Effects of Tides on Maximum Tsunami Wave Heights: Probability Distributions*

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HAROLD O. MOFJELD,FRANK I. GONZÁLEZ,VASILY V. TITOV,ANGIE J. VENTURATO, AND JEAN C. NEWMAN NOAA/Pacific Marine Environmental Laboratory, Seattle, Washington

(Manuscript received 29 November 2004, in final form 26 April 2006)

ABSTRACT

A theoretical study was carried out to understand how the probability distribution for maximum wave ␩ heights ( m) during tsunamis depends on the initial tsunami amplitude (A) and the tides. It was assumed that the total wave height is the linear sum of the tides and tsunami time series in which the latter is decaying exponentially in amplitude with an e-folding time of 2.0 days, based on the behavior of observed Pacific- wide tsunamis. Direct computations were made to determine the statistics of maximum height for a suite of different arrival times and initial tsunami amplitudes. Using predicted tides for 1992 when the lunar nodal f factors were near unity during the present National Tidal Datum Epoch 1983–2001, the results show that when A is small compared with the tidal range the probability density function (PDF) of the difference ␩ Ϫ ␩ Ϫ m A is closely confined in height near mean higher high water (MHHW). The m A PDF spreads in ␩ Ϫ height and its mean height o A decreases, approaching the PDF of the tides and MSL, respectively, when A becomes large compared with the tidal range. A Gaussian form is found to be a close approximation to ␩ Ϫ the m A PDF over much of the amplitude range; associated parameters for 30 coastal stations along the U.S. West Coast, Alaska, and Hawaii are given in the paper. The formula should prove useful in probabilistic mapping of coastal tsunami flooding.

1. Introduction tude tsunami wave is not the first one but occurs later in the wave train. When studying past tsunamis and mak- The tides can have a major effect on the maximum ing probabilistic height forecasts for future ones, it is wave heights experienced during a tsunami. By wave therefore essential to take the tides into account. height, we mean the total wave height (tsunami plus Observations have shown that Pacific-wide tsunamis tide) relative to a fixed reference level, such as mean form long wave trains that persist over several tidal lower low water (MLLW). Even when the first wave in cycles in which the envelopes encompassing the tsu- a tsunami wave train striking a coastal location has the nami energy and successive peak amplitudes decay ex- largest amplitude (height of the wave peak relative to ponentially in time (Miller et al. 1962; Van Dorn 1984, the background water level at the time of the peak), a 1987; Mofjeld et al. 2000). Mofjeld et al. (1997) used higher tide may combine with a smaller amplitude tsu- this fact to develop a short-term tsunami forecasting nami wave in the same wave train to produce a greater scheme for the total wave heights of later waves in net wave height. Whether this occurs depends on the tsunami wave trains. To better understand the influ- amplitudes of the successive waves in the tsunami wave ence of tides on the maximum tsunami wave heights train and the height of the tide at the time of each wave from a probabilistic point of view, the present theoret- peak. Also, it sometimes occurs that the largest ampli- ical study was carried out in which analytic tsunami wave trains that decay exponentially in time are com- * NOAA/Pacific Marine Environmental Laboratory Contribu- bined linearly with tidal time series. By varying the tion Number 2768. arrival time of the wave trains sequentially through a long, representative time series of the tides at a given location, it is then possible to generate probability den- Corresponding author address: Dr. Harold O. Mofjeld, NOAA/ Pacific Marine Environmental Laboratory, 7600 Sand Point Way sity functions (PDFs) that characterize the behavior of NE, Seattle, WA 98115-6349. the maximum wave height as a function of the initial E-mail: [email protected] tsunami amplitude and the local tides. This is very simi-

