Abstract
We have developed a method to obtain a misfit function for robust waveform inversion. In this method, called adaptive traveltime inversion (ATI), a matching filter that matches predicted data to measured data is computed. If the velocity model is relatively accurate, the resulting matching filter is close to a Dirac delta function. Its traveltime shift, which characterizes the defocusing of the matching filter, is computed by minimization of the crosscorrelation between a penalty function such as [Formula: see text] and the matching filter. ATI is constructed by minimization of the least-squares errors of the calculated traveltime shift. Further analysis indicates that the resulting traveltime shift corresponds to a first-order moment, the mean value of the resulting matching filter distribution. We extend ATI to a more general misfit function formula by computing different order moment of the resulting matching filter distribution. Choosing the penalty function in adaptive waveform inversion (AWI) as [Formula: see text], the misfit function of AWI is the second-order moment, the variance of the resulting matching filter distribution with zero mean. Because our ATI method is based on a global comparison using deconvolution, such as AWI, it can resolve the “cycle skipping” issue. We evaluate our ATI misfit function and compare it with state-of-the-art options such as least-squares inversion (L2 norm), wave-equation traveltime inversion, and AWI using schematic examples before moving to more complex examples, such as the Marmousi model. For the Marmousi model, starting with a 1D [Formula: see text] model, with data without low frequencies (no energy below 3 Hz), a meaningful estimation of the P-wave velocity model is recovered. Our ATI misfit function (first-order moment) indicates comparable performance with the AWI misfit function (the second-order moment). We also include a real data example from the Gulf of Mexico to demonstrate the effectiveness of our method.