A Study of Tropical Cyclone Track Sinuosity in the Southwest Pacific

CLIMATIC VARIABILITY: A STUDY OF TROPICAL CYCLONE TRACK SINUOSITY IN THE SOUTHWEST PACIFIC

Arti Pratap Chand

A supervised research project submitted in partial fulfillment of the requirements for the degree of Masters of Science (M.Sc.) in Environmental Sciences

School of Geography, Earth Science and Environment Faculty of Science and Technology and Environment The University of the South Pacific

October, 2012

DECLARATION

Statement by Author

I, Arti Pratap Chand, declare that this thesis is my own work and that, to the best of my knowledge, it contains no material previously published, or substantially overlapping with material submitted for the award of any other degree at any institution, except where due acknowledgement is made in the text.

Signature……………………………………… Date…18th October 2012… Arti Pratap Chand Student ID No.: S99007704

Statement by Supervisors The research in this thesis was performed under our supervision and to our knowledge is the sole work of Ms Arti Pratap Chand

Signature……………………………………… Date…18th October 2012….. Principal Supervisor: Dr M G M Khan Designation: Associate Professor in Statistics, University of the South Pacific

Signature…… ……….Date……18th October 2012…… Co - supervisor: Dr James P. Terry Designation: Associate Professor in Geography, National University of Singapore

Signature……………………………………… Date…18th October 2012…… Co - supervisor: Dr Gennady Gienko Designation: Associate Professor in Geomatics, University of Alaska Anchorage

DEDICATION

To all tropical cyclone victims.

i ACKNOWLEDGEMENTS

I am heartily thankful to my co – supervisor, Dr Gennady Gienko, whose trust, encouragement and initial discussions lead me to this topic. I would like to gratefully acknowledge my Principal Supervisor, Dr MGM Khan and my co – supervisor Dr James Terry for their advice, guidance and support from the initial to the final level enabling me to develop an understanding of the subject and statistical techniques.

I would also like to acknowledge and thank Dr Gennady and Dr Shingo Takeda for helping me with displaying my results using ArcGIS software. My sincere thanks to them for their time and patience.

My sincere thanks and appreciation goes to Dr MGM Khan and his student for helping me with C++ programming technique.

I would also like to thank Dr Tony Weir and Mr Rajendra Prasad (former Director of the Fiji Meteorological Services) for discussions I had with them regarding my thesis topic.

Special thanks to my family for their encouragement and moral support.

ABSTRACT

Tropical cyclones (TCs) are one of the most destructive natural hazards in the tropical Pacific, with large impacts on socio-economic and environmental sectors of island nations. Improved understanding of the characteristics of these intense storms is critical. A continuing problem lies in forecasting TC movement after formation. One way to add to existing knowledge in this area is to analyse available data on cyclone track shape, in order to identify any special patterns. In this context, this study examines statistical characteristics of several TC track parameters, using archived data from 1970 to 2008 for the South Pacific region. The dataset includes information on 292 TCs, which includes all storms with wind intensity of 35 knots and above that have their genesis in tropical waters.

TC paths are analysed within the geographical grid covered by 0 – 25°S and 160° E – 120° W. The particular focus of this study is on track sinuosity values and how these may be characterised and grouped. River sinuosity has contributed a lot in understanding fluvial geomorphology (Terry and Feng, 2010) and therefore extending the technique to study TC track maybe useful. A sinuous track having loops and curves will affect many more islands than a TC moving along a straight path. Some Islands may be affected more than once or may be exposed to a TC for a longer time period if the TC makes a loop during its journey. Sinuosity values for all TC tracks were calculated by measuring the total distance travelled by each TC and then dividing this by the vector displacement between cyclogenesis and decay positions.

In this study, the problem of categorising the TCs based on sinuosity index (SI) values obtained by transformation of sinuosity values allows the grouping of similar TCs. The SI categories are so constructed that the variance of groups is as small as possible. Thus in this thesis a technique is developed to construct the SI categories of the TCs that seek minimization of the sum of weighted deviations of SI from the mean of group. Then the problem is solved for determining the optimum boundary points of the groups by using a dynamic programming technique.

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Three TCs from the dataset were found to have very high SI values and therefore were grouped in a separate SI category as an outlier category. Then the remaining TCs were grouped into five homogeneous sinuosity index categories using proposed method within which the TCs were very similar.

The results from above method were compared with the SI categories obtained by hierarchical cluster analysis with Ward’s method. The comparison results show that the SI categories constructed by the proposed method are more homogenous with respect to the sinuosity index values of the TC tracks.

The homogenous SI categories obtained was further explored using GIS tool to study the geographical distribution of these SI categories in the study area.

Keywords: Track Sinuosity, Cyclogenesis and decay positions, Homogeneous Categories

ABBREVIATIONS

IPCC Intergovernmental Panel on Climate Change SI Sinuosity Index TC Tropical Cyclones

TABLE OF CONTENTS DEDICATION i ACKNOWLEDGMENT ii ABSTRACT iii ABBREVIATIONS v TABLE OF CONTENTS vi LIST OF FIGURES ix LIST OF TABLES x LIST OF APPENDICES xi

CHAPTER 1: INTRODUCTION 1 1.1. Tropical cyclones in the Pacific Region 1 1.2. Tropical cyclone variability 3 1.3. Tropical cyclone classification 4 1.4. Tropical cyclone tracks 6 1.5. Sinuosity of cyclone tracks 6 1.6. Research objectives 12 1.7. Chapter organizations 12

CHAPTER 2: LITERATURE REVIEW 14

CHAPTER 3: DATA AND METHODS 20 3.1 Study area and data collection 20 3.2 Sinuosity calculation 22 3.3 Distribution of Sinuosity values 22

3.3.1 Analysis of extreme Tropical Cyclones from sinuosity data 24 3.3.2 Sinuosity index 26 3.3.3 Analysis of extreme Tropical Cyclones from sinuosity index data 26 3.4 Correlation of sinuosity index with other parameters 28 3.5 Methodology for grouping the sinuosity index: a proposed technique 29

3.5.1 Estimate of the distribution of sinuosity index values 31 3.5.2 Estimate of the parameters of distribution 32 3.5.3 Determination of optimum grouping using dynamic programming technique 32 3.6 Alternative methodology for grouping the sinuosity index using Hierarchical Cluster Analysis 34 3.7 A comparison study of grouping methods 35

CHAPTER 4: RESULTS AND INTERPRETATIONS 36 4.1 Tropical Cyclone frequency 36 4.2 Average sinuosity index 36 4.3 Correlation of average sinuosity index with southern oscillation index 37 4.4 Correlation of sinuosity with other tropical cyclone parameters 39 4.4.1. Correlation of sinuosity index with start latitude 39 4.4.2. Correlation of sinuosity index with start longitude 39 4.4.3. Correlation of sinuosity index with end longitude 39 4.4.4. Correlation of sinuosity index with time 40 4.4.5. Correlation of sinuosity index with duration 40 4.5 Grouping the sinuosity index values 40 4.6. Geographical distribution of the tropical cyclone genesis and decay positions 41 4.7 Tropical Cyclone frequency and percentages in different tropical cyclone months for the five categories 45 4.8 Mean values for other parameters of the tropical cyclone tracks in relation to the sinuosity index category mean 46

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CHAPTER 5: DISCUSSION 47

5.1 Tropical cyclone genesis position and sinuosity index 48 5.2 Tropical cyclone decay position and sinuosity index 49 5.3 Tropical cyclone journey and sinuosity index 50 5.4 Sinuosity Index categories 50

CHAPTER 6: CONCLUSIONS 52

REFERENCES 55

APPENDICES 60

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LIST OF FIGURES

Figure 1 An aerial photograph of Nadi during March 2012 flooding 2 Figure 2 Flooding in Nadi in April 2012 2 Figure 3 Tropical Cyclone Henrieta track of sinuosity value 1.01 7 Figure 4 Tropical Cyclone Daman track of sinuosity value 1.07 8 Figure 5 Tropical Cyclone Tomas track of sinuosity value 1.16 8 Figure 6 Tropical Cyclone Gavin track of sinuosity value 1.34 9 Figure 7 Tropical Cyclone Xavier track of sinuosity value 1.75 9 Figure 8 Tropical Cyclone Rewa track of sinuosity value 4.36 10 Figure 9 Tropical Cyclone Rewa (28 December 1993 – 21 January 1994) 16 Figure 10 Tropical Cyclone Zaka (1995) 17 Figure 11 Tropical Cyclone Rae, Olaf, Meena, Percy and Nancy 18 Figure 12 Map of study area 21 Figure 13 Map of study area with 291 TC tracks during 1969/70 – 2007/08 cyclone seasons 21 Figure 14 Sinuosity values for each tropical cyclone track was calculated 22 Figure 15 Histogram for the sinuosity values 23 Figure 16 Boxplot analysis of sinuosity values 24 Figure 17 Boxplot analysis of sinuosity index 27 Figure 18 Dotplot of the sinuosity index 28 Figure 19 P-P plots of sinuosity index 31 Figure 20 Frequency distribution of sinuosity index 32 Figure 21 Tropical Cyclone frequency against tropical cyclone seasons (1969/70 – 2007/08) 36 Figure 22 Graph of average sinuosity index against tropical cyclone seasons 37 Figure 23 Tropical Cyclone displacement tracks for the 291 tropical cyclones that occurred between (1969/70 – 2007/08) 42 Figure 24 Tropical Cyclone displacement tracks for the five sinuosity index categories 43

Figure 25 Tropical Cyclone frequency and percentages for cyclone months for different sinuosity index categories 46

LIST OF TABLES

Table 1 Saffir Simpson Scale for categories of hurricane force tropical cyclones 5 Table 2 Outlier cyclones and their sinuosity values 25 Table 3 Outlier cyclones and their sinuosity index values 28 Table 4 Correlation of sinuosity index with cyclone variables 29 Table 5 Homogeneous categories based on sinuosity index using dynamic programming approach 34 Table 6 Homogeneous categories based on sinuosity index using hierarchical cluster analysis 35 Table 7 Number of cyclones, sinuosity and sinuosity index average and average SOI 38 Table 8 Suggested names for the five sinuosity index categories 41 Table 9 Tropical cyclone frequency and percentages for tropical cyclone months in each sinuosity index category 45 Table 10 Comparison of mean values of tropical cyclone parameters with the mean for sinuosity index 46

LIST OF APPENDICES

Appendix 1 Cyclone dataset for the years 1969/70 – 2007/08 59

Appendix 2 Southern oscillation index (SOI) archives 1969 – 2008 68

Appendix 3 C++ Program for finding the optimum group of cyclones using Dynamic Programming Technique 69

CHAPTER I

INTRODUCTION

1.1 Tropical Cyclone in the Pacific region

Tropical cyclones (TCs) are one of the most destructive natural hazards for the tropical Pacific, and have a large impact on socio-economic and environmental sectors in island nations therein (Terry, 2007). More than half the population of tropical Pacific lives in coastal environment making them more vulnerable to the impacts of the TC events. River flooding, storm surge, landslides, strong winds, heavy rainfall and coastal erosion are the consequences of TCs that have the capability of destroying properties and claiming lives of people and livestock.

Water sources in the Pacific are mostly from ground water, rain, river and dams and therefore are extremely vulnerable to changes and variations in climate, particularly rainfall because of their limited size, availability, geology and topography (The Global Mechanism and IFAD). Pipes are run from these sources to households and factories and all these are affected by flooding. Flooding is usually huge during TCs and it contaminates water sources and destroys pipes transporting water. It takes authorities months to restore services back to normal. The same problem lies in the electricity sector. Electricity is distributed to households and industries via cables hanging in air supported by posts. The system is able to withstand winds up to category 3 TCs but lot of damage is done to the posts and the power lines during category 4 and 5 TCs (Table 1) and the time it takes to bring services back to normal is several weeks to months.

In the Pacific Island countries, agriculture is the main source of income for rural dwellers where majority of people still live and depend on subsistence agriculture (The Global Mechanism and IFAD). While subsistence agriculture provides local food security, cash crops (such as sugar cane, banana and copra) are exported for foreign exchange. These farmers are mostly located along the coast or rivers for 1 fertile soil but it also makes these areas highly vulnerable to flooding resulting from heavy rain and high seas associated with TCs. TCs are capable of destroying vast areas of farms and also buildings with strong winds and flooding. Figures 1 and 2 below show the extent of water level during two major floods in Nadi, Fiji in 2012.

Figure 1: An aerial photograph of Nadi Figure 2: Flooding in Nadi in April 2012 during March 2012 flooding Photo courtesy Photo courtesy of Mohammed Ashiq, taken of Helene Muller, taken 30March, 2012. 12 April, 2012.

There have been some very destructive TCs to strike the study area. TC Tomas in 2010 was the most intense TC to strike Fiji since TC Bebe in 1972 (Gopal, A. 2012). It proved to be very destructive leaving many homeless and entire villages under water. Many homes were destroyed and washed away by strong winds and storm surges. Electricity and running water was disrupted in the main land and numerous outer islands (Gopal, A. 2012). TC Uma in 1987 struck Vanuatu and resulted in a very destructive cyclone claiming 48 lives and affected 48 000 people and the damage from the cyclone totaled to around USD 25 million (A Special Submission to the UN Committee for Development Policy on Vanuatu’s LDC Status, 2009). These are two examples of destructive TCs experienced in the study area but there are many other TCs that had a great impact on the study area.

1.2 Tropical Cyclone Variability

The patterns of TC variability are strongly affected by large-scale modes of interannual variability. Interannual variability in this context refers to any mechanism that can modulate the location and intensity of the monsoon troughs affects the genesis location and frequency of tropical cyclones (Chen et al., 2006). For example, the eastward shift in tropical cyclone formation positions over the western North Pacific in response to large-scale circulation changes during an El Nino – Southern Oscillation event is a particular example of the interannual variability of TC characteristics (Harr and Elsberry, 1991). The stronger storms (categories 3-5) tend to show stronger relationships to ENSO than do weaker storms (tropical storm through category 2 strength) (Frank and Young, 2007).

The studied dataset (Figure 13) shows that each TC track is unique in its own way, that is, no two cyclones have followed exactly the same path or same distance covered or caused the same degree of flooding. All these depend on various parameters including strength, longevity, position of the TC and the track they follow. One of the requirements for a TC to form and survive is the moisture from the sea because as soon as the TC moves on land, the moisture source is cut off and as a result the cyclone dies out. However, it was seen with two TCs namely TC Bebe in 1972 and TC Mick in 2009 that passed over Vitilevu Island in Fiji but they survived and continued their journey. DeMaria et al (2006) modified the method developed by Kaplan and DeMaria (1995) on TC and wind decay model that move over narrow landmasses. In the modified model the decay rate is proportional to the current intensity times the fraction of the storm circulation area that is over land. In another report by De Velde (2007), it is reported that smaller land masses lay in the path of TCs. Therefore, it can be said that Island landmasses are small and narrow for the TCs to pass through and still maintain its journey. This factor makes the Islands more vulnerable owing to the nature of the islands being small and narrow in the way that intense TCs will not easily decay and may affect many Island countries.

1.3 Tropical Cyclone Classification

TC is a generic name for a tropical depression or low pressure system. At its very early stage it is called a tropical depression and as the wind force increases it is categorized accordingly (gale force wind 34-47 knots); a tropical storm (storm force wind 48-63 knots); and a hurricane or typhoon (hurricane force wind 64 knots and above) (Terry, 2007).

On the Saffir - Simpson Scale, hurricane force category of tropical cyclones is further divided into five categories according to the maximum sustained winds (Table 1).

Table 1: Saffir Simpson Scale for categories of hurricane –force tropical cyclones is a standard for all tropical cyclones worldwide

Category Winds & Effects Surge

74-95mph (64-82 kt) 1 4-5 ft No real damage to building structures. Damage primarily to

unanchored mobile homes, shrubbery, and trees. Also, some coastal flooding and minor pier damage.

96-110mph (83-95 kt) 2 Some roofing material, door, and window damage. 6-8 ft Considerable damage to vegetation, mobile homes, etc. Flooding damages piers and small craft in unprotected moorings may break their moorings.

111-130mph (96-113 kt) Some structural damage to small residences and utility 9-12 3 buildings, with a minor amount of curtainwall failures. ft

Mobile homes are destroyed. Flooding near the coast destroys smaller structures with larger structures damaged by floating debris. Terrain may be flooded well inland.

131-155mph (114-135 kt) 13-18 4 More extensive curtainwall failures with some complete roof ft

structure failure on small residences. Major erosion of beach areas. Terrain may be flooded well inland.

