Investigation on relationship between epicentral distance and growth curve of initial P-wave propagating in local heterogeneous media for earthquake early warning system
© Okamoto and Tsuno. 2015
Received: 1 June 2015
Accepted: 6 October 2015
Published: 13 October 2015
In the earthquake early warning (EEW) system, the epicenter location and magnitude of earthquakes are estimated using the amplitude growth rate of initial P-waves. It has been empirically pointed out that the growth rate becomes smaller as epicentral distance becomes far regardless of the magnitude of earthquakes. So, the epicentral distance can be estimated from the growth rate using this empirical relationship. However, the growth rates calculated from different earthquakes at the same epicentral distance mark considerably different values from each other. Sometimes the growth rates of earthquakes having the same epicentral distance vary by 104 times. Qualitatively, it has been considered that the gap in the growth rates is due to differences in the local heterogeneities that the P-waves propagate through. In this study, we demonstrate theoretically how local heterogeneities in the subsurface disturb the relationship between the growth rate and the epicentral distance. Firstly, we calculate seismic scattered waves in a heterogeneous medium. First-ordered PP, PS, SP, and SS scatterings are considered. The correlation distance of the heterogeneities and fractional fluctuation of elastic parameters control the heterogeneous conditions for the calculation. From the synthesized waves, the growth rate of the initial P-wave is obtained. As a result, we find that a parameter (in this study, correlation distance) controlling heterogeneities plays a key role in the magnitude of the fluctuation of the growth rate. Then, we calculate the regional correlation distances in Japan that can account for the fluctuation of the growth rate of real earthquakes from 1997 to 2011 observed by K-NET and KiK-net. As a result, the spatial distribution of the correlation distance shows locality. So, it is revealed that the growth rates fluctuate according to the locality. When this local fluctuation is taken into account, the accuracy of the estimation of epicentral distances from initial P-waves can improve, which will in turn improve the accuracy of the EEW system.
where A and B are constants, and t represents time. A and B are determined by a least-squares method. From the B value, epicentral distances are estimated using an empirical relationship. Odaka (2003) pointed out that the B value decreases as the epicentral distance increases. The decrease in the B value is mainly due to the geometrical spreading and scattering and anelastic attenuation of seismic waves. As the amplitude of P-waves decreases by propagating through the medium, the amplitude growth rate of the early portion of the P-waves also decreases. The decrease seemed to be independent of the magnitude of the earthquakes. Thus, the B value can be used to estimate the epicentral distance. However, in practice, the B values of different earthquakes show quite different values even if the epicentral distances are same. The gap in the B values is sometimes by 104 times. So, the epicentral distances estimated from B values include error to some extent, sometimes the error reaches to 102 km order. Possible causes of the gap in the B values are differences in the radiation pattern, source function, conditions of the subsurface, etc. However, Odaka (2003) pointed out that differences in radiation patterns do not result in an order-of-magnitude difference of the B values, except for the case when the seismic station is located near the nodal plane. Moreover, the B values of the observed earthquakes in this study are calculated from high-frequency (higher than 10 Hz) seismic waves. It has been pointed out that the radiation pattern of high-frequency seismic waves (higher than several hertz) seems to be isotropic (e.g., Liu and Helmberger 1985 for S-wave and Kobayashi et al. 2014 for P-wave). Source functions whose duration times reach several seconds or more, i.e., earthquakes bigger than magnitude M7, may cause a significant change to the B value. This is because the source duration time is longer than the length of the time window to calculate the B value, which is 2 s in the present EEW system. However, the B value fluctuates considerably even when the magnitude of earthquakes is relatively small, like M4 or M5. In that case, the source duration time is on the order of around 10−2−10−1 s. So, the source function, especially the source duration time, cannot be the main cause for the B value fluctuation, especially for relatively low-magnitude earthquakes. The remaining potential cause for the B value fluctuation is seismic attenuation and scattering due to heterogeneities in the subsurface. Tsukada et al. (2004) conducted a theoretical calculation on the broadening of scalar wave envelopes based on three-dimensional von Kármán type random medium. They found that the gap in the B values of different earthquakes having the same epicentral distance can be attributed to differences in the scattering condition. In their study, it is assumed that conversion waves between P and S waves (i.e., PS and SP conversions) are negligible. However, P coda waves seem to be dominated by the SP conversion waves (Yoshimoto et al. 1997a).
