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Full-waveform seismic inversions based on minimizing the distance between observed and predicted seismograms are, in principle, able to yield better-resolved earth models than those minimizing misfits derived from traveltimes alone. Adjoint-based methods provide an efficient way of calculating the gradient of the misfit function via a sequence of forward-modeling steps, which, using spectral-element codes, can be carried out in realistically complex media. Convergence and stability of full-waveform-difference adjoint schemes are greatly improved when data and synthetics are progressively presented to the algorithms in a constructive multiscale approximation using a (bi)orthogonal wavelet transform. Wavelets provide the nonredundant spectral decomposition that paves the way for the inversion to proceed successively from long-wavelength fitting to detailed exploration of the phases in the seismogram. The choice of wavelet class and type, the initial depth of the multiscale decomposition, and the minimization algorithms used at every level continue to play crucial roles in our procedure, but adequate choices can be made that test successfully on 2C elastic seismograms generated in toy models, as well as in the industry-standard 2D Marmousi model. Although for simplicity our inversion ignored surface waves by prior tapering and filtered removal, those also appeared to be very well matched in the final model.

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