Master of Science Thesis Report by JAMEL D. BANTONParametric Models and Methods of Hindcast Analysis for Hurricane Waves by Jamel D. Banton May, 2002 IHE / ALKYON M.Sc. Thesis Report

PARAMETRIC MODELS ANDMETHODS OF

HINDCAST ANALYSISFOR HURRICANE WAVES

Spatial Distribution of Peak Vmax for Hurricanes of The North Atlantic Basin

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VariationinOccurrenceofTropicalStormsandHurricanesinTheAtlanticBasin

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PolynomialFit

Submitted To:

Prof. Han Vrijling

Ir. David P. Hurdle

Ir. M. van der Wegen

Master of Science Thesis Report

by

JAMEL D. BANTON

May, 2002

Parametric Models and Methods of Hindcast Analysis for Hurricane Waves by Jamel D. Banton May, 2002 IHE / ALKYON M.Sc. Thesis Report Page I

EXECUTIVE SUMMARY

ES-1.1 INTRODUCTION The North Atlantic Basin has seen recent increases in Tropical Cyclone frequency and in-tensity. Researchers are proposing that this is a result of global warming. Whether global warming is simply a cyclic climatic trend or a human induced event is still being argued. Meanwhile, the ravaging impacts of the generated storm waves on the islands and coast-lines in and around the Caribbean continue. Consequently, it is left up to the Engineers to determine and prepare for the probable impacts of these waves. Scientists and Engineers around the world have naturally developed a fascination with these cyclones (called Hurricanes in the North Atlantic Basin). With this fascination, we have de-veloped a range of prediction models for cyclone-generated winds and waves. In addition, we have adapted statistical procedures to analyze the probable wind and wave conditions for different time intervals. These conditions are vital in determining the potential shoreline impacts such as storm surge levels and coastline erosion. There are several prediction models and similarly several commonly used statistical meth-ods. Unfortunately, these models and methods often give varying results and as such it be-comes unclear as to which are most suitable for application to a particular situation. In addi-tion, there is only a handful of measured data to inter-compare and calibrate these models. The prediction models include complex numerical models and also simplified parametric models. The numerical models supposedly give more accurate results than the simpler and more parameterized models. However, they are far more financially and computationally ex-pensive. The common statistical methods of analysis are the modified extremal methods, traditionally used in river/rainfall flooding designs. To a far lesser extent, a more recently developed ap-proach, the Monte Carlo Method, has been applied. There are several ways of applying the historical methods, which sometimes give varying results. As for the Monte Carlo Approach, there is no widely accepted published method. The beautiful sandy beaches of the Caribbean Sea are the prime source of foreign ex-change income in most of the islands. In addition, the most developed and populated areas are the coastal regions. The financial resources and the luxury of time are often not avail-able to employ the use of complex numerical prediction models to study the probable im-pacts of hurricane waves on these islands. Furthermore, the Monte Carlo approach requires a large number of computations, which are usually not possible with these models. There-fore, an evaluation of the available parametric models and an assessment of the various statistical procedures is now a most relevant and necessary assignment. Even more essen-tial, is the development of an appropriate tool to easily apply the best methods.

Parametric Models and Methods of Hindcast Analysis for Hurricane Waves by Jamel D. Banton May, 2002 IHE / ALKYON M.Sc. Thesis Report Page II

ES-1.2 AIMS AND OBJECTIVE It is the aim of this study to garner the available storm data, evaluate its reliability, and then use it in the application of the various parametric prediction methods both for inter-comparative purposes and for comparisons with measurements. Thereafter, the most widely used statistical procedures were applied and their validity to the relevant storm hindcast procedures examined. It was also the intention to examine the properties of the various characteristic components (wind speed, central pressures, radius of maximum winds, etc.) of storms in the North Atlantic Basin. For example, the inter-correlations between the asso-ciated pressures and wind speeds, and the temporal distributions of storm frequency and intensity were evaluated. Finally, a computer program should be developed to allow for quick application of relevant procedures. The study’s objective was then to provide Engineers and Scientists in and around the Carib-bean with insight to the hindcast procedures for hurricane waves, through comparisons of the available parametric wave models and the commonly used extremal statistical methods and a suitable computer program to perform hurricane wave hindcasting.

ES-1.3 SCOPE OF WORK The study was undertaken at Alkyon Hydraulic Consultants and Research, The Netherlands, under the supervision of David P. Hurdle. A number of topics were covered ranging from the availability of hurricane records to the application of the appropriate statistical procedures used to find return values of wave heights. In addition to this wide coverage of topics a computer program with an advanced windows interface, named HURWave, was developed. The program incorporates all the parametric models and statistical procedures presented in the study. It therefore has the capacity to compute return wave heights for any given loca-tion within the North Atlantic Basin for any model or statistical procedure of choice. The study also presents a Monte Carlo approach for Hurricane Statistical Analysis. The ap-proach involves random generation of the various statistically independent components of hurricanes to derive a synthetic database of storm tracks. This procedure will be most useful in cases when limited availability of data does not allow for a complete definition of a particu-lar statistical distribution. This procedure is also included in HURWave and as such is easily re-applied to any area of interest. The following is a list of the tasks outlined for the study: • Compile the available hurricane records from various organizations and evaluate the

consistency of these records over the entire period for which they are available. • Summarize the statistics of hurricane frequency and intensity and derive the statistical

relations between the characteristic components of hurricanes. • Compare the results of hurricane wave prediction models to available wind and wave

measurements from data buoys and remote sensing techniques.

Parametric Models and Methods of Hindcast Analysis for Hurricane Waves by Jamel D. Banton May, 2002 IHE / ALKYON M.Sc. Thesis Report Page III

• Evaluate the various statistical methods used in hindcast analysis of extreme events and establish their relevance and applicability to hurricane waves.

• Develop a Monte Carlo Approach to carry out hindcast analysis for storm waves. • Develop an all-encompassing computer program to carry out hindcast analysis any-

where in the North Atlantic Basin.

ES-1.4 SUMMARY OF STUDY TOPICS HURRICANE PARAMETER VARIATION It is essential to first understand the properties of the characteristic components, before any form of analysis is carried out on their generated waves. We need to establish their individ-ual statistical properties as well as their inter-correlations. For example, it is useful to know 1) whether the central pressure/wind speed of a storm is dependent on the storms geo-graphic location, 2) if a particular location is likely to have more extreme storms than another or 3) whether there have been temporal patterns of storm occurrence. As such, for this section examined the distributions and established the relationships, if any, between the components involved. Specifically, we examined: a) the spatial distribution of hurricane occurrence; b) the temporal distribution of intensity and occurrence; c) the spatial distribution of the peak Vmax of each storm, and; d) the correlations that exist among the central pressure, the maximum wind speed, the latitude of occurrence, the forward speed and the peak wind speed along each storm track. The Best Track database of storms was used to evaluate these aforementioned distributions and correlations. All storms occurring between 1900 and 2000 were included. The following are a number of the findings:

• Hurricanes do have a preferred track that is geographically dependent. While in the Caribbean Sea, they tend to move in a west to northwesterly direction. In the Gulf of Mexico and on the east coast of the USA they tend to track north-northwesterly to northeasterly. While present understanding of the complex climatic patterns that propel these storms does not allow for accurate predictions of these tracks, reason-able predictions of storm tracks can be made from these patterns.

Parametric Models and Methods of Hindcast Analysis for Hurricane Waves by Jamel D. Banton May, 2002 IHE / ALKYON M.Sc. Thesis Report Page IV

Variation in Occurrence of Tropical Storms and Hurricanes in The Atlantic Basin

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Figure A: Temporal Distribution of North Atlantic Storm Occurrence

• There seem to be a multi-decadal cycle in the number of hurricane occurrence and

intensity as shown in Figure A above. The figure shows the variance of the annual number of storm occurrences from 1900 to 2000. In addition, a polynomial fit is shown. The fit shows a sort of cyclic trend in the variation of the number of storm occurrences. The plot suggests that the recent increase in Tropical Cyclone activity may in fact be part of this multi decadal cycle, rather than a manifestation of an un-natural Global Warming as some researchers speculate.

• The central pressure and Vmax are the only two parameters that show an obvious

correlation between each other. However, there is still a wide spread in the relation-ship between the two components. A formula was derived expressing the central pressure as a function of Vmax and including a random variable, α, to account for the spread. This formula is given by:

Pc a bVc

d= + --

( ) max( )

αα

(Pc in mbars and Vmax in knots), where a = 1014, b = 0.029,

c = 1.626, d = 200 and α is a random variable uniformly distributed in the interval [-10, 10]. α=0 for the expected value of Pc.

PARAMETRIC WIND AND WAVE MODELING The use of parametric models to describe the hurricane wind and wave field is widely used in the field of Ocean and Coastal Engineering. Researchers have developed both wind and wave models that give relatively accurate predictions. Most of these models have been de-veloped empirically. For the wave models, the data set for the empirical developments have come from either actual measurements or outputs of complex numerical models. In either case, the theoretical basis is similar for most of the models. However, when applied to a par-ticular case the models tend to give varied predictions. With a varied choice of models, it is then useful to examine the performance of each model. A number of wind and wave models were considered. The wind models include the Holland (1980) model, the Hydromet (1990) model, the Bret Model-X (1990) and the Fujita model (1962). The wave models include Ross (1976), Cooper (1988), Young (1988), Bretschneider (1990) and an Improved Young (1995). In order to apply the models an expression was de-

Parametric Models and Methods of Hindcast Analysis for Hurricane Waves by Jamel D. Banton May, 2002 IHE / ALKYON M.Sc. Thesis Report Page V

veloped to determine the radius to maximum winds for each storm. This is essential to de-fine a hurricane’s wind field. An expression was derived from measurements. This is given by: R x e Pc

max.= -63 10 0 017 ; where Rmax is in km and Pc is in mbars.

The available measured data were buoy measurements for four (4) hurricanes as well as Synthetic Radar Altimeter (SRA) spatial readings for Hurricane Bonnie (1998). Futile at-tempts were made to secure satellite data. Table A below shows the results for the wave models with buoy measurements from four (4) hurricanes. The RMS error and the bias are shown for each case.

Hurricane Felix'95 Erin'95 Bertha'96 Opal'95Buoy ID 41001 42036 41010 42001

Mesured Max Hs 7.6 4.6 6.26 8.3Ross 7.68 4.42 3.93 9.12 1.24 -0.40Ross_Holland 9.59 8 9.05 14.92 4.10 3.70Ross_Hydromet 11.51 8.44 9.24 15.18 4.64 4.40Ross_Bret Model X 9.35 7.92 7.61 14.71 3.77 3.21Ross_Fujita 9.89 8.02 9.45 14.84 4.18 3.86Cooper 12.07 6.58 4.88 9.71 2.64 1.62Cooper_Holland 9 7.71 7.71 12.95 2.97 2.65Coop_Hydromet 10.21 7.82 7.81 13.02 3.24 3.03Coop_Bret Model X 9 7.71 6.9 12.94 2.90 2.45Cooper_Fujita 9 7.71 7.91 12.95 3.00 2.70Young 9.5 6.13 5.75 11.9 2.19 1.63Improved Young 9.49 6.29 5.56 10.69 1.78 1.32Bretschneider 9.82 8.25 6.84 12.24 2.92 2.60

RMS Error Bias

Table A – Wave Height Model Predictions vs Buoy Measurements The concept of “mixing models” was introduced whereby the parent wind model of a wind-wave model was replaced by each of the others being studied. It is evident from the above table that with the exception of Ross, all the wave models predicted waves higher than measured. The models of Ross and Young gave the best performances. Table B below shows the results for the wave models with SRA data from a NASA Goddard flight into Hurricane Bonnie (1998). The model predictions are shown for 13 locations rela-tive to the hurricane’s center. For example, 150E is a point 150km east of the hurricane’s center. The concept of “mixing models” was also applied.

Parametric Models and Methods of Hindcast Analysis for Hurricane Waves by Jamel D. Banton May, 2002 IHE / ALKYON M.Sc. Thesis Report Page VI

Point Location Relative to Average Position of Hurricane Center (km)150 W 100 W 50 W 50 E 100 E 150 E 0 150 N 100 N 50 N 50 S 100 S 150 S

SRA Data 7.1 7.4 9.8 9.8 9.8 9.5 6.9 9.5 10.4 9.5 5.8 6.4 5.8 RMS Error BiasRoss 2.7 4.7 8.1 8.1 8.1 3.8 4.1 3.6 5.2 6.4 5.5 5.5 3.7 3.47 -3.1Ross_Holland 7.2 10.5 15.4 15.4 15.4 7.4 5.9 8.0 11.0 12.6 10.1 11.6 8.3 3.96 2.8Ross_Hydromet 9.7 11.1 12.1 12.1 12.1 9.8 6.7 10.0 11.2 10.8 9.0 11.4 10.2 2.74 2.4Ross_Bret Model-X 6.5 10.4 15.6 15.6 15.6 6.9 8.6 7.5 11.0 13.4 11.6 11.6 7.8 4.27 3.1Ross_Fujita 11.2 13.6 16.0 16.0 16.0 11.4 9.1 11.8 13.9 14.2 12.1 14.2 12.1 5.52 5.4Cooper 3.8 5.6 8.8 8.8 8.8 6.8 12.2 5.4 6.3 8.8 10.3 6.4 5.4 2.87 -0.9Cooper_Holland 6.3 8.6 12.2 12.2 12.2 6.5 6.1 6.9 9.0 10.7 8.8 9.4 7.1 2.23 0.8Cooper_Hydromet 7.7 9.0 10.4 10.4 10.4 7.8 7.9 8.0 9.1 9.9 9.0 9.3 8.2 1.67 0.8Cooper_BretModel-X 6.0 8.6 12.3 12.3 12.3 6.2 9.3 6.6 9.0 11.4 10.6 9.4 6.9 2.65 1.2Cooper_Fujita 8.5 10.2 12.5 12.5 12.5 8.6 9.8 8.9 10.5 11.8 11.0 10.7 9.1 2.91 2.4Young 7.4 9.6 9.2 13.9 12.7 11.3 9.9 8.0 10.0 11.9 11.1 7.8 7.1 2.66 1.9Improved Young 6.6 8.6 8.2 12.4 11.3 10.1 8.9 7.1 8.9 10.6 9.9 7.0 6.3 1.89 0.7Bretschneider 6.9 9.7 10.3 10.3 10.3 7.4 5.8 8.9 10.2 9.1 7.3 7.5 6.3 1.13 0.2RMS Error for all models 2.13 2.85 3.23 3.46 3.32 2.59 2.39 2.63 2.23 2.42 4.27 3.82 2.70RMS Error for best 3 models 0.46 1.74 1.05 1.57 1.00 1.61 1.43 1.67 1.13 0.71 3.11 1.82 1.42

Right-Front Quadrant of Hurricane Table B – Wave Height Model Predictions vs SRA Measurements The closest predictions to actual measurements are shown in red italics. The models of Bretschneider (1990), Young (1995) and Cooper (using the Hydromet model) gave the best performances. These three models also gave a very good performance in the front right quadrant of the hurricane. The models tend to over predict the measurements for most cases. A number of conclusions were drawn from the comparisons. These include:

• Both the wind and wave parametric predictions far exceed the NDBC buoy meas-urements.

• The parametric wave predictions compared better with the SRA spatial data for Hur-

ricane Bonnie’98.

• The significant difference between the buoy measurements and wind and wave pre-dictions questions the integrity of either one. However, given the performance of the models with the SRA data, it is recommended that further studies be carried out to test the accuracy of the buoy data.

• The parametric wind wave models performed better in a few cases when its parent

wind model was replaced by another wind model. This demonstrates that weak-nesses in the wind wave models are sometimes inherent to the wind models and re-placing a wind model with another (mixing models) may indeed improve the overall performance of a wind wave model.

• The number of comparisons carried out was limited by data availability and are not

sufficient to definitively say which model is the best. As such, further data sources should be explored such as satellite data.

DATA BASE SELECTION CRITERIA Before a hindcast analysis is carried out, one is faced with a number of choices. Available, is

Parametric Models and Methods of Hindcast Analysis for Hurricane Waves by Jamel D. Banton May, 2002 IHE / ALKYON M.Sc. Thesis Report Page VII

a database of storm track information from which one has to extract the relevant data (storm positions, central pressure, wind speed, etc.) set on which to perform statistical analyses. The database is not consistently accurate hence one may choose to ignore that unreliable part of the database. But on the other hand, there is the need to have a sufficiently robust data set for statistical analyses. Another question is whether all the storms in the database can be regarded as having the same statistical properties. That is, can the least intense storms and the most intense be treated as being from the same statistical population? In addition, we discussed the limiting radius for inclusion of storm occurrence. It is postulated that storms occurring far away from the point of interest will not produce any extreme waves, but is this true? Also, the parametric models may not be applicable for wave predictions at great distances from the storm center. Finally, we discussed which points along a storm track are most likely to produce the maximum wave height. This is relevant, as often one can only use the wave height from one point along the storm track to limit the required com-putational effort. The wave models were applied for storms at five (5) locations within the basin. In response to the questions raised, the following are recommended:

• The inconsistency in hurricane intensity for the past 100 years is evident. It is un-known whether this is a true representation of the multi-decadal trend or whether it is a manifestation of errors in the database. For return periods of 50 years and over one should consider the use of the 100 years of data while for lesser return periods such as 25 years, the latter 50 years of data should be used. Further studies should be conducted to look on the variation in return values for different time intervals over the past 100 years.

• The cyclones of the North Atlantic, regardless of intensity, may be treated as belong-ing to the same statistical population.

• There are limiting radii beyond which storm effect may be ignored. This is however dependent on the model and geographic location. A maximum radius of 400km is recommended for use.

• The maximum Hs from a point will mostly occur when a storm is at its closest to the point in question. This will be most evident for points close to land masses, for ex-ample within the Gulf of Mexico, and will be less evident for points over open ocean. For the latter, the maximum Hs will occur at the point of occurrence of the peak Vmax with a probability of just less than 50%. It is recommended that an attempt be made to always use all the points in the record.

THE HISTORICAL APPROACH TO HINDCASTING This “Historical Method” for extremal analysis may be broken down into four steps. The first is the selection of the appropriate data series of wave heights upon which to perform statis-tical formulations. Two data series, namely, the peak value series and the annual maximum series were applied. The second is to determine whether the data series should include the entire population of waves. In other words, should the data series be censored to get rid of the smallest values? The third is to select the best method of data fitting for the censored or uncensored data series from a range of methods. Finally, a statistical distribution has to be chosen to fit the given data set. Among these distributions are the Fischer Tippet and

Parametric Models and Methods of Hindcast Analysis for Hurricane Waves by Jamel D. Banton May, 2002 IHE / ALKYON M.Sc. Thesis Report Page VIII

Weibull distributions. Given the choice of two data series, varying acceptable limits of censoring, more than three established methods of data fitting and a number of distribution functions, the Engineer could be easily overwhelmed with too many choices. It is the aim of this part of the study to limit the number of choices by means of elimination of those that are not well suited for hurricane hindcast analyses. This was carried out by means of comparisons between the various data series, censoring limits, methods of data fitting and statistical distributions. The comparisons were carried out with the program HURWave. The program was used to perform runs for a total of 100 Atlantic basin deep-water sites. The sites were spaced no less than 300km apart. For each site, runs were carried out for two (2) data series; the peak value series and the annual maximum series. For each data series, runs were carried out for three (3) censoring limits; Hs>0, Hs>3m and Hs>5m. For each censoring limit, runs were carried out for three (3) methods of data fitting; the method of least squares, the method of moments and the least squares method of Hurdle. This means that for each site 18 com-parisons were carried out, resulting in an overall of 1800 points for comparison. Two sets of these comparisons were done using the results of the parametric wave models of Cooper and Bretschneider. From this set of comaparisons, a number of conclusions and recommendations were drawn. These are summarized below:

• The Annual Maximum Data Series should not be censored. Researchers such as Goda [Ref 9] have found that the difference between the two data series (Peak Value and Annual Maximum) becomes small as the return values increase. The evi-dence found here show that this is true. But, they also show that the difference in-creases for both the low and high return periods as the threshold value increases. Therefore when the data series is censored then there is a significant difference be-tween the return values given by the two data series.

• The Least Squares method of Hurdle is indeed a promising method for deriving the best-fitting distribution function for the Weibull distribution. However, the perform-ance of this method has room for improvement. Similarly, to the other methods used here, the method fits the tail of the distribution first. As such, the tail governs the fit of the entire distribution. This sometimes results in a poor fit for the extreme values in the data series.

• The Method of Moments should never be applied for censored data as the theoreti-cal mean and variance estimates do not account for censoring.

• The Annual Maximum data series is not recommended for use to find low return pe-riods.

• The underlying implication is that the Peak Value series is a more reliable data se-ries as it allows censoring without much variation in results for either the low or high return periods.

Parametric Models and Methods of Hindcast Analysis for Hurricane Waves by Jamel D. Banton May, 2002 IHE / ALKYON M.Sc. Thesis Report Page IX

The recommended approach is shown in the flow chart below:

Predict Wave Heights withParametric Model

Extract Peak Wave HeightsTo Form Peak Value Series

Extract Annual Maximum Wave Heights To Form Annual Maximum Series

Censor the Data Series, if desired, but for Maximum

Threshold Value of 3m

Use Method of Least Sqaures To Fit Distributions

Select Best-Fitting Distribution

Compute Return ValuesAnd Confidence Levels

Do Not Censor the Data Series

Do Not Use Annual Maximum Data Series For Low Return

Periods

For A Censored Data Series, Do Not Use Method of Moments

To Fit Distributions

(Use Method of Hurdle for Weibull Distributions)

The Model of Bretschneider Predicts Wave Heights with

Weibull Distributions Having Unusually High Shape Factors

F l o w C h a r t A : R e c o m m e n d e d A p p l i c a t i o n o f t h e H i s t o r i c a l A p p r o a c h t o H i n d c a s t i n g MONTE CARLO APPROACH The method requires the random generation of a number of variables, which are statistically independent of each other. This allows for the simulation of any number of statistically inde-pendent events. Hurricane occurrence can be considered to be a random event. Further-more, the relevant characteristics of hurricanes can be described with a number of variables for which the statistical distributions are known or can be found. In addition, the correlation between these components is easily established. We have previously established that whereas there is a relationship between the Vmax, Pc and Rmax, there are none noticeable between the other parameters of interest. This makes the Monte Carlo simulations suitable for application here. A Monte Carlo procedure was developed and is summarized in the following flow chart.

Parametric Models and Methods of Hindcast Analysis for Hurricane Waves by Jamel D. Banton May, 2002 IHE / ALKYON M.Sc. Thesis Report Page X

Plot the values of the pVmaxand Determine the Best-fitting

Theoretical Distribution Function.

Generate Random Values of pVmax From the Distribution, Equal

to Number of Synthetic Storms Required

Choose a Storm at RandomFrom the Parent Population

Determine the Peak Vmax (pVmax) and the Minimum Distance (Dmin) for each Storm

Search Synthetic DatabaseAnd Re-adjust the New Values of pVmax that

Exceed the Random pVmax for Each Synthetic Storm

Plot the values of the Dmin and Determine the Best-fitting


Extract Parent Population ofStorms From Database For Site Location

Generate Random Values of Dmin From the Distribution, Equal


Loop

for

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f Re

q. S

ynth

etic

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rms

Set the Dmin of this Storm to the Randomly Selected Dmin

Perform Extremal Analysis on Wave Heights

Calculate Wave Heights for Each Synthetic Storm

Set the pVmax of this Storm to the Randomly Selected pVmax

For each Synthetic Storm:Generate a Random Number [-10,10]

Calculate Pc from Equation 3-1Compute Rmax

For example, if this procedure were to be applied to an area at the south coast of Jamaica then first the center point of interest and the radius of inclusion would have to be specified. The database of storms would then be searched and all the storms within the specified ra-dius selected. The peak Vmax (pVmax) within the area and the minimum distance (Dmin) to the point of interest is extracted or calculated from the database. The series of Dmins and pVmax are then plotted and fitted to a number of statistical distributions. The best fitting dis-tributions are selected and their theoretical functions established. For each synthetic storm to be generated, a Dmin and pVmax is selected at random from their respective theoretical functions. Thereafter, a storm is selected at random from the parent population. This storm is moved towards or away from the center point by moving the position of its real Dmin to

Parametric Models and Methods of Hindcast Analysis for Hurricane Waves by Jamel D. Banton May, 2002 IHE / ALKYON M.Sc. Thesis Report Page XI

the position of the randomly selected Dmin. Its real pVmax is also changed to the randomly selected pVmax and all the other values of Vmax along its track are scaled accordingly. Therefore, if the real pVmax is increased by a factor of 1.2 then all the other Vmax values along the storm’s track is increased by this said factor. . When each storm is moved, for a few cases, the new pVmax within the area of interest may be greater than the original pVmax. To counter this, the pVmax of each synthetic track is scaled to the previously ran-domly selected pVmax for that storm. For each of these new tracks, the central pressures are computed with the formula given above for Pc in terms of Vmax. In applying this formula the random parameter α is used so a random Pc (within the measured range of values) is always computed. The Rmax is then computed from this randomly selected value of Pc. A wave model is then applied to derive the data series of wave heights for the synthetic tracks. An extremal analysis follows to determine the return values. A number of simulations were carried out for two deep-water sites. For each site, 5 synthetic populations were generated of 50, 100, 200, 500 and 1000 storms. For each population the return wave heights were calculated using extremal analysis. This was done with the signifi-cant wave height predictions from two parametric models: Bretshneider and Cooper. An ex-ample of the return values for the 5 and 50 year return periods are shown in the plot below. These are compared to those of the parent population (the first pair).

Variation in 5 & 50-Year Wave Height for 5 Synthetic Databases

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9

10

11

12

13

14

15

23 Average 50 100 200 500 1000

Number of Storms

Hs

(m)

5-year50-year

real database

The return values for the 5-year return period seem to increase with increasing number of storms. On the other hand, there is no such pattern in the variation of the 50-year values, which is expected for a random simulation procedure as this one. Hence, the change in the shape of the distribution seems to affect the low return periods but then these are not impor-tant values for design. The 50-year values for the synthetic populations seem to lie within 1.5 m of the parent population for all cases that were done. The average return values of all the synthetic values are shown. Of the four cases, only one showed an average value that differs from that of the parent population by more that 0.3m. This is a very good agreement between the synthetic and real cases and shows the validity of the Monte Carlo Approach presented.

Parametric Models and Methods of Hindcast Analysis for Hurricane Waves by Jamel D. Banton May, 2002 IHE / ALKYON M.Sc. Thesis Report Page XII

ES-1.5 DISCUSSION In concluding, it must be said that this work sets the stage for numerous possibilities in Tropical Cyclone Wave Design Engineering as well as further development into more so-phisticated modeling techniques. The former has already taken effect as HURWave and the study’s various findings have been applied to find deepwater wave heights for several is-lands in the Caribbean Sea. These islands include St. Kitts, Nevis, Antigua, Trinidad, The Bahamas and Jamaica. The works carried out have been vital contributions to Coastal Zone Management Plans and Hurricane Mitigation studies for these islands. The latter is bound-less. The potential developments and improvements to this study include more comprehen-sive comparisons of the wave models with satellite data, comparisons between parametric and numerical wave models, more detailed filtering of hurricane activity data to establish po-tentially hidden trends, further development of the Monte Carlo to generate independent random storm tracks using auto-correlation formulations, etc. Evidently, the study does not cover the most sophisticated and state of the art formulations and techniques, even though it does present interesting scientific findings and recommendations. On the contrary, it ex-plores a vast range of known techniques and offers a pragmatic guide for “preliminary” ex-treme wave design. These are computationally inexpensive, reasonably accurate and as such are well suited for application within the Caribbean and the rest of the developing world, which are frequently ravaged by “the greatest storms on Earth”.

Parametric Models and Methods of Hindcast Analysis for Hurricane Waves by Jamel D. Banton May, 2002 IHE / ALKYON M.Sc. Thesis Report Page XIII

ABSTRACT Hurricanes are often called “the greatest storms on Earth”. They have earned this title through their devastating impacts on the people and economies of many nations including those in and around the Caribbean Sea and the Gulf of Mexico. Annually, coastal areas of these nations undergo inundation from the storm surges and severe wave attack induced by passing hurri-canes. In order to evaluate the effects of these extreme storm waves, the engineer must first derive the probable wave heights for particular return periods. This process requires the proper hurri-cane-wave predictors and knowledge of the suitable statistical methods. There are a number of wave predictors and statistical methods. These often give different results and as such, engi-neers generally have different preferences. The wave models in some cases are unreliable and the statistical methods are often misused. A study was carried out with the objective of giving insight to hindcast analysis of hurricane waves, through comparisons of the available parametric wave models and the commonly used extremal statistical methods. In addition, the study aimed to develop a Monte Carlo simulation capable of generating a synthetic database of storms from the statistical properties of the par-ent population. This method may offer a viable alternative in cases where the number of rele-vant historical records is too low to be able to carry out reliable statistical analysis. This may occur where the tropical cyclones are infrequent, they have been intermittently recorded or a point is highly influenced by the local coastal geometry. The study entails the following: 1) A compilation and validation of available hurricane records in the North Atlantic Basin; 2) Examination of the correlation between, and the statistical proper-ties of the characteristic hurricane components; 3) Comparisons of parametric hurricane wind and wave predictions with measured data. The wave models applied were those of Ross (1976), Cooper (1988), Young (1988 & 1995) and Bretschneider (1990); 4) An evaluation and inter-comparison of the various methods used in the approach for analysis of storm waves. These comparisons focused on the use of i) two types of data series; ii) three methods of data fitting; iii) various threshold values and iv) a number of extremal distributions; and 5) Develop-ment of a Monte Carlo simulation procedure to derive a synthetic historical database. To carry out the necessary computations, a programming tool, named HURWave, was developed. The program was developed in Visual Basic and is capable of carrying out hindcast analysis of tropical cyclone anywhere within the North Atlantic Basin. The following are just a few of the conclusions drawn: 1) The historical variation in Tropical Cyclone occurrence and intensity in the North Atlantic

Basin implies a multi-decadal cycle, which is being manifested in the increased number of occurrences for the past few years;

2) The distribution of hurricane wind speeds prior to 1950 is inconsistent with those after 1950. Whether this is a result of data inaccuracy or multi decadal variations in hurricane in-tensity is unsure. However, the latter seems more plausible from concurrent correlations with sea surface temperatures.