In this study, we theoretically calculated P-waves in a heterogeneous medium considering the conversion waves between P and S waves and obtained theoretical B values. We showed that variation in the condition of heterogeneities (correlation distance) changes the B values and causes fluctuation in the B values even if the epicentral distances are the same. Then, we compared the theoretically calculated B values with those calculated from earthquakes in 1996–2011 observed by K-NET and KiK-net in Japan. The B values calculated from various earthquakes showed different values even when their epicentral distances were the same. We calculated regional parameters of heterogeneities (correlation distances) in Japan that can account for the fluctuation in the observed B values. As a result, it was found that the spatial distribution of the correlation distances showed a similar tendency with that of parameters reflecting local heterogeneities, like coda-Q. The observed B values also showed similar distribution to that of the correlation distance.
In this study, we hypothesize that the variation in the condition of heterogeneities causes significant B value fluctuation, and verify this hypothesis. The fluctuation in the B values causes error in estimation of epicentral distances. Thus, if we address the fluctuation in B values by considering the local condition of heterogeneities, the estimation of epicentral distances becomes more accurate, which in turn improves the accuracy of the EEW system.
Calculation of theoretical P-wave
We calculated theoretical P-waves and its scattering waves (PP, PS, SP, and SS conversions) using the method by Sato (1984). P-waves were calculated on the basis of the first-ordered Born approximation. The P-wave radiating from a source fault in the subsurface is scattered by heterogeneous structures, and the P- and scattered waves are observed at a receiver on the surface. The heterogeneous structure is characterized by an exponential autocorrelation function with the fractional fluctuation ε and the correlation distance a. Gao et al. (1983) and Hoshiba (1991) pointed out that the first-ordered Born approximation is appropriate within a time window whose length is shorter than the mean free time of the medium by about 0.57 times. When P-wave velocity is 5.0 km/s and the mean free path is 102−103 km, the time appropriate to the first-ordered Born approximation is 101−102 s from the origin of earthquakes, which corresponds to hypocentral distance of ~102 km. Frankel and Clayton (1986) numerically showed that the amplitude growth of P-waves is well accounted for by the first-ordered Born approximation when ε is less than 10 %. In this study, we used a 2-s length time window from the onset of P-waves, and considered several % of ε, so that we could ignore multiple scatterings. Source faults have parameters, which are strike, dip-and-slip angles, depth, epicentral distance, and moment magnitude, and these are all considered in the theoretical calculation. So, the geometry of source-receiver configuration, the size, and the radiation pattern of earthquakes are considered in the theoretical calculation. Though the radiation pattern is taken into account for the theoretical calculation, the directivity effect of earthquakes is not considered because we adopt a point source. The source function we adopted is the ω-squared model. The calculated waves for a given frequency are velocity envelopes, and simply multiplying the angular frequency ω to the velocity envelopes, we get the acceleration envelope waveforms. Waveforms were calculated for 4–20 Hz at 1-Hz intervals, and the waves are converted to the time-domain by the Fourier transform.
Example of theoretical calculation
Equation (2) was obtained from the core samples of the boreholes down to the depth of the seismic bedrock in Chiba prefecture (Chiba prefecture 2004). Here, ρ = 2.73 g/cm3 was obtained from Eq. (2). The heterogeneous medium was characterized by a = 800 m and ε = 8 %. The theoretical P-wave envelope matched the observed seismogram well. Thus, P-wave envelopes are calculated, considering the focal mechanism, location, and magnitude of earthquakes and the parameters (V p , V s , ρ, a, ε) of the medium that the seismic waves propagate through. Those parameters, except a and ε, are determined when the focal mechanism and location of the hypocenter and the station are given. The theoretical B value is obtained by fitting Eq. (1) to the theoretical envelope. To fit Eq. (1), a least-squares method is used for a 2-s envelope from the onset of the P-wave.
