We propose a method for near-field tsunami forecasting from data acquired by cabled offshore ocean bottom tsunami meters (OBTMs) in real time. We first invert tsunami waveforms recorded at OBTMs to estimate the spatial distribution of initial sea-surface displacements in the tsunami source region without making any assumptions about fault geometry and earthquake magnitude. Then, we synthesize the coastal tsunami waveforms from the estimated sea-surface displacement distribution. To improve the reliability of the tsunami forecasting, we use updated OBTM data to repeat the forecast calculation at 1-min intervals. We tested our method by simulating the 1896 Sanriku tsunami earthquake, which caused a devastating tsunami with maximum runup height of 38 m along the Pacific coast of northeastern Japan. Instead of real OBTM records, proxies were used. The simulation demonstrated that our method provided accurate estimations of coastal arrival times and amplitudes of the first peak of the tsunami more than 20 min before the maximum amplitude wave reached the coastal site nearest to the source. We also applied the method to real data of a small tsunami that was caused by a local earthquake and successfully forecasted the tsunami at coastal tide stations. We found that accuracy of our estimated coastal tsunami amplitudes can be affected by the spatial relationship between the tsunami source and the offshore observation stations. Our numerical simulation showed that even more accurate tsunami amplitude forecasts would be achieved by deployment of additional offshore stations separated by a distance comparable to the trench-parallel length of the tsunami source.
 Submarine earthquakes or landslides can generate tsunamis that cause loss of life as well as severe damage to houses and infrastructures in coastal communities. The 2004 Sumatra earthquake (Mw 9.1) generated a catastrophic tsunami that killed more than 220,000 people. One of the reasons for the immense scale of this disaster was lack of information about the impending tsunami, and inadequate procedures for warning communities around the Indian Ocean. This tragedy has driven the international community to reaffirm the importance of mitigating the effects of tsunamis. Forecasting tsunami events and providing timely evacuation warnings to communities that may be affected is one of the most effective ways to reduce the loss of human lives and the damage to communities. Accurate and rapid dissemination of information about impending tsunamis is essential for successful reduction of damage to property and loss of life.
 In Japan today, tsunami warnings for both of near-field and far-field tsunamis are issued by the Japan Meteorological Agency (JMA) [Tatehata, 1997]. For near-field events, warnings are issued within 3 min of the earthquake. The warning is transmitted by radio and television stations to people in the affected coastal areas. However, for “tsunami earthquakes” the warnings may be unreliable because they are based only on seismic data, not on tsunami observation data. “Tsunami earthquake” is a specific type of earthquake as opposed to “tsunamigenic earthquake” and can generate tsunamis that are much larger than predictions based on analysis of seismic data alone [Kanamori, 1972]. Moreover, the present warning system is less reliable for tsunamis caused by nonseismic events.
 The sense of crisis in a coastal community can be weakened by the provision of inaccurate tsunami warnings. For the 2006 Kuril earthquake, JMA issued a warning for a tsunami predicted to have a wave height of about 2 m on the coastlines bordering the Okhotsk Sea, but the wave height of the tsunami that followed was only about 60 cm. Experiences such as this can foster a lack of attention to tsunami warnings in coastal communities and may result in evacuation orders being tragically ignored. Although disastrous tsunami earthquakes are rare, they must be the critical targets for tsunami forecasting, and the accuracy and reliability of tsunami forecasting must be improved.
 Far-field tsunami forecasting based on recorded tsunami data is operational in the United States. The Pacific Marine Environmental Laboratory (PMEL) of the National Oceanic Atmospheric Administration (NOAA) developed a worldwide network of offshore tsunami observatories called the Deep-ocean Assessment and Reporting of Tsunamis (DART) system [Titov et al., 2005]. This system uses ocean bottom pressure sensors to measure offshore tsunamis and then transmits the observations to data centers in real time by using moored buoys and communication satellites [González et al., 2005]. PMEL successfully used the DART system to experimentally produce far-field forecasts of the tsunamis generated by the 2003 Rat Island earthquake [Titov et al., 2005], and the Tonga, Kuril, Solomon, and Peruvian earthquakes in 2006 and 2007 [Wei et al., 2008].
 For near-field tsunamis, however, there are currently no warning systems that use tsunami data. Near-field tsunamis in areas close to subduction zones can reach the coast in a few tens of minutes, or less, and can cause catastrophic damage. Therefore, the time window for issuing warnings of arrival times and amplitudes of near-field tsunamis is much shorter than that for far-field tsunamis. Despite this difficulty, development of a system to provide reliable forecasts of tsunamis caused by local earthquakes is important for effective mitigation of damage and loss of life in coastal communities. The purpose of this study is to develop a method that uses offshore tsunami data recorded in real time to provide timely and reliable forecasts of near-field tsunamis.
 Here we focus on near-field tsunamis that are generated by local earthquakes along the Kuril Trench and Japan Trench subduction zones. Two cabled ocean bottom observation systems have been installed on the landward slopes of these trenches to monitor earthquakes and tsunamis [Hirata et al., 2002; Kanazawa and Hasegawa, presented paper, 1997] (Figure 1). The 2003 Off-Fukushima earthquake, which was a moderately strong interplate underthrust earthquake (M 6.8) that occurred west of the Japan Trench (Figure 1a), generated a small tsunami. The tsunami was observed to have an amplitude of 2 cm (Figure 1b) at two ocean bottom tsunami meter (OBTM) stations (TM1 and TM2 of Figure 1a) [Hino et al., 2005]. Even though the fluctuation of sea level due to this tsunami was only a few centimeters, it was clearly detected by the OBTMs. It is noteworthy that this tsunami was detected at OBTM station TM1 18 min earlier than at the nearest coastal tide station (AY, Figure 1b), even though the distance from the epicenter to the OBTM station (∼160 km) was greater than that for the tide station (∼130 km) (Figure 1a). This is because tsunamis propagate faster in the deep-water area where the OBTMs are deployed than in shallow areas near the coast. Rapid observations such as this allow the issue of early warnings of near-field tsunamis. Furthermore, the tsunami waveform observed at the OBTM is simpler than that observed at the tide station AY (Figure 1b) and thus provides a tsunami source signature without distortions due to complex coastal topography [e.g., González et al., 1991]. Few studies [Baba et al., 2004] have attempted to use OBTM records for near-field tsunami prediction, even though they are used to investigate earthquake source parameters [Hino et al., 2001; Tanioka et al., 2004a; Matsumoto and Mikada, 2005]. In this paper, we propose a method for near-field tsunami forecasting in which the offshore OBTM data play a significant role. We investigate how well our method works for tsunami earthquakes by carrying out a numerical simulation and also apply the method to real tsunami data.
