A method of real-time prediction of ground motion is proposed for application for the Earthquake Early Warning (EEW). In many methods of the present EEW systems, hypocenter and magnitude are determined quickly, after which ground motions are predicted. Although these methods can predict the strength of ground motions by using a few parameters (e.g., hypocenter, magnitude, and site factors), error in the parameters leads directly to error in the prediction, and it is not easy to take the effects of rupture directivity and source extent into account. During the aftershock activity of the 2011Tohoku earthquake (Mw 9.0), multiple events occurred simultaneously, which made it difficult to accurately determine the hypocenters and magnitudes, and led to some false alarms. To address these problems, a new method is proposed that applies the Kirchhoff-Fresnel boundary integral equation. Ground motion is predicted from real-time ground motion observation at front stations in the direction of incoming seismic waves. The real-time monitoring of wavefield and propagation direction are important for this method, but a hypocenter and magnitude are not required. It is possible to predict ground motion without a hypocenter and magnitude, and the precision of the prediction is not affected by error in the source parameters. The effects of rupture directivity, source extent, and simultaneous multiple events are substantially included in this method. The method is a quantitative extension of the front detection technique of EEW.
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 The aim of the Earthquake Early Warning, EEW, is to mitigate earthquake disasters by giving people enough time to take appropriate safety measures in advance of strong ground motion. EEW systems have been researched and developed in Japan, Mexico, the United States, Taiwan, Italy, Turkey, and other countries [e.g., Nakamura, 1988; Hoshiba et al., 2008; Kamigaichi et al., 2009; Nakamura et al., 2009; Allen et al, 2009a; Espinosa Aranda et al, 2009; Hsiao et al., 2009; Zollo et al., 2009; Alcik et al., 2009].
 Prediction of ground motion is an important element for EEW. The prediction methods for EEW are classified by their basic concepts, such as onsite, front detection, and network method [Allen et al., 2009b]. Techniques of EEW are basically based on one or more of these concepts.
 The principle of the onsite (or single-station) method is to detect initial seismic waves (P waves) at a single station and provide warning of coming large ground motions (S waves) at the same station. Many onsite methods are based on the empirical relation of P wave amplitude and peak ground motion [e.g., Wu and Kanamori, 2005; Kanda et al., 2009], or on the use of the ground motion period of P waves to estimate earthquake magnitude, M [e.g., τpmax and τc method; τpmax is the dominant ground motion period and τc is the average ground motion period [Nakamura, 1988; Allen and Kanamori, 2003; Wu et al., 2007].
 The front detection method triggers a warning ahead of seismic ground motion when strong ground shaking above a certain level is detected at one or more front stations between seismic source and populated city. In the front detection method, it does not predict numerically the strength of the ground shaking (that is, wave amplitude) at the target cities, so that the warning is usually qualitative—the warnings are issued using a few categories. When the sources are distant from the cities, the front station provides long warning times. The Seismic Alert System (SAS) for Mexico City uses the front detection method, and it can provide approximately 60 s for earthquakes at the Pacific coast (approximately 300 km from the city) [Espinosa Aranda et al., 2009]. Alcik et al.  described an EEW system for Istanbul based on the front detection method using seismometers along the northern shore of the Marmara Sea. Aoi et al. [2011a] and Kanjo  proposed to issue EEWs when neighboring stations detect strong ground motion.
 The network method relies on seismic signals from a seismic observation network. Many network methods are based on rapid estimation of source location and M, followed by quick prediction of ground shaking strength using empirical attenuation relations of ground motion with distance (e.g., peak ground acceleration, PGA; peak ground velocity, PGV; and seismic intensity). The nationwide EEW system of Japan, which has been operated by the Japan Meteorological Agency (JMA) since 2007, adopts the network method using the seismic signals from more than 1100 stations [Hoshiba et al., 2008]. Elsewhere, Allen et al. [2009a], Hsiao et al. , Zollo et al. , and Cua et al.  have described the development of EEW systems for northern California, Taiwan, southern Italy, and southern California, respectively, based on the network method.
 Performance of the JMA EEW system for the 2011 off the Pacific coast Tohoku earthquake (the Tohoku earthquake, Mw 9.0; 11 March 2011) has been reported by Hoshiba et al.  and Hoshiba and Ozaki . Everywhere in the Tohoku district, the JMA EEW was earlier than the S wave arrival and more than 15 s earlier than the strong ground motion. That is, the system performed as expected for the Tohoku district. However, for the Kanto district around Tokyo (approximately 400 km from the epicenter), the predicted intensity was smaller than the observed intensity. This under-prediction can be attributed to the large extent of the later fault rupture [Aoi et al., 2011b; Kurahashi and Irikura, 2011]. For several weeks after the mainshock, aftershocks sometimes occurred simultaneously over the wide source region. When this happened, the system became confused and did not always determine the location and magnitude correctly, which led to some false alarms. The experience of the Mw 9.0 Tohoku earthquake revealed weak points of the network method based on rapid estimation of hypocenter and M: large errors in hypocentral location and M led directly to large errors in predicted ground motion, and the system cannot proceed to a prediction of ground motion without hypocenter and M information.
