Modelling directivity of strong ground motion with a fractal, k−2, kinematic source model

Abstract

The kinematic fractal source model presented in this study is able to simulate broad-band accelerograms with spectral amplitudes proportional to a fraction of the directivity coefficient Cd in the far-field approximation. This approach is based on a composite source description, where subevents are generated using a fractal distribution of sizes, and, by summation, produces k-square space distribution of the slip. Each elementary source is described as a crack-type slip model growing circularly from a nucleation point when the rupture front reaches it. In order to better control the directivity effect, the location of the nucleation point for an elementary source is assumed to be scale-dependent. For the larger sources, the nucleation point is located near the intercept of the crack with the rupture front, whereas for smaller sources, it is randomly chosen within the crack. For simplicity, a constant rupture velocity is assumed. Each subevent is set up with scale-dependent rise-time, assuming a boxcar source–time function, hence filtering out its own high frequency radiation. The resulting mean slip-velocity functions are very similar to the ones derived from dynamic rupture modelling. Ground motion synthetics are computed by convolving the slip-velocity functions with the Green's functions. It is demonstrated that, in the far-field approximation, accelerogram spectra follow the ω2 model with amplitudes controlled by the frequency-dependent directivity effects. In particular, spectral amplitudes at high-frequencies are proportional to a fraction of Cd. These results were verified for few earthquake magnitudes. In addition, a validation exercise was made in the near-fault region by modelling the complete wavefield of strong ground motion at a few receivers and for several rupture scenarios. The synthetic strong-motion parameters are compared to the ones predicted by empirical attenuation relationships. It is shown that calculated standard deviations are in good agreement with the empirical ones, as well as the ground-motion parameter amplitudes predicted as a function of distance for whole interval of source distance considered in modelling. Minor differences were found in peak ground-accelerations computed at large distance from the fault, a problem related to the simplified response of the medium.