2. Cabled Ocean Bottom Pressure Data and Data Processing
 There are two cabled ocean bottom observation systems in the area of the Kuril and Japan Trench subduction zones covered by this study (Figure 1a). Each of these systems comprises two OBTMs. One system (OBTM stations TM1, 1563 m water depth; TM2, 990 m) was deployed near the Japan Trench and has been operated by the Earthquake Research Institute (ERI) of the University of Tokyo since 1996 (Kanazawa and Hasegawa, presented paper, 1997). The other system (stations PG1, 2218 m; PG2, 2210 m) was installed near the Kuril Trench by the Japan Agency for Marine-Earth Science and Technology (JAMSTEC) in 1999 [Hirata et al., 2002]. The OBTM data recorded by the two systems are transmitted in real time to their respective data centers at ERI and JAMSTEC. The OBTMs of both systems are pressure sensors that utilize quartz oscillators. The resolution and full scale of the Japan Trench sensors are 0.5 mm and 3000 m, respectively, and the sampling interval is 0.1 s [Hino et al., 2001]. The resolution of the data obtained by the Kuril Trench system is 0.3 mm and the resampling interval is 10 s [Hirata et al., 2002].
 The pressure data recorded by the OBTMs include the effects of ocean tides, atmospheric and oceanographic disturbances, vertical deformation of the seafloor, and seismic waves, as well as tsunami data. The pressure variations due to nontsunami phenomena have very different time scales from those of tsunamis and can therefore be easily separated.
 The data processing we applied to the real-time bottom pressure time series recorded by the OBTMs can be accomplished immediately after recording, so the processed data can be used to produce real-time tsunami forecasts. The results of the data processing applied to the tsunami signal recorded at OBTM (TM1) (Figure 2) show that large pressure fluctuations (∼3 cm relative sea level) caused by seismic waves (acoustic waves which are converted from body waves at the seafloor) are still visible after applying a 60-s moving average to the data. To remove these fluctuations, we applied a low-pass digital filter with a cutoff period of 60 s [Saito, 1978]. The ocean tide components were removed by subtracting the sea level variation computed from a theoretical tide model developed by Matsumoto et al. . This subtraction was implemented during real-time waveform data processing. Because both the high-frequency seismic wave component and the long-period effect of oceanic tides have been removed, the tsunami signal is more pronounced after processing (Figure 2b) than in the unprocessed record (Figure 2a). After real-time processing of the OBTM waveforms they are ready to be used for tsunami forecasting.
3. Methodology for Near-Field Tsunami Forecast
 Our method for near-field tsunami forecasting presented here has two stages: (1) estimation of the distribution of the initial sea-surface displacement in the tsunami source region by using the offshore OBTM data and (2) forward calculation of tsunami waveforms along coastal regions by using the estimate of the sea-surface displacement field from the first stage to forecast arrival times and amplitudes.
 In the first stage, we invert the OBTM data of sea-surface displacement distribution as in some previous studies [e.g., Baba et al., 2005], but not for slip distribution along an assumed fault plane as has been the case in many previous tsunami source inversion studies [e.g., Satake, 1987]. Our approach allows us to estimate a tsunami source without knowing the geometry of the fault that initiated the tsunamigenic earthquake. Most large tsunamis are generated by megathrust earthquakes. It is generally assumed that their sources are at the upper surface of the subducting plate, where the geometry is well defined. However, tsunamis generated by earthquakes originating from normal faults in the trench outer slope region must also be taken into account, such as the 1933 Sanriku earthquake and the 2007 Kuril earthquake. Even for megathrust earthquakes, splay faults may produce a substantial amount of tsunami energy [e.g., Fukao, 1979], but little is known about the geometry and orientation of the splay faults. As well as fault geometry, we do not make any assumptions on the size of an earthquake (e.g., a seismic moment) in the methodology. Use of such parameters may bias the estimation of tsunami sources. It is undesirable for tsunami forecasts in the cases of tsunami earthquakes. Another benefit of our approach is that the consideration of the effect of deformation of the seafloor on the sea surface during tsunami generation process is not needed [Kajiura, 1963, 1970] and, therefore, only the effects of tsunami propagation need to be considered.
 In both stages of our tsunami forecasting procedure, it is necessary to calculate tsunami waveforms caused by an initial sea-surface displacement; these calculations must be completed quickly enough to be useful for near-field tsunami forecasting.
 In our procedure, tsunami waveforms are obtained by superposition of Green's functions as follows:
where fj (t) is the tsunami waveform at the jth location and Gij(t) is Green's function, which is the response to a unit displacement of the ith element of the sea surface at the jth location. ai is the amount of displacement of the ith sea-surface segment, and M is the total number of sea-surface segments. Green's functions can be calculated in advance and stored in a database. In the tsunami forecasting procedure, appropriate Green's functions can be extracted from the database to calculate their linear superposition. These calculations can be completed quickly enough to be applied in real-time forecasting.