 The characteristics of the initial waveforms from the Tohoku earthquake were reported by Hoshiba and Iwakiri . The amplitude of both acceleration and displacement was quite small during the first several seconds. These were smaller than those from the Mw 7.3 foreshock of 9 March, and even Mw 6 events observed at the same sites. This suggests that the Mw 9.0 Tohoku earthquake started with a very small rupture. The small amplitude resulted in a large deviation from the empirical relation of P wave amplitude and peak ground motion, which was studied previously for the onsite method. Hoshiba and Iwakiri  also reported that the predominant period (τc, τpmax) of the initial part of the waveform was smaller than that of the Mw 7.3 foreshock, which is also a large deviation from the empirical τc-M and τpmax-M relations. They pointed to strong radiation of high-frequency waves as the reason for this deviation from the empirical relation. The source duration of the Mw 9.0 Tohoku earthquake was more than 150 s, and the moment release was greatest at 70–80 s after the origin time [Yoshida et al., 2011], which was much longer than the S-P time in the Tohoku district (S-P time was 15–30 s). These observations explain the difficulties in applying the onsite method, especially for sites at which S-P time is much smaller than the source rupture duration. They also suggest that an updating procedure is necessary for EEW, using ongoing waveform data.
 In Japan, JMA operates approximately 200 strong-motion seismometers, and the National Research Institute for Earth Science and Disaster Prevention (NIED) deploys 1800 instruments in K-NET and KiK-net [Okada et al., 2004]. In the Kanto district around Tokyo, KiK-net sensors at 30 sites are installed at depths of 500–3500 m in boreholes and at the ground surface. In addition, JMA and municipalities deploy seismic intensity meters at approximately 400 and 2800 sites, respectively. Although the main purpose of the seismic intensity meters is the measurements of seismic intensity based on the analysis of accelerograms and quick reports of the analyzed results (definition of seismic intensity on JMA scale is explained in Hoshiba et al. ), many intensity meters have a function to record waveforms of ground motion because their sensor is a three-component accelerometer. Some other organizations and private companies perform their own observations. Although each network is composed of many stations, integrating them would create an even denser strong ground motion observation network.
 In this paper, we propose a new method of ground motion prediction that extends the front detection method based on the Kirchhoff boundary integral technique (or the Kirchhoff-Fresnel integral technique for high frequency approximation). This method can predict ground motion at a specific target station in real time without requiring hypocenter and M data, using a dense seismometer network. The analyses at many target stations can be applied for EEW purpose.
2 Method for Ground Motion Prediction in Real Time
 A dense seismometer network that transmits ground motion data in real time makes it possible to monitor seismic wave propagation in real time. Figure 1 shows an example of the monitoring for the case of the Mw 9.0 Tohoku earthquake. Comparing snapshots several seconds apart gives an intuitive sense of the spreading of the waves, and wave propagation in the near future is easily visualized even without knowing the location of the hypocenter and M. This image is a basic concept of the proposed method in this paper. Unlike network methods of EEW based on rapid estimation of hypocenter and M, in which the causes (hypocenter and magnitude) are first identified and the prediction is then performed based on the identified causes, the proposed method is based on real-time monitoring of wave propagation, and the prediction of ground motion skips the steps of estimating the hypocenter and M.
 In general, wave motion can be predicted when initial condition and boundary condition are given. Here the initial condition does not necessarily mean the information of source characteristics (e.g., source location, M, direction and extent of rupture); instead wavefield at arbitrary moment can be used as the initial condition. In the proposed method, current wavefield observed by the real-time monitoring is used as the initial condition, and then wave propagation is predicted based on time evolutional approach.
 Site amplification factors are important factors in seismic ground motion in addition to source factors and propagation effects. This method assumes that site amplification factors can be evaluated beforehand.
2.1 Introduction of the Concept of Timing Into EEW
 To be more useful than today, EEW should if possible answer user questions such as “when will ground motion become large?”, “has peak ground motion gone, and will larger shaking be unexpected again?”, and “how long will ground motion continue?” EEW methods based on hypocenter and M provide information that is limited to the inception of strong ground motion, and even that may not be reliable. For instance, the S wave arrival time is implicitly assumed to coincide with the start of strong shaking. However, that is not always true, especially for large or distant earthquakes. Figure 2 shows accelerograms during the Mw 9.0 Tohoku earthquake observed at IYASAT in the Kanto district, whose epicentral distance is 315 km. There the strong ground motion started much later than the S wave arrival, and peak acceleration was observed more than 70 s after the S wave arrival. The late strong ground motion can be attributed to the later rupture, which extended toward the south from the initial rupture location [e.g., Aoi et al., 2011b; Kurahashi and Irikura, 2011].
 For many forecast subjects such as weather forecasts, predictions for near future are relatively precise and those for the distant future are less precise. Some users may demand rapid determination regardless of its preciseness, and others may need precise information rather than rapid determination. The trade-off between a rapid determination and a precise one varies depending on the users’ purposes. EEW is a similar case; thus, the introduction of more sophisticated timing information into predictions of ground motion is important to meet the varying demands of users. As seen in Figure 2, timing information of strong shaking, rather than the arrival times of phases, is informative for EEW.
