maximum annually recurring wave heights in hawai‘i · 541 maximum annually recurring wave heights...

541

Maximum Annually Recurring Wave Heights in Hawai‘i1

Sean Vitousek2,3 and Charles H. Fletcher2

Abstract: The goal of this study was to determine the maximum annually recur-ring wave height approaching Hawai‘i. The motivation was scientific as well asadministrative: to enhance understanding of the recurring nature of dominantswell events, as well as to inform the Hawai‘i administrative process of deter-mining the ‘‘upper reaches of the wash of the waves’’ (Hawai‘i Revised Statutes[H.R.S.] § 205-A), which delineates the shoreline. We tested three approachesto determine the maximum annually recurring wave, including log-normal andextremal exceedance probability models and Generalized Extreme Value (GEV)analysis using 25 yr of buoy data and long-term wave hindcasts. The annual re-curring significant wave height was found to be 7.7G 0.28 m (25 ftG 0.9 ft),and the top 10% and 1% wave heights during this annual swell was 9.8G 0.35m (32.1 ftG 1.15 ft) and 12.9G 0.47 m (42.3 ftG 1.5 ft), respectively, for openNorth and Northwest Pacific swell. Directional annual wave heights were alsodetermined by applying hindcasted swell direction to observed buoy data lackingdirectional information.

The islands of Hawai‘i lie in the midst of alarge swell-generating basin, the North Pa-cific. Tropical storms tracking to the north-west and north of the Islands produce winterswell with breaking face heights exceeding 5m several times each year. These swell eventslead to concerns over coastal erosion, coastalflooding, and water safety for the large popu-lation of ocean communities in Hawai‘i.

Runup generated by the largest of thesewaves poses a hazard to coastal developmentby flooding roadways, undermining struc-tures, and causing erosion. According toHawai‘i state law (Hawai‘i Revised Statutes[H.R.S.] § 205-A) the highest runup of theseannual swells sets the legal position of theshoreline.

In Hawai‘i, the shoreline serves as a ref-erence line used to delineate public beachaccess, construction setbacks, state conserva-tion land, submerged lands, and the borderof management jurisdiction. Several statesdefine the shoreline differently; for example,California uses the mean high water markand Massachusetts uses the mean low watermark based on tidal water levels (not includ-ing wave setup or runup). In 1968, the Stateof Hawai‘i changed the definition of theshoreline from the mean high water mark tothe highest reach of the waves (Ashford v.State of Hawai‘i). The State of Hawai‘i defini-tion of the shoreline is ‘‘the upper reaches ofthe wash of the waves, other than storm andseismic waves, at high tide during the seasonof the year in which the highest wash of thewaves occurs, usually evidenced by the edgeof vegetation growth, or the upper limit ofdebris left by the wash of the waves’’ (H.R.S.§ 205-A). In October 2006, the Hawai‘i Su-

Pacific Science (2008), vol. 62, no. 4:541–553: 2008 by University of Hawai‘i PressAll rights reserved

1 This paper is funded by a grant/cooperative agree-ment from the National Oceanic and Atmospheric Ad-ministration, Project No. R/EP-26, which is sponsoredby the University of Hawai‘i Sea Grant CollegeProgram, SOEST, under Institutional Grant No.NA05OAR4171048 from NOAA Office of Sea Grant,Department of Commerce. The views expressed hereinare those of the authors and do not necessarily reflectthe views of NOAA or any of its subagencies. UNIHI-SEAGRANT-JC-06-17. Manuscript accepted 28 No-vember 2007.

2 Coastal Geology Group, Department of Geologyand Geophysics, School of Ocean and Earth Science andTechnology, University of Hawai‘i at Manoa, 1680 East-West Road, POST 721, Honolulu, Hawai‘i 96822.

3 Corresponding author (e-mail: [email protected]).

preme Court issued a ruling (Diamond v. Stateof Hawai‘i) that the shoreline should be estab-lished ‘‘at the highest reach of the highestwash of the waves.’’

