Hyperbolic reflections and convolutional wavelets are fundamental models for seismic data processing. Each sample of a 'stacked' zero-offset section can parameterize an impulsive hyperbolic reflection in a midpoint gather. Convolutional wavelets can model source waveforms and near-surface filtering at the shot and geophone positions. An optimized inversion of the combined modeling equations for hyperbolic traveltimes and convolutional wavelets makes explicit any interdependence and nonuniqueness in these two sets of parameters.I first estimate stacked traces that best model the recorded data and then find nonimpulsive wavelets to improve the fit with the data. These wavelets are used for a new estimate of the stacked traces, and so on. Estimated stacked traces model short average wavelets with a superposition of approximately parallel hyperbolas; estimated wavelets adjust the phases and amplitudes of inconsistent traces, including static shifts. Deconvolution of land data with estimated wavelets makes wavelets consistent over offset; remaining static shifts are midpoint-consistent. This phase balancing improves the resolution of stacked data and of velocity analyses.If precise velocity functions are not known, then many stacked traces can be inverted simultaneously, each with a different velocity function. However, the increased number of overlain hyperbolas can more easily model the effects of inconsistent wavelets. As a compromise, I limit velocity functions to reasonable regions selected from a stacking velocity analysis--a few functions cover velocities of primary and multiple reflections. Multiple reflections are modeled separately and then subtracted from marine data.The model can be extended to include more complicated amplitude changes in reflectivity. Migrated reflectivity functions would add an extra constraint on the continuity of reflections over midpoint. Including the effect of dip moveout in the model would make stacking and migration velocities equivalent.

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