DOI: 10.1175/JTECH1955.1

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JTECH1955 118 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 24 lar to the method used by Houston and Garcia (1978) in their statistical study of tsunamis along the U.S. West Coast. Of particular interest in the present study are where the maximum wave heights tend to cluster in elevation above MLLW and how the spread in elevation of the PDFs depend on the initial tsunami amplitude relative to the tidal range. It is found that a simple Gaussian form fits the PDFs reasonably well. Furthermore, modified exponential laws are found to closely approximate the dependences of the PDF mean heights and standard deviations on the initial tsunami amplitude; these two parameters are sufficient to quantitatively determine the Gaussian PDF. The coefficients in the exponential laws provide a concise characterization of the tidal effects on the maximum tsunami heights for a given location. In the next section, theoretical tsunamis are superimposed on predicted tides on the open coast at Sea- side, Oregon. These mixed semidiurnal tides are typical FIG. 1. Example of a water level time series consisting of a of those along the U.S. West Coast. This serves to il- theoretical tsunami adding to predicted tides at Seaside, OR. The lustrate the influence of the tides on the maximum of tsunami arrives at 0000 UTC 2 Jan 2004 and then decays expo- the total wave height (tsunami wave plus tide) for wave nentially in time after the first wave. At the time resolution trains of various initial amplitudes. This height is la- shown, the rapidly oscillating (20-min period) time series appears as a solid distribution between decaying envelopes. beled by the arrival time of the first tsunami wave peak so that time series (loci) of maximum wave heights can be plotted as functions of the arrival time. In section 3, (NOAA), and the U.S. Geological Survey (USGS). De- it is convenient to use the differences between these caying exponentially in amplitude, the maximum tsu- heights and the initial tsunami amplitude in order to nami height (4.91 m above MLLW) for this event then compare PDFs when the initial amplitude is varied. occurs at 1604 UTC, the next higher high water. The Subsequent sections justify the modified exponential use of an exponentially decaying envelope to charac- laws for the coefficients in the Gaussian approximation terize the maximum wave heights along sections of tsu- to the tsunami PDFs and present values of these coef- nami time series is justified by the case studies and ficients for 29 U.S. Pacific tide stations plus the Seaside stochastic modeling of Mofjeld et al. (1997, 2000). In location. The discussion addresses the influence of the Fig. 1 as well as elsewhere in the study, the decay co- 18.6-yr nodal cycle of the tides, the insensitivity to using efficient (␶ ϭ 2.0 days) is set by observations of Pacific- observed or predicted tides for fitting the coefficients, wide tsunamis (e.g., Mofjeld et al. 2000). The predicted and limiting the application of the theory to situations tides are based on 37 harmonic constants, where those where nonlinear interactions between tsunamis and the for O1, K1, N2, M2, and S2 are from the Eastern North tides can be neglected. Pacific 2003 (ENPAC 2003) tide model (for details, see Spargo 2003; Mofjeld et al. 2004a); the others are in- 2. Tsunami time series ferred from observed relationships at South Beach, Or- Figure 1 shows a typical tsunami time series used in egon (44°37.5ЈN, 124°02.6ЈW). this study. The wave period of 20 min is within Moving the tsunami arrival time to progressively the middle range (10–40 min) of major transpacific later times then generates time series for the maximum tsunamis striking the U.S. West Coast. The theoretical wave height as functions of tsunami amplitude and ar- tsunami arrives at Seaside, Oregon (46Љ00.1°N, rival time. Figure 2 shows the maximum wave height for 123Љ55.7°W), at 0000 UTC 2 January 2004. The tides at successive time series when arrival time is moved for- Seaside are typical of those along the northwestern sec- ward every 15 min over 1-min sampled predicted tides tion of the West Coast; this is also the site for a proba- and the tsunamis have initial amplitudes ranging from bilistic tsunami pilot study recently carried out by the 0.5 to 9.0 m. The result is a set of serrated patterns that Federal Emergency Management Agency (FEMA), the grow in amplitude as the tsunami amplitude increases. National Oceanic and Atmospheric Administration For small amplitude tsunamis, the maximum heights