 Tsunamis caused by local earthquakes may reach coastal regions less than 20 min after the earthquakes. To issue a timely tsunami warning for such a near-field tsunami, the tsunami forecasting procedure must start as soon as the tsunamigenic earthquake occurs. Therefore, in our tsunami procedure it is desirable to perform the inversion using only the early part of the tsunami records observed at OBTMs. However, the wavelength of a great tsunami can be comparable to the distance between the OBTM stations and the nearest coast, and it may require more than 10 min to obtain sufficient tsunami waveform data to estimate source parameters. Tsunami forecasting based on insufficient tsunami waveform data may result in inaccurate predictions, but we believe that an early report based on limited waveform data is still useful. Another problem is that we do not know exactly how many data are needed to make an accurate prediction.
 In our forecasting procedure, we begin the calculations as soon as an earthquake occurs. We can expect that the accuracy of the prediction improves as time passes and more OBTM data are processed. Our forecasting procedure performs the inversion and forward calculation repeatedly by progressively updating the OBTM data. Because individual predictions (calculated using a Core2 Duo 1.8 GHz CPU personal computer) take less than 1 min to calculate, we can update the tsunami prediction regularly at intervals of 1 min or less, and thus provide successive tsunami predictions with improved accuracy.
3.1. Observation Equation
 Observed offshore tsunami waveform data contain direct information about the tsunami source [e.g., González et al., 1991]. Thus, the OBTM data provide useful information for real-time estimation of a tsunami source. In our method, we invert OBTM tsunami waveform data for an initial sea-surface displacement distribution as follows:
where fkobs(T) is the observed tsunami waveform at the kth offshore OBTM station and ai is the amount of displacement of the ith point on the sea surface, which is the unknown parameter we wish to calculate.
 Here we defined the elapsed time after an earthquake as T. The waveform data that we use to make our forecast come from the OBTM records between the origin time of the tsunamigenic earthquake (T = 0) and the time at which the forecast is made (T = T0).
where fobs(T) is the vector of the observed OBTM tsunami waveform, G(T) is a matrix whose elements are Green's functions Gik(T), and a is a vector representing the initial sea-surface displacement ai.
3.2. Sea-Surface Area Incorporated in the Inversion
 As explained earlier, in order to make timely forecasts the inversion for the initial sea-surface displacement must be performed by using only the early part of the records obtained at OBTMs. In this early period there may be insufficient data to estimate the displacement distribution accurately. Instability of the inversion because of insufficient data is an important issue. In our method, we limit the area over which sea-surface displacement is estimated (the “influence area” hereafter) to reduce that instability. Specifically, we invert the OBTM records only for the displacements that potentially affect the tsunami waveforms for 0 ≤ T ≤ T0. This approach allows us to reasonably reduce the number of unknowns in the inversion.
 We define the influence area at T = T0 by drawing back-propagated wavefronts from each of the OBTM locations. We regard the area surrounded by the back-propagated wavefronts at T = T0 as the influence area at T = T0, and estimate surface displacements within that area by inverting the OBTM records for 0 ≤ T ≤ T0. To define the wavefronts, we must know the origin time of the tsunami. We assume that the origin time of the earthquake estimated from seismic data analysis is the origin time for the tsunami. The influence area is defined according to elapsed time T (Figure 3). As elapsed time increases, the influence area becomes larger according to the enlargement of the back-propagated wavefronts from the OBTM locations, which means that tsunamis generated at long distances can reach the OBTMs.
3.3. Constraints on the Inversion
 Another difficulty arises for tsunami source estimations from only the early part of OBTM records. If tsunami estimations are made soon after the occurrence of the earthquake, we encounter cases where the tsunami is observed at only one of the OBTMs. For these cases, the location of the tsunami source cannot be constrained and the estimated sea-surface displacement is distributed along the back-propagated wavefront from the OBTM. This incorrect estimation of the sea-surface displacement degrades the accuracy of the prediction of both arrival times and amplitudes of tsunamis reaching the coast.
 To overcome this difficulty we introduce a priori information to the inversion about the location of the tsunami source. In this study, we imposed a constraint on the spatial distribution of sea-surface displacements by applying the constraint that the initial sea-surface deformation due to an earthquake should be zero if the epicenter is far enough away. This constraint can be formulated by adding the condition
to equation (3). Here w(ri) is a weighted function of the horizontal distance ri from the epicenter to the ith point on the sea surface. Equation (4) can be written in vector form as
where D is a diagonal matrix representing the damping constraint whose components dij are
where δij is the Kronecker delta.
 In this study, we used
as the weighting function to impede sea-surface displacement in areas distant from the earthquake epicenter. In equation (7), r and rmax are the epicentral distance and cutoff distance, respectively. We defined rmax as the horizontal distance between the epicenter and the most distant point within the influence area. The epicenter location used for damping constraint was fixed at that estimated from seismic data analysis. Since influence area enlarges as time elapses, rmax also becomes larger. Thus, at a certain point (r = ri) the value of w(ri) becomes smaller in the later forecasting and the effect of the damping constraint becomes weaker. This constraint does not depend on earthquake magnitude estimates.
 Additionally, we imposed a smoothness constraint on the spatial distribution of the sea-surface deformation as follows:
where al,m is the amount of initial sea-surface displacement at the position defined by l and m on the easting and northing axes, respectively. In vector form, equation (8) is
where S is a matrix representing the operator that measures smoothness.
3.4. Equations Used in the Inversion and the Forward Calculation
 By incorporating the constraints of equations (5) and (9) in equation (3), the equation to be solved in the inversion can be expressed as
where α and β are weighting quantities (or hyperparameters) of the smoothness and the damping constraints, respectively. Solving this least squares problem by using the singular value decomposition method [Press et al., 1992], we estimated the initial sea-surface displacement distribution. In this study, we fixed the weighting quantities to our preferred values of α = 10 and β = 100, which we obtained by trial and error. By using these values, sea-surface displacement distribution can be well estimated in numerical simulation in section 4.3.
