Extreme wave predictions around New Zealand from hindcast data

Transcription

1 New Stephens Zealand & Gorman Extreme Journal of Marine wave and Freshwater predictions Research, around New 2006, Zealand Vol. 40: /06/ The Royal Society of New Zealand Extreme wave predictions around New Zealand from hindcast data Scott A. Stephens Richard M. Gorman National Institute of Water and Atmospheric Research Limited P.O. Box Hamilton, New Zealand Abstract A recently implemented wave hindcast for the New Zealand region was used in conjunction with wave-buoy data to evaluate extreme significant wave height at multiple sites around New Zealand, for the first time. Hindcast storm wave heights were under-predicted compared with wave-buoy measurements at three inshore sites, and a method for scaling the hindcast data to improve the comparison of predicted extreme wave heights was explored. Different statistical methods for predicting extreme wave heights were also compared. Offshore, extreme wave heights displayed a north south and an east west gradient that is in keeping with the mean wave climate, with larger waves in the south and in the west. However, the variation of extreme wave heights between sites was less than the mean wave climate would suggest, because mid-latitude depressions generate comparatively large waves on the generally more sheltered northeast coast. At the most energetic site to the southwest of the South Island, a 1 in 100-year return significant wave height H s(100) of 19.3 m and maximum wave height H max(100) of 45 m were predicted. At the least energetic site to the northeast of the North Island, estimates of H s(100) = 13.9 m and H max(100) = 33 m were obtained. Keywords extreme wave height; wave hindcast; wave climate; New Zealand M05073; Online publication date 7 July 2006 Received 23 November 2005; accepted 21 February 2006 INTRODUCTION Estimates of the most energetic wave conditions to be expected over various time scales are fundamental to navigation, the design of offshore structures and ships, and broadly applicable to many aspects of coastal management and engineering, e.g., wave overtopping hazards and appropriate mitigation (e.g., Clauss 2002; Wang & Swail 2002; Alves & Young 2003; Stansell 2004). This information is most commonly presented as occurrence statistics for extreme values of the significant wave height H s, which describes the average of the highest 1 / 3 of the waves in a measured burst. The extreme significant wave height, denoted H s(r), can be described as the threshold H s value exceeded once during a lifetime interval, or return period, R. New Zealand is an island nation exposed to a considerable range in wave climate (Laing 2000), but little is known about the surrounding extreme wave climate. The recent implementation of the wave evolution model WAM (WAve Model) to provide a 20-year hindcast of wave conditions for the New Zealand region (Gorman et al. 2003a,b) has provided a new tool for the evaluation of H s(r). Determination of H s(r) traditionally involved the statistical analysis of a historical time series of wave heights measured by wave-buoys. Although wavebuoys may underestimate the extreme individual wave heights occurring in their vicinity (Magnusson et al. 1999), they provide adequate temporal resolution to capture the peak significant wave heights produced during storm events (e.g., Alves & Young 2003). Therefore, extreme significant wave heights can be reliably predicted from buoy data, providing the buoy records are of sufficient duration (capture enough storms) to reliably fit a statistical distribution and to account for variability in the wave climate (e.g., Mathiesen et al. 1994). Unlike some wellstudied areas in the northern Atlantic and Pacific oceans, wave-buoy records around New Zealand are sparse and typically of short duration, and generally do not provide adequate historical coverage of wave data for the purpose of estimating H s(r). Furthermore,

2 400 New Zealand Journal of Marine and Freshwater Research, 2006, Vol. 40 Fig. 1 New Zealand landmass and locations of wave-buoys (+) and offshore hindcast sites (o) for which extreme wave height statistics were derived. those buoy records that do exist are often located relatively close to shore, so the measured wave records tend to be location-specific and not easily transferred to other sections of coast (Gorman et al. 2003a). Methods for describing wave climatologies over wide areas have become available more recently. Satellite altimeter data have been used to provide a wave climatology for the New Zealand region (Laing 2000). In addition to requirements for verification and bias correction, altimeter data may be under-sampled in both space and time so that storm peaks are underrepresented in the record (e.g., Laing 2000). Altimeter data can be used in characterising extreme wave statistics, but this under-sampling presents difficulties that must be treated with care (e.g., Cooper & Forristall 1997; Alves & Young 2003). Similar reservations apply to wave measurements from satellite-borne Synthetic Aperture Radar, although the radar offers promise in detecting individual extreme wave events (Schulz-Stellenfleth & Lehner 2004). Numerical simulation of the wave spectrum using advanced third-generation