Preliminary Work on the Prediction of Extreme Rainfall Events and Flood Events in Australia

Prepared by Kevin Fergusson

Presented to the Actuaries Institute General Insurance Seminar 13 – 15 November 2016 Melbourne

This paper has been prepared for the Actuaries Institute 2016 General Insurance Seminar. The Institute’s Council wishes it to be understood that opinions put forward herein are not necessarily those of the Institute and the Council is not responsible for those opinions.

 Kevin Fergusson, Centre for Actuarial Studies, University of Melbourne

The Institute will ensure that all reproductions of the paper acknowledge the author(s) and include the above copyright statement.

Institute of Actuaries of Australia ABN 69 000 423 656 Level 2, 50 Carrington Street, Sydney NSW Australia 2000 t +61 (0) 2 9239 6100 f +61 (0) 2 9239 6170 e [email protected] w www.actuaries.asn.au

Preliminary Work on the Prediction of Extreme Rainfall Events and Flood Events in Australia

K. Fergusson Centre for Actuarial Studies The University of Melbourne 15th November 2016

Abstract

Among many scientific discoveries over the centuries, the pioneering work in Richardson [1922] has provided the mathematical theory of weather forecasting used today. The subsequent technological advances of high speed computers have allowed Richardson’s work to be exploited by modern day meteorological teams who coordinate their efforts globally to predict weather patterns on Earth, particularly extreme weather events such as floods. This paper describes preliminary work in applying some known predictors of rainfall in Australia to forecasting extreme rainfall events and linking these to flood events.

Keywords: Floods, rainfall intensity-frequency-duration curves, sunspot numbers, El Nino-Southern Oscillation, Southern Oscillation Index, Indian Ocean Dipole, Southern Annular Mode, Madden-Julian Oscillation, regression trees, bootstrapped aggregation.

1 1. Introduction

2 2. Background on Australia’s Weather

A good coverage of the subject of Australia’s weather is given by Whitaker [2010], where causes of weather, measurement of weather, prediction of weather and notable weather disasters are discussed. A fascinating history of weather prediction and ﬂood prediction in Australia is given in Day [2007]. Also, in Risbey et al. [2009] several drivers of rainfall, such as the Indian Ocean Dipole, Southern Annular Mode and Madden-Julian Oscillation, are described. In this section a background is presented on such drivers of weather as the Sun, the El Nino Southern Oscillation measured by the Southern Oscillation Index, The Indian Ocean Dipole, the Southern Annular Mode and the Madden-Julian Oscillation. The aim is to employ data pertaining to these drivers to predict extreme rainfall events, which are associated with ﬂood events.

2.1. Solar Inﬂuences of Weather on Earth

Weather on Earth is driven by the Sun’s heating of Earth, the transference of this heat from the equator to the poles and accompanying interactions between oceans, the atmosphere and land masses throughout this process. Therefore, it would reasonable that the level of solar energy reaching Earth is associated with the level of sun spot activity, the distance of Earth from the Sun and the tilt of Earth’s axis, among other causes. For example, in Milankovic [1920], the Serbian astronomer Milankovic theorised that variations in the tilt of the earth, the precession of the earth and the eccentricity of Earth’s orbit around the Sun inﬂuence climate on Earth. Also, the connection between the sunspot number and solar irradiance is discussed in Boucher et al. [2001] and the connection between solar cycle length and temperatures on Earth in the subsequent cycle is discussed in Solheim et al. [2012].

2.1.1. Sunspot Number

Sunspots are dark spots on the photosphere of the Sun which are concentrations of magnetic ﬁeld ﬂux and are of relatively lower temperatures than their surrounds. The sunspot number has been observed for several centuries, waxing and waning in number, as shown in Figure 1 and is a proxy for the level of solar inactivity.