The State of Hawai‘i has established acoastal management system that relies onthis definition of the shoreline, not only as ademarcation of public shoreline access, butalso to establish a baseline for constructioncontrol and development setbacks. The dis-cord of private landowners seeking to pre-serve or develop the economic value of theirproperty and public ocean users wishing toaccess and preserve pristine coastal environ-ments is responsible for continuing debateover shoreline laws. Resolving the annuallyrecurring maximum wave height around theIslands would improve the scientific basis tounderstanding the shoreline definition, be-cause this line is set by the upper limit ofwave runup resulting from the largest or setof largest annually recurring waves (underoptimal runup conditions). Runup is not asimple function of wave height; rather theinteraction of different wave conditions andtopography determine runup. However, waveheight is a fundamental determinant ofrunup; contrary to topography, which is fixedas a data set, wave height is not. This paperprovides the description of maximum annu-ally recurring wave heights in Hawai‘i forsuch purpose.

Local consulting engineers as part of theirrespective design projects on the coast arerequired to describe the regional and localwave climate. This analysis typically consistsof specifying the largest characteristic rangesand scatter tables or rose diagrams of waveheight, period, and direction of the dominantswell regime for the area of interest. Such en-gineering reports (e.g., Noda and Associates1991, Bodge and Sullivan 1999, Bodge 2000)do not give detailed statistical analyses of therecurring nature of swells in Hawai‘i.

To provide a comprehensive analysis, wewish to resolve the annually recurring maxi-mum wave based on the best records andstate-of-the-art methods. The determinationof the annually recurring wave height is alsothe first step to ensuring a sound scientific ba-sis for policy-based decision making involved

in the determination of the shoreline as de-fined by H.R.S. § 205-A.

Previous Work

The seasonal wave cycle in Hawai‘i has beenexplored in several different publications.Moberly and Chamberlain (1964) outlinedthe wave cycle in terms of four swell regimes:North Pacific swell, northeast trade windwaves, Kona storm waves, and southern swell.We have added a wave rose to their originalgraphic depicting annual swell heights and di-rections (Plate 1).

The seasonal wave cycle in Hawai‘i ischaracterized by large North Pacific swelland decreased trade wind waves dominatingin winter months and southern swell accom-panied by increasing trade wind waves domi-nating in summer months. However, large-scale oceanic and atmospheric phenomenaincluding El Nino Southern Oscillation(ENSO) and Pacific Decadal Oscillation(PDO) are thought to control the numberand extent of extreme swell events (Seymouret al. 1984, Caldwell 1992, Inman and Jenkins1997, Seymour 1998, Allan and Komar 2000,Wang and Swail 2001). Extreme wave eventshave been argued to control processes such ascoral development (Dollar and Tribble 1993,Rooney et al. 2004) and beach morphology(Moberly and Chamberlain 1964, Ruggieroet al. 1997, Storlazzi and Griggs 2000).

Although there are several factors thatcontribute to annual variability in maximumwave height in Hawai‘i, including the ENSOand PDO cycles, we seek to resolve the meanmaximum annual signal from both highs andlows of these cycles. Ruggiero et al. (1997)evaluated extreme runup using empiricalequations as a means of calculating frequencyof dune impact. This empirical approach or amore robust process-based numerical model-ing approach could similarly be used to eval-uate the extent of extreme runup in Hawai‘ibased on the annual maximum wave height.This study could provide boundary condi-tions for a more sophisticated wave transfor-mation and runup model for identification ofthe shoreline in Hawai‘i for a particular loca-tion.

542 PACIFIC SCIENCE . October 2008

Plate 1. Hawai‘i dominant swell regimes after Moberly and Chamberlain (1964), and wave-monitoring buoy locations.

materials and methods

To determine the annually recurring maxi-mum wave height we used the record ofwave buoys from the National Oceanic andAtmospheric Administration (NOAA) Na-tional Data Buoy Center (NDBC) and Wave-Watch III (WWIII) model hindcasts fromthe Coastal and Hydraulics Laboratory’s(CHL) Wave Information Studies (WIS)program (Vicksburg, Mississippi).

Hourly reports of significant wave height(average of the largest one-third of waveheights, Hs) and other meteorological infor-mation from monitoring buoys are availablefrom NOAA’s NDBC Web site (http://www.ndbc.noaa.gov/Maps/Hawaii.shtml). Basedon the observation that wave heights follow aRayleigh distribution, we can use the signifi-cant wave height to estimate other statisticsof a swell, such as the mean wave height orthe top 10% wave height, based on the signif-icant wave height. These buoys have an in-strument precision of 0.2 m, which results insmall errors (less than 5% for all waves above4 m).