155mph+ (135+ kt) Complete roof failure on many residences and industrial 5 18 ft buildings. Some complete building failures with small utility + buildings blown over or away. Flooding causes major damage to lower floors of all structures near the shoreline. Massive evacuation of residential areas may be required.

Source: Governor’s Office of Homeland Security & Emergency Preparedness, 2009.

A tropical storm officially becomes a hurricane once it reaches winds of 64 knots or greater (Terry, 2007). Once this happens the hurricane is then given a category based on how powerful the winds are. The category also gives an idea of likely

5 damages caused by flooding and structural damage once the hurricane hits land as shown in Table 1.

1.4 Tropical Cyclone Tracks

Atmospheric circulation is the dominant influence on storm properties. As TC moves poleward, it loses its tropical characteristics when it moves over cooler water and encounters the increasing vertical wind shears associated with the mid- latitude westerlies (Sinclair, 2002). The tracks studied in this research are confined to 0° to 25° south and therefore are restricted to the tropical climate. The reason to analyse TC tracks without the extratropical atmospheric system influence is to avoid confusion introduced by tropical and extratropical climates.

Tropical systems, while generally located equatorward of the 20 - 25th parallel, are steered primarily westward by the east to west winds on the equatorward side of the subtropical ridge – a persistent high pressure area (Landsea, 2010). The coriolis force defined as the apparent deflection of objects (such as airplanes, wind, missiles, and ocean currents) moving in a straight path relative to the earth’s surface causes cyclonic systems to turn towards the poles in the absence of strong steering currents ( Briney, 2013). The poleward portion of a tropical cyclone contains easterly winds (Sinclair, 2002), and the coriolis effect pulls them slightly more poleward. The general movement of TCs, therefore, is from the equator towards the poles.

1.5 Sinuosity of Cyclone Tracks

TCs tend to display various track shapes from straight - curvy - single loop - multiple loops. In this study, I chose to represent the different shapes of cyclone tracks, the sinuosity value was then correlated with several other tropical cyclone parameters, including cyclone genesis and decay positions, duration, displacement, distance travelled and time (in years).

Sinuosity of the TC track simply means how straight or not straight a cyclone track is (Terry and Gienko, 2011). A straight moving TC has a sinuosity value of 1, the minimum value for sinuosity. When a sinuosity value exceeds 1, the TC track becomes more curvy or loopy. Figures 3 - 8 below show cyclone tracks of six cyclones of differing sinuosity.

S = 1.01 Latitude

Longitude Figure 3: Tropical Cyclone Henrietta track of sinuosity value 1.01

S = 1.07 Latitude

Longitude

Figure 4: Cyclone Daman track of sinuosity value 1.07

S = 1.16 Latitude

Longitude

Figure 5: Cyclone Tomas track of sinuosity value 1.16

S = 1.34 Latitude

Longitude

Figure 6: Cyclone Gavin track of sinuosity value 1.34

S = 1.75 Latitude

Longitude Figure 7: Cyclone Xavier track of sinuosity value 1.75

S = 4.63 Latitude

Longitude Figure 8: Cyclone Rewa track of sinuosity value 4.36

TC Henrietta in figure 3 above having sinuosity value of 1.01 has a fairly straight track. TC Daman and Tomas have similar track shape but cyclone Daman has lower sinuosity value than TC Tomas. One reason could be that TC Tomas has travelled a longer distance and covered greater latitudes than TC Daman and was therefore more sinuous. TC Gavin has a variety of turns and a sinuosity value of 1.34 and falls in sinuosity category 4. It can be seen that the shape of the track has various turns which gives it a high sinuosity value. Figure 7 shows track for TC Xavier from sinuosity category 5 which has sinuosity value of 1.75. TC Xavier travelled a long distance and had a loop in its journey which contributed to its high sinuosity value. TC Rewa travelled a great distance and has various turns and loops making it a highly sinuous track.

The forecasting of TCs is very challenging owing to the complexity of the contributing factors and the diverse nature of the event. The situation may worsen with climate change scenarios in terms of future distribution and characteristics of TCs (IPCC, 2011). For example, large amplitude fluctuations in the frequency and intensity of TCs can greatly complicate both the long-term

10 trends and their attribution to rising levels of atmospheric greenhouse gases (Knutson et al., 2010).

1.6 Research Objectives Specific objectives of the research are:

1. Explore available tropical cyclone data set for the Southwest Pacific region to develop sinuosity and sinuosity index data.

2. Investigate whether there is a correlation between sinuosity values with other tropical cyclone parameters and Southern Oscillation Index.

3. Implement statistical analyses of the tropical cyclone tracks and develop a technique to group the tropical cyclone tracks into different categories according to their sinuosity values.

4. Employ GIS techniques to map and study the distribution of the resulting sinuosity categories in the study area.

1.7 Chapter Organizations

This study presents the outcome of the cyclone groups, categorised based on sinuosity index values of cyclones for the period 1969/1970 to 2007/2008 for the Southwest Pacific.

The thesis is structured in six chapters as follows;

Chapter 1: Introduction This chapter introduces to the tropical cyclones in the Pacific, classification of the cyclones, sinuosity of the cyclone tracks and tropical cyclone tracks. The chapter also describes the objectives of the research carried out and presented in the thesis.

Chapter 2: Literature Review

The focus of this chapter was to study similar work done on this topic in in the region and in other regions (basically western North Pacific) and to study the nature of some tropical cyclones based on their tracks.

Chapter 3: Data and Methods The chapter introduces the study area and the nature of the data. It also describes the various preprocessing steps for the data normalization process and the categorization methods.

Chapter 4: Results and Interpretation This chapter analyzes and interprets the results obtained from the categorization of the cyclone data.

Chapter 5: Discussion The chapter involves discussing the two methods used for categorization process and the sinuosity index categories obtained.

Chapter 6: Conclusions The final chapter summarizes the key findings of this study with recommendation for further research

CHAPTER 2 LITERATURE REVIEW

Tropical Cyclone season in the South Pacific is from November to April (Terry, 2007). Based on the studied dataset, the following statistics were calculated (refer to table 9). The number of TCs varies significantly ranging from 2 – 12 per season. The average number of TCs per season for 1969/1970 – 2007/2008 periods is 7.5. More than 80% of TCs occur between December – March within the cyclone season. About 6% of TCs from the total 291 TCs studied, occurred outside the cyclone season and 81% of these TCs occurred in the El Niño years. It is evident from the 39 cyclone seasons studied that Southern Oscillation has an impact on the number of cyclones and off season TCs in the Southwest Pacific.

There are relatively few previous investigations that focus on sinuosity of TC tracks in the South Pacific. A difficult part of this research was finding unpublished/ published studies that focus on sinuosity of TC track analysis. Studies have been conducted to enhance understanding of TC patterns and behavior and have been used to improve the understanding of the cyclone characteristics. Most of these studies have been undertaken for the North Atlantic and Western North Pacific cyclone basins due in part to the reliable record (Landsea, 1999). However, recently Diamond (2010) has developed an enhanced Tropical Cyclone Track database for the Southwest Pacific which would attract researchers to utilize this opportunity to study climatology of TCs in the Southwest Pacific.

There are some studies done on shapes and trajectories of TC tracks (e.g., Chen et al. 2006; Camargo, et al, 2007; Harr & Elsberry, 1991; Lander, 1996). Two principal track types identified in previous studies (e.g., Sandgathe, 1987; Harr and Elsberry, 1991; Lander, 1996; Camargo et al., 2007) are recurving and straight - moving track types. Another study by Elsner and Liu (2003) analyzed typhoon tracks based on the typhoon’s position at maximum intensity and its final intensity and obtained three clusters; (1) straight – moving; (2) recurving

14 and (3) north – oriented tracks. The study employed the K- means cluster analysis. Camargo et al. (2007) used new probabilistic clustering technique based on a regression mixture model to categorize the cyclone trajectories in the western North Pacific. Seven different clusters were obtained and then analyzed in terms of genesis location, trajectory, landfall, intensity, and seasonality. Only two studies have focused on the sinuosity of TCs is by Terry and Feng (2010) for western North Pacific and Terry and Gienko (2011) for the Southwest Pacific. The calculation of sinuosity values for cyclone tracks for this study is consistent with the method employed by Terry and Feng (2010) but the categorization method is different. In Terry and Feng (2010) the categories for track sinuosity was based on quartile ranges due to the strong skew in the data. Our study is built on the study of Terry and Feng (2010) but uses a different method of categorization. In this study, a proposed method using a dynamic programming technique and Hierarchical Cluster Anaylysis with Ward’s method was used for categorization.

The greater the sinuosity of a cyclone track is, the greater the potential area covered during its journey. There are many small islands in the South Pacific. TCs that tend to curve or loop, are more likely to involve landfall, for example, TC Rewa (28 December 1993– 21 January 1994) which lasted for 25 days and underwent several major changes in direction during its lifetime (Bureau of

Meteorology, 2012). TC Rewa track has a sinuosity value of 4.36 (Figure 9).

Figure 9: Cyclone track for TC Rewa (28 December 1993– 21 January 1994) Source: Bureau of Meteorology, 2012

“Tropical cyclone Rewa was formed and situated to the north of Vanuatu and moved in the western direction before moving in the west – southwest direction, it crossed the southern tip of the island of Malaita before passing south of Guadalcanal Island in its passage through the Solomons. The system then recurved to the south and continued in a south – southeasterly direction followed by southeast and then more easterly direction. Along its path the cyclone passed over central New Caledonia heading in a northeasterly direction then changed its course and started moving in a northwest direction for a short while then continued in a more western direction. It again started moving in a northwest direction before moving in a northerly direction towards the north – west tip of Tagula Island in the Louisiade Archipelago. The cyclone then executed a sharp clockwise turn just off the northern side of Tagula Island and continued in the southest direction before recurving to the west – southwest approaching the Queensland coast. Cyclone Rewa then turned south on the track before moving towards the southeast away from the coast towards north of Lord Howe Island. The cyclone then moved southeast across the Tasman Sea towards the north of

16 the South Island of New Zealand before dying out” (Bureau of Meteorology, 2012).

TC Zaka has a sinuosity value of 1, it formed south of Tonga and moved in the west direction for a little while before moving in the northwest direction towards north of New Zealand and died out (Figure 10). It was a very weak category one cyclone which brought some pesky rain and occasional roaring gusts (Natural Hazards Spring, 2012).

Figure 10:Ttropical Cyclone track for Cyclone Zaka (1995) Source: Natural Hazards Spring, 2012

TC tracks for five different TCs are shown in figure 11 below. These five TCs occurred during 2004/05 within a period of five weeks. Sinuosity values of these TCs are Rae (1.08), Olaf (1.13), Meena (1.15), Percy (1.18) and Nancy (1.57). TCs Nancy, Percy, Olaf and Meena having more sinuous tracks than TC Rae also caused more damage and lasted longer. They brought storm surges, huge waves which destroyed buildings in coastal areas, seawater inundated buildings along coastal areas and rubble and trees were strewn on buildings. TC Rae only lasted for ten hours with no damages to the Island. (Ngari, 2005).

Figure 11: Cyclone tracks for Cyclone Rae, Olaf, Meena, Percy and Nancy Source: (Ngari, 2005)

When TCs tend to recurve, it has its peak power and highest sustained wind speeds (De Velde, 2007). A slow moving TC stays longer in an area and therefore will have more impact than a cyclone that moves in a straight line. Tropical cyclones normally (about 70%) recurve to the east, at latitude of approximately 20° to 30° N/S, following the general air circulation (westerlies) around the globe (De Velde, 2007). The remaining 30% of the tropical cyclones continue to travel west, northwest, north, or have an erratic track, or start to loop back or remain stationary (De Velde, 2007). Therefore, it is important to study why cyclones tend to differ so much in the way they travel and in order to do this, the best way will be to study the long term trend of cyclone tracks for sinuosity and also correlating the sinuosity trend with other parameters such as,

18 cyclogenesis and decay positions, displacement, total distance travelled by TCs, wind speed and duration.

CHAPTER 3

DATA AND METHODS

3.1 Study Area and Data Collection This study examines TC track parameters using data from 1969/70 to 2007/08. Appendix 1 gives the date, atmospheric pressure, wind speed and location of the TCs at starting and ending phase. It also gives the name, duration, azimuth and sinuosity of the TCs of interest in this study. The primary sources of the data are the Fiji Meteorological Service and the Tropical Cyclone Warning Centre in New Zealand. The studied dataset lists the TCs which includes the portion of the TC tracks with intensity of 35 knots and above and the TCs which have their genesis in the tropics. The portion of the TC track which was below 35 knots was eliminated from the analysis. The data set records 6- hourly centre location and intensities and therefore the track plotted joining the recorded positions. TC paths analysed are within the geographical grid covered by 0 – 25°S and 160° E – 120° W. (Figure 12). This area falls under the responsibility of the Fiji Meteorological Services. In the 39 TC seasons during 1969/70 to 2007/08, the study area experienced 291 TCs (Figure 13). Data from 1969/70 onwards were analyzed when satellite observation was introduced so that sinuosity categories are constructed based on reliable dataset. However, extensive work has been done by International Best Tracks for Climate Stewardship (IBTrACS) project, under the auspices of the World Data Centre for Meteorology in compilation of TC best track data from 12 TC forecast centres around the globe, producing a unified global best track data set (Diamond, H, 2010). Diamond, H (2010) then developed an enhanced TC tracks database for the Southwest Pacific for 1840 – 2009. Having such a long term reliable data set would be very useful for this kind of study, however, it was not possible to incorporate this dataset in this study due to unavailability of the dataset at the commencement of this study in 2008.

0º

10º S

20º S

30º S

180º 160º W 140º W 140º E 160º E Figure 12: Map of Study Area Source: Modified from Bureau of Meteorology, 2009

Figure 13: Map of study area with 291 TC tracks during 1969/70 – 2007/08 cyclone seasons

3.2 Sinuosity Calculation In this study TC tracks are studied based on sinuosity values of cyclone tracks. Sinuosity values for the TC tracks were calculated by dividing displacement of the track by distance travelled by the cyclone. Only the portion of the cyclone track which is in the study area and is 35 knots and above was used to calculate the sinuosity. The following illustration in Figure 14 illustrates how sinuosity of TC tracks was calculated:

Figure 14: Sinuosity values for each cyclone track was calculated

The red portion of the cyclone track was included in this study as it falls in the study area and has wind speeds > 35 knots. The distance (in red) and the displacement (in black) of the TC track were measured using GIS tool. Sinuosity was calculated as the ratio of the two;

Distance of tropical cyclone Sinuosity = 4 Displacement of tropical cyclone

3.3 Distribution of Sinuosity Values In this section, frequency distribution of sinuosity data is studied as shown in Figure 15 below.

Figure 15: Histogram for the sinuosity data

The histogram shows that the distribution is skewed towards left as the bulk of the data lies between 1 and 2 and a very extreme value at around 52. Thus, statistical analysis on this distribution will not be much useful because of the presence of large extreme value which is making the distribution very skewed. The option of eliminating three extreme values 52.74, 4.51 and 4.36 was also tested but distribution was still skewed towards right. Thus, a statistical analysis using boxplot was conducted to eliminate outliers from the dataset. However, in the following section, a statistical analysis is carried out to identify extreme cyclones that can be considered as outliers with respect to the sinuosity values.

3.3.1 Analysis of Extreme Tropical Cyclones from sinuosity data

Figure 16: Boxplot analysis of the sinuosity dataset

The boxplot in figure 16 clearly shows that there was one TC with an extreme sinuosity value. To identify TCs with extreme values in the distribution of sinuosity, the following quantities are computed:

1. Lower inner fence: QIQ1 1.5 2. Upper inner fence: QIQ3 1.5 3. Lower outer fence: QIQ1 3 4. Upper outer fence: QIQ3 3

th Where, Q1 is the 25 percentile = 1.028

th Q3 is the 75 percentile = 1.300 IQ is the interquartile range = QQ31= 0.27

Substituting the values of Q1 , Q3 and IQ into the equations above we get: 1. Lower inner fence: 1.02790 1.5 0.27150 = 0.62

2. Upper inner fence: 1.29940 1.5 0.27150= 1.71 3. Lower outer fence: 1.02790 3 0.27150 = 0.21 4. Upper outer fence: 1.29940 3 0.27150 = 2.11

Outlier detection criteria shows sinuosity value beyond an inner fence on either side is considered as a mild outlier and therefore sinuosities below 0.62 and above 1.71 are outliers and the value beyond an outer fence is considered as extreme outlier, which implies all values below 0.21 and above 2.11 are extreme outliers in this case. Table 2 shows that there are 28 outliers in total and the last 13 out of 28 cyclones are considered as extreme outliers.

