 Article
 Open Access
Empirical relationship of tsunami height between offshore and coastal stations
 Yutaka Hayashi^{14}
https://doi.org/10.5047/eps.2009.11.006
© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB. 2010
 Received: 1 July 2009
 Accepted: 11 November 2009
 Published: 4 March 2010
Abstract
This study compares tsunami height data obtained by coastal tidal stations and offshore wave stations of the Nationwide Ocean Wave information network for Ports and HArbourS (NOWPHAS) with data obtained from realtime kinematic global positioning system (RTKGPS) buoys. NOWPHAS wave stations and RTKGPS buoys are typically installed off the coast—the former within several kilometers of the coast and the latter 2–20 km offshore. The ratio of initial tsunami height observed at a coastal tidal station to that observed at an offshore station was found to be approximately proportional to the fourth root of the ratio of the seabottom depths to the mean sea level at the respective offshore and coastal station. This approximation can also be applied to maximum tsunami amplitudes. The relationship derived in this paper will enable the initial tsunami height to be forecast by realtime applications using detected tsunami initial height from offshore stations, such as the seabottom pressure gauges of NOWPHAS stations and RTKGPS buoys.
Key words
 Green’s law
 NOWPHAS
 realtime tsunami forecast
 RTKGPS buoy
1. Introduction
Although “sunami” etymologically means “harbor wave”, there have been many occasions in the very recent past to observe tsunamis outside harbors. In the coastal waters off Japan, such opportunities have been made possible through the use of cabled oceanbottom pressure gauges in the deep ocean (e.g., Seismology and Volcanology Research Division, Meteorological Research Institute, 1980), wave stations of the Nationwide Ocean Wave information network for Ports and HArbourS (NOWPHAS) (e.g., Nagai, 1998), and realtime kinematic global positioning system (RTKGPS) buoys (e.g., Kato et al., 2000). NOWPHAS has recently incorporated six RTKGPS buoy observatories into its network, but these have not yet detected a tsunami. Consequently, in this paper, “NOWPHAS wave station” indicates a wave station equipped with a sensors other than a GPS buoy. Tsunamis in the open ocean can potentially be detected both by buoys of the U.S. Deepocean Assessment and Reporting of Tsunamis (DART) array (e.g., González et al., 2005) and by altimeters installed on satellites (e.g., Okal et al., 1999; Gower, 2005).
At the present time, a tsunami early warning issued by the Japan Meteorological Agency (JMA; Tatehata, 1997) is based on the seismic information available just after an earthquake and the precomputed tsunami scenario database. However, it is believed that the accuracy of tsunami prediction may be significantly improved if tsunami early warnings could utilize realtime offshore tsunami observation data (e.g. Tsushima et al., 2009). Such an improvement could potentially contribute to tsunami disaster mitigation.
Within this framework, in 2009, the Headquarters for Earthquake Research Promotion (HERP), a special governmental organization of Japan, drew up the report entitled “The next Promotion of Earthquake Research—Comprehensive Basic Policies for the Promotion of Earthquake Research through the Observation, Survey, and Research” (HERP, 2009). This report establishes “Upgrading tsunami prediction technology by realtime utilization of seaarea tsunami data and/or sophisticated tsunami source modeling” as one of the basic earthquake research targets to be addressed in the upcoming 10 years.
RTKGPS buoys, which are installed 2–20 km off the coast of Japan, have only sampled two tsunami events to date for which the clarity of the data was sufficient data for use in assessing tsunami height: the 2003 Tokachioki earthquake tsunami (Nagai and Ogawa, 2004) and the 2004 earthquake tsunami off the Kii peninsula (Kato et al., 2005; Nagai and Satomi, 2005). Due to this limited amount of data and sampled tsunami events, it is therefore impossible to directly discuss the relationship between observed tsunami height as recorded by the RTKGPS buoys and coastal tidal gauges. In contrast, NOWPHAS wave stations have recorded a significant amount of tsunami data since Takayama et al. (1994) first reported the detection of a tsunami using data from these stations—the 1993 southwest Hokkaido earthquake tsunami. However, no general formula between offshore tsunami observation data acquired by NOWPHAS wave stations and those acquired by coastal tidal stations has been developed.
Here, we present a formula explaining the relationship between tsunami height data recorded by offshore NOWPHAS wave stations or RTKGPS buoys and coastal tidal stations in Japan.
2. Offshore and Coastal Tsunami Data
Offshore and coastal observatories.

Tsunami events.

3. Experimental Equation Relating Offshore and Coastal Tsunami Height
One equation derived from Green’s law was tested to determine if it could explain the observed data on initial tsunami heights and maximum tsunami amplitudes that were retrieved as described in Section 2.
The assumption that the initial tsunami height should be partially affected by the reflected wave was based on the following evidence recorded by both an offshore NOWPHAS seabottom pressure gauge and a coastal tide gauge at Tomakomai (no. 2 in Fig. 1 and Table 1) located in southern Hokkaido during the 2003 Tokachioki earthquake tsunami (Nagai and Ogawa, 2004). As shown in Fig. 2, the crest of the initial tsunami wave at the NOWPHAS offshore wave station appeared after the arrival of tsunami recorded at the nearby coastal tide gauges.
The seabottom depth is obtained by reading the average water depth within a certain distance (r) from a coastal tidal station along a waterway and using the largestscaled sea charts (mostly 1:10,000) available. In this study, r = 100 m is used, and the average water depth read by the charts is indicated in the Table 1.