2.2 Application of Huygens’ Principle and Kirchhoff-Fresnel Boundary Integral
 Humans can envision the wave propagation for the near future when looking at a map view of real-time monitoring. This intuitive form of prediction is probably based on extrapolation of the apparent velocity of wave propagation, and also Huygens’ principle, which considers the points on wavefront to be the source of secondary waves aligned along the wavefront. Huygens’ principle is a qualitative description of the physics of wave propagation, and Kirchhoff's boundary integral is the quantitative one, generally expressed as
for scalar waves in three-dimensional space, in which u(r, t) is the wavefield at location r and time t, r is the location of the target point of the prediction, S is the surface enclosing r, r1 is a location on S, ∂/∂n is the derivative with respect to the normal vector to S at r1, and G is the Green's function. We can take S arbitrarily. Here the integrations are performed with respect to r1 on S (that is, 2-D integral), and with respect to τ (convolution integral). Equation ((1)) is valid when there are no radiations (that is, no sources) inside of surface S.
 When the wave length is much smaller than the spatial fluctuation of absolute amplitude of u(r, t) and G(r, t–τ, r1, 0), that is, in high-frequency cases, equation ((1)) is approximated as Kirchhoff-Fresnel Integral [e.g., Born and Wolf, 1980; Shearer, 1999],
where θ (= θ(r1, τ)) is the angle of incoming ray paths from the surface normal, and θ ′ (=θ ′(r1)) is the angle between the surface normal and the ray to r (Figure 3), and v(r1) is the velocity at r1. Here θ′ is determined by the velocity structure, v(r), and geometries of S and r, that is, bend of the ray, irrespective of the wavefield, so that θ′ is a given parameter. The image of equation ((2)) is that the contributions from the secondary sources located on S produce the waveform at r, which is qualitatively similar to Huygens’ principle. Information on the original source location (that is, hypocenter) and the number of the original sources is not used in ((1)) or ((2)), but local curvature of wavefront is important instead, as is the same as Huygens’ principle. Because the waveform at r is determined by the interference of many waves from the secondary sources, phase delays given by the curvature affect the amplitude. When all the secondary waves arrive at r in-phase the amplitude becomes quite large, but when the phases are random it remains to be small. Equations ((1)) and ((2)) make it possible to predict the waveforms at r and t when the time derivative of u(r1, τ) on S and their propagation directions, θ, are known on the condition that the Green's function, G, and velocity, v, can be evaluated beforehand.
 Green's function, G, is determined by the velocity structure, and it generally has a complicated form. When velocity v is approximated as homogeneous, that is, v(r)≈v0, the Green's function is simply expressed as
where θ=θ(r1, t – |r – r1|/v0). The ray from r1 to r is the straight line and θ ′ is the angle between the surface normal and the straight line. The waveform at time t is the summation of waveforms at time, t-|r-r1|/v0, so that the lead time for the prediction is |r – r1|/v0. Equation ((4)) means that the wavefield near the target point, r, is used for the prediction of the near future (that is, small |r – r1|/v0), and the wavefield far from r is used for the distant future. Therefore, small S and large S are used for predictions of the near and distant future, respectively. Impulsive Green's function, ((3)), is an approximation for case where |r – r1| is small and inhomogeneity of velocity structure is weak. When the inhomogeneity is strong, we should consider appropriate Green's function for large |r – r1| instead of ((3)); the discussion will be given in section 5.
 In this paper, we attempt to predict wave motions based on ((1)), ((2)), or ((4)). Because this method uses real-time observation of waves approaching the target point and provides the strength of ground shaking quantitatively, this method corresponds to a quantitative extension of the conventional front detection technique of EEW. In the conventional front detection technique, it is not easy to predict quantitatively the strength of ground shaking, as explained in section 1.
 During the Mw 9.0 Tohoku earthquake, the JMA EEW system under-predicted the seismic intensity in the Kanto region because of the large extent of the fault rupture, and the system issued some false alarms because of confusion when multiple aftershocks occurred simultaneously. Addressing these problems is important for EEW. Equation ((1)) and its approximations ((2)) and ((4)) are derived from physics of wave propagation, and they are valid even for cases in which multiple sources exist (that is, multiple simultaneous earthquakes), or in which the waves are radiated from large areas (that is, large source extent), or in which the radiation is not isotropic (that is, strong directivity), when locations of all sources are outside of S. Therefore, the extended front detection method is applicable to these cases.
 The concept of the above equations is based on the boundary integral equation method [e.g., Zhou et al., 2010], and is similar to the phase screen method [e.g., Wu, 1994; Hoshiba, 2000] for high-frequency waves approximation, which has been used for simulation of seismic wave propagation. The proposed method is the application of the boundary integral equation method to the real ground motion.
2.3 Area Affecting Wave Motions
 In this section, the area that affects wave motions is discussed as compared with methods based on the hypocenter and M.