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FIG. 3. Probability distribution functions (PDFs) of the maxi- FIG. 2. Locus of the maximum tsunami wave heights (tsunami mum tsunami wave heights at Seaside, OR, based on exponen- plus tide) as a function of the arrival time of the first wave peak tially decaying tsunamis with the indicated amplitudes. The tsu- in each tsunami wave train. For each initial amplitude A (height of nami amplitudes have been subtracted from the heights of the the first wave peak relative to the background water level), the corresponding PDF. Also shown is the PDF for the predicted time series are generated by moving the arrival time sequentially tides. in time. In this example, exponentially tsunami wave trains are superimposed linearly on predicted tides on the open coast at Seaside, OR. 1983–2001. Whereas Houston and Garcia (1978) assumed that the later waves in a tsunami had constant amplitude (set to 40% of the first wave amplitudes from (Fig. 2) are determined by the next higher high water or their numerical tsunami model) for 24 h after which the possibly even later ones. At the larger amplitudes, the amplitude was zero, we use time series that decay ex- maximum heights are determined by the first few waves ponentially over five days, based on the observed be- in the tsunamis and the tidal stage during which they havior of Pacific-wide tsunamis (e.g., Mofjeld et al. occur. As a result, the maximum heights of small tsu- 2000). We also use single period (20.0 min) tsunamis, namis tend to occur near mean higher high water which, as they are moved along the tidal time series, (MHHW) plus the tsunami amplitude, with little varia- provide equivalent statistical results to an ensemble of tion in height as a function of arrival time. In contrast, tsunamis with randomly distributed periods (see the maximum heights of large tsunamis form a distri- Mofjeld et al. 2000, for a statistical discussion of tsu- bution centered more closely to mean sea level (MSL) nami time series). plus the tsunami amplitude, with a spread in heights Histograms of heights, binned every 0.1 m, are com- approaching the diurnal tidal range. puted from each series that are then normalized (sum over the bins equal to unity) and interpolated with a 3. Probability density functions cubic spline to form continuous PDFs. The harmonic constants for Seaside are from the tidal model of To get a quantitative description of the height distri- Spargo (2003) and Spargo et al. (2004). All others are butions of maximum tsunami heights, we adopt a modi- official NOAA harmonic constants posted on the fied version of the procedure used by Houston and NOAA/National Ocean Service (NOS)/Center for Garcia (1978) in their study of expected 100-yr and Operational Oceanographic Products and Services 500-yr tsunami heights along the U.S. West Coast. Time (CO-OPS) Web site (see online at http://tidesandcurrents. series like those in Fig. 2 are computed for a full year noaa.gov). that corresponds to a time in the 18.6-yr lunar nodal When the initial tsunami amplitude A is subtracted cycle in which the nodal factors are near their average from the PDF heights, the distributions (Fig. 3) all have values. In our case, 1992 is chosen since it is in the an upper limit in height corresponding to the maximum middle of the present National Tidal Datum Epoch predicted tide for 1992. With increasing tsunami ampli-

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FIG. 4. PDFs computed from the time series of maximum tsunami wave heights and a Gaussian PDF with the same mean ␩ ␴ height o and standard deviation .

␩ Ϫ tude A, the widths of the m A PDFs spread and their centers decrease in height from MHHW toward MSL ␩ FIG. 5. Analytic fits to (a) the mean height o and (b) standard (datums for Seaside were computed from harmonic deviation ␴ as functions of tsunami amplitude A. constants by the method of Mofjeld et al. 2004b). The decrease in the maximum values occurs as the PDFs broaden because the height integral of each PDF needs diak, Crescent City, and Hilo. The latter three stations to be unity. As expected, the PDFs converge to that of have a long history of damaging tsunamis, and numeri- the predicted tide as A becomes large compared with cal tsunami models are often tested and calibrated us- the tidal range. ing these sites. Their tides are typical of the coastal tides Figure 4 gives an example of how a Gaussian PDF in the northern Gulf of Alaska, northern California, ␩ ␴ often closely resembles the shape of the empirical and Hawaii, respectively. While o and at Seaside, ␩ PDFs, where the mean height o and the standard de- Kodiak, and Crescent City vary throughout the range viation ␴ are computed from the latter A Յ 10 m (Fig. 5), these parameters reach their asymp- totic values much more quickly at Hilo because the ͑ ͒ ϭ ͓Ϫ͑ Ϫ ␩ ͒2ր ␴2͔ Ϫ1 ϭ ͌ ␲␴ P y B exp y o 2 , B 2 , tidal range is relatively small in Hawaii as compared with those in the other North Pacific regions. ͑1͒