Monthly mean total sunspot numbers, sourced from the World Data Centre - Sunspot Index and Long-Term Solar Observations (WDC-SILSO) at the Royal Observatory of Belgium, Brussels, having the website address:

3 Sunspot Numbers 1749 - 2016 450

400

350

300

250

200

Sunspot Sunspot Number 150

100

0 1700 1750 1800 1850 1900 1950 2000 2050 Year

Crude Sunspot Number Savitzky-Golay Filter 40,4

Figure 1: Sunspot numbers over the period 1749 to 2016

http://sidc.oma.be/silso/dataﬁles, are used in the analysis. Smoothing of the data has employed the Savitzky- Golay ﬁlter, given in Savitzky and Golay [1964], with half-width 40 and degree 4 polynomial. In this paper, the sunspot number is not modelled and instead the actual observed numbers are used in the predictions of extreme rainfall levels.

It is purported in Solheim et al. [2012], for example, that the length of the solar cycle is associated with lower temperatures on Earth in the subsequent solar cycle.

2.1.2. Earth’s Tilt

Earth’s tilt relative to the Sun is the angle that the Earth’s rotational axis makes with the perpendicular to the Sun’s rays hitting the Earth. It is what causes the seasons on Earth and is a signiﬁcant driver of rainfall.

It can be measured by the latitude over which the Sun is overhead, described

4 mathematically as

d − 18000321 L (d) = 23.5◦ × sin 2π × , (1) SUN 365.25 where d is the date in YYYYMMDD format and the difference of dates is computed in calendar days. At the solstices the Sun is directly overhead at the north and south latitudes of 23.5◦, namely the Tropics of Cancer and Capricorn respectively. Also, at the equinoxes the Sun is directly overhead at the equator. It is assumed here that the Vernal Equinox occurs on the 21st of March each year, as is evident in (1).

2.1.3. Earth’s Distance from the Sun

A well known law of physics states that the intensity of radiation is inversely proportional to the square of the distance from the source. This law was discovered by many physicists, including Kepler, as mentioned in Gal and Chen-Morris [2005], and is applicable to the intensity of solar radiation hitting the Earth.

The distance the Earth is from the Sun can be computed from its elliptical orbit around the Sun, given by the mathematical formula for an ellipse in polar coordinates A r = , (2) φ 1 + e cos φ where e is the eccentricity of the ellipse, A is a constant. It is assumed here that the ellipse has major and minor axes of lengths a = 149.600 million km and b = 149.579 million km respectively. The relations A A a = , b = √ (3) 1 − e2 1 − e2 hold. Kepler’s second law of planetary motion states that a planet’s orbit around its focus sweeps out a constant angular area per unit time. As an approximation, it is assumed here that the angular speed is constant and therefore the value of φ on day d is taken to be d − 18000103 φ(d) = × 2π, (4) 365.25 which models the Earth reaching its perihelion (closest point to the Sun, a distance of 147.1 million km) on about the 3rd of January and its aphelion (farthest point from the Sun, a distance of 152.1 million km) on about the 6th of July each year.

5 The further the Earth is from the Sun, the less intense is the solar radiation reaching the Earth and therefore, the lower the energy levels associated with storms and cyclones.

2.2. Southern Oscillation Index

The El Nino climatic event refers to a sustained warming of the tropical areas of the central and eastern Pacific Ocean which results in lower than average rainfall over Eastern and Northern Australia. On the other hand, the La Nina climatic event refers to a sustained warming of the tropical area of the Western Pacific, which results in higher than average rainfall over Northern and Eastern Australia and potentially Central Australia. The La Nina event involves trade winds blowing westward along the surface of the Pacific Ocean, resulting in moisture laden air rising over the warmer area in the Western Pacific. In normal conditions, the rising air is the Western Pacific is blown eastward at higher altitudes, leading to an air circulation known as the Walker Circulation. The strength of the Walker Circulation is measured by the monthly Southern Oscillation Index (SOI).

Monthly data for the SOI sourced from the Bureau of Meteorology’s website

ftp://ftp.bom.gov.au/anon/home/ncc/www/sco/soi/soiplaintext.html for the period from 1876 to 2016 is used in the analysis and is shown in Figure 2. The quadratic variation is shown in Figure 3, which shows that the volatility of the SOI is fairly constant, demonstrating the homoskedasticity of the SOI and simplifying the modelling of the SOI in further work.