 Once the spatial distribution of sea-surface displacements a have been determined from the waveform inversion of stage 1 of our method, tsunami waveforms can be calculated using superposition of Green's functions. If we calculate the Green's function at the nth point along the coast corresponding to the ith point of sea-surface displacement Gin(T) in advance, the tsunami waveform at the nth points fncalc(T) is
3.5. Database of Green's Functions
 To compute the Green's functions, we calculated the finite difference approximation of the linear long-wave equations [Satake, 1995] in the area shown in Figure 4. The grid size we used was 20 arc seconds (about 600 m, as shown in Figure 4a), except for the regions near coastal tide stations, where it was set to 4 arc seconds (Figures 4b and 4c) as nested grids. The time step for the computation was 1 s in order to satisfy the stability condition.
 The initial sea-surface displacement model for the computation of Green's functions is shown in Figure 5a. A single element of the tsunami source was represented by a square of dimensions 700 × 700 arc seconds. The spacing between elements was set to 540 arc seconds (0.15 arc degrees). This was set to less than the size of the element so that neighboring elements overlapped at their margins (Figure 5b). We varied the amount of displacement within an element as shown in Figure 5a. The use of overlapping elements with different displacements provides a way to express smooth variations of sea-surface displacement with a finite number of discrete elements [Aida, 1972].
 The elementary tsunami sources were distributed in the sea area shown in Figure 4 (∼1698 elementary sources) and Green's functions corresponding to 14 coastal tide stations (locations shown in Figure 4) and the four OBTMs were computed for each of them. All of these Green's functions were calculated in advance and stored in a database, from which appropriate functions were extracted and used for the inversion (equation (10)) and forward calculation (equation (11)).
4. Numerical Simulation
4.1. The 1896 Sanriku Tsunami Earthquake
 To test the effectiveness of the method proposed here for near-field tsunami forecasting we performed a numerical simulation of the 1896 Sanriku earthquake, which was an interplate underthrust earthquake near the Japan Trench [Tanioka and Satake, 1996]. This earthquake is considered to represent a typical tsunami earthquake: its tsunami magnitude (Mt) was 8.6 [Abe, 1979], but its surface wave magnitude (Ms) was only 7.2 [Abe, 1994]. It generated a disastrous tsunami with maximum runup height of 38 m and caused the death of 22,000 people, even though the ground motions were weak and caused little damage. To assist with tsunami disaster mitigation, forecasts of disastrous tsunamis such as this must be reliable. However, forecasts based only on seismic wave data underestimate the height of tsunamis caused by anomalous tsunamigenic earthquakes such as the 1896 Sanriku earthquake. To provide reliable estimates of tsunami arrival times and amplitudes it is very important that real-time tsunami observations are used.
4.2. Simulation Procedure
 The simulation was performed as follows:
 1. Tsunami waveforms were computed for offshore and coastal observation points by assuming a fault model corresponding to the 1896 tsunami earthquake. For the simulation, these waveforms were regarded as the tsunami waveforms observed at the offshore and coastal stations.
 2. Coastal tsunami waveforms were then estimated by using the algorithm presented in this paper. The waveforms “observed” at offshore stations were inverted for sea-surface deformation and coastal tsunami waveforms were synthesized from the inverted source model.
 3. The performance of the forecasting method was evaluated by comparing the “observed” waveforms at coastal observation points obtained in step 1 with the predicted waveforms obtained in step 2.
 For calculation of the “observed” tsunami waveforms, we assumed the fault model presented by Tanioka and Satake  (Figure 6a). This model was obtained by analyzing the tsunami waveforms recorded at coastal tide stations during the 1896 earthquake. Assumed slip on the fault plane was uniform (5.7 m). The vertical displacement field of the seafloor at the fault was calculated by means of the algorithm of Okada . We then computed the tsunami waveforms at the 4 OBTMs and 14 coastal tide stations by assuming that the initial sea-surface displacement was the same as the seafloor vertical displacement. The first simulated tsunami arrival at the coast was at station KM, 20 min after the earthquake.
 We repeated the forecast calculation at 1-min intervals between 5 and 20 min after the earthquake event on the basis of the “observed” waveforms simulated at the OBTMs. As time elapses after an earthquake, the lengths of the OBTM records used in our forecasting scheme increase. We examined the effect that the length of OBTM records had on the accuracy of our forecasts. For this test, we did not add ocean tide components to the “observed” waveforms because tidal signals can be removed from observation data, as explained earlier. We also did not add noise from various sources to the “observed” records because the wave height of this tsunami was large enough to override the effect of noise on the forecast.
 In evaluating our method, we focused on the results obtained during the period up to 20 min after the earthquake; that is, the time between the earthquake event and the earliest arrival of a tsunami wave at the coast. We chose this period because for tsunami forecasts to be useful they must be available before the tsunami arrives at the coast. In evaluating the results of our forecast, we paid most attention to the arrival times and amplitudes of the first peak of the tsunami at coastal tide stations; these are the critical components of a tsunami warning. In the simulation, we defined an arrival at a coastal tide station as the time at which the tsunami caused a sea level change of more than 0.5 m. This accords with the criteria of the present JMA tsunami warning protocol [Tatehata, 1997]. To quantitatively evaluate the accuracy of the predicted tsunami amplitudes we scored the accuracy of the tsunami forecasting as follows:
where Anobs is the maximum positive amplitude of the first tsunami wave observation at the nth coastal tide station, and Anpred is the maximum positive amplitude of the predicted tsunami waveform. Anpred is measured in a time window that has a central time corresponding to the arrival time of the first positive peak of the observed tsunami waveform and a length twice that of the positive part the first tsunami waveform observation. Flooding by the positive part of the tsunami wave tends to cause the greatest amount of damage. N is the number of coastal tide stations from which tsunami waveforms were used to evaluate accuracy. A high score indicates accurate prediction of tsunami amplitude: a score of 100% indicates absolute accuracy.
 A comparison of the simulated tsunami forecast 15 min after the 1896 Sanriku tsunami earthquake with the observed data is shown in Figure 7. The predicted tsunami waveforms match the observations very well, as do the arrival times and amplitudes of the first peaks of the tsunami wave at most of the coastal tide stations on both Honshu and Hokkaido islands.
