Some of the gross characteristics of the rupture processes for the 1966 Parkfield, 1979 Imperial Valley, and 1980 Mexicali Valley earthquakes are determined by waveform inversion of near-source data employing a steady state dislocation model in a layered half-space. The forward model involves a piecewise-linear rupture front moving with a constant horizontal rupture velocity on a fault of infinite length and finite width. The inferred shapes for the rupture front and for the distribution of slip as a function of depth are consistent with previous results obtained by use of more general models. The results obtained show that a strong velocity pulse observed in the near-source region can be modeled as the passage of the rupture front phase and that the supershear propagation of the rupture front in the sedimentary layers of the medium provides a mechanism for the generation of the observed large amplitudes. ¿ American Geophysical Union 1990