A Theoretical Method for Computing Near-Fault Ground Motions in Layered Half-Spaces Considering Static Offset Due to Surface Faulting, with a Physical Interpretation of Fling Step and Rupture Directivity

An efficient mathematical method is presented for computing the near-fault strong ground motions in a layered half-space, giving explicit consideration to the static offset due to surface faulting. In addition, the combined effects of "fling step" and "rupture directivity" on the near-fault ground motions are investigated. First, after checking the fault integration in the representation theorem, it is found that when an observation point is close to the fault plane, Greenʹs functions exhibit near singularities, which consist of extremely sharp peaks in a narrow band close to the observation point. Therefore, direct numerical integration becomes quite onerous for computing near-fault ground motions, because the dynamic Greenʹs functions must then be distributed very densely in order to evaluate accurately the effects of the near singularities. Instead, a new form of the representation theorem is introduced, which exploits the property that the dynamic Greenʹs functions can be approximated by the corresponding static Greenʹs functions in the vicinity of the singularities. The modified theorem, which involves the device of adding and subtracting the static Greenʹs functions from the dynamic ones, is the sum of two fault integrations. The first integration involves the difference of the dynamic and the corresponding static Greenʹs functions, while the second contains only the static Greenʹs functions. This formulation requires much less CPU time than the original one when near-fault ground motions are considered, because the near singularities of the dynamic Greenʹs functions in the first integration are completely eliminated by subtracting the static Greenʹs functions. While the second integration does require a densely distributed set of points to capture the near-singular behavior of the static Greenʹs function, it needs to be performed only once, as it is valid for all frequencies. Subtraction of the static Greenʹs functions from the dynamic functions has the added benefit of making the integration over the wavenumber in the determination of the Greenʹs functions much more efficient, especially for surface faulting. This is because the difference of the dynamic and static integrands converges rapidly to zero with increasing wavenumbers, whereas the original integrands diverge in the case of a source point on the free surface. The proposed methodology is used to investigate the two most important effects in near-fault ground motions, fling step (e.g., Abrahamson, 2001) and rupture directivity (e.g., Somerville et al., 1997), by paying special attention to the contribution of static and dynamic Greenʹs functions. It is found that the fling effects stem mainly from the second integral in the modified representation theorem, which involves the static Greenʹs function. The fling effects are dominant in the slip direction only in the vicinity of the surface fault and are negligible for buried faults, because the static Greenʹs function attenuates rapidly with distance from the fault, r, as the order of (1/r^2). Also, more importantly, when an observation point is located above a buried fault, the medium has to remain continuous, and thus cannot "fling." By contrast, the directivity effects stem mainly from the first integral, which involves the dynamic Greenʹs function, and attenuate much more slowly than the fling, on the order from 1/r to 1/(square root)r. The directivity effects are dominant in the fault-normal direction, especiallyin the forward rupture direction, not only for the surface fault, but also for the buried fault. Due to the combined effects of fling and directivity in the vicinity of the surface fault, the directions of the maximum velocities and displacements are inclined with respect to the fault plane. On the other hand, when softer surface layers are added to the medium, the directivity effects become more significant than the fling effects, because the dynamic Greenʹs functions are more pronounced than the static ones.