The SOI is calculated as the difference in monthly sea level barometric pressures at Tahiti and Darwin, adjusting for standard deviation. When the SOI maintains a value of roughly +8 or more, it is indicative of a La Nina event. In contrast, a value of roughly −8 or less is indicative of an El Nino event, as mentioned in Davis [2012].

An example of the impact of SOI values of around −8 is when in 2010 and 2011 parts of Australia experienced record-breaking rainfall, due to a La Nina event, which resulted in major ﬂooding in South Eastern Queensland and Northern and Western Victoria, as mentioned in Davis [2012].

6 Southern Oscillation Index 40

0 1870 1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 2020

-10 SOI SOI Value

-20

-30

-40

-50 Year

Figure 2: The Southern Oscillation Index

Quadratic Variation of Southern Oscillation Index 140,000

120,000

100,000

80,000

60,000

40,000

20,000

Cumulative Variance of SOI (units of SOI (units ofSOI squared) Variance Cumulative 0 1870 1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 2020 Year

QV Fitted Line (Slope = 975.86 p.a.)

Figure 3: The quadratic variation of the Southern Oscillation Index

7 2.3. Indian Ocean Dipole

The Indian Ocean Dipole (IOD) is an oscillation of warm and cold sea surface temperatures in the Indian Ocean and strongly inﬂuences rainfall in Australia.

Speciﬁcally, the IOD measures the difference between sea surface temperatures of one pole located in the Arabian Sea and another pole in the eastern Indian Ocean. Monthly data sourced from the Japan Agency for Marine-Earth Science and Technology (JAMSTEC) is used in the analysis here, the website address being

http://www.jamstec.go.jp/frcgc/research/d1/iod/DATA/dmi_HadISST.txt and the period being from 1958 to 2010. The historical series is shown in Figure 4 and the quadratic variation is shown in Figure 5. The quadratic variation of the IOD shows a periodic increase in volatility, roughly every 14 years, and demonstrates the heteroskedasticity of the phenomenon which will need to be incorporated in the volatility process when modelling the IOD and predicting rainfall.

The three phases of the IOD are: neutral, positive and negative. In the negative phase, westerly winds blowing along the Equator allow a concentration of warmer water near Australia, resulting in higher than average rainfall over parts of southern Australia during Winter and Spring. The opposite happens in a positive phase, with lower than average rainfall over southern Australia.

2.4. Southern Annular Mode

The Southern Annular Mode (SAM), also known as the Antarctic Oscillation (AAO), describes the north-south movement of the westerly wind belt that circles Antarctica, dominating the middle to higher latitudes of the southern hemisphere.

Monthly data sourced from the website of the British Antarctic Survey,

http://www.nerc-bas.ac.uk/public/icd/gjma/newsam.1957.2007.txt, for the period 1957 to 2016 is used in the analysis here. The historical series is shown in Figure 6 and the quadratic variation is shown in Figure 7. The quadratic variation illustrates the periodicity of increased volatility of the SAM, roughly every 40 years. In subsequent work, the heteroskedasticity will be

8 Indian Ocean Dipole 4

0 IOD IOD Value -1

-2

-3

-4 1/07/1957 1/07/1967 1/07/1977 1/07/1987 1/07/1997 1/07/2007 Date

Figure 4: The Indian Ocean Dipole

Quadratic Variation of the Indian Ocean Dipole 70

10 QV of IOD (units of IOD (units IOD ofsquared units) QV

0 1/07/1957 1/07/1967 1/07/1977 1/07/1987 1/07/1997 1/07/2007 Date QV Fitted Line (slope = 1.32 p.a.)

Figure 5: The quadratic variation of the Indian Ocean Dipole

9 Southern Annular Mode 1957 - 2016 6

-2 SAM Value SAM

-4

-6

-8 15/02/1956 14/02/1966 14/02/1976 13/02/1986 13/02/1996 12/02/2006 12/02/2016 Date

Southern Annular Mode 1Y Average 5Y Average

Figure 6: Southern Annular Mode modelled by the SAM volatility process when determining the SAM’s inﬂuence on rainfall.