3) The wave model of Young showed the most consistent agreement with measured wave data. However, the number of comparisons carried out was not sufficient to definitively conclude that this is the best of the parametric models;

4) The use of the peak over threshold method of statistical analysis is demonstrated to give more reliable results than the annual extreme value method; and

5) The Monte Carlo simulation is valid for deriving synthetic storm populations and extreme values of the design parameters.

Parametric Models and Methods of Hindcast Analysis for Hurricane Waves by Jamel D. Banton May, 2002 IHE / ALKYON M.Sc. Thesis Report Page XIV

The study was carried out at Alkyon Hydraulic Consultancy and Research, under the supervi-sion of Alkyon’s David P. Hurdle and IHE’s Mic van der Wegen.

Parametric Models and Methods of Hindcast Analysis for Hurricane Waves by Jamel D. Banton May, 2002 IHE / ALKYON M.Sc. Thesis Report Page XV

ACKNOWLEDGEMENT This work could not have been carried out without the financial support provided by Alkyon Hy-draulic Consultancy and Research and The Laminga Foundation. I therefore would like to ex-tend my sincerest gratitude to these parties. I am indebted to Verhagen and Mic van der Wegen, my IHE mentors, who played significant roles in securing this funding. The study was undertaken at the office of Alkyon Hydraulic Consultancy and Research. I there-fore express my appreciation to the directors and members of staff for allowing me this oppor-tunity and for lending me the much-needed technical support during the period of study. I must make special mention of David Hurdle, my supervisor, who has provided expert advice and guidance. I am also compelled to mention Peter van der Bosch, Peter Santbergen and Joost Hoekstra for providing unwearied support with Visual Basic. In addition, I must applaud Jeanne de Visscher, the librarian, for her usually quick response in finding requested literature and Gijs van Banning and Marco Westra for unfailing interest in my progress. For responding to my request for data and information, I am grateful to Ed Walsh of NASA Goddard Space flight center, Ian Young, Stephen Stichter, Smith Warner International and staff members of the NDBC, USA. Life in a Dutch polder would have been far less fascinating had it not been for the kind assis-tance of Ewout van der Reijden, Jan van Overeem, and Anneke Mooiweer, who have aided me in one way or another with my social affairs. Ewout, dank u wel, for putting up with me for 5 months. My friends and classmates (Chia, Mouricio, Patricio, Roberto, Richard, Jimmy, Peter, Alex, Toni, Denise, Carlos, Anand Sasa, Juan, Itai, Paula and Martha), with whom I shared many wonderful experiences, thank you for your companionship. Ana Maria Garavito Rojas, muchas gracias for being my “dearest sister“. Tara Smith, your long distance support has not fallen short. To my family, relatives, friends and colleagues in Jamaica, your love and support has encour-aged me to keep the mid-night oil burning. Lisa Franklin, you stand out! Your outpouring affec-tion has been a constant source of comfort and inspiration. Finally, to God be the glory, great things he has done!

Parametric Models and Methods of Hindcast Analysis for Hurricane Waves by Jamel D. Banton May, 2002 IHE / ALKYON M.Sc. Thesis Report Page XVI

TABLE OF CONTENTS EXECUTIVE SUMMARY..............................................................................................I

ES-1.1 INTRODUCTION.....................................................................................I ES-1.2 AIMS AND OBJECTIVE..........................................................................II ES-1.3 SCOPE OF WORK .................................................................................II ES-1.4 SUMMARY OF STUDY TOPICS ...........................................................III

1.1.1 Hurricane Parameter Variation......................................................................... III 1.1.2 Parametric Wind and Wave Modeling .............................................................. IV 1.1.3 Data Base Selection Criteria ............................................................................ VI 1.1.4 The Historical Approach to Hindcasting .......................................................... VII 1.1.5 Monte Carlo Approach...................................................................................... IX

ES-1.5 DISCUSSION ...................................................................................... XII

ABSTRACT ............................................................................................................. XIII

ACKNOWLEDGEMENT...........................................................................................XV

TABLE OF CONTENTS ..........................................................................................XVI

INTRODUCTION .........................................................................................................1

NOMENCLATURE ......................................................................................................5

BACKGROUND...........................................................................................................6

1.1 CHARACTERISTICS OF HURRICANES ..................................................................6 1.2 THE STUDY AREA.............................................................................................7 1.3 HURRICANE OCCURRENCE ...............................................................................7 1.4 HURRICANE RECORDS......................................................................................7 1.5 WIND AND WAVE DATA .....................................................................................8 1.6 CENTRAL PRESSURE AND RADIUS TO MAXIMUM WIND ........................................8 1.7 WIND AND WAVE PARAMETRIC MODELS ............................................................9 1.8 METHODS OF HINDCAST ANALYSIS ..................................................................10

1.8.1 The Historical Approach .................................................................................. 10 1.8.2 The Monte Carlo Approach ............................................................................. 14

PROGRAMING TOOL - HURWAVE.........................................................................16

2.1 THE SINGLE GRID MODULE.............................................................................16 2.2 THE SINGLE STORM MODULE..........................................................................16 2.3 THE WAVE MODULE .......................................................................................17 2.4 THE EXTREMAL STATISTICAL MODULE .............................................................17 2.5 THE MONTE CARLO MODULE ..........................................................................17 2.6 THE MULTIPLE GRID MODULE .........................................................................17 2.7 TECHNIQUES AND FORMULATIONS...................................................................17

2.7.1 Searching the Database .................................................................................. 18 2.7.2 Minimum Distance Computation...................................................................... 18 2.7.3 Random Number Generation .......................................................................... 18

HURRICANE PARAMETER VARIATION.................................................................19

3.1 SPATIAL DISTRIBUTION OF HURRICANE OCCURRENCE ......................................19

Parametric Models and Methods of Hindcast Analysis for Hurricane Waves by Jamel D. Banton May, 2002 IHE / ALKYON M.Sc. Thesis Report Page XVII

3.2 TEMPORAL AND INTENSITY DISTRIBUTION ........................................................20 3.3 SPATIAL DISTRIBUTION oF PEAK VMAX ............................................................20 3.4 CORRELATION BETWEEN COMPONENTS ..........................................................21

3.4.1 Central Pressure and Latitude......................................................................... 21 3.4.2 Forward Speed and Peak Vmax...................................................................... 21 3.4.3 Central Pressure and Vmax ............................................................................ 21

3.5 CONCLUSIONS AND RECOMMENDATIONS..........................................................22

PARAMETRIC WIND AND WAVE MODELLING .....................................................23

4.1 WIND MODELS ...............................................................................................25 4.1.1 Holland (1980) ................................................................................................. 25 4.1.2 Hydromet Model .............................................................................................. 26 4.1.3 Fujita Model ..................................................................................................... 27 4.1.4 Bret Model ....................................................................................................... 27

4.2 WAVE MODELS ..............................................................................................27 4.2.1 Ross (1976) ..................................................................................................... 28 4.2.2 Young (1988) ................................................................................................... 30 4.2.3 The Improved Young (1995)............................................................................ 32 4.2.4 Cooper (1988).................................................................................................. 32 4.2.5 Bretschneider (1990) ....................................................................................... 34

4.3 APPLICATION OF MODELS ...............................................................................36 4.3.1 Estimation of Unknown Parameters (Pc & Rmax)........................................... 36 4.3.2 Modeling with HURWave................................................................................. 36

4.4 MEASURED WIND AND WAVE DATA .................................................................37 4.4.1 Buoy Measurements........................................................................................ 37 4.4.2 SRA Measurements ........................................................................................ 38

4.5 BUOY DATA VS WIND & WAVE MODEL PREDICTIONS......................................38 4.5.1 Methodology .................................................................................................... 38 4.5.2 Analysis and Results ....................................................................................... 39 4.5.3 Discussion of Results ...................................................................................... 39

4.6 SRA DATA VS WAVE MODEL PREDICTIONS......................................................40 4.6.1 Methodology .................................................................................................... 40 4.6.2 Results and Analysis ....................................................................................... 41 4.6.3 Discussion of Results ...................................................................................... 41


DATABASE SELECTION CRITERIA .......................................................................44

5.1 RELIABILITY OF RECORDS...............................................................................44 5.1.1 Methodology .................................................................................................... 45 5.1.2 Discussion of Results ...................................................................................... 45

5.2 SELECTION OF STORM POPULATION ................................................................46 5.2.1 Methodology .................................................................................................... 46 5.2.2 Discussion of Results ...................................................................................... 47

5.3 EFFECTIVE RADIUS ........................................................................................47 5.3.1 Methodology .................................................................................................... 47 5.3.2 Discussion of Results ...................................................................................... 48

5.4 OCCURRENCE OF MAXIMUM HS ......................................................................48 5.4.1 Methodology .................................................................................................... 49 5.4.2 Discussion of Results ...................................................................................... 49


Parametric Models and Methods of Hindcast Analysis for Hurricane Waves by Jamel D. Banton May, 2002 IHE / ALKYON M.Sc. Thesis Report Page XVIII

THE HISTORICAL APPROACH ...............................................................................52

6.1 THE DATA SERIES..........................................................................................52 6.2 CENSORING THE DATA ...................................................................................53 6.3 METHODS OF DATA FITTING............................................................................53 6.4 DISTRIBUTION FUNCTIONS ..............................................................................53 6.5 METHODOLOGY..............................................................................................53 6.6 COMPARISON OF METHODS AND DISTRIBUTIONS ..............................................56

6.6.1 Methodolgy ...................................................................................................... 56 6.6.2 Discussion of Results ...................................................................................... 57


THE MONTE CARLO APPROACH ..........................................................................63

7.1 THE CHOICE OF VARIABLES ............................................................................64 7.1.1 Vmax and Peak Vmax ..................................................................................... 64 7.1.2 Central Pressure, Pc ....................................................................................... 64 7.1.3 Radius To Maximum Winds, Rmax ................................................................. 64 7.1.4 Minimum Distance To Point of Interest ........................................................... 65 7.1.5 Track Heading ................................................................................................. 65 7.1.6 Forward Velocity .............................................................................................. 65

7.2 THE SIMULATION PROCEDURE ........................................................................66 7.2.1 Selection of Parent Population ........................................................................ 66 7.2.2 Determination of Distribution Functions .......................................................... 66 7.2.3 Generation of Random Variables .................................................................... 67 7.2.4 Construction of Synthetic Storm Track............................................................ 67 7.2.5 Retaining the Distribution of Peak Vmax......................................................... 68 7.2.6 Summary of Monte Carlo Procedure............................................................... 68

7.3 DISCUSSION OF SIMULATION RESULTS .............................................................70 7.4 RECOMMENDATIONS FOR APPLICATION............................................................70

SUMMARY OF CONCLUSIONS AND RECOMMENDATIONS ...............................72

DISCUSSION.............................................................................................................75

REFERENCES ..........................................................................................................78

LIST OF FIGURES ....................................................................................................80

Parametric Models and Methods of Hindcast Analysis for Hurricane Waves by Jamel D. Banton May, 2002 IHE / ALKYON M.Sc. Thesis Report Page 1

INTRODUCTION Hurricanes are an amazing feature of tropical weather and are often called “the greatest storms on Earth”. They have earned this infamous title through their devastating im-pacts on the people and economies of nations in and around the Caribbean Sea and the Gulf of Mexico. Annually, coastal areas of these nations undergo inundation from storm surges induced by the associated high water levels and severe wave attack from pass-ing hurricanes. The various mitigation projects, particularly for the Caribbean, have focused on the main components contributing to an increase in water level during a storm surge. These com-ponents are 1) Inverse Barometric Pressure Rise, which is a result of increased water level due to the low-pressure system associated with a hurricane. ; and 2) Wind Set-up, which is a result of water being piled up against the coastline from the high wind speeds of a hurricane. On the contrary, there has been significantly less focus on the associated wave component during storm surges. Although not a significant contributor to increased water level, waves do have a very damaging effect on coastal defense systems (sea walls, breakwaters and dunes). It is necessary for the engineer to be able to determine wave forces, wave run-up and wave over topping levels so that reasonable predictions can be made of coastal flooding and inundation limits. Hurricanes produce the most ex-treme wave events. As such, it is absolutely important to be able to predict these condi-tions and to further statistically manipulate these results to produce design conditions. The available methods for statistical manipulation of extreme events are wide and var-ied. Researchers have compiled these methods for use in analysis of storm waves, but it is left to be discovered whether all these cases are applicable for North Atlantic hurri-cane wave analysis. Furthermore, when compared to other parts of the world, there are significantly more and better records of hurricane activity within the North Atlantic Basin. This lends the opportunity to develop and validate “new” methods of statistical analysis, which are necessary for application in other areas with insufficient records. In these ar-eas the more traditional statistical approaches would not be applicable because of insuf-ficient records. Hurricane Wave Prediction A hurricane’s low-pressure center leads to high-speed rotating winds that serve as the generation mechanism for a complex wave system of both swell and sea waves. The randomness involved in the formation and movement of hurricanes and the complex processes involved in the wave generation, particularly due to the rapidly turning winds, presents a huge challenge to scientists and engineers. Modelling theories and techniques are continually being improved for the wind and wave


generation processes associated with a hurricane. Extensive data collection programs are making it possible for us to substantiate these theories and techniques but still there is a far way to go, especially since we still lack the complete understanding of the com-plex climatic systems that propel these storms. Hurricane wave prediction models have taken different forms. They range from complex spectral models to simpler parametric models. The term “parametric” refers to simple models requiring the input of only a few parameters. The parametric models, in most cases, are the simplification or the parameterization of numerical formulations related to wind-wave generation theories in combination with results of complex spectral models. The verification and improvement of both parametric and their more sophisticated coun-terparts, the spectral models, have been hindered by the lack of sufficient measured data. Often, they have been calibrated against only a few measured data points from a handful of storms. As such, particularly the parametric models, they sometimes predict incorrect results when applied to different hurricanes. Although the spectral models, understandably, will outperform their parametric counter-parts, the complexity involved in their application requires far more computational time, effort and money. Parametric models have the advantage that they may be applied to a much larger database of storms. This means that one can achieve a much more com-prehensive data set on which to perform statistical analyses. In this way, a more robust statistics can be achieved than is easily possible from the use of a spectral model. Statistical Methods of Analysis Forecasting of hurricane activity is one of the developing areas of Meteorology, but while there is still so much to be learnt, hindcasting seems to be a more viable alternative for estimating the damaging effects from probable hurricane ac-tivity. Presently, there are several hindcasting methods used by engineers. While several of these methods have been proven to be applicable for many different situations in various engineering disciplines, their application to hurricane-associated activity is still being studied. Hindcasting methods for hurricane waves differ in the manner in which the historical data (measured and modelled) are statistically manipulated. The traditional methods (termed The Historical Approach) require only the fitting of the data to various known statistical distributions and thereafter deriving the excedence levels for desired return periods. Others such as the Monte Carlo Approach, (which arguably, should not be referred to as a method of hindcasting), require the generation of a synthetic database from the prop-erties of the existing historical population. The historical approach has the advantage that it represents the real storm data set while the Monte Carlo gives a more reliable sta-tistical interpretation of the probable storm occurrences and intensity. Such a case, as the latter, is applicable only to situations where the correlations be-


tween the main parameters of a random event are easily established. The randomness associated with hurricanes makes this method very applicable. It also provides a plat-form for comparison with The Historical Approach. The Monte Carlo Approach may not be necessary for application in the North Atlantic Basin. This is because of the high fre-quency of storm occurrence and the relatively good records of hurricane activity. For these same reasons, it is possible to attempt to verify an approach which could then be applied elsewhere in the world were storm occurrence may be less frequent or where the quantity and quality of available data leaves a lot to be desired. Hurricane Records and Wave Data The accuracy of both the wave prediction models and the statistical methods depend largely on the accuracy of the available database of hurricane records. Since the 1960’s data logging of hurricane parameters has improved tremendously. Prior to this, the accu-racy of recorded data is questionable. There are hurricane databases, which contain re-cords of hurricane tracks and their main components, the central pressure and the maximum wind speed. Also available, but on a much smaller scale, are recorded hurri-cane wind and wave conditions. These wind and wave records are mostly in the form of point measurements from buoys that give the variation in time but not in space. Other remote sensing methods from aircraft flights (using Scanning Radar Altimeters) and sat-ellite imagery provides the spatial variation in the generated wave conditions. However, these data are scarce and the measuring techniques are still in their developing stages. Scope of Work The study was undertaken at Alkyon Hydraulic Consultants and Research under the su-pervision of David P. Hurdle with the following aims: 1. Compile the available hurricane records from various organizations and evaluate the

consistency of these records over the entire period for which they are available. 2. Summarize the statistics of hurricane activity and derive the statistical relations be-

tween the characteristic components of hurricanes. 3. Compare the results of hurricane wave prediction models to available measured

wind and wave data. 4. Evaluate the various statistical methods used in hindcast analysis of extreme events

and establish their relevance and applicability to hurricane waves. 5. Develop a Monte Carlo Approach to carry out hindcast analysis for storm waves. 6. Develop an all-encompassing computer program to carry out hindcast analysis any-

where in the North Atlantic Basin. This study attempts to recommend 1) the use of available hurricane records 2) the best wave-prediction parametric models 3) the use of the most sound statistical methods for hindcast analysis and 4) the procedure for application of The Monte Carlo Approach.


This work will be most relevant to the engineers and scientists carrying out hindcast analysis for storm events. The results can be further used to determine the probable impacts on offshore and coastal structures by hurricane waves. Further studies of the consequential storm surge levels, damage to coastal structures and coastal erosion may follow.

Parametric Models and Methods of Hindcast Analysis for Hurricane Waves by Jamel D. Banton

Page 5 IHE/Alkyon M.Sc. Thesis Report May, 2002

NOMENCLATURE Symbol Definition Units Vmax Maximum cyclostrophic wind speed. m/s, knots Rmax Radius to maximum wind speed, Vmax. km, NM Pc Minimum Central Pressure at the eye of a storm. mb, in. Hg P0 Ambient Pressure – atmospheric pressure at infi-

nite distance from storm center. mb, in. Hg

ΔP Difference between Ambient Pressure and Central Pressure, Pc.

mb, in. Hg

Vfd Velocity of Forward Motion of a storm track be-tween recorded points.

m/s, knots

Hs Significant Wave Height m, ft Tp Peak Wave Period s Vc 10m/20m – Elevation wind speed at a distance r

from the centre of a storm m/s, knots

R Distance from storm center to a point of interest. km Dmin Minimum distance from point of interest to storm

track. km

PVmax Peak Vmax for a given storm track m/s, knots f Cariolis Parameter ω Angular velocity of the earth’s rotation rad/s φ Latitude of storm occurrence degrees, rads i The ith orded data with the data set arranged in

ascending order.

m The mth orded data with the data set arranged in descending order.

Xi, Xm The ith & mth data value m PI & Pm Plotting probabilitity of the ith & mth data value YR Reduced variate Rp Return period years XR Return wave height m



BACKGROUND

1.1 CHARACTERISTICS OF HURRICANES A tropical cyclone is a vortex of air circulating around a center of low pressure such that the force caused by the pressure gradient pushing the air towards the low pressure is balanced by the sum of the deflective force (due to the Earth’s rotation) and the centrifugal force (James B. Elsner and A. Birol Kara (1999)). Tropical cyclones are referred to by different names. For ex-ample, in the Pacific Ocean they are referred to as typhoons, whereas in the Indian Ocean they are called Monsoons. For the area of interest, the North Atlantic Basin (shown in Figure 1.1), these storms are called Hurricanes. These storms differ from their counterparts occurring in the Southern Hemisphere, in that, their rotating wind field is in an anti clockwise direction. This dif-ference is of course due to the difference in the direction of the earth’s rotational force which is referred to as the Coriolis effect. Tropical cyclones vary in intensity and as such have been classified into different categories. In the North Atlantic basin, a cyclone is only referred to as a hurricane after it has reached a 1-minute maximum sustained near-surface (10m) winds in excess of 33m/s (64 knots). Before this stage, they are referred to as Tropical Storms. Hurricanes are commonly classified into catego-ries according to the Saffir Simpson Scale shown below in Table 1.

Category 1 2 3 4 5 Vmax

(knots) 64-83 84-95 96-113 114-135 >135

Vmax(km/hr) 119-154 155-178 179-210 211-250 >250 Vmax(m/s) 33-43 44-49 50-58 59-70 >70

Table 1 : The Saffir Simpson Hurricane Intensity Scale

Hurricane tracks in the North Atlantic basin can more or less be described as a parabolic sweep. They form between latitudes 5o and 25o north of the Equator. Those formed at the lower latitudes are usually pushed on a westerly track by the Northeast Trade winds whereas those of the higher latitudes track more to the north and northwest. Figures 1.2 and 1.3 show the charac-teristic storm tracks and all the tracks of the last 50 years, respectively. Figure 1.4 shows the schematics of the characteristic components of a hurricane. These severe weather systems range in diameters from 200-1300 km and have life spans lasting between one and 30 days. Winds in a hurricane increase from their lowest speeds at the eye (low-pressure center) to a maximum, immediately beyond the eye. The wind speed then decreases gradually outwards from the eye. The radius at which maximum wind speeds (Vmax) occur is termed as the radius to maximum winds (Rmax). The maximum wind speeds around the eye are the same order even though the overall distribution of the wind speed over the entire body of the hurri-

1.



cane varies. Particularly, winds in the right front quadrant are strongest because of the addi-tional forward component due to the movement of the hurricane. The wind speed indicates the intensity of the hurricane but even more indicative of the intensity, is the central pressure (Pc). This is the pressure at the eye of the hurricane. Half of all hurricane observations have central pressures of 980 mb or lower (James B. Elsner and A. Birol Kara (1999)). The hurricane moves along its track with a forward velocity (Vfd) ranging from 2m/s to 12 m/s.

1.2 THE STUDY AREA This study focuses on hurricane occurrence in the North Atlantic Basin. This area represents the part of the Atlantic Ocean, which is confined between latitudes 5o and 40o north of the equa-tor. The area includes the Caribbean Sea, The Gulf of Mexico, the Sargasso Sea and the North Atlantic Ocean. The area is shown in Figure 1.1. The Caribbean Sea may be further sub divided into the Eastern and Western Caribbean. The Eastern Caribbean reefers to the region which is east of the island of Hispaniola.

1.3 HURRICANE OCCURRENCE Hurricane occurrence varies within the area of interest. For instance, the islands of the eastern Caribbean have a higher frequency of hurricane activity than those of the western Caribbean. This is because the hurricanes which are formed off the coast of west Africa and move west-wards towards the Caribbean sea are then usually propelled in a northwesterly direction just north of the island chain and then north of the western Caribbean islands. The hurricanes, which form within the Caribbean sea usually, progress in a northwesterly direction towards ei-ther the Gulf of Mexico or the east coast of the USA. Hurricanes tend to avoid the US mainland into the north Atlantic but often progress inland where the eventually die out.

1.4 HURRICANE RECORDS The National Oceanic and Atmospheric Administration (NOAA) of the USA, through the National Hurricane Center (NHC), maintain much of the historical records of hurricanes. The US National Weather Service was the first to provide details of individual tropical cyclone tracks for the North Atlantic Basin, including statistics for the period 1886-1958. The tracks and statistical data have since then been updated several times and the latest version was issued under the subtitle Tropical Cyclones of the North Atlantic Ocean, 1871-1992. This so-called best track data set represents the most comprehensive and reliable data source for all North Atlantic hurricane in-formation. The best-track data represents a post-season analysis of all tropical cyclone intensi-ties and tracks at 6-hour intervals. The data set includes the 6-hour track positions and the equivalent measurements or approximations of the maximum wind speed, Vmax, and the cen-tral pressure, Pc. Prior to 1970, there are scarce records of the central pressure. Data reliability is not uniform throughout the period of record, as there are uncertainties in the wind and pressure estimates. According to Neumann and McAdie (1997), there are both sys-



tematic and random errors mostly confined to the years before 1968. The random errors were introduced largely through mistakes made as the data were entered on punch cards. The sys-tematic errors are a result of the interpolations necessary to provide 6-hourly reports from 12- and 24-hourly observations. These errors include wind speeds that are not always consistent with the pressures, peripheral pressures rather than central pressures and forward speeds es-timated to be too fast at the beginning and end of a hurricane track, among others.

1.5 WIND AND WAVE DATA Improvement in the ability to log hurricane occurrences has come about with technological ad-vancement. Hurricane tracks and its characteristics are now surveyed by aircraft reconnais-sance, satellite tracking, radar systems (Synthetic Aperture Radar (SAR)), and data buoys. SAR and data buoys provide the most comprehensive data set of hurricane-generated waves. Most records of ocean waves are visual observations from ships. Since ships usually avoid se-vere weather conditions such as hurricanes, this source of data is not very useful for this study. We then have to rely on observations from instruments. These data are generally far less bi-ased than the visually observed data but they do have their peculiar problems. For example, spherical data buoys tend to avoid very steep waves and due to power requirements, the sam-ple period is usually quite short. Remote sensing techniques from airplanes and satellites are being experimented for wave measurements and these data are generally scarce and expen-sive. They do however have the advantage in providing spatial distribution of wave conditions which in-situ measurement techniques cannot.

1.6 CENTRAL PRESSURE AND RADIUS TO MAXIMUM WIND Hurricanes are characterized by the central pressure (Pc), maximum wind speed (Vmax), veloc-ity of forward motion (Vfd) and the radius to maximum wind (Rmax). These parameters are the basis for the prediction of a hurricane wind and wave field. Whereas the Vmax and Vfd are gen-erally available from measurements, the Pc and Rmax are in most cases unavailable. In fact, for the database of hurricane tracks there are only a few records of Pc prior to 1970. The Rmax can be deduced from measured wind or pressure profiles. Such profiles have only been measured for a handful of storms and as such the derivation of the Rmax component generally poses a problem. Many models have been proposed for the estimation of the central pressure from the Vmax. These include the following:

Vmax = 3.44(1010-Pc) ^0.644 ...Equation 1-1 by Atkinson and Holliday (1977) [Ref. 1] from measured Tropical Cyclone data in the Pacific Ocean;

• Vmax = -1.37Pc + 1423, ...Equation 1-2 derived from measured hurricane data in the Western Caribbean given by Stephen Lamming (1975) [Ref. 15];

• 21

c0max10 )PP(7V −= ...Equation 1-3



by Kroft (1961) from measured data in the Atlantic Ocean.

• )]f575.0(R)PP(73[868.0V 21

c0max35 −−= ...Equation 1-4

where: P0 = 1013 mb, f = 0.2 to 0.3 for latitude 5o-30o North and Vmax = 0.865V35max. This expression was given by Bretschneider (1957) and differs from the others as it includes the Coriolis effect and has the Rmax as an additional variable.

Whereas there have been many models proposed for estimation of the central pressure from the Vmax, there are far fewer for the estimation of Rmax. With the exception of Bretschneider (1990) [Ref. 3], researchers have rarely discussed the methods used in deriving the Rmax com-ponent used in their studies. In this paper Bretschneider describes a method of deriving Rmax from the measured pressure profile whereby the Rmax is given at the maximum product of the radius from the hurricane center and the pressure gradient.

ie. Rmax = Max[r(dp/dr)] ...Equation 1-5

A few cases have been found where a relationship between Rmax and another of the main components was given. For example, Hurdle et.al. (2000) gives the following expression derived for Tropical Storms of the Andaman Sea:

Rmax= 1190/Vmax. ...Equation 1-6

Another case is the derived expression from data given by Lamming (1975) [Ref. 15] for the western Caribbean:

Rmax = 2500/(Vmax – 5)^1.1 ...Equation 1-7

1.7 WIND AND WAVE PARAMETRIC MODELS The past three decades have been marked by active theoretical and experimental research concerned with the development of mathematical models that attempt to describe the wave en-vironment associated with storm systems. They range from complex spectral models to simpler parametric models. The parametric models, in most cases, are the simplification or the parame-terization of numerical formulations related to wind-wave generation theories. The accuracy of these models are very much dependent on the accuracy in the prediction of the wave-generation mechanism, the wind field. The fact that this characteristically intense wind field is spatially inhomogeneous and directionally varying makes it a challenge to consistently achieve accurate predictions. The large gradients in wind speed and the rapidly changing wind direc-tions generate extremely complex wave fields. Attempts to model the wind and wave field have been further challenged by the significant variations in the intensity, the spatial scale and veloc-ity of forward motion for different hurricanes. In spite of the complexity involved, researchers have managed to propose several parametric models, which have been proven to achieve very good results for a number of cases. The pa-rametric wave models range from being largely empirical (Bretschneider (1959) [Ref. 3]; Ross (1976) [Ref. 19]; Young (1988) [Ref. 24]) to the analytical ones based on the Radiative transfer equation by Hasselman et.al. (1973), (Cardone et. al., (1979) [Ref. 20]; Young (1987) [Ref. 24]). These models are parameterized in terms of the Pc, Vmax, Rmax and Vfd of a hurricane. Thus



given these parameters the models predict the Hs and in several cases, the Tp at any distance from the center of a hurricane. The details of these models shall be given in preceding sections. There are spectral numerical models, which do predict accurate wave conditions for a hurricane wind field. These models are excellent for application to a handful of storms when there are adequate measurements to represent the hurricane wind and pressure field. In such cases ac-curate wave predictions can be made. In the more common cases when there is insufficient data, these models, however accurate, inherit the inaccuracies associated with the hurricane records. A number of these models are commercially available many of which have been used in the Petroleum industry for the design of oil platforms against extreme hurricane waves.

