English translation of the paper published on B25 of “PROGRAMME and ABSTRACTS”,
Seismological Society of Japan, Vol. 2, 1997 (in Japanese).

Revision of the empirical Green's function method
Kojiro IRIKURA, Takao KAGAWA, and Haruko SEKIGUCHI

1. Introduction

The empirical Green’s function method proposed by Irikura (1986) is one of the most powerful
methods for predicting strong ground motions from large earthquakes. The method does make anautomated correction of the difference of slip velocity time function between large and small
earthquakes. However, there remain some problems to improve it to obtain more accurate groundmotions. One is to revise the correction function to get the slip velocity of the target large
earthquakes. The other is to be applicable for simulating ground motions at near-fault sites. In thisstudy, we propose revision of the empirical Green’s function method to solve those two problems,correction of slip velocity time function in simulation procedure and application to near-source
ground motions..

2. Empirical Green’s function method by Irikura (1986) and correction function of slipvelocity
Ground motions U ( R, t ) from a large event as a target are estimated with a superposition ofobserved records from a small event used as empirical Green’s functions (Irikura, 1986; Irikura andKamae, 1994).
NxN

U ( R, t ) = ∑ ( R0 / Rn ) F (t − ξ n / v r − Rn / v s )u ( R0 , t )

,(1)

n =1

where R and R0 are distance from the large event and that from the small event, Rn and ξn are
distance from n-subfault to site and that from rupture starting point to n-subfault, and vr and vs arerupture velocity and S-wave velocity. Moment ratio of the target and small event is N3. Then, thefault plane of the target event is divided into N x N subfaults. Here, stress drop is assumed to beconstant in the large and small event because equations are simplified. We can easily treat the casewhere stress drop of the large event is different from that of the small event F (t ) in (1) is a function
to correct the difference of slip velocity between the target and small event.
In Irikura (1986), F (t ) is given as

F (t ) = δ (t ) + D ⋅ bT (t ).

(2)

The first term of (2) is a delta function and the second one is a boxcar function with width T andamplitude D. The integration of F (t ) from 0 to T is defined to be N. Then D is (N-1)/T and T is
NΔt (Δt: time sampling rate).
Synthetic ground motions by (1) have spectral contents following the ω2 model in overall shape.

~~ (ω ) at low frequency end and N ⋅ u~ (ω ) at
That is, the amplitude spectrum of U (ω ) has N ⋅ u3high frequencies. However, for details the amplitude spectrum of F (t ) has significant troughs atfrequency 1/T and its higher harmonics shown in Fig. 1. Resultantly, the spectra of the synthetic
ground motions by (1) have troughs at the same frequencies as F (t ) .

Fig. 1. Fourier spectrum of correction function F(t) [Eq. (3)] for slip velocity inestimating ground motions from large earthquake by Irikura (1986).

The peculiar characteristics of F (t ) mentioned above are avoided using a revised function asfollows.

F (t ) = δ (t ) + D ⋅ eT (t ),(3)where eT (t ) is an exponential function defined aseT (t ) = exp(t / T )eT (t ) = 00≤t ≤Tt ≤ 0 and t ≥ T .

(4)where D in (3) is defined the same as (2), that is, the integration of F (t ) from 0 to T is N.
The amplitude spectrum of F (t ) given by (3) has less troughs at frequency 1/T and its higher
harmonics shown in Fig. 2. Therefore, the synthetic ground motions using the correction function (3)have better spectral shapes following the ω2 model.

Fig. 2. Fourier spectrum of revised correction function F(t) [Eq. (3)] for slip velocity proposed in thisstudy.

3. Application to simulation of ground motions at near-fault sites
The empirical Green’s function method has successfully applied in simulating strong groundmotions from a large event even using observed records of just one or two small events. The records
are used as the empirical Green’s functions over the fault plane of the large event by correcting only
attenuation effects due to difference in distance from each subfault to site. One of problems insimulating near-fault ground motions from the large event is that the geometrical spreading factor
1/Rn (Rn: distance from n-subfault to site) at near-fault sites becomes unrealistically large as Rn isclose to 0. Actually, the attenuation-distance curves of peak ground motions are saturated atnear-fault distances even for relatively small events with magnitude 4 to 5 often used as theempirical Greens functions for large earthquakes with more than magnitude 7 (e.g. Cambell, 1997).
Practically, the attenuation-distance curves are approximately given to be 1/(Rn + Ra), where Ra
correspond to source radius of the small event. We can simulate near-source ground motions fromthe large event by replacing ( R0 / Rn ) in (1) with R0 /( Rn + Ra ) .
References
Cambell, K. W. (1997): Empirical near-source attenuation relationships for horizontal and verticalcomponents of Peak Ground Acceleration, Peak Ground Velocity, and Pseudo-absoluteacceleration response spectra, Seismological Research Letters, 68, 154-179.
Irikura, K. (1986): Prediction of strong acceleration motion using empirical Green's function, Proc.
7th Japan Earthquake Engineering Symposium, Tokyo, 151-156.
Irikura, K. and K. Kamae (1994): Estimation of strong ground motion in broad-frequency band
based on a seismic source scaling model and an empirical Green's function technique, ANNALIDI GEOFISICA, 37, 1721-1744, 1994.