Our focus concerned buoy 51001 ( buoy1), which is located 315 km northwest of theisland of Kaua‘i and is moored at a depth of3.25 km. The buoy has recorded 25 yr ofwave height and period data, since 1981.

Buoy 1 is ideally located to record Northand Northwest Pacific swell without interfer-ence from neighboring islands. Only recentlyhas buoy 1 been able to record swell direc-tion. Thus the time series recorded in themajority of buoy 1 data and by all of the re-maining buoys lack swell direction. The lackof observations of wave direction meansthat any analysis of open North Pacific andNorthwest Pacific swell is limited to datafrom buoy 1 because all the remaining buoysare significantly affected by island blockage.However, hindcasts using WaveWatch IIIcan be used to recover directional informa-tion.

Long-term statistical analysis was appliedto the simple case of the 1-yr recurring sig-nificant swell height. Statistics of extremescan usually be extrapolated to approximatelythree times the length of the time series. Be-

cause we were primarily interested in the an-nually recurring maximum wave height, our25-yr time series was more than adequate toresolve this value. Long-term statistical mod-els are typically applied to long-return-periodevents such as the 50- to 100-yr events;such methods were originally developed todefine stream flood heights or return periodsfrom discharge records (Gumbel 1941). Al-though typically applied to long-return peri-ods, they can also be applied for short- andintermediate-return periods. The followingprocedure was used to construct our long-term statistical model:

(1) Large swell events (n per year) fromthe buoy record were assigned an ex-ceedance probability (see equation).

(2) Log-normal and extremal models usedlinear regressions to determine the re-lationship between large swell eventsand exceedance probability (the prob-ability that a larger swell event willoccur during the return period). Tocorroborate this analysis GeneralizedExtreme Value (GEV ) probabilitymodels also determined the relation-ship between large swell events and ex-ceedance probability using MaximumLikelihood Estimates (MLE).

(3) Methods including removing outliersand the peak over threshold (van Vled-der et al. 1993) method were evaluatedto improve model performance.

(4) These statistical models assigned prob-abilities to a full range of swell heightsand were modified to give the relation-ship between swell heights and returnperiod (particularly the 1-yr return pe-riod).

(5) The maximum annually recurringwave height is determined from thetail of a Rayleigh distribution.

The log-normal statistical model was con-structed on the assumption that maximumswell events will plot as a linear function ona horizontal logarithmic scale of exceedanceprobability. Exceedance probability ðQ ¼1� pÞ was given by the probability that thenext swell will be greater than the sorted

Maximum Annually Recurring Wave Heights in Hawai‘i . Vitousek and Fletcher 543

wave events on record as if drawing from ahat containing all the maximum swell heightsand the next swell event.

WaveHeight

Q ¼ Exceedanceprobability ¼ ð1� pÞ

Hs1 !N

N þ 1 High probability

the next swellwill be larger.

Hs2 !N � 1

N þ 1

Hs3

N � 2

N þ 1

..

. ...

HsN !1

N þ 1 Low probability

the next swellwill be larger.

In this procedure, Hs1 is the smallest signif-icant wave height, HsN is the largest signifi-cant wave height, and N is the total numberof waves in the analysis. Selecting differentnumbers of events ðnÞ per year, such as thesingle maximum significant wave height eachyear or the top 25 significant wave heights,can yield different results as discussed later.

Our long-term statistical analyses were

performed using the significant wave height.To determine the maximum wave heightthat occurs with a given significant waveheight requires further statistical analysis ona probability distribution of random waves.

results

Our results fell into two categories: resultsusing the log-normal and extremal exceed-ance probability models and results using theGEV models.

Log-Normal and Extremal Models

The log-normal model of exceedance proba-bility versus wave height, as seen in Figure 1,is quite linear on a log (x-axis)-linear (y-axis)scale.