1.8 METHODS OF HINDCAST ANALYSIS Presently, there are several hindcasting methods used by engineers to calculate extreme wave statistics. These methods of extremal analysis have taken different forms, as the choice of the type of data set, the method of data fitting and the distribution functions vary. Hindcasting methods differ in the manner in which the historical data (measured and modelled) are statistically manipulated. For instance, the historical approach requires the fitting of the data to various known statistical distributions and thereafter deriving the excedence levels for desired return periods. Others such as the Monte Carlo approach require the generation of a synthetic database from the properties of the existing historical population. Such a case, as the latter, is applicable only to situations where the main parameters of the parent population are regarded as statistically independent of each other. The randomness associated with hurricanes, and the very weak correlation between the main parameters, makes this method very applicable.

1.8.1 The Historical Approach This method of analysis has long been used in the field of flood design. The statistical proce-dures for use in extremal analysis of storm waves are varied. Some of the procedures are not recommendable from a statistical point of view while others are quite complicated to use. Re-searchers such as Goda [Ref. 10] have compiled the various procedures in an attempt to arrive at clear and sound statistical procedures. These various methods, however statistically sound, give different results when applied for a particular problem. It is usually up to the engineer to explore the various procedures and in the end make his own judgement on which to use. It is true that there may not be one “correct” procedure applicable for all situations. However, there are some procedures that may be eliminated for application to hurricane wave statistics, which will in effect, save computational time and effort. It is then necessary for inter-comparison of the various procedures. 1 . 8 . 1 . ( a ) T o t a l S a m p l e A n a l y s i s The total Sample Analysis uses the complete distribution of storm waves. This was once a popular method to compute return values of storm waves. By knowing the time interval between successive wave observation hours Δt, the excedence probability was estimated as 1/(Rp/Δt), where Rp is the return period, and then the return value was read from an extrapolated distribu-



tion curve. This method of analysis is however statistically unsound for two main reasons, namely:

1. 1Since the wave observations are taken over just a few hours in a storm, they are in fact mutually correlated and thus do not constitute an independent random data set. This vio-lates the requirements of most statistical theories for independence between the random variables; and

2. There is no guarantee that data in the upper quartile fit well to the tail of the distribution that governs the whole range of wave heights.

1 . 8 . 1 . ( b ) P e a k V a l u e A n a l y s i s This refers to the criterion of selection of waves from the data set. This is generally divided into two areas: the partial duration series and the annual maximum series. The partial duration se-ries shall hereafter be termed the peak value series. These two methods of classification (the peak value series and annual maximum series) are more statically sound than the Total Sample Analysis. The peak value method uses only the peak wave height from each storm. Since each storm is an independent event then each data point is a random independent variable. The an-nual maximum method uses only the maximum wave height from all storm occurrences within each year. This has been widely used for river flood design but here the annual frequency of storm occurrence may well be much more varied than that for floods. As such one may argue that apart from limiting your data set, you may ignore high wave heights occurring in very active years. For example, the eastern Caribbean Islands, such as Antigua and Barbuda, were experi-enced extreme waves from two severe hurricanes two weeks in succession in 1999. This method is however still employed by many engineers and its validity for storm wave analysis has not been disproved. In fact, Goda [Ref. 9] has stated that the difference between the return values by the peak value method and the annual maximum method becomes negligible for re-turn periods longer than 5 years. 1 . 8 . 1 . ( c ) C e n s o r i n g t h e D a t a S e t Another common practice is to censor the data set of waves. In this way, the high excedence values are excluded. Thus the tail of the distribution, which may be of a different form and is the least interesting part of extreme value distribution, is ignored. But one may argue that in doing this you are just limiting the data set and not representing the true distribution of the entire population. The censoring parameter is given as v= Nc/Nt where Nc and Nt are the total number of data points for the censored and uncensored data set respectively. For an uncensored data set then Nc = Nt and v = 1. 1 . 8 . 1 . ( d ) F i t t i n g o f D a t a s e t t o a D i s t r i b u t i o n F u n c t i o n The commonly employed distribution functions for both annual maximum series and peak value series are given below. These distribution functions are listed in the form of excedence probabil-ity.

1. The Fisher-Tippett type I (FT- I) or Gumbel Distribution

1 “Distribution of Sea State Parametersand Data fitting”, Yoshima Goda (1988)



P[H > x] = 1- exp[-exp{-(x-B)/A}] Mean: E[x] = B + γA (γ = 0.5772 .... Euler’s constant) Variance: E[{x – E(x)}2] = π2A2/6

2. The Fisher Tippett type II ( FT- II) or Frechet Distribution P[H > x] = 1- exp[-(x/A)-k]

Mean: E[x] = A Γ(1-1/k) Variance: E[{x – E(x)}2] = = A 2[Γ(1-2/k) - Γ2(1-1/k)]

3. The three-parameter Weibull Distribution P[H > x] = exp[-{(x - B)/A}k]

Mean: E[x] = B + A Γ(1+1/k) Variance: E[{x – E(x)}2] = = A 2[Γ(1+2/k) - Γ2(1+1/k)]

4. The Log-normal Distribution

∫ −−−=>x

0

2 /2]dtB)/A}{(lntexp[At1

211π

x] P[H

Mean: E[x] = exp[B + A2/2] Variance: E[{x – E(x)}2] = exp[2B + A2]{exp(A2)-1} Γ denotes the Gamma function and E represents the ensemble mean of the expected value. The parameters A, B and k are called the scale, location and shape parameters respectively. The FT-I and the Weibull distributions are the most commonly used for extreme wave analysis. The Lognormal distribution has been mostly used in the total sample method discussed earlier. According to Goda [Ref. 9], the log-normal distribution has a tendency for the rate of increase of return wave height to decrease as the return period becomes long, when compared with the FT-I or the Weibull distributions. As such, it is not recommended for peak value analysis. 1 . 8 . 1 . ( e ) M e t h o d s o f D a t a F i t t i n g The data series must be fitted to some distribution function to enable estimates of the return wave heights for various return periods, there are four methods of data fitting namely:

1. The graphical method using a specially devised plotting paper on which the data from a particular distribution are plotted on a straight line.

2. Method of Least Squares 3. Method of Moments 4. Method of Maximum Likelihood

GRAPHICAL METHOD AND THE METHOD OF LEAST SQUARES The graphical method was used in the past when calculations of standard deviations was a te-dious job. Presently, the least square method replaces the graphical method. For the least square method, the peak value data are first rearranged in descending order of magnitude (from the largest to the smallest).The mth largest data is denoted as Xm. Then a plotting probability Pm is assigned to Xm and the reduced variate Ym is calculated from the distribution function being used. For the FT-I and the Weibull distributions the formulae for the reduced variate are



as follows: FT-I : Ym = -ln[-ln(Pm)] Weibull: Ym = [-ln(1 - Pm)]1/k

The following linear relationship exists and the estimates of the parameters A and B are easily obtained by graphical fitting or by least squares method. Xm = A Ym + B The shape parameter k cannot be estimated by this method. Patruaskas and Aagaard [Ref. 18] proposed to fix k to one of the seven values ( 0.75, 0.85, 1.0, 1.25, 1.5, 2.0) to fit the data to seven alternative Weibull distributions and the FT-I distribution and to choose an acceptable distribution from the eight alternatives. Hurdle [Ref. 13], on the other hand, developed a method to optimize the fit of the data to the Weibull distribution by deriving the k value that gives the best fit. THE METHOD OF MOMENTS The method of moments uses the mean and variance of the peek value series data which are equated to the theoretical values for each distribution so that estimates of A and B or A and k can be obtained. For this method, the mean and variance of the data series is equated to the theoretical mean and variance. Each theoretical distribution expresses the mean and variance in terms of A & B (see Section 1.8.1(d)). A & B are the scale and location parameters, respec-tively. THE METHOD OF MAXIMUM LIKELIHOOD The maximum likelihood method makes estimates of parameter values by maximizing the likeli-hood that the data are from the distribution function under investigation. This method seems to yield more reliable estimates than the other methods (Carter and Challenor ) [Ref. 4]. Its use in extreme wave data analysis is however, not yet popular which is probably because of the com-plexity of the numerical analyses required. No census has yet been established on the best method for data fitting. One must either as-sume a prior that a certain distribution should fit the data or try several distribution functions to find the most appropriate one. 1 . 8 . 1 . ( f ) P l o t t i n g P o s i t i o n s The most commonly known plotting position formula is that of Gumbel Pi = i /(Nc+1). The follow-ing plotting formulas are recommended by Goda [Ref. 9] for extreme wave analysis :

• FT-I : Gringorton Formula

Pi = (i – 0.44)/ ( Nc + 0.12) ...Equation 1-8

Barnett Formula

Pi = exp [-exp (γ + 3 (ln2)[2i/ Nc+1) - 1])] ...Equation 1-9



• Weibull: Petruaskas and Aagard Formula

Pi = [i - (0.49 - 0.50/k)]/[Nc + (0.21 + 0.32/k)] ...Equation 1-10

Goda Formula

Pm = 1 - [m - (0.20 + 0.27/k0.5)]/[Nt + (0.20 + 0.23/ k0.5)] ...Equation 1-11

In the case of censored data , the plotting positioned should be assigned by using the total number of peak wave data that must have occurred in the time period considered, 1 . 8 . 1 . ( g ) R e t u r n V a l u e s a n d T h e i r C o n f i d e n c e I n t e r v a l s The return values may be simply calculated from the distribution function in question. If the pe-riod over which the data set spans is given as K years then the mean rate of storm wave gen-eration is λ = Nt /K. Hence the probability of excedence for a given return period, Rp, is 1/(λRp). The following equations may be used:

XR =AYR + B ...Equation 1-12

where:

YR = -ln {-ln[1 - 1/(λRp)]} for FT-I ...Equation 1-13

YR = [ln(λRp)]1/k for Weibull ...Equation 1-14

Goda [Ref. 9] derived an empirical standard error for return values:

σ [XR] = (σ/Nc0.5)[1.0 + AR(YR - CR)2]0.5 ...Equation 1-15

where AR = a1exp[a2(N)-1.3] ...Equation 1-16

CR, a1 and a2 are empirical coefficients given by: FT-I : a1 = 0.64, a2 = 9.00, CR = 0.0 Weibull (k = 0.75) : a1 = 1.65, a2 = 11.4, CR = 0.0 Weibull (k = 1.0) : a1 = 1.92, a2 = 11.4, CR = 0.3 Weibull (k = 1.4) : a1 = 2.05, a2 = 11.4, CR = 0.4 Weibull (k = 2.0) : a1 = 2.22, a2 = 11.4, CR = 0.5

1.8.2 The Monte Carlo Approach This method requires the generation of a synthetic database from the properties of the existing historical population. It is applicable only to situations where the main parameters of the parent population are regarded as statistically independent of each other. The randomness associated with hurricanes, and the very weak correlation between the main parameters, makes this method very applicable. Monte Carlo Simulations have for some time been used as a tool for statistical analysis. In the field of storm wave analysis it has been applied by various researchers two of which are most



relevant to this study: Donosso et. al. (1987) [Ref. 7]; Hurdle et. al. (2001).[Ref. 13]. Donosso et. al. carried out a study for an area in the Gulf of Mexico. The central pressure deficit ΔP, Vfd, and Rmax were treated as random variables. Only the minimum distance, ro , to each storm track was considered and also treated as a random variable. The sample distribution for each variable was fitted to known statistical distributions and the best fit distribution selected. It was found that the log normal distribution provided the best fit for the Rmax and ΔP while the uniform and the normal distributions provided the best fit for ro and Vfd respectively. Random values were generated from the derived distribution functions. The storms were simulated such that the frequency of storm points to the left or right of the point of interest, was the same as that of the parent population. Up to 2000 simulations were done. Hurdle et. al. carried out over 2000 simulations of synthetic storm tracks for an area in the An-daman Sea. Each track was drawn at random from the parent population of 13 tracks. Each randomly drawn track was then moved to a randomly drawn crossing position along a line of latitude from the established distribution for the crossing positions. A Vmax was also drawn at random from its parent distribution, and each value of Vmax at each step of the storm was scaled by a constant such that the peak Vmax of the storm track corresponds to the random value.



PROGRAMING TOOL - HURWAVE

The study aims to compare the results from a number of parametric wind-wave models and from various statistical methods. To do this, a large database of hurricane records has to be manipulated. The most effective way of carrying out such tasks is by employing the use of com-puter programs. Whereas there are no known available programs to carry out parametric wind-wave prediction, there are commercial programs for handling hurricane databases and others for performing extremal analysis of waves. Employing these various models separately, would not only be a cumbersome procedure, but since the details of the methods applied in these pro-grams are not known, the credibility of the results could not be defended. Consequently, a pro-gram was developed that encompass all the necessary tasks to achieve the aims of this study. The tool was created in Visual Basic. It consists of 6 main modules, namely: The Single Grid Module; The Single Storm Module; The Wave Module; The Extremal Statistical Module; The Monte Carlo Module; and The Multiple Grid Module. Figure 2.1 shows a basic flow chart of the program. The arrows indicate how the modules link to each other as output values from one module are used as input in the linking module. A summary of the input and output parameters associated with each module is also given. Advantage was taken of the capability of Visual Ba-sic to produce user-friendly interfaces. Figure 2.2 shows examples of the interfaces for a num-ber of the input and output modules.

2.1 THE SINGLE GRID MODULE This module uses a search routine to return the storm points of all storm tracks passing within the specified area. The module offers the choice of using either the hurricane database before 1950, after 1950 or the entire database of records. The user can also specify the range of hurri-cane categories that are to be searched. It computes the minimum distance of approach from each storm track and the point along the track of occurrence of the peak Vmax. Further to this, each track can be divided into segments of a specified interval (for example 30km). For this, the program performs linear interpolation in order to determine the storm characteristics for the in-terpolated points between the recorded storm points.

2.2 THE SINGLE STORM MODULE This module uses the same search routine as the single grid module but searches only the re-cords of a single storm as opposed to an entire historical database of storms. This is useful for evaluation of a single storm.

2.



2.3 THE WAVE MODULE Here significant wave heights are computed for a number of parametric models namely: Ross (1976), Cooper (1988), Young (1988), Improved Young (1995) and Bretschneider (1990). The 10m-elevation (assumed to be 10-minute averaged) wind speeds are computed by four (4) wind parametric models, namely: Holland, Hydromet, Bret Model-X and Fujita models. One of these four models may be used as the default wind model to replace the parrent wind model used in either the Ross or the Cooper wind-wave models. This is useful in attempting to improve the results of these parametric wave models. The points on which to carry out the computations may be any of or all of the output points of the Single Grid or the Single Storm Module (i.e. Re-corded Points, Points of peak Vmax, etc). Additionally, this module is capable of filtering out de-sired wave directions. Hence, for instances when approaching wave directions are limited by land masses, HURWave is capable of filtering only the relevant directions.

2.4 THE EXTREMAL STATISTICAL MODULE Here the various data series, censoring limits, methods of data fitting, and distribution functions, associated with extremal wave analysis, may be employed. This module is most useful for inter comparisons. For a choice of data series and method, the program computes the return values for Hs and its associated standard deviations, confidence limits and encounter probabilities. Plots of the data fitted to the selected distribution functions are available, along with the correla-tion values.

2.5 THE MONTE CARLO MODULE This module employs a two-step approach to performing Monte Carlo simulations. First, the pa-rameters (the minimum distance and the peak Vma) are fitted to various statistical distribution functions. The best-fit distribution may be selected for each parameter. Secondly, a synthetic database of storms is developed from the chosen statistical distribution. Hereafter, extremal analysis may be carried out on this database by use of the previously discussed modules.

2.6 THE MULTIPLE GRID MODULE This is basically a batch process for the computation of the return wave heights for a range of points. Computations may be carried out either for the historical approach or for the Monte Carlo simulations. The program outputs an array of Hs values for each data series, censoring limit, method of data fitting and best-fit distribution function. Hence a vast number of comparisons can be made, not only between geographical locations, but also between the various extremal pro-cedures.

2.7 TECHNIQUES AND FORMULATIONS The credibility of the results of this study is, to some extent, dependent on the programming techniques and the mathematical formulations employed in the program. As such, a number of the most relevant points shall be discussed here.



2.7.1 Searching the Database The hurricane database of historical records is searched for the storm points, which are within the user-defined geographical area. The area may be defined by either a circle (a single point and a radius) or a rectangle (the upper left and lower right coordinates). In the case of a circle, the perimeter is schematized as an octagon. In both cases, for each segment of each storm, a cross product routine is applied to determine whether the track crosses each segment of the polynomial (octagon or rectangle). Another routine is employed to determine whether each storm point lies within the defined area. This accounts for the segments of a track, the whole of which may lie within the area of interest. If a tack is found to cross any of the segments of the polynomial then both its end points are considered as crossing points. Therefore, sometimes the search returns points that are just beyond the boundaries of the specified area.

2.7.2 Minimum Distance Computation This is determined for each storm, which is found to lie within the specified area. The cross-product routine is applied to each segment of the storm track and the minimum distance to the center point of interest determined. Distances are determined by the following expression.

22Re latlonr Δ+Δ= , ...Equation 2-1

where Re = 6400km, the radius of the earth; and Δlon and Δlat are the difference in longitude and latitude (in radians), respectively, between the storm point and the point of interest. The shortcomings in this expression are not significant. This is because distances larger than 400km never come into play and as such inaccuracies due to the earth’s curvature or differences in the longitudinal spacing are minimal over such distances.

2.7.3 Random Number Generation In generating random numbers for the Monte Carlo simulations, the visual basic random number generator function is used. The function can either generate a random number between 0 and 1 (inclusive) or between any two specified bounds. Both cases were applied.



HURRICANE PARAMETER

VARIATION

Before any form of analysis is carried out on hurricanes and their generated waves, it is impor-tant to understand the properties of the characteristic components. We need to establish their individual statistical properties as well as their correlations. With this knowledge, we will be able to make sound assumptions and define relationships for the forthcoming analyses. For example, it will be useful to know whether 1) the central pressure/wind speed of a storm is dependent on the storms geographic location, 2) a particular location is likely to have more extreme storms than another or 3) there have been temporal patterns of storm occurrence. As such, for this section we will attempt to examine the distributions and establish the relation-ships, if any, between the components involved. Specifically, we shall examine: a) the spatial distribution of hurricane occurrence; b) the chronological and intensity distribution; c) the spatial distribution of the peak Vmax of each storm; d) the correlation between: i) the central pressure and the latitude; ii) the forward speed and the peak Vmax and; iii) the central pressure and the Vmax along each storm track.

3.1 SPATIAL DISTRIBUTION OF HURRICANE OCCURRENCE In Figure 1.3 we see the tracks of all hurricanes of the North Atlantic Basin for 1950-2000. Fig-ure 3.1 shows the spatial distribution of hurricane occurrence on a 2x2-degree grid of the study area. Hence, Figure 3.1 is a numerical representation of the distribution of the tracks shown in Figure 1.3. It is clear that the islands of the eastern Caribbean have a larger number of occurrences than those islands of the western Caribbean. Noticeable, is the typical northwesterly heading of hur-ricanes from the area east of the Caribbean Sea towards the east coast of the USA or north-wards into the North Atlantic Ocean. One of the few exceptions was hurricane Lenny(1999) which tracked in an easterly direction across the Caribbean Sea, the first to do so in 112 years. In the Gulf of Mexico, there is relatively large number of occurrences with a markedly higher number in the northern part of the gulf. Just east of the US coastline, there are a large number of occurrences but only a few on the coastal mainland. This it is believed may be a reflection of the tendency of hurricanes to avoid continental landmasses or the effect of oceanic winds/currents (for example the Gulf Stream) on hurricane tracks. By far, the largest number of occurrences are over the North Atlantic Ocean. We therefore see that the frequency of storm occurrence is greater in some areas, however,

3.



their randomness in time, frequency and trajectory does not allow us to predict their occurrence at any particular time or place.

3.2 TEMPORAL AND INTENSITY DISTRIBUTION Figures 3.2(a & b) show the variation of all the recorded storm occurrences in the North Atlantic Basin for 1900-2000. The first figure shows the occurrences for all the recorded Topical cy-clones while the second shows only the most intense storms (higher than Category 3). The number of occurrences is given as variances to the mean for the entire period. A polynomial fit is also shown. The fit, in both cases, shows a multi decadal variation in the number of storm oc-currences. The plot suggests that the recent increase in Tropical Cyclone activity may in fact be part of this multi decadal cycle, rather than a manifestation of unnatural Global Warming as some researchers speculate. It is also evident that the number of intense storms is directly cor-related with the total number of storm occurrences. This variation in hurricane occurrence may be attributed to the concurrent variations in sea sur-face temperature. As the source of energy for the formation and growth of tropical cyclones the sea surface temperature is a most valid clue to hurricane frequency. This evidence is shown in Figure 3.2 (c). It is evident that there is a clear correlation as from 1930 to 1970 when tempera-tures went above average, so did the hurricane frequency and intensity. Hurricane formation is most common in the months of September to October. This is illustrated in Figure 3.2(d). This occurs because it is within these months that the atmospheric conditions (water temperature, pressure, humidity, etc.) are most suitable for the formation of hurricane systems. A hurricanes wind speed is generally the measure of its intensity. The Saffir Simpson scale de-fines the equivalent wind speeds for the five (5) hurricane categories. Figure 3.2(e) shows the number of occurrences for each hurricane category. The figure shows that there is an almost even distribution among categories 1 to 4. Category 5 hurricanes are, fortunately, not as com-mon as their less intense counterparts.

3.3 SPATIAL DISTRIBUTION oF PEAK VMAX Figure 3.3 shows the spatial distribution of the peak Vmax from each storm track for the period 1950 to 2000. Most occurrences have been over the Atlantic Ocean for two main reasons. The first is that there have been more occurrences over this area. The second is that storms will generally attain higher wind speeds far away from the damping effects of land. A large number of the hurricanes that track across the Gulf of Mexico are formed within the gulf or within the Caribbean Sea. As these systems progress, they usually intensify until they make landfall. As such, they attain their peak wind speeds just before reaching the continental land-mass. This is why there is such a concentration of peak values along the coastline of the Gulf of



Mexico. The point to note here is that there is some degree of dependence of the occurrence of the peak Vmax, for a particular storm track, on the geographic location. This point will be referred to in later sections for the Monte Carlo Approach.

3.4 CORRELATION BETWEEN COMPONENTS The correlation between three of the characteristic components (Vfd, Vmax, and Pc) as shown in Figure 3.4, is investigated here.

3.4.1 Central Pressure and Latitude The variation of central pressure with latitude of occurrence is plotted in Figure 3.4(a). The cen-tral pressure at the point of occurrence of the peak Vmax of each track is used. The corriolis force, which is one of the balancing forces in the circulating wind field, increases with increasing latitude. The central pressure (or rather its deficit from the ambient pressure) is the main driving force for the wind field. As such, it may be expected that some form of correla-tion exists between the latitude and the central pressure of a storm. Many researches, as will be seen in following sections, have included a coriolis force parameter for the prediction of hurri-cane wind fields while others have neglected its influence. The plot of Figure 3.4(a) shows why this is the case as there seems to be only a weak correlation between the two components. Each storm track is generally at a different stage of development across a particular line of lati-tude. As such, while the Coriolis force is the same, its effect on each storm’s intensity is then storm dependent. Therefore, while we are aware of the Coriolis effect on an individual storm, there can be no relationship established for a population of storms which are at different stages of development and at different geographic locations. Consequently, the central pressure, Pc, may in fact be treated independent of the latitude (or geographic location) when considering a population of storm tracks within a particular area.

3.4.2 Forward Speed and Peak Vmax The variation of forward speed, Vfd, with peak wind speed, for each storm track, is plotted in Figure 3.4(b). Hurricane forward speeds range from 1 to 15 m/s. Over this range of values, there is a variation in the equivalent wind speed. The figure demonstrates that there is no no-ticeable relationship between these parameters.

3.4.3 Central Pressure and Vmax The variation of Central Pressure, Pc, with both the peak wind speed of each storm track and the wind speeds at all points of each storm track are shown in Figure 3.4(c) and 3.4(d). The plots show that there is a relationship between the Pc and Vmax. This is expected as with low-



ering of the central pressure, the pressure gradient increases. The pressure gradient is the wind-generating mechanism and thus an increase results in an intensification of the wind field. There is however, an obvious variation in the value of central pressure for a particular value of Vmax. An expression was derived to represent the relationship between Pc and Vmax. This is shown in Figure 3.5. The expression is given by:

Pc a bVc

d= + --

( ) max( )

αα

...Equation 3-1 (Pc in mbars and Vmax in knots)

where a = 1014, b = 0.029, c = 1.626, d = 200 and α is a random variable uniformly distributed in the interval [-10, 10]. The random variable, α, is used to represent the spread in the relation-ship. α = 0 gives the expected value of Pc with a correlation of 0.886 between Pc and Vmax. Using this relationship, the value of the central pressure may be derived for historical storm points where this data is missing. This is applicable to almost all storm records prior to 1970.

3.5 CONCLUSIONS AND RECOMMENDATIONS From the above discussions, we draw the following conclusions:

• Hurricanes do have a preferred track that depends on the geographic region. While in the Caribbean Sea, they tend to move in a west to northwesterly direction. In the Gulf of Mexico and on the east coast of the USA they tend to track north-northwesterly to north-easterly. While present understanding of the complex climatic patterns that propel these storms does not allow for accurate predictions of these tracks, reasonable estimates can be made from these patterns.

• There seem to be a multi-decadal cycle in hurricane activity. • Storms occurring around the Gulf of Mexico or off the east coast of North America usu-

ally reach their peak Vmax as they near these landmasses. This shows some geo-graphic dependence of the peak Vmax but is only applicable around large landmasses.

• The central pressure and Vmax are the only two parameters that show an obvious corre-lation. However, there is still a wide spread in the relationship between the components. The derived formula may be used to estimate central pressures for instances where there are no measured central pressure values.



PARAMETRIC WIND AND

WAVE MODELLING

The use of parametric models to describe the hurricane wind and wave field is widely used in the field of Ocean and Coastal Engineering. Researchers have developed both wind and wave models that give relatively accurate predictions. Most of these models have been developed empirically. For the wave models, the data set for the empirical developments have come from either actual measurements or outputs of complex numerical models. In either case, the theo-retical basis is similar for each model. However, when applied to a particular case the models tend to give varied predictions. With a varied choice of models, it is then useful to examine the performance of each model. Before we focus on the wave models it is necessary to study the available wind models. Most of the wave models are actually, wind-wave models. This means that they predict the wave condi-tions from predicted wind field. The accuracy of the wave prediction is thus dependent on the accuracy of the predicted wind condition. In this section, the background of a number of wind and wave models will be given. Firstly, a number of wind models will be presented followed by a summary of several wind wave models. The model of Young, which is not a wind wave model, is also presented. The steps for applica-tion of each model are given for most cases. Thereafter, the model predictions are compared to measured wind and wave data for a number of recent hurricanes. Wind and wave data were gathered from a number of sources. The wind and wave models will be applied for each case and the resulting wind and wave magnitudes compared to the measured values. The flow chart overleaf presents a summary of the subjects covered in this chapter:

4.



Overview of Wind Models: Holland, Fu-jita, Hydromet and Bret Model-X

Overview of Wave Models: Ross, Young, The Improved Young, Cooper, Bretschneider

Overview of the Application of the Wind and Wave Models: The estimation of the unknown parameters (Pc and Rmax) is discussed and the concept of “mixing models” is introduced. The use of HURWave to apply the models is described.

Description of Available Wind and Wave Data: Two sources are described, which are: 1) NDBC Data buoys within the Carib-bean Sea and the Gulf of Mexico and; 2) SRA spatial data for Hurricane Bonnie (1998).

Comparisons between Predictions of Wind and Wave Mod-els and Buoy Measurements: The methodology used is de-scribed, followed by a discussion of the results.

Comparisons between Predictions of Wave Models and SRA Measurements for Hurricane Bonnie: The methodology used is described, followed by a discussion of the results.

Conclusions and Recommendations: A summary of the areas covered in the chapter is given along with the conclusions reached and the recommendations stemming from the comparisons between model results and measurements.

Flow Chart of Chapter Contents:



4.1 WIND MODELS A number of wind models have been proposed for prediction of the instantaneous wind speed within a hurricane. Examples of such models are Holland (1980) [Ref 11], Fujita [Ref 3], Hydro-met ()[Ref. 3], Bret Model-X [Ref. 3]. These models belong to one of the two general types of hurricane wind models; 1) The modified Rankine Vortex model by Holland, of which the Hydro-met model is a special case and 2) the Bret-General model by Bretschneider, of which the Bret Model – X and the Fujita models are special cases. The mathematical forms of both are:

Modified Rankine Vortex: P PP P

Aer c

n c

B R r--

= - [ max/ ] ; ...Equation 4-1

Bret General: P PP P

ar

Rr c

n c

b--

= - + -1 1 2[ ( ) ]max

; ...Equation 4-2

where A = B-1 and a = b-1 The differences between the models is a result of: 1) the size of the hurricane as governed by the radius of maximum cyclostrophic wind; 2) the wind intensity of the hurricane as governed by the central pressure reduction; and 3) the assumption that there is a latitude effect as governed by the Coriolis parameter. Each model will be further discussed hereafter.

4.1.1 Holland (1980) The logarithmic/linear pressure profiles for 9 Florida hurricanes were approximated by:

rBln[(Pn-Pc)/(Pr-Pc)] = A; ...Equation 4-3

where A and B are scaling parameters. On taking the antilogarithm and rearranging we get:

Pr=Pc+(Pn-Pc) exp(-A/rB). ...Equation 4-4

Applying the gradient wind equations, the wind profile is:

Vg = [AB(Pn-Pc)exp(-A/rB)/ρrB+r2f2/4]0.5 – rf/2; ...Equation 4-5

where Vg is the gradient wind at radius r. f is the coriolis parameter and ρ the air density (as-sumed constant at 1.15kg/m3). In the region of maximum winds, the Coriolis force is small in comparison to the pressure gradi-ent and centrifugal forces. Here the air is said to be in cyclostrophic balance. These winds are given by:

VAB P P

Ar

rc

n c B

B=- -

[( ) exp( )

] .