 Let us assume that waves are radiated from a point source O (Figure 4), and suppose that there is a transparent infinite screen, Σ, between the source and target point, r. Here surface S is composed of Σ and Σ ′, which is a hemisphere of infinite radius extending to the right of Σ. Based on Fresnel's theory, the area that mainly affects the wave motions at r is approximately given by the relation of “(Path A)–(Path B) ≤ (half wavelength)” in Figure 4, that is,
where L1 and L2 are the distance between O and Σ, and between r and Σ, respectively. The area is the first Fresnel zone. When L1>>D, and L2>>D, ((5)) is approximately expressed as
 The waves passing inside the area arrive at r in-phase. When the hypocentral distance is 100 km, for example, and Σ is located just at the midpoint (L1=L2=50 km), D ≤ 10 km is obtained for 1 Hz waves when v0=4 km/s. The range up to 10 km is important for wave motion at r. When the passing waves within that area are well observed, it becomes possible to predict precisely the wave motion at r. The waves passing outside of the area are negligible. Waves that do not approach r are therefore not so important for prediction of wave motion.
D becomes small with increasing frequency. In the real application of ((1)), ((2)), or ((4)), the whole range of surface S is not necessarily required; contribution from outside much larger than D is negligible.
 Prediction of ground motion based on hypocenter and M depends on the assumption of isotropic radiation from the source, in which it is supposed that precise estimation of M directly leads to precise prediction of ground motion. Because M is usually estimated using as many stations as possible for precise determination, information not only from waves approaching the target station but also from waves propagating in the opposite direction are taken into account. When directivity effects are large and the isotropic radiation assumption does not hold, the estimation of M may lead to lower precision in the prediction of wave motion. Precise estimation of M does not necessarily lead to precise prediction of ground motion, if the waves inside the range of ((6)) are well observed.
 When waves can be observed midway to the target point, prediction based on source parameters leads to applying a Green's function, G, for the relatively long distance from the hypocenter to the target point, instead of the shorter distance from the observing stations to the target point. The use of G for the long distance may also reduce precision in the prediction, because estimating G for a long distance is usually less accurate than doing so for a short distance.
 Figure 5 shows a comparison of the proposed method with the current method based on hypocenter and M. In the current method, observations are used to infer the source parameters (e.g., location of hypocenter, M, source extent), and then the strength of ground motion is predicted from the source parameters. In a sense, the information is sent first backward then forward. On the other hand, in the proposed method the information observed at front stations is sent forward directly to the target point. Kuruk and Motosaka  and Nagashima et al.  have also discussed precise prediction of ground motion based on the forward use of front stations. Hoshiba et al.  concluded that prediction using observations from neighboring front stations is more precise than that based on the hypocenter and M by analyzing the observed seismic intensity data. Even when the hypocenter and M are determined precisely, uncertainty in attenuation relation with distance (e.g., difference of frequency content in source factor even if M is the same, regional difference of path factor) gives the intrinsic precision limit of the prediction. When midway front stations are used for the prediction, the uncertainty can be minimized.
2.4 Lead Time and the Size of Surface S
 By introducing the concept of timing into the prediction in addition to strength, it becomes possible to predict time-dependent strength of waveforms. A large size of surface S corresponds to the prediction of the distant future, because the distance to the target point r is long, and thus |r – r1|/v0 is large. On the other hand, a small S corresponds to the prediction of the near future. That is, lead time for a prediction is controlled by the size of S. With expanding S, waveform data are required from a wider area, in turn requiring a Green's function, G, for the longer distance. Furthermore, the possibility of new sources (that is, new earthquakes) inside S increases, which leads to imprecision in the prediction. The imprecision usually increases with expanding S and increasing lead time.
 By using stations near the target point (small S), it is possible to predict precisely, but lead time is short. On the other hand, far stations (large S) give us long lead times but less precision. There is the trade-off between a rapid determination and a precise one.
2.5 Approximation for Small Lead Time
 When the distance from the observing station to target point, |r – r1|, is much smaller than the distance from the sources, that is, when S is small and the lead time is small, equations ((1)), ((2)), and ((4)) can be expressed as a simpler approximation.
 Let us consider two parallel infinite planes (Σ1 and Σ2) as shown in Figure 6, in which the observing station (r1) and the target point (r) are located on Σ1 and Σ2, respectively. Surface S is Σ1, and an upper hemisphere of infinite radius. Contributions from the hemisphere are negligible, so those from Σ1 are evaluated. Because the distance to the sources is much larger than |r – r1|, plane wave incidence can be assumed locally around r1, equation ((4)) is given by
where ((3)) is used for the Green's function. This means that the waveform at r and that at r1 is the same after correction for travel time, and the waveform at r is predicted from the single station located at r1. This relation is easily inferred because of the plane wave propagation. To derive ((7)), we use an assumption of local plane wave incident; that is, the phase is not fluctuated locally around r1. For real applications in seismology and earthquake engineering, however, site amplification factors should be taken into account,
where f (t) is the time series representing the site amplification factors, and * denotes the convolution integral with respect to t.