Figure 1A shows a log-normal model ofthe data, with the following equation:

Hs ¼ A½�log Q� þ B

where A and B are regression coefficients.An extremal model of the same data is

shown in Figure 1A and uses the followingequation, which differs from the log-normalmodel by the 1/k exponent term ( below),which serves to limit the occurrence of ex-

Figure 1. Log-normal and extremal probability models for the top 70 largest wave height events per year recorded bybuoy 1. The largest wave heights recorded by buoy 1 are plotted versus the negative log of exceedance probability inðAÞ and plotted versus the return period in ðBÞ. In these models the largest event (a 12.3 m significant wave height)outlier has been removed and the peak over threshold method is used with a threshold of 5 m.


tremely large events (when k > 1) and give abetter fit:

Hs ¼ A½�log Q�1=k þ B

Figure 1B shows the same data in terms ofsignificant wave height versus return periodinstead of exceedance probability. This allowsdetermination of the annually recurring sig-nificant wave height. The relationship be-tween the exceedance probability ðQÞ andthe return period ðTRÞ is

TR ¼r:i:

Q

where r:i: is the recurrence interval.The return period is simply the recurrence

interval, r:i: (1/70 yr because we used the top70 events each year) divided by the exceed-ance probability ðQÞ. In this case 70 eventswas the smallest number of recorded waveheights in a year of buoy data, because thebuoy was not operating for the majority of1983 due to maintenance issues.

The 1-yr return period is given in Figure1B as 7.67G 0.014 m. The confidence levels,CI, shown in Figure 1A and B are given bythe typical confidence interval equations fora linear regression:

CI ¼ tN�2; 1�a=2ðSEÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

Nþ ðx� mÞ2P

ðx� mÞ2

s

where t is given by the student-t statistic, ais the significance level, m is the mean, andSE is the standard error given by the equation

SE ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

N � 2

Xðy� yyÞ2

rThis 1-yr return swell event has an annual

return probability percentage based on therecurrence interval of the time series givenby the equation:

E ¼ 1� 1� r:i:

TR

� �ðL=r:i:Þ

where E is the probability that we will en-counter the event (¼ 64% for the followingconditions), r:i: is the recurrence interval(1/70 yr), TR is the return period (1 yr), andL is the life span (1 yr). According to thebuoy 1 time series, significant wave heightsexceeding 7.7 m have occurred in 16 of the

25 yr on record (i.e., 68% of the time, whichis consistent with the encounter probabilityof 64% calculated earlier).

Log-normal models tend to overpredictlarge events because physical processes exertnatural limitations on event magnitude thatare not accounted for in the model. For ex-ample, flood height is limited by the rainfallamount, wave height is limited by energy dis-sipation, and hurricane intensity is limited byheat transfer to fuel propagation. Thus, ex-treme events (long-return-period events) areoften not best fit with a log-normal relation-ship, and other models such as the extremalmodel should be considered. A particular ex-ample of this concerns the largest significantwave height in the 25-yr record of buoy 1:a 12.3 m event that occurred at 0400 hourson 5 November 1988. The second largest onrecord is 10.1 m (1985). These are the onlytwo events with significant wave heights ex-ceeding 10 m and, notably, the largest swellon record is more than 2 m greater than thenext largest swell. In the analysis just de-scribed this 12.3 m event was removed. Wemust consider the possibility that the 12.3 msignificant wave height event was an extraor-dinary swell and perhaps unlikely to occurduring a period of 25 yr. In exceedance prob-ability models the largest event (12.3 m) pro-vides information about the longest returnperiod of recorded data (25 yr in this case).In reality, this 12.3 m event could very wellbe the 50- or 100-yr swell event, and includ-ing this event overestimates the frequency oflarge events in the model as well as affects thevalue for the annually recurring wave height.A simple procedure is to test potential over-estimation to determine the best fit withoutthe outlier event and determine the expectedreturn period of the removed outlier. Usingthe model described earlier, the return periodof a 12.3 m event is approximately 150 and700 yr using log-normal and extremal mod-els, respectively. Typically, forecasts longerthan three to four times the data collectionperiod (25 yr in this case) are not realistic,and furthermore they are not the focus ofthis paper. However, we performed suchanalysis to confirm our suspicion that a verylarge event occurred in 1988 with a recur-


rence interval exceeding 100 yr and justify itsremoval from the analysis.

Returning to the annual return period, alog-normal model would perhaps be appro-priate, but for completeness we investigatedthe behavior of swell events using both log-normal and extremal models as well as theGEV model (described next). The GEV sta-tistical model returns results very similar tothose of the log-normal and extremal models,which focus on our estimates of the annuallyrecurring significant wave height.