ρ0 5 ...Equation 4-6



By setting dVc/dr = 0, then:

Rmax = A1/B ...Equation 4-7

and by substituting Rmax in the equation for Vc we get:

Vmax = C(Pn – Pc)0.5; ...Equation 4-8

where C = (B/ρe)0.5 ...Equation 4-9

Here e is the base of natural logarithm and the limits of B are 1 and 2.5 Two methods may be used for the application of this model; 1)The direct application and 2) the climatological application. The former relates to fitting the model to observed profiles of either wind or pressure. The values of parameters A and B are those that give the best fit to the measured profiles. Since in many cases, the observations are not available, this method of ap-plication is mostly not feasible. The second method requires that there be an established rela-tionship between Vmax and Pc. In this case climatological estimates of B may be determined by substitution into equation 4-9. This value of B can then be used to determine A for a given Rmax. The values of A and B are then used to establish the wind profile according to equation 4-6. Holland [Ref. 11] showed that the model gives good approximations for wind profiles for 9 Flor-ida hurricanes. However he states that the model requires very accurate estimate of the Rmax and that even small errors in Rmax will result in large errors in estimating the maximum winds. The method of application of the model is as follows:

1. Substitute Pc (Pa), Pn (Pa) and Vmax (m/s) into equation 4-8 to find C 2. Find B from equation 4-9 with the value of C 3. Determine Rmax (km) from wind data or from a parametric relationship 4. Determine A from equation 4-7 5. Compute the 10m-elevation, Vc (m/s) wind speed from equation 4-6

4.1.2 Hydromet Model This is a special case of the Holland modified Rankine Vortex model. For A = B = 1 the model is reduced to the Rankine Vortex which is:

P PP P

Rr

r c

n c

--

= -exp( )max ...Equation 4-10

From this the pressure gradient is obtained as follows:

dpdr

P PR

Rr

Rr

n c=-

-max

max max( ) exp( ) ...Equation 4-11

and the cyclostrophic wind equation is given by:



V rdpdr

P PR

rR

rc n c= = - -[ ] [ ( )( )exp( )]. max max .1 10 5 0 5

ρ ρ...Equation 4-12

where ρ is the air density. In applying this model, the values of Pn (mb), Pc (mb), ρ, r (km) and Rmax (km) are substituted in the equation to determine Vc (m/s).

4.1.3 Fujita Model This is a special case of the Bret General Model, A = B-1= 2. The cyclostrophic wind equation is given by:

V V

rRr

R

c =+

maxmax

max

.

.[( )

( ( ) )]

3 3

1 2

2

2 15

0 5 ...Equation 4-13

In applying this model, the values of Vmax (knots), r (n mi) and Rmax (n mi) are substituted in this equation to determine Vc (knots).

4.1.4 Bret Model This is also a special case of the Bret General Model, A = B= 1. The cyclostrophic wind equa-tion is given by:

V VrR

R rc =+maxmax

max

.[( )

( )]

4 2

2 20 5 ...Equation 4-14

In applying this model, the values of Vmax (knots), r (n mi) and Rmax r (n mi) are substituted in this equation to determine Vc (knots).

4.2 WAVE MODELS The parametric wave models that will be discussed here all have similar theoretical back-grounds. They are based on the principles of wave generation accounting for the various source and sink mechanisms for wind wave growth. As previously mentioned, it is the data set with which these models are calibrated and the method of calibration that makes them different. The models are parameterized with different variables. The main variables are the distance from the hurricane center, r, radius to maximum winds, Rmax, the maximum winds, Vmax, the veloc-ity of forward motion, Vfd, the central pressure or pressure difference, Pc or ΔP, the coriolis pa-rameter, f, and the position within the hurricane. Given these parameters, the significant wave height may be calculated at any point within the hurricane.



4.2.1 Ross (1976) 2The model is based on the physical concepts embodied in the Kitaigorodski similarity theory (1962) and the radiation transfer theory of wave growth (with its associated sink-source mecha-nisms and non-linear wave–wave interactions) and on the experimental support given these theories by the JONSWAP data by Hasselmann et al (1973). The one dimensional energy spectrum given by Kitaigorodski (1962) is of the form:

E f g f E f x( ) ( , )= -52 ;...Equation 4-15

in which g = acceleration due to gravity f= wave frequency, E is a universal dimensionless

function of two parameters: the dimensionless frequency f and the dimensionless fetch, x . For example,

fVg

fc= and xg

Vx

c

= 2 ...Equation 4-16

The JONSWAP experiment provided the first opportunity to test Kitaigorodski’s theory and to

establish an approximate form for the universal function E . The JONSWAP experiment repre-sented the one dimensional frequency spectrum by:

E f g fff

f ffm

m

m

( ) ( ) exp{ ( ) ln exp[( )

}= - + --- - -α π γσ

2 4 5 42

2 2254 2

; ...Equation 4-17

where fm and α are scaling parameters and γ and σ are shape parameters. fm= the spectral peak frequency; α = Phillip’s constant; γ = peak enhancement factor (the ratio of the peak en-ergy to that of the corresponding Pierson-Moskowitz spectrum; and σ is the width of the spec-tral peak (different for f<fm and f>fm). This parametric form was fitted to measured spectra and the dependence of the parameters on fetch established. It was found that the scale parameters obeyed simple power laws. An integral parameter characterizing the total wave energy (wave variance) namely,

∫=ε¥

0

E(f)df ...Equation 4-18

also obeyed a simple power law given by:

ε = -716 10. x x ...Equation 4-19

in which ε ε=gVc

2

2 ...Equation 4-20

Ross (1976,1979) proposed that the wave field in a hurricane might be largely determined by the local wind speed and an equivalent fetch determined by the local radius of curvature of the

2 Data Base of Maximum Sea States by Donoso, M. C. et al, 1987 [Ref#]



wind streamlines. This, in turn, is proportional to the radial distance from the eye of the storm, r. If so, then the dimensionless distance from the eye,

ξrc

gV

r= 2 ...Equation 4-21,

should play the same role for hurricane wave fields as dimensional fetch plays for fetch-limited growth. Ross then established a correlation between the JONSWAP shape and scale parameters and

this parameter ξr , from 40 measurements of one dimensional frequency spectra obtained dur-

ing hurricanes Camille ( Gulf of Mexico, June 1969) and Ava ( eastern Pacific , June 1973). Through regression analysis, the following power laws were established:

fm r= -00 97 21. .ξ ...Equation 4-22

ε ξ= -52 25 10 0 45. .x r ...Equation 4-23

α = -00 035 82. .fm ...Equation 4-24

γ ξ= -04 7 13. .r ...Equation 4-25

By combining equations x and y, the energy of the wave field in dimensional form becomes:

ε = 2 25 10 55 3 1 0 45. . . .x g V rc-5 -1 ...Equation 4-26

The significant wave height is related to this wave variance by:

Hs = 4 ε ...Equation 4-27

To specify the surface wind field, Vc, within the hurricane, Ross adopted a model proposed by Overland (1977) which defined the 10m , 15-minute average wind speed as

V VrR

R rrR

R rV Sinc fd=

++

+maxmax

max

max

max

22 2 2 2 θ (units are in standard SI units); ...Equation 4-28

in which θ is the direction of the radial from the eye of the storm to the point of interest (meas-ured clockwise from the direction of storm travel). With measured data from a number of hurri-canes, Ross re-scaled a model proposed by Patterson (1971) to obtain the following relation-ship:

V p R Cmax.

max. ( ) .= -9 05 0 4180 50Δ ;...Equation 4-29

where Vmax is in knots, ΔP in millibars and Rmax in nautical miles and C0 = 0.525Sinφ. The steps for application of this model, given Vfm and Vmax are as follows:



1. Determine Rmax either from data or from a parametric relationship and substitute into equation 4-29 to find Vmax

2. Determine the value of θ and substitute into equation 4-28 to find Vc 3. Compute the wave variance, ε, from equation 4-26 4. Calculate Hs from equation 4-27.

4.2.2 Young (1988) First Young developed an extensive synthetic database by running a numerical wave prediction model for a wide range of hurricane parameters. The data from these numerical experiments were then used to clarify the wave generation process within hurricanes and further to develop a simple parametric model suitable for wave prediction in deep water. The model used for the numerical experiments is called ADFAI. This is a second-generation spectral wave model based on a numerical solution of the radiation transfer equation. Details of this model shall not be presented here. Reference may be made to Young (1987 a, 1987 c). It should be mentioned that the model has a simple but very flexible treatment of the non-linear source term, which is of particular importance under rapidly turning winds of hurricanes. The model results have been validated with measured data from several storms on the north west coast of Australia. A total of 43 experiments were performed for a range of hurricane parameters. These allowed for the evaluation of the importance of both parameters, Vfd and Vmax, on the hurricane gener-ated wave height and frequency. It was shown that for a given Vmax, if Vfd is relatively slow, the dominant wave would outrun the storm and appear as swell ahead of the storm. Conversely if Vfd is relatively fast, the waves will be left behind the storm and no swell will be present ahead of the storm. Further, for a given value of Vmax, Hs gradually increases as a function of Vfd until a peak is reached. After this, the wave height decreases rapidly. Young states that the maximum wave conditions occur when the waves have a group velocity slightly greater than Vfd. Bretschneider (1957) proposed that maximum wave conditions would occur when Vfd = Cgmax. Under such conditions, waves would move forward with the hurricane and experience an extended fetch. Due to the effect of non-linear wave-wave interactions, however, there is a con-tinual migration of the spectral peak to lower frequencies (Hasselmann et al 1973). Conse-quently there will be a tendency for the dominant waves to continually move to lower frequen-cies and outrun the storm. Applying the concept of an equivalent fetch within a hurricane to fetch limited wave growth rela-tionships of JONSWAP (Hasselmann et al 1973) we get:

gHsV

gFV

max

max max

.. ( )2 20 50 0016= ...Equation 4-30 and



gfm V

gFV2

0045 20 33

π max max max

.. ( )= ...Equation 4-31

Young applied equation (Hsmax) to data from ADFAI for storms with Rmax = 30km to derive a relationship for the fetch dependence:

F aV bV V cV dV eV ffm fm fm302 2= + + + + +max max max ...Equation 4-32

where a, b, c, d, e and f are constants and F30.is the equivalent fetch (in meters) for a storm of Rmax= 30km. Young established that for a given Vmax, there is a value of Vfm that will give the maximum equivalent fetch and thus the maximum wave conditions. 3As Vmax increases, the value of Vfm that produces the maximum equivalent fetch also increases. This occurs since higher values of Vmax generate waves with higher group velocities and therefore a more rap-idly moving storm is required to maximize the equivalent fetch. Young then evaluated the radial dependence by comparing the equivalent fetch for the 30km radius storm to those of a number of other radii for the same values of Vfm and Vmax. The fol-lowing expression was derived:

FF

Rx30

30 7530 10

1= +. log( )max ; ...Equation 4-33

where F is the equivalent fetch for a hurricane with radius to maximum winds, Rmax. Both Rmax and F are in units of meters. Now in order to develop a simple, but flexible parametric model, an effective radius (R’) was de-rived from Equation 4-33: R’=22.5x103 logRmax-70.8x103 , ...Equation 4-34 where both R’ and Rmax have units of meters. Using this R’, Vfm and Vmax, the equivalent fetch was determined from Equation 4-32 :

FR

aV bV V cV dV eV ffm fm fm' max max max= + + + + +2 2 ...Equation 4-35

where a = -2.175 x 10-3; b = 1.506 x 10-2; c = -1.223 x 10-1; d = 2.19 x 10-1; e = 6.737 x 10-1; and f = 7.98 x 10-1, Again values of Vfm, Vmax, R’ and F are given in standard SI units. With these expressions, the maximum Hs (Hsmax) may be determined for any hurricane, given the Rmax, Vfm and Vmax by applying the modified JONSWAP expression for equivalent fetch (Equation 4-35). The spatial variation of Hs is given by Young in the form of diagrams for Hs/Hsmax and r/R’ for a

3 Parametric Hurricane Prediction Model by Ian Young (1988)



range of value of Vfm and Vmax. The digitized versions of these diagrams were used for this study. The spatial plots give values of Hs/Hsmax up to values of r/R’ of 7.5. This becomes a limit-ing factor as for values of r > 250km, the ratio Hs/Hsmax cannot be found. This model has a significant advantage over other simple parametric models as it recognizes the important role played by both Vfm and Vmax in determining the spatial distribution of wave pa-rameters within a hurricane. The steps for application of this model, given Vfm and Vmax are as follows:

1. Determine Rmax (m) either from data or from a parametric relationship and substitute into Equation 4-34 to find the effective radius to maximum winds R’ (m)

2. Determine F/R’ and thus F(m) by substituting Vmax (m/s) and Vfm (m/s) into Equation 4-35

3. Substitute Vmax (m/s) and F(m) into Equation 4-30 to determine Hsmax (m). 4. Select the appropriate spatial distribution diagram for the values of Vmax (m/s) and Vfm

(m/s) and read Hs/Hsmax

4.2.3 The Improved Young (1995) Young and Burchell (1995) [Ref. 25] compiled measured wave data from 100 hurricanes by the GEOSAT satellite. The data was collected over a three-year period and includes wind observa-tions as well. This data was used to calibrate the model of Young discussed above. From this, a dimensionless correction factor was given for the equivalent fetch given by: C = -0.015Vmax + 0.0431Vfm + 1.30 ...Equation 4-36 This correction factor is to be applied to Equation 4-35 for the calculation of the equivalent fetch, F.

4.2.4 Cooper (1988) Cooper developed parametric wind and wave models by statistically analyzing the output from numerical wind and wave models for six Gulf of Mexico hurricanes. The storms were assumed to move along a straight-line path with constant intensity, forward speed and size. The storms used, covered a wide cross-section of hurricane conditions, recorded for the Gulf of Mexico. Equations for the parametric model were derived by fitting the numerical model results with an algebraic expression. The spatial dependence was incorporated by developing a separate ex-pression for 8 radials for each storm. The final expression was derived by fitting the set of 8 ex-pressions to a Fourier series. The parametric models were developed from a numerical model of Cardone et. al. (1976). The derived expression for the wind parametric model is as follows:

V V r Rca

20= max max( / ) ...Equation 4-37



where a = -0.38 + 0.08Cosθ...Equation 4-38

and V p R f V Cosfdmax max. ( . . )= - +0 885 5 6 0 5Δ θ ...Equation 4-39

Vc20 is the wind speed at 20m elevation and θ is the angle measured anti-clockwise from the radial at right angle to the direction of forward movement of the storm. The units are given as standard SI units except for the central pressure difference, which is given in millibars. The ex-pression for Vmax is a slightly modified form of one given by Ho et. Al (1975). A correlation coefficient of 0.95 and an average RMS error ranging from 1.6 to 2.1m/s (for the different quadrants of each storm) were achieved between the results of the parametric model and the results of its parent model of Cardone et. Al (1976). A model for the wind direction was also proposed and is given by:

β= θ + α + 90o ...Equation 4-40

where α is the deflection angle given by:

α = 22 + 10Cosθ ...Equation 4-41

In this case a mean RMS difference of 10 degrees was achieved. The model of Cardone et. al. was also used to develop the parametric wave model. This nu-merical model is based on the numerical integration of the wave energy balance equation:

dSdt

v ÈS Fg

rr r r

= - -. ...Equation 4-42

where rS is the directional wave spectrum,

rvg is the group velocity and

rF is the so-called

source function representing all processes that can transfer energy to or from the spectrum. The equation is solved by successive simulation at each time step on a regular grid array of points covering the spatial domain of interest. Reece and Cardone (1982) compared the modeled and measured peak significant wave height for 60 storms. They found negligible bias and an RMS error of less than one meter. For the peak period they found that the numerical model has an RMS error of 1s and that it underesti-mates the average period by 0.22s. The parametric wave equation for significant wave height is expressed as a 25% rule. i.e.

Hs=0.25Vc20 ...Equation 4-43

where again Vc20 is the local wind speed at 20m-elevation. The mean RMS error for this expression, is less than 1m for all the quadrants in the 6 storms used for comparison with the numerical model results. The equation for the peak period, is:



Tp = a Vc20 b ...Equation 4-44

Where Vc20 is the 20m-elevation wind speed.

a = 8 – 3.5 Cosθ + 2.7 Sinθ ...Equation 4-45

b = 0.143 + 0.138 Cosθ – 0.074Sinθ ...Equation 4-46

Here the mean RMS error was found to be 1.5s. The parametric model performed best for the right front quadrant of the hurricane where the mean RMS error was 1s. The equation for the average wave direction is:

φ α θ= + + -ar

Rb( )

max

90 ...Equation 4-47

where a = 144 + 39Cosθ – 25Sinθ – 15cos2θ ...Equation 4-48

b = −0.08 The comparison with the numerical model shows a range of RMS errors from 10 to 35 degrees. The steps for application of this model, given Vfm (m/s), are as follows:

1. Determine Rmax (m) either from data or from a parametric relationship 2. Find the values of r (m) and θ and the coriolis parameter and substitute into Equation

4-39 to determine Vmax (m/s). 3. Calculate the value of Vc20 (m/s) from Equation 4-37 and then applying the appropriate

equations to determine Hs (m) and Tp (s).

4.2.5 Bretschneider (1990) The hurricane wind model is first considered as a stationary mode and as such can be made into two directional wind stress models which are coupled with the two directional significant wave-forecasting mode. Then by simple geometric means, both the stationary wind model and the corresponding stationary wave model can be obtained. This procedure assumes that steady state is achieved for the stationary wind and wave conditions. The wind and wave models are then coupled with a forward speed with an increase in wind speed and wave height to the right of the hurricane path and a decrease to the left. The procedure seemed to work very well for hurricanes with forward speed less than about 7.8m/s and probable failed for forward speeds greater than 10 to 13 m/s. At such high forward speeds, the hurricane begins to move faster than the group velocity of the waves, resulting in reduced wave heights. Bretschneider [Ref. 3] gives a quick method of obtaining significant wave height conditions us-ing the Hydromet Wind Model. This method shall be presented here. The maximum cyclostrophic wind speed (URc in knots) representing the balance between the centrifugal force and the pressure gradient, for the stationary hurricane, is given by:



U K pRc = Δ ...Equation 4-49

where Δp is in in. of Hg and

Kcons t

a

=tan

ρ ...Equation 4-50

where ρa is the air density and the constant depends on the wind model being used For the moving hurricane, the maximum cyclostrophic wind speed is given by:

U K p fR VRv fd= - +Δ 0 5 0 5. .max ...Equation 4-51

where f = 2ωSinφ, the Coriolis force parameter and URv and Vfd are given in knots and Rmax in nautical miles. Here K=66 for the Hydromet Wind model. For the wave height at any distance within a stationary hurricane, Bretschneider presents a graph of the ratio of this significant wave weight to the significant wave height at the radius of maximum winds. These values are given in relation to the ratio of the distance to the point of interest (r/Rmax) and a dimensionless factor called the Rankine vortex number. The Rankine vortex number is given by:

fRU

R SinK pRc

max max.=

0 525 φΔ

...Equation 4-52

The significant wave height at the radius to maximum winds is given by: Stationary:

H K R pRs = ' maxΔ ...Equation 4-53

Moving:

H HV

URv Rsfd

Rc

= +[.

]10 5 2 ...Equation 4-54

where HRs and HRs are in feet and K’ is given as a function of the Rankine vortex number. Finally the significant wave height calculations can be made for the moving hurricane by the fol-lowing:

H HV Cos

Vr rsfd

c

= +[. [ ( )]

]10 5 2β

...Equation 4-55

where Hrs is the significant wave height (in feet) at a distance r (n mi.) from the center of the hur-ricane. B is taken as the total angle between the Vfd (knots) and Vc (knots) and is positive on the right side and negative on the left side of the moving hurricane. Vc ( knots) is the 10-minute av-eraged – 10m-elevation wind speed at the distance, r, from the center of the hurricane. Pc is given in. of mercury.



4.3 APPLICATION OF MODELS These models were programmed in HURWave. However, before these models can be applied, the relevant parameters must be determined. Parameters such as Vmax are available from measured data in all instances but others like the central pressure and Rmax have to be deter-mined from parametric equations. Here we will look on the parametric formulations adopted to find the unknowns, after which we will describe how the models were applied.

4.3.1 Estimation of Unknown Parameters (Pc & Rmax) The central pressure and radius to maximum winds are two parameters essential for the para-metric estimation of hurricane wave conditions. However, these data are often not available, and as such an estimation has to be made for use in the parametric models. 4 . 3 . 1 . ( a ) C e n t r a l P r e s s u r e , P c The central pressure data for North Atlantic storms were not recorded until reconnaissance flights started in 1968. Prior to this, this data is not available. This is a very relevant parameter in the application of wind and wave parametric models as it is the pressure difference that de-fines the wind speed. Hence it is necessary to estimate this parameter. The expression given by Equation 3-1 was used to estimate this parameter. The value of α = 0 was used so that the ex-pected value of Pc was given for the measured value of Vmax. 4 . 3 . 1 . ( b ) R a d i u s t o M a x i m u m W i n d s , R m a x The radius to maximum winds, Rmax. This parameter governs the spatial variation in wave height and is one of the most relevant parameters for application of all the parametric models. Unlike the other parameters, it is not easily measured. The radius to maximum winds is the point at which there is cyclostrophic balance between the pressure gradient and the centrifugal force of the rotating wind field. Given a pressure profile, Rmax is given by: Rmax = Max[rdp/dr], where p is the atmospheric pressure at a radius r from the storm center. Numerous attempts were made to collect measured pressure profiles of hurricanes but this was not successful. Eventually, data was taken from the paper “Determination of Oceanographic Risks from Hurri-canes on the Mexican Coast” [Ref. 21]. The original source of this data is unknown. A plot is given of the central pressure versus the radius to maximum winds. The data is presented in Figure 3.6. Also shown is an exponential relationship derived for Rmax in terms of Pc: R x e Pc

max.= -63 10 0 017 ; where Rmax is in km and Pc is in mbars.

This expression has a correlation of 0.720 and was used to compute the radius to maximum winds at each storm point.

4.3.2 Modeling with HURWave The formulations of all the aforementioned wind and wave models were programmed into the



Wave Module of the program HURWave. With this program, computations can be made for any of the models for either one of or all of the following data points: the recorded storm points, the points of minimum distance, the points of maximum Vmax and the interpolated points between the recorded points. The interface for the Wave module is shown in Figure 2.2(b) (i). The wave models with the exception of Young are wind-wave models. The models of Ross and Cooper, both employ a wind model, other than one of those given in the section on Wind mod-els (4.1), to compute the wave field. The performance of the wave models is influenced by the results of their given wind models. Another wind model may very well give better performance for a particular wave model. As such, the wave module of HURWave was programmed to be able to replace the original wind model with one of the aforementioned wind models, a concept that will be referred to as “mixing models”. The program was applied to model different cases, which will be presented in the following sec-tions.

4.4 MEASURED WIND AND WAVE DATA In this section, a brief description of available storm wind and wave data will be given. Two sources of wind and wave data were used for this study. The first is the NOAA database of meteorological and wave conditions available on the Internet. Here there are a number of buoy measurements for instances when hurricanes have tracked within their vicinity. The second is a documentation of measurements carried out by NASA with a Scanning Radar Altimeter (SRA) mounted on an aircraft for Hurricane Bonnie’98. Other sources such as satellite data were not available for this study.

4.4.1 Buoy Measurements The NOAA organization for oceanic wave and meteorological data collection, The National Data Buoy Center (NDBC), monitors a number of buoys within the North Atlantic Basin. These buoys are mostly found off the coast of the USA and in the Gulf of Mexico. This data is available on the NDBC website: http://www.ndbc.noa.gov/data/dataindex.shtml. The website has an archive of data files which are concatenated daily with meteorological and wave data. Of interest here are data files of wind and wave data for a number of buoys for which a hurricane has passed within its vicinity. In addition to a range of meteorological data, these files contain hourly records of the following: • Atmospheric pressure • Wind speed averaged over an 8-minute period • Wind direction averaged over an 8-minute period • Significant wave height for 20-minute sampling period • Average wave period over the sampling period • Dominant wave period or the period with the maximum wave energy, Tp.



• Mean wave direction corresponding to energy of the dominant wave period.

4.4.2 SRA Measurements Scanning Radar Altimeters (SRA) are being mounted on aircrafts and used to measure ocean wave characteristics. The application is still in its early stages of development. The technique, however, provides the spatial variation in wave conditions, which is not possible with wave data buoys. NASA/Goddard Flight Center provided the first documentation of the sea surface directional wave spectrum in all quadrants of a hurricane in open water in 1998 when a hurricane research aircraft equipped with the NASA scanning radar altimeter (SRA) flew a mission into hurricane Bonnie. The data has been documented in the paper “Hurricane Directional Wave Spectrum Spatial Variation in Open Ocean” by Walsh et.al (2001) [Ref.23]. Figure 4.6 shows spatial varia-tion in significant wave height over the period of record. It is assumed that the values given rep-resent the average conditions over the 5-hour period of record. Measurements were carried out over a 5-hour period. Bonnie was at the time a category 3 hurricane and the data presented of-fers an opportunity to compare measurements with predicted values of parametric wave models on a spatial scale. Figure 4.5 shows the aircraft’s ground track and the positions of the storm center during the period of record.

4.5 BUOY DATA VS WIND & WAVE MODEL PREDICTIONS In this section we will first describe the methodology carried out to compare measurements from data buoys and predictions from the parametric models. Following this, the results will be pre-sented along with a discussion of the comparisons.

4.5.1 Methodology The comparisons of the wind and wave predictions and the buoy measurements for four hurri-canes will be presented here. These four hurricanes are Erin’95, Felix’95, Opal’95 and Ber-tha’96. The hurricanes passed within 300 km of buoy numbers 42036, 41001, 42001 and 41010, respectively. Figure 4.1 shows the location of these buoys and the storm tracks of the four hurricanes. The tracks of these hurricanes were extracted from the historical database. Each point is given every 6-hours and as such, between these points, only a straight-line track could be assumed. The Single Storm module of HURWave was used to interpolate points at 10km interval along each track within a 300km radius of the buoy, for each of the three cases. The wave character-istics given by each wind and wave model were computed with the storm at each of the interpo-lated positions. As a result, the time-varying wind conditions and the maximum Hs were calcu-lated. The results are compared to the measured data as presented in Figures 4.2(a) to 4.2(d) and Figure 4.3.



4.5.2 Analysis and Results 4 . 5 . 2 . ( a ) P a r a m e t r i c W i n d M o d e l s Figures 4.2(a) to 4.2(d) show the plots of the measured and predicted wind profiles for the pa-rametric wind models. The four parametric wind models discussed previously (Holland, Hydro-met, Bret Model –X and Fujita) were applied along with the wind models used for the wave models of Ross and Cooper. The obvious observation in all cases is that the model results ex-ceed the measurements. With the exception of Cooper, the models show excellent correlation with each other at the peak wind speeds (at the closest points to the storm center) but tend to differ at further distances away from the storm center. This is difference is probably due to the fact that the wind model used by Cooper computes the Vmax from a parametric equation whereas the others use the Vmax given in the NOAA best track data. 4 . 5 . 2 . ( b ) P a r a m e t r i c W a v e M o d e l s Figure 4.3 shows the comparison between the measured and the predicted values of the maxi-mum significant wave height, Hsmax. This Hsmax is simply the maximum Hs measured at the buoy location during the period the hurricane is within a 300km radius. It is evident that the pre-dictions exceed the measurements. Figure 4.4 shows the RMS error and the bias for each of the parametric wave models. The idea of “mixing models” was also applied here as the four other parametric wind models were used instead of the original wind models for the wave mod-els of Ross and Cooper. A positive bias suggests that the values are over-predicted and a negative bias suggests that the values are under-predicted. The bias for each model demon-strates the level of over-prediction. The model of Ross compares best with the measurements with an RMS of 1.24 and a bias of –0.4. The models of Young also show good comparisons with RMS errors of 2.19 and 1.78.

4.5.3 Discussion of Results The poor comparisons between the measurements and the predictions for both the winds and waves are disappointing but not surprising. The possible reasons for the differences are given here. • The tracking positions of each hurricane used in analysis were taken from the historical da-

tabase of 6-hourly track positions. There is no choice but to assume a straight line track be-tween the given points and to apply the values of Vmax and Pc at all points along the track. If such comparisons are to be done properly, then more detailed track positions are re-quired. This argument will be validated in the following section where accurate track posi-tions are compared with those from the historical database for Hurricane Bonnie’98.

• The measured wind and wave periods are relatively short. The wind speed is measured over 8-minute periods hourly and the significant wave height over 20-minute periods. This is necessary because of power requirements for recording the data. Whereas this period of re-cord may be sufficient for determining daily wave climate, they are too short for measuring extreme waves for hurricanes. Therefore, it is likely that peak values may have been



missed. As such, regardless of the accuracy of the hurricane records used in the parametric models, the measurements are likely to be lower than the predicted values.

• During a storm, high winds and waves create chaos of splash and wave breaking. There will not be a clear interface between water and air and hence wind speeds are likely to be measured incorrectly.

The comparisons here do not provide the grounds to acknowledge the level of performance of the models. It is recommended that further comparisons be carried out with more accurate re-cords of storm track information. In addition, the integrity of the data buoy measurements should be examined. Only then could there be conclusive evidence on the performance of the paramet-ric models. The associated shortcomings in the buoy measurement techniques do not lend much credibility to the parametric models, which were mostly calibrated with buoy data. In the advent of remote sensing technology we may consider the buoy data as a first step calibration of these models. Researches such as Ian Young (Ref. 25) have gone further to calibrate his parametric model of 1988 (Ref. 24) with satellite measurements. The following section presents a case of compari-son between the parametric models and measurements using remote sensing technology.

4.6 SRA DATA VS WAVE MODEL PREDICTIONS The preceding section discussed the possible inaccuracies in data buoy measurements. In this section, we will first describe the methodology carried out to compare SRA spatial measure-ments for Hurricane Bonnie (1998) and predictions from the parametric models. Following this, the results will be presented along with a discussion of the comparisons.