where B is chosen so that the height integral of P is 4. Formulas for ␩ and ␴ unity. This leads to a compact characterization (Fig. 5) o ␩ ␴ ␩ of the PDFs in terms of o and as functions of the It is possible to compute the mean height o and initial tsunami amplitude A for each coastal location. standard deviation ␴ for each tsunami amplitude A and ␩ Note that o is the most probable height for the com- location of interest. However, this becomes very combined tsunami and tide, while Ϯ1.96␴ gives the 95% putationally intensive in numerical tsunami modeling confidence limits for Gaussian distributions. The inte- when tsunami heights are estimated at high resolution gral of (1) from a height y to ϱ is the cumulative density along a section of coastline. It is therefore useful to look ␩ ␴ function that provides the probability that the maxi- for empirical formulas for o and as functions of A, mum tsunami height of an event will exceed the value which can be put into the Gaussian PDF (1) when this of y. approximation is acceptably close to the PDF for a ␩ ␴ Figure 5 shows curves of o and for Seaside, Ko- given purpose.

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␩ ␴ TABLE 1. Parameters for o and in Eqs. (2) and (3) fitted to the PDFs as functions of the tsunami amplitude A, for selected Pacific ␴ tide stations. Also shown are the MHHW and MSL tidal datums relative to MLLW, as well as the standard deviations o of the tides at the stations. These results were based on predicted tides for 1992, when the lunar 18.6-yr nodal f factors are near unity.

␩ ␴ Mean height o Std dev Tidal quantities (m) ␣␤ Ј ␣Ј ␤Ј ␴ Lat (N) Lon (W) C C MHHW MSL o Alaska Adak 51°51.8Ј 176°37.9Ј 0.959 0.122 1.130 0.585 0.015 1.755 1.131 0.650 0.408 Unalaska 53°52.8Ј 166°32.2Ј 0.928 0.129 1.016 0.650 0.016 1.595 1.098 0.635 0.370 Sand Point 55°20.2Ј 160°30.1Ј 1.070 0.190 0.826 0.690 0.070 1.048 2.204 1.181 0.680 Kodiak 57°43.9Ј 152°30.7Ј 1.113 0.219 0.749 0.717 0.095 0.918 2.675 1.370 0.825 Seward 60°07.2Ј 149°25.6Ј 1.145 0.252 0.674 0.711 0.094 0.867 3.239 1.694 1.006 Yakutat 59°32.9Ј 139°44.1Ј 1.121 0.230 0.712 0.730 0.082 0.936 3.070 1.612 0.951 Sitka 57°03.1Ј 135°20.5Ј 1.119 0.228 0.705 0.710 0.080 0.927 3.028 1.610 0.947

Washington Neah Bay 48°22.1Ј 124°37.0Ј 1.082 0.196 0.828 0.691 0.070 1.050 2.425 1.315 0.744 Port Townsend 48°06.7Ј 122°45.5Ј 1.022 0.176 0.846 0.811 0.059 1.102 2.596 1.522 0.842 Toke Point 46°42.5Ј 123°57.9Ј 1.080 0.199 0.771 0.718 0.063 1.040 2.719 1.458 0.834 Oregon Astoria 46°12.5Ј 123°46.0Ј 1.032 0.175 0.805 0.747 0.060 1.072 2.624 1.373 0.790 Seaside 46°00.1Ј 123°55.8Ј 1.068 0.201 0.774 0.726 0.068 1.026 2.740 1.460 0.833 South Beach 44°37.5Ј 124°02.6Ј 1.064 0.186 0.804 0.729 0.064 1.052 2.542 1.358 0.784 Charleston 43°20.7Ј 124°19.3Ј 1.052 0.178 0.829 0.719 0.059 1.092 2.323 1.244 0.709 Port Orford 42°44.4Ј 124°29.8Ј 1.060 0.179 0.844 0.725 0.063 1.086 2.221 1.199 0.679