Generalized Extreme Value (GEV ) Model

The GEV distribution is applied to deter-mine relationships between wave height andreturn period with particular focus on theannually recurring wave height. Introducedby Jenkinson (1955), the GEV distributionuses Gumbel (type I), Frechet (type II), andWeibull (type III) distributions for differentvalues of the shape parameter, k ¼ 0, k < 0,k > 0, respectively. Iterative maximum-likelihood estimates (MLE) fit the observed

data to find the best estimates of the shapeðkÞ, scale ðsÞ, and location ðmÞ parameters ofthe GEV cumulative distribution function,FðxÞ, given by:

FðxÞ ¼ exp � 1þ kx� m

s

� �� 1=k( )

for k00

exp �exp �ðx� mÞs

� �� for k¼ 0

Based on given probability distributionsand the return period probability equation

pTR¼ 1� r:i:

TR

, wave height for an arbitrary

return period is found. The GEV model ismore robust than the previous approach be-cause it combines the Gumbel, Frechet, andWeibull extreme value distributions, althoughit yields very similar results to our log-normaland extremal analysis (Figure 2). The GEVanalysis, being a more robust model, remainslargely unaffected by the presence of the 12.3m event.

Figure 2. The Generalized Extreme Value probability model used to determine the annually recurring significantwave height.


Recovering Swell Directionality from ModelHindcasts

Thus far the annually recurring wave heightanalysis is applicable to open North andNorthwest Pacific swells because no informa-tion of wave direction has been considered.By using WaveWatch III (WWIII) modelhindcasts concurrent with buoy data to re-cover the directionality of wave heights, wecan determine the annually recurring maxi-mum wave heights for a particular directionwindow. WWIII is an ocean-scale spectralwave model on a 0.5-degree grid that com-putes open swell generation and propagationbased on spatial wind fields. WWIII hasbeen well validated using buoy and altimetrydata (Tolman 2002, Baird and Associates2005, Tracy et al. 2006). Figure 3 and Table1 show the annually recurring maximum sig-nificant wave heights according to buoy datafor extremal and GEV models for swell direc-tion windows of 30 degrees. Applying mod-eled direction to the actual buoy data mayseem problematic, but it is the best optionto ensure preference toward observed over-modeled wave height. As can be seen from

Figure 3, observed directional annually re-curring maximum significant wave heightsproduced by both models are similar as areresults from the observed versus modeledwave heights shown in Table 1.

The north to northwest windowed annualsignificant wave heights found in Table 1 aresmaller than the previously determined valueof 7.7 m because the largest events used inthe analysis do not fall into the same 30-degree windows. This limiting effect is causedby the directional variability of north swells,which typically range clockwise from 270 to90 degrees (W–E). Analysis of all northern-facing swell directions (270 to 90 degrees)recovered this 7.7 m annual wave height.Southern swell occurs in much more narrow-banded directions, typically ranging clockwisefrom 150 to 210 degrees (SSE–SSW ) andtherefore should be much less affected by thelimiting effect of the directional variability.

Significant Wave Height versus MaximumProbable Wave Height

It is important to keep in mind that 7.7 m isthe annually recurring significant wave height

Figure 3. Observed directional annually recurring maximum significant wave heights ðHsÞ given from ðAÞ extremaland ðBÞ GEV models.


or the average of the highest one-third of thewave heights. Each wave height data pointfrom the buoy record used in the analysiswas part of a swell train that certainly con-tained larger waves. The significant waveheight represents the typical observationalstate of the ocean, not the very largest wavesthat occur during a swell event. To determinethe largest 10% and 1% of wave heights, weassumed that the probability distribution ofthe waves followed a Rayleigh distribution,which has the following probability distribu-tion function (pdf ):

pðHÞ ¼ 2H

H 2rms

exp � H

Hrms

� �2" #

where pðHÞ is the probability of encounter-ing a wave of a given height, H, and Hrms isthe root mean square wave height, which isequal to Hs/

ffiffiffi2p

.The assumption that random waves follow

a Rayleigh distribution has been shown tobe quite good for deep-water waves with anarrow-banded wave spectrum (Longuet-Higgins 1952) (i.e., waves created by a singleswell event rather than two converging swellevents). With a given pdf, one can determineseveral parameters of interest, such as the

average of the 10% largest waves or the max-imum probable wave. Maximum probablewave, Hmax, can be solved using the followingequation: ðy

Hmax

pðHÞ dH ¼ 1/N

where N is the number of waves in the swellevent. Solving for Hmax yields the followingequation:

