As opposed to the conventional 2D (circular/elliptical crack) and 1D (linear crack) models for earthquake rupture, which are, respectively, for small-to-large earthquakes and mega-earthquakes, Vere-Jones' branching crack model describes the extreme case when the entire rupture process is completely controlled by randomness. This article discusses the similar features between the conventional source models and Vere-Jones' branching crack model, which include (a) the magnitude-frequency relation, (b) the source time function, and (c) the rupture duration-seismic moment relation. We proved that the Gutenberg-Richter magnitude-frequency relation holds as an asymptote for more general cases of branching processes. The branching crack model shows that the variance of observed rupture durations is relatively larger for small magnitudes and gives a scaling relation of T proportional to M-1/2 between the rupture duration T and the seismic moment M of an earthquake. Moreover, the stochastic property of the branching process explains why the earthquake magnitude cannot be determined until the entire source process is completed. The results in this article imply that the randomness caused by many unknown factors in the complex environments of earthquake sources cannot be ignored, and it can be effectively described by Vere-Jones' branching crack model. We conclude that the earthquake rupture is at least an intermediate process between the conventional deterministic model and the stochastic branching crack model if its mechanism is not completely controlled by randomness, as described by the latter.