wave models now represents the main source of data for wave climate studies (Lopatoukhin et al. 2000). When driven by accurate wind fields, accurate simulations of the principal scale and shape properties of the gravity wave field are produced (Laing et al. 1997; Lopatoukhin et al. 2000; Swail & Cox 2000; Wang & Swail 2002). However, to accurately specify storm events often requires some form of detailed reanalysis of the input wind fields to add the necessary fine-scale high-energy wind gusts (Cardone et al. 1996; Wang & Swail 2002). Without this labourintensive re-analysis, wave hindcasts tend to underpredict H s during storms (e.g., Laing et al. 1997; Sterl et al. 1998; Gorman et al. 2003a). Here the hindcast model was used to calculate extreme wave heights around New Zealand. We initially compared hindcast H s(r) with those derived from buoy data. To improve the accuracy of hindcast H s(r) we tried different methods of analysing hindcast data and tested a simple method for scaling the hindcast data. We then used the hindcast to characterise H s(r) at six deep-water sites located off different sectors of the New Zealand coast, and compared the trends with those from the mean wave climate (Gorman et al. 2003a). materials and METHODS Wave records were taken from four deployments around the New Zealand coast (Fig. 1, Table 1). These sites are subject to a wide range of wave conditions

3 Stephens & Gorman Extreme wave predictions around New Zealand 401 Table 1 Buoy deployments referred to in the comparison with wave hindcast results. Columns describe the location, the deployment period for which the data were available (notwithstanding any gaps in the records), the depth of deployment, and the funder/operators of the buoy (NIWA, National Institute of Water and Atmospheric Research Limited). Site Longitude Latitude Data available Depth (m) Funded/operated by Baring Head, Wellington Apr 1995 present 45 TranzRail, Wellington Regional Council, NIWA Katikati, Bay of Plenty Mar Dec NIWA Maui platform, Taranaki Sep Apr Shell, BP, Todd Petroleum, University of Auckland Mokohinau Island, Hauraki Gulf May 1998 present 100 Auckland Regional Council experienced on the New Zealand coast. Two of the deployments (Mokohinau Island and Katikati), were on the northeast coast of the North Island and sheltered from the energetic Southern Ocean, the Maui site was affected by wave conditions in the Tasman Sea to the west as well as from Southern Ocean swell, and the Baring Head site in eastern Cook Strait was exposed to the south and southeast. Other buoy records around New Zealand (e.g., Gorman et al. 2003a) are not suitable for extreme wave comparison with the hindcast model, owing to limited duration or sheltering effects. The wave generation model WAM (Wave Model) was implemented over a domain covering the southwest Pacific and Southern oceans with a rectangular grid of resolution latitude/ longitude (Gorman et al. 2003a). The model was used to hindcast the generation and propagation of deep-water waves incident on the New Zealand coast over a 20-year period ( ), using winds from the European Centre for Medium-Range Weather Forecasts (ECMWF) supplied at horizontal resolution and at 10 m elevation. The wind data are supplied at 6-h intervals, between which the WAM model uses linear interpolation of wind speed and direction. Gorman et al. (2003a) compared the hindcast with data from wave-buoy deployments at eight representative sites around the New Zealand coast and found it to provide a satisfactory simulation of wave conditions at sites on exposed coasts (notwithstanding under-prediction of H s during storms). An interpolation routine that included correction for the effects of limited fetch and sheltering by land (Gorman et al. 2003a) was used to extract hindcast wave spectra and statistics at the buoy sites at 3-h intervals. Wave statistics from the buoy records and the matching hindcasts, discussed above, apply to specific nearshore locations that may not fully characterise the wave climate in New Zealand offshore waters at a broader scale. We therefore used the hindcast to predict H s(r) at six sites off the New Zealand coast (Fig. 1) to examine the spatial distribution of extreme waves in relation to the general wave climate presented in Gorman et al. (2003a,b). The lower South Island, particularly the west coast, is exposed to long wind fetches and storms often arise from elongated trough depressions (Steve Reid, NIWA pers. comm.). Conversely, storm waves on the east coast of the North Island tend to be generated over comparatively short fetches by mid-latitude depressions, which often originate in the Tasman Sea before moving across to the east coast (Steve