California Crescent City 41°44.7Ј 124°11.0Ј 1.044 0.170 0.858 0.707 0.056 1.119 2.095 1.130 0.638 North Spit 40°46.0Ј 124°13.0Ј 1.067 0.170 0.874 0.743 0.065 1.100 2.090 1.128 0.639 Arena Cove 38°54.8Ј 123°42.5Ј 1.029 0.159 0.923 0.686 0.057 1.163 1.792 0.960 0.540 Pt. Reyes 37°59.8Ј 122°58.5Ј 1.041 0.159 0.927 0.700 0.058 1.159 1.758 0.946 0.536 San Francisco 37°48.4Ј 122°27.9Ј 1.005 0.141 0.959 0.729 0.050 1.219 1.780 0.951 0.542 Monterey 36°36.3Ј 121°53.3Ј 1.032 0.149 0.972 0.677 0.058 1.193 1.626 0.862 0.492 Port San Luis 35°10.6Ј 120°45.6Ј 1.056 0.153 0.957 0.658 0.060 1.177 1.623 0.853 0.499 Santa Monica 34°00.5Ј 118°30.0Ј 1.061 0.167 0.916 0.628 0.073 1.102 1.653 0.849 0.500 Los Angeles 33°43.2Ј 118°16.3Ј 1.059 0.170 0.906 0.634 0.074 1.090 1.673 0.861 0.505 La Jolla 32°52.0Ј 117°15.5Ј 1.071 0.168 0.914 0.626 0.075 1.094 1.624 0.832 0.496 San Diego 32°42.8Ј 117°10.4Ј 1.138 0.184 0.867 0.642 0.086 1.025 1.745 0.897 0.551

Hawaii Nawiliwili 21°57.3Ј 159°21.4Ј 1.022 0.132 0.989 0.559 0.089 1.037 0.558 0.252 0.173 Honolulu 21°18.4Ј 157°52.0Ј 1.082 0.126 1.009 0.618 0.109 1.027 0.580 0.251 0.193 Kahului 20°53.9Ј 156°28.3Ј 1.020 0.140 0.934 0.511 0.026 1.327 0.686 0.339 0.215 Hilo 19°43.8Ј 155°03.4Ј 1.048 0.138 0.939 0.546 0.043 1.193 0.731 0.349 0.231

Average 1.056 0.173 0.875 0.677 0.065 1.120

␩ By trying a variety of formulas, it is found that the added to the right-hand side of the o equation (2) to following modified exponential forms provide close ap- give the actual mean height. proximations (e.g., Fig. 5) to these parameters: The nondimensional parameters C, ␣, ␤, CЈ, ␣Ј, and ␤Ј (Table 1), together with the tidal quantities MSL, ␩ ͑A͒ ϭ A ϩ MSL ϩ C͑MHHW Ϫ MSL͒ ␴ o MHHW, and o, provide a concise summary of the ϫ exp͓Ϫ␣͑Aր␴ ͒␤͔, ͑2͒ probabilistic tidal effects on maximum tsunami heights o at a given coastal location as functions of the tsunami ␴͑ ͒ ϭ ␴ Ϫ Ј␴ ͓Ϫ␣Ј͑ ր␴ ͒␤͔ ͑ ͒ amplitude A. A o C o exp A o , 3 The parameters in Table 1 provide insight into their ␴ where o is the standard deviation of the predicted consistency and variations between representative U.S. tides. Note that the tsunami amplitude A has been locations along the West Coast, Alaska, and Hawaii.