 An example of such a case is the prediction of waveforms of body waves at the ground surface using borehole observations. In Figure 6, Σ1 and Σ2 correspond to an infinite horizontal plane at the depth of the borehole sensor and the ground surface, respectively. As described in section 1, 30 stations of KiK-net of NIED have borehole accelerometers at depths of 500–3500 m in the Kanto district. In this case, the borehole stations are located in direction of incoming waves. The borehole enables us to predict easily and robustly ground motion at the surface.
3 Examples of the Application
 As described in section 1, quite dense seismic networks of strong ground motion stations can be realized by combining the existing networks of some organizations. At present, some stations transmit the data continuously in real time, but others record the waveform data using triggers. While trigger-type stations do not send out waveforms data in real time, some do continuously transmit representative parameters of waveforms such as seismic intensity [Aoi et al., 2008], because their data volume is much smaller than that of the full waveform. In Japan, the JMA scale is usually used for seismic intensity, which is obtained from three-component accelerograms. JMA intensity is defined as the logarithm of vector amplitude after filtering of around 0.5 Hz (approximately first-order for low-pass and 0.5th for high-pass); details have been explained by Hoshiba and Ozaki  and Hoshiba et al. . Even when stations that send out waveform in real-time are few, those that send the representative parameters of ground motion may be densely distributed. Dense station distribution is an important factor for the proposed method. For a sample application of the extended front detection method, in this section we use these seismic intensity data instead of waveform data, taking the current situation of available data into account.
 Let us consider here the correction of site amplification factor. When we focus on a certain frequency band of seismic waves and f (t) does not have a long tail along t for high frequencies, ((8)) is approximated as
where f0 is a scalar value, which represents the site amplification factor around angular frequency of ωc. Seismic intensity on the JMA scale is basically evaluated from the logarithm of waveform amplitude; thus ((9)) may be described as an intensity expression,
where I(r, t) is the seismic intensity, and F0 corresponds to log10f0.
3.1 Case of Small S
 As described in section 2.5, a borehole is an example of small S, in which we may assume a vertical plane wave incidence (θ ≈ 0). Seismic intensity at the surface will be predicted using borehole observation. Figure 7 (right) shows the three-component accelerograms at station TKYH11 (KiK-net) from the M6.0 event (23 July 2005; focal depth is 73 km) observed in borehole (depth: 3000 m) and surface sensors at an epicentral distance of 30 km. F0 for this station was empirically evaluated to be 1.8 using past data [Iwakiri et al., 2012]. Borehole sensors are installed directly below the surface sensors, so that θ′=0, and S wave arrival times differ by approximately 3.0 s between the borehole and surface sensors [Iwakiri et al., 2012], that is |r – r1|/v0 ≈ 3.0 s.
 Figure 7 (left) indicates the seismic intensity at the surface and in the borehole. Because the site amplification factor is 1.8, the seismic intensity at the surface is predicted by adding 1.8 to that in the borehole. The borehole data yield a precise prediction of ground motion at the surface with a short lead time, approximately 3.0 s. Figure 8 indicates the histograms of the difference between predicted and observed seismic intensities, in which 61 earthquakes in 2005 to 2009 are analyzed. The intensity is predicted from the hypocenter and M using empirical attenuation relation of ground motion adopted in JMA EEW (Figure 8a), and from the seismic intensity observed in the borehole (Figure 8b). Borehole observation leads to more precise prediction of ground motion at the surface than that from the hypocenter and M.
 In the above paragraphs, we discussed the case when θ is known, or can be assumed. In the following paragraph, we will describe again the case of small S, but for unknown θ. Here |r – r1| is assumed again to be small compared to the distance from the source, so that plane wave incidence is assumed.
 With small aperture arrays, it is possible to estimate propagation direction θ, but many arrays are not realistic at present. In this case, EEW should be based on the most severe scenarios as found by assuming various values of θ. For example,
is a candidate, where ri is the location of the i-th observing station located near r (for example, |r – ri |≤ v0T, where T is lead time to be required) and F0i is the site amplification factor relative to the i-th station in terms of seismic intensity. Here it is assumed that at least one of the neighboring stations can detect strong ground motion before it arrives at r. This is similar to the idea proposed by Aoi et al. [2011a] and Kanjo . Because wave propagation direction, θ, is not used, this idea corresponds to the application of Huygens’ principle instead of Kirchhoff-Fresnel integral of ((2)).
 For cases of small S described in this section, waveforms at r can be predicted in accordance with the observation at ri. Time required for the prediction is quite small, once the intensity, I(ri, t – |r – ri|/v0), is evaluated, because the procedure is merely the addition of the site factor, ((10)), or, the search of the maximum value, ((11)). The intensity can be evaluated in real-time manner using Kunugi et al.’s  technique.
 Let us consider the special case of |r – ri|→0. This means that the location of observing station is the same as that of the target point. In this case, EEW will be issued just after its criterion is satisfied at the target point. The lead time is 0, but it indicates that we can avoid a complete missed alarm. Here “missed alarm” means that EEW is not issued despite the actual occurrence of strong ground motion.