4 402 New Zealand Journal of Marine and Freshwater Research, 2006, Vol. 40 Reid, NIWA pers. comm.). H s(r) predictions using the IAHR recommended practice The statistical analysis of extreme waves can be undertaken following several approaches (see Muir & El-Shaarawi 1986 for an early review). In analysing both buoy and hindcast data we followed the methods recommended by the International Association for Hydraulic Research (IAHR) for the statistical analysis of extreme waves (Mathiesen et al. 1994). We also trialled the use of the totalsample method (TSM also known as the initial distribution method, Mathiesen et al. 1994) for predicting H s(r) from hindcast data, following Alves & Young (2003). The IAHR recommended practice can be summarised as follows (e.g., Alves & Young 2003): (1) select a subset of H s using the peaks-overthreshold (POT) method; (2) separate the subset of H s maxima into new subsets with data from different storm populations; (3) fit the three-parameter Weibull (3PW) cumulative distribution function (CDF) to the data from each subset; (4) compute the H s(r) values associated with a chosen return-period, for each storm population; and (5) compute confidence intervals for the predicted H s(r). The accuracy of the extreme wave predictions depends on (e.g., Mathiesen et al. 1994; Alves & Young 2003): (1) the quality of the input data, including: (a) the accuracy of the measured or hindcast wave maxima this is fundamental; accurate extreme values rely on high-quality storm data; (b) suitable historic coverage the longer the available record the more reliable the estimates. Increased reliability results from improved statistical precision of the estimates and from decreased error associated with climate variability; and (2) the degree of fit between the true distribution of the storm heights, and the fitted statistical distribution (e.g., 3PW) used to extrapolate to the extreme values. Although buoy records provide adequate storm resolution, they are location specific and very few buoy deployments around New Zealand have sufficient historical coverage for reliable H s(r) predictions. As the model hindcast systematically under-predicts peak wave heights during storms, H s(r) predictions based on hindcast data are likely to be under-predicted also. However, the hindcast has a relatively long record (20 years) compared with available buoy data, so the extreme values predicted from the hindcast have a higher degree of statistical precision, and uncertainty owing to climate variability is minimised. An attempt to scale the hindcast data to improve the accuracy of extreme value predictions is described below. Separation of the data into storm populations that reflect the different forcing mechanisms allows the fitting of the chosen statistical model (e.g., 3PW) to the data and provides confidence that the extrapolation to a chosen return period will be a reliable estimate of H s(r) (Mathiesen et al. 1994). Fitting a statistical model to all H s maxima may lead to unreliable extreme-wave estimates, since the probability distribution does not necessarily fit all values of H s (e.g., Muir & El-Shaarawi 1986; Teng 1998). The hindcast extreme value predictions were made from six sites around New Zealand that experience considerable natural sorting of wave conditions (Gorman et al. 2003a). Based on this natural sorting, and given the aforementioned uncertainty in hindcast storm heights, we chose to forego the considerable effort involved in separating storm origins. The wave-buoy deployment at the Maui platform (Table 1) is the only buoy record of sufficient duration to warrant the separation of storm populations, but it cannot provide an extreme wave climatology by itself. The wave hindcast records consisted of time series of wave statistics at 3-h intervals, including significant wave height, calculated from output energy spectra. The buoy records consisted of wave statistics calculated from 20-min measurement bursts spaced 1 3 h apart depending on the site. The POT method was used to select peak H s values from the wave records and to separate them into individual storm events. First, all values of H s more than one standard deviation above the mean were selected from the record. Second, the local H s maxima were selected from this group, such that the minimum time-interval between local maxima was 3 days (e.g., Mathiesen et al. 1994; Teng 1998). The 3-day separation ensured that the maxima were from separate storms; wave observations at synoptic time-scales are correlated, on average, for days (Lopatoukhin et al. 2000). At most sites this resulted in one storm every days, or a rate of λ = storms per year. This storm rate is similar to other studies that have used partial-duration maxima to predict extreme waves (e.g., Cooper & Forristall 1997, λ = 30.11; Goda et al. 2000, λ = ; Alves & Young 2003, λ = ). The 3PW distribution was fitted to each of the storm peak distributions using the method of leastsquares (Mathiesen et al. 1994). The goodness of fit

5 Stephens & Gorman Extreme wave predictions around New Zealand 403 was assessed using both the Cramér-von Mises and the Anderson-Darling statistics at 95% confidence (Stephens 1974). Generally speaking the Anderson- Darling statistic is effective at detecting departures in the middle of the distribution, whereas the Cramérvon Mises statistic is more effective at the tail of the distribution and is in general more powerful (Muir & El-Shaarawi 1986). Return values were then calculated based on the fitted 3PW distributions following Mathiesen et al. (1994) Monte Carlo simulations were undertaken for each fitted distribution, yielding mean and standard deviation statistics for the calculated return values (e.g., Goda 1988). The Monte Carlo technique provides a better representation of the asymmetry of the confidence intervals about