Table 2: Shows the outlier cyclones and their sinuosity values. Tropical cyclones Sinuosity LENA 1.72 ERICA 1.73 NORMAN 1.75 XAVIER 1.75 KERRY 1.77 ZOE 1.77 IMA 1.78 VEENA 1.79 NAMELESSB 1.80 ZUMAN 1.83 BENI 1.84 ABIGAIL 1.85 DANI 1.86 CYC1981 1.86 BETTY 2.11 ESAU 2.21 WATI 2.22 IVY 2.23 FIONA 2.24 CARLOTTA 2.26 HALI 2.33 BOLA 2.44 YANI 2.69 HARRY 2.76 JUNE 2.80 REWA 4.36 TRINA 4.50 KATRINA 52.74

It is true that a dataset needs to be free of outliers before any statistical analysis could be done on the dataset to concentrate on the bulk of the data. This would

25 allow identifying the most likely response of the sinuosity values but the outliers in this case are significant as it represents a population and not sample and therefore it is not appropriate to eliminate 28 cyclones as outliers. An option of calculating sinuosity index, which is discussed in Terry and Gienko (2010), was then considered for categorizing the TCs in this research.

3.3.2 Sinuosity Index From the sinuosity values, sinuosity indexes were calculated using the following formula (Terry and Gienko, 2010):

3 SI = S – 1 10 .

Where, SI = Sinuosity Index value, S = calculated sinuosity. Sinuosity Indexes are cubed – root transformation of sinuosity values in order to normalize the sinuosity values. The subtraction (S – 1) allows the transformed distribution for SI to start at zero and product x10 is introduced in order to avoid dealing with decimal numbers.

The need to calculate sinuosity indexes was to reduce number of outliers from the dataset so that maximum number of cyclones could be included in the analysis. It was important to include maximum number of cyclones in the categorization analysis so that the categories obtained based on sinuosity index values are true representation of the dataset.

3.3.3 Analysis of Extreme Tropical Cyclones from Sinuosity Index Data The boxplot in figure 17 still shows some extreme cyclones based on the sinuosity index values.

Figure 17: Boxplot analysis of the sinuosity index values

To identify these outliers, as discussed in Section 3.3.1, the inner and outer fences for SI values are obtained as follows:

1. Lower inner fence: 3.0330 1.5 3.6569 = -2.45 2. Upper inner fence: 6.6899 1.5 3.6569 = 12.18 3. Lower outer fence: 3.0330 3 3.6569 = -7.94 4. Upper outer fence: 6.6899 3 3.6569 = 17.66

Thus, TCs whose sinuosity index falls below -2.45 and above 12.18, are considered to be outliers in this case. There are three outliers found and one of them is considered to be an extreme outlier, which is tropical cyclone Katrina with SI value of 37.26 as shown in the Dot-plot (Figure 18).

Figure 18: Dotplot of the sinuosity index values

Converting the sinuosity values into sinuosity index values brings distribution closer to normal and reduces number of outliers from 28 to only 3 TCs (Table 3). Table 3: Outlier cyclones and their sinuosity index values. Tropical Sinuosity cyclones Index values REWA 14.97 TRINA 15.19 KATRINA 37.26

3.4 Correlation of Sinuosity Index values with other Parameters SI values were correlated with other TC parameters. Sinuosity index value was negatively correlated with start latitude, start longitude and end longitude and positively correlated with duration, distance and time (in years). TC displacement and end latitude did not show any significant correlation with sinuosity index values. Results from correlation tests are presented in Table 4. Correlations of sinuosity index with all the parameters are significant at the 0.01 level except with start longitude which is significant at 0.05 level. Thus, correlation of SI with six other TC parameters is significant and, therefore, it may be appropriate to use sinuosity index values to categorize the TCs into different groups or categories.

Table 4: Correlation of sinuosity index with tropical cyclone variables. Other Variable Correlation (r) p-value Start Latitude -0.20 .001 Start Longitude -0.15 .011 End Longitude -0.21 .000 Distance 0.46 .000 Duration 0.54 .000 Time 0.15 .009

3.5 Methodology for Grouping the Sinuosity Index values: A proposed Technique

In this section a method is proposed to categorize the TCs based on the sinuosity index values.

Let the sinuosity index ( x ) of size N is to be classified into G mutually exclusive and homogeneous groups consisting Nhh; ( 1,2,..., G ) units in hth group so as to

NN12... NG N and the variance of the sinuosity index within the group is as minimum as possible. That is, in order to make the groups internally homogenous, the groups should be constructed in such a way that the variance of the groups be as small as possible. A reasonable criterion to achieve this is,

Let x0 and xG be the smallest and largest values of sinuosity index x respectively and xx12, ,..., xG 1 denote the set of intermediate optimum boundary points of the groups. If xhi are the values of sinuosity index of i th cyclone that fall in h th group, then the problem of optimum grouping can be described as to find the intermediate group boundaries xx12,..., xG 1 such that the sum of weighted variance due to the grouping, that is,

G 2 BWhh (1) h1 is minimum.

N Where W h = the proportion of cyclones that falls in h th group, h N

N B h x 2 2 i1 hi h h = the variance of h th group, Nh

N B h x i1 hi and h = the mean of h th group. Nh

It should be noted that the values of Nh and xhi are unknown as the groups are yet to be constructed. Further, the problem is to determine the best boundaries that make groups internally homogeneous by minimizing (1), which is not a function of boundary points. Therefore, a way to achieve the optimum boundary points effectively is, if (1) can be expressed as the function of boundary points which is possible when the distribution of sinuosity index known and then create groups by cutting the range of the distribution at suitable points (See Khan, et al. 2002, 2005, 2008).

Let fx() denotes frequency function of the sinuosity index ( x ). Then the values

2 of weights Wh and the variance h of h th group are obtained as the function of boundary points ( xh1 , xh ) by

xh Wfxdxh O () (2)

xh1 1 xh 22O xfxdx() 2 (3) h W h h xh1 1 xh Where O xf() x dx (4) h W h xh1

Therefore, when the frequency function fx() is known and is integrable, using

2 (2), (3) and (4) Whh in (1) could be expressed as a function of xh and xh1 , and hence the optimum boundary points are obtained. (Khan et al., 2008).

3.5.1 Estimate of the Distribution of Sinuosity Index values

P-P plot (using SPSS):

A probability - probability (P-P) plot of sinuosity index ( x ) is obtained to determine whether the distribution of x matches a particular distribution. Figure 19 shows that x match the gamma distribution as the points cluster around a straight line.

Figure 19: P-P plot for sinuosity index values (x)

Figure 20: Frequency distribution of sinuosity index.

Also figure 20 of relative frequency histogram reveals that x is assumed to follow Gamma distribution with a probability density function given by

x 1 fx xer 1 ;0;,0 x r . (5) r()r

Where r is the shape parameter and is the scale parameter.

3.5.2 Estimate of the Parameters of Distribution

Using the maximum likelihood estimate (MLE) method for the sinuosity index data, the parameters of Gamma distribution given in (5) are found to be

Shape, rˆ =3.822976 and scale, ˆ =1.351949 (6)

3.5.3 Determination of Optimum Grouping using Dynamic programming Technique

2 Using (2), (3) and (4), we obtain Wh , h and h as follow:

CSCxxl S WQrDTD,,hhh11 Qr T h EUE U (7)

FVCSCSxxl rQrGWDTDT1,hhh11 Qr 1, HXEUEU h FVCSCxxl S GWQrDTD,,hhh11 Qr T HXEUE U and FVFVCSCSCSCSxx xxl 2rr(1)GWGW QrDTDTDTDT 2,hh11 Qr 2,22 r Qr 1, hhh 11 Qr 1, HXHXEUEUEUEU 2 (8) h 2 CSCSxx FVCSC S hh11 xxlhhh11 QrDTDT,, Qr GWQrDTD,, Qr T EUEU HXEUE U Where

lxxhhh1 (9) is the width of h th group and

1 Qrx(, ) trt1 e dt ; rx , 0;() r 0 O ()r x denotes the upper incomplete Gamma function.

Therefore, from (7) and (8), the expression (1) reduces to

FVCSCSxxl 22rQrGWDTDT1,hhh11 Qr 1, G FVCSCSxxHXEUEU B 2rr(1)GW QrDTDT 2,hh11 Qr 2, (10) HXEUEUFVCSCxxl S h 1 GWQrDTD,,hhh11 Qr T HXEUE U

To obtain the optimum boundary points xxhh1, of the groups, the optimum widths lh are obtained by formulating a nonlinear optimization problem as given below (See Khan, et al. 2002, 2005, 2008):

Minimize FVCSCS 22 xxlhhh11 G rQrGWDTDT1, Qr 1, FVCSCSxxHXEUEU B 2rr(1)GW QrDTDT 2,hh11 Qr 2, HXEUEUFVCSCxxl S h 1 GWQrDTD,,hhh11 Qr T HXEUE U

G Subject to Bldh . (11) h1

Where d is the range of the sinuosity indexes, that is, dxL x0 12.1561 0 12.1561.

If five groups, that is G 5, are to be formed, then the proposed method using a dynamic programming technique by extending Khan, et al. (2002, 2005, 2008) gives the optimum boundary points for each group by executing a computer program coded in C++ (See Appendix 3) for Problem (11) as shown in Table 5:

Table 5: Five homogeneous categories using the proposed dynamic programming approach.

Group Sinuosity Index No. of cyclones Weight Variance Weighted Variance ( h ) ( x ) 2 2 ( Nh ) (W ) ( h ) (Whh) h

1 0 – 3.03 73 0.25 0.82 0.20 2 3.03 – 4.64 71 0.16 0.12 0.02 3 4.64 – 6.40 65 0.31 0.38 0.12 4 6.40 – 8.84 53 0.18 0.48 0.087 5 8.84 – 12.16 26 0.089 1.05 0.096 288 0.52

3.6 Alternative Methodology for Grouping the Sinuosity Index values using Hierarchical Cluster Analysis

Hierarchical Cluster Analysis was also used to identify relatively homogeneous groups of TCs. Using SPSS with Ward’s method, five homogeneous groups of TCs were determined based on sinuosity index values (Table 6).

Table 6: Five homogeneous categories based on sinuosity index using hierarchical cluster analysis.

Group Sinuosity Index No. of Weight Variance Weighted ( h ) ( x ) cyclones 2 Variance (Wh ) ( h ) ( ) 2 Nh (Whh) 1 0 – 3.20 103 0.36 1.04 0.37 2 3.20 – 5.01 60 0.21 0.094 0.020 3 5.01 – 7.27 72 0.25 0.43 0.11 4 7.27 – 9.51 42 0.14 0.49 0.071 5 9.51 –12.16 11 0.038 0.37 0.014 288 0.59

3.7 A Comparison Study of Grouping Methods

In Section 3.5 and Section 3.6, the cyclones are categorized into five groups based on their sinuosity index values using the following two methods, respectively:

1. A proposed method using a dynamic programming technique by extending Khan et al. (2002, 2005, 2008). 2. Hierarchical Cluster Analysis method with Ward’s method.

Table 5 and 6 show the results of five SI categories for TC dataset obtained by the proposed method and Hierarchical Cluster Analysis method, respectively. The tables also show variance of each group and sum of the weighted variance. SI categories one, three and four in proposed method have smaller variance as compared to Hierarchical Cluster Analysis method. Moreover, the sum of weighted variance (0.52) is also smaller for the proposed method as compared to Hierarchical Cluster Analysis method (0.58). Thus, on the basis of these comparisons, it can be concluded that categorization using proposed dynamic programming technique is a more appropriate approach since it produces more homogenous SI categories.

4.0 RESULTS and INTERPRETATIONS

4.1 Tropical Cyclone frequency

The graph below (Figure 21) shows that number of TCs has slightly decreased for the study period (1969/70 – 2007/08). The 1997/98 season shows the greatest frequency of TCs and other seasons recording high frequencies (ten and above TCs per season) include seasons 1980/81, 1982/83, 1986/87, 1988/89, 1991/92, 1992/93, 1996/97 and 2002/03.

Figure 21: Tropical Cyclone frequency against cyclone seasons (1969/70 – 2007/08)

4.2 Average sinuosity index

Figure 22 obtained from the average sinuosity index calculated in Table 7 shows that average sinuosity index have slightly increased for the study period. The three seasons having high sinuosity index average are 1993/94, 1997/98 and 2001/02. All these seasons also include cyclones from outlier category.

Figure 22: Graph of sinuosity index averages against cyclone seasons (1969/70 – 2007/08)

One clear observation from the two graphs above is that 1997/98 season has the highest number of TCs and also highest sinuosity index average.

4.3 Correlation of Average Sinuosity Index with Southern Oscillation Index (SOI)

Average for sinuosity indexes for each thirty nine TC seasons were calculated and correlated with Southern Oscillation Index (SOI) averages. SOI was obtained from the archives of Australian Government Bureau of Meteorology (Appendix 2) and the average of each tropical cyclone season was calculated as shown in Table 7.

Table 7: Number of cyclones, sinuosity, sinuosity index and SOI averages for cyclone seasons

Cyclone Sinuosity Sinuousity Average Southern Seasons No. of Cyclones Index Average Average Oscillation Index 1969 - 70 6 5.31 1.21 -3.3 1970 - 71 6 5.55 1.35 36.4 1971 - 72 9 4.73 1.25 2.9 1972 - 73 8 4.11 1.12 -5.5 1973 - 74 7 2.76 1.03 48.2 1974 - 75 5 5.86 1.34 4 1975 - 76 5 3.53 1.07 12 1976 - 77 9 4.81 1.34 -1.4 1977 - 78 9 4.52 1.18 -11 1978 - 79 6 4.66 1.21 -1.4 1979 - 80 7 2.51 1.03 -4.8 1980 - 81 12 4.79 1.17 -4.4 1981 - 82 6 6.78 1.37 2.6 1982 - 83 14 6.07 1.30 -26.8 1983 - 84 7 3.93 1.16 0.4 1984 - 85 9 3.49 1.07 3 1985 - 86 7 4.4 1.20 0 1986 - 87 12 4.65 1.14 -14.5 1987 - 88 5 6.25 1.41 -1.8 1988 - 89 11 5.86 1.38 13.6 1989 - 90 6 5.57 1.20 -5.7 1990 - 91 2 5.80 1.20 -4.2 1991 - 92 12 5.73 1.19 -16.9 1992 - 93 10 4.79 1.17 -9.7 1993 - 94 5 7.66 1.86 -5.3 1994 - 95 3 4.12 1.13 -6.3 1995 - 96 5 3.83 1.11 3.2 1996 - 97 11 5.52 1.24 0 1997 - 98 16 7.25 4.44 -19.9 1998 - 99 9 5.26 1.33 12.9 1999 - 00 6 3.34 1.07 11.6 2000 - 01 4 4.64 1.15 9.6 2001 - 02 5 6.21 1.75 0 2002 - 03 10 5.51 1.32 -6.3 2003 - 04 3 5.93 1.26 -1.9 2004 - 05 9 5.40 1.23 -9.2 2005 - 06 5 6.11 1.36 6.6 2006 - 07 6 6.00 1.45 -3.1 2007 - 08 4 7.23 1.42 12.7

Correlation test of average sinuosity index and average SOI gives (r = -0.273, p– value = 0.46 at 0.05 level). Therefore it can be said that sinuosity index has a significant relationship with average SOI but the degree of association is weak.

4.4 Correlation of sinuosity index with other tropical cyclone parameters

Table 4 shows that SI of cyclone tracks have significant correlation with six other cyclone parameters. However, due to unavailability of any literature, it was not possible to do any comparison of these correlation results with other studies.

4.4.1 Correlation of sinuosity index with start latitude

From Table 4, it can be seen that the correlation between SI and latitude is -0.20, which is statistically significant at 0.01 level (p-value = 0.001). Although the degree of association is weak, the relationship is significant and negative correlation which means that the TCs forming at higher latitudes are less sinuous compared to cyclones forming in low latitudes.

4.4.2 Correlation of sinuosity index with start longitude

The correlation between SI and start longitude is weak (-0.15) but significant at 0.05 level (p-value = 0.011). It is a negative correlation meaning that the TCs forming in the east of the study area are less sinuous.