3.2 Case of Large S
 When a long lead time is desired, a large S is used. In this case, |r – r1| is not assumed to be small relative to the hypocentral distance, and plane wave incidence cannot be assumed. At first in this section, we assume that wave propagation direction θ is known or can be assumed.
 In this case, equation ((2)) or ((4)) is used for real-time prediction of ground motion, and we need to introduce some assumptions for performing the integration, considering the current situation of data availability as described above.
 Although the wavefield at the ground surface is observable when dense stations of strong ground motion are deployed at the surface, the wavefield underground cannot be observed at depths below a few kilometers. Some assumptions are required for the underground wavefield. For simplicity we assume ground motion that does not vary with depth, that is, u(r, t)=u(x, y, z, t)=u(x, y, 0, t), in this paper for the application of ((4)). Here z represents the depth.
 As described above, many stations send out seismic intensity data at present in real time instead of the full waveform itself. In this section, we will try to use the seismic intensity data of the dense observation stations. Here it is required to reconstruct waveforms from the intensity data for the application of ((4)). We use the empirical relation of PGV and JMA intensity given by Midorikawa et al. ,
where PGV is measured in cm/s. Note that JMA intensity is measured from some manipulations using filtered accelerograms as explained in section 3, and ((12)) is the empirical relation. Using ((12)), the PGVs are evaluated from the seismic intensity every second, and the waveform is constructed of pulses as shown in Figure 9, in which the amplitude of each pulse is given by the PGV for each second. The waveform is composed as the sequence of pulses with various amplitudes. Using the composed waveforms, equation ((2)) or ((4)) is applied to predict the waveform at the target point. From the predicted waveform, seismic intensity is obtained from ((12)) again. The obtained seismic intensity is the prediction at the target point. In this paper, we use 1 Hz pulses taking into account the frequency band for JMA intensity.
 In the real application of equations ((2)) and ((4)), the surface S is replaced with a mesh, and the integral is performed numerically using the waveforms at each grid on the mesh. Here the mesh size should be much smaller than Fresnel zone described in section 2.3, and also smaller than the wave length, and correlation lengths of spatial fluctuations of both amplitude and phase. The distribution of amplitude on the mesh is assumed to be given by interpolation of the amplitudes observed at nearby representative stations. The correlation length of phase is also assumed to be long in comparison with the mesh size. Because phase information is not available in the seismic intensity information, we need some assumption for the distribution of phase. Assuming no phase delay in the interpolations leads to a locally plane wavefront around the representative stations (Figure 10a). When the observation network is not dense enough, the hypocentral distance of an observing station may be much smaller than the station interval. In this case, the integral of equations ((2)) and ((4)) relies heavily or exclusively on data from the single station closest to the source, and the assumption of a locally plane wavefront may lead to an over-prediction of the ground motion. To avoid this, a phase delay, δ/v0, is introduced, using an artificial secondary source located slightly behind the real location by d (Figure 10b), where δ is distance between the wavefront of plane wave and sphere of radius d, that is δ= d(1 – cosφ) where φ is the angle between the wave propagating direction and that to the grid on the mesh. The phase delay acts as a spherical wave when the contribution of a single station is large, reducing the over-prediction of ground motion. As more stations contribute to the integration, this phase delay gradually becomes approximated with a plane wavefront (Figure 10c) that corresponds to the assumption of no phase delay. Here artificial source location is assumed to be 10 km behind from the real location in the following application, considering the typical station interval in the network. When the real source is located much nearer than the artificial source from the station, this assumption may lead to over-prediction; however, the quite near source is not so usual even when the location is just beneath the station because depth of the seismogenic zone is 5–20 km, in which actual phase delay is distributed according to the focal depth.
 Figure 11 is an example from the Mw 7.0 Iwate Miyagi Nairiku earthquake (14 June 2008; focal depth 8 km). The target point is MYG013 (K-NET), and an incoming angle from due north is assumed. The figure compares the real observation at MYG013 and the prediction using front stations 75–95 km from the target point in direction of the incoming waves. Site amplification factors f0 of 1 are assumed for all stations. The surface S is taken as a plane perpendicular to the assumed incoming direction, in which two axes are taken horizontally along the front stations and vertically. As described in section 2.3, we need not consider the whole range of S when limited information from direction of the incoming waves yields a suitable approximation. Contribution from less than 45° measured from the incoming direction is considered in this paper, and seven stations are used for the prediction in Figure 11. The mesh is taken with 0.2 km interval on the plane. The amplitude observed at the seven front stations is superimposed on the plane at z=0, and the amplitude on the mesh at z=0 is given by that of nearby stations using simple linear interpolation. Using assumption of invariability with depth, amplitude on the mesh is given along z-direction. The case of Figure 11 is an example in which the hypocentral distance to the closest front station is much smaller than the station interval. With no phase delay, the prediction is overestimated because of the large contribution of the closest front station (IWTH25) under the plane wavefront assumption (Figure 11, center). By introducing the phase delay, the overprediction is reduced (Figure 11, right).