 We also examined the accuracy of forecasts made earlier than 15 min after the earthquake. The accuracy scores obtained by applying equation (12) to forecasts generated at 1-min intervals between 5 and 20 min after the earthquake event are shown in Figure 8. For the first forecast (at 5 min), the accuracy score is low (∼10%), but it increases as time elapses and shows a sharp increase to almost 80% at 11 min. The accuracy score increases gradually after 11 min and stabilizes at about 90% at around 15 min. The accuracy of the forecast improves as the length of the OBTM records used to generate it increases: for the early forecasts, the data length is too short to estimate the tsunami source accurately.
 For the forecast made at 11 min (Figure 9), immediately after the sharp increase in the accuracy score (Figure 8), the estimated tsunami wave height was reasonably accurate at most tide stations on Honshu Island (Figure 9c). The good predictions of the coastal tsunamis on Honshu Island are attributed to the observation of the positive peak amplitude of the tsunami at TM1 (Figure 9a). The peak amplitude, even if observed at only a single OBTM, helps to constrain the forecast magnitude of the tsunami. On the other hand, there is no significant improvement of the accuracy score after 15 min, which corresponds to the arrival of the peak amplitude at station TM2 (Figure 7a).
 This numerical simulation shows that the amplitudes and arrival times of the peak of the tsunami observed at the offshore stations provide the most important information for accuracy of the forecasts. The contribution of the later part of the first wave of the offshore tsunami records is less substantial. In this simulation, the observed amplitudes at stations PG1 and PG2 were very small and made only a small contribution to the forecast.
 Our simulation has shown that if the early tsunami forecasts are inaccurate, the accuracy improves as time elapses and very accurate forecasts are possible before the earliest arrival of a tsunami at the coast. These results demonstrate that the tsunami forecasting method presented here can provide reliable tsunami predictions for earthquakes such as the 1896 Sanriku tsunami earthquake by using tsunami waveform data from the four existing OBTMs.
5. Application to Real Tsunami Data
 To investigate the validity of our forecasting method for real tsunami data containing noise from various sources (e.g., ocean tides, oceanographic phenomena, seismic waves, etc.), we applied the method to a small tsunami that was generated by the 2003 Off-Fukushima earthquake (Figure 1). We made forecasts by using tsunami records at TM1 and TM2 as input data for the inversion. Because the tsunami was small, it was observed at only a few tide stations, so we compared our tsunami predictions with observations at the three coastal tide stations (MY, KM, OF).
Figure 10 shows the forecasts made 20, 25, and 30 min after the earthquake. Table 1 shows a comparison of the observed and predicted tsunami arrival times and amplitudes of the first peaks at each tide station. In this case, we defined the tsunami arrival time at a tide station as the time at which the sea level change caused by the tsunami first appeared in the tide records. The earliest coastal arrival time was at tide station OF (and the other tide station AY, indicated in Figure 1), 37 min after the earthquake.
Table 1. Results of Tsunami Forecast for the 2003 Off-Fukushima Earthquake
Arrival Time, T (min)
Amplitude of the First Peak (cm)
Forecast at T = 25 min
Forecast at T = 30 min
Forecast at T = 25 min
Forecast at T = 30 min
 At 20 min, the tsunami had not reached either TM1 or TM2 (Figure 10a). The amount of sea-surface displacement was not substantial at this time (Figure 10b) and the amplitudes of the predicted coastal tsunami waveforms were very small (Figure 10c). This indicates that false alarms are unlikely when our tsunami forecasting calculation is applied to observational noise in OBTM records.
 At 25 min, the first peak of the tsunami had been observed at both TM1 and TM2 (Figure 10d). A substantial sea-surface displacement (∼4 cm) was estimated in the area around the earthquake epicenter (Figure 10e). The predicted tsunami waveforms at tide stations MY and KM matched the observations within 3 min for arrival times and within a factor of 1.5 for first peak amplitudes (Table 1). At station OF, the first peak amplitude of the predicted tsunami was less than a third of that of the observed tsunami (Figure 10f).
 At 30 min (Figures 10g–10i), one full wavelength of the tsunami had been observed at both OBTM stations. As shown in Table 1, the tsunami predictions were almost the same as those made at 25 min. At tide stations MY and KM, the amplitudes of the first peaks and arrival times were well estimated; at tide station OF, the estimated tsunami amplitude was a quarter of the observed amplitude. The lack of improved accuracy of the forecasts after 25 min can in this case be explained by saturation of the forecasting accuracy as we saw in the numerical simulation (Figure 8). The peak tsunami amplitudes were observed about 20 and 25 min after the origin time at TM1 and TM2, respectively (Figure 1); therefore, the accuracy of the forecasts was not greatly improved beyond 25 min after the earthquake (Table 1).
 Although we used the tsunami waveforms from only two OBTMs for our forecasts, the method provided accurate tsunami predictions at two of the three tide stations. As the accurate predictions obtained at 25 min preceded the earliest tsunami arrival time at the tide stations by 11 min, they provided timely and accurate information that could be used for tsunami warnings. The underestimation of the tsunami amplitude at tide station OF may have been related to harbor response, or the directivity of tsunami energy radiation [Kajiura, 1970]. The former problem might be overcome by recalculating the Green's functions for this station using more detailed bathymetric data than we used in this study. The latter issue is discussed later. Although there remains potential for adjustment and improvement to our tsunami forecasting method, this case study demonstrates that it can provide reliable arrival times and amplitudes of coastal tsunamis before they reach the coast.
6.1. Contribution of Our Method to Tsunami Disaster Mitigation
 We have shown that our forecasting method would have worked well for the 1896 Sanriku tsunami earthquake by carrying out the simulation in which uniform slip distribution was assumed and proxies were used for the OBTM records. In this section, we discuss how the present tsunami warning system in Japan can be improved by incorporating tsunami data recorded offshore. We use the 1896 Sanriku tsunami earthquake as a theoretical example.