the mean value than standard error techniques (Mathiesen et al. 1994), and provides confidence estimates for those distributions that have no available formulae for return value confidence intervals (Goda 1988). Hindcasts based on spectral wave models simulate significant wave heights for a certain time interval, but do not directly provide estimates of the maximum height H max of individual waves for the corresponding time. Also, the available buoy records do not always include such data. The maximum wave height can, however, be estimated if the distribution of wave heights within a recording interval are assumed to satisfy the Rayleigh distribution, which can be demonstrated to apply for a stationary Gaussian process (Cartwright & Longuet-Higgins 1956). With this assumption, the maximum expected wave height can be estimated as: Hmax = Hs( R) ln( NR) 2 (1) where N R = number of storm waves during return period R and H s(r) is the mean significant wave height with return period R, obtained from the Monte Carlo analyses. We note, however, that return height maxima calculated in this way may be exceeded by rogue or freak waves, such as described by Clauss (2002) and Stansell (2004). However the concept of a rogue wave applies to situations in which the Rayleigh distribution as assumed above does not apply (Haver 2000). H s(r) predictions using TSM data Alves & Young (2003) used satellite altimeter observations of H s to estimate H s(r). The satellite sampling density means that not all extreme events are sampled, hence methods such as POT will result in an underestimation of H s(r) (Alves & Young 2003). To overcome this limitation, Alves & Young (2003) used the TSM to select H s, in conjunction with the Fisher-Tippett 1 (FT1) distribution for extrapolating H s to H s(r), following Carter (1993). The TSM uses all available data (e.g., Goda 1988; Mathiesen et al. 1994), i.e., a statistical distribution is fitted to the whole data set. Alves & Young (2003) observed that although the FT1 distribution did not necessarily provide a good statistical fit, when combined with TSM data it provided a better match to H s(r) predicted from buoy records than when the IAHR method was followed. They suggested that the TSM method combined with the FT1 (or another suitable) distribution, might also be useful for estimating H s(r) from wave hindcast data, which also commonly under-samples storms. We therefore compared the use of both TSM and POT data for the prediction of H s(r) from hindcast H s, and comment on the comparison between different statistical distributions for the representation of the TSM data. The 3PW, FT1, and lognormal CDFs were fitted to the TSM data from each site, using the method of least-squares (Mathiesen et al. 1994). These three statistical distributions have been commonly applied for extreme wave prediction (e.g., Muir & El-Shaarawi 1986; Goda 1988). The Generalised Pareto distribution is suited for the extrapolation of heavily censored data (e.g., van Gelder et al. 2000; Stansell 2004) but did not fit the TSM data well. The goodness of fit was assessed using both the Cramérvon Mises and the Anderson-Darling statistics at 95% confidence (Stephens 1974, 1977). Return values were then calculated based on the fitted distributions following Goda (1988) and Mathiesen et al. (1994). RESULTS We derived extreme wave statistics from wave-buoy records obtained at four sites, and compared these with corresponding estimates from hindcast wave records for the same locations. First we compared methods of analysis for determining H s(r) from hindcast H s and then tested a method for scaling the hindcast extreme wave distributions. We then predicted H s(r), to investigate spatial variations in extreme wave climate around New Zealand. Comparison of methods for predicting extreme waves from hindcast data A number of wave records are available from instruments deployed around the New Zealand coast

6 404 New Zealand Journal of Marine and Freshwater Research, 2006, Vol. 40 Fig. 2 Scatter plots of matching storm peak H s from the buoy records and from hindcast data extracted at the four buoy sites (Table 1). Lines of best linear fit and of equivalence are shown by black and grey lines, respectively. Table 2 Predicted H s(100) (m) from buoy and hindcast H s data. Combinations of data treatment (peaks-over-threshold (POT) and total-sample method (TSM)) and CDF fit (three-parameter Weibull (3PW), lognormal, and Fisher-Tippett 1 (FT1)) were compared for hindcast data. Values in parentheses show the difference between hindcast and buoy predictions as a percentage of the buoy prediction. Goodness of fit to the Anderson-Darling (left) and Cramer-Von Mises statistics (right) are indicated by or (following Stephens 1974, 1977). Statistical distributions were fitted using least-squares and H s(100) calculated directly (i.e., not from Monte Carlo analyses, c.f. Table 3). Buoy Model pot/3pw PoT/3PW (IAHR) (IAHR) TSM/Lognormal TSM/3PW TSM/FT1 Baring Head ( 15%) (5%) 7.34 ( 26%) 7.34 ( 26%) Katikati (40%) (114%) 7.72 (41%) 6.29 (15%) Maui ( 6%) (10%) 8.53 ( 17%) 9.61 ( 6%) Mokohinau Island ( 18%) 9.90 ( 4%) 8.65 ( 16%) 6.97 ( 32%)