 Figure 12 shows examples of the Mw 9.0 Tohoku earthquake for TKY013 (epicentral distance, 382 km), where 35° east of due north is assumed for the direction of wave propagation. Figures 12a and 12b, respectively, show the prediction using 10 stations 50–70 km and 5 stations 30–50 km from TKY013, corresponding to a lead time of approximately 15 and 9 s, respectively. The start time of the ground motion and its peak strength were predicted using ((4)).
 For the case of large S described in this section, procedures for reconstruction of waveforms and integration of equation ((2)) or ((4)) take time for their processing. However, it depends on CPU power of the analyzing machine; processing time can be reduced by using powerful CPU machines.
 We assumed in the above discussion that θ is known (or that data exist to specify θ). However, it is not always possible to specify θ. If θ is unknown, the possible strength of the predicted ground motion may be found by maximum strength using various values of θ. When we have many arrays that give us the information of θ, ((2)) and ((4)) could be applicable with the observed θ (without an assumption of θ).
3.3 Interpolation With the Method of Hypocenter and M
 Although knowledge of source parameters (e.g., hypocenter and M) is not required in the proposed method, the source information can be useful when it is well determined. To predict ground motion, the proposed method uses real-time data from stations along the ray path between the source and the target point. When these stations are few (e.g., the target point is on a peninsula or cape, the source is offshore, and ocean bottom seismometers are scarce), precise prediction is difficult. In such cases, the source information may be a useful complement to the proposed method.
 Figure 13 schematically shows an example based on the Mw 9.0 Tohoku earthquake. Using the hypocenter location and M determination of 7.7 as estimated by the JMA EEW system 60 s after the origin time [Japan Meteorological Agency, 2011a], the intensity distribution may be estimated as shown on the left side of the figure. The intensity distribution of concentric circles is derived from an attenuation relation with distance of intensity. The predicted distribution can be merged and replaced with the observation in regions with a dense observation network (right panel of Figure 13). The resulting distribution can be used for ground motion prediction in the extended front detection method. For example, the offshore distribution of the intensity can be used for target points on a peninsula or cape.
 Although the precision of the prediction may be reduced, interpolation with the method of hypocenter and M enables prediction of ground motion even where observations are sparse, when the hypocenter and M are well determined.
4 Avoidance of False Alarm
 After the Mw 9.0 Tohoku earthquake, aftershock activity and induced seismic activity were quite high for several weeks [Hirose et al., 2011]. When multiple aftershocks occurred simultaneously within the large source region, the JMA EEW system did not always determine locations and magnitudes correctly, which led to overprediction of ground motion and some false alarms [Hoshiba et al., 2011].
 For example, JMA issued an EEW at 05:34 (Japan Standard time) on 15 March 2011 predicting strong shaking around central Japan, based on the estimated hypocentral location, shallow focal depth (10 km) and M of 6.3 [Japan Meteorological Agency, 2011b]. The EEW was issued 11.3 s after the origin time. However, the real magnitude of the event was 1.3, and the shaking was quite weak (seismic intensity was less than 1 on the JMA scale everywhere, as weak as people did not detect the shaking). The false alarm was caused by the coincidence of this event with two other small events, one off Kanto (approximately 300 km away) and the other off Tohoku (500 km away); that is, in total three earthquakes occurred simultaneously. The simultaneous multiple earthquakes caused confusion for hypocenter and M determination in the JMA EEW system.
 Based on the estimated hypocentral location and M=6.3, the distribution of strong ground motion was expected to spread approximately 50 km from the hypocentral location at the time of EEW issuance (11.3 s after the origin time), but in reality the observed intensity was less than 1 everywhere at the time. This example shows that it may make it possible to avoid such overpredictions by monitoring the distribution of ground motion in real time. When ground motion is predicted from the real distribution of ground motion using the extended front detection method, false alarm might be avoided.
5 Further Steps to Real Time Prediction
 To apply the method proposed in this paper, the following are required in advance:
 (A) Deployment of dense observation stations that send data in real time;
 (B) Evaluation of site amplification factors of target points and observation stations;
 (C) Evaluation of velocity structures v(r) and Green's functions
 (D) Real-time estimation of spatial distribution of u(r, t) at any moment, including interpolation and extrapolation of observed ground motions;
 (E) Real-time estimation (or assumption) of the direction of wave propagation θ;
 (F) Construction of waveforms, based on equations ((1)), ((2)), or ((4)), or equivalent method;
 are to be manipulated for prediction of ground motions.
 Regarding (A), dense observation stations are important for this method. The distribution should be so dense to be capable of recognizing the wavefront expansion and propagation. In section 3, we used seismic intensity instead of the waveform itself owing to the current limits of real-time data transmission. However, the waveform is required for prediction of long-period ground motions, which include peaks and troughs of the waveform, that is, phase information. As described in section 2.2, phase delay is important information for this method. Although discussion in this paper is based on the physics of scalar waves, manipulations of vector waves of three components will improve the method by enabling consideration of P waves and S waves and their interactions. In that case, the onsite method for a single station can be regarded as a specific case of |r – ri|→0, as explained in section 3.1.
