Rapid tsunami models have recently been developed to forecast far-field tsunami amplitudes from initial earthquake information (magnitude and hypocenter). Earthquake source parameters that directly affect tsunami generation as used in rapid tsunami models are examined, with particular attention to local versus far-field application of those models. First, validity of the assumption that the focal mechanism and type of faulting for tsunamigenic earthquakes is similar in a given region can be evaluated by measuring the seismic consistency of past events. Second, the assumption that slip occurs uniformly over an area of rupture will most often underestimate the amplitude and leading-wave steepness of the local tsunami. Third, sometimes large magnitude earthquakes will exhibit a high degree of spatial heterogeneity such that tsunami sources will be composed of distinct sub-events that can cause constructive and destructive interference in the wavefield away from the source. Using a stochastic source model, it is demonstrated that local tsunami amplitudes vary by as much as a factor of two or more, depending on the local bathymetry. If other earthquake source parameters such as focal depth or shear modulus are varied in addition to the slip distribution patterns, even greater uncertainty in local tsunami amplitude is expected for earthquakes of similar magnitude. Because of the short amount of time available to issue local warnings and because of the high degree of uncertainty associated with local, model-based forecasts as suggested by this study, direct wave height observations and a strong public education and preparedness program are critical for those regions near suspected tsunami sources.