 The tsunami warning system currently used by JMA is based on rapid analysis of seismic observations [Tatehata, 1997]. The surface wave magnitude (Ms) of the 1896 Sanriku earthquake was only 7.2 [Abe, 1994], which means that the current JMA method would have predicted a tsunami wave height of about 2 m. Although this prediction can be obtained relatively quickly, within 3 min of the earthquake, the tsunami wave height is grossly underestimated compared to the coastal tsunami amplitudes used in our numerical experiment. Because of the underestimation of tsunami amplitudes, the accuracy score for the JMA forecast (Figure 8) is less than 30%. In Japan today, the score would improve because much work has been done to establish rapid Mw estimates [e.g., Tsuboi, 2000] which would be much better than Ms. Even if such magnitudes provide no improvement with the forecasting accuracy, the information of the JMA forecast would have been available only 3 min after the earthquake. For the period up to about 7 min after the earthquake, the accuracy score for the JMA forecast is higher than that of our method. On the other hand, the current tsunami warning system does not provide updated predictions.
 Combining our forecasting method with the current JMA method would greatly improve tsunami forecasts. For a tsunamigenic earthquake, this would allow a tsunami warning based on seismic data (the JMA method) to be issued around 3 min after the earthquake, thus taking advantage of the rapid availability of the seismic data. This initial warning could be updated by a second more accurate warning based on the real-time offshore tsunami observation data of our method. It is important to note that the underestimated tsunami amplitudes derived from the seismic data are available much earlier than the time at which the tsunami reached the coast. At 11 min, the more accurate forecast would be available and still early enough to urge people in Kamaishi (KM; the nearest city to the earthquake source) to evacuate around 8 min before the first arrival of the tsunami 20 min after the earthquake. Considering that the arrival of the peak amplitude of the tsunami would be 35 min after the earthquake, people would have nearly 20 min to reach safety before the maximum seawater level was reached.
 As discussed in section 4.1, the 1896 Sanriku earthquake was not a typical tsunamigenic earthquake. Improvement of tsunami forecasting based on offshore observations of tsunami data is also important for typical tsunamigenic earthquakes. For typical events, the seismic-data-oriented method can provide accurate tsunami information, but that accuracy cannot be evaluated until the tsunami has been observed at the coast. The tsunami information obtained by our method could be used to confirm the accuracy of tsunami predictions obtained by the JMA method. The confirmed, or if necessary adjusted, tsunami information would encourage people to respond appropriately to tsunami warnings.
6.2. Scenario for a Tsunami Source at a Different Location
 Future disastrous tsunamigenic earthquakes are unlikely to occur at exactly the same locations as past events such as the 1896 Sanriku earthquake. Thus, it is important to understand how the reliability of the tsunami forecasting changes for different source locations. We used the same procedure as that described in section 4.2 to carry out a numerical simulation for an earthquake at a different location.
 The assumed fault model for this simulation was the same as that used in the previous one, except that its location was moved 1 arc degree to the north of the 1896 Sanriku earthquake location. We used the initial sea-surface displacement field due to the assumed fault slip (Figure 11) to compute synthetic observations. We used the synthetic observations at the four OBTMs as input data for the inversion and we used tsunami waveforms at 14 coastal tide stations to evaluate the forecast. The earliest arrival time of the tsunami at the coastal tide stations was again at station KM, but 25 min after the earthquake.
 The forecast 20 min after the earthquake (Figure 12) shows that one full wavelength of the tsunami had been observed at all OBTMs at this time. This means that the accuracy of the tsunami forecast had already reached its maximum. In fact, the accuracy score reached its maximum of about 50% at around 15 min (Figure 13). The arrival times of the tsunami were well estimated at all tide stations on the both islands (Figures 12c and 12d). The first peak amplitudes of the predictions also match those of the observations at some tide stations (KM, OF, KN, IS, HN, AK, KS). At several stations (HCM, HCS, MY, HR, UR, TKE, TKW), however, the predicted amplitudes are less than half the observed amplitudes. This underestimation of the coastal tsunami amplitude resulted in a low-accuracy score (Figure 13). This experiment shows that the accuracy of forecasts of coastal tsunami amplitudes by our method can vary, depending on the relationship between the location of the tsunami source and the locations of offshore tsunami observation points.
6.2.2. Improvement of Forecast Accuracy by Deploying More OBTMs
 Our underestimation of coastal tsunami amplitudes can be explained by the azimuthal dependence of tsunami energy radiation. According to Kajiura , for a tsunami source of aspect ratio greater than one, most of the tsunami energy is radiated perpendicular to the major axis of the source. In our study, the major axis of the source was parallel to the Japan Trench and much larger than the minor axis (see Figure 11). Thus, the tsunami energy would be strongly radiated perpendicular to the Japan Trench, that is, westward. Consequently, the tsunamis that reached the coastal tide stations west of the source would be larger than those at stations farther to the north or south. However, the OBTMs are located almost along the strike direction of the trench, the direction in which weaker amplitude tsunami waves are radiated. This means that the data input to our forecasting scheme lacked information about the maximum tsunami energy generated by the earthquake. Thus, the initial sea-surface displacement was underestimated (Figure 12b) and led to the substantial underestimation of the tsunami amplitude at several tide stations, especially those west of the source.
 The underestimation of the tsunami amplitude at tide station OF in the forecast for the 2003 Off-Fukushima earthquake (see section 5) may also be a function of the pattern of energy radiation from the source. According to the Global Central Moment Tensor catalog (http://www.globalcmt.org), the strike angle of the fault that generated the 2003 Off-Fukushima earthquake was 196°. The seismological scaling law of Geller  indicates that the major axis length (length of the fault) of the source is usually twice the minor axis length (width of the fault). Thus, the largest tsunami wave generated by the 2003 earthquake would have propagated northwestward from the source toward coastal tide station OF. There is no OBTM on the path of the tsunami from its source to station OF, so the maximum tsunami data would not have been recorded and incorporated in our forecast.