4.6.1 Methodology The available data presents the rare opportunity to test the parametric wave models in all quad-rants of a hurricane. With the given positions of the storm track at approximately one-hour in-tervals, a more accurate schematisation of the storm track can be attained than with the histori-cal best track records. (You should recall that these storm tracks are given at 6-hour intervals.) Unfortunately, the equivalent wind speeds and central pressure are not available and those given by the best track records for the time of measurements were used. A best-fit track was drawn for the period of record. The heading of this track was taken as the North direction. (Refer to Figure 4.5.) A perpendicular bisector was drawn to the schematized track to give the East to West directions relative to the storm track. Three points were marked along the track and on the perpendicular bisector north and south and east and west, respec-tively. The points were taken at 50km intervals such that the furthest point is at 150 km away from the center of the track in all 4 directions. The center point was also included to give a total of ((4 x 3 ) + 1 =) 13 points.



The Single Grid followed by the Wave Module of HURWave was used to compute the significant wave heights at each of the 13 points over the 5-hour recording period. The average wave height is taken for the period and compared to that given in the plot shown in Figure 4.6.

4.6.2 Results and Analysis The table of Figure 4.7 shows the predicted wave height for each of the 13 points. The results of each of the parametric wave models are given. In addition, for the Ross and Cooper models, the concept of “mixing models”, was applied. The original wind models were replaced with each of the wind models discussed in Section 4.1 (Holland, Hydromet, Bret Model-X and Fujita). Hence, there are 5 sets of results for the Ross and Cooper wave models. The RMS error is shown for each model and is used to judge the performance of each model. Also given is the bias for each model. A positive bias means that the predictions are higher than the measurements and lower for a negative bias. The RMS error is calculated at each of the 13 points for all the models. This is also given for the three models with the lowest RMS error (Bretschneider, Cooper and Young) The following points are noted: • The model of Ross has a high negative bias. The model performed best when applied with

the Hydromet wind model. Here the RMS error is 2.74m and the positive bias is 2.4m. • The model of Cooper predicted lower values of Hs for 11 of the 13 points giving a small

negative bias of -0.9m. This model also performed best when applied with the Hydromet model. In this case the second lowest RMS error was achieved (1.67m) and a positive bias of 0.8.

• The Improved model of Young outperformed its parent model and gained the third lowest RMS error.

• The model of Bretschneider gave the lowest RMS error of 1.13m with a bias of only 0.2m. • The three models with the lowest RMS errors seem to perform reasonably well in the right-

front quadrant of the hurricane. Here the average RMS error is 1.3m.

4.6.3 Discussion of Results The results clearly demonstrate that the parametric models do indeed give varied predictions. The RMS errors for the three best performing models range from 0.46m to 1.82m. This must be regarded as a fair agreement with the measured data and does justify the methodology used in comparing the measurements to the models. The fact that for both the Ross and Cooper mod-els, the Hydromet wind model gave better results than their parent wind model, suggests that indeed the idea of “mixing models” could prove to better the performance of these parametric wave models. This comparison has given valuable insight into the performance of both the wind and wave



models but it cannot be treated as conclusive evidence for the simple fact that it is only one storm.

4.7 CONCLUSIONS AND RECOMMENDATIONS A number of widely used parametric wind and wave models have been presented. Measured wind and wave data from NDBC buoys were compared to the predictions of these parametric models. Similarly, spatial wave data available for Hurricane Bonnie’98 was compared to para-metric predictions. From the comparisons, the following are the important conclusions and rec-ommendations:

1. Both the wind and wave parametric predictions far exceed the NDBC buoy measure-ments.

2. The parametric wave predictions compared well with the SRA spatial data for Hurricane Bonnie’98.

3. The significant difference between the buoy measurements and predictions questions the integrity of either one. One argument given for the differences between the buoy measurements and the predictions was that the historical tracks may not have provided sufficiently detailed track positions and storm characteristics. Figure 4.8 shows the track of hurricane Bonnie taken from the historical database with the points given at almost hourly intervals. The schematized historical track seems to fit well to the measured points. For the application of the parametric models, in both comparisons (buoy and SRA) we had to use the hurricane characteristics, Pc & Vmax, given in the historical database. The point here is that since the basis for comparison in both cases are similar, the differences can only be blamed on either the poor performance of the pa-rametric models or the inaccuracies in the measured data. In the case of the SRA data we have seen acceptable differences between the measurements and the parametric predictions. On the other hand, we noticed that the buoy data was consistently much lower than the parametric predictions. Consequently, one may postulate that the reli-ability of the buoy measurements is to be questioned. However, further studies are re-quired to validate such a claim.

4. The parametric wind wave models performed better in some cases when its parent wind model was replaced by another wind model. This demonstrates that weaknesses in the wind wave models are sometimes inherent to the wind models and replacing a wind model with another (mixing models) may indeed improve the overall performance of a wind wave model.

5. In spite the fact that some models out performed others when comparisons were made to the SRA data, no definitive conclusion could be drawn as to which model is most reli-able for parametric wave predictions. This is because there weren’t enough compari-sons made and it is further believed that the accuracy of the available buoy data is questionable.

6. The wave models seemed to perform well in the right front quadrant of Hurricane Bon-nie’98, which is the region with the generally highest wave heights.



7. Further data sources should be explored such as satellite data so that more compre-hensive comparisons can be carried out. Only then could a recommendation be made as to which is the most reliable parametric wind, wave or wind-wave model.



DATABASE SELECTION

CRITERIA

Before a hindcast analysis is carried out, one is faced with a number of choices. Available, is a database of information from which one has to extract the relevant data set on which to perform statistical analyses. The database is not consistently accurate hence one may choose to ignore that unreliable part of the database. But on the other hand, there is the need to have a suffi-ciently robust data set for statistical analyses. Another question is whether all the storms in the database can be regarded as having the same statistical properties. That is, can the least in-tense storms and the most intense be treated as being from the same statistical population? These are two of the points that shall be discussed in this section. In addition, we shall discuss the limiting radius for inclusion of storm occurrence. It is postulated that storms occurring far away from the point of interest will not produce any extreme waves. Also, the parametric models may not be applicable for wave predictions at great distances from the storm center. Finally, we shall discuss which points along a storm track are most likely to produce the maxi-mum wave height. This is relevant, as often one can only use the wave height from one point along the storm track, usually to limit computational effort. Hence, the following questions will be addressed:

1. Are the storm data from the historical database prior to the era of aircraft reconnais-sance consistent with the data in more recent periods?

2. May all storms, regardless of intensity, be regarded as being from the same statistical population?

3. Is there a limiting radius beyond which storm effect must be or may be ignored? 4. Is the peak Hs at a point of interest most likely to occur when the storm is at its minimum

distance from this point or when the storm reaches its peak Vmax within the vicinity of the point?

5.1 RELIABIL ITY OF RECORDS There are over 850 records of storm occurrence in the North Atlantic Basin from 1900 to 2000. This represents the most comprehensive database of storm occurrences. However, data reli-ability is not uniform throughout the period of record, as there are uncertainties in the wind and pressure estimates. According to Neumann and McAdie (1997), there are both systematic and random errors mostly confined to the years before 1968. The random errors were introduced largely through mistakes made as the data were entered on punch cards. The systematic errors are a result of the interpolations necessary to provide 6-hourly reports from 12- and 24-hourly observations. These errors include wind speeds that are not always consistent with the pres-

5.



sures, peripheral pressures rather than central pressures and forward speeds estimated to be too fast at the beginning and end of a hurricane track, among others. n light of this, one may be inclined to using only the last 30 years of data. However, we have seen in Section 3.2 that there are temporal variations in storm intensity and frequency for the last 100 years. Similarly, storm occurrence is not uniform over the entire North Atlantic Basin and as such for an area of interest, there may not be sufficient measurements to aptly carry out statistical analyses.

5.1.1 Methodology The maximum wind speed, Vmax, shall be used as the basis of comparing the reliability of the database of storms. The Vmax is indicative of storm intensity and as such is appropriate for use here. But for climatic variations, the intensity of hurricanes should be similar for both the first and second half of the last century. If not, then the errors in the records should be evident. The database was divided into two parts: prior to 1950 and post 1950. The study area was di-vided into 5 areas as shown in Figure 5.1. For each area the exponential distribution (Weibull distribution with a shape factor of 1) was fitted to the peak Vmax of each storm track which passed within the area. This was done separately for three databases 1900 to 1949, 1950 to 1999 and 1900 to 1999. The 25 and 50 year return values were extrapolated from the plots shown in Figures 5.2(a) to 5.2(e).

5.1.2 Discussion of Results The tables of Figures 5.2(a) to 5.2(e) show that for both the 25 and 50 year return values, the first 50 years show a lower return value for all the areas, with the exception of Area 4. With the advent of aircraft reconnaissance into hurricanes since 1968, it may be assumed that the data prior to 1950 has been under estimated or that storms have become more intense for the last 50 years. The former reason is supported by the fact that Area 4 shows a contrasting pattern to the other three areas. Area 4 (Figure 5.1) is east of the Florida peninsular. In this area, one may presume that more accurate records have been kept of hurricane measurements when com-pared to the other areas of the North Atlantic Basin. It might then have been a case that the val-ues of Vmax were under-estimated in the other areas. However, the latter reasoning is not im-plausible given the evidence of there being more intense storm since the 1950’s. The higher values for the latter half of the century may then be simply a manifestation of the multi-decadal variations in hurricane frequency (Section 3.2). The supporting evidence of concurrent sea sur-face temperatures lends credibility to this point. The reasons given suggest that the values of Vmax given in the database prior to 1950 may or may not have been under-estimated. It is a fact that there are errors in this database as well as there is strong evidence supporting the observed trends.



Consequently, one will have to determine whether the inclusion of the first 50 years of storm data is worthy of improving the accuracy of ones statistics. Obviously, for long return periods (such as greater than 50 years), it will be more prudent to include the entire database of storms in spite of the possible inaccuracies.

5.2 SELECTION OF STORM POPULATION Tropical storms, according to the Saffir Simpson Scale (Table 1), is a tropical cyclone with a maximum one-minute sustained wind speed less than 33m/s. Above this speed the cyclone is called a hurricane. This is the basic difference between a Tropical storm and a hurricane. Both sets of storms are characterized by the same variables and their generation mechanism is the same. As such, there is no physical evidence that there should be any difference in the statisti-cal properties between the two sets of storms. However errors in historical records may show otherwise. In a hindcast analysis, one may or may not choose to include the waves from Tropical storms. These waves are generally smaller than waves generated by hurricanes but this is dependent on the distance from the storm to the point of interest. Evidently, a tropical storm tracking rela-tively close to a point of interest could produce higher waves than a hurricane passing farther away. Given this, it is pointless to exclude these storms unless there are other reasons to deter their inclusion. A possible reason is that the mass of lower values may dominate the fit to a par-ticular distribution and thus the fit to the extreme values could be adversely affected. In such a case, the question of censoring the data set then comes into play. But then some methods of data fitting may not perform well for censored data. Another possible reason is that the paramet-ric models having been calibrated against data for the more intense storms and may not be ap-plicable for tropical storms.

5.2.1 Methodology Firstly, we shall examine how the 50-year return value of the peak Vmax of each storm track varies as one excludes the storms of lower intensity. For each of the 5 areas shown in Figure 5.1, the best fitting distribution function was found for four data sets. The first includes tropical storms and hurricanes. The second excludes the tropical storms. The third contains hurricanes only above category one (i.e. categories 2 to 5) and the fourth contains hurricanes above cate-gory 2 (i.e. categories 3 to 5). HURWave was used to determine the best-fitting distribution func-tion for each population. Secondly, we shall examine the dependence of Hs on Vmax for a wide range of storms. HUR-Wave was used to perform runs for all the parametric wave models for a deep-water location. All storms within a 500km radius were included. The predicted Hs were plotted against Vmax at each recorded storm point. These are shown in Figure 5.4.



5.2.2 Discussion of Results Figure 5.3 shows the plots of the distributions for all four different storm populations for Area 5 (shown in Figure 5.1). Shown also are the 50 year return values of peak Vmax. The values are 77, 77, 78 and 76 m/s for the four populations (tropical storms and hurricanes, all hurricane categories, hurricane categories 2 to 5 and hurricane categories 3 to 5, respectively). This con-sistency proves that all storms in the North Atlantic can indeed be considered as belonging to the same statistical population. Figure 5.4 shows the variation of Hs with Vmax for a wide range of storms. All four plots show that there are predicted values of Hs of up to 6m for Tropical storms (Vmax < 33m/s). For each model, there are subtle differences but the point is that Tropical storms cannot be ignored as within small distances from a point of interest, they do produce significantly high waves. Whether one excludes the tropical storms is a matter of the degree of censoring that is desired for the data set. The four plots of Figure 5.4 presents a guide for selection of the minimum storm intensity for a desired censoring limit. This, of course, is model-dependent.

5.3 EFFECTIVE RADIUS For extreme wave analysis, the waves generated by storms far away from the point of interest can be ignored. This is because these waves would have become swell waves over the long distance of travel. The intention here is not to down play swells, as it is often the case that these low-long-period waves have significant damaging effects on coastal structures. But extremal analysis focuses on the highest wave heights and as such these swells are ignored. A storm that is far away, say 400km, may also generate higher waves than one closer, say 200km, by virtue of it being more intense. But the parametric models may not be suitable for predicting wave heights at large distances beyond the center of the storm for two reasons: 1) they were not calibrated for distances far from the storm center; and 2) there are other wind sys-tems besides the storm winds which are influencing the wave conditions. This may be most evi-dent for smaller storms. Another point of relevance is that one may need to increase the number of storms to have a more robust data set. Consequently a suitable radius must be determined so that the following are satisfied: 1) All storms with the potential of producing extreme waves are included; 2) The parametric models are still applicable for predicting wave heights; 3) There are sufficient storms in the database for statistical analyses.

5.3.1 Methodology HURWave was used to perform runs for all the parametric wave models for a deep-water loca-tion. All storms within 500 km were included. This offered the opportunity to evaluate the varia-tion of Hs with radius for a wide range of storms. A total of 36 storms were found. The plots of



Figure 5.5 shows the variation of Hs with distance from each recorded point of each storm. Hence there is a lot more than 36 points for each plot.

5.3.2 Discussion of Results The plots of Figure 5.5 show that the dependence of Hs on r varies for the different wave mod-els. The following points are noted: • Ross has a very strong dependence as observed from the small spread in the data points. • Cooper is less dependent but there is a noticeable exponential decrease of Hs with increas-

ing r. • The model of Young shows also the expected decrease but there are no values beyond a

radius of 250 km. This is because the spatial plots for calculating Hs for Young’s model do not give values beyond a certain radius. This then defines a limiting radius for the model of Young.

• The model of Bretschneider predicts unexpectedly high values of Hs at distances greater than 400km. These values are predicted for very intense storms but even for the most in-tense of storms, these high values of Hs are unrealistic. The model may not be applicable at such large distances from the storm center.

Therefore, the radius that will satisfy the above listed conditions is model-dependent. One may use the plots of Figure 5.5 as a guide for selection of a suitable radius depending on the choice of parametric model. For each model, the most suitable radius will depend on the desired cen-soring limit for the wave height.

5.4 OCCURRENCE OF MAXIMUM HS The records of storm tracks are given at 6hr intervals. In analysis one has no choice but to as-sume straight line paths of constant forward speed between the given points. With hurricane forward speeds ranging from 1 to 7m/s, the distance between the points range from 20 to 150 km. If the values of Hs are computed only at these recorded points then it is likely that none will represent the point of occurrence of the maximum Hs. There are several points that are often taken in analysis, namely: the point of minimum distance, the point of peak Vmax, the recorded points and the interpolated points between the recorded points. Using the point of peak Vmax or the recorded points require the least computational ef-fort. On the other hand, the point of minimum distance, at first site, should predict higher waves than either of these two. But this may not be the case for all situations as the occurrence of maximum Hs is a function of Vmax and Vfd (Young [Ref 26]). Consequently, it is likely that the maximum Hs could be generated at a point further away from that of the minimum distance and possibly at the point of peak Vmax. Interpolating between the recorded points then seems the best suited method to represent as many points along the storm track. However, this requires a lot more computational effort and sometimes using a single storm point is necessary for doing



hindcast analyses. For example in using a Monte Carlo approach, it is far easier to describe the statistical properties of the variables at a single storm point than for the entire storm track. As such, even though not relevant for this study, it is important to have an insight of the most prob-able point of occurrence of maximum Hs. The aim is then to determine whether either of the closest point of a storm’s approach or the storm’s point of peak Vmax is more probable in producing the maximum wave height at a point of interest.

5.4.1 Methodology Four sites (Areas 1 to 4 of Figure 5.1) will be compared here. For each area, HURWave was used to search the database of storms for points of closest approach of each storm and simi-larly for the points of peak Vmax for each storm. The Wave module was used to compute the wave heights at the center point with the each storm at its equivalent minimum distance and peak Vmax point.

5.4.2 Discussion of Results The observations and discussions for each area are noted below: Area 1 (Figure 5.6(a)) • The first plot shows that the points of occurrence of peak Vmax in the eastern Caribbean are

uniformly distributed over the entire area. This is because the small islands have little or no effect on the hurricanes’ intensity and so the point at which a hurricane reaches its maxi-mum wind speed is not constrained in any way. The second plot shows that there is only a small bias for the highest wave heights at the point of minimum distance. This is a conse-quence of the distribution of the peak Vmax.

Area 2 (Figure 5.6(b)) • The first plot shows that the points of occurrence of peak Vmax in the western Caribbean

are not as uniformly distributed as in the eastern Caribbean. Here they tend to lie on the boundaries of the area considered as hurricanes increase or decrease in intensity. The sec-ond plot shows that there is a noticeable bias towards the minimum distance point producing higher wave heights. This is a consequence of the tendency of the points of peak Vmax to be on the boundaries of the area.

Area 3 (Figure 5.6(c)) • The first plot shows that the points of occurrence of peak Vmax are concentrated around the

land boundaries of the Gulf of Mexico and the Yucatan Peninsular. This is because the storms tend to intensify over open water until they near the large continental landmass. Hence they reach the peak Vmax just before landfall. The second plot shows that there is a significant bias towards the minimum distance point producing higher wave heights. This is a consequence of the tendency of the points of peak Vmax to be closer to the land bounda-ries.



Area 3 (Figure 5.6(c)) • The first plot shows that the points of occurrence of peak Vmax are randomly distributed.

This is because this area is basically an open-ocean area. It is true that the US landmass tend to stare the storms in a north to northeasterly direction but with the high number of storm occurrences and the wide variation in storm intensity, the point of occurrence of peak Vmax is not constrained to any particular area. The second plot shows that there is only a slight tendency for the minimum distance point to produce the maximum Hs.

In closing, it is evident that the maximum Hs at a point of interest is more likely to occur when the storm is at its closest to the point than when the storm reaches its peak Vmax within the vi-cinity of the point of interest. However, it is clear that the supporting evidence for this argument is dependent on the geographic location being considered. For example, in open sea such as in the eastern Caribbean where the effects of land mass are negligible, the random distribution of the position of the peak Vmax increases the likelihood of the maximum Hs occurring when the storm reaches its peak Vmax. In contrast, within the Gulf of Mexico where most of the peak Vmax points are located along the land boundaries, the maximum Hs will almost always occur when the storm is at its closest to the point of interest.

5.5 CONCLUSIONS AND RECOMMENDATIONS In response to the questions posed in the introduction to this chapter, the following are recom-mended:

• The inconsistency in hurricane intensity for the past 100 years is evident. It is unknown whether this is a true representation of the multi-decadal trends or whether it is a mani-festation of errors in the database. For return periods of 50 years and over one should consider the use of the 100 years of data while for lesser return periods such as 25 years, the latter 50 years of data should be used. Further studies should be conducted to look on the variation in return values for different time intervals over the past 100 years.

• The cyclones of the North Atlantic, regardless of intensity, may be treated as belonging to the same statistical population. Tropical storms should then not be ignored especially because the sometimes produce higher waves than their more intense counterparts at a particular location.

• There are limiting radii beyond which storm effect may be ignored. This is however de-pendent on the model and geographic location. The Bretscneider model, for example, does not perform well beyond a 400km radius and the model of Young does not provide results beyond 250km. For countries such as Trinidad where storm frequency is low, one may have to use a larger area radius than for Antigua that has a far greater number of occurrences. The effects of external wind fields have not been addressed in this study and although hurricanes span up to 1000km in diameter, these external effects could further give reason to question the applicability of the models at large distances from the storm center. A maximum radius of 400km is recommended for use.



• The maximum Hs from a point will mostly occur when a storm is at its closest to the point in question. This will be most evident for points close to land masses, for example within the Gulf of Mexico, and will be less evident for points over open ocean. For the latter, the maximum Hs will occur at the point of occurrence of the peak Vmax with a probability of just less than 50%. Therefore for instances when only one storm point can be taken as that which produces the maximum Hs, the point of minimum distance to the point of in-terest is recommended. However, as shown this is not always the point that generates the maximum Hs at a point of interest and as such an attempt should always be made to use all the points in the record.



THE HISTORICAL

APPROACH

The statistical methods of analysis for this approach have long been used by Engineers, hence the title “The Historical Approach”. Its application to storm waves have been compiled by re-searchers such as Goda [Ref. 10]. Section 1.8.1 presents a summary of that compilation. The methods presented are generally applicable for extreme wave analysis from any storm popula-tion. Their application to hurricane waves shall now be examined. This “Historical Method” for extremal analysis may be broken down into four steps. The first is the selection of the appropriate data series of wave heights upon which to perform statistical formulations. The two previously discussed series (Section 1.8.1(b)), the peak value series and the annual maximum series, shall be applied here. The second is to determine whether the data series should include the entire population of waves. That is, should the data series be cen-sored to get rid of the smallest values? The third is to select the best method of data fitting for the censored or uncensored data series from a range of methods. Finally, a statistical distribu-tion has to be chosen to fit the given data set. Among these distributions are the Fischer Tippet and Weibull distributions. Given the choice of two data series, varying acceptable limits of censoring, more than three es-tablished methods of data fitting and a number of distribution functions, the Engineer could be easily overwhelmed with too many choices. It is the aim of this part of the study to limit the number of choices by means of elimination of those that are not well suited for hurricane hindcast analyses. This shall be carried out by means of comparisons between the various data series, censoring limits, methods of data fitting and statistical distributions. The comparisons were carried out with the program HURWave. We shall therefore first describe the application of each step of the “Historical Approach” in the program, HURWave.

6.1 THE DATA SERIES The peak value series is formed from the peak significant wave height predicted for each storm at the point of interest. The peak Hs for each storm is determined from the time series of waves as the hurricane progresses pass the point of interest. The hurricane tracks were segmented into 30km intervals between each pair of 6-hourly recorded storm positions. In this way, we were able to get a better representation of the wave heights while the storm was in the vicinity of the point of interest. The peak Hs was chosen from the set of Hs values so that there is a peak Hs for each storm.

6.



The Annual Maximumseries is taken from the set of peak values. Within a year, there might have been more than one storm. In such a case, the maximum peak Hs is taken from all the storms within that year. The end result is a set of annual maximums, which forms the Annual Maximum data series. The number of points in each series is denoted as Nt.

6.2 CENSORING THE DATA Three sets of censored data series were used: Hs > 0 (i.e. the uncensored data set), Hs > 3m and Hs > 5m. The number of data points in the censored data set is denoted as Nc and the cen-soring parameter is denoted as v, and given by: as v=Nc/Nt.

6.3 METHODS OF DATA FITTING Three methods were applied. The first two are well-known methods of data fitting; the method of least squares and the method of moments. The third is a least squares method being developed by David Hurdle of Alkyon Hydraulic Constancy and Research. This method uses a different approach towards getting the best-fit Weibull distribution. Instead of fixing the shape factor to a number of values as has been commonly done, the method utilizes a programming routine to optimize the fit of the three-parameter Weibull distribution to the data set in question. In doing so, the optimum shape factor is determined. Further details of this method shall not be pre-sented here as it is still under development. The well-known method of maximum likelihood was not applied due to the programming requirements to apply the associated numerical formula-tions.

6.4 DISTRIBUTION FUNCTIONS Two extremal distributions were used: The Fisher Tippet-Type I and the Weibull 3-parameter distribution. For the Weibull distribution, the 7 shape parameters recommended by Patruaskas and Aagaard [Ref. 18] were used (0.75, 0.85, 1.0, 1.25, 1.5, 2.0). Included in the program HURWave, is the option to specify another value for the shape factor (besides any of the 7 men-tioned above). The Hurdle Least Squares method fits the data with a Weibull distribution but as previously discussed, the shape factor is not fixed but rather the optimum shape factor is found.

6.5 METHODOLOGY The Historical Approach followed the following basic procedure:

1. The wave data is prepared for both the peak value and the annual maximum series from the predictions of the selected wave model.

2. The data set is sorted in descending order with the largest being assigned the order number m = 1. The total number of storms in the database is denoted as Nt.



3. The plotting probability, Pm, is calculated from the Gringorton formula (Equation 1-8) for the FT-I distribution and from the Petruaskas and Aagard formula (Equation 1-10) for the Weibull distribution.

4. In the case of censored data, the values below the threshold are excluded from the data set. The number of data points is in the censored data set is denoted as Nc. Here the plotting probability, Pm, is calculated using the total number of storms in the uncensored data set, Nt.

5. The Method of Least Squares a. The reduced variate is calculated from equations given in Section 1.8.1(e) for

both the FT-I and Weibull distributions. b. Regression analysis is carried out to obtain estimates of the parameters A and B

to determine the linear relationship between the data and the known distributions. 6. The Method of Moments

a. The mean and the variance are computed for data set of wave heights. b. These computed values are equated to the estimated mean and variance rela-

tionships given for each distribution function. The equations for the estimated mean and variance for each distribution function are given in Section 1.8.1(d). The unknown parameters in the expressions (A and B) are then determined al-gebraically.

7. Each wave height in the dataset is plotted on probability paper against its plotting prob-ability, Pm. The plotting probability according to each distribution function is then also plotted for each wave height. The fit of each distribution function is determined by the correlation coefficient. is computed between the probability given for each distribution function In the case of the Hurdle least squares method, the plotting probability is taken as that given by Gumbel (Pi = i/(Nc+1)), where i = Nc – m +1, the order number from the lowest value of the data series (as opposed to m, from the highest value in the data se-ries).

8. The return wave heights are determined for the 2, 5, 10, 20, 25, 50 and 100 year return periods (Equations 1-13 and 1-14).

The flow chart overleaf summarises this procedure for the Historical Approach.



Predict Wave Heights with

Parametric Model


Extract Annual Maximum Peak Wave Heights To Form

Annual Maximum Series

Sort Data in Descending Order

Determine the Plotting Probabilities

Censor the Data Series (if desired)

Apply Data Fitting MethodsTo Fit Real and Known

Distributions





6.6 COMPARISON OF METHODS AND DISTRIBUTIONS

6.6.1 Methodolgy The program HURWave was used to perform runs for a total of 100 Atlantic basin deep-water sites. The sites were spaced no less than 300km apart. For each site, runs were carried out for two (2) data series; the peak value series and the annual maximum series. For each data se-ries, runs were carried out for three (3) censoring limits; Hs>0, Hs>3m and Hs>5m. For each censoring limit, runs were carried out for three (3) methods of data fitting; the method of least squares, the method of moments and the least squares method of Hurdle. This means that for each site 18 comparisons were carried out, resulting in an overall of 1800 points for compari-son. Two sets of these comparisons were done using the results of the parametric wave models of Cooper and Bretschneider. For each data series, censoring parameter and method of data fitting, the best-fitting distribution was selected from one of the following; the Fisher-Tippet Type-I or Gumbel distribution and four Weibull distributions with shape parameters k=0.75, k=1.0, k=1.25, k=1.4 and k=2. In the case of the Hurdle Least Squares method, the shape factor providing the best fit was used. The best-fitting distribution was determined by the highest correlation between the distribution of the data set and the distribution functions. The flowchart below further illustrates the methodology applied.



Predict Wave Heights withBretschneider Wind-Wave Model

Form Peak Value SeriesOf Wave Heights

Censor Data forHs>0 (Uncensored)

Fit Data by Method of

Least Squares

Select Best Fit Distribution From:FT-I, Weibull (k=0.75, 1, 1.4, 2)

Predict Wave Heights withCooper Wind-Wave Model

Extract Storms From DatabaseFor Site Location

Form Annual MaximumSeries Of Wave Heights

Censor Data for Hs>3m


Fit Data by Method of Moments

Fit Data by Method of Hurdle’s

Least Squares

Return k-Value for Best-Fit Weibull Distribution

Predict Wave Heights withBretschneider Wind-Wave Model

Form Peak Value SeriesOf Wave Heights

Censor Data forHs>0 (Uncensored)

Fit Data by Method of

Least Squares

Select Best Fit Distribution From:FT-I, Weibull (k=0.75, 1, 1.4, 2)

Predict Wave Heights withCooper Wind-Wave Model

Extract Storms From DatabaseFor Site Location

Form Annual MaximumSeries Of Wave Heights



Fit Data by Method of Moments

Fit Data by Method of Hurdle’s

Least Squares

Return k-Value for Best-Fit Weibull Distribution

6.6.2 Discussion of Results From the methodology described above, 1800 sets of computations were carried out for each parametric wave model. This presents an opportunity to make fair comparisons and draw con-clusions about the distribution functions, the methods of data fitting, the censoring limits (threshold values), the data series and the parametric model predictions. The results were illus-trated in different formats to highlight different observations. These formats are given in Figures 6.1 to 6.5(c). Hereafter, we shall discuss the figures separately. 6 . 6 . 2 . ( a ) F i g u r e 6 . 1 - T h e B e s t - F i t D i s t r i b u t i o n This table shows the number of times each distribution function was the best-fit for all the com-binations of methods of data fitting, threshold values and data series. The first column shows the distribution functions. These are the FT-I and the Weibull distributions with shape factors of 0.75, 1, 1.4 and 2.0. The second column also shows the distribution functions but for the Weibull Distribution only. Here, only the ranges for shape factors for the Weibull distribution are given. This second column facilitates Hurdle’s Least Squares Method, which, as described pre-



viously, returns the optimum shape factor for the Weibull distribution. Each column thereafter shows the number of times each distribution function was the best fit for the corresponding combination of data fitting method, threshold value and data series. Each column of numbers adds up to 100, which is the total number of sites used. The measure of fit is determined from the correlation. The mean correlation for each method of data fitting is given in the bottom row of the tables. The first table shows the results for the Bret-schneider model and the second shows the results for the Cooper model. In the second table, for example, the FT-I function was the best fit 36 times (out of 100) for an uncensored peak value series when the data was fit with the Method of Least Squares. In contrast, the FT-I func-tion was the best fit 13 times for an annual maximum series with a threshold value of 5 (Hs>5) when the data was fit with the Method of Least Squares. A number of observations are discussed below:

• The Bretschneider model (first table) predicts wave heights with very steep distributions. This is evident as the Weibull distributions with high shape factors are preferred. For both the Method of Least Squares (MOLS) and the Method of Moments (MOM), the best-fit distribution was the Weibull with k = 2. This is true for more than 88% of the time for both types of uncensored data series. For the Least Squares Method of Hurdle (MOH) the situation is similar. Here the preferred shape factor is greater than 1.4 for more than 70% of the time and greater than 2.0 up to 87% for some cases. The MOM shows contrasting evidence for the censored data series. However, these will be ignored here due to the very low mean correlation observed for these cases.