4.4.3 Correlation of sinuosity index with end longitude

The correlation between SI and end longitude is -0.21, which is significant at 0.01 level (p–value = 0.001). It is a weak and negative correlation meaning that the TCs that decay more eastward are less sinuous.

4.4.4 Correlation of sinuosity index with time

The correlation between SI and time (in years) is 0.15, which is weak but significant at 0.01 level (p–value = 0.009). It is a positive correlation which means that the TCs have become more sinuous with time.

4.4.5 Correlation of sinuosity index with duration

The correlation of SI with duration is strong (0.54), which is significant at 0.01 level (p-value < 0.001). The relationship is positive which implies that longer lived TCs have a tendency to be highly sinuous.

4.4.6 Correlation of sinuosity index with distance travelled by cyclone

The correlation of sinuosity index with total distance of cyclone travel is 0.46 at 0.01 level (p-value < 0.001). The relationship is positive which means that cyclones that travel greater distance have a chance of being more sinuous than TCs having short paths.

4.5 Grouping the Sinuosity Index values

Two methods were used in this study for grouping the sinuosity index values. The first method was a proposed technique using dynamic programming based on Khan et al (2002, 2005, 2008) where the optimum boundary points for each group were obtained by executing a computer program coded in C++( see Appendix 3). An alternative method of hierarchical cluster analysis in SPSS with Ward’s method was also employed for comparison purpose. The two methods resulted in comparable categorization but the proposed method provided a better grouping of the categories as the total weighted variance was small for this method as compared to the other method. It also met the objective of this study to categorize the TCs into similar groups so that the variance within the groups is minimum.

There are six SI categories formed from all the 291 TCs reported during the study period, which also includes the outlier category. Out of the total 291 TCs, 288 TCs were statistically categorized into five homogeneous SI categories and the sixth was treated as an outlier category which consisted of three TCs of extreme sinuosity index values. The reason for grouping the TCs into five SI categories was to have a middle category with above representing the straight moving cyclones and the below representing the sinuous tracks. However the

40 straight moving cyclones were divided into two categories to separate the perfect straight tracks from not so straight tracks and the same was done for the sinuous tracks. Table 8 below gives the five SI categories and the outlier category with the suggested category names.

Table 8: Suggested names for the five sinuosity Index categories.

Sinuosity Index Sinuosity Description Categories Index 1 0 – 3.03 Straight Tracks 2 3.033 – 4.64 Near Straight Tracks 3 4.6431 – 6.40 Curving Tracks 4 6.4045 – 8.84 Sinuous Tracks 5 8.8374 – 12.16 Wiggly Tracks 6 14.9714, 15.19 Extreme Sinuous Tracks and 37.26 (Outlier)

4.6 Geographical Distribution of the cyclone genesis and decay positions

The different SI categories coded in different colours were represented graphically using arcGIS to display the visual differences between the SI categories. Figure 23 below shows the map of the study area with all TC displacement line from the start point of the cyclone to the end point. When looking at separate maps (Figure 24) for each category containing cyclone displacement tracks, more clear distribution of the cyclone genesis (position where tropical cyclone first attained 35 knots intensity) and decay (position where tropical cyclone had 35 knots before weakening to depression intensity) can be seen.The displacement line is just used to show the start (genesis) and end (decay) latitude and longitude point of the cyclone tracks.

Category 1 Category 2 Category 3 Category 4 Category 5 Category 6

Figure 23: Cyclone displacement tracks for the 291 TCs that occurred between 1969/70 – 2007/08.

SI category 1 TCs are seemed to have their cyclogenesis and decay positions quite evenly spread across the study area and SI category 3 TCs seem to be concentrated somewhere in the middle of the study area around 170° west while SI categories 2, 4 and 5 TCs are clustered far west of the study area.

Sinuosity Index Category 1 (a)

Sinuosity Index Category 2 (b)

Sinuosity Index Category 3(c)

Sinuosity Index Category 4 (d)

Sinuosity Index Category 5 (e)

Sinuosity Index Category 6 (f)

Figure 24: (a – f). Tropical Cyclone displacement tracks for the five sinuosity Index categories and the one outlier category.

A clear contrast exists between the first two (a and b) and the last two (d and e) categories (excluding the outlier category). TCs in first two SI categories (straight and near straight) are distributed quite evenly between 160° east to 130° west. TCs in SI categories 4 (d) and 5(e) (sinuous and convoluted) are formed and geographically limited to far west of the study area between 160° east to 180°. Overall, the sinuosity index categories get more sinuous, TC genesis shifts westward.

4.7 Tropical Cyclone frequency and percentages in different cyclone months for the five sinuosity index categories.

From Table 9 and Figure 25, it can be seen for all SI categories that the number of TCs occurring in the months of January, February and March are greater than other months. No obvious trend is observed as the SI categories get more sinuous, percentage of TCs occurring in the months of December, January, February and March is greater for sinuosity index categories (category three, four and five).

Table 9: Tropical Cyclone frequency and percentages for cyclone months in each SI category

Month SI Category 1 SI Category 2 SI Category 3 SI Category 4 SI Category 5 No. of No. of No. of No. of Cyclone Cyclone Cyclone Cyclone No. of s % s % s % s % Cyclones % October 1 1.4 1 1.4 1 1.5 0 0 1 3.8 Novembe r 2 2.8 6 8.5 4 6.2 2 3.8 1 3.8 Decembe r 9 12.5 10 14 9 13.8 9 17 1 3.8 January 14 19.4 18 25 21 32.3 10 18.9 6 23 February 19 26.4 18 25 14 21.5 10 18.9 8 30.8 March 17 23.6 9 13 10 15.4 12 22.6 7 26.9 April 10 13.9 5 7 4 6.2 6 11.3 1 3.8 May 1 1.4 3 4.2 2 3.1 2 3.8 1 3.8 June 2 3.8 Total 73 71 65 53 26

Figure 25: Tropical Cyclone frequency and percentages for cyclone months in each SI category

4.8 Mean values for other parameters of the tropical cyclone tracks in relation to the sinuosity index category mean

Table 10: Comparison of the mean values of tropical cyclone parameters with the sinuosity index mean

Av. Av. index speed speed latitude Av. end Av. end Av. end pressure pressure Av. start Av. start Av. start travelled Av. track Av. track longitude longitude longitude longitude Categories end latitude atmospheric atmospheric

Av. distance Av. distance Av. duration Av. duration displacement Av. end wind wind Av. end Av. Sinuosity Av. Sinuosity Av. start wind Av. start wind 178. 187. 1 2.01 16.9 23.8 9 3 991.7 991.3 38.4 38.3 2.7 1377 1357 179. 186. 2 3.94 15.6 23.5 4 1 993.3 988.6 36.8 41.5 3.2 1577 1482 177. 184. 3 5.41 13.2 24.4 7 3 993.1 984.5 36.9 46.5 4.8 2051 1761 176. 180. 4 7.42 13.3 23.8 6 5 993.3 984.9 36.7 45.9 5.5 2016 1430 174. 5 10.03 15 23.4 170 7 992.5 986.9 38.3 43.7 7.6 2544 1295

When averages of all the TC parameters are calculated and compared with the mean for the five SI categories (Table 10), there are some obvious patterns seen. The duration and the total distance of TCs increases as the sinuosity index categories get more sinuous as expected. The TC displacement track, start and end longitude decreases as the sinuosity index categories get more sinuous. More sinuous TCs occur closer towards the equator and further west.

5.0 DISCUSSION

Sinuosity is a measure of linear shape, specifically how much a TC track deviates from a straight line (Terry and Feng, 2010). Sinuosity of TCs is a study of the shape of TC tracks assigning a value to the shape in order to incorporate this useful parameter in TC studies. Thus, sinuosity of TC track is calculated as a ratio of the total distance travelled by the TC against the displacement line between the start and the end points of the TC track (Terry and Gienko, 2011). Total length of TC track (distance travelled) and displacement can easily be calculated using GIS tool. Including sinuosity of TC as one of the parameters to study TCs can be helpful in understanding TC climatology especially since it correlates well with other TC parameters such as, duration and distance (cyclone track length). Island Countries in the study area are scattered, small and narrow and therefore sinuous TC tracks as tested to live longer and travel greater distance may affect many Islands and may be more than once if TC curves and makes loops in its journey.

Two categorization methods were used to categorize the TC dataset containing 291 cyclones based on their sinuosity index values with three outliers excluded from the comprehensive analysis. Both methods involved categorizing as such that each SI category is homogeneous which means that the variance within a group is minimum. Three TCs with extreme sinuosity index were not included in the categorizing process as they were found to be outliers and therefore they formed a separate SI category. Including these outliers would have greatly affected the categorization process resulting in skewness in the distribution of data and obtaining homogenous SI categories.

The two categorization methods used in this study were: a proposed dynamic programming approach where the optimum boundary points for each group was obtained by executing a computer program coded in C++ and a hierarchical cluster analysis in SPSS with Ward’s method. The weighted variance for the former

method was 0.52 and the latter method gave 0.58. Since the aim of this study was to obtain homogeneous SI categories, the proposed dynamic programming approach was found to be more useful for the categorization of TCs as it produced lower value for weighted variance. Thus each category obtained contains the TCs statistically more similar to each other than the TCs from other SI categories.

Previous work on western North Pacific Typhoon tracks categorized the TCs based on their sinuosity values into four sinuosity categories and used similar category names such as straight, quasi-straight, curving and sinuous (Terry and Feng, 2010). However for this study, five SI categories were formed and named straight, near straight, curving, sinuous and convoluted. Five SI categories were chosen to group TCs so that comparisons could be made more convenient. Having a central category dividing the straight moving and the sinuous cyclones gives an even distribution of categories. However, the straight moving and sinuous categories were further divided into two different sinuosity index categories in order to minimize the difference among the variables.

From a total of thirty nine seasons studied, nine seasons experienced ten and more TCs (Figure 22). The 1997/98 season experienced the most number of TCs having sixteen TCs altogether. This season also coincided with the El Niño phase. Other seasons having high frequencies of cyclones were 1980/81, 1982/83, 1986/87, 1988/89, 1991/92, 1992/93, 1996/97 and 2002/03. One notable observation is that all seasons having higher frequencies are mostly in a consecutive season and either one of them coincides with the El Niño event except for 1986/87 and 1988/89 seasons. Therefore, based on this observation, it is very likely that El Niño years and years before and after El Niño years are expected to bring more frequent cyclones in the Southwest Pacific which may be due to favorable conditions provided by the El Niño phase for TCs formation. A study by Diamond et al., (2012), it was investigated based on the new South Pacific Enhanced Archive for Tropical Cyclones dataset, that positive relationships exist among TCs, sea surface temperature, and atmospheric circulation which is consistent 48 with previous studies. The same study also revealed that statistically significant greater frequency of major TCs was found during the latter half of the study period (1991 – 2010) compared to the 1970 – 90 period.

The seasons having higher sinuosity index averages are 1993/94, 1997/98 and 2001/02. These are also the three seasons having cyclones with extreme sinuosity values and are categorized in the outlier category. The 1997/98 season had the highest number of cyclones and also has the highest sinuosity index average. It also coincided with the El Niño phase. The correlation test of average sinuosity index with SOI also shows significant correlation and therefore it can be justified that El Niño events do have a weak but significant effect on the sinuosity of TC tracks.

5.1 Tropical Cyclone Genesis Position and Sinuosity Index

The correlation of sinuosity index with initial latitudes and longitudes shows negative but significant relationship. The mean values for start latitude decreases as the SI categories get more sinuous. One exception was SI category 5 which did not follow the trend and increased from SI category 4. The mean value for start longitude (except for SI Category 2) also decreased as the SI category got more sinuous. Therefore, TC depressions which are intensified into cyclone intensity at lower latitude and more eastward tend to display straight tracks when compared with the TCs forming in high latitudes and more westward of the study area.

5.2 Tropical Cyclone Decay Position and Sinuosity Index

The correlation of sinuosity index with end longitude was tested to be negatively significant and the mean longitude of the five SI categories also decreased as the SI categories got more sinuous. Therefore TCs which decay further east in the southwest Pacific region follow more straight tracks than TCs decaying in the west. There was no significant correlation of sinuosity index with end latitudes. One reason for this could be because cyclone tracks were cut off at 25°S for the purpose of this study and so the decay position studied may not have been the actual decay position of the TC which may have decayed beyond 25°S.

5.3 Tropical Cyclone Journey and Sinuosity Index

The correlation of sinuosity index with both TC duration and total distance travelled by the TC is positive at 0.01 level. Also, mean for both the parameters increased as the SI categories became more sinuous (exception was SI category 4 for distance travelled which did not follow the increasing trend). It can be concluded that longer lived TCs tend to travel longer distance but are sinuous which means they do not travel far from where they are formed but form loops and curves and finishes closer to the genesis position. The average displacement for the straight moving SI categories increases (SI category 1 – SI category 3) but starts decreasing for the sinuous categories from SI category 3 to SI category 5. TCs tend to travel further away from the genesis positions as TCs tend to recurve from straight track but the trend is reversed when ‘curving cyclone tracks’ become more sinuous and convoluted. One possible reason could be that straight moving TCs are short lived and therefore do not travel long distances and convoluted tracks finish close to the formation point in process of forming loops and therefore not travelling far from the formation point.

5.4 Sinuosity Index Categories

The displacement tracks for the five SI categories in Fig 23 show that straight moving categories (SI category 1 and 2) which comprises 50 % of the TCs are distributed quite evenly across the study area. The two sinuous categories (SI category 4 and 5) comprising of 27.4 % of the total TCs are concentrated in the far west of the study area. SI category 3 somewhat seems to be in the transition from straight and sinuous tracks. The TC displacement tracks in SI category 3 are concentrated between 180° and 170° west. This SI category comprises of 22.6 % of the total 291 cyclones in the study period.

Thus, from this research it was found that 50 % of the TC tracks can be classified as straight moving cyclones during the study period between 1969/70 and 2007/08 and the sinuous tracks accounts for 27.4% while the curving tracks which lays between the straight and sinuous tracks accounts for 22.6%. A cyclone forming in the Southwest Pacific has therefore 50% chances that it will follow a fairly

50 straight line which could mean that having only 22.6% of sinuous tracks. TCs following straight tracks may be under normal circumstances and as the conditions change, tracks become more sinuous.

51 6.0 CONCLUSIONS

This study analyzes TC sinuosity variability from 1970 to 2008 for the Southwest Pacific. Sinuosity index was calculated from sinuosity values in order to reduce the number of outliers from the dataset. Gamma distribution provided the best fit to the sinuosity index data. Five categories were formed using a dynamic programming approach where the optimum boundary points for each group was obtained by executing a computer program coded in C++. The sinuosity index categories were named: straight, near straight, curving, sinuous and convoluted tracks. Three cyclone outliers indentified through boxplot analysis were categorized into an additional outlier category as category 6.

This study shows that average sinuosity index has slightly increased from 1969/70 to 2007/08 suggesting an increase in more sinuous cyclones could occur in the Southwest Pacific in the future. The 1997/98 cyclone season had outstanding values of number of cyclones and sinuosity average. The year also marked a strong El Niño year. Furthermore, trend analysis of average sinuosity index over time (years) and comparison with the Southern Oscillation Index show significant relationship suggesting that climate change may have an effect on the cyclone track. A report on Tropical Cyclone Trends by Australian Government Bureau of Meteorology suggests based on substantial evidence from theory and model experiments that the large-scale environment in which tropical cyclones form and evolve is changing as a result of Greenhouse Warming (Bureau of Meteorology, 2012). Therefore, these changes in the environment may also have an influence on TC tracks.

The 288 TCs were categorized based on their sinuosity index due to the fact that it is well correlated with other parameters and therefore can be used to categorize TCs. Sinuosity Index of the TC tracks studied correlated weakly with other parameters such as: start latitude, start longitude, end longitude, time (year), and Southern Oscillation Index and strongly correlated with duration and distance travelled by the TCs. Sinuous tracks tend to affect larger number of islands and believed to stay longer at places where it curves or make loops and therefore more

52 damages. It is evident from this study that sinuous cyclones are formed at lower latitudes closer to the equator and in the west in the Southwest Pacific. Sinuous tracks have longer lifetimes and travel greater distance.

Three cyclones which were grouped in the outlier category were not included in the categorization process as the sinuosity index values for these cyclones were very large. Several different methods were used to identify the outliers. The dataset represented a population and any analysis should involve the full dataset as to give realistic results. However statistical analyses do have its limitations and therefore three outlier cyclones were grouped in a separate category to avoid extreme skewness in the distribution of the sinuosity index data to be statistically categorized into homogenous groups.