The changing latitudinal position of this westerly wind belt affects the strength and position of cold fronts and mid-latitude storm systems. For example, in a positive SAM event, the belt of strong westerly winds contracts towards Antarctica, resulting in weaker westerly winds and higher pressures over southern Australia, thus hindering cold fronts moving inland. On the other hand, in a negative SAM event, the belt of strong westerly winds shifts towards the equator, resulting in stronger storms and low pressure systems over southern Australia and can result in southern Australia avoiding rainfall in Autumn and Winter but south eastern Australia receiving more rainfall during Spring and Summer.

2.5. Madden-Julian Oscillation

The Madden-Julian Oscillation (MJO) was discovered by Madden and Julian [1972], whereby a pulse of cloud and rainfall, called an enhanced convective phase, and an associated suppressed convective phase, move eastward over

10 Quadratic Variation of Southern Annular Mode 4000

3500

3000

2500

2000

1500

1000 QV of SAM (squared units (squared ofSAM) of SAM QV 500

0 15/02/1956 14/02/1966 14/02/1976 13/02/1986 13/02/1996 12/02/2006 12/02/2016 Date

QV Fitted Line (Slope 57.10 p.a.)

Figure 7: The quadratic variation of the Southern Annular Mode the tropics with a periodicity of roughly 30 to 60 days. This dipole consisting of the enhanced convective phase and the suppressed convective phase moves eastward over the topics, eventually circling the globe and returning to its original position. A two-dimensional index has been developed in Wheeler and Hendon [2004] which captures the MJO effect, based on combinations of outgoing long-wave radiation, 850-hPa zonal winds (lower altitude winds) and 200-hPa zonal winds (higher altitude winds) averaged over latitudes 15◦ S to 15◦ N.

Daily data sourced from the Bureau of Meteorology’s website

http://www.bom.gov.au/climate/mjo/graphics/rmm.74toRealtime.txt over the period 1st June 1974 to 3rd August 2016 is used in the analysis here. The two components of the MJO effect are denoted as RMM1 and RMM2 and when plotted on a two-dimensional graph they can be assigned a phase number between one and eight, according to the octant in which they lie. The Cartesian coordinates (RMM1,RMM2) can be converted to polar

11 coordinates (r, θ) using the formulae

r2 = (RMM1)2 + (RMM2)2 (5) RMM1 = r cos θ RMM2 = r sin θ and the phase number is determined as

θ − π 1 + b × 8c mod 8 . (6) 2π

The phase numbers generally correspond to locations along the Equator around the globe.

A Markov Chain model applied to the time series of phases gives the transition probability matrix

 0.753 0.183 0.006 0.003 0.002 0.001 0.002 0.048   0.058 0.741 0.183 0.009 0.001 0.001 0.001 0.006     0.004 0.070 0.715 0.197 0.008 0.003 0.001 0.003     0.002 0.004 0.046 0.765 0.170 0.007 0.003 0.002  T =   (7)  0.003 0.004 0.008 0.061 0.733 0.187 0.004 0.001     0.003 0.001 0.002 0.003 0.061 0.721 0.200 0.010     0.007 0.004 0.003 0.002 0.008 0.063 0.732 0.184  0.194 0.007 0.002 0.001 0.003 0.005 0.058 0.730 where it is evident that transitions from Phase X mostly end up in Phase X + 1. Here the phases are 1, 2,..., 8 and the (i, j)-th entry of the transition probability matrix T shows the probability of moving from Phase i on a given day to Phase j on the following day. So the transition matrix demonstrates that the typical phase pattern is from 1 through all numbers to 8 and then repeating, namely that the MJO circumscribes the Equator periodically. The transition probability matrix is a useful tool for predicting future phases of the MJO given a knowledge of its current phase, and therefore useful for rainfall prediction, as discussed in Donald [2004].

12 3. Rainfall Data

Ever since British settlement of Australia, weather data has been monitored because of its relevance to agriculture and other industries. Particularly, rainfall levels have been observed at rainfall stations located in various parts of Australia for more than a century.

Rainfall data sourced from the Bureau of Meteorology’s website

http://www.bom.gov.au/watl/rainfall/observations/ is used in the analysis here. The 181 locations of the rainfall stations used in the analysis are shown in Figure 8 and in Appendix B.