7 Stephens & Gorman Extreme wave predictions around New Zealand 405 Table 3 Comparison between H s(100) statistics predicted from buoy records (Table 1) and from hindcast data extracted at the buoy sites. Columns describe the number (n) of storms and the mean and standard deviation (σ) of the H s(100) (m), calculated from a population of 2500 Monte Carlo simulations. m is the difference (hindcast buoy) between the predictions of mean H s(100) from buoy and hindcast data, as a percentage of the hindcast prediction. Distribution fits were applied to all available storm data and did not account for overlap between the buoy and hindcast records. Buoy (IAHR POT / 3PW) hindcast (IAHR POT / 3PW) hindcast (TSM / lognormal) Site n Mean H s(100) σ H s(100) n Mean H s(100) σ H s(100) m (%) n Mean H s(100) σ H s(100) m (%) Baring Head Katikati Maui Mokohinau Is during the hindcast period. We have selected four deployments where the buoy record contained a sufficient number of storms to reasonably fit CDFs (Table 1, Fig. 1). Fig. 2 shows the scatter between matched storm peak H s from the buoy records and from hindcast data extracted at the buoy sites. The hindcast generally under-predicted H s during storms. Table 2 shows a comparison of H s(r) predicted from both buoy and hindcast data. The IAHR recommended practice was followed for the analysis of the buoy data, but several analysis methods were compared for the hindcast data, including the IAHR recommended practice and the fitting of the 3PW, FT1, and lognormal distributions to hindcast TSM data. In making these comparisons we assumed that the IAHR recommended practice gives the best estimate of H s(r) for buoy data, as recommended by Mathiesen et al. (1994) and implemented by Alves & Young (2003). This seems reasonable given the good fit of the 3PW distribution to the POT data (Table 2). The predicted H s(100) values from the buoy record generally exceeded those predicted by the hindcast. The exception was the Katikati site, where the hindcast H s(100) was considerably higher than that from the buoy. The relatively small value of H s(100) obtained from the Katikati buoy data does not appear to relate to the scatter in storm H s for this site (Fig. 2), which would lead us to expect higher return heights predicted from the buoy record than from the hindcast. However, comparison of the 20-year H s time series with the 4-year buoy time series revealed that the buoy deployment ( ) coincided with a period of relatively low wave energy on this coast. Therefore, the buoy record missed some of the largest events occurring at the site since 1979, i.e., the buoy record does not provide a sufficient historical record to accurately extrapolate return wave heights. Thus, the use of short-term records can lead to substantial errors in predicted return values, by not accounting for climate variability. The same constraint was expected to apply to the other buoy records, particularly at the Mokohinau Island site where the record is shortest. However, the timing (and location) of the other buoy deployments meant that they captured some storms of comparable size to the largest storm predicted by the hindcast. We thus omitted the Katikati site from the comparisons that follow. Not considering the Katikati site, the methods can be ranked according to the best to worst match with buoy H s(100) values as follows TSM/lognormal,

8 406 New Zealand Journal of Marine and Freshwater Research, 2006, Vol. 40 Fig. 3 Scatter plots of H s for 24 matching storm peaks at the Mokohinau Island buoy site. Black lines represent lines of best fit from linear regression, grey line represents the line of equivalence. Hindcast values adjusted by linear scaling H s = mh s + c: A, raw data (m = 1, c = 0); B, model data scaled by adjusting m and c; C, model data scaled by adjusting m only (fit constrained through (0,0)); D, model data scaled by adjusting c upward by 0.80 m, so that the mean storm H s values are equal between model and buoy. POT/3PW (IAHR), TSM/3PW, TSM/FT1. Thus, the lognormal distribution fitted to TSM data provided a better match to the buoy predictions than the IAHR method, but the latter provided the next-best match. The TSM/FT1 combination used to analyse satellite altimeter data by Carter (1993) and Alves & Young (2003) provided the worst match. Table 3 shows the comparison of the mean and standard deviation of H s(100) predicted from Monte Carlo analyses of both buoy and hindcast data for the two most accurate analysis methods. Comparing mean H s(100) values obtained from buoy and hindcast data through the relative bias m = (H s(100) (hindcast) H s(100) (buoy))/h s(100) (hindcast) (2) we found that hindcast H s(100) under-predicted buoy H s(100) by mean( m) = 18% when using the IAHR