The greatest number of cyclones in the Southwest Pacific occurred between January to March and the higher sinuosity index categories have greater percentage of cyclones occurring in the months of December, January, February and March. However there is no significant trend seen in the number of cyclones per month as the categories become more sinuous.

The only existing studies done on TC tracks based on sinuosity are by Terry and Feng in 2010 and Terry and Gienko in 2011. Apart from these, studies done on TC track were based on other parameters. Utilizing sinuosity index to categorize TC tracks is a convenient method as sinuosity index has significant relationship with many other TC parameters.

The study concludes that SI of TCs has slightly increased over time during the study period. Also based on this study, it is observed that the study area experienced more TCs during El Niño phase. In special report by the IPCC (2012), it is reported that average TC maximum wind speed is likely to increase, although increases may not occur in all ocean basins but it is likely that the global frequency of TCs will either decrease or remain essentially unchanged. The same study also reports that continued use of climate models to make projections of TC behavior includes frequency, location, intensity, rainfall and movement remains a 53 high priority for Pacific Climate Change Science Program (PCCSP) region. Sinuosity of TC can also be included with all these parameters as it is an important parameter to consider for PCCSP region based on the fact that more sinuous TC has potential to affect greater number of Islands. Also sinuosity index categories should be correlated with all the phases of ENSO to study how cyclone track sinuosity may response to the different phases. The recently developed South Pacific Enhanced Archive for TCs dataset by Diamond (2010) which archives information on TC from 1840 – 2009 and the method for categorization developed in this study provide great opportunity to explore TC dataset based on sinuosity to study climatology and long term trends in the Southwest Pacific.

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APPENDICES

APPENDIX 1 – CYCLONE DATASET FOR THE YEAR 1969/70 – 2007/08

Start Start Pressure Start Start Pressure Start Speed End Year End Month End Latitude End Longitude End Pressure End Speed Azimuth Sinuosity (days) duration distance travelled Sinuosity Index Displacement Name Name Start Year Start Month Start Latitude Longitude

PRISCILLA_1970 1970 12 18.1 176.4 990 40 1970 12 23.1 182.8 997 30 130.7729 1.0228 1.5 886 2.8356 866.2495 GILLIAN_1969 1970 4 16.5 182.3 990 40 1970 4 28 195 980 55 136.4688 1.0274 2.5 1874 3.0147 1824.022 EMMA_1969 1970 2 14.7 200 990 40 1970 3 27.1 212.3 980 55 139.0616 1.0374 4.5 1944 3.3442 1873.916 ISA_1969 1970 4 10 163.3 990 40 1970 4 9 154.7 990 40 275.9569 1.0703 4 1018 4.1272 951.1352 ROSIE_1970 1970 12 16.3 164.4 990 40 1971 1 28.8 166.6 990 40 171.1212 1.092 3 1532 4.5144 1402.93 DOLLY_1969 1970 2 14.9 162.4 990 40 1970 2 27.1 207.8 980 55 113.7099 1.2627 12 6163 6.4045 4880.811 HELEN_1969 1970 4 16.1 184.9 990 40 1970 4 17.8 186.9 997 30 131.7372 1.3632 1.5 387 7.1348 283.8908 DAWN_1969 1970 2 12.2 145.9 990 40 1970 2 25 159 990 40 137.6271 1.4799 7 2925 7.8292 1976.485 VIVIENNE_1971 1971 12 18.1 206.1 990 40 1971 12 20 208.5 997 30 130.1645 1 0.5 329 0 329 DORA_1970 1971 2 21.5 156.1 990 40 1971 2 25.1 160.8 990 40 130.5775 1.0017 1 626 1.1935 624.9376 CYC19711104_1971 1971 11 21.9 166.7 990 40 1971 11 25.1 170.6 990 40 132.4223 1.0067 1 537 1.8852 533.426 IDA_1970 1971 2 16.6 156.9 990 40 1971 2 23.3 164.2 997 30 135.3401 1.0126 3.5 1078 2.327 1064.586 URSULA_1971 1971 12 8.4 165 990 40 1971 12 25.2 176.6 980 55 147.8892 1.3266 7 2958 6.8866 2229.76 LENA_1970 1971 3 15.1 157.5 990 40 1971 3 25.2 168.7 997 30 135.4929 1.7217 6.5 2785 8.9699 1617.587 FIONA_1970 1971 2 22.3 158.8 990 40 1971 2 21 161.8 997 30 65.6902 2.241 3 767 10.7463 342.2579 COLLETTE_1972 1972 11 11.1 183.8 990 40 1972 11 15 182.6 990 40 196.6546 1.0013 1 451 1.0914 450.4145 IDA_1971 1972 5 6.7 156.9 980 55 1972 6 27.8 173.4 980 55 145.239 1.0263 4.5 2990 2.9738 2913.378 YOLANDE_1971 1972 3 16.5 174.2 990 40 1972 3 25.4 164.6 980 55 223.748 1.0472 4 1468 3.6139 1401.833 AGATHA_1971 1972 3 16.8 199.7 990 40 1972 3 25.1 200 990 40 178.1075 1.0825 4 995 4.3533 919.1686 WENDY_1971 1972 2 10 176 990 40 1972 3 25.5 159.8 980 55 222.813 1.0955 7.5 2654 4.5709 2422.638 BEBE_1972 1972 10 7.5 180.6 990 40 1972 10 27.5 194 997 30 149.0681 1.2575 7.5 3303 6.362 2626.64 DIANA_1972 1972 12 9.7 167.6 990 40 1972 12 26.8 164.7 997 30 188.7795 1.2694 8.5 2434 6.4585 1917.441 GAIL_1971 1972 4 13.7 155 990 40 1972 4 15.7 169.2 997 30 100.0019 1.4232 8 2199 7.5078 1545.11 CARLOTTA_1971 1972 1 14.6 158 980 55 1972 1 25.9 172.4 970 65 132.0652 2.2568 13 4409 10.7917 1953.651 HENRIETTA_1972 1973 3 12.3 172.7 990 40 1973 3 18.4 183.2 997 30 122.2212 1.0071 1.5 1323 1.922 1313.673 GLENDA_1972 1973 1 18.9 195.1 990 40 1973 2 22.8 198 990 40 145.5527 1.0232 1 539 2.8521 526.7787 JULIETTE_1972 1973 4 15.1 174.1 990 40 1973 4 26.3 192.6 997 30 125.7987 1.0242 2.5 2343 2.8925 2287.639 60

CYC19731106_1973 1973 11 16.7 189.9 990 40 1973 11 26.6 191.8 997 30 170.1672 1.0427 2.5 1161 3.4952 1113.455 LOTTIE_1973 1973 12 15.5 172.3 990 40 1973 12 28 192 990 40 127.5719 1.0574 4.5 2598 3.8575 2456.97 FELICITY_1972 1973 1 16.4 195.8 990 40 1973 1 25.2 202.2 997 30 146.72 1.0872 3 1283 4.4344 1180.096 ELENORE_1972 1973 1 12 184 990 40 1973 2 26.6 185.2 990 40 175.7258 1.3188 5.5 2138 6.8313 1621.171 NESSIE_1973 1974 1 24.9 170 990 40 1974 1 27.2 172.2 990 40 139.6501 1 0.5 337 0 337 VERA_1973 1974 1 17.7 151 990 40 1974 1 26 167 997 30 121.8821 1.0113 2.5 1911 2.244 1889.647 TINA_1973 1974 4 17 179.2 990 40 1974 4 21.7 193.1 997 30 111.8283 1.0148 1.5 1572 2.4552 1549.074 MONICA_1973 1974 1 20 167 990 40 1974 1 25.5 172.4 990 40 138.6924 1.0329 1.5 851 3.2043 823.8939 PAM_1973 1974 1 12.2 178.8 990 40 1974 2 26.3 159.3 955 80 229.7867 1.0681 5 2745 4.0837 2569.984 FLORA_1974 1975 1 15.8 158.7 990 40 1975 1 25.4 183 990 40 116.7763 1.0188 5 2793 2.659 2741.461 GLORIA_1974 1975 1 17.5 148.7 990 40 1975 1 25 168 990 40 115.8422 1.0505 4 2275 3.6963 2165.635 VAL_1974 1975 1 13.8 183.5 980 55 1975 2 25.7 173.8 960 75 216.1232 1.1655 6 1938 5.4903 1662.806 ALISON_1974 1975 3 16.2 172.5 990 40 1975 3 25.5 165.8 970 65 212.9679 1.3581 4.5 1688 7.1012 1242.913 BETTY_1974 1975 5 14 168.6 990 40 1975 5 27 175 997 30 156.1758 2.1121 8 3349 10.3605 1585.626 LAURIE_1976 1976 12 14 188.5 990 40 1976 12 19 201 997 30 114.2156 1.006 1 1453 1.8171 1444.334 JAN_1975 1976 4 19.3 168.5 990 40 1976 4 24.3 174.7 997 30 131.929 1.0073 1 853 1.9399 846.8182 KIM_1976 1976 12 14.5 189 990 40 1976 12 24 211 997 30 117.8374 1.0102 2.5 2562 2.1687 2536.131 DAVID_1975 1976 1 15.4 168 990 40 1976 2 25.5 142.3 997 30 243.161 1.0227 39 2964 2.8314 2898.211 HOPE_1975 1976 3 20 164.5 990 40 1976 3 25.7 157.1 990 40 228.8801 1.0275 2 1014 3.0184 986.8613 FRANCES_1975 1976 2 22 220 990 40 1976 2 25.5 208.2 965 70 249.8168 1.0631 3 1343 3.9812 1263.287 ELSA_1975 1976 1 14.2 167.5 990 40 1976 1 26.5 160 990 40 208.6711 1.205 4.5 1892 5.8964 1570.124 TESSA_1977 1977 12 12.5 212 990 40 1977 12 14.5 217 997 30 112.8031 1.0012 1.5 585 1.0627 584.2988 CYC19770219_1976 1977 2 21 198 990 40 1977 2 27 201 997 30 155.9251 1.0027 1 733 1.3925 731.0262 CYC19770202_1976 1977 2 18.3 176.2 990 40 1977 2 21.2 180.3 997 30 127.4407 1.0038 1.5 538 1.5605 535.9633 STEVE_1977 1977 11 6.7 176 990 40 1977 11 19 175 997 30 184.4577 1.0295 4.5 1405 3.0899 1364.74 PAT_1976 1977 3 19.5 185 990 40 1977 3 28 197 997 30 129.8833 1.031 2 1590 3.1414 1542.192 ANNE_1977 1977 12 13.8 181.5 990 40 1977 12 24 194.5 997 30 131.465 1.1531 5 2044 5.3496 1772.613 MARION_1976 1977 1 15.3 167 990 40 1977 1 23 181 997 30 122.2288 1.1666 6.5 1983 5.5025 1699.811 ROBERT_1976 1977 4 13 205 990 40 1977 4 24 218 980 55 133.4841 1.2744 5.5 2336 6.4982 1833.019 NORMAN_1976 1977 3 12.3 165.8 980 55 1977 3 25.5 171.5 990 40 158.5272 1.7522 9 2767 9.0945 1579.158 JUNE_1976 1977 1 17.5 161 990 40 1977 1 21.5 168 997 30 122.2064 2.7963 4.5 2398 12.1561 857.5618 GWEN_1977 1978 3 21 155.5 990 40 1978 3 25 161 997 30 129.2019 1.0008 1 717 0.9283 716.4269 BOB_1977 1978 2 11 178.3 990 40 1978 2 26.5 165.5 990 40 216.3078 1.0156 5.5 2213 2.4987 2179.007 FAY_1978 1978 12 10 175 990 40 1978 12 26 184 980 55 152.9912 1.0172 3 2044 2.5813 2009.438 HAL_1977 1978 4 13 145 990 40 1978 4 27.5 162 990 40 134.8065 1.1699 6 2794 5.5386 2388.238 61

DIANA_1977 1978 2 14 200 990 40 1978 2 23 207.5 997 30 142.6403 1.2421 5 1580 6.2325 1272.039 CHARLES_1977 1978 2 14.5 194 990 40 1978 2 27 204.3 980 55 143.9321 1.5068 10.5 2635 7.9728 1748.739 ERNIE_1977 1978 2 14 175 990 40 1978 2 24 182 997 30 147.4217 1.5148 4.5 2013 8.0146 1328.888 LESLIE_1978 1979 2 20 187.3 990 40 1979 2 29 194 980 55 147.0751 1.0005 1.5 1206 0.7937 1205.397 OFA_1979 1979 12 14.3 181 995 35 1979 12 22.4 202 997 30 115.1022 1.0095 2.5 2413 2.1179 2390.292 HENRY_1978 1979 1 15.5 169 990 40 1979 2 28 171.5 990 40 169.8612 1.0401 3 1464 3.4228 1407.557 GORDON_1978 1979 1 8.5 172 990 40 1979 1 19.7 152 997 30 237.9748 1.117 7 2777 4.891 2486.124 MELI_1978 1979 3 15.5 184.5 990 40 1979 3 26 176.5 970 65 214.2846 1.3559 4.5 1938 7.0867 1429.309 KERRY_1978 1979 2 7.8 166 990 40 1979 3 16.5 147.6 997 30 242.5894 1.7708 17.5 3930 9.1688 2219.336 RAE_1979 1980 2 14.9 170.2 990 40 1980 2 16.9 172.4 997 30 133.5067 1 0.5 323 0 323 SINA_1979 1980 3 17.5 159.7 995 35 1980 3 25.5 167.5 970 65 138.9891 1.0106 2 1211 2.1967 1198.298 TIA_1979 1980 3 15 176.8 990 40 1980 3 28.8 200 997 30 126.504 1.0254 3.5 2906 2.9395 2834.016 PENI_1979 1980 1 12 173.5 990 40 1980 1 19.2 178 997 30 149.3722 1.0271 2 957 3.0037 931.7496 VAL_1979 1980 3 13 180.5 990 40 1980 3 17.1 188.5 997 30 118.804 1.0277 1.5 999 3.0257 972.0736 DIOLA_1980 1980 11 18.1 220.7 995 35 1980 11 24 216 997 30 215.9713 1.0766 2 878 4.2469 815.5304 WALLY_1979 1980 4 15.3 178.5 995 35 1980 4 17 177.8 997 30 201.6049 1.0779 1.5 218 4.2708 202.2451 CYC1981_23RD_1980 1981 3 22 187.4 995 35 1981 3 26 192.5 995 35 131.5026 1.0107 1 689 2.2036 681.7057 FRAN_1980 1981 3 14.9 201.6 995 35 1981 3 27.1 214 990 40 138.4119 1.0142 4 1891 2.4216 1864.524 CYC1981_16TH_1980 1981 2 19.5 197 990 40 1981 2 28 202 987 45 152.5398 1.0227 1 1095 2.8314 1070.695 ESAU_1980 1981 3 11.2 179.6 995 35 1981 3 23 198 997 30 126.1642 1.0575 3 2485 3.8597 2349.882 CLIFF_1980 1981 2 14.5 168 990 40 1981 2 25.6 156.4 987 45 222.7792 1.1026 3.5 1902 4.6815 1725.014 TAHMAR_1980 1981 3 20 205 990 40 1981 3 25.9 217 980 55 120.2162 1.1095 2.5 1545 4.7841 1392.519 DAMAN_1980 1981 2 17.1 193 990 40 1981 2 25.5 201.4 987 45 138.3143 1.1395 1.5 1452 5.1863 1274.243 BETSY_1980 1981 1 17 189.8 995 35 1981 2 20.5 186.8 997 30 218.7619 1.1706 2.5 585 5.5462 499.7437 ARTHUR_1980 1981 1 13.5 179 990 40 1981 1 26 178 990 40 184.17 1.1712 4 1625 5.5527 1387.466 FREDA_1980 1981 2 15.5 144.6 995 35 1981 3 27.5 167 975 60 123.6007 1.2962 9.5 3457 6.6659 2667.027 GYAN_1981 1981 12 11.6 168.4 990 40 1981 12 22.6 165.3 997 30 194.7311 1.4907 6.5 1880 7.8875 1261.152 CYC1981_18TH_1980 1981 2 12 188 995 35 1981 3 22.5 193.5 997 30 154.0532 1.8606 5.5 2420 9.5119 1300.656 JOTI_1982 1982 11 10 170.9 995 35 1982 11 16.7 164 990 40 224.4919 1.1111 5 1169 4.8073 1052.111 ISAAC_1981 1982 2 13.5 190.2 995 35 1982 3 25.5 184.8 960 75 202.2547 1.1137 3.5 1608 4.8446 1443.836 BERNIE_1981 1982 4 7.8 158 990 40 1982 4 25.5 164.6 975 60 161.0956 1.1243 5.5 2339 4.9906 2080.406 KINA_1982 1982 11 12.1 172.1 995 35 1982 11 17 171.5 997 30 186.7282 1.1407 1.5 623 5.2011 546.1559 HETTIE_1981 1982 1 18.1 172 995 35 1982 2 27.4 178 980 55 150.2194 1.2461 6.5 1495 6.2667 1199.743 LISA_1982 1982 12 14.7 206 990 40 1982 12 23.9 205.9 997 30 180.5753 1.3214 4 1346 6.8499 1018.617 CLAUDIA_1981 1982 5 13 156.5 995 35 1982 5 11.7 161.3 997 30 75.1209 1.3736 3 744 7.2023 541.6424 62