 Regarding (B), Iwakiri et al.  estimated site amplification factors in terms of seismic intensity at about 1300 seismic intensity stations, and Takemoto et al.  evaluated the frequency-dependent site factors of K-NET and KiK-net stations using the coda normalization method [Aki, 1980]. These estimates will be used for improving the method. However, the frequency-dependent site factors are usually estimated in the frequency domain in many previous studies. For the application of EEW, representation in frequency domain is not suitable because the analysis should be performed in real time. Representation in time domain is useful for EEW, as indicated in ((8)). The correction of the site amplification factors will be performed in time domain before application of the prediction of ((1)), ((2)), or ((4)).
 Regarding (C), the impulsive shape of ((3)) gives a good approximation for the Green's function when |r – r1| is small, but it is not realistic when |r – r1| is large; it is well known that the seismogram has longer coda waves than the source duration and the attenuation of amplitude with distance differs from 1/|r – r1|. Empirical Green's function [e.g., Irikura, 1986] and stochastic Green's function [e.g., Kamae et al., 1998] methods will be applicable for large |r – r1|. For long period waves, the Green's function can be evaluated using 3-D simulation of wave propagation. Using these realistic Green's functions, it is possible to produce the waveforms that have long coda and amplitude decay different from 1/|r – r1|. Coda wave shapes have been investigated for high frequency waves by many researchers [Sato and Fehler, 1998], and they have interpreted the long duration of coda waves as the effect of scattering [e.g., Aki and Chouet, 1975]. The duration of the seismogram envelope increases with hypocentral distance, which is controlled by the scattering strength and intrinsic absorption attenuation [e.g., Hoshiba, 1991; Saito et al., 2002]. This knowledge will aid estimation of Green's functions in advance. The Green's function having long tail leads to the precise prediction for both amplitude and timing, especially for large |r – r1|.
 Regarding (D), dense observation stations are important for this method, as described above. Data assimilation technique, which is widely used in numerical weather forecast and oceanography, is applicable for producing the artificial dense network. The current spatial distribution predicted from past data is merged into the current real observation, and the merged distribution is used for application of the prediction of ((1)), ((2)), or ((4)). The data assimilation technique will lead to the precise prediction.
 Regarding (E), array analysis is a tool for estimation of θ. In addition to that, the array analysis gives tools for discrimination of P and S waves by comparing particle motion and propagation direction, and also for estimation of apparent velocity, which make it possible to estimate the wavefield of underground. When apparent velocity is comparable to v(r1), the wavefront is vertical (that is, the ray is horizontal), and large apparent velocity means that the wavefront is horizontal (that is, vertical incidence). In section 3.2, we assumed depth independence for the wavefield, but it may be too simple especially for shallow earthquakes of short distance. The estimation of the wavefield of underground is quite useful for the process of (D). Ideally, each station should be a small array for this method.
 Regarding (F), we show some examples using equations ((1)), ((2)), or ((4)). For high frequency wave, the propagation is well approximated by energy propagation using radiative transfer theory [Sato and Fehler, 1998; Ishimaru, 1997]. Monte Carlo approach is a powerful tool to simulate the energy propagation in 3-D media [Hoshiba, 1997; Yoshimoto, 2000]. Even when the phase information cannot be observable, the application of the radiative transfer theory may be useful for the prediction of high-frequency ground motion, instead of the direct application of ((1)), ((2)), or ((4)), which makes ease the process of (F).
 In the extended front detection method proposed in this paper, “trigger” is not necessarily required, so that the mistriggering can be avoided. Whether strong ground motion is detected or not, analyses should be performed continuously. Prediction such as “strong shaking is NOT expected” is also useful information as well as “strong shaking is expected”. Although at least a few seconds of ground motion is required for the estimation of M in many network methods, real-time observation of ground motions is immediately used at any moment in the proposed method, which will lead to quickening EEW.
 We have proposed a method for real-time prediction of ground motion using real-time monitoring of ground motion, extending the front detection method for application to EEW. The basic idea is from Huygens’ principle as quantified using Kirchhoff-Fresnel boundary integral. Although this method requires a dense observation network, it becomes possible to predict ground motion without waiting for a hypocenter and magnitude to be determined, because these source parameters are not necessary. The effects of rupture directivity, source extent and simultaneous multiple events can also be substantially included in this method.
 The proposed method enables predictions to include more information about the timing of ground motion. When predictions 3 s in advance are required, borehole observations are useful for precise predictions, as described in section 3.1, which is applicable in the Kanto district around Tokyo where 30 existing borehole stations are at depths of 500–3500 m. When a longer lead time (~10 s) is required, neighbor stations around the target point can be used at distances less than approximately 35 km, as explained in section 3.1. For longer lead times, information from more distant stations can be used, as discussed in section 3.2.
 We thank the anonymous reviewers and Editor, R. Nowack, for their useful comments for revising the manuscript. We also thank A. Katsumata, K. Iwakiri, and N. Hayashimoto for their help in completing the figures. Seismic intensity data were provided by JMA, NIED and local governments and municipalities. Waveform data were obtained from the JMA network, K-NET and KiK-net of NIED. The unified hypocenter catalog and CMT catalog of JMA were used in this analysis. Figures were made using Generic Mapping Tools [Wessel and Smith, 1995].