Hmax ¼ Hrms

ffiffiffiffiffiffiffiffiffiln Np

¼ Hs

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2ln N

r

The average of the 10% largest waves isgiven by the following equation:

Avg: of H10% ¼Ðy

H10%pðHÞH dHÐy

H10%pðHÞ dH

or more generally;

Avg: of Hpct% ¼Ðy

Hpct%pðHÞH dHÐy

Hpct%pðHÞ dH

where H10% or ðHpct%Þ is given by the equa-

tion: Hs

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2ln

r1

pct, where pct is the percentage

TABLE 1

Observed and Modeled Directional Annually Recurring Maximum Significant Wave Heights Using Extremal andGEV Exceedance Probability Models

WindowAnnual Hs (m): Extremal

Model Annual Hs (m): GEV Model

Lower Upper Source Observed Modeled Observed Modeled

0 30 Buoy1 5.75 6.1 5.85 6.1530 60 Buoy1 6 6.5 6 6.4560 90 Buoy4 5 5.25 5.1 5.3790 120 Buoy4 4.1 4.2 4.3 4.32

120 150 Buoy4 3.1 3.1 2.75 2.65150 180 Buoy2 2.5 2.4 3 2.95180 210 Buoy2 2 2.1 2.35 2.35210 240 Buoy1 1.5 1.5 1.6 1.65240 270 Buoy1 2.1 2.2 1.5 1.6270 300 Buoy1 4 4.5 3.7 3.9300 330 Buoy1 6.2 6.7 5.9 6.4330 360 Buoy1 6 6.5 5.8 6.2

Note: Wave hindcasts of buoy 3 did not return more than one swell event per year in the southerly and westerly directional win-dows; hence buoy 1 was used instead.


of interest. By integrating the equation justgiven, the averages of the top percentages ofwave heights in relation to the significantwave height are shown in Figure 4.

Consistently, according to the Coastal En-gineering Manual (U.S. Army Corps of Engi-neers 2002), the average of the largest 10%and 1% of wave heights in a swell event isgiven by the following:

H10% highest ¼ 1:27Hs ¼ 9:8 m

H1% highest ¼ 1:67Hs ¼ 12:9 m

Hmax ¼ 1:86Hs ¼ 14:3 m

This analysis, for Hmax, is based on 1,000waves and represents wave conditions thatwould occur for a peak swell duration ofaround 4.44 hr assuming the typical wave pe-riod is 16 sec. Note: Wave hindcasts of buoy3 did not return more than one swell eventper year in the southerly and westerly direc-tional windows; hence buoy 1 was used in-stead.

Table 2 shows the maximum annuallyrecurring significant wave heights and thelargest 10% and 1% wave heights for variousdirections in 30-degree windows aroundHawai‘i.

discussion

The number of maximum swell events (threeper year versus some other number) used inthe exceedance probability model influencesthe determination of the annually recurringsignificant wave height. This motivates a sen-sitivity test to determine the optimal numberof events to use. Notably, as more events areselected per year the annual recurring wave

Figure 4. Top percentage of waves versus relation to significant wave height ðHsÞ.

TABLE 2

Observed Maximum Annually Recurring SignificantWave Heights and the Largest 10% and 1% Wave

Heights for Various Directions around Hawai‘i

Window Annual Hs (m): GEV Model

Lower Upper Observed-Hs H1/10 H1/100

0 30 5.9 7.4 9.830 60 6.0 7.6 10.060 90 5.1 6.5 8.590 120 4.3 5.5 7.2

120 150 2.8 3.5 4.6150 180 3.0 3.8 5.0180 210 2.4 3.0 3.9210 240 1.6 2.0 2.7240 270 1.5 1.9 2.5270 300 3.7 4.7 6.2300 330 5.9 7.5 9.9330 360 5.8 7.4 9.7


height is higher for the log-normal and ex-tremal models and lower for the GEV modelafter a peak at about 10 events per year.However when large numbers of events areselected ðN > 10Þ the return wave heightprediction for the log-normal and extremalmodels begins to stabilize (Figure 5).

This increasing annual recurring waveheight result for the log-normal and extremalmodels is due to the trend that as more eventsare selected per year, the data become satu-rated with lower swell events, which tend todominate the behavior of the regression. Be-cause a majority of data points behave witha return period of less than 1 yr, the log-normal model predominantly captures thebehavior of the frequency of short-return-period events at the expense of long-perioddata. This can be seen in Figure 6: the log-normal fit is quite good for short-return-period data and problematic for long-return-period data.