ABIGAIL_1981 1982 1 17.8 154.4 995 35 1982 2 22.6 174.5 997 30 107.556 1.8495 12 4004 9.4708 2164.909 PREMA_1982 1983 2 12.4 197.6 990 40 1983 2 14 207 997 30 100.9087 1.0076 2 1042 1.9661 1034.141 SABA_1982 1983 3 15.8 223.8 990 40 1983 3 26.1 231 995 35 147.9336 1.011 3 1378 2.224 1363.007 SARAH_1982 1983 3 13.2 177.5 995 35 1983 3 26.2 181.1 985 50 165.8808 1.1358 5 1689 5.14 1487.058 ATU_1983 1983 12 15.8 170.3 995 35 1983 12 21.3 173.6 997 30 150.7282 1.1475 2.5 805 5.2836 701.5251 NANO_1982 1983 1 13.4 220.4 990 40 1983 1 27 235 990 40 136.9455 1.171 4.5 2506 5.5505 2140.051 TOMASI_1982 1983 3 11.5 199.7 990 40 1983 4 26.3 193.9 990 40 199.5744 1.181 5.5 2064 5.6567 1747.671 OSCAR_1982 1983 2 13.5 173.5 990 40 1983 3 27 183 990 40 147.9494 1.1862 7 2126 5.7103 1792.278 NISHA_1982 1983 2 13.8 216.8 995 35 1983 2 24 219 990 40 168.7409 1.4288 5.5 1647 7.5408 1152.716 REWA_1982 1983 3 11.7 212.7 995 35 1983 3 26 223 995 35 147.1001 1.4519 8 2784 7.6739 1917.487 WILLIAM_1982 1983 4 10.9 227.3 995 35 1983 4 25 237 997 30 148.0185 1.6425 7 3066 8.6289 1866.667 MARK_1982 1983 1 12 174 990 40 1983 1 19 175.1 997 30 171.478 1.6624 5.5 1303 8.7171 783.8065 VEENA_1982 1983 4 12.2 221.5 990 40 1983 4 25 218.8 987 45 190.9633 1.7939 6 2592 9.2595 1444.897 NAMELESSA_1983 1984 2 25 175 990 40 1984 2 26.6 175 987 45 180 1 0.5 177 0 177 BETI_1983 1984 2 16.3 161.2 995 35 1984 2 22.2 172.4 997 30 120.7807 1.0105 3.5 1360 2.1898 1345.868 CYRIL_1983 1984 3 17.9 175.7 990 40 1984 3 25 186.4 997 30 127.1981 1.0164 2.5 1381 2.5407 1358.717 HARVEY_1983 1984 2 16.3 154.7 990 40 1984 2 21 163.4 997 30 120.8825 1.0398 4.5 1097 3.4142 1055.011 MONICA_1984 1984 12 12 146 990 40 1984 12 28 163 990 40 137.5274 1.0759 4 2694 4.234 2503.95 CYC_1984_58TH_1984 1984 12 8.4 178.4 995 35 1984 12 10.1 180 997 30 137.0486 1.0781 1.5 278 4.2745 257.8611 GRACE_1983 1984 1 15 150.5 995 35 1984 1 24.4 160.3 997 30 136.9228 1.1072 5.5 1618 4.7504 1461.344 NAMELESSB_1983 1984 3 14.9 175.1 990 40 1984 3 17.4 187 997 30 103.8803 1.796 7 2339 9.2677 1302.339 GAVIN_1984 1985 3 15.9 170.5 995 35 1985 3 27.4 184.2 990 40 134.288 1.0053 4.5 1913 1.7435 1902.915 ERIC_1984 1985 1 15.6 165.5 990 40 1985 1 25.5 198 987 45 113.3105 1.0136 4 3599 2.387 3550.71 ODETTE_1984 1985 1 14.8 150.5 987 45 1985 1 21 173.5 997 30 109.0992 1.0136 4 2563 2.387 2528.611 FREDA_1984 1985 1 19.1 199 985 50 1985 1 26 188.7 960 75 232.3068 1.0177 2 1328 2.6061 1304.903 NIGEL_1984 1985 1 16.2 156 990 40 1985 1 21.4 194.9 997 30 104.1809 1.0224 6.5 4223 2.8189 4130.477 DRENA_1984 1985 1 12.1 185 995 35 1985 1 18.7 188 997 30 156.5868 1.0678 3 852 4.0776 797.9022 HINA_1984 1985 3 13.9 165.9 990 40 1985 3 29.8 182 955 80 139.3741 1.3222 5 3196 6.8555 2417.183 MARTIN_1985 1986 4 12.9 171.4 995 35 1986 4 19.9 185.9 997 30 118.495 1.0068 3 1742 1.8945 1730.234 ALFRED_1985 1986 3 16.5 154.6 995 35 1986 3 21.5 171.3 997 30 110.0882 1.0074 2.25 1856 1.9487 1842.366 LUSI_1985 1986 3 18.4 161 995 35 1986 3 24.4 178.6 997 30 113.0864 1.0231 3.75 1985 2.848 1940.182 KELI_1985 1986 2 19 168.1 995 35 1986 2 24.7 189.8 997 30 109.6145 1.0353 3.25 2409 3.2804 2326.862 OSEA_1986 1986 11 13.1 168.2 995 35 1986 11 17.3 174.5 997 30 125.2649 1.042 2 855 3.476 820.5374 JUNE_1985 1986 2 21.5 219.9 995 35 1986 2 25.4 225.5 980 55 148.9139 1.0602 1.25 538 3.9192 507.4514 SALLY_1986 1986 12 13.3 195.8 995 35 1987 1 25.1 206.5 980 55 140.8973 1.2731 8.25 2193 6.4879 1722.567 63

PATSY_1986 1986 12 11.5 170 995 35 1986 12 25.8 165.6 990 40 195.6838 1.2994 6 2143 6.6899 1649.223 RAJA_1986 1986 12 11.4 177.5 995 35 1987 1 25.2 181.7 980 55 164.4117 1.4252 9.25 2267 7.5197 1590.654 NAMU_1985 1986 5 8.3 163.1 990 40 1986 5 18.6 163.1 997 30 180 1.4563 4.5 1660 7.6987 1139.875 IMA_1985 1986 2 17.5 192.2 990 40 1986 2 26.2 206.1 975 60 126.2895 1.7814 8.25 3078 9.2107 1727.854 NONAME_1986 1987 2 16.6 198.6 995 35 1987 3 25.5 204.4 990 40 149.5418 1.0021 1.5 1157 1.2806 1154.575 UMA_1986 1987 2 13.2 162.6 995 35 1987 2 21.5 174 995 35 128.7609 1.0217 3.75 1552 2.7892 1519.037 BLANCH(E)_1986 1987 5 11.7 160.7 995 35 1987 5 16.3 157 997 30 217.7153 1.0734 2.75 695 4.187 647.4753 ZUMAN_1986 1987 4 11.5 186.4 995 35 1987 4 23.8 199.2 997 30 136.8506 1.0856 3.5 2085 4.4072 1920.597 VELI_1986 1987 2 14 155.3 995 35 1987 2 24.5 180 997 30 117.896 1.0866 4.25 3084 4.4242 2838.211 YALI_1986 1987 3 16.6 163.7 995 35 1987 3 18.7 164.2 997 30 167.2162 1.0959 2.25 261 4.5773 238.1604 TUSI_1986 1987 1 9.1 187.9 995 35 1987 1 25.5 199.4 990 40 147.5901 1.1054 6.5 2416 4.7237 2185.634 WINI_1986 1987 2 12.5 180.2 995 35 1987 3 25.1 198.3 965 70 128.8388 1.1465 4.75 2704 5.2716 2358.482 CILLA_1987 1988 2 18.2 200.6 995 35 1988 3 26.5 211.5 980 55 131.2779 1.0114 2 1466 2.2506 1449.476 DELILAH_1988 1988 12 17.6 154.4 995 35 1989 1 25.1 170 985 50 119.8669 1.0652 2.75 1935 4.0248 1816.56 ESETA_1988 1988 12 19.1 171.5 995 35 1988 12 25.3 174.7 985 50 154.9188 1.1104 2.5 846 4.7972 761.8876 AGI_1987 1988 1 11.3 153.9 995 35 1988 1 19.2 162.3 997 30 135.1254 1.163 3.25 1460 5.4626 1255.374 ANNE_1987 1988 1 6.1 178.8 995 35 1988 1 24.7 165.5 990 40 213.1885 1.1974 6.75 2994 5.8226 2500.418 DOVI_1987 1988 4 17.5 169.8 995 35 1988 4 25.9 174.3 980 55 154.2081 1.2677 5.75 1318 6.4449 1039.678 BOLA_1987 1988 2 15.1 177.5 995 35 1988 3 27 178.5 970 65 175.6621 2.4403 8.5 3225 11.2932 1321.559 FILI_1988 1989 1 18.8 189 995 35 1989 1 25.6 195.4 985 50 139.9337 1.001 2.25 1002 1 1000.999 MEENA_1988 1989 5 13.4 160.6 995 35 1989 5 12.2 143.7 997 30 272.2228 1.0786 4.25 1984 4.2836 1839.421 LILI_1988 1989 4 12.5 162.5 995 35 1989 4 24.8 169.1 997 30 153.9041 1.09 4.75 1666 4.4814 1528.44 UNNAMED_1988 1989 2 21.4 180.7 995 35 1989 2 26.8 191.1 990 40 121.548 1.0957 2 1330 4.5741 1213.836 FELICITY_1989 1989 12 15.8 139.2 990 40 1989 12 22.2 161 997 30 110.5407 1.15 5 2759 5.3133 2399.13 GINA_1988 1989 1 14.6 187.3 995 35 1989 1 19.8 187.1 997 30 182.0882 1.1723 2.25 675 5.5645 575.7912 JUDY_1988 1989 2 19 208 995 35 1989 2 26.7 199.1 990 40 225.3136 1.2226 4.75 1527 6.0605 1248.978 KERRY_1988 1989 3 20.1 178.8 995 35 1989 4 25.7 187.7 997 30 125.8554 1.3287 3.25 1466 6.9013 1103.334 IVY_1988 1989 2 17.2 167.7 995 35 1989 3 23.6 169.7 995 35 163.9229 2.227 7.75 1645 10.7057 738.6619 HARRY_1988 1989 2 17.7 161.4 995 35 1989 2 25.8 165.2 987 45 157.0148 2.7648 10.75 2707 12.0846 979.0943 RAE_1989 1990 3 20 173 990 40 1990 3 25.1 188.2 990 40 112.6793 1.0434 2.25 1733 3.5142 1660.916 PENI_1989 1990 2 10.2 199 990 40 1990 2 25.5 207.1 970 65 154.2637 1.1347 4.25 2153 5.1261 1897.418 OFA_1989 1990 1 8 180.2 995 35 1990 2 25.8 190.3 970 65 152.686 1.1386 7.5 2553 5.1751 2242.227 SINA_1990 1990 11 10.3 173.8 995 35 1990 12 25.4 200.5 988 40 124.2306 1.1836 6.25 3877 5.6836 3275.6 NANCY_1989 1990 1 15.3 158.5 995 35 1990 2 25.2 154 980 55 202.4722 1.345 2 1604 7.0136 1192.565 HILDA_1989 1990 3 19.4 153.2 995 35 1990 3 26 165 985 50 123.2666 1.385 3.5 1959 7.2748 1414.44 64

ARTHUR_1991 1991 12 22.7 217.5 995 35 1991 12 18 228.1 997 30 66.7391 1.031 4.25 1260 3.1414 1222.114 VAL_1991 1991 12 9.5 181.9 995 35 1991 12 25.5 197.5 975 60 139.0197 1.1915 8 2883 5.764 2419.639 LISA_1990 1991 5 9 155.3 995 35 1991 5 20 169.8 996 30 129.5484 1.207 4.75 2388 5.9155 1978.459 WASA_1991 1991 12 11 201 995 35 1991 12 23.4 215.7 995 35 133.2574 1.2302 7.75 2555 6.1287 2076.898 TIA_1991 1991 11 8.6 170.1 995 35 1991 11 16.4 171.4 997 30 170.8335 1.6517 4.5 1444 8.6699 874.2508 HETTIE_1991 1992 3 14 210 995 35 1992 3 26 218.5 990 40 147.5626 1.0374 2.75 1657 3.3442 1597.262 CLIFF_1991 1992 2 11.3 216 995 35 1992 2 25.6 226 995 40 147.7456 1.0658 3 2026 4.0372 1900.919 DAMAN_1991 1992 2 12.6 170 995 35 1992 2 26 158.5 975 60 217.4124 1.0684 3 2042 4.0896 1911.269 KINA_1992 1992 12 11.6 170.6 995 35 1993 1 25 200 997 30 119.7067 1.0712 9.5 3677 4.1447 3432.599 NINA_1992 1992 12 14.6 150 985 50 1993 1 17.2 191 997 30 99.4384 1.0744 6.25 4719 4.2059 4392.219 GENE_1991 1992 3 14.5 194 995 35 1992 3 26 197.8 985 50 163.3085 1.1054 2.75 1474 4.7237 1333.454 FRAN_1991 1992 3 13.5 184 990 35 1992 3 25.3 153.1 990 40 243.3248 1.1623 11 4053 5.4547 3487.052 JONI_1992 1992 12 10.2 180 995 35 1992 12 27 185.5 985 50 163.4815 1.207 6.5 2350 5.9155 1946.976 INNIS_1991 1992 4 11.7 171.5 995 35 1992 5 26 181 990 40 149.1114 1.2322 3.5 2306 6.1464 1871.449 BETSY_1991 1992 1 9.5 169.6 995 35 1992 1 25.3 157.9 975 60 213.8177 1.2924 7 2769 6.6373 2142.526 ESAU_1991 1992 2 15.5 167.3 995 35 1992 3 26.5 165.8 975 60 187.0384 2.2121 8.5 2716 10.6622 1227.793 NISHA_1992 1993 2 17.5 196.6 995 35 1993 2 26 210 980 55 126.53 1.0071 2.5 1686 1.922 1674.114 OLI_1992 1993 2 16.8 176.7 995 35 1993 2 25.5 182.4 990 40 149.4112 1.0118 1.75 1144 2.2766 1130.658 POLLY_1992 1993 2 16.4 158.2 990 40 1993 3 25.9 167.9 945 85 137.8921 1.0754 5 1565 4.2246 1455.272 MICK_1992 1993 2 18.2 186.7 990 40 1993 2 25.3 178.7 990 40 225.0461 1.0802 2.75 1232 4.3125 1140.53 LIN_1992 1993 1 13.2 186.6 995 35 1993 2 26 193.2 995 35 155.0015 1.0978 4.5 1730 4.6073 1575.879 PREMA_1992 1993 3 13.7 171.8 995 35 1993 4 25.4 177.4 975 60 156.4832 1.434 5 2039 7.5712 1421.897 ROGER_1992 1993 3 12 155.8 990 40 1993 3 25.5 172.5 995 35 132.7712 1.668 9.75 3845 8.7416 2305.156 REWA_1993 1993 12 10.3 164.5 995 35 1994 1 25.3 155.1 987 45 209.6523 4.3564 23 8427 14.9724 1934.395 THEODORE_1993 1994 2 10.5 154.5 995 35 1994 2 25.8 171.4 968 65 135.8211 1.0533 4 2588 3.7634 2457.04 TOMAS_1993 1994 3 12.5 171.4 995 35 1994 3 25.4 189.5 987 45 129.5272 1.1555 4 2746 5.3775 2376.46 USHA_1993 1994 3 12.4 160.5 995 35 1994 3 25.5 167.4 997 30 154.4336 1.3166 4.25 2134 6.8156 1620.842 VANIA_1994 1994 11 12.5 169.2 995 35 1994 11 19 166.2 997 30 203.692 1.3576 3.75 1069 7.0979 787.419 SARAH_1993 1994 1 15 164 995 35 1994 1 25.4 177.1 975 60 132.1799 1.404 8 2508 7.3925 1786.325 WILLIAM_1994 1995 1 16 198 990 35 1995 1 27.5 214 987 45 130.3433 1.017 2.75 2120 2.5713 2084.562 VIOLET_1994 1995 3 15.8 152 995 35 1995 3 25.5 160.5 965 70 141.9053 1.0193 2.75 1418 2.6824 1391.151 ZAKA_1995 1996 3 22.2 169.5 992 35 1996 3 24 174 996 30 114.2554 1 0.5 502 0 502 CELESTE_1995 1996 1 19.5 148 990 40 1996 1 16.9 162.5 997 30 81.7068 1.0188 3.5 1589 2.659 1559.678 YASI_1995 1996 1 22.3 187.6 996 35 1996 1 26.5 198.5 996 30 115.0115 1.0666 2.25 1279 4.0534 1199.137 CYRIL_1996 1996 11 14.8 160.5 995 35 1996 11 19.8 162.1 997 30 163.1447 1.1119 2.5 644 4.8188 579.1888 65

