Analytical representation of the fault slip velocity from spontaneous dynamic earthquake models

Abstract

We have analyzed the most relevant features of three different analytical representations of the time evolution of the cosesimic slip velocity derived from theoretical basis; the so-called modified Yoffe function (MY), which pertains to a singular crack solution, the solution for a nonspontaneous crack obeying a position-weakening governing equation (PR) and the solution for a 1-D fault model subject to a linear slip-weakening friction law (B). By considering the same input parameters, we quantitatively compare these slip velocity functions (SVF) and we found that the time evolutions of the velocity and the correspondent slip predicted by the MY and B functions are very similar, while the PR predicts a very sharp peak. Correspondingly, the PR SVF is richer in high frequency and the fall off of its spectrum at high frequencies goes roughly as w–1.5, while those of MY and B more closely follow w–2. Then we select two spontaneous, 3-D, dynamic, subshear models, representing a crack-like or a pulse-like rupture and we account for both homogeneous and heterogeneous configurations. We then compare the three SVF in order to see how they are able to reproduce the 3-D solutions; we also show how the input parameters of the SVF can be constrained from the results of the dynamic models. In the homogeneous cases our results indicate that the MY and the PR SVF reproduce adequately well the main features of a dynamic solution in the case of a crack-like rupture. The PR function overestimates vpeak and the MY SVF predicts a too rapid deceleration. In the case of a pulse-like rupture both the MY and the B SVF tend to underestimate vpeak , but all of them capture very well the final cumulated fault slip. Moreover, the B function fits better that the MY the overall behavior of the fault slip. The considered SVF are able to reproduce the spectral fall off of a 3-D solution at intermediate frequencies (for w < 20 Hz), the MY and the PR for a crack-like rupture and the MY and the B SVF for a pulse-like rupture. In particular, for w < 10 Hz the spectral content of the B function is practically indistinguishable from that of the spontaneous pulse-like solution. In the heterogeneous configurations the analytical functions cannot reproduce all the spectral details of the numerical solutions, but we see how it is possible to fit the overall behavior of a single pulse in fault the slip velocity time history. The thorough analysis performed in this work can contribute to the discussion about the debated choice of the source time function to be used in the kinematic models, which in turn is extremely important in the contest of hazard assessment and ground motions generation, although stress heterogeneities, geometrical irregularities, attenuation and free surface effects can definitively smear the details of the analytical functions.