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␩ For the mean height o, the scaling parameter C is only TABLE 2. Percentage changes (%) in the parameters for the ␩ ␴ slightly greater than unity (average ϭ 1.056 m), show- mean height o and standard deviation at Seaside, OR, during ing that the PDF is centered near MHHW for small the 18.6-yr nodal cycle of the tides. Also shown are the percentage deviations from the mean of nodal f factors (multiplying the mean tsunami amplitudes (A Ͻ 1 m). The values of the coef- tidal amplitudes) for the major tidal constituents O1, K1, and M2. ficient ␣ are less consistent, ranging from 0.122 to 0.252. The years shown are within the National Tidal Datum Epoch The outliers in a region are often associated with sta- 1983–2001. tions (e.g., Adak, Alaska; and Port Townsend, Washing- 1987 1992 1997 ton) that are closer to tidal amphidromes than the others ␩ and therefore have somewhat different tidal character- o C Ϫ0.4 0.0 Ϫ2.9 istics. The values of the exponent ␤ tend to be consis- ␣ 1.0 0.0 Ϫ1.4 tent within regions, ranging from 0.705 to 1.130, with an ␤ Ϫ0.4 0.0 0.1 overall average (0.875) that is slightly less than unity. ␴ For the standard deviation ␴ in (3), the scaling pa- CЈ 0.0 0.0 Ϫ0.3 rameter CЈ (Table 1) ranges from 0.559 to 0.811. The ␣Ј 0.3 0.0 Ϫ1.3 ␤ЈϪ1.5 0.0 4.8 coefficient ␣Ј is smaller (average ϭ 0.065 m) and shows f factor considerably greater variations between stations than Ϫ O1 15.8 0.0 21.6 ␣ ␩ ϭ ␤Ј Ϫ its counterpart for o (average 0.173 m), while K1 9.7 0.0 13.2 Ϫ (average ϭ 1.120) is systematically larger than ␤. The M2 3.6 0.0 3.8 ␣ ␤ ␩ ␴ differences in the and parameters of o and re- flect the different rates at which they respond to changes in the tsunami amplitude A (e.g., Fig. 5). b. Effects of using observed versus predicted tides A cautionary note is in order with respect to using To assess the effects of using predicted tides rather Eqs. (2) and (3) for very small tsunami amplitudes com- than observed water levels in computing the ␩ and ␴ pared with the local tidal range. This is because the o statistics, each was computed for 1992 time series at underlying time series procedure for computing the Crescent City. This location is often used as a calibra- maximum tsunami heights breaks down since the pro- tion and test site for numerical tsunami models. The cedure simply selects the highest tidal high water in the results (Fig. 6) show that the ␩ difference is equal next five days irrespective of the tsunami amplitude. o within a few millimeters to the difference (7.85 cm) in Still, Eqs. (2) and (3) do provide well-defined estimates annual mean sea level (AMSL) except at small tsu- of ␩ and ␴ as A → 0. o nami amplitudes (A Ͻ 1 m) where the differences are ⌬(␩ ) Ͻ 3 cm. For ␴, the difference (Fig. 6) is less than 5. Discussion o 3 cm (except for A ϭ 0.1 m) and decreases with increas- a. Influence of the 18.6-yr nodal cycle ing tsunami amplitude. Hence, it makes little difference in ␩ and ␴ whether observed or predicted tides are The results in this paper have been based on tides o used at Crescent City, which has typical tides for the during 1992, when the lunar nodal effects were close to West Coast. This is fortunate since observed tides are their mean conditions (Table 2) for the National Tidal available at only a limited number of locations along Datum Epoch 1983–2001. Using a single year during most coastlines. the 18.6-yr nodal cycle follows Houston and Garcia (1978), who argue that the small variations induced by c. Issues of linear superposition of tides and the nodal cycle on the tidal probabilities cancel out on tsunamis average. This makes the computations more efficient since they involve 1 yr rather than 19. As a check of the This study is based on the assumption that tsunamis variations induced by the nodal cycle on the parameters and tides can be combined linearly, at least to the ex- ␩ ␴ o and , the same computations for the predicted 1992 tent of computing the first-order effects of the tides on tides at Seaside were carried out for 1987 and 1997. the maximum tsunami wave heights. The probability These years have the largest nodal deviations from the density function for the local tides can then be con- mean during the NTDE 1983–2001. The results (Table volved with that of tsunami heights from sources of 2) show that the values of the parameters vary by only various magnitudes and location to get the total height a few percent in contrast to the variations in the tidal distribution. This procedure was followed by Houston nodal f factors. This provides confidence in the results and Garcia (1978) to estimate 100- and 500-yr tsunami based on the 1992 series as representative of the long- exceedance heights along the West Coast. To under- term average conditions. stand the nonlinear interaction between tsunamis and

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U.S. West Coast, Alaska, and Hawaii. Having PDFs in closed form greatly facilitates the inclusion of tidal effects into probabilistic studies of coastal tsunami flooding.