Australian Rainfall Station Locations -10

-15

-20

-25

-30 South)

-35

-40

-45 Degrees Latitude (Negative Indicates Indicates (Negative Latitude Degrees 110 115 120 125 130 135 140 145 150 155 160 Degrees Longitude East

Figure 8: Location of rainfall stations in Australia

The logarithm of the probability density function of daily rainfall amounts in shown in Figure 9. Taking the logarithm allows a comparison to be made with other distributions in the tails of the distributions. It is evident that a Generalised Pareto distribution provides a good ﬁt to the distribution of observed rainfall amounts, which is to be used in subsequent research.

Based on the available data, several predictive factors in Appendix A pertain

13 Comparison of Logarithm of Empirical Probability Density Function for Daily Rainfall Amounts with those of other Distributions 10 5 0 Function

‐5 ‐10 Density ‐15 ‐20 ‐25 Probability ‐30 of 0 50 100 150 200 250 300 350 400 450 500 550 Daily Rainfall Amount (mm)

Logarithm Log Density Log Normal (mu=‐0.2665, sigma=1.0470) Log Student t (mu=‐0.6928, sigma=0.0003, nu = 0.4265) Generalised Pareto (mu=2.5420, sigma=1.1812, xi = 0.7147)

Figure 9: Logarithm of probability density function of daily rainfall amounts at all station locations to the location itself, such as the longitude, latitude, height above sea level, distance to nearest river, distance to the coast and intensity of the Sun at that location. The intensity of the Sun at the location is computed using the formula

ISUN (θLOC ) = cos |θLOC − θSUN |, (8) where θLOC is the latitude of the location and θSUN is the latitude at which the Sun is overhead.

The proximity of a location to a river is a key driver of the likelihood and severity of a ﬂood event at that location. However, in this paper, this factor is used as a general predictor of rainfall, because naturally a river requires rainfall at its source to maintain its ﬂow. Australia’s major rivers are shown in Appendix C.

14 4. Flood Event Data

Floods in Australia are typically associated with either tropical cyclones, for example in the the northern areas of Australia, or slow moving low pressure cells, such as in southern areas of Australia. There are other pertinent factors associated with floods, such as the topography of the location, proximity to rivers, heights of river banks and hydrology of the location. So, it would be ideal to have river gauge data for the major rivers shown in Appendix C, topographical data and hydrological data to determine precisely how extreme rainfall events lead to flood events. However, in this paper, rainfall levels which correspond to flood events are deduced by looking at the following historical flood events, sourced from Whitaker [2010] and Davis [2012]:

• Brisbane, 1839 - the gauge height of the Brisbane River at the Post Ofﬁce was about 8.4m;

• Brisbane, March 1890 - the gauge height of the Brisbane River at the Post Ofﬁce was about 5.3m;

• Brisbane, February 1893 - the gauge height of the Brisbane River at the Post Ofﬁce was about 8.4m;

• Northern Tasmania, April 1929 - an intense slow moving low pressure cell over central Victoria brought heavy rainfall;

• Hunter Valley, February 1955 - a low pressure cell over north-eastern New South Wales brought heavy rainfall;

• Brisbane, January 1974 - Tropical Cyclone Wanda dumped heavy rainfall across Brisbane, causing major ﬂooding of the Brisbane River, resulting in its highest gauge reading for the century;

• Brisbane, January 2011 - heavy intense rainfall resulted in ﬂoods, with the gauge height of the Brisbane River at the Post Ofﬁce being about 4.5m;

• Katherine, December 2011 - Tropical Cyclone Grant brought heavy rainfall which caused ﬂoods north of Katherine;

Flood peak data for Brisbane is shown in Figure 10 and rainfall amounts during flood events are shown in Table 1, which corroborate some of the aforementioned flood events. Each flood event has involved at least one day of rainfall of around 90 mm or more.