 Because the underestimation is caused by a lack of observations of large amplitude tsunami energy radiated normal to the strike direction of the fault, the simplest way to solve the problem is to improve the azimuthal coverage of the stations by deploying more OBTMs along the trench. If the OBTMs were deployed with spacing comparable to the length of the major axis of the source, the strongly radiated tsunami energy would be detected at least one of them, regardless of the location of the source. To investigate whether the underestimation problem can be solved with such an OBTM array, we performed another numerical simulation with two additional cabled observation pairs deployed between the existing pairs (TM3, TM4, TM5, and TM6; Figure 14b) with each cable separated by about 200 km; this distance is equivalent to the major axis length of the tsunami source discussed here.
 The forecast made 20 min after the earthquake (Figure 14) shows that strongly radiated tsunami energy is observed at the additional OBTMs TM5 and TM6 (Figure 14a). When we used these observations for the source inversion, the estimated sea-surface displacement (Figure 14b) was closer to that caused by the assumed fault motion than in the experiment using only the data from the original four OBTM stations. Because the tsunami source was better estimated, the accuracy of the coastal tsunami predictions was considerably improved. For the coastal tide stations where tsunami amplitudes were underestimated in the previous test with fewer OBTMs (i.e., HCM, HCS, MY, HR, UR, TKE, TKW), the predicted coastal tsunami amplitudes now match very well the observed amplitudes (Figures 14c and 14d). This improvement is confirmed by the accuracy score (Figure 13): the maximum value of the accuracy score increased from ∼50% to ∼90%.
 To evaluate the contribution of the additional OBTM arrays to the estimation of tsunami source, we introduced the recovery index R, which measures how well the assumed initial sea-surface displacement is recovered by the inversion of the current simulation as follows:
where aiass is the assumed initial sea-surface displacement of the jth sea-surface segment, and aiest is the estimated sea-surface displacement by the inversion. A large value of R indicates that the displacement of a sea-surface segment is well retrieved and R of 100% indicates complete estimation.
 In Figure 15, the spatial variations of R for two cases, with the additonal OBTMs data and without them, are compared. We calculated the R values for the sea-surface segments within the rectangular area with substantially large sea-surface displacement. The results of the inversion performed 20 min after the earthquake (Figures 12b and 14b) were used to calculate R. Without the additional OBTMs, the recovery index is less than 50% in almost entire part of the source region (Figure 15a), indicating that it is difficult to estimate the initial tsunami height accurately from the records of the existing OBTMs. After adding the OBTM arrays, significant improvements of R in the source region as shown in Figure 15b, demonstrating that the additional OBTMs help significantly for reliable tsunami forecasting through the improvement in the initial wave height estimation.
6.2.3. Suggestions for Improving Forecast Accuracy Without Deploying Additional OBTMs
 Increasing the density of OBTM stations (with an appropriate spatial distribution) is one way to overcome the problem of inaccurate tsunami forecasts related to the spatial relationship between the tsunami source and OBTMs. But it is difficult to deploy a dense cabled tsunami observation network. Here we propose an alternative solution.
 The directivity of tsunami energy is determined by the different lengths of the major and minor axes of the tsunami source. Therefore, this problem might be solved if we can introduce information about the aspect ratio of the tsunami source into the forecasting scheme. A tsunami waveform inversion for slip distribution on an assumed fault plane may be an effective way to constrain directivity. NOAA's PMEL succeeded in forecasting far-field tsunamis by inverting tsunami data for slip distribution based on assumed fault geometry, although the distribution of the buoys of the DART system is not everywhere optimal; they are mostly distributed along strike direction in trench systems. Consequently, many of them record tsunami waveforms of less than the maximum amplitude [Titov et al., 2005]. The PMEL results support the view that a fault slip inversion is an effective way to overcome the difficulties inherent in poor station coverage. However, tsunami forecasts based on assumed fault geometry may break down when the source of the tsunami and the assumed fault geometry differ.
 As long as the limitations of the present distribution of offshore tsunami observation stations remains, it may be practical to simultaneously undertake two tsunami inversions: one for slip distribution based on assumed fault geometry, and the other for sea-surface displacement distribution. Both inversions can be completed quickly and the forward calculation of coastal tsunami waveforms can be made immediately after source estimations are completed.
6.3. Scenarios for Tsunamis due to Nonuniform Fault Slips
 In the previous numerical simulations, we assumed uniform distribution of fault slip. In realistic situations, however, coseismic slip would have nonuniform distribution, especially for large earthquakes [e.g., Piatanesi and Lorito, 2007]. The rupture complexity of earthquakes affects the near-field tsunami [Geist, 2002]. Therefore, it is important to know how well initial sea-surface displacement due to nonuniform slip distribution can be estimated by our inversion. We used the same procedure as that described in section 4.2 to perform numerical simulations for earthquakes whose slip distributions are nonuniform.
 We assumed two fault models which were the same as that of the 1896 Sanriku earthquake, except for the slip distribution: different amount of fault slip were assumed for the northern and the southern halves. The slip amounts were assumed to be 6 and 3 m for the two subfaults and we performed two numerical simulations by changing the slip distribution: the larger slip on the southern subfault (case A) and the smaller slip in the southern half (case B). We used the two different initial sea-surface deformation patterns (Figures 16a and 17a) to compute the synthetic observations for the two cases. In both cases, the synthetic observations at the existing four OBTMs were fed into the inversion and those at 14 coastal tide stations were used for evaluating the accuracy of the forecasts.