ATU_1995 1996 3 21.5 168.5 996 35 1996 3 25.2 172.5 997 30 135.829 1.2021 2.5 696 5.8684 578.9868 BETI_1995 1996 3 13 169 996 35 1996 3 25.5 168.1 980 55 183.7703 1.2805 5.75 1776 6.546 1386.958 FERGUS_1996 1996 12 12.8 160 995 35 1996 12 26 173.8 965 70 137.3458 1.4113 5.5 2900 7.4368 2054.843 HINA_1996 1997 3 12.8 180.7 990 40 1997 3 25.2 188.5 970 65 150.2978 1.0134 2 1620 2.3752 1598.579 IAN_1996 1997 4 20.1 176.1 995 35 1997 4 23 187 997 30 107.8358 1.0159 1.75 1192 2.5146 1173.344 LUSI_1997 1997 10 8.6 169.6 995 35 1997 10 23.8 178.2 997 30 152.4324 1.021 2.75 1956 2.7589 1915.769 MARTIN_1997 1997 10 10.1 194.2 992 35 1997 11 26 218.7 995 35 127.5473 1.0301 4.88 3218 3.1107 3123.969 FREDA_1996 1997 1 22 175.9 995 35 1997 1 25.5 177.7 980 55 155.0251 1.0314 1.5 442 3.1548 428.5437 NUTE_1997 1997 11 12 163.4 995 35 1997 11 20.6 158.5 1000 25 208.1692 1.0651 2.5 1157 4.023 1086.283 OSEA_1997 1997 11 12.3 202.1 995 35 1997 11 21.4 212 998 30 135.0092 1.0669 4.25 1555 4.0595 1457.494 PAM_1997 1997 12 11.5 197.3 995 35 1997 12 24.9 204.6 997 30 153.5606 1.1067 4.25 1849 4.743 1670.733 EVAN_1996 1997 1 13.6 190.5 995 35 1997 1 27.2 192.2 975 60 173.5612 1.1257 3 1707 5.0093 1516.39 HAROLD_1996 1997 2 14.8 156.8 995 35 1997 2 26.5 164.5 994 35 149.5074 1.1905 4.25 1813 5.7539 1522.89 JUNE_1996 1997 5 14 174.5 990 40 1997 5 17.6 176.9 996 30 147.4774 1.1961 2.25 567 5.8098 474.0406 GAVIN_1996 1997 3 9.6 173.7 995 35 1997 3 26.2 176.8 940 85 170.3084 1.3388 5.71 2498 6.9713 1865.85 KELI_1996 1997 6 8.6 183.5 995 35 1997 6 20 202 997 30 124.3364 1.4977 5.5 3531 7.9248 2357.615 DRENA_1996 1997 1 14.7 164.1 995 35 1997 1 25.5 167.5 970 65 163.9975 1.7066 5.25 2128 8.9069 1246.924 BART_1997 1998 4 17.2 220.2 995 35 1998 5 19.9 224.6 997 30 123.4411 1.0188 1.5 563 2.659 552.6109 URSULA_1997 1998 1 14.1 208 999 35 1998 2 25.2 224.1 975 60 128.5578 1.0415 2.5 2171 3.4622 2084.494 VELI_1997 1998 2 13.7 207.2 995 35 1998 2 23.2 216.7 997 30 137.7517 1.0632 2.75 1544 3.9833 1452.22 WES_1997 1998 2 11.7 168.4 995 35 1998 2 17.3 158.5 997 30 238.6887 1.0671 3.5 1316 4.0636 1233.249 CORA_1998 1998 12 15.2 181.8 994 35 1998 12 25.2 196.9 970 65 127.4894 1.0707 3.75 2061 4.135 1924.909 SUSAN_1997 1998 1 12.4 172.9 987 45 1998 1 26.4 183.6 940 90 145.6993 1.251 5.5 2392 6.308 1912.07 TUI_1997 1998 1 13.3 187.5 995 35 1998 1 14.6 187.7 995 35 171.4788 1.2537 1.25 182 6.3305 145.1703 RON_1997 1998 1 9.6 192.3 990 40 1998 1 28.2 191.5 975 60 182.223 1.4027 6.75 2891 7.3846 2061.025 ALAN_1997 1998 4 11.8 201.4 995 35 1998 4 16.4 208 997 30 126.2986 1.5436 4.75 1352 8.1613 875.8746 YALI_1997 1998 3 13.3 163.7 995 35 1998 3 25.1 162.1 990 40 187.0889 1.5707 5.25 2069 8.2947 1317.247 ZUMAN_1997 1998 3 13.9 170.2 996 35 1998 4 27 171 1000 25 176.8394 1.8265 7 2653 9.3846 1452.505 KATRINA_1997 1998 1 16.9 152.3 995 35 1998 1 17.9 152.5 997 30 169.1602 52.7437 21.5 5944 37.2637 112.6959 GITA_1998 1999 2 24.5 204 995 35 1999 2 25.5 204.5 990 40 155.6083 1 0.25 122 0 122 ELLA_1998 1999 2 11 163 995 35 1999 2 25 170 999 30 155.4413 1.0202 2.25 1751 2.7234 1716.33 OLINDA_1998 1999 1 17.2 158.3 995 35 1999 1 25.4 168.5 990 50 132.3873 1.0524 2.88 1466 3.7421 1393.006 26F_1998 1999 5 20 162.5 992 40 1999 5 25.5 162 992 40 184.7246 1.0548 1.75 645 3.7983 611.4903 PETE_1998 1999 1 15 151.8 994 35 1999 1 23.8 169.6 997 30 120.294 1.1055 5.25 2327 4.7252 2104.93 FRANK_1998 1999 2 20.2 160 995 35 1999 2 25.2 164.8 987 45 139.21 1.4516 2.75 1076 7.6722 741.251 66

DANI_1998 1999 1 15.9 164.9 995 35 1999 1 26.2 172.5 975 60 146.5885 1.857 7 2574 9.4986 1386.107 HALI_1998 1999 3 20.2 199.8 995 35 1999 3 24.6 198.9 997 30 190.6002 2.332 5.5 1157 11.0028 496.1407 LEO_1999 2000 3 24.7 196.6 995 35 2000 3 25.6 195.4 990 40 230.2579 1 0.25 157 0 157 IRIS_1999 2000 1 15.5 164.3 996 35 2000 1 19.4 177.7 998 30 108.8114 1.0279 3.25 1528 3.033 1486.526 JO_1999 2000 1 17.9 173.1 995 35 2000 1 25.1 179 975 60 143.5582 1.032 2.5 1036 3.1748 1003.876 KIM_1999 2000 2 23.2 224.4 994 35 2000 2 25.7 220.1 935 90 236.7032 1.0443 1.75 539 3.5384 516.1352 NEIL_1999 2000 4 20 178.4 995 35 2000 4 22.7 179.4 997 30 161.0448 1.0722 1 339 4.164 316.1724 MONA_1999 2000 3 18.8 185.5 995 35 2000 3 25.5 187.8 960 75 162.6893 1.2305 2.5 958 6.1314 778.5453 OMA_2000 2001 2 21.6 196.5 990 40 2001 2 26 202.8 987 45 128.4387 1.0057 1 810 1.7863 805.4092 VICKY_2001 2001 12 12.6 202.5 996 35 2001 12 13.6 202.7 997 30 168.9309 1.055 0.5 119 3.803 112.7962 PAULA_2000 2001 2 12.2 164.9 997 35 2001 3 25.6 185.3 970 65 127.6244 1.0832 5 2822 4.3656 2605.244 WAKA_2001 2001 12 11.3 185.5 995 35 2002 1 25.7 191.4 960 75 159.5265 1.1138 3.25 1905 4.846 1710.361 SOSE_2000 2001 4 14 165.5 995 35 2001 4 25.5 169.7 990 40 161.602 1.1693 5 1575 5.532 1346.96 RITA_2000 2001 2 19.3 223.7 998 35 2001 3 25.1 223.6 990 40 180.9013 1.3266 3.17 852 6.8866 642.2433 TRINA_2001 2001 11 21.5 201 995 35 2001 12 21.4 201.3 996 30 70.4561 4.5039 1.25 149 15.1886 33.08244 YOLANDE_2002 2002 12 20.4 185.8 995 35 2002 12 21.7 187.9 996 30 123.7762 1.0002 0.5 261 0.5848 260.9478 DES_2001 2002 3 19.4 159.5 993 40 2002 3 24.4 168.1 997 30 123.4744 1.0439 1.88 1093 3.5277 1047.035 CLAUDIA_2001 2002 2 20.5 156.5 995 35 2002 2 25.1 162 970 65 133.091 1.0508 1.25 799 3.7036 760.373 ZOE_2002 2002 12 10.8 175.5 995 35 2003 1 20.3 175.1 997 30 182.2862 1.7724 6.13 1865 9.1752 1052.246 FILI_2002 2003 4 20.4 188.4 995 35 2003 4 27 190 985 50 167.7308 1.0001 0.38 749 0.4642 748.9251 CILLA_2002 2003 1 18 182 995 35 2003 1 25 194.7 1002 35 122.7182 1.0977 3.5 1675 4.6057 1525.918 AMI_2002 2003 1 10.8 180.6 995 35 2003 1 26.8 190.2 970 70 151.6846 1.1134 3 2269 4.8403 2037.902 ESETA_2002 2003 3 15.5 172.4 997 35 2003 3 25.3 194.7 965 70 118.6301 1.1192 4 2868 4.9214 2562.545 DOVI_2002 2003 2 14 197.3 995 35 2003 2 26 191 986 50 205.3735 1.1346 5.04 1682 5.1249 1482.461 GINA_2002 2003 6 11.3 169.1 995 35 2003 6 16.5 162 997 30 232.33 1.3404 4 1285 6.9823 958.6691 ERICA_2002 2003 3 19.7 149.7 1001 40 2003 3 26.5 174 960 75 111.4187 1.7333 12 4499 9.0177 2595.627 BENI_2002 2003 1 13.2 161.2 992 35 2003 1 24.3 163.5 997 30 169.187 1.8389 6.75 2303 9.4313 1252.379 JUDY_2004 2004 12 19.5 214.7 995 35 2004 12 27 212.7 993 40 193.4572 1.0205 2 873 2.7369 855.463 GRACE_2003 2004 3 16.5 148.4 993 35 2004 3 22 166 997 30 110.997 1.0956 4.25 2132 4.5725 1945.966 HETA_2003 2004 1 8 185.8 995 35 2004 1 25.6 195.6 955 80 153.0796 1.114 5.25 2459 4.8488 2207.361 IVY_2003 2004 2 15 172.5 995 35 2004 2 26.6 171.8 950 80 183.1282 1.583 4.88 2036 8.3539 1286.166 SHEILA_2004 2005 4 17.4 189.4 995 35 2005 4 20.9 195.2 997 30 123.341 1.0078 0.75 728 1.9832 722.3655 RAE_2004 2005 3 20.5 195.3 995 35 2005 3 22.9 198.7 997 30 127.6835 1.0797 0.75 476 4.3035 440.8632 OLAF_2004 2005 2 9 182.4 995 35 2005 2 26.6 199.2 965 70 139.8598 1.1305 5.75 2977 5.0723 2633.348 MEENA_2004 2005 2 14.4 191.8 995 35 2005 2 25.4 205.5 950 80 132.5007 1.1496 4.25 2160 5.3086 1878.914 67

PERCY_2004 2005 2 8.2 180.7 995 35 2005 3 25.2 204.5 980 50 129.5516 1.1759 8 3703 5.603 3149.077 KERRY_2004 2005 1 13.3 171.6 995 35 2005 1 25 158.2 997 30 225.3855 1.2713 8.5 2430 6.4737 1911.429 NANCY_2004 2005 2 12.8 194.2 995 35 2005 2 25.1 195.2 994 35 175.7315 1.5725 4.75 2147 8.3034 1365.342 LOLA_2004 2005 1 22.6 183.8 995 35 2005 2 24.8 184 998 35 175.2547 1.6902 1.5 413 8.8374 244.3498 URMIL_2005 2006 1 14.6 185.6 995 35 2006 1 25.3 189.8 989 40 160.3219 1.0387 1.5 1312 3.3825 1263.117 TAM_2005 2006 1 14.5 181.5 995 35 2006 1 27 192 988 45 143.4297 1.0858 2 1913 4.4106 1761.835 JIM_2005 2006 1 18.1 148.4 995 35 2006 2 26.4 173.3 987 45 114.1889 1.1566 4.75 3146 5.3901 2720.042 VAIANU_2005 2006 2 17.4 185.1 995 35 2006 2 25.1 186.8 980 55 168.6061 1.2982 3.5 1130 6.6809 870.436 XAVIER_2006 2006 10 10.5 167.8 995 35 2006 10 15.2 170.6 987 45 149.9916 1.7537 4 1056 9.1005 602.1554 WATI_2005 2006 3 15.7 164.5 995 35 2006 3 25.1 161.6 965 75 195.7311 2.2211 5.5 2407 10.6885 1083.697 YANI_2006 2006 11 12.3 162.4 995 35 2006 11 13.5 162 987 45 198.0635 2.6888 2.5 376 11.9086 139.8393 CLIFF_2006 2007 4 17.2 180.6 995 35 2007 4 25.2 186.2 985 50 147.691 1.019 1.75 1079 2.6684 1058.881 ZITA_2006 2007 1 14.2 202.7 995 35 2007 1 25.4 209.9 990 45 149.8348 1.0366 1.75 1504 3.3202 1450.897 ARTHUR_2006 2007 1 14.5 192.5 995 35 2007 1 26.3 211.5 980 55 126.441 1.0712 2.75 2539 4.1447 2370.239 BECKY_2006 2007 3 13.1 163 995 35 2007 3 20.8 167.6 995 35 150.7431 1.1126 2.5 1093 4.8289 982.3836 DAMAN_2007 2007 12 12.1 177.7 995 35 2007 12 18.5 181.9 997 30 148.0349 1.586 4.17 1331 8.3682 839.2182 ELISA_2007 2008 1 21.4 184.4 994 35 2008 1 24.8 191.4 997 30 119.0445 1.1001 1.75 891 4.6431 809.9264 FUNA_2007 2008 1 14.8 164.8 990 35 2008 1 25.2 172.4 945 85 146.5953 1.4713 3.5 2057 7.7821 1398.083 GENE_2007 2008 1 17.4 178.4 995 35 2008 2 25.2 173 970 65 212.0305 1.5374 6.5 1582 8.1302 1029.01

APPENDIX 2 – SOURTHERN OSCILLATION INDEX (S. O. I) ARCHIVES 1969 - 2008.