9 Stephens & Gorman Extreme wave predictions around New Zealand 407 extrapolation method (averaging m across all sites except Katikati owing to the under-sampling of storms by the buoy at this site). The standard deviation of the absolute value of m across the three sites was 12%. Following the same procedure using the TSM/lognormal method, we found that hindcast H s(100) agreed with buoy H s(100) to mean( m) = 1% with standard deviation 3%. The various statistical distributions were also assessed against the quality of their fit to the underlying data. The IAHR method provided a good statistical representation of the hindcast data, but the chosen CDFs did not fit the TSM data well. Thus, although the TSM/lognormal method provided the best match to buoy H s(100), there remains some doubt as to the validity of the approach. It may be possible to use the IAHR recommended practice to predict H s(r) from both buoy and hindcast data, and scale the hindcast predictions according to their relationship with buoy predictions. Fig. 3 shows the effects of linear scaling H s = mh s + c (3) adjusting either the multiplicative factor m, the offset c, or both. A multiplicative factor is required to align the line-of-best-fit with the line of equivalence (Fig. 3B,C). However, applying a multiplicative factor caused the data to be stretched along the line-of-best-fit and this caused a stretching of the distribution (compare Fig. 3A and B), effectively lengthening the distribution tail. Therefore, when the 3PW distribution (or another statistical distribution) was fitted to the data and used to predict return wave heights, there was a disproportionately large increase in return height relative to the original scaling. The problem is manifest through a change to the scale parameter of the Weibull distribution. We rated the percentage difference in H s(100) predicted from buoy and model data, both before and after the linear offset was applied. Comparing data from all sites but Katikati, the best scaling technique was to apply an additive offset (+c) only, adjustments to m generally caused considerable over-prediction of model-derived H s(100) relative to buoy-derived H s(100) values. Applying an 18% additive scaling factor to hindcast H s(100) led to a mean difference of 0%, or 1% (as a percentage of model- or buoy-derived H s(100), respectively) across the three comparison sites. Thus, the linear scaling method shows some potential for improving the match between H s(100) predicted from the hindcast data and buoy data, notwithstanding the remaining scatter between the two sets of storm data. Predictions of extreme wave heights require some form of error analysis to be useful for engineering applications. Fig. 4 indicates the spread of the predicted H s(100) values calculated from Monte Carlo simulations for the four buoy sites, whereas the standard deviations presented in Table 3 indicate the measure of statistical precision. The effect of sample size can be clearly seen in Fig. 4; the buoy at the Maui site measured 332 storms, whereas only 48 storms were measured at Mokohinau Island. H s(100) values predicted from Maui buoy data show a relatively narrow spread, the standard deviation is only 2% of the mean value. Conversely, H s(100) values predicted from the Mokohinau buoy show a relatively wide spread, with a standard deviation of 12% of the mean value. Likewise, the return heights predicted from the hindcast data generally have substantially less spread than the shorter buoy records. Spatial distribution of extreme wave climate estimated from hindcast storms Based on the above comparisons, we chose to use the TSM/lognormal method to obtain H s(r) estimates around New Zealand from hindcast data. The rationale for this choice is discussed further below, but relates to the under-prediction of H s(r) when applying the IAHR method to hindcast data. In general, the predicted H s(r) followed the trend shown by mean H s, with smaller values in the upper North Island and larger values in the lower South Island, with the smallest (H s(100) = 13.9 m) occurring on the northeast coast of the North Island (NI NE) and largest values (H s(100) = 19.3 m) on the southwest coast of the South Island (SI SW) (Table 4). However, the H s(r) showed less spatial variation relative to the mean waves. For example, the mean H s of NI NE was 55% of SI SW, but the mean H s(100) of NI NE was 72% of SI SW. Therefore, although the SI SW environment is considerably more energetic on average than the NI NE environment, mid-latitude depressions are capable of generating storm waves on NI NE that are larger than the mean wave climate would suggest. Engineering design often requires knowledge of the highest force that will be placed on a ship or structure. For example, rogue waves have been known to disable ships and are implicated in the occasional unexplained disappearance of ships in the deep ocean (e.g., Clauss 2002). Table 4 shows the maximum return wave heights (H max(50), H max(100) ) calculated using Equation 1. The predicted H max(r) are > 2H s(r), which is the expected minimum threshold to

10 408 New Zealand Journal of Marine and Freshwater Research, 2006, Vol. 40 Fig. 4 Box and whisker plot of H s(100), obtained from a population of 2500 Monte Carlo simulations. Plots corresponding to the storm data presented in Table 3, from buoy data analysed using the IAHR method and from hindcast data analysed using the TSM/lognormal method. Box lines at the lower quartile (25%), median (50%), and upper quartile values (75%). Whisker lines extending from each end of box mark data within range [q (q 75 q 25 ) : q (q 75 q 25 )], where q 25 and q 75 are the lower and upper quartiles respectively. (+, outlying data.) meet the definition of a rogue wave given by Stansell (2004). Although large, the reported H max could still under-represent the true rogue wave heights associated with the H