15 Flood Event Rf Stn Day 1 Day 2 Day 3 Day 4 Northern Tasmania 19290403 19290404 19290405 19290406 (April 1929) YOLLA 6.1 mm 114.3 mm 101.6 mm 83.8 mm Hunter Valley 19550223 19550224 19550225 19550226 (February 1955) NEWC 0 mm 83.8 mm 24.6 mm 40.1 mm ATTU 22.9 mm 38.9 mm 89.4 mm 109 mm Brisbane 19740125 19740126 19740127 19740128 ( January 1974) TOOW 104.6 mm ALDE 62 mm 224 mm 220 mm 43 mm Brisbane 20110109 20110110 20110111 20110112 (January 2011) ALDE 23.4 mm 123.2 mm 28.2 mm 54.2 mm MT MEE 64.6 mm 189.6 mm 185 mm 176 mm Katherine 20111225 20111226 20111227 20111228 (December 2011) KATH 0 mm 0 mm 255 mm 1 mm KOOL 143 mm 0 mm

Table 1: Daily rainfall amounts during various historical ﬂood events at rainfall stations listed in Table 2

Abbreviation Rainfall Station Details YOLLA YOLLA (SEA VIEW) (TAS) Lat -41.14 Long 145.71 NEWC NEWCASTLE NOBBYS SIGNAL STATION AWS (NSW) Lat -32.9 Long 152 ATTU ATTUNGA (GARTHOWEN) (NSW) Lat -30.9 Long 151 TOOW TOOWONG BOWLS CLUB (QLD) Lat -27.5 Long 153 ALDE ALDERLEY (QLD) Lat -27.4 Long 153 ALDE ALDERLEY (QLD) Lat -27.4 Long 153 MT MEE MT MEE (QLD) Lat -27.1 Long 153 KATH CDU KATHERINE RURAL CAMPUS (NT) Lat -14.37 Long 132.16 KOOL KOOLPINYAH (NT) Lat -12.4 Long 131

Table 2: List of details of rainfall stations corresponding to various ﬂood events

16 Figure 10: Highest annual ﬂood peaks for Brisbane (source: BOM, with permission)

17 5. Methodology

The aim here is to predict ﬂood events by using a combination of weather indices such as IOD, SAM, MJO and SOI and solar indicators.

Extreme rainfall events are characterised here by n-day rainfall intensities in excess of a 90 mm per day threshold, for n = 1, 2, 3, 4, 5. These levels roughly correspond to ﬂood events in Brisbane and other cities.

The predictive factors which are best able to identify extreme rainfall events, out-of-sample, using regression trees, as described by Berk [2008] are determined. The in-sample data set for the analysis covers the period from 1st January 1800 to 31st December 1989. The out-of-sample period is from 1st January 1990 to 30th June 2016. An explanation of regression trees and bootstrapped aggregation of regression trees is provided here by ﬁrst looking at ordinary least squares regression.

5.1. Ordinary Least Squares Regression

Given the independent data matrix X and the dependent data vector Y it is sought to predict the values of Y using the data matrix X. Here X is an n × p matrix and Y is an n × 1 matrix, or equivalently a column vector of length n. The value of n is the number of data points observed and for the in-sample data set n = 1, 108, 286. The value of p is the number of predictors in Table 9 in Appendix A, that is n = 23. The (i, j)-th entry of X, denoted by Xi,j, corresponds to the i-th data observation of the j-th independent variable or predictor variable. For example, X2,1 is the second observation of the sunspot number. Also the i-th entry of Y , denoted by Yi, corresponds to the i-th data observation of the response variable, namely the i-th rainfall observation.

Ordinary Least Squares Regression (OLS) models the response variable, which is the daily rainfall amount, as a constant term β0 plus a linear combination of the predictor variables, that is

Yi = β0 + β1Xi,1 + ... + βpXi,p + i, (9) where the sum of the squares of the error terms i is minimised. In matrix notation, the equation can be written as

Y = [1|X]β + , (10) e e e T where 1 is a column vector of ones of length n, β = (β0, . . . , βp) and T = (1, .e . . n) . A statistical measure of the explanatorye power of the model is e

18 Statistic 1D 2D 3D 4D 5D R-squ (in-sample) 8.2% 12.1% 15.3% 18.2% 20.8% R-squ (out-of-sample) 7.5% 10.4% 13.1% 15.4% 17.6% R-squ (out-of-sample for -10.2% -18.6% -28.7% -36.9% -46.0% predicted RF≥15mm)