4. Discussion
4.1 Accuracy of tsunami height correction
Proportional constants (α) are 1.065 for initial tsunami height and 1.170 for maximum tsunami amplitude (Eqs. (6) and (7)). The standard deviation of the logscaled residual error of initial tsunami heights and maximum tsunami amplitudes are 0.135 and 0.169, respectively (Fig. 3). Here, we discuss the accuracy of forecasting initial tsunami heights and maximum tsunami amplitudes at coastal tidal stations, when a tsunami at a coast is forecast by applying Eqs. (6) and (7) to offshore tsunami data.
If the range of the abovementioned standard deviation of the logscaled residual error is considered, the initial tsunami height at a coastal tidal station is likely to be within 10^{0.135} (73%) to 10^{0.135} (136%) of the corrected tsunami height calculated by Eq. (6) with offshore tsunami data. If a range of twice the standard deviation is considered, then most of the initial tsunami height at a coastal tidal station can be expected to be within 54–186% of the corrected tsunami value. In the same way, if the range of the standard deviation (or twice the standard deviation) is considered, the maximum tsunami amplitudes at a coastal tidal station is likely to be within 68–148% (or 46–218%) of the corrected tsunami amplitude calculated by Eq. (7) with offshore tsunami data.
Thus, the formulas derived in this paper (Eq. (6)) can correct initial tsunami height data at offshore NOWPHAS wave stations and RTKGPS buoys so that the data can be treated in the same way as data recorded at a nearby coastal tidal station. This correction brings the difference between the two data sets to approximately 30%. If the height of the offshore tsunami at these offshore observatories can be obtained in realtime, tsunami height at the nearby coasts can potentially be forecasted with equal accuracy by applying the formula derived in this paper to the obtained offshore data.
4.2 Dependency of definition of h_{c}
In Section 3, the depth at a coastal site (h_{c}) is defined as the average depth of the seabottom to mean sea level within r (= 100 m) from the site along the waterway. However, it is not selfevident that the average depth within 100 m can represent the depth near the coast or that it can be applied to Eq. (4) derived from Green’s law because the typical tsunami wavelength near a coast exceeds the kilometer order. In order to check whether parameter r, which is necessary in the definition of h_{c}, is appropriate, the proportional constants (α) and standard deviations of logscaled residual errors (σ) are calculated by using various r (= 50, 100, 200, 400, 600, 800 m).
4.3 Application to tsunami data by RTKGPS buoys
RTKGPS buoys have to date sampled only two tsunami events for which they have recorded data with sufficient clarity to use in a discussion of tsunami height: the 2003 Tokachioki earthquake tsunami at the Off Muroto GPS buoy, which is 2 km off the coast (Nagai and Ogawa, 2004), and the 2004 earthquake tsunami off Kii peninsula at the Off Muroto GPS buoy, which is 13 km off the coast (Kato et al., 2005; Nagai and Satomi, 2005). Although RTKGPS buoys are generally installed further offshore than NOWPHAS wave stations, as indicated in Fig. 3, tsunami data obtained by these two systems seem to fit the same regression lines for correcting both the initial tsunami height and maximum amplitude data recorded by the RTKGPS buoys. This correlation suggests that Eqs. (6) and (7) can be used to correct tsunami observation data recorded not only by NOWPHAS wave stations but also by RTKGPS buoys, with the aim of comparing or predicting initial tsunami heights or maximum tsunami amplitudes at nearby tidal stations. If the initial tsunami height from RTKGPS buoys can be detected in realtime, the realtime application of Eq. (6) enables us to forecast initial tsunami height at the nearby coast. Such an advanced tsunami forecast may probably be achieved in the near future because RTKGPS buoys have been adopted as a new sensor for NOWPHAS and have been installed gradually since 2007.
5. Summary
Initial tsunami height and maximum tsunami amplitude data obtained by coastal tidal stations and offshore sites of NOWPHAS wave stations or RTKGPS buoys (Fig. 1; Table 1) were retrieved from technical reports published by PARI or JMA (Table 3). Pairs of tsunami data observed at offshore observatories and nearby coastal stations during eight tsunami events (Tables 2 and 3) were compared (Section 3). The ratio of initial tsunami height or maximum amplitude observed at a coastal tidal station to that at the offshore site was found to be proportional to the fourth root of the ratio of the seabottom depths from the mean sea level at the offshore sites to the coastal station (Eqs. (6), (7); Fig. 3). Based on the standard deviations of the logscaled residual errors (Fig. 4(b)) of Eqs. (6) and (7), differences between the initial tsunami height or maximum tsunami amplitude corrected by these equations and the observed value at the nearby coastal tidal station are expected to be on the order of about 30% (Section 4.1). If offshore NOWPHAS wave stations or RTKGPS buoys successfully detect initial tsunami height in real time, the relationship (Eq. (6)) derived in this paper enables us to realize realtime forecasting of initial tsunami height at a nearby coast with high accuracy.
Declarations
Acknowledgments
Thanks are due to three anonymous reviewiers for their valuable comments and helpful suggestions. This work was supported in part by Japan Society for the Promotion of Science (GrantinAid for Scientific Research No. 20510173). Some of the figures were prepared using the Generic Mapping Tools (Wessel and Smith, 1998). I gratefully acknowledge helpful discussions with Dr. K. Hirata (Meteorological Research Institute), Mr. Y. Hasegawa, Mr. Y. Nishimae, and Mr. K. Nakata (Japan Meteorological Agency) on several points in the paper.
Authors’ Affiliations
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