s(r) distribution, because the Rayleigh distribution under-predicts the occurrence of extreme waves (Stansell 2004). Nonetheless, the predicted H max provide a conservative estimate of maximum return wave heights, in keeping with the H s(r) values. DISCUSSION We have used a 20-year wave hindcast (Gorman et al. 2003a,b) to predict extreme wave heights around New Zealand. The hindcast under-predicted peak H s during storms, similar to other wave hindcasts where kinematically re-analysed wind fields were not included (e.g., Cardone et al. 1996; Sterl et al. 1998; Swail & Cox 2000). This leads to under-prediction of H s(r) using hindcast data, when compared to predictions from buoy data. Considerable effort is required to re-analyse wind fields for all major storms across the hindcast model domain (e.g., Laing et al. 1997; Wang & Swail 2002) and is beyond the scope of the present paper. One advantage of the hindcast is that the H s(r) estimates were predicted with a high degree of precision compared with those available from shorter buoy records, and the extended historical coverage may also provide more accurate predictions if buoy records are of insufficient duration. For example, use of the 4-year Katikati buoy record resulted in a considerable under-prediction of return heights, because the buoy was deployed during a relatively calm period. The 20-year hindcast accounts for considerable short-term climatic variability such

11 Stephens & Gorman Extreme wave predictions around New Zealand 409 Table 4 Mean, standard deviation (σ) and maximum values of hindcast significant wave height H s (m) hindcast extreme significant (H s(50), H s(100) ), and extreme maximum (H max(50), H max(100) ) wave height values at six selected grid cells in New Zealand: northeast of the North Island (NI NE), northwest of the North Island (NI NW), southeast of the North Island (NI SE), west of the South Island (SI W), southeast of the South Island (SI SE), and southwest of the South Island (SI SW) (see Fig. 1). Maximum H s values are the maximum value output in the 20-year hindcast, whereas maximum return heights H max(r) are those predicted from H s(r) assuming that wave heights within a recording interval follow a Rayleigh distribution (e.g., Equation 1). Except for the right-most column, return values were calculated using the TSM/lognormal method. For comparison, the right-most column shows H s(100) calculated using the IAHR method after scaling up by 18%. Mean Mean Mean Scaled Site Lon ( E) Lat ( S) H s σ H s Max H s H s(50) σ H s(50) H max(50) H s(100) α H s(100) H max(100) IAHR H s(100) NI NE NI NW NI SE SI W SI SE SI SW as that associated with the Southern Oscillation (Gorman et al. 2003a). Around New Zealand there is a paucity of buoy data adequate for extreme wave prediction; available buoy records tend to be of short duration and often located close to shore so that the recorded waves are specific to that location. This paucity of data also makes it difficult to assess the hindcast performance for predicting extreme waves; there is limited data for comparison and the hindcast is affected by nearshore topographic and wave-shoaling effects close to land (a nested wave refraction model can be used to improve the predictions close to land for specific areas (e.g., Macky et al. 2000; Gorman et al. 2003b), but at considerable effort). Given the paucity of buoy data, the predictions based on hindcast data provide a useful guide to extreme wave heights around New Zealand. Our analysis suggests that when the IAHR method was applied to hindcast data, H s(r) values were initially underestimated relative to the buoy data by c. 18% (before scaling), based on comparisons at the Mokohinau, Maui, and Baring Head sites (Fig. 1). This underestimation is directly related to the quality of the hindcast storm H s. Applying an 18% additive factor showed some potential for improving the match between H s(100) predicted from the hindcast and buoy data, but a major limitation to the method s general applicability is the requirement for buoy records to obtain the scaling factors. Only three suitable buoy records were available, and this led to considerable uncertainty (standard deviation 12%) in the applied scaling factor. One reason for this uncertainty is that the mean bias includes some geographic variability. It is recommended to separate H s data into separate storm populations before extrapolating H s(r) because the different populations are likely to have different distribution shapes that will affect the fitting of extrapolation CDFs (Mathiesen et al. 1994). The geographical separation between sites does this naturally to an extent because the sites have different wave exposures. Thus, strictly speaking, buoy/model comparisons should be made on a siteby-site basis, and the application of an 18% mean between-site bias as a scaling factor is in violation. This violation, combined with the apparent continual under-estimation of H s(r) following scaling, led us to choose the TSM/lognormal method to obtain the key results. Alternatively, and had the data supported it, the location of the three sites on different coastlines could be seen as an advantage, in that it produces a scaling factor that more generally applies to all areas around the New Zealand coast.