• The Cooper model (second table) predicts wave heights, which fit best to the FT-I distri-bution. The dominance of this function is not as overwhelming as the case was for the Bretschneider model (with the steep Weibull distributions). In addition, the preference changes to Weibull distributions, with low shape factors, as the threshold value in-creases. Figure 6.2 shows the difference in the distributions for the two models.

• For the MOH, the best-fitting distribution is more widely spread over the entire range of shape factors. This is because the values are not constrained as is the case for the other two data fitting methods. This highlights the strong point of this method of data fit-ting. It is most evident for the Bretschneider table. Here the method shows that for the uncensored Peak Value Series, the best fit distribution has a shape factor between 1.4 and 2, 50% of the time and greater than 2, 28% of the time. In comparison, the MOLS and the MOM just show that the best fit distribution has a shape factor equal to 2, 90% of the time. This is point is not as evident in the Cooper model as this model fits best to FT-I distribution.

• The distribution shape for an Annual Maximum Series is more sensitive to censoring than the distribution shape for the Peak Value Series. This is evident as variation be-tween the values for Hs>0 and Hs>3m is not significant for Peak Value Series whereas they are for the Annual Maximum Series. For example, the FT-I is the best fit for 36 of the 100 sites for both Hs>0 and Hs>3m for the Cooper Peak Value Series. In contrast,



there is a difference of 8 (Hs>0 = 44 and Hs>3 =36) for the Cooper Annual Maximum Series. This point will be further discussed in upcoming sections.

• The shape of the distribution changes significantly at a threshold value of 5m. For both data series, the threshold value of 5m has an obvious difference in distribution shape to the threshold values of 0 and 3m.

6 . 6 . 2 . ( b ) F i g u r e 6 . 3 – M e t h o d o f D a t a F i t t i n g This figure shows the variation in the mean correlation for each method of data fitting. The mean correlation is given for all the combinations of threshold values and data series.

A number of observations are discussed below: • The MOM does not handle the censored data set very well. As shown, the mean correla-

tion decreases drastically from above 0.950 to much lower values when the data is cen-sored. This is because the mean and variance of the data set changes when the data set is censored. This change is not accounted for in the calculation of the scaling parameters (in the form of equations – Section 1.8.1(d)) given for the various distributions.

• The MOLS provides the most consistent performance. For both models, both data series and the three censoring parameters, the method has an average correlation above 0.950.

• The mean correlation consistently decreases as the censoring parameter increases. This decrease is expected as the number of points in the data series is reduced.

• The method of Hurdle consistently provides the highest mean correlation for the model of Bretschneider but not so for the model of Cooper. This is due to the previous observation that the model of Cooper predicts wave heights that best fit to FT-I. The MOH is applied for the Weibull distributions only.

6 . 6 . 2 . ( c ) F i g u r e s 6 . 4 ( a ) - 6 . 4 ( d ) – C e n s o r i n g t h e D a t a S e r i e s The plots of Figure 6.4 show the return wave heights for 50 of the 100 sites. The wave heights for the Peak Value and Annual Maximum Data Series are given for the 5 and 50 year return pe-riods. Figures 6.4(a) and 6.4(b) show the plot for the 5 year return values for the Peak Value Series and the Annual Maximum Series, respectively. Similarly, the following two figures show the 50-year return values for the respective data series. Each figure has one plot for each of the two wave models used. For all the plots, the return wave heights are given for three threshold values (Hs>0, Hs>3m, Hs>5m). Also, the RMS Error is given for each threshold value. This RMS Error is computed relative to the mean of all three return wave heights for each site. On the y-axis of each plot, the significant wave height is given and on the x-axis, the site number is shown. Shown in the figures are the wave heights for the last 50 sites of the 100 sites for which comparisons were done. A number of observations are discussed below: • The return wave heights decrease as the threshold value increases. This is due to the

shape of the distributions and method of fitting the data. Particularly for high shape factor Weibull distributions, when the tail is taken out (after censoring) the distribution’s shape changes slightly. Since each distribution is fitted from its tail (smallest values), its overall shape is very much governed by the shape of the tail. (This is so as the fit is never uniform



throughout the entire distribution.) So when the tail is taken out, the strong impression it has on the distribution’s shape is also removed. This change in shape consequently changes the return wave heights. The fact that there is a consistent decrease implies that the extreme values are being fitted on their “high side”. In other words, the fit predicts a higher exce-dence probability for the highest wave height(s) in the data set. This is then a conservative approach.

• The return values for the Annual Maximum series are obviously more affected by censoring than those for the Peak Value Series. This is evident for both the 5-year and the 50 year re-turn values. The RMS Errors are much greater for all plots of Annual Maximum Series. For example, in Figure 6.4a(ii) the RMS Error for the 5m-threshold is 0.36m while in Figure 6.4b(ii) the RMS Error for the 5m-threshold is 2.61m. Indeed, using an Annual Maximum Se-ries is a means of bringing uniformity to the data set. This is in effect, a type of censoring. However, in this case, the censoring is not discrete. In other words, the data is not removed from only one side of the distribution but possibly, from across the entire distribution. Hence, when this data series is further censored, the change in the shape of the distribution is far more significant than when the Peak Value Series is censored.

• Again, it is observed that a 5m-threshold introduces a far more dramatic change to the shape of the distribution than the 3m-threshold value. In all 8 plots it is visually evident that the uncensored and the 3m-threshold return values are much closer together than the 5m-threshold values. Congruently, the evidence is reflected in the relatively high RMS errors for the 5m-threshold values.

• The RMS Errors for the Bretschneider model are less than the RMS Errors for Cooper. This is because the Bretschneider model, in most cases, predicts higher wave heights than Coo-per. Therefore, the number of data points and consequently the shape of the distribution for the censored data series are less affected by censoring. In other words, fewer wave heights will be removed from a Bretschneider data series.

6 . 6 . 2 . ( d ) F i g u r e 6 . 5 ( a ) t o 6 . 5 ( c ) - T h e D a t a S e r i e s These figures show a direct comparison between the return wave heights for the Peak Value Series and the Annual Maximum Series. The plots are given for the 5 and 50-year return peri-ods and for each threshold value. The line of direct proportionality between the return values for both data series is shown on each plot. The plots illustrate the bias and the spread between the return values for both data series. A number of observations are noted below:

• For all threshold values, the bias and the spread is less for the 50-year return values than they are for the 5-year return values. This is presumably because the most severe storms are more infrequent. Therefore, when the annual maximum wave heights are ex-tracted from the series of maximums, more of the smaller values (the more frequent small storm waves) are taken out. Consequently, the change in shape of the distribution is most evident at its tail (smallest values).

• There are consistently higher return wave heights for the Peak Value Series. This is evi-



dent as the data points are mostly to the right of the line of direct proportionality. • As censoring increases, so do the spread and the bias. This is because censoring has a

more significant impact on the shape of the Annual Maximum distribution than it does on the shape of the Peak Value distribution. The reason for this is explained in Section 6.6.2(c)

6.7 CONCLUSIONS AND RECOMMENDATIONS After this extensive set of comparisons, we are now at liberty to draw conclusions and make recommendations regarding the application of the “Historical Approach”. The above discussions may seem juvenile to a statistician. In contrast, it will be of practical importance to even the well experienced practicing engineer. The following conclusions and recommendations may be taken as guidelines for applying The Historical Approach.

1. The Annual Maximum Data Series should not be censored. Researchers such as Goda [Ref 9] have found that the difference between the two data series (Peak Value and An-nual Maximum) becomes small as the return values increase. The plots of Figure 6.5 do show that this is true. But, they also show that the difference increases for both the low and high return periods as the threshold value increases. Therefore when the data se-ries is censored then there is a significant difference between the return values given by the two data series.

2. The method of Hurdle is indeed a promising method for deriving the best-fitting distribu-tion function for the Weibull distribution. However, the performance of this method has room for improvement. Similarly, to the other methods used here, the method fits the tail of the distribution first. As such, the tail governs the fit of the entire distribution. This sometimes results in a poor fit for the extreme values in the data series.

3. The Method of Moments should never be applied for censored data. Its poor perform-ance goes back to how the scaling parameters are determined. The estimates for the expected value and variance of the distribution functions do not account for censoring and as such will vary significantly from the expected value and variance of the censored data set.

4. The model of Bretschneider predicts values of Hs that fit best to Weibull distributions with values of shape parameter unusually higher than the traditional maximum of k = 2. (i.e. k>2).

5. The model of Cooper predicts values of Hs which fit best to either the Fisher-Tippet (Type I) distribution or a Weibull distribution within the traditional limits of shape factors (0.75 < k < 2.0)

6. The Annual Maximum data series is not recommended for use to find low return periods. 7. The underlying implication is that the Peak Value series is a more reliable data series as

it allows censoring without much variation in results for either the low or high return peri-ods.



The flow chart overleaf, shows the recommendations as they should be applied in carrying out the Historical Approach to doing Extremal Analysis.

Predict Wave Heights withParametric Model


Extract Annual Maximum Wave Heights To Form Annual Maximum Series

Censor the Data Series, if desired, but for Maximum

Threshold Value of 3m

Use Method of Least Sqaures To Fit Distributions



Do Not Censor the Data Series

Do Not Use Annual Maximum Data Series For Low Return

Periods

For A Censored Data Series, Do Not Use Method of Moments

To Fit Distributions

(Use Method of Hurdle for Weibull Distributions)

The Model of Bretschneider Predicts Wave Heights with

Weibull Distributions Having Unusually High Shape Factors



THE MONTE CARLO

APPROACH

The method requires the random generation of a number of variables, which are statistically in-dependent of each other. This allows for the simulation of any number of statistically independ-ent events. As previously discussed, hurricane occurrence can be considered to be a random event. Furthermore, the relevant characteristics of hurricanes can be described with a number of variables for which the statistical distributions are known or can be found. In addition, the correlation between these components is easily established. We have previously established that whereas there is a relationship between the Vmax, Pc and Rmax, there are none noticeable between the other parameters of interest. This makes the Monte Carlo simulations suitable for application here. The method is most useful in cases where 1) there are either sparse or poor records of histori-cal data or 2) where the frequency of storm occurrence does not provide adequate number of data points for statistical analysis in the historical approach. These two conditions do not apply to the present study area and as such would not be preferred to the historical approach. But, this presents the opportunity to validate a Monte Carlo approach that could be applied to other areas of the world where storm frequency or data availability is less. This is precisely the objec-tive of this section. The two methods summarised in Section 1.8.2 , given by Donosso et.al. and Hurdle et.al. are good examples of how the Monte Carlo simulation can be applied to hindcast studies. However they both have their shortcomings.

• The method of Donosso accounts for only one point of the storm track as opposed to considering the entire track. The point considered is the storm point of minimum distance to the point of interest. In doing so, it is assumed that the peak wave conditions occur when the storm is at this point. This is not always the case as discussed earlier in Sec-tion 5.4. The method also treats the radius to maximum winds, Rmax, and the central pressure deficit ΔP, (Pc – P0), as two independent random variables. This pressure dif-ference is the driving wind force of a hurricane. Hence there must be some kind of rela-tionship between this pressure deficit and Vmax, which we have shown in Section 3.4.3 . The occurrence of this maximum wind speed, Rmax, is dependent on this value of Vmax, hence there has to be a relationship between the value of Rmax and the value of Pc. This we have shown in Section 4.3.1.(b). Therefore treating each of these variables as randomly independent of each other is not statistically correct.

• The method of Hurdle requires that all storms pass across a particular line. This method is not applicable for general cases as tracks traveling parallel or nearly parallel to this line would be excluded. In addition, the distribution of the crossing position was estab-lished. However, for the general case, this may not be possible. There may be unlikely

7.



storm tracks crossing at an uncommon point. This presents a problem as defining the fi-nite distribution between this rare point and the rest of the more likely points in the distri-bution may not be possible.

7.1 THE CHOICE OF VARIABLES The first step in developing a Monte Carlo simulation procedure is to establish whether each characteristic component may be treated as a random variable. The main components of con-cern are: 1) The Vmax and the Peak Vmax; 2) The Central Pressure; 3)Radius to Maximum Winds; 4) Minimum Distance to Point of Interest; 5)Track Heading ; and 6) Forward Velocity. These will be discussed in the following sections.

7.1.1 Vmax and Peak Vmax The first point to note is that there is mutual correlation between adjacent values of Vmax in a storm track. As such the Vmax cannot be treated as a random variable along the track. We have shown that the location of peak Vmax may be geographically dependent. This was discussed at length in Sections 3.3 & 5.4.2 . This dependence is a result of preferred storm tracks coupled with the presence of large landmasses. However, the actual value of this peak Vmax is a random occurrence. This is because storms vary in magnitude and intensity and as such the effect of landmass is similarly varied. Consequently, the peak Vmax within a defined area, may be treated as a random variable.

7.1.2 Central Pressure, Pc Figure 3.5 shows the relationship between Vmax and Pc. As shown there is an almost consis-tent spread in the relationship. An expression has been given for the relationship between these variables and is repeated here:

Pc a bVc

d= + --

( ) max( )

αα

(Pc in mbars and Vmax in knots) ...Equation 7-1

where a = 1014, b = 0.029, c = 1.626, d = 200 and α is a random uniformly distributed between the intervals [-10, 10]. A value of α =0 represents the expected value of Pc. Using this relationship, the value of the central pressure may be derived for historical storm points where this data is missing. This is the case for almost all storm records prior to 1970. Hence, for a given Vmax the value of Pc may be determined within the limits given by α = -10 and α = 10. In treating α as a random variable, we are indeed randomizing the value of Pc, but within the given limits. In essence, Pc may be termed a quasi-random variable.

7.1.3 Radius To Maximum Winds, Rmax This parameter is estimated from the established relationship between Rmax and Pc given in



Section 4.3.1.(b).: Rmax = 3x10-6exp(0.017Pc) Hence, if the measured value of Pc is available then the value of Rmax is fixed by that given by the relationship. However, in the case where Pc is unknown, a degree of randomness is inher-ited from the randomness incorporated in the quasi-random variable, Pc. Rmax, like Pc, then becomes a quasi-random variable.

7.1.4 Minimum Distance To Point of Interest For a particular geographical location, there may be noticeable patterns in hurricane track and direction. This could be due partly to the presence of land boundaries and partly due to external climatic factors such as the trade winds and the warm ocean currents (the Gulf Stream). These systems may steer a hurricane in a particular direction. For example, hurricanes occurring off the east coast of Florida tend to progress northward avoiding the continental landmass. In spite of such patterns, the distance a hurricane passes a given point is still a random component. This can be explained by the fact that the influence of these external systems is very much de-pendent on a hurricane’s physical dimension and characteristic properties. We have previously discussed the importance of the minimum distance for the point of occur-rence of the peak significant wave height in Section 5.4. Given the importance of this parameter, and given that it is indeed a random component, it becomes a prime candidate for use as one of the random variables of a Monte Carlo simulation.

7.1.5 Track Heading Hurricanes in North Atlantic do have an obvious preferred directional heading which is depend-ent on the geographic area(refer to Figure 5.6. In the eastern Caribbean they tend to have a west to northwesterly heading. In the Gulf of Mexico, they range from moving westward to north easterly. Off the east coast of the USA they tend to move in a north to northeasterly direction. It would then seem that the tracks could be randomized within the directional limits of west to north easterly (clockwise direction). However, this would exclude the exceptional cases such as Lenny in 1999, which tracked in an easterly direction across the Caribbean sea. Treating this parameter as a random variable would then require that there be no directional limits as estab-lishing the finite distribution between the rare track directions and the rest of the distribution would be impossible.

7.1.6 Forward Velocity We have seen from Figure 3. that there is no correlation between the velocity of forward motion and the maximum wind speed. Since the other two parameters, Pc and Rmax, are strongly cor-related with Vmax, then it is unlikely that Vfd will have any correlation with either of them. How-ever, similarly to the value of Vmax, there is a mutual correlation between the contiguous values of Vfd. As such, this variable cannot be treated as a random independent variable. In any case,



with the exception of the model of Young, the forward velocity is generally not an important vari-able for determining the significant wave height.

7.2 THE SIMULATION PROCEDURE We have thus far established that the peak Vmax and the minimum distance can both be treated as independent random variables. Similarly, Pc can be randomized but within the limits given by the corresponding value of Vmax. Consequently, the Rmax has an inherent random-ness by virtue of being estimated from the quasi-random value of Pc. The program HURWave includes a Monte Carlo Module, discussed in Section 2.5 Here a syn-thetic database of any specified number of hurricanes can be generated. The procedure in-volves three main steps. The first is to select the parent population of storms. The second is to derive the best-fitting distribution function for the random variables. The third is to generate the random variables from the distribution functions and the final step is to apply these variables to construct a synthetic track. These steps are discussed below.

7.2.1 Selection of Parent Population HURWave uses the search routine described in Section 2.7.1 to extract, from the selected historical database, the tracks for the specified area. The search routine returns the storm pa-rameters (tracking positions and times, Vmax, Vfd, Pc). In addition, the point of minimum dis-tance and the value of Dmin along with the point of peak Vmax and the value of pVmax are computed. These two parameters Dmin and peak Vmax, are of most interest.

7.2.2 Determination of Distribution Functions The two variables of interest here are the peak Vmax, pVmax, and the minimum distance , Dmin. Dmin is expected to be uniformly distributed. On the other hand the case for pVmax is not as clear. Researchers have reported that Vmax frequently fits an exponential distribution. How-ever, due to measurement inaccuracies, this may not always the case. In any case,4 there is no physical reason for preferring one distribution over another and one should use the whichever distribution fits the data best Therefore, rather than pre-selecting one distribution function, a number of distributions were fit-ted to the data and the best-fitting one chosen. The distribution functions used are, the Fisher-Tippet type I (FT-I), the Weibull (using several shape parameters), the Exponential and the Uni-form distributions. Also, the three methods of data fitting used in the Historical Approach are used; the method of least squares, the method of moments and the least squares method of Hurdle. 4“On the Calculation of Extreme Wave Heights: A Review” by L.R. Muir an A. H. El-Shaarawi (1986) [Ref 17 ].



7.2.3 Generation of Random Variables Given the two (2) best-fitting distributions functions for the variables Dmin and Vmax, we can now use a random number generation process to produce a database of random variables. But before this is done, we must establish the limits within which the variables must be generated. This is necessary as we are using the best-fitting distribution function, which is different for each simulation. This is particularly important for the extreme values of the peak Vmax. Even the best-fitting distribution does not always fit the extreme-value end of the distribution. As such de-pending on the shape of the distribution, the equivalent excedence probability for the most ex-treme value in the distribution may be appreciably higher or lower than the real value. With this happening we could very well end up with “unrealistically” high extreme values for pVmax. One approach to dealing with this is defining the physical upper limit of pVmax. But then the question is whether there is a physical limit for hurricanes in the particular geographical area of interest, or for that matter, storms in general. Researchers have proposed probable maximum wind speeds in relation to North Atlantic surface water temperatures. Unfortunately, attempts to gather these publications were futile. As a result, the lower and upper excedence limits were set to the equivalent values given by the distribution function for the lowest and highest values of the parent population. This was applied for both Dmin and pVmax. This approach is arguable, an unrealistic constraint on these distributions. With a large number of parent storms it may not be but would be for just a few known storms. The ideal approach would be to set the upper limit of Vmax according to the probable upper limit given by some other component such as the sur-face water temperatures. However, here again, this could be also a constrained if the available sea surface temperature data is limited. This point is then the most likely weakness in this ap-proach. For each synthetic storm, the random generator was used to select a probability of excedence value between the established lower and upper limits. The values of pVmax and Dmin are then calculated using the pre-selected best-fitting distribution.

7.2.4 Construction of Synthetic Storm Track Each synthetic track is constructed from modifying one of the tracks in the parent population. A track is drawn at random and assigned a random Dmin. The entire storm track is then translated so that its minimum distance from the point of interest becomes this random value of Dmin. The translation occurs for the entire track along a straight line (towards or away from the point of in-terest) along the radial drawn from the point of interest to the minimum distance point of the each storm tack. This is illustrated in Figure 7.1. In doing so the tack maintains its directional properties in terms of it’s heading and position (left or right) to the point in question. The spatial and directional distribution of the storm tacks in the parent population is thus maintained as each track has an equal probability of being selected.



7.2.5 Retaining the Distribution of Peak Vmax When a storm track is moved away or towards the point of interest, then its peak Vmax within the bounds of the area being considered may be different. Therefore, the distribution of peak Vmax for the synthetic storms would be different to that of the parent distribution. To avoid this, after construction of the synthetic storm tracks, the database is re-searched and the new peak Vmax of each storm track within the area of interest is re-scaled with the equivalent random Vmax. This may be viewed as constraining the possible higher wind speeds in the distribution and indeed it does and rightly so. However, it does ensure that the distribution is generic for the particular area of interest is maintained. For example, if the area of interest is close to a mas-sive land mass, then moving a track may result in there being a greater Vmax close to land than is physically “possible”. In spite of this, it does not constrain the distribution of wave heights as the storms at different locations with different wind speeds have the potential to generate very different distributions of wave heights.

7.2.6 Summary of Monte Carlo Procedure The flow chart below presents a summary of this Monte Carlo procedure.



Plot the values of the pVmaxand Determine the Best-fitting


Generate Random Values of pVmax From the Distribution, Equal


Choose a Storm at RandomFrom the Parent Population

Determine the Peak Vmax (pVmax) and the Minimum Distance (Dmin) for each Storm

Search Synthetic DatabaseAnd Re-adjust the New Values of pVmax that

Exceed the Random pVmax for Each Synthetic Storm

Plot the values of the Dmin and Determine the Best-fitting


Extract Parent Population ofStorms From Database For Site Location

Generate Random Values of Dmin From the Distribution, Equal


Loop

for

Num

ber o

f Re

q. S

ynth

etic

Sto

rms

Set the Dmin of this Storm to the Randomly Selected Dmin

Perform Extremal Analysis on Wave Heights

Calculate Wave Heights for Each Synthetic Storm

Set the pVmax of this Storm to the Randomly Selected pVmax

For each Synthetic Storm:Generate a Random Number [-10,10]

Calculate Pc from Equation 3-1Compute Rmax

For example, if this procedure were to be applied to an area at the south coast of Jamaica then first the center coordinates of interest and the radius of inclusion would have to be specified. The database of storms would then be searched and all the storms within the specified radius selected. The peak Vmax (pVmax) within the area and the minimum distance (Dmin) to the point of interest is extracted or calculated from the database. The series of Dmins and pVmax are then plotted and fitted to a number of statistical distributions. The best fitting distributions are selected and their theoretical functions established. For each synthetic storm to be gener-



ated, a Dmin and pVmax is selected at random from their respective theoretical functions. Thereafter, a storm is selected at random from the parent population. This storm is moved to-wards or away from the center point by moving the position of its real Dmin to the position of the randomly selected Dmin. Its real pVmax is also changed to the randomly selected pVmax and all the other values of Vmax along its track are scaled accordingly. Therefore, if the real pVmax is increased by a factor of 1.2 then all the other Vmax values along the storm’s track is in-creased by this said factor. When each storm is moved, for a few cases, the new pVmax within the area of interest may be greater than the original pVmax. To counter this, the pVmax of each synthetic track is scaled to the previously randomly selected pVmax for that storm. For each of these new tracks, the central pressures are computed with the formula given for Pc in terms of Vmax. In applying this formula the random parameter α is used so a random Pc (within the measured range of values) is always computed for a given Vmax. The Rmax is then computed from this randomly selected value of Pc. A wave model is then applied to derive the data series of wave heights for the synthetic tracks. An extremal analysis follows to determine the return values.

7.3 DISCUSSION OF SIMULATION RESULTS A number of simulations were carried out for two deep-water sites. For each site, 5 synthetic populations were generated of 50, 100, 200, 500 and 1000 storms. For each population the re-turn wave heights were calculated using extremal analysis. This was done with the significant wave height predictions from two parametric models: Bretshneider and Cooper. The distribu-tions and the return values for the 5 and 50 year return periods are shown in Figures 7.2 and 7.3. These are compared to those of the parent population. The figures show a shift in the distribution as the number of synthetic storms increases. This is expected as the plotting probability changes with the number of storms. The shape of the distri-butions varies slightly as is evident in the variation of return values of the 5 and 50-year wave heights. The return values for the 5-year return period seems to increase with increasing num-ber of storms. On the other hand, there is no such pattern in the variation of the 50-year values which is expected for a random simulation procedure as this one. Hence, the change in the shape of the distribution seems to affect the low return periods but then these are not important values for design. The 50-year values for the synthetic populations seem to lie within 1.5 m of the parent population. In each plot of the variation of the return values, the average return val-ues of all the synthetic values are shown. Of the four cases, only one shows an average value that differs from that of the parent population by more that 0.3m. This is a very good agreement between the synthetic and real cases and shows the validity of the Monte Carlo Approach pre-sented.

7.4 RECOMMENDATIONS FOR APPLICATION The above discussion shows that the developed procedure is valid for computing return values



of significant wave height for tropical cyclones. It is recommended that for a given application, the procedure given in the preceding sections be applied. In the cases where there is limited availability of accurate records, then it will be difficult to es-tablish the limits for the peak Vmax discussed in Section 7.2.3 . If other evidence suggests that there may be storms of greater peak Vmax, for example sea surface temperatures, then the highest probable peak Vmax should be adjusted accordingly. Hereafter, a number of synthetic storm populations should be determined. These populations may vary in number as well as one population may be run a number of times. The average of the return wave heights of either set of synthetic storm population should then be taken as the design value.



SUMMARY OF

CONCLUSIONS AND

RECOMMENDATIONS The following are the main conclusions and recommendations drawn from the study:

• Hurricanes do have a preferred track that is geographically dependent. While in the Car-ibbean Sea, they tend to move in a west to northwesterly direction. In the Gulf of Mexico and on the east coast of the USA they tend to track north-northwesterly to northeasterly.

• There seem to be a multi-decadal cycle in the frequency of hurricane occurrence and in-

tensity. This cycle correlates well with the multi-decadal variation in sea surface tem-perature for the equatorial regions of the Atlantic Ocean.

• The central pressure and Vmax are the only two parameters that show an obvious cor-

relation between each other. A formula was derived expressing the central pressure as a function of Vmax and including a random variable, α, to account for the spread in the relationship. This formula is given by:

Pc a bVc

d= + --

( ) max( )

αα

(Pc in mbars and Vmax in knots), where a = 1014, b = 0.029, c

= 1.626, d = 200 and α is a random variable uniformly distributed in the interval [-10, 10]. α = 0 for the expected value of Pc.

• The radius to maximum winds (Rmax) may be expressed in terms of the central pres-sure (Pc) as follows: R x e Pc

max.= -63 10 0 017

• Both the wind and wave parametric model predictions far exceed the NDBC buoy

measurements while the parametric wave predictions compared well with the SRA spa-tial data for Hurricane Bonnie’98.

• The significant difference between the buoy measurements and wind and wave predic-

tions questions the integrity of either one. However, given the performance of the mod-els with the SRA data, it is recommended that further studies be carried out to test the accuracy of the buoy data.

• The parametric wind wave models performed better in a few cases when its parent wind

model was replaced by another wind model. This demonstrates that weaknesses in the wind wave models are sometimes inherent to the wind models and replacing a wind

8.



model with another (mixing models) may indeed improve the overall performance of a wind wave model.

• The number of comparisons carried out between parametric models and measurements

was limited by data availability. We were therefore not able to definitively say which model is the best. As such, further data sources should be explored such as GEOSAT measurements of hurricane waves.

• The inconsistency in hurricane intensity for the past 100 years is evident. It is unknown

whether this is a true representation of the multi-decadal trends or whether it is a mani-festation of errors in the database. For return periods of 50 years and over one should consider the use of the 100 years of data while for lesser return periods such as 25 years, the latter 50 years of data should be used. Further studies should be conducted to look on the variation in return values for different time intervals over the past 100 years.

• The cyclones of the North Atlantic, regardless of intensity, may be treated as belonging

to the same statistical population.

• There are limiting radii beyond which storm effect may be ignored. This is however de-pendent on the model and geographic location. A maximum radius of 400km is recom-mended for use.

• The maximum Hs from a point will mostly occur when a storm is at its closest to the point

in question as opposed to the point when the storm reaches its peak wind speed.

• The Annual Maximum Data Series should not be censored. Researchers such as Goda [Ref 9] have found that the difference between the two data series (Peak Value and An-nual Maximum) becomes small as the return values increase. The evidence found here show that this is true but they also show that the difference increases for both the low and high return periods as the threshold value increases. Therefore when the data se-ries is censored then there is a significant difference between the return values given by the two data series.