Table 3: R-squared statistics of the OLS regression ﬁts given by the R-squared statistic, which is calculated using the formula V AR() R2 = 1 − , (11) V AR(Ye) where n 1 X 2 V AR(Y ) = Y − E(Y ) (12) n i i=1 is the variance of Y and V AR() is the variance of . The R-squared statistics of OLS regression ﬁts are shown ine Table 3. It is clear that the ability of the OLS regression model to predict daily rainfall levels out-of-sample is poor and its ability to predict rainfall intensities over higher durations is better, due to aggregate rainfall statistics being less volatile. However, the OLS regression model is incapable of predicting rainfall intensities in excess of 15mm per day, for any duration, let alone rainfall intensities in excess of 90mm per day as was hoped. The problem with predicting 90mm or more of rainfall is that the OLS model fails to predict any such levels out of sample, and therefore no R-squared statistic can be calculated.

The OLS regression when performed on quantiles of the predictor variables gives normalised coefﬁcients, shown in Table 4, whose magnitudes indicate the importance of the predictor variables. Based on this, the top four predictor variables in order of importance are:

• Average daily rainfall over previous years in that month at location;

• Intensity of Sun at rainfall station;

• Latitude of rainfall station;

• Standard deviation of daily rainfall over previous years in that month at location; for each of the durations of one to ﬁve days. The following remarks are pertinent. The intensity of the Sun at a location with latitude θLOC is given by (8) and this intensity varies with the latitude θSUN at which the Sun is overhead. For locations in Australia, the intensity of the Sun is a maximum when

19 Factor 1D 2D 3D 4D 5D Constant unity 2.9129 2.8307 2.7784 2.7178 2.6570 Sunspot number 0.4524 0.4643 0.4716 0.4717 0.4796 SG ﬁlter 0.4785 0.4751 0.4728 0.4669 0.4685 Time in current SS cycle 0.4840 0.4348 0.4247 0.4147 0.4065 Lat. Sun overhead 2.4794 2.4536 2.5015 2.5856 2.7049 Time in current tilt cycle 0.8009 0.7701 0.7551 0.7599 0.7759 Distance from Sun 1.5227 1.4678 1.5015 1.5746 1.6861 Time in current orbital period 0.9854 0.9756 0.9746 0.9668 0.9581 SOI 1.0982 1.0804 1.0630 1.0464 1.0340 IOD 0.0680 0.0833 0.0909 0.0977 0.1030 SAM 0.3260 0.3161 0.3124 0.3069 0.3020 MJO: RMM1 0.4692 0.5111 0.5392 0.5620 0.5802 MJO: RMM2 1.2216 1.2132 1.1653 1.0974 0.9963 MJO: r 0.0475 0.0517 0.0570 0.0618 0.0661 MJO: theta 0.1480 0.1776 0.2009 0.2166 0.2277 MJO: phase 0.2271 0.2946 0.3058 0.2991 0.2542 Long. of RF stn. 0.1131 0.1282 0.1273 0.1257 0.1242 Lat. of RF stn. 5.4938 4.6366 4.4322 4.2790 4.1548 RF stn. hgt. above SL 0.0159 0.0410 0.0600 0.0790 0.0972 Distance RF stn. from river 0.0287 0.0002 0.0142 0.0248 0.0349 Distance RF stn. from coast 0.1930 0.1443 0.1245 0.1080 0.0931 Intensity of Sun 8.4262 7.5314 7.2902 7.1061 6.9542 Avg. daily RF for month 10.1640 9.9415 9.8200 9.7333 9.6585 Std. dev. daily RF for month 3.8977 3.8312 3.7797 3.7425 3.7115

Table 4: Regression coefﬁcients for various durations

◦ θSUN = max{−23.5 , θLOC }, which roughly corresponds to Australia’s Summer. In contrast, the latitude of the rainfall station is, for a given rainfall station, ﬁxed. So the intensity of the Sun is a proxy for the closeness of the season to Summer, concurring with the tropical wet season in northern Australia, and the latitude of the rainfall station is simply a measure of distance to the Equator. The two remaining predictor variables, the average daily rainfall and the standard deviation of daily rainfall, are obvious indicators of daily rainfall levels and are only associated to the extent that higher average daily rainfall correlates with a higher standard deviation of daily rainfall. The rationale for including both as predictive factors, despite their potential correlation, is that, ignoring third order and higher moments, the daily rainfall is approximately the sum of the average daily rainfall and a noise term which is the product of the standard deviation of daily rainfall and a normalised random variable, that is one having mean zero and variance one.