Year Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 1969 -13.5 -6.9 1.8 -8.8 -6.6 -0.6 -6.9 -4.4 -10.6 -11.7 -0.1 3.7 1970 -10.1 -10.7 1.8 -4.6 2.1 9.9 -5.6 4 12.9 10.3 19.7 17.4 1971 2.7 15.7 19.2 22.6 9.2 2.6 1.6 14.9 15.9 17.7 7.2 2.1 1972 3.7 8.2 2.4 -5.5 -16.1 -12 -18.6 -8.9 -14.8 -11.1 -3.4 -12.1 1973 -3 -13.5 0.8 -2.1 2.8 12.3 6.1 12.3 13.5 9.7 31.6 16.9 1974 20.8 16.2 20.3 11.1 10.7 2.6 12 6.6 12.3 8.5 -1.4 -0.9 1975 -4.9 5.3 11.6 14.4 6 15.5 21.1 20.7 22.5 17.7 13.8 19.5 1976 11.8 12.9 13.2 1.2 2.1 0.2 -12.8 -12.1 -13 3 9.8 -3 1977 -4 7.7 -9.5 -9.6 -11.4 -17.7 -14.7 -12.1 -9.4 -12.9 -14.6 -10.6 1978 -3 -24.4 -5.8 -7.9 16.3 5.8 6.1 1.4 0.8 -6.2 -2 -0.9 1979 -4 6.7 -3 -5.5 3.6 5.8 -8.2 -5 1.4 -2.5 -4.7 -7.5 1980 3.2 1.1 -8.5 -12.9 -3.5 -4.7 -1.7 1.4 -5.2 -1.9 -3.4 -0.9 1981 2.7 -3.2 -16.6 -5.5 7.6 11.5 9.4 5.9 7.5 -5 2.6 4.7 1982 9.4 0.6 2.4 -3.8 -8.2 -20.1 -19.3 -23.6 -21.4 -20.2 -31.1 -21.3 1983 -30.6 -33.3 -28 -17 6 -3.1 -7.6 0.1 9.9 4.2 -0.7 0.1 1984 1.3 5.8 -5.8 2 -0.3 -8.7 2.2 2.7 2 -5 3.9 -1.4 1985 -3.5 6.7 -2 14.4 2.8 -9.6 -2.3 8.5 0.2 -5.6 -1.4 2.1 1986 8 -10.7 0.8 1.2 -6.6 10.7 2.2 -7.6 -5.2 6.1 -13.9 -13.6 1987 -6.3 -12.6 -16.6 -24.4 -21.6 -20.1 -18.6 -14 -11.2 -5.6 -1.4 -4.5 1988 -1.1 -5 2.4 -1.3 10 -3.9 11.3 14.9 20.1 14.6 21 10.8 1989 13.2 9.1 6.7 21 14.7 7.4 9.4 -6.3 5.7 7.3 -2 -5 1990 -1.1 -17.3 -8.5 -0.5 13.1 1 5.5 -5 -7.6 1.8 -5.3 -2.4 1991 5.1 0.6 -10.6 -12.9 -19.3 -5.5 -1.7 -7.6 -16.6 -12.9 -7.3 -16.7 1992 -25.4 -9.3 -24.2 -18.7 0.5 -12.8 -6.9 1.4 0.8 -17.2 -7.3 -5.5 1993 -8.2 -7.9 -8.5 -21.1 -8.2 -16 -10.8 -14 -7.6 -13.5 0.6 1.6 1994 -1.6 0.6 -10.6 -22.8 -13 -10.4 -18 -17.2 -17.2 -14.1 -7.3 -11.6 1995 -4 -2.7 3.5 -16.2 -9 -1.5 4.2 0.8 3.2 -1.3 1.3 -5.5 1996 8.4 1.1 6.2 7.8 1.3 13.9 6.8 4.6 6.9 4.2 -0.1 7.2 1997 4.1 13.3 -8.5 -16.2 -22.4 -24.1 -9.5 -19.8 -14.8 -17.8 -15.2 -9.1 1998 -23.5 -19.2 -28.5 -24.4 0.5 9.9 14.6 9.8 11.1 10.9 12.5 13.3 1999 15.6 8.6 8.9 18.5 1.3 1 4.8 2.1 -0.4 9.1 13.1 12.8 2000 5.1 12.9 9.4 16.8 3.6 -5.5 -3.7 5.3 9.9 9.7 22.4 7.7 2001 8.9 11.9 6.7 0.3 -9 1.8 -3 -8.9 1.4 -1.9 7.2 -9.1 2002 2.7 7.7 -5.2 -3.8 -14.5 -6.3 -7.6 -14.6 -7.6 -7.4 -6 -10.6 2003 -2 -7.4 -6.8 -5.5 -7.4 -12 2.9 -1.8 -2.2 -1.9 -3.4 9.8 2004 -11.6 8.6 0.2 -15.4 13.1 -14.4 -6.9 -7.6 -2.8 -3.7 -9.3 -8 2005 1.8 -29.1 0.2 -11.2 -14.5 2.6 0.9 -6.9 3.9 10.9 -2.7 0.6 2006 12.7 0.1 13.8 15.2 -9.8 -5.5 -8.9 -15.9 -5.1 -15.3 -1.4 -3 2007 -7.3 -2.7 -1.4 -3 -2.7 5 -4.3 2.7 1.5 5.4 9.8 14.4 2008 14.1 21.3 12.2 4.5 -4.3 5 2.2 9.1 14.1 13.4 17.1 13.3 Source: Australian Bureau of Meteorology, 2011

APPENDIX 3 - C++ PROGRAM FOR FINDING THE OPTIMUM GROUP OF CYCLONES USING DYNAMIC PROGRAMMING TECHNIQUE.

/*This program finds the optimum group of cyclones based on sinuosity index values with Gamma distribution*/

#include #include #include #include #include using namespace std; typedef double Number;

/************************************************************** ******* Returns the imcomplete gamma function P(a,x) = (int_0^x e^{-t} t^{a-1} dt)/Gamma(a) , (a > 0). C.A. Bertulani May/15/2000 *************************************************************** ******/

Number gammp(Number a, Number x) { voidgcf(Number *gammcf, Number a, Number x, Number *gln); voidgser(Number *gamser, Number a, Number x, Number *gln); Number gamser,gammcf,gln;

if (x < 0.0 || a <= 0.0) cerr<< "Invalid arguments in routine gammp"; if (x < (a+1.0)) { gser(&gamser,a,x,&gln); returngamser; } else { /* Use the continued fraction representation */ gcf(&gammcf,a,x,&gln); /* and take its complement. */ return 1.0-gammcf; } }

/************************************************************** ******* Returns the imcomplete gamma function Q(a,x) = 1-P(a,x) = (int_x^infinity e^{-t} t^{a-1} dt)/Gamma(a) , (a > 0). 70

C.A. Bertulani May/15/2000 *************************************************************** ******/

Number gammq(Number a, Number x) { voidgcf(Number *gammcf, Number a, Number x, Number *gln); voidgser(Number *gamser, Number a, Number x, Number *gln); Number gamser,gammcf,gln;

if (x <= 0.0 || a <= 0.0) cerr<< "Invalid arguments in routine gammq"; if (x < (a+1.0)) { /* Use the series representation */ gser(&gamser,a,x,&gln); return 1.0-gamser; /* and take its complement. */ } else { /* Use the continued fraction representation. */ gcf(&gammcf,a,x,&gln); returngammcf; } }

/************************************************************** ******* Returns the imcomplete gamma function P(a,x) evaluated by its series representation as gamser. Also returns ln(Gamma(a)) as gln. C.A. Bertulani May/15/2000 *************************************************************** ******/

#define ITMAX 1000 #define EPS 3.0e-7 voidgser(Number *gamser, Number a, Number x, Number *gln) { Number gamma_ln(Number xx); int n; Number sum,del,ap;

*gln=gamma_ln(a); if (x <= 0.0) { if (x < 0.0) cerr<< "x less than 0 in routine gser"; *gamser=0.0; return; } else { ap=a; del=sum=1.0/a; 71

for (n=1;n<=ITMAX;n++) { ++ap; del *= x/ap; sum += del; if (fabs(del)

#undef ITMAX #undef EPS

/************************************************************** ******* Returns the imcomplete gamma function Q(a,x) evaluated by its continued fraction representation as gammcf. Also returns ln(Gamma(a)) as gln. C.A. Bertulani May/15/2000 *************************************************************** ******/

#define ITMAX 1000 /* Maximum allowed number of iterations. */ #define EPS 3.0e-7 /* Relative accuracy */ #define FPMIN 1.0e-30 /* Number near the smallest representable */ /* floating point number. */ voidgcf(Number *gammcf, Number a, Number x, Number *gln) { Number gamma_ln(Number xx); int i; Number an,b,c,d,del,h;

*gln=gamma_ln(a); b=x+1.0-a; /*Setup fr evaluating continued fracion by modified Lent'z*/ c=1.0/FPMIN; /* method with b_0 = 0. */ d=1.0/b; h=d; for (i=1;i<=ITMAX;i++) { /* Iterate to convergence. */ an = -i*(i-a); b += 2.0; 72

d=an*d+b; if (fabs(d) < FPMIN) d=FPMIN; c=b+an/c; if (fabs(c) < FPMIN) c=FPMIN; d=1.0/d; del=d*c; h *= del; if (fabs(del-1.0) < EPS) break; } if (i > ITMAX) cerr<< "a too large, ITMAX too small in gcf"; *gammcf=exp(-x+a*log(x)-(*gln))*h; /* Put factors in front. */ }

#undef ITMAX #undef EPS #undef FPMIN

/************************************************************** ****** Returns the value of ln[Gamma(xx)] for xx > 0 *************************************************************** *****/

Number gamma_ln(Number xx) { Number x,y,tmp,ser; static Number cof[6]={76.18009172947146,-86.50532032941677, 24.01409824083091,-1.231739572450155, 0.1208650973866179e-2,-0.5395239384953e-5}; int j;

y=x=xx; tmp=x+5.5; tmp -= (x+0.5)*log(tmp); ser=1.000000000190015; for (j=0;j<=5;j++) ser += cof[j]/++y; return -tmp+log(2.5066282746310005*ser/x); }

/************************************************************** *******/

/*Program written by Karuna G. Reddy as per the MPP for Gamma Study Variable*/

# define z 100 //(refine to 5 dp ) 73

# define b 1 //beta value-will be found from data # define r 3.822976 //shape parameter-will be found from data # define t 1.351949 //theta-scale parameter-will be found from data //# define ts 8.630574831 // theta square //# define m 4.9028809776729 /*complete gamma function with only one argument: //Gamma(3.836157) value*/ //# define m1 18.808221182667 // gamma(r+1) //# define m1s 353.749184 // [gamma(r+1)]^{2} //# define m2 90.959510530101 //gamma(r+2) //# define ms 24.03824187 //[gamma(r)]^{2}

/*Recursive function receives the parameter k and dk,yk to calculate f.*/ doubleRootVal(int k, double d, double y); /*calculates the value of the minimal elements*/ double fun(int,int,double ,int,int ,bool ); double Minimum(double val1,double val2) // returns minimum of 2 numbers { if(val1<=val2) { return val1; } else { return val2; } }

//Change here for the number of stages and the distance g and initial value x0 int h ; // number of stages const double g = 12.1561; // g is the distance double s; // s=x0, the initial value const doubleinc = 0.001; //PRECISION AMMOUNT const double inc2 = 0.00001; //PRECISION AMMOUNT const doubleprec = 1/inc; constint stages = 8; constint points = 1000 ; //Keep this to be 1/inc constint factor =4;

/* eg. function(3,1) will be passed as function(3,1000), your value divided by inc to make it precise*/ intylimits[10]; //stores the 3dp values for refining constint e = (int)(g*points*z+1); 74 constint p=(int)(g*points); double minkf2[stages][e]; //stores minimum f to 6dp double dk2[stages][e]; //stores minimum d for the 6dp calculations main() { //initialize minkf cout<<"Initializing points ...."<> h; cout<<"enter s = Initial value " <> s; double f=fun(h,p,inc ,0,p ,true);

float d6,d5,d4,d3,d2,d1, y6,y5,y4,y3,y2,y1; int temp;

//backward calculation for the 3dp results d6 = g; y6 = dk2[6][p]; d5=d6-y6; temp = (int)(d5*points); y5=dk2[5][temp]; d4=d5-y5; temp = (int)(d4*points); y4=dk2[4][temp]; d3=d4-y4; temp = (int)(d3*points); y3=dk2[3][temp]; d2=d3-y3; temp = (int)(d2*points); y2=dk2[2][temp]; d1=d2-y2; y1=d1;

//setup the limits for the 6dp calculations

75 temp = (int)(y6*points*z); ylimits[6] = temp; temp = (int)(y5*points*z); ylimits[5] = temp; temp = (int)(y4*points*z); ylimits[4] = temp; temp = (int)(y3*points*z); ylimits[3] = temp; temp = (int)(y2*points*z); ylimits[2] = temp; temp = (int)(y1*points*z); ylimits[1] = temp;

f=fun(h,e-1,inc2 ,ylimits[h]-factor*z,ylimits[h]+ factor*z ,false);//for k>=2 cout<<"stage: h = " << h << " distance: g = " << g<

//Backward calucation for the 6 dp d6=g; y6 = dk2[6][(e-1)]; d5=d6-y6; temp = (int)(d5*points*z); y5=dk2[5][temp]; d4=d5-y5; temp = (int)(d4*points*z); y4=dk2[4][temp]; d3=d4-y4; temp = (int)(d3*points*z); y3=dk2[3][temp]; d2=d3-y3; temp = (int)(d2*points*z); y2=dk2[2][temp]; d1=d2-y2; y1=d1; printf("\nd6: %f y6: %f",d6,y6); printf("\nd5: %f y5: %f",d5,y5); printf("\nd4: %f y4: %f",d4,y4); printf("\nd3: %f y3: %f",d3,y3); printf("\nd2: %f y2: %f",d2,y2); printf("\nd1: %f y1: %f",d1,y1); getch(); } //end main doubleRootVal(int k, double d, double y)/*calculate the root value of the 76 current distribution*/ { doublertval; doublecalc;

/*double c1 = (gammq(r,(s/t))-gammq(r,(d+s)/t))/(gammq(r,(d-y+s)/t) -gammq(r,(d+s)/t));//error case 1 (being used now) double c2 = (t*r*(gammq(r+1,(d-y+s)/t)-gammq(r+1,(d+s)/t)))/(gammq(r,(d- y+s)/t) -gammq(r,(d+s)/t));//error case 2 double c3 = (pow(t,2)*r*(r+1)*(gammq(r+2,(d-y+s)/t) -gammq(r+2,(d+s)/t)))/(gammq(r,(d-y+s)/t)-gammq(r,(d+s)/t))*/

//double c1 = 1;//error case 1 (being used now) //double c2 = 11.26991907;//error case 2 //double c3 = 164.4962888;//error case 3 - out of range double c = 0;//error calculated from sugar mill data calc=(pow(b,2)*pow(t,2))*(r*(r+1)*(gammq(r+2,(d-y+s)/t)- gammq(r+2,(d+s)/t))*(gammq(r,(d-y+s)/t)-gammq(r,(d+s)/t)) -pow(r,2)*pow((gammq(r+1,(d-y+s)/t)-gammq(r+1,(d+s)/t)),2)) +(c)*pow((gammq(r,(d-y+s)/t)-gammq(r,(d+s)/t)),2); if(calc<0) { //cout<<"\nError: Negative Root\n"; //rtval = -1; } else { calc = sqrt(calc); } rtval = calc; returnrtval; } double fun(intk,intn,doubleincf,intminYk,intmaxYk,boolisFirstRun) /*this functions performs the same actions as "function". It only defers in terms of the iterations of the for loop.*/ { assert (k>=1); //Abort if k is negative doubledblRetVal; double d =n*incf; //d value for the function double y; double min; doubleval; doubleminy; 77 int col; if(k==1) //base case { y = d; dblRetVal = RootVal(k,d,y); } else { for(int i=minYk;i<=maxYk;i++)/*iterate over the interval allowed to calculate the 6dp results*/ { y = i*incf;//this sets to precission of y to 6dp double root; root = RootVal(k,d,y); //calculate the root. if(root != -1) //if root is valid { col =n-i;//get the current d value

if(minkf2[k-1][col]==-9999) {/*check if the result has been previously calculated*/ if(isFirstRun){ val = root+ fun((k-1),col,incf,0,col,true);//if not, // calculate the result } else{ val = root+ fun((k-1),col,incf,ylimits[k-1]- factor*z,ylimits[k-1]+ factor*z,false);//if not, //calculate the result } } else val = root+ minkf2[k-1][col];//if result exists, use it for calculations } if (i==minYk) { min =val;//base case } else { min = Minimum(min,val);//get the minimum if the result and the current mininmum } if(min == val){miny=y;}//get the position of the current minimum }//end for

dblRetVal = min;

}//end else

//store the f and the d value of the minimum calculated. col = n; minkf2[k][col] = dblRetVal; dk2[k][col]=miny; returndblRetVal;

}//end function