• The Least Squares method of Hurdle is indeed a promising method for deriving the best-

fitting distribution function for the Weibull distribution. However, the performance of this method could be improved by first fitting the most extreme values in the database rather than the tail. The tail would then not govern the shape of the distribution.

• The Method of Moments should never be applied for censored data as the estimates for

the theoretical mean and variance do not account for censoring.



• For the four cases presented for the comparison between the Monte Carlo simulation re-sults and the results of the Historical approach, the maximum difference between the 50-year return values is 1.5m. This is a very good agreement between the synthetic and real cases and shows the validity of the Monte Carlo Approach presented.



DISCUSSION

The study, for the most part, has achieved its objectives. These are discussed hereafter. 1) Compilation and Validation of Hurricane Records The hurricane records have been compiled and their consistency has been established. We have seen that there are missing data of the central pressure measurements prior to 1970. The consistency of the records of maximum wind speed, Vmax, before and after 1950, was evalu-ated through comparisons of the 25 and 50-year return values of peak wind speed, peakVmax. The conclusion was drawn from this comparison that both data sets are inconsistent with each other. On the other hand, we have also shown evidence of multi-decadal variations in hurricane activity along with concurrent sea surface temperatures. This evidence is overwhelming and leads us to question whether ongoing arguments that the recent increases in hurricane activity is a cause of unnatural global warming are valid. But still, given that the inconsistencies in the data prior to 1970 could, to an extent, be a result of data inaccuracy, we are left to ponder. 2) Summary of Statistics and Parameter Variation The hurricane statistics have been summarized and presented. Among others, the frequency of storm occurrence, both spatially and temporally, has been given. In addition, the spatial distribu-tion of the peak Vmax has been shown. It was concluded that the location of occurrence of peak Vmax is geographically dependent due to the presence of large landmasses and the preferred track direction in certain areas. We have further established the relationships between the characteristic components. Two expressions were derived between Pc and Vmax and Rmax and Pc. The former includes a random variable to account for the spread in the relationship. It was found that whereas there is an existing correlation among Pc, Vmax and Rmax, there is no noticeable correlation with the forward speed of the hurricane. We also presented evidence to show that the highest waves do occur at the storm’s minimum distance to the point of interest. But we also showed that sometimes the point of peak Vmax within the area of interest does generate the highest waves. We therefore recommended the use of all the recorded storm points to find the highest wave height, when computational time and effort is not a burden. 3) Comparison of Wind and Wave Models In this area, the study has failed to say definitively, which parametric wind or wave model is best for calculation of hurricane significant wave heights. This was a result of lack of adequate measured data for comparison. The most suitable measurements would have been satellite measurements recorded from NASA’s GEOSAT. This data is only commercially available and as such was beyond the budget of the study. In spite of this, the few comparisons carried out with measured buoy and SRA data did give some valuable insight on the performance of the models. The comparisons with both sets of data also raised the question of the accuracy of the

9.



buoy measurements during periods of extreme hurricane waves. 4) The Historical Approach Here, the various statistical procedures commonly used by engineers and scientists were ap-plied to extreme storm wave analysis. The peak value series (peak over threshold) was com-pared to the annual maximum series for selecting the data series of waves. A total of 1800 comparisons were carried out over 100 sites (18 per site) across the North Atlantic basin. In do-ing so, we were able to evaluate the difference between the two mentioned data series, the ef-fect of censoring these data series, the performance of the methods of data fitting and the fits to a number of extremal distribution functions. This part of the study concluded that the annual maximum series should not be censored and should not be preferred to the more consistent peak value series. We also recommended that the least squares method of Hurdle should be improved to first fit the most extreme values in the distribution. The set of comparisons was done for two parametric wave models. This gave insight to the variation in the best-fitting distri-butions and the performance of the data fitting methods for different parametric model results. 5) The Monte Carlo Approach A simulation procedure was developed capable of generating any number of synthetic storms from the properties of the parent storm population. This procedure was applied to 2 sites in the study are and for 2 parametric models. The return values given by this approach were com-pared to those given by the historical approach. The synthetic values show only small deviations from the values for the real population. It was then concluded that the Monte Carlo Approach is valid and may be applied instead of the historical approach in areas were storm records are sparse. 6) HURWave This programming tool is capable of carrying out quick hindcast analysis for hurricane waves and can be applied for use any where within the North Atlantic Basin. This tool is extremely handy for application to hindcast analysis for the islands of the Caribbean. It is equally user-friendly and so can be easily mastered. There is presently, no widely used tool that offers such a complete set of parametric models and statistical methods for hindcast analysis. Both design engineers and researchers may use the tool. In concluding, it must be said that this work sets the stage for numerous possibilities in Tropical Cyclone Wave Design Engineering as well as further development into more sophisticated modeling techniques. The former has already taken effect as HURWave and the study’s various findings have been applied to find deepwater wave heights for several islands in the Caribbean Sea. These islands include St. Kitts, Nevis, Antigua, Trinidad, The Bahamas and Jamaica. The works carried out have been vital contributions to Coastal Zone Management Plans and Hurri-cane Mitigation studies for these islands. The latter is boundless. The potential developments and improvements to this study include more comprehensive comparisons of the wave models with satellite data, comparisons between parametric and numerical wave models, more detailed



filtering of hurricane activity data to establish potentially hidden trends, further development of the Monte Carlo to generate independent random storm tracks using auto-correlation formula-tions, etc. Evidently, the study does not cover the most sophisticated and state of the art formu-lations and techniques, even though it does present interesting scientific findings and recom-mendations. On the contrary, it explores a vast range of known techniques and offers a prag-matic guide for “preliminary” extreme wave design. These are computationally inexpensive, rea-sonably accurate and as such are well suited for application within the Caribbean and the rest of the developing world that are frequently ravaged by “the greatest storms on Earth”.



REFERENCES

Ref. 1 Atkinson, G. D. and Holliday, C. R., 1977, “Tropical Cyclone Minimum Sea Level Pressure Maxi-mum sustained Wind Relationship for Western North Pacific,” Monthly Weather Review, 105, pp. 421-427.

Ref. 2 Bain, l. J. and Engelhardt, M., 1992, “An Introduction to Probability and Mathematical Statistics.”, Duxbury Press, Belmont California, USA, ISBN 0-534-985633-7

Ref. 3 Bretschneider, C. L., “Tropical cyclones”, 1990, Handbook of Coastal and Ocean Engineering. Volume I: Wave Phenomena and Coastal Structures, editor John B. Herbich, pp 249-370. Gulf Publishing Co.

Ref. 4 Carter, D. J. T., and Challenor, P. G., 1983, “Methods of Fitting the Fisher-Tippet Type I Extreme Value Distribution,” Ocean Eng., Vol. 10, No. 3, pp. 191-199 .

Ref. 5 Cooper, C. K., 1988, “Parametric Models of Hurricane-Generated Winds, Waves and Currents in Deep Water.”, proc. 20th Annual OTC, Houston, Texas, USA, May, pp. 475-484.

Ref. 6 Donoso, M. C. I., Mehaute, B. L. and Long, R. B., 1987, Data Base of Maximum Sea States Dur-ing Hurricanes.”, paper, Journal of Waterway, Port, Coastal and Ocean Engineering, Vol. 113, No. 4, July, pp. 311-326

Ref. 7 Donoso, M. C. I., Mehaute, B. L. and Long, R. B., 1987, Long Term Statistics of Maximum Sea States During Hurricanes” paper, Journal of Waterway, Port, Coastal and Ocean Engineering, Vol. 113, No. 6, November, pp 636-646

Ref. 8 Elsner, J.B., and Kara, A. B., 1999, “Hurricanes of the North Atlantic: Climate and Society”, Ox-ford University Press, New York, ISBN 0-19-512508-8.

Ref. 9 Goda, Y., 1988, “On the Methodology of Selecting Design Wave Height,” Proc. 21st Int. Conf. On Coastal Engineering., June, pp. 899-913.

Ref.10 Goda, Y., (1990) Distribution of Sea State Parameters and Data Fitting. Handbook of Coastal and Ocean Engineering. Volume I: Wave Phenomena and Coastal Structures, editor John B. Herbich, pp 371-408. Gulf Publishing Co.

Ref.11 Holland, G. J., 1980, “Analytical Model of the Wind and Pressure Profiles in Hurricanes.” Monthly Weather Review, 108, pp. 1212-1218.

Ref.12 Holthuijsen, L. H. and Battjes, J. A., 2000, “Wind Waves”, Lecture Notes for IHE course in Ocean Waves, IHE, Delft, The Netherlands.

Ref.13 Hurdle, D. P., van Vledder, G. Ph., 2001, “A Hybrid Approach to Determine Extreme Wave and Wind Conditions in the Andaman Sea due to Tropical Storms,” prepared for: 20th Int. Conference on Offshore Mechanics and Arctic Engineering., June, 2001



Ref.14 Isaacson, M. de St. Q., and Mackenzie, 1981. “Long Term Distributions of Ocean Waves: A Re-view.”, paper, Journal of Waterway, Port, Coastal and Ocean Engineering, Vol. 107, No. WW2, May, pp 93-109

Ref.15 Lamming, S. D., Depradine, C. A. and Rudder G. M., 1973, “Some Characteristics of Hurricanes in the Eastern Caribbean.”, report, Caribbean Meteorological Institute, Barbados, West Indies.

Ref.16 Lamming, S. D., 1975, “Some Characteristics of Hurricanes in the Western Caribbean.”, report, Caribbean Meteorological Institute, Barbados, West Indies.

Ref.17 Muir, L. R. and Shaarawi, A. H., 1986, “On The Calculation of Extreme Wave Heights: A Review.” paper, Ocean Engineering., Vol. 13, No. 1, pp. 93-118.

Ref.18 Patruaskas, C. and Aagaard, P. M., 1970, “Extrapolation of Historical Storm Data for Estimating Design Wave Heights,” Prepr. 2nd Annual Offshore Tech. Conf., OTC 1190.

Ref.19 Ross, D. (1976). “A simplified for Forecasting Hurricane Waves” (abstract). Bull. Am. Meteoro-logical Society 57(1).

Ref.20 Ross, D., and Cardone, V. (1977). “ A Comparison of Parametric and Spectral Hurricane Wave Prediction.” NATO Symp. Of Turbulent Fluxes through the Sea Surface, Wave Dynamics and Predictions, Marseilles, France.

Ref.21 Silva, R. et. al., 2000, “Determination of Oceanographic Risks from Hurricanes on the Mexican Coast

Ref.22 Sylvester, R., Hsu, J. R. C., 1997, Coastal Stabilization, book, World Scientific, ISBN 981-02-3137-7.

Ref.23 Walsh, E.J., Wright, C.W. et al., “Hurricane Directional Wave Spectrum Spatial Variation in Open Ocean”, 2000, paper, Journal of Physical Oceanography.

Ref.24 Young I. R., 1988., “Parametric Wave Prediction Model.”, paper, Journal of Waterway, Port, Coastal and Ocean Engineering, Vol. 114, No. 5, September, pp 637-652.

Ref.25 Young, I. R. and Burchell, G. P., 1996, “Hurricane Generated Waves as Observed By Satellite.” Pap. Ocean Engineering, Vol. 23, No. 8, pp 761-776.

Ref.26 Young, I. R. “Wind Generated Ocean Waves”, 1999, book, University of Adelaide, Australia.

Ref. 27 Goldberg, S. B., Landsea, C. W., et. al., 2001, “ The recent Increase In Atlantic Hurricane Activity: Causes and Implications”, pap. Science, Vol. 293, No. 5529, pp 381-560.



LIST OF FIGURES Figure # Title

1.1

The Study Area – North Atlantic Basin

1.2 Characteristic Hurricane Tracks of the North Atlantic Basin

1.3 1950 to 2000 Hurricane Tracks for The North Atlantic Basin

1.4 The Characteristic Components of a Hurricane

2.1 Flow Diagram for the Program HURWave.

2.2 Examples of HURWave’s Interfaces.

3.1 Spatial Distribution of North Atlantic Hurricane Occurrence on a 2ox2o Grid from 1950-

2000.

3.2 Distribution of North Atlantic Hurricane Occurrences and Intensities.

3.3 Spatial Distribution of Peak Vmax of Each Storm Track from 1950-2000.

3.4 Correlation Between Characteristic Components of North Atlantic Hurricane Occur-

rences.

3.5 Derived Relationship for Central Pressure in terms of Vmax.

3.6 Derived Relationship for Rmax in terms of Central Pressure.

4.1 Hurricane Tracks and Data Buoy Locations for Wave Height Comparisons.

4.2(a) Measured and Predicted 10m-Elevation Wind for Hurricane Felix’95 at Buoy 41001.

4.2(b) Measured and Predicted 10m-Elevation Wind for Hurricane Bertha’96 at Buoy 41010.

4.2(c) Measured and Predicted 10m-Elevation Wind for Hurricane Erin’95 at Buoy 42036.

4.2(d) Measured and Predicted 10m-Elevation Wind for Hurricane Opal’95 at Buoy 42001.

4.3 Chart of Measured Hsmax and Predicted Hsmax for 5 Wave Models.

4.4 Table of Comparisons Between Measured Hsmax and Predicted Hsmax.

4.5 NOAA Aircraft Ground Tracks and Normalized Axes for Comparisons.

4.6 Spatial Variation in Hs from SRA Measurements for Hurricane Bonnie’98.

4.7 Comparison of Parametric Model Predictions with SRA Data for Hurricane Bonnie.

4.8 Detailed Track Positions and 6-hourly Track Records given in Historical Database.

5.1 Areas for the Comparison of Distribution of Peak Vmax

5.2(a) Statistical Distribution of Peak Vmax for Area 1

5.2(b) Statistical Distribution of Peak Vmax for Area 2



5.2(c) Statistical Distribution of Peak Vmax for Area 3

5.2(d) Statistical Distribution of Peak Vmax for Area 4

5.2(e) Statistical Distribution of Peak Vmax for Area 5

5.3 Best-Fitting Statistical Distribution of Peak Vmax Four Storm Populations

5.4 Variation in Predicted Hs with Vmax for Four Wave Models

5.5 Variation in Predicted Hs with Distance from Storm Center, r, for Four Wave Models.

5.6(a) Occurrence of Hsmax (Point of Minimum Distance vs Point of Peak Vmax) – Area 1.

5.6(b) Occurrence of Hsmax (Point of Minimum Distance vs Point of Peak Vmax) – Area 2.

5.6(c) Occurrence of Hsmax (Point of Minimum Distance vs Point of Peak Vmax) – Area 3.

5.6(d) Occurrence of Hsmax (Point of Minimum Distance vs Point of Peak Vmax) – Area 4.

6.1 Tabulation of Best-Fit Distributions for Bretschneider and Cooper Models

6.2 Typical Shape of Distributions and Return Values For Bret. And Cooper Models.

6.3 Performance of the Methods of Data-Fitting.

6.4(a) 5-Year Return Hs for Bret. and Coop Models using Peak Value Series.

6.4(b) 5-Year Return Hs for Bret. and Coop Models using Annual Maximum Series.

6.4(c) 50-Year Return Hs for Bret. and Coop Models using Peak Value Series.

6.4(d) 50-Year Return Hs for Bret. and Coop Models using Annual Maximum Series.

6.5(a) Comparison between Peak Value and Annual Maximum Series for Hs > 0m.

6.5(b) Comparison between Peak Value and Annual Maximum Series for Hs > 3m.

6.5(c) Comparison between Peak Value and Annual Maximum Series for Hs > 5m.

7.1 Variation in Distributions and Return Hs for different Synthetic Databases (29oN, 65oW).

7.2(a) Variation in Distributions and Return Hs for different Synthetic Databases (16oN, 60oW).

7.2(b) Variation in Distributions and Return Hs for different Synthetic Databases (16oN, 60oW).

7.3(a) Variation in Distributions and Return Hs for different Synthetic Databases (29oN, 65oW).

7.3(b) Variation in Distributions and Return Hs for different Synthetic Databases (29oN, 65oW).

Figure 1.1: The Study Area - North Atlantic Basin

Caribbean Sea

Gulf of Mexico

North Atlantic Ocean

PARAMETRIC MODELS AND METHODS OF HINDCAST ANALYSIS FOR HURRICANE WAVES

Figure 1.2: Characteristic Hurricane Tracks of The North Atlantic Basin

Notes:The figure shows the typical tracks followed by hurricanes. Most are formed over the Atlantic Ocean between latitudes 5 and 20o N. They then progress in a westerly or north-westerly direction. There are also frequent formations within the Caribbean Sea.


Figure 1.3: 1950 to 2000 Hurricane Tacks of The North Atlantic Basin

Notes:The figure shows the tracks of all hurricanes between 1950 and 2000. It can be seen that most occurrences have been over the Atlantic Ocean. The west to northwesterly heading across the Caribbean Sea to the Gulf of Mexico can also be observed. The tracks are hardly going beyond the continental land boundaries as here they are usually weakened to Tropical Storms.

Hurricane Tracks for The North Atlantic Basin from 1950 - 2000

10

15

20

25

30

35

-100 -95 -90 -85 -80 -75 -70 -65 -60 -55 -50Longitude

Latit

ude


Figure 1.4: The Characteristic Components of a Hurricane

WIN

D S

PE

ED

(m

/s)

1010 –1015

MAX. WIND SPEED (VMAX)RADIUS TO MAX. WINDS (RMAX)

CENTRAL PRESSURE (PC)

PRESSURE

WIND SPEED

PR

ES

SU

RE

(m

bs)

ANTI CLOCKWISE ROTATING WINDS

33 -110

880 –980

0

200 -1300 kmNotes:The figure shows the schematized wind and pressure profiles of a hurricane. The wind speed rises from almost zero at the center of the storm (called the eye) to the radius of Maximum Winds, Rmax. The maximum winds are denoted as Vmax. The central pressure, Pc, is at its lowest at the eye and increases to ambient pressure outside of the hurricane system. The atmospheric pressure deficit (difference between Pc and ambient pressure) is the generating mechanism for the winds. The winds rotate in an anti-clockwise direction in the Northern Hemisphere. This is due to the direction of the Coriolisforce in the northern hemisphere.


Flow Diagram For the Program - HURWave

Single Grid ModuleInput:

Area Coords & Cat. RangeOutput:

Hurricane Stats, Measured and Interpolated Pts;Pts. of Max Vmax, & Min Dist

Single Storm ModuleInput:

Area Coords & Cat. RangeOutput:

Measured and Interpolated Pts;Pts. of Max Vmax & Min Dist,

Monte Carlo ModuleInput:

Dist Functions & No. of SimulationsOutput:

Fit of Peak Vmax and Min Dist toa number of Dist. Functs.;

Synthetic Data Base of Storms

Wave ModuleInput:

Parametric Wind & Wave Models (Cooper 1988), Young(1995), Bretschneider (1990), Ross(1976))

& DataSet (Recorded Pts., Min Dist, Max Vmax, Interpolated Pts)

Output:Rmax, Pc, Vc, Hs, Tp, Wave Dir.

Extremal Statistical ModuleInput:

Hs Results, Series Type,Data Fitting Method , Des. Life & CI

Output:Return Values of Hs, std. Dev. Hs%,

Encounter Prob. and Distribution Plots

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Notes:The arrows indicate how the modules link to each other as output values from one module are used as input in the linking module. The Multiple Grid Module is a loop of runs for a range of locations. This can be done for either the Historical Approach or the Monte Carlo Approach.

Figure 2.2(a): Examples of HurWave’s Interfaces

i) The Single Grid ModuleHere the search parameters are defined for the program to searchthe selected database of hurricane records.. ii) The Monte Carlo Module

Here a number of data fitting methods and distribution functions are available from which the best-fitting one can be selected. Hereafter, the synthetic storm tracks are generated.

iii) The Single Storm ModuleHere one storm may be studied in detail. The output points from this module are fed to the wave module were wave calculations can be made.


Figure 2.2(b): Examples of HurWave’s Interfaces

i) The Wave ModuleHere a choice of wave models are available. A wind model may also be selected to replace the original wind model of the wind-wave models. Any or all of the data points (the recorded points, the points of occurrence of peak Vmax, the minimum distance points and the interpolated points between the recorded points) may be used for the calculations.

ii) The Extremal Statistical ModuleHere a range of data fitting methods and distribution functions are available. The selected data series may also be censored at any desired limit.

iii) Results from the Extremal Statistical ModuleHere the return values for the selected data series, method of data fitting, distribution function and censoring limit are presented. The equivalent confidence intervals and encounter probabilities are also given.


Figure 3.1: Spatial Distribution of North Atlantic Hurricane Occurrence on 2ox 2o Grid from 1950-2000

The North Atlantic Basin, The Caribbean Sea and the Gulf of MexicoNotes:There is a noticeable larger number of occurrences above the chain of Caribbean Islands. The northern islands of the Eastern Caribbean (Anguilla, Barbuda, St. Martin and Antigua seem to have the highest frequency of occurrence. The coast of the USA at the north western part of the Gulf of Mexico also has a high number of occurrences.

0 0 0 0 1 2 1 1 0 4 12 22 16 13 16 13 12 12 8 16 21 20 15

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4 6 4 5 7 7 8 9 9 8 8 7 12 15 19 16 8 11 12 11 5 6 6

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Figure 3.2: Distribution of North Atlantic Hurricane Occurrences and Intensities

a) Total Storm Occurrence The plot shows the variation of all the recorded storm occurrences in the North Atlantic Basin for 1900-2000. The number of occurrences are given as variances to the mean(8) for the entire period. A polynomial fit is also shown. The fit shows a multi decadal variation in the number of storm occurrences. The plot suggests that the recent increase in Tropical Cyclone activity may in fact be part of this multi decadal cycle, rather than a manifestation of unnatural Global Warming as some researchers speculate. The period of 1930 to 1970 had the most storms similar to the recent period since 1992.

b) Intense Storm Occurrence The plot shows the variation of occurrence of the most intense Tropical Cyclones in the North Atlantic Basin for 1900-2000. As with the above plot, there seem to be a multi decadal cycle. (The mean for the entire period is 2)


Variation in Occurrence of Tropical Storms and Hurricanes in The Atlantic Basin

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Variation in Occurrence of Intense Hurricanes (Category 3 and Greater) in The Atlantic Basin

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c) Atlantic Sea Surface Temperatures The plot shows the variance of the annual sea surface temperatures around 5-15 degrees North of the Equator. It is within this area that most hurricanes are formed. A five year moving average is also shown. The plot shows the same type of temporal distribution as the hurricane intensity and frequency. The temperatures were above average between 1930 and 1970 as they have been since 1994. Source: Science, Vol.293, No.5529, pg. 475



Monthly Distribution of Storm s

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Cat 1 Cat 2 Cat 3 Cat 4 Cat 5 All Cats Total

d) Monthly Distribution September has the highest number of occurrences

e) Category Distribution An almost even distribution among categories 1 to 4. Fewer category 5’s.


Figure 3.3: Spatial Distribution of Peak Vmax of Each Storm Track from 1950-2000

Spatial Distribution of Peak Vmax for Hurricanes of The North Atlantic Basin

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Notes:The point of occurrence of the peak Vmax from each storm track from 1950-2000 is shown. Most of these peaks are over the Atlantic Ocean which is owed to the fact that here hurricane occurrence is greatest. There is a concentration of points along the coastline of the Gulf of Mexico. This a result of storms intensifying within the open waters of the gulf and then quickly dying out as soon as they make landfall.


Figure 3.4: Correlation Between Characteristic Components of North Atlantic Hurricane Occurrences

a) Variation of Central Pressure Difference with LatitudeThere seem to be a very weak correlation between these components.

Fwd speed vs Max Vmax

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b) Variation of Vfwd with Max VmaxThere seem to be no correlation between these components.

c) Variation of Central Pressure with Max Vmax for each StormThere seem to be some kind of linear relationship.

Lattitude vs Central PressDiff for Max Vmax Search Criterium

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d) Variation of Central Pressure with VmaxThere seem to be some kind relationship which may be expressed in the form P=Po - (R)*a*Vmax^b where R is a random variable.


Figure 3.5: Derived Relationship for Central Pressure in terms of Vmax

Pc vs Vmax for All Tropical storms and Hurricanes of the North Atlantic Basin

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d)= + − −α

α a = 1014b = 0.029c = 1.626d = 200alpha is a random variable such that:For alpha = 0, R^2 = 0.886

− < <10 10α


Figure 3.6: Derived Relationship for Rmax in terms of Central Pressure

Central Pressure vs. Radius of Maximum Winds

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ax (k

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Figure 4.1: Hurricane Tracks and Data Buoy Locations for Wave Height Comparisons

Hurricane Tracks and Data Buoy Locations

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Opal'95 Erin'95Bertha'96

Felix'95

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Figure 4.2(a): Measured and Predicted 10m-Elevation Wind for Hurricane Felix’95 at Buoy 41001

Comparison of 10-m Elev. Wind Profiles for Hurricane Felix'95

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Time

Vc_Holland Vc_Hydromet Vc_Bret Mod X Vc_Fujita U10_Ross W10_Cooper Measured Data (NOAA41001)

Notes:•The predictions of the four parametric wind models are shown plus those used in the wave models of Ross and Cooper•All the model predictions exceed the measured values.•The models compare with each other very well at the peak value •The modified Rankine Vortex Models (Holland and Hydromet) have similar profiles•The model of Ross and Bret Model-X have very similar profiles


Comparison of 10-m Elev. Wind Profiles for Hurricane Bertha'96

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7/10/96 19:12 7/11/96 0:00 7/11/96 4:48 7/11/96 9:36 7/11/96 14:24 7/11/96 19:12 7/12/96 0:00 7/12/96 4:48 7/12/96 9:36 7/12/96 14:24

Time


Figure 4.2(b): Measured and Predicted 10m-Elevation Wind for Hurricane Bertha’96 at Buoy 41010

Notes:•The predictions of the four parametric wind models are shown plus those used in the wave models of Ross and Cooper•All the model predictions exceed the measured values.•The wind model of Cooper shows the best comparison with the measurements•The model of Ross and Bret Model-X have very similar profiles


Comparison of 10-m Elev. Wind Profiles for Hurricane Erin'95

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8/2/95 4:48 8/2/95 9:36 8/2/95 14:24 8/2/95 19:12 8/3/95 0:00 8/3/95 4:48 8/3/95 9:36 8/3/95 14:24 8/3/95 19:12

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Figure 4.2(c): Measured and Predicted 10m-Elevation Wind for Hurricane Erin’95 at Buoy 42036



Comparison of 10-m Elev. Wind Profiles for Hurricane Opal'95

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10/3/95 9:36 10/3/95 14:24 10/3/95 19:12 10/4/95 0:00 10/4/95 4:48 10/4/95 9:36 10/4/95 14:24 10/4/95 19:12 10/5/95 0:00

TimeVc_Holland Vc_Hydromet Vc_Bret Mod X Vc_FujitaU10_Ross W10_Cooper Measured Data (NOAA42001)

Figure 4.2(d): Measured and Predicted 10m-Elevation Wind for Hurricane Opal’95 at Buoy 42001



Figure 4.3: Chart of Measured Hsmax and Predicted Hsmax for 5 Wave Models

Comparison Between The Measured Hsmax from 4 Hurricanes and The Predicted Hsmax for 5 Wave Models

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Felix'95 Erin'95 Bertha'96 Opal'95

Hurricane

Hsm

ax (m

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Notes:•The predicted Hsmax by each of the 5 parametric wave models is presented here. •The predictions exceed the measurements with the exception of a few cases


Figure 4.4: Table of Comparisons Between Measured Hsmax and Predicted Hsmax

Notes:•The predicted Hsmax of each parametric wave model is presented here. Four other wind models wereused instead of the generic wind models for Ross and Cooper.

•The RMS error and the bias is given for each model. A positive bias means an over-prediction of the measurement.•There is a strong positive bias for all except one model. •The models of Ross and Young have the lowest RMS errors.

Hurricane Felix'95 Erin'95 Bertha'96 Opal'95Buoy ID 41001 42036 41010 42001

Mesured Max Hs 7.6 4.6 6.26 8.3Ross 7.68 4.42 3.93 9.12 1.24 -0.40Ross_Holland 9.59 8 9.05 14.92 4.10 3.70Ross_Hydromet 11.51 8.44 9.24 15.18 4.64 4.40Ross_Bret Model X 9.35 7.92 7.61 14.71 3.77 3.21Ross_Fujita 9.89 8.02 9.45 14.84 4.18 3.86Cooper 12.07 6.58 4.88 9.71 2.64 1.62Cooper_Holland 9 7.71 7.71 12.95 2.97 2.65Coop_Hydromet 10.21 7.82 7.81 13.02 3.24 3.03Coop_Bret Model X 9 7.71 6.9 12.94 2.90 2.45Cooper_Fujita 9 7.71 7.91 12.95 3.00 2.70Young 9.5 6.13 5.75 11.9 2.19 1.63Improved Young 9.49 6.29 5.56 10.69 1.78 1.32Bretschneider 9.82 8.25 6.84 12.24 2.92 2.60

RMS Error Bias


Figure 4.5: NOAA Aircraft Ground Tracks and Normalized Axes for Comparisons

Notes:The solid black lines show the ground tracksof the NOAA aircraft during the 5-hour period of measurements

The circles show the track positions recordedby the NOAA aircraft. Those filled are the positions during which measurements weremade.

The dotted lines show the axes used used to mark the 13 points used for comparisonof parametric predictions with SRA measurements. These axes are assumedto match those of Figure 4.6 where the SRAspatial measurements are given relative to the direction of forward motion of the hurricane.

EAST

NO

RTH


Figure 4.6: Spatial Variation in Hs from SRA Measurements for Hurricane Bonnie’98

Notes:Shown here is the spatial variation in Hsmeasured over the 5-hour recording period.Contours for integer values of wave heights are solid and contours for integer values plus0.5m are dashed.

