to be expected. During the inversion the total fitted seismogram length increases by 46 per cent, providing a clear indication of the efficiency and consistency of the iterative optimization algorithm. The resulting SV -wave velocity model reveals structural features of the Australasian upper mantle with great detail. We confirm the existence of a pronounced low-velocity band along the eastern margin of the continent that can be clearly distinguished against Precambrian Australia and the microcontinental Lord Howe Rise. The transition from Precambrian to Phanerozoic Australia (the Tasman Line) appears to be sharp down to at least 200 km depth. It mostly occurs further east of where it is infe...

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S U M M A R YWe 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 L 2 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.