A Generalization of the Stochastic Summation Scheme of Small Earthquakes to Simulate Strong Ground Motions

Abstract

A generalization of the stochastic summation scheme of small earthquakes following a general form of the source spectrum (ω⁻ⁿ) model is developed to estimate the proper probability density functions for rupture propagation over the fault and to simulate site-specific strong ground motions in two independent stages. The probability density functions for rupture propagation are first developed for a single-corner-frequency source model, and then extended to multiplicative and additive double-corner-frequency source models to partially account for the effects of finite-fault rupture such as fault geometry, rise time, and rupture time on ground motions, and to better match observed Fourier amplitude spectra (FAS). The generalized two-stage stochastic summation scheme developed in this study can be used to simulate realistic strong-ground-motion time histories from small events, represented as empirical Green’s functions (EGFs), by specifying only two source parameters of the seismic moment and stress drop for a given site. The proposed two-stage stochastic summation schemes are tested using numerical examples following the theoretical source models as well as observed ground motions of the 2011 M_w 9.0 Tohoku earthquake in Japan and the 1994 M_w 6.7 Northridge earthquake in California to verify the applicability of the approach. Comparisons between the observed and simulated time histories for these two earthquakes, their response spectra, and FAS show satisfactory performance of the proposed approach for engineering purposes. The results of these comparisons indicate that the proposed stochastic summation of small events as EGFs can adequately replicate observed ground motions and evaluate uncertainties in ground motions with the generation of many time history realizations corresponding to a multitude of possible rupture processes . The proposed approach can also be used to generate multiple realistic acceleration time histories matched on average to the target response spectra for use in earthquake performance design, and to develop ground-motion models for low-seismicity regions, where there is a lack of recording of strong ground motions.

Appendix

This appendix explains how the user can generate random numbers with the probability distribution functions, p(t), proposed in this study. In this regard, the first step is to obtain the cumulative distribution function, P(t), as

$$ 0 \le P\left( t \right) = \mathop \int \limits_{ - \infty }^{t} p\left( \tau \right){\text{d}}\tau \le 1, $$

where τ is time. Then, in the second step, a random number with the desired pdf, t_j, is obtained from the inverse of cdf, P⁻¹, as

$$ t_{j} = P^{ - 1} \left( {u_{j} } \right) $$

in which u_j is a random number with a uniform distribution in [0, 1].

When the cdf cannot be analytically derived, numerical integration is used instead. Note that because the proposed pdfs are symmetric, p(−t) = p(t) and P(−t) = 1 − P(t). The user can create a table of t and its corresponding P(t) with a sufficient time-step resolution instead of solving a parametric equation. Since the t − P(t) is constructed, its inverse function can be numerically obtained by finding the closest t satisfying u = P(t).

Three MATLAB functions “SCF_distribution.m”, “ADCF_distribution.m”, and “MDCF_distribution.m”, as well as a script with some examples, are provided as supplementary materials to generate random numbers following single-, additive double-, and multiplicative double-corner-frequency source models, respectively. In addition, the Python code is provided to produce random numbers and to replicate the numerical examples performed in this paper.