 In the case A, the initial wave height distribution estimated 20 min after the earthquake (Figure 16b) coincides well to the assumed pattern in the southern part of the source region. This feature is displayed more clearly in the distribution of the recovery index R (Figure 16c): R is more than 50% in the southern subfault. On the other hand, R in the northern subfault is much lower. This spatial pattern of R would be caused by the spatial relationship between the tsunami source and the OBTMs as explained in section 6.2. The strong tsunami energy from the southern subfault is radiated westward and observed at TM1 and TM2. Owing to the observations of large tsunami, large R is obtained for the southern subfault in the case A, and accordingly, large amplitude tsunamis at the coastal sites are successfully forecasted.
 In the case B, the spatial pattern of R (Figure 17c) is almost the same as that in the case A: R is larger in the southern part than that in the northern part of the assumed source area. In this case, however, low R in the northern subfault resulted in poor forecasting in large coastal tsunami because the large initial tsunami height was not estimated correctly. In spite of this, the score of forecasting accuracy defined by equation (12) is good (∼80%) as shown in Figure 18. This discrepancy was caused by the inhomogeneous distribution of the coastal tide stations used for the score calculation. In the case B, the large tsunami radiated from the northern part is not observed by the OBTMs and the initial sea-surface displacement in the northern part is underestimated as shown in Figures 17b and 17c. This makes the forecasting scores low at tide stations HCM and HCS, located to the west of the northern subfault emitting large tsunami (Figure 18). At those stations, observed tsunami amplitudes are expected to be 1 to 3 m in the first peak amplitudes. However, the score shown in Figure 18 was obtained as the value averaged over the entire costal stations and the misfits at the two stations, HCM and HCS may not affect the averaged score substantially. Considering the expected tsunami disaster caused by the large amplitude tsunami at stations HCM and HCS, the improvement of the forecasting accuracy would be needed.
 These simulations demonstrate that, when the coseismic slip distribution is ununiform, the forecasting accuracy depends on the spatial relationship between the location where fault slip is large and the locations of OBTMs. The cause of low forecasting accuracy at two stations in the case B of the current simulations is lack of the observations of large tsunami by the OBTMs as well as in the section 6.2. Therefore, the suggestion mentioned in sections 6.2.2 and 6.2.3 is also applicable for improving the forecasting accuracy in the case of tsunamigenic earthquakes with heterogeneous slip distribution.
6.4. Tsunami Forecast When an OBTM is Located Within the Tsunami Source Region
 The largest tsunamigenic earthquake occurring after the current cabled observations systems has been deployed in the study area (Figure 1a) is the 2003 Tokachi-oki earthquake (M 8.0). This interplate underthrust earthquake occurred at the southern coast of Hokkaido Island and generated large tsunami. The largest tsunami runup height was 4 m [Tanioka et al., 2004b] and caused damages along the coasts of Hokkaido and Honshu islands. The tsunamis were well recorded at the four OBTMs in this study area (TM1, TM2, PG1 and PG2) [Tanioka et al., 2004a; Mikada et al., 2006]. Considering its magnitude, it is important to investigate whether our forecasting scheme works well for the Tokachi-oki event. However, it is difficult to directly apply our current forecasting scheme to the event because of its exceptional circumstance. According to Mikada et al. , the large pressure variation (equivalent to the sea level change of 0.1–0.4 m) caused by the static seafloor deformation due to the fault motion of the main shock was contained in the records of the OBTMs (PG1 and PG2) located within the tsunami source region, as well as the tsunami signals. However, the effect of such seafloor deformation on the OBTM records is not taken into consideration in our current procedure. To forecast tsunamis from the OBTM data in such exceptional cases as the 2003 Tokachi-oki tsunami, another improvement will be required.
 We developed a method for near-field tsunami forecasting based on real-time processing of offshore tsunami observation data. By virtue of the early and accurate determination of the size of a tsunami, the method can provide reliable estimates of arrival times and amplitudes of coastal tsunamis, even for tsunami earthquakes. The results of numerical simulations of the disastrous 1896 Sanriku tsunami earthquake demonstrate that our method can provide reliable tsunami predictions for the Pacific coastal area of northeastern Japan about 5 min or more before the earliest arrival of the tsunami at the coast.
 We also applied our forecast method to real tsunami data that included observational noise. The arrival times and amplitudes of the tsunami at the coast were successfully estimated from the small (∼2 cm) amplitude tsunami data recorded at OBTMs from a local M 6.8 earthquake. Although the tsunami amplitudes recorded at coastal tide stations were only about 10 cm, our predictions matched the observations at two out of three tide stations. This demonstrates that our method reliably forecasts near-field tsunamis from offshore observed tsunami data.
 Our method updates tsunami forecasts at short regular intervals from offshore tsunami data acquired in real time. These updates successively improve the accuracy of the forecast. Integration of our method with the current JMA tsunami warning system, which is based on seismic observations, will provide the residents of coastal areas of Japan with reliable tsunami information that will help them make rational decisions about evacuation.
 We found that the accuracy of forecasts of coastal tsunami amplitude can be affected by the spatial relationship between the tsunami source and the OBTMs. Our numerical simulation showed that the effect of the radiation pattern of the tsunami source on forecast accuracy would be considerably reduced by deploying additional OBTMs separated by a distance equivalent to the trench-parallel length of the tsunami source; this distance for northeastern Japan is roughly 200 km, at most.
 We thank A. Hasegawa, T. Matsuzawa, S. Miura, M. Kido, T. Okada, T. Baba, Y. Osada, M. Nishino, Y. Ito, T. Iinuma, Y. Yamamoto, and I. Abe for useful discussions and suggestions. We also thank T. Kanazawa and S. Sakai for allowing us to use tsunami data observed by the cabled ocean bottom pressure sensors of the observatory off Kamaishi as well as Japan Meteorological Agency and Japan Coast Guard for providing tsunami data observed at tide stations. We are grateful to two anonymous reviewers for their critical and constructive comments and suggestions which improved the manuscript. This work was promoted by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists, and the 21st COE program of Tohoku University. The figures in this paper were prepared using Generic Mapping Tools (GMT) [Wessel and Smith, 1991].