Comparison of theoretical and observed B values
Locality of observed B value
Relation between a and γ values
−6.71 × 10−1
−4.42 × 10−1
−2.94 × 10−1
−2.23 × 10−1
The shorter the correlation distance is, the higher the observed attenuation is due to heterogeneities. Here, higher attenuation corresponds to higher absolute value of γ. In the areas where correlation distance is short, the initial P-wave amplitude grows more gradually because the energy in the incident wave scatters into the subsequent wave trains and is absorbed. From the gradually growing initial P-waves, smaller B values are observed. On the other hand, higher B values are observed at the area where the correlation distance is longer. It is because the incident wave is not scattered so much and has more energy in its initial part. So, the initial P-wave amplitude grows rapidly.
In the conventional EEW method, especially for railways, the B-Δ equation has the term of α only, which is related to geometrical spreading mainly. Moreover, the coefficient α is calculated using all the B obs. values. The B obs. values at a given epicentral distance have a considerable extent of variance (sometimes 104 times) due to difference in heterogeneous conditions as we showed. So, the epicentral distances estimated by the B-Δ equation also have errors. The errors in the estimated epicentral distances can be on the order of 102 km. The errors decrease the accuracy of the EEW system. When the B-Δ equation is defined considering local heterogeneities properly, the errors in the estimated epicentral distances will decrease.
We propose a method to consider the local heterogeneities, that is, changing the γ value in Eq. (3) according to the station’s a value. The γ value of the target a value was calculated using the B obs. values obtained at the station characterized by the target a value. Other choices to consider the local heterogeneities are changing the β value or both the β and γ values in Eq. (3). The β value represents effects around source and receiver, and the γ value represents effects along path. In this study, although we determined the γ values according to the station’s a value, the determination of the γ values according to the geological conditions (e.g., kind of plates, tectonic line) could be a more simple choice in practical use.
Summary and discussion
In this study, we hypothesized that B values for a given epicentral distance fluctuate strongly due to differences in local heterogeneous conditions. Firstly, we theoretically calculated B values for earthquakes occurring in March 1996 to April 2011. In this calculation, we chose parameters controlling the conditions of heterogeneities, that is, the correlation distance a and fractional fluctuation ε, to account for the observed B values. As a result, we found that differences in the a value are one of the major causes that make the B values fluctuate. The spatial distribution of the a value has a similar tendency with that of Q c -1, which is said to have a correlation with heterogeneities in the subsurface. This means that the observed B values have locality according to the local heterogeneities.
Currently in the EEW system, epicentral distances are estimated from the empirical relationship with B values (B-Δ equation), which contains errors due to heterogeneities in the subsurface. Although the B values for a given epicentral distance fluctuate very much, the empirical relationship is based on the assumption that the B values decrease against epicentral distances monotonically. Thus, the epicentral distance estimated by the empirical relationship has a considerable error, sometimes on the order of 102 km. As a result of our study, we found that the distribution of B values shows tendencies according to the regional heterogeneities. So that, by considering local conditions of heterogeneities, the accuracy of epicentral distance estimation will be improved. In this study, we proposed that the B-Δ equation optimized by the station’s a value could improve the estimation accuracy.
The estimation is done by a single station. An advantage of the single-station method is that information on earthquakes is quickly obtained. This advantage is especially effective for warning high-speed railways, like the Shinkanse, which need longer lead time, to decrease its speed as much as possible before strong motions hit trains. Recently, an approach for improving the accuracy of back-azimuth estimation to estimate epicentral direction was studied (Noda et al., 2012). So, both epicentral distances and directions can be obtained more accurately. The accuracy of the EEW system will be improved using those new techniques.
We would like to thank the National Research Institute for Earth Science and Disaster Prevention (NIED) for making K-NET and KiK-net data available and the Japan Meteorological Agency (JMA) for making the JMA unified earthquake catalog available. Some of the figures were drawn by Generic Mapping Tools (GMT) software.
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