As seen in Figure 6A a large number of

short-return-period (high exceedance proba-bility) data points force the slope of the log-normal fit to be higher than it would be forlong-return-period data points, which seemto have a lower slope. This suggests that theinherent behavior or occurrence of short-return-period events differs from that oflong-return-period events, and thus the en-tire data set is not appropriately representedby a log-linear model.

In dealing with estimates of the annuallyrecurring significant wave height, it is some-what unrealistic to give a confidence level onthe order of centimeters when dealing withwaves exceeding 7 m, although by this analy-sis it may be statistically acceptable to do so.Rather, we make a recommendation that ac-counts for the variability by using differentnumbers of events selected each year thattypically rangeG 0.2 m and the instrumentprecision, which is alsoG 0.2 m. Summingthe error in quadrature gives a confidencelevel of 0.28 m and still represents a very

Figure 5. Sensitivity of models to number of largest events selected per year.


narrow band of about 3.5% of the waveheight.

Thus our recommendation for the annu-ally recurring significant wave height in Ha-wai‘i is 7.7G 0.28 m (25 ftG 0.9 ft), and thetop 10% and 1% wave heights during this an-nual swell are 9.8G 0.35 m (32.1 ftG 1.15 ft)and 12.9G 0.47 m (42.3 ftG 1.5 ft), respec-tively. For good measure we also multiplythe confidence level by the coefficient givenon the y-axis of Figure 4, so that the confi-dence levels represent the same percentageof the final value. The difference betweenselecting the annually recurring significantwave height as 7.5 or 7.7 or somewhere inbetween is fairly trivial, especially for engi-neering calculations, because the differencebetween selecting one or the other results ina maximum difference of only 3.5%.

It is important to note that this analysisconsiders only deep-water wave heights,which are not the same as wave heights nearthe shoreline. There are several physical pro-cesses that can cause deep-water wave heights

to increase or decrease when propagating intoshallower water, such as shoaling, refraction,diffraction, and nonlinear interactions. Thereare a number of theories and methods thatare used to model the transformation fromdeep-water wave heights to nearshore waveheights, including linear wave theory, spectraland phase-resolving wave models, and empir-ical equations. Caldwell (2005) applied thisapproach for predicting the observed break-ing wave height at Waimea Bay, Hawai‘i,from the deep-water wave height recordedby buoy 1.

It is also important to keep in mind thatHmax represents the single largest wave thatwould occur during a swell event, and per-haps the 10% or 1% highest wave conditions(depending on the acceptable risk tolerance)would be more representative of all of thelargest waves in a swell event. Another benefitof using the top 10% or 1% of wave heightsis that information about the number ofwaves in a particular swell is not required.The analysis performed in Figure 4 has no in-

Figure 6. When a large number of events are used in analysis, the log-normal model is strongly fit to the high-frequency events (which represent the bulk of the data) at the expense of the extreme events. The fit of the waveheights versus the negative log of exceedance probability is shown in ðAÞ, and the fit of wave heights versus the returnperiod is shown in ðBÞ.


put on the number of waves in a swell event;only the maximum probable wave requiresthe number of waves as input. These valuesshould be considered the maximum annuallyrecurring wave height for open north- andnorthwest-facing shores such as Kaua‘i andO‘ahu, where swell is directly incident tothe shoreline and blocking from neighboringislands is minimized. For shorelines not di-rectly exposed to North and Northwest Pa-cific swell the annually recurring maximumwave height may be estimated from Figure 3or Table 1 and the relationship of Hs to Hmax

given in Figure 4 or Table 2.

Future Work

The determination of the maximum annuallyrecurring wave height is just the first step in aprocess of formulating a sound scientific basisto evaluate the physical processes involved inwave runup. The next step in the process in-volves propagation of this deep-water waveinto the nearshore and resolving the spatialvariability of wave heights due to shoaling,refraction, diffraction, convergence, diver-gence, nonlinear interactions, and breaking.The spatial and physical properties of near-shore waves can be determined throughmodeling or empirical approaches. Finally, toevaluate runup, these nearshore wave prop-erties can be used as boundary conditions ina runup model, and observations of runupshould be recorded during wave events withdeep-water wave heights around the annuallyrecurring maximum level.

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