One way to improve the accuracy and reliability of kinematic earthquake source imaging is to investigate the origin of uncertainty and to minimize their effects. The difficulties in kinematic source inversion arise from the nonlinearity of the problem, nonunique choices in the parameterization, and observational errors. We analyze particularly the uncertainty related to the choice of the source time function (STF) and the variability in Earth structure. We consider a synthetic data set generated from a spontaneous dynamic rupture calculation. Using Bayesian inference, we map the solution space of peak slip rate, rupture time, and rise time to characterize the kinematic rupture in terms of posterior density functions. Our test to investigate the effect of the choice of STF reveals that all three tested STFs (isosceles triangle, regularized Yoffe with acceleration time of 0.1 and 0.3 s) retrieve the patch of high slip and slip rate around the hypocenter. However, the use of an isosceles triangle as STF artificially accelerates the rupture to propagate faster than the target solution. It additionally generates an artificial linear correlation between rupture onset time and rise time. These appear to compensate for the dynamic source effects that are not included in the symmetric triangular STF. The exact rise time for the tested STFs is difficult to resolve due to the small amount of radiated seismic moment in the tail of STF. To highlight the effect of Earth structure variability, we perform inversions including the uncertainty in the wavespeed only, and variability in both wavespeed and layer depth. We find that little difference is noticeable between the resulting rupture model uncertainties from these two parameterizations. Both significantly broaden the posterior densities and cause faster rupture propagation particularly near the hypocenter due to the major velocity change at the depth where the fault is located.