12 410 New Zealand Journal of Marine and Freshwater Research, 2006, Vol. 40 The hindcast model provides a good description of the mean wave climate, but tends to under-predict H s during storms (Gorman et al. 2003a). In situations where storms are under-sampled, an extrapolation method that makes use of the body of the data such as the TSM can be applied (e.g., Carter, 1993; Mathiesen et al. 1994; Lopatoukhin et al. 2000; Alves & Young 2003). Alves & Young (2003) found that the FT1 distribution combined with TSM data provided reasonable H s(r) estimates from altimeter data, despite the FT1 distribution having a poor statistical fit to the data. Our analyses showed the TSM/FT1 method to provide the poorest estimate of H s(r), with the best estimate obtained using a TSM/lognormal combination. However, like Alves & Young (2003), we found that the TSM data was not exactly represented by the lognormal distribution as assessed graphically and by EDF statistics (Stephens 1974). Alves & Young (2003) suggest that this may be because the much larger number of data points used in the TSM, relative to POT data, leads to relatively narrower tolerance limits at a given statistical confidence level. Although graphical analysis showed that the three CDFs did not fit the tail of the TSM hindcast data exactly, it also showed that the lognormal distribution provided a reasonable if not an exact match. The TSM/lognormal combination is the method that was originally applied in the early days of extreme wave analysis (Mathiesen et al. 1994). Muir & El-Shaarawi (1986) pointed out that there is no physical reason for preferring one distribution over another and recommended use of the distribution which fits the data best. Here, we have chosen the lognormal distribution because it provided a reasonable match to the data, but mainly because it provided the closest match of H s(r) when compared to the buoys. There was some scatter between the predicted H s(r) when different extrapolation CDFs were applied to the TSM data, which leaves doubt as to which CDF should be applied if comparison data were absent, and suggests that further investigation is warranted on a bigger data set. Our analysis is far from exhaustive with only three buoy model comparison sites available and three statistical distributions applied. Thus, the suggestion by Alves & Young (2003) that TSM data might provide useful estimates of H s(r) from wave hindcast data remains to be fully explored. It would be interesting to see if the TSM/lognormal combination or TSM with another statistical distribution performed well on a data set with more comparison sites (e.g., Cardone et al. 1996; Swail & Cox 2000). The only truly robust method that is currently available for predicting extreme waves from hindcast data is to accurately hindcast the storm waves through the use of kinematically re-analysed wind fields input to the model (e.g., Wang & Swail 2002). Our results show H s(100) values ranging from 14 m in relatively sheltered waters to the north of New Zealand to >19 m in exposed waters to the south. These values are similar to those predicted using altimeter data by Alves & Young (2003). Examples of maximum H s(100) predictions from buoy data in the northern Pacific are 16 m (Yamaguchi & Hatada 1997) and 17.9 m (Alves & Young 2003), whereas H s c. 16 m have been measured (Cardone et al. 1996) and >17 m hindcast have been predicted (Wang & Swail 2002) in the northern Atlantic. Goda et al. (2000) provide H s(100) estimates of m along exposed sites of the Japanese coast that have similar extreme wave probability to our northern output sites; the northern Pacific and Atlantic oceans have similar extreme wave probability to our southern output sites (e.g., Alves & Young 2003). These comparisons suggest that our estimates obtained using the TSM/lognormal method are reasonable. ACKNOWLEDGMENTS This work was carried out with support from the Public Good Science Fund (Contract No. C01X0401) of the New Zealand Foundation for Research, Science and Technology. Thanks to Dr Murray Smith of NIWA and two anonymous reviewers for insightful comments that led to improvements in the manuscript. references Alves JHGM, Young IR Extreme significant wave heights from combined satellite altimeter data. In: Edge BL ed. Proceedings of the 27th International Conference on Coastal Engineering. Sydney, ICCE. Pp Alves JHGM, Young IR On estimating extreme wave heights using combined Geosat, Topex/ Poseidon and ERS 1 altimeter data. Applied Ocean Research 25: Cardone VJ, Jensen RE, Resio DT, Swail VR, Cox AT Evaluation of contemporary ocean wave models in rare extreme storm events: the Halloween storm of October and the storm of the century of March Journal of Atmospheric and Oceanic Technology 13: Carter DJT Estimating extreme wave heights in the NE Altlantic from GEOSAT data. Offshore Technology Report prepared by Institute of Oceanic Sciences Laboratory for the Health and Safety Executive, OTH p.

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