The July 8, 1986, North Palm Springs earthquake is used as a basis for comparison of several different approaches to the solution for the rupture history of a finite fault. The inversion of different waveform data is considered; both teleseismic P waveforms and local strong ground motion records. Linear parameterizations for slip amplitude are compared with nonlinear parameterizations for both slip amplitude and rupture time. Inversions using both synthetic and empirical Green's functions are considered. In general, accurate Green's functions are more readily calculable for the teleseismic problem where simple ray theory and flat-layered velocity structures are usually sufficient. However, uncertainties in the variation in t* with frequency most limit the resolution of teleseismic inversions. A set of empirical Green's functions that are well recorded at teleseismic distances could avoid the uncertainties in attenuation. In the inversion of strong motion data, the accurate calculation of propagation path effects other than attenuation effects is the limiting factor in the resolution of source parameters. The assumption of a laterally homogeneous velocity structure is usually not a good one, and the use of empirical Green's functions is desirable. Considering the parametrization of the problem, any degree of fault rupture complexity can be described in terms of a linear parametrization for slip amplitudes. However, a nonlinear parametrization for rupture times and slip amplitudes can have a distinct advantage over a simple linear one by limiting the number of unknown parameters. Regardless of the choice of data or the type of parametrization, the model or solution will be affected by the choice of minimization norm and the type of stabilization used. ¿ American Geophysical Union 1989