Considering the complex nonlinearity between seismic data and perturbations in the model, a waveform optimization problem based on the least square sample-to-sample comparison has proven to be inadequate in dealing with such nonlinearity. A slew of more global comparison based optimizations have proven their values in circumventing such nonlinearity. Among these methods in particular, adaptive waveform inversion (AWI), which is based on the deconvolution operator, offers the right balance between phase matching and amplitude normalization. The key element in the AWI success is the normalization factor, which is inherently offered by the instantaneous traveltime measure used to unwrap the phase. Thus, we recast the AWI problem using a misfit function based on the original generalized instantaneous travel time, which highlights some of the features of AWI and traces their roots. We demonstrate that the misfit function of AWI is actually the least square averaged instantaneous travel time over multiple frequencies. We use the Marmousi model and Chevron 2014 FWI benchmark dataset to verify the effectiveness of the proposed method.


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