Acknowledgments. The authors wish to thank Emily Spargo and Ed Myers (NOAA/NOS/Coastal Survey Development Laboratory) for providing tidal harmonic constants for Seaside, Oregon. The authors also wish to thank J. William Lavelle for helpful comments. This study was funded in part by FEMA as part of a pilot study to develop improved coastal flood maps along the U.S. West Coast. ␩ FIG. 6. Differences between the mean heights o and standard ␴ deviations for the 1992 observed and predicted tides at Crescent REFERENCES City, CA. Houston, J. R., and A. W. Garcia, 1978: Type 16 flood insurance study: Tsunami predictions for the West Coast of the conti- tides near and at the coast, it will be necessary to carry nental United States. USACE Waterways Experimental Sta- out a detailed numerical study in which both the tsu- tion Tech. Rep. H-78-26, 67 pp. Miller, G. R., W. H. Munk, and F. E. Snodgrass, 1962: Long- namis and tides are varying in time. Such a study is period waves over California’s borderland. Part II: Tsunamis. beyond the scope of the present one. J. Mar. Res., 20, 31–41. Mofjeld, H. O., F. I. González, and J. C. Newman, 1997: Short- term forecasts of inundation during teletsunamis in the east- 6. Conclusions ern North Pacific Ocean. Perspectives on Tsunami Hazard Reduction, G. Hebenstreit, Ed., Kluwer, 145–155. For Pacific tsunami wave trains, this study shows that ——, ——, E. N. Bernard, and J. C. Newman, 2000: Forecasting the maximum wave heights encountered during small the heights of later waves in Pacific-wide tsunamis. Nat. Haz- tsunamis tend to occur on average (most probable oc- ards, 22, 71–89. currence) near mean higher high water (MHHW): ——, A. J. Venturato, F. I. González, and V. V. Titov, 2004a: there is little spread (i.e., standard deviation ␴)inthe Background tides and sea level variations at Seaside, Oregon. NOAA Tech. Memo. OAR PMEL-126, 15 pp. probability density functions (PDFs) with height. At ——, ——, ——, ——, and J. C. Newman, 2004b: The harmonic larger initial tsunami amplitudes A, the most probable constant datum method: Options for overcoming datum dis- ␩ value o of the maximum tsunami height increases less continuities at mixed-diurnal tidal transitions. J. Atmos. Oce- than the sum of A and MHHW, and the PDF spreads in anic Technol., 21, 95–104. height: ␩ approaches the sum of A and the MSL when Spargo, E. A., 2003: Using a finite element model of the shallow o water equations to model tides in the eastern North Pacific the amplitude is much greater than the tidal range, and Ocean. M.S. thesis, University of Notre Dame, 224 pp. the PDF approaches that of the tides themselves. Over ——, J. Westerink, R. Luettich, and D. Mark, 2004: Developing a almost all of the tsunami amplitude range, the maxi- tidal constituent database for the Eastern North Pacific mum height PDFs can be approximated by Gaussian Ocean. Proc. Eighth Int. Conf. on Estuarine and Coastal ␩ Modeling, Monterey, CA, ASCE, doi:10.1061/40734(145)15. forms in which the mean height o and the standard ␴ Van Dorn, W. G., 1984: Some tsunami characteristics deducible deviation are simple formulas of the tsunami ampli- from tide records. J. Phys. Oceanogr., 14, 353–363. tude A and local tidal parameters. Values for these ——, 1987: Tide gage response to tsunamis. Part II: Other oceans parameters are given for 30 coastal stations along the and smaller seas. J. Phys. Oceanogr., 17, 1507–1516.

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