It is assumed that the values represent the average Hs over the 5-hour period of record andthat the direction north of the eye is taken as thedirection of forward motion of the hurricane.


Figure 4.7: Comparison of Parametric Model Predictions with SRA Data for Hurricane Bonnie

Notes:• For the Ross and Cooper models, four other wind models are also used in place of the original wind model:Holland, Hydromet Rankine Vortex, Bret Model - X and Fujita

• A positive bias means that the measurement has been over predicted• The wave height values in bold italics have the least error for each point.•The models of Young gave the best performance.•The model of Cooper performed best with the Hydromet model as its wind predictor.•The models over predict the measurements in most cases.

Point Location Relative to Average Position of Hurricane Center (km)150 W 100 W 50 W 50 E 100 E 150 E 0 150 N 100 N 50 N 50 S 100 S 150 S

SRA Data 7.1 7.4 9.8 9.8 9.8 9.5 6.9 9.5 10.4 9.5 5.8 6.4 5.8 RMS Error BiasRoss 2.7 4.7 8.1 8.1 8.1 3.8 4.1 3.6 5.2 6.4 5.5 5.5 3.7 3.47 -3.1Ross_Holland 7.2 10.5 15.4 15.4 15.4 7.4 5.9 8.0 11.0 12.6 10.1 11.6 8.3 3.96 2.8Ross_Hydromet 9.7 11.1 12.1 12.1 12.1 9.8 6.7 10.0 11.2 10.8 9.0 11.4 10.2 2.74 2.4Ross_Bret Model-X 6.5 10.4 15.6 15.6 15.6 6.9 8.6 7.5 11.0 13.4 11.6 11.6 7.8 4.27 3.1Ross_Fujita 11.2 13.6 16.0 16.0 16.0 11.4 9.1 11.8 13.9 14.2 12.1 14.2 12.1 5.52 5.4Cooper 3.8 5.6 8.8 8.8 8.8 6.8 12.2 5.4 6.3 8.8 10.3 6.4 5.4 2.87 -0.9Cooper_Holland 6.3 8.6 12.2 12.2 12.2 6.5 6.1 6.9 9.0 10.7 8.8 9.4 7.1 2.23 0.8Cooper_Hydromet 7.7 9.0 10.4 10.4 10.4 7.8 7.9 8.0 9.1 9.9 9.0 9.3 8.2 1.67 0.8Cooper_BretModel-X 6.0 8.6 12.3 12.3 12.3 6.2 9.3 6.6 9.0 11.4 10.6 9.4 6.9 2.65 1.2Cooper_Fujita 8.5 10.2 12.5 12.5 12.5 8.6 9.8 8.9 10.5 11.8 11.0 10.7 9.1 2.91 2.4Young 7.4 9.6 9.2 13.9 12.7 11.3 9.9 8.0 10.0 11.9 11.1 7.8 7.1 2.66 1.9Improved Young 6.6 8.6 8.2 12.4 11.3 10.1 8.9 7.1 8.9 10.6 9.9 7.0 6.3 1.89 0.7Bretschneider 6.9 9.7 10.3 10.3 10.3 7.4 5.8 8.9 10.2 9.1 7.3 7.5 6.3 1.13 0.2RMS Error for all models 2.13 2.85 3.23 3.46 3.32 2.59 2.39 2.63 2.23 2.42 4.27 3.82 2.70RMS Error for best 3 models 0.46 1.74 1.05 1.57 1.00 1.61 1.43 1.67 1.13 0.71 3.11 1.82 1.42

Right-Front Quadrant of Hurricane


Figure 4.8: Detailed Track Positions and 6-hourly Track Records given in Historical Database

Notes:The circles show the track positions recordedby the NOAA aircraft. Those filled are the positions during which measurements weremade. The dotted line shows the 6-hourly track positions given by the NOAA historical database


Area 1

Area 3

Area 5

Area 2

Area 4

Figure 5.1: Areas for the Comparison of Distribution of Peak Vmax


Figure 5.2(a): Statistical Distribution of Peak Vmax for Area 1

i) Exponential Distribution of Peak Vmaxfor 1900-1949; Number of storms = 41

ii) Exponential Distribution of Peak Vmaxfor 1950-2000; Number of storms = 43

iii) Exponential Distribution of Peak Vmaxfor 1900-2000; Number of storms = 84

Period 1900-1949 1950-2000 1900-200025-year peak Vmax (m/s) 61 80 7550-year peak Vmax (m/s) 66 90 80

iv) 25 and 50 year Return Values of Peak Vmax


Figure 5.2(b): Statistical Distribution of Peak Vmax for Area 2







Figure 5.2(c): Statistical Distribution of Peak Vmax for Area 3







Figure 5.2(d): Statistical Distribution of Peak Vmax for Area 4







Figure 5.2(e): Statistical Distribution of Peak Vmax for Area 5







Figure 5.3: Best-Fitting Statistical Distribution of Peak Vmax Four Storm Populations

Notes:The plot shows the distributions for the peak Vmax for four storm populations. The first population includesall storms with the area considered (Area 5). The second includes only hurricanes and the third and fourthincludes hurricanes from categories 2 to 5 and categories 3 to 5 respectively. The 50 year return periods are given which vary by only 1 m/s for all populations. This illustrates the point that these storms, regardless of intensitymay be treated as belonging to the same statistical population

Variation in Distribution of peak Vmax for Different Storm Populations

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Trop Storms and Hurricanes

Hurricanes Cats. 1-5

Hurricanes Cats.2-5

Hurricanes > Cats. 3-5

50-yr pVmaxTS + Hur = 77m/sCats 1 -5 = 77m/sCats 2-5 = 78 m/sCats 3-5 = 76m/s


Figure 5.4: Variation in Predicted Hs with Vmax for Four Wave Models

Notes:The plots show the variation of Hs with Vmax. A range of storms have been used over a large area hence the plotsare representative of the models’ performance for a wide range of storms and storm conditions. The same area and hencethe same storms, are used for each model. Tropical Storms are included. The plots demonstrate the difference in themodel predictions and the dependence of each model on Vmax. The model of Young which has Vmax and Vfd as its two mainparameters does show the highest dependence on Vmax. The main point here is that the tropical storms do produce significantlyhigh waves.

Predicted Hs by Breschneider vs Vmax

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Predicted Hs by Cooper vs Vmax

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Predicted Hs by Improved Young vs Vmax

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Figure 5.5: Variation in Predicted Hs with Distance from Storm Center, r, for Four Wave Models

Notes:The plots show the variation of Hs with radius from storm center, r. A range of storms have been used over a large area hence the plotsare representative of the models’ performance for a wide range of storms and storm conditions. The same area and hencethe same storms, are used for each model. The plots demonstrate the difference in the model predictions and the dependence of each model on r. The model of Ross shows a strong dependence on r which is evident if one examines the formulations used for the parametric model.The model of Young has a limit of 250km beyond which predictions are not available. The model of Bretschneider seems to give unrealistic values at large distances.

Predicted Hs by Ross vs Distance from Storm Center

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Predicted Hs by Cooper vs Distance from Storm Center

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Predicted Hs by Improved Young vs Distance from Storm Center

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Predicted Hs by Bretschneider vs Distance from Storm Center

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Figure 5.6(a): Occurrence of Hsmax (Point of Minimum Distance vs Point of Peak Vmax) – Area 1

Hurricane Tracks and Point of Occurrence of Peak Vmax for Area 1 - Eastern Caribbean

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ii) Comparison of predicted Hs between point of minimum distance and point of peak Vmax for Area 1

Notes:The first figure shows that the points of occurrence of peak Vmax in the eastern Caribbean are randomly distributed. This is because the small islands have little or no effect on thehurricanes intensity and so the point at which a hurricane reaches its maximum wind speedis not constrained in any way. The second figure shows that there is no significant biasto either of the points. This is a consequence of the random distribution of the peak Vmax.

i) Distribution of Hurricane Tracks and Occurrence of Peak Vmax for Area 1

Comparison of Predicted Hs for Point of Minimum Distance and Point of Peak Vmax for Area 1

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Figure 5.6(b): Occurrence of Hsmax (Point of Minimum Distance vs Point of Peak Vmax) – Area 2


Hurricane Tracks and Point of Occurrence of Peak Vm ax for Area 2 - Western Caribbean

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i) Distribution of Hurricane Tracks and Occurrence of Peak Vmax for Area 2


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Notes:The first figure shows that the points of occurrence of peak Vmax in the western Caribbean are not as randomly distributed as in the eastern Caribbean. Here they tend to lie on the boundaries of the area considered as hurricanes increase or decrease in intensity. The second figure shows that there is a noticeable bias towards the minimum distance point producing higher wave heights. This is a consequence of the tendency of the points of peak Vmax to be on the boundaries of the area.


Figure 5.6(c): Occurrence of Hsmax (Point of Minimum Distance vs Point of Peak Vmax) – Area 3


Notes:The first figure shows that the points of occurrence of peak Vmax are concentrated around the land boundaries. This is because the storms tend to intensify over open water until they near the large continental land mass. Hence they reach the peak Vmax just before land fall. The second figure shows that there is a significant bias bias towards the minimum distance point producing higher wave heights. This is a consequence of the tendency of the points of peak Vmax to be closer to the land boundaries.

Hurricane Tracks and Point of Occurrence of Peak Vmax for Area 3 - Gulf of Mexico

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i): Distribution of Hurricane Tracks and Occurrence of Peak Vmax for Area 3


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Figure 5.6(d): Occurrence of Hsmax (Point of Minimum Distance vs Point of Peak Vmax) – Area 4


Notes:The first figure shows that the points of occurrence of peak Vmax are randomly distributed. This is because this area is basically an open-ocean area. It is true that the US landmass will stare the storms in a north to northeasterly direction but with the high number of storm occurrences and the wide variation in storm intensity, the point of occurrence of peak Vmax is not constrained to any particular area. The second figure shows that neither point is more likely to producer higher waves than the other.

i): Distribution of Hurricane Tracks and Occurrence of Peak Vmax for Area 4

Hurricane Tracks and Point of Occurrence of Peak Vmax for Area 4 - The USA Easdt Coast

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Figure 6.1: Tabulation of Best-Fit Distributions for Bretschneider and Cooper Models

Peak Value Series Annnual Maximum SeriesUncensored Hs > 3 Hs > 5 Uncensored Hs > 3 Hs > 5

Meth of LS and meth of Mom

Meth of Hurdle LS

Meth. of LS

Meth. of

Mom

Meth of Hur.

LSMeth. of LS

Meth. of

Mom

Meth of Hur.

LSMeth. of LS

Meth. of

Mom

Meth of Hur.

LSMeth. of LS

Meth. of

Mom

Meth of Hur.

LSMeth. of LS

Meth. of

Mom

Meth of Hur.

LSMeth. of LS

Meth. of

MomMeth of Hur. LS

FT - I 0<k<0.75 4 2 5 4 1 3 2 0 10 6 5 1 2 1 1 1 0 0k=0.75 0.75<k<1.0 1 1 3 1 1 3 1 67 5 1 0 0 1 0 0 0 47 3

k=1 1.0<k<1.4 1 1 11 1 13 9 2 27 11 1 2 0 1 5 1 0 30 2k=1.4 1.4<k<2 4 5 50 4 35 31 3 4 18 3 3 11 3 24 17 0 16 15k=2 k>2 89 90 28 89 48 52 91 1 55 88 89 87 92 68 80 98 6 79

0.964 0.961 0.980 0.955 0.873 0.975 0.938 0.598 0.963 0.964 0.961 0.977 0.954 0.892 0.977 0.937 0.660 0.959Mean r2

Peak Value Series Annnual Maximum SeriesUncensored Hs > 3 Hs > 5 Uncensored Hs > 3 Hs > 5

Meth of LS and meth of Mom

Meth of Hurdle LS

Meth. of LS

Meth. of

Mom

Meth of

Hur. LS

Meth. of LS

Meth. of

Mom

Meth of

Hur. LS

Meth. of LS

Meth. of

Mom

Meth of

Hur. LS

Meth. of LS

Meth. of

Mom

Meth of

Hur. LS

Meth. of LS

Meth. of

Mom

Meth of

Hur. LS

Meth. of LS

Meth. of

Mom

Meth of Hur.

LSFT - I 0<k<0.75 36 30 67 36 9 67 18 7 58 44 35 11 36 20 14 13 13 22

k=0.75 0.75<k<1.0 16 9 2 9 8 5 32 8 10 10 3 11 10 2 4 10 2 3k=1 1.0<k<1.4 19 22 1 17 25 3 21 26 4 13 14 13 11 13 16 6 14 11

k=1.4 1.4<k<2 16 20 15 12 39 12 12 40 4 16 21 26 8 39 29 9 47 24k=2 k>2 12 18 14 25 18 10 16 18 15 16 26 38 34 25 36 61 23 35

0.970 0.971 0.905 0.966 0.657 0.910 0.951 0.007 0.507 0.971 0.972 0.931 0.964 0.770 0.933 0.928 0.066 0.524Mean r2

Notes:The table shows the number of times each distribution function was the best-fitting distribution for each combination of data series, data fitting method and threshold value. Each column sums to 100, the total number of sites. The mean correlation is also given for each method of data fitting (in the bottom row). 1) The Bretschneider model (above) shows a preference for Weibull distributions with unusually high shape factors. 2) The Cooper Model (below) exhibits a preference for the FT-I distribution. 3) The method of Hurdle is very useful in providing the precise best-fit Weibul shape factor. 4) The shape of the distribution changes significantly at a threshold value of 5m.

i) Best-Fitting Distributions for Bretschneider Wave Model for 100 sites

ii) Best-Fitting Distributions for Cooper Wave Model for 100 sites


Figure 6.2: Typical Shape of Distributions and Return Values For Bret. And Cooper Models

Typical Distributions and Return Values of Hs for the Bretschneider and Cooper Models

0.1

1

10

1000 2 4 6 8 10 12 14 16 18

Hs (m)

Exc

eden

ce P

roba

bilit

y (%

)

0

10

20

30

40

50

60

70

80

90

100

Ret

urn

Per

iod

(yea

rs)

Bretsch Hs Dist

Cooper Hs Dist

Bretsch Return Hs

Coop Return Hs

Notes:Shown here are the typical distributions of Hs for the two models. There is a noticeable difference between the shape of the 2 distributions. The Bretschneider generally fits to Weibull distributions with high shape factors (> 2) where as the Cooper fits to a range of shape factors, usually less than 1.4. The return values are also shown. Bretschneider usually predicts slightly higher values, most apparent for the low return periods.


Figure 6.3: Performance of the Methods of Data-Fitting

Mean Correlation for Data Series, Censoring Limits and Methods of Data Fitting from Cooper Predicted Hs

0.00

0.10

0.200.30

0.40

0.50

0.60

0.70

0.800.90

1.00

PV - Hs >0 PV - Hs >3 PV - Hs >5 AM - Hs >0 AM - Hs >3 AM - Hs >5Data Series - Censoring Limit

Mea

n C

orre

latio

n fo

r 99

site

s

Meth of Least Squares Meth of Moments Hurdle-W eibull - Meth of LS

M e a n C o rre la tio n fo r D a ta S e r ie s , C e n s o r in g L im its a n d M e th o d s o f D a ta F ittin g fro m B re ts c h n e id e r P re d ic te d Hs

0 .55

0 .60

0 .65

0 .70

0 .75

0 .80

0 .85

0 .90

0 .95

1 .00

PV - Hs>0 PV - Hs>3 PV - Hs>5 AM - Hs>0 AM - Hs>3 AM - Hs>5D ata S eries - C en so rin g L im it

Mea

n C

orre

latio

n fo

r 100

si

tes

Meth of Leas t S quares Meth of Mom ents Hurdle-W eibull - Meth of LS

i) Mean Correlation For Bretschneider Wave Model

ii) i) Mean Correlation For Cooper Wave Model

Notes:The mean correlation is calculated from the correlations of the best-fit functions for each method of data fitting. A set of three bars is given for each combination of data series and threshold value. 1)The method of Hurdle consistently provides the highest correlation for Bretshneiderbut not for Cooper. 2) The method of moments does not work for censored data. 3) The mean correlation consistently decreases as the censoring parameter increases. 4) The Method of Least Squares provides the most consistent performance.


Figure 6.4(a): 5-Year Return Hs for Bret. and Coop Models using Peak Value Series

Variation in 5-Year Return Wave Height for Peak Value Series Using the Best Fitting Distribution from the Method of Least Squares

6.0

7.0

8.0

9.0

10.0

11.0

12.0

50 55 60 65 70 75 80 85 90 95 100

Site Number

5 ye

ar R

etur

n W

ave

Hei

ght

Uncensored Hs > 3m Hs > 5m

Mean RMS ErrorHs >0 : 0.15mHs >3 : 0.13mHs >5 : 0.20m


0

2

4

6

8

10

12

50 55 60 65 70 75 80 85 90 95 100

Site Number

5 ye

ar R

etur

n W

ave

Hei

ght



i) 5-Year Return Periods for Peak Value Series- Bretschneider

i) 5-Year Return Periods for Peak Value Series- Cooper

Notes:The figures show the 5-year return values for 50 sites using the peak value series. The return values are given for 3 threshold values (0, 3 and 5m). The RMS difference is calculated relative to the mean for the three values.

1) The differences in the return values are clear but not significant. 2) The return value decreases with increasing censoring limit. 3)The 5m-threshold values show a slightly greater RMS difference than either of the other two thresholds. 4) The RMS Errors for theBretschneider model are less than the RMS Errors for Cooper.


Figure 6.4(b): 5-Year Return Hs for Bret. and Coop Models using Annual Maximum Series

Variation in 5-Year Return Wave Height for the Annual Maximum Series Using the Best Fitting Distribution from the Method of Least Squares

0.0

2.0

4.0

6.0

8.0

10.0

12.0

50 55 60 65 70 75 80 85 90 95 100

Site Number

5 ye

ar R

etur

n W

ave

Hei

ght



Variation in 5-Year Return Wave Height for the Annual Maximum Series Using the Best Fitting Distribution from the Method of Least Squares

0

2

4

6

8

10

12

50 55 60 65 70 75 80 85 90 95 100Site Number

5 ye

ar R

etur

n W

ave

Hei

ght



i) 5-Year Return Periods for Annual Maximum Series -Bretschneider

i) 5-Year Return Periods for Annual Maximum Series -Cooper


1) Unlike the case for the peak value series, there are significant differences between the return values (larger RMS Errors). 2) The return value decreases with increasing censoring limit. 3) Whereas there are not significant differences in the return values for the uncensored and 3m-censored data sets, there is for the 5m-censored data.


Figure 6.4(c): 50-Year Return Hs for Bret. and Coop Models using Peak Value Series


0

2

4

6

8

10

12

14

16

18

50 55 60 65 70 75 80 85 90 95 100

Site Number

50 y

ear R

etur

n W

ave

Hei

ght



Variation in 50-Year Return Wave Height for Peak Value Series Using only the Best Fitting Distribution from the Method of Least Squares

6.0

8.0

10.0

12.0

14.0

16.0

18.0

50 55 60 65 70 75 80 85 90 95 100

Site Number

50 y

ear R

etur

n W

ave

Hei

ght



i) 50-Year Return Periods for Peak Value Series -Bretschneider

i) 50-Year Return Periods for Peak Value Series -Cooper


1) There are significant differences between the return values (greater RMS Errors than observed with the equivalent 5-year return values) . 2)The return value decreases with increasing censoring limit for the Bretschneider models but this is not apparent for the Cooper Model.


Figure 6.4(d): 50-Year Return Hs for Bret. and Coop Models using Annual Maximum Series


6.0

8.0

10.0

12.0

14.0

16.0

18.0

50 55 60 65 70 75 80 85 90 95 100

Site Number

50 y

ear R

etur

n W

ave

Hei

ght




0

2

4

6

8

10

12

14

16

18

50 55 60 65 70 75 80 85 90 95 100

Site Number

50 y

ear R

etur

n W

ave

Hei

ght



i) 50-Year Return Periods for Annual Maximum Series -Bretschneider

i) 50-Year Return Periods for Annual Maximum Series -Cooper


1)The differences are greater than the 50-year Peak Value Series. 2) The return value decreases with increasing censoring limit. 3) Whereas there are not significant differences in the return values for the uncensored and 3m-censored data sets, there is for the 5m-censored data. However, this difference is not as significant as they are for 5-year return values.


Figure 6.5(a): Comparison between Peak Value and Annual Maximum Series for Hs > 0m

Comparison of 5-Yr Return Values for Best Fitting Distribution of the Method of Least Squares for the Uncensored Peak Value and


6.0

7.0

8.0

9.0

10.0

11.0

12.0

6.0 7.0 8.0 9.0 10.0 11.0 12.0

5 - Year Hs for PEAK VALUE SERIES

5-Ye

ar H

s fo

r AN

NU

AL

MA

XIM

UM

SER

IES

Hspeak = Hsannmax

Comparison of 50-Yr Return Values for Best Fitting Distribution of the Method of Least Squares for the Uncensored Peak Value and


10.0

11.0

12.0

13.0

14.0

15.0

16.0

17.0

18.0

10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0


50-Y

ear H

s fo

r AN

NU

AL

MA

XIM

UM

SER

IES

Hspeak = Hsannmax

i) 5-Year Hs for Uncensored Annual Maximum Series vs 5-Year Hs for Uncensored Peak Value Series -Bretschneider

Notes:The figures show the 5-year return values for the annual maximum series plotted against the equivalent 5-year return values for the peak value series. The line of direct proportionality (Hs_annual max = Hs_peak val, is also shown. The values of Hs have been predicted with the Model of Bretschneider.

1) The peak value series show higher values of Hs at all sites for the 5-year return periods. 2) In contrast, there is small and an almost even spread of the data about the line of direct proportionality for the 50-year return periods.

ii) 50-Year Hs for Uncensored Annual Maximum Series vs 50-Year Hs for Uncensored Peak Value Series - Cooper


Figure 6.5(b): Comparison between Peak Value and Annual Maximum Series for Hs > 3m

Comparison of 5-Yr Return Values for Best Fitting Distribution of the Method of Least Squares for the Censored (Hs>3m) Peak

Value and Annual Maximum Series

4.05.06.0

7.08.09.0

10.0

11.012.0

4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0


5-Ye

ar H

s fo

r AN

NU

AL

MA

XIM

UM

SER

IES

Hspeak = Hsannmax

Comparison of 50-Yr Return Values for Best Fitting Distribution of the Method of Least Squares for Censored (Hs>3m) Peak Value s

and Annual Maximum Series

10.0

11.0

12.0

13.0

14.0

15.0

16.0

17.0

18.0

10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0


50-Y

ear H

s fo

r AN

NU

AL

MA

XIM

UM

SER

IES

Hspeak = Hsannmax

i) 5-Year Hs for Censored(Hs>3m)Annual Maximum Series vs 50-Year Hs for Censored(Hs>3m) Peak Value Series - Bretschneider

Notes:The figures show the 5-year return values for the annual maximum series plotted against the equivalent 5-year return values for the peak value series. The line of direct proportionality (Hs_annual max = Hs_pesk val, is also shown. The values of Hs have been predicted with the Model of Bretschneider.

1) The peak value series show higher values of Hs at all sites for the 5-year return period. 2) In contrast, there is small spread of the data about the line of direct proportionality. However, now there is a more noticeable bias for the peak value series as the cloud of data is shifted to the right.

ii) 50-Year Hs for Censored(Hs>3m)Annual Maximum Series vs 50-Year Hs for Censored(Hs>3m) Peak Value Series -Cooper


Figure 6.5(c): Comparison between Peak Value and Annual Maximum Series for Hs > 5m

Comparison of 5-Yr Return Values for Best Fitting Distribution of the Method of Least Squares for the Censored (Hs>5m) Peak Value


4.0

5.0

6.0

7.08.0

9.0

10.0

11.0

12.0

4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0


5-Ye

ar H

s fo

r AN

NU

AL

MA

XIM

UM

SER

IES

Hspeak = Hsannmax

Comparison of 50-Yr Return Values for Best Fitting Distribution of the Method of Least Squares for Censored (Hs>5m) Peak Value s


10.0

11.0

12.0

13.0

14.0

15.0

16.0

17.0

18.0

10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0


50-Y

ear H

s fo

r AN

NU

AL

MA

XIM

UM

SER

IES

Hspeak = Hsannmax

i) 5-Year Hs for Censored(Hs>5m)Annual Maximum Series vs 5-Year Hs for Censored(Hs>5m) Peak Value Series - Bretschneider

Notes:The figures show the 5-year return values for the annual maximum series plotted against the equivalent 5-year return values for the peak value series. The line of direct proportionality (Hs_annual max = Hs_pesk val, is also shown. The values of Hs have been predicted with the Model of Bretschneider.

1) The peak value series show higher values of Hs at all sites for the 5-year return period. 2) The spread in the data about the line of direct proportionality is greater than for the previous two threshold values and again there is a more noticeable bias for the peak value series as the cloud of data is shifted further to the right.

ii) 50-Year Hs for Censored(Hs>5m)Annual Maximum Series vs 50-Year Hs for Censored(Hs>5m) Peak Value Series -Cooper


Figure 7.1: Variation in Distributions and Return Hs for different Synthetic Databases (29oN, 65oW )

f * Vmaxi

Vmaxi

Peak Vmax

f * Vmaxi

Vmaxi

Randomly drawnpeak Vmax

maxVmax

Randomly drawnpeak Vmax

Real storm track

Synthetic storm track

f = rand(maxVmax)/maxVmax Randomly drawn minimum distance

Real minimum distance

Real minimum distanceRandomly drawn minimum distance

Notes:A track is selected at random from the parent population of tracks. The peak Vmax of this track is scaled to the value of the randomly drawn peak Vmax. The other values of Vmax along this storm track are then scaled by the same scaling factor. The entire storm track is then moved along a straight line towards or away from the point of interest. This is done by moving the minimum distance point of the real storm track to the minimum distance selected at random.


Figure 7.2(a): Variation in Distributions and Return Hs for different Synthetic Databases(16oN, 60oW )

Distributions for 5 Synthetic Databases and the Real Database for Bretschneider Model

0.1

1

10

1005 7 9 11 13 15

Hs (m)

Prob

abilt

y of

Exc

eden

ce (%

)

Real Database 50

100 200

500 1000

Number of Storms in Synthetic Database


8

9

10

11

12

13

14

15

16

24 Average 50 100 200 500 1000

Number of Storms

Hs

(m)

5-year50-year

real database

i) Extremal distributions for Hs predicted by Bretschneider model for the real and synthetic databases for a location with coordinates 16oN, 60oW .

ii) 5 & 50-Year Return Periods predicted by Bretschneider for the Real and Synthetic Databases for a location with coordinates 16oN, 60oW . Notes:The average of the synthetic return periods is shown. The 50 year average is 0.01m less than the the return period for the real population

Notes:The distributions vary only slightly in shape.


Figure 7.2(b): Variation in Distributions and Return Hs for different Synthetic Databases (16oN, 60oW )

i) Extremal distributions for Hs predicted by Cooper model for the real and synthetic databases for a location with coordinates 16oN, 60oW .

ii) 5 & 50-Year Return Periods prediced by Cooper model for the Real and Synthetic Databases for a location with coordinates 16oN, 60oW . Notes:The average of the synthetic return periods is shown. The 50 year average is 1.72m more than the return period for the real population

Notes:The distributions vary only slightly for the synthetic bases with 100 or more storms .

Distributions for Hs for 5 Synthetic Databases and the Real Database for the Cooper Model

0.1

1

10

1005 7 9 11 13 15 17

Hs (m)

Prob

abilt

y of

Exc

eden

ce (%

)

Real Database 50

100 200

500 1000


Variation in 5 & 50-Year Wave Height for 5 Synthetic Databases using the Results of the Cooper Model

8

10

12

14

16

18

20

24 Average 50 100 200 500 1000

Number of Storms

Hs

(m)

5-year50-year


Figure 7.3(a): Variation in Distributions and Return Hs for different Synthetic Databases (29oN, 65oW )

i) Extremal distributions for Hs predicted by Bretschneider model for the real and synthetic databases for a location with coordinates 29oN, 65oW .

ii) 5 & 50-Year Return Periods prediced by Bretschneider model for the Real and Synthetic Databases for a location with coordinates 29oN, 65oW . Notes:The average of the synthetic return periods is shown. The 50 year average is 0.34m less than the return period for the real population

Notes:The shape of the distributions vary only slightly.

Distributions for Hs for 5 Synthetic Databases and the Real Database for Bretschneider Model

0.1

1

10

1005 7 9 11 13 15

Hs (m)

Prob

abilt

y of

Exc

eden

ce (%

)

Real Database 50

100 200

500 1000



8

9

10

11

12

13

14

15

23 Average 50 100 200 500 1000

Number of Storms

Hs

(m)

5-year50-year

real database


Figure 7.3(b): Variation in Distributions and Return Hs for different Synthetic Databases (29oN, 65oW )

i) Extremal distributions for Hs predicted by Cooper model for the real and synthetic databases for a location with coordinates 29oN, 65oW .

ii) 5 & 50-Year Return Periods prediced by Cooper model for the Real and Synthetic Databases for a location with coordinates 29oN, 65oW . Notes:The average of the synthetic return periods is shown. The 50 year average is 0.2m are than the return period for the real population

Notes:The shape of the distributions vary only slightly for synthetic databases with more than 100 storms

Distributions for Hs for 5 Synthetic Databases and the Real Database for the Cooper Model

0.01

0.1

1

10

1005 7 9 11 13 15

Hs (m)

Prob

abilt

y of

Exc

eden

ce (%

)

Real Database 50

100 200

500 1000


Variation in 5 & 50-Year Wave Height for 5 Synthetic Databases using the Results of the Cooper Model

8

9

10

11

12

13

14

15

23 Average 50 100 200 500 1000

Number of Storms

Hs

(m)

5-year

50-year

real database


Documents

Master of Science Thesis Report by JAMEL D. BANTONParametric Models and Methods of Hindcast Analysis for Hurricane Waves by Jamel D. Banton May, 2002 IHE / ALKYON M.Sc. Thesis Report