Simulation of Near-Fault High-Frequency Ground Motions from the Representation Theorem

Abstract

“What is the maximum possible ground motion near an earthquake fault?” is an outstanding question of practical significance in earthquake seismology. In establishing a possible theoretical cap on extreme ground motions, the representation integral of elasticity, providing an exact, within limits of applicability, solution for fault radiation at any frequency, is an under-utilized tool. The application of a numerical procedure leading to synthetic ground displacement, velocity, and acceleration time histories to modeling of the record at the Lucerne Valley hard-rock station, uniquely located at 1.1 km from the rupture of the M _w 7.2 Landers, California event, using a seismologically constrained temporal form of slip on the fault, reveals that the shape of the displacement waveform can be modeled closely, given the simplicity of the theoretical model. High precision in the double integration, as well as carefully designed smoothing and filtering, are necessary to suppress the numerical noise in the high-frequency (velocity and acceleration) synthetic motions. The precision of the integration of at least eight decimal digits ensures the numerical error in the displacement waveforms generally much lower than 0.005% and reduces the error in the peak velocities and accelerations to the levels acceptable to make the representation theorem a reliable tool in the practical evaluation of the magnitude of maximum possible ground motions in a wide-frequency range of engineering interest.

Appendix

We need to show that the convolution integral

$$\mathop \int \limits_{R/\alpha }^{R/\beta } t^{'} \Delta u({\varvec{\upxi}},t - t^{'} ){\text{d}}t^{'} ,$$

(A1)

used in the first term of the integrand in Eq. (1) is the same as the function

$$\begin{aligned}&\left[ {F\left( {t - \frac{R}{\alpha }} \right) - F\left( {t - \frac{R}{\beta }} \right) + \frac{R}{\alpha }\dot{F}\left( {t - \frac{R}{\alpha }} \right) - \frac{R}{\beta }\dot{F}\left( {t - \frac{R}{\beta }} \right)} \right],\, \\ & \quad {\rm where}\, F\left( t \right) = \mathop \int \limits_{0}^{t} {\text{d}}t'\mathop \int \limits_{0}^{{t^{'} }} \Delta u\left( {{\varvec{\upxi}},t''} \right){\text{d}}t'', \end{aligned}$$

(A2)

appearing instead in the original Eq. (14.37) of Aki and Richards (1980).

We start with the convolution (A1) and switch to a new integration variable $t^{''} = t - t'$, transforming (A1) to

$$\mathop \int \limits_{t - R/\beta }^{t - R/\alpha } (t - t^{''} )\Delta u({\varvec{\upxi}},t^{''} ){\text{d}}t^{''} .$$

(A3)

With the use of Barrow’s theorem, $\frac{\text{d}}{{{\text{d}}x}}\mathop \int \limits_{a}^{x} f\left( t \right){\text{d}}t = f(x)$ (e.g., Harris and Stocker 1998, p. 552), (A3) is re-written as

$$\mathop \int \limits_{{t - \frac{R}{\beta }}}^{{t - \frac{R}{\alpha }}} \left( {t - t^{''} } \right)\left[ {\frac{\text{d}}{{{\text{d}}t^{''} }}\mathop \int \limits_{0}^{{t^{''} }} \Delta u\left( {{\varvec{\upxi}},t} \right){\text{d}}t} \right]{\text{d}}t^{''} = \mathop \int \limits_{{t - \frac{R}{\beta }}}^{{t - \frac{R}{\alpha }}} (t - t^{'} )\left[ {\frac{\text{d}}{{{\text{d}}t^{'} }}\mathop \int \limits_{0}^{{t^{'} }} \Delta u\left( {{\varvec{\upxi}},t^{''} } \right){\text{d}}t^{''} } \right]{\text{d}}t^{'} ,$$

(A4)

where we renamed the variables of integration in the right-hand side. Integrating (A4) by parts, noting that ${\text{d}}t/{\text{d}}t^{\prime } = 0$, and observing that several terms cancel, we transform the right-hand side of (A4) to

$$\frac{R}{\alpha }\mathop \int \limits_{0}^{t - R/\alpha } \Delta u\left( {{\varvec{\upxi}},t^{''} } \right){\text{d}}t^{''} - \frac{R}{\beta }\mathop \int \limits_{0}^{t - R/\beta } \Delta u\left( {{\varvec{\upxi}},t^{''} } \right){\text{d}}t^{''} + \mathop \int \limits_{{t - \frac{R}{\beta }}}^{{t - \frac{R}{\alpha }}} {\text{d}}t'\mathop \int \limits_{0}^{{t^{'} }} \Delta u\left( {{\varvec{\upxi}},t^{''} } \right){\text{d}}t^{''} .$$

(A5)

Equation (A5) can be re-cast as

$$\begin{aligned} \frac{R}{\alpha }\mathop \int \limits_{0}^{t - R/\alpha } \Delta u({\varvec{\upxi}},t^{''} ){\text{d}}t^{''} - \frac{R}{\beta }\mathop \int \limits_{0}^{t - R/\beta } \Delta u({\varvec{\upxi}},t^{''} ){\text{d}}t^{''} + \mathop \int \limits_{0}^{{t - \frac{R}{\alpha }}} {\text{d}}t^{'} \mathop \int \limits_{0}^{{t^{'} }} \Delta u({\varvec{\upxi}},t^{''} ){\text{d}}t^{''} \hfill \\ - \mathop \int \limits_{0}^{{t - \frac{R}{\beta }}} {\text{d}}t^{'} \mathop \int \limits_{0}^{{t^{'} }} \Delta u({\varvec{\upxi}},t^{''} ){\text{d}}t^{''} . \hfill \\ \end{aligned}$$

(A6)

If we now introduce the function $F(t)$ as in (A2) and note that, by Barrow’s theorem,

$$\dot{F}\left( t \right) = \frac{\text{d}}{{{\text{d}}t}}\mathop \int \limits_{0}^{t} {\text{d}}t^{\prime } \mathop \int \limits_{0}^{{t^{'} }} \Delta u\left( {{\varvec{\upxi}},t^{\prime \prime } } \right){\text{d}}t^{\prime \prime } = \mathop \int \limits_{0}^{t} \Delta u\left( {{\varvec{\upxi}},t^{\prime \prime } } \right){\text{d}}t^{\prime \prime } ,$$

(A6) becomes

$$\frac{R}{\alpha }\dot{F}\left( {t - \frac{R}{\alpha }} \right) - \frac{R}{\beta }\dot{F}\left( {t - \frac{R}{\beta }} \right) + F\left( {t - \frac{R}{\alpha }} \right) - F\left( {t - \frac{R}{\beta }} \right),$$

(A7)

which is the desired equation (A2).