20 5.2. Regression Trees

The simplest regression tree consists of a decision node and two emanating branches. The decision node involves the choice of a factor, say the i1-th, and a demarcation value, say θ1, such that the partitioned response variables are as homogeneous as possible. Mathematically, the variables i1 and θ1 are chosen such that

X 2 X 2 (Yi − E[Yj|Xj,i1 ≤ θ1]) + (Yi − E[Yj|Xj,i1 > θ1]) (13)

i∈{k:Xk,i1 ≤θ1} i∈{k:Xk,i1 >θ1} is minimised, where i1 ∈ {1, 2, . . . , p}, θ1 ∈ (−∞, ∞) and

X E[Yj|Xj,i1 ≤ θ1] = Yj /|{k : Xk,i1 ≤ θ1}| (14)

j∈{k:Xk,i1 ≤θ1}

X E[Yj|Xj,i1 > θ1] = Yj /|{k : Xk,i1 > θ1}|. (15)

j∈{k:Xk,i1 >θ1} This procedure can be extended to each branch’s partition of the data set to form another two nodes each of which has two emanating branches, resulting in a tree with three nodes and six branches. Recursive application of this procedure can be continued until a desired level of ﬁt has been achieved or until a prespeciﬁed maximum number of nodes is reached. Given a set of predictor variables, the predicted response variable is simply the average of the response variables in the partition at the base of the tree corresponding to the sequence of decisions taken at preceding nodes.

The R-squared statistics of the regression tree ﬁts having 100 nodes (or splits) are shown in Table 5. It is clear that the ability of the model to predict daily rainfall levels out-of-sample is poor and its ability to predict rainfall intensities over higher durations is better, due to aggregate rainfall statistics being less volatile. Comparison of Table 5 with Table 3 indicates that the out-of-sample performance of the regression tree model is inferior to that of the OLS regression model, despite its better in-sample ﬁt. However, unlike the OLS regression model, the regression tree model is able to provide predictions of rainfall intensities in excess of 75mm per day and 90mm per day, for a two-day duration, as is evident in Table 5.

The R-squared statistics for regression tree models having 50,000 nodes are shown in Table 6, where the high R-squared statistics of in-sample performances contrast with the lowly valued statistics for the out-of-sample performances and is symptomatic of overﬁtting. We overcome this drawback in the next section.

21 Statistic 1D 2D 3D 4D 5D R-squ (in-sample) 11.2% 16.1% 19.7% 22.9% 26.0% R-squ (out-of-sample) 2.6% -13.3% 12.6% 14.1% 16.7% R-squ (out-of-sample -39.7% -172.8% -7.9% -20.8% -15.7% for predicted RF≥15mm) R-squ (out-of-sample -112.9% -400.5% -1033.2% -327.5% -346.7% for predicted RF≥30mm) R-squ (out-of-sample -294.2% -66.8% 0.0% -65.5% -73.6% for predicted RF≥45mm) R-squ (out-of-sample -179.7% -66.8% 0.0% 0.0% 0.0% for predicted RF≥60mm) R-squ (out-of-sample -179.7% 0.3% 0.0% -100.0% -100.0% for predicted RF≥75mm) R-squ (out-of-sample -110.7% 0.3% -100.0% -100.0% -100.0% for predicted RF≥90mm)

Table 5: R-squared statistics of the regression tree ﬁts with 100 splits

Statistic 1D 2D 3D 4D 5D R-squ (in-sample) 68.8% 78.7% 83.6% 87.4% 89.7% R-squ (out-of-sample) -121.6% -121.3% -113.0% -86.1% -83.6% R-squ (out-of-sample -328.6% -242.0% -230.8% -183.7% -152.8% for predicted RF≥15mm) R-squ (out-of-sample -426.9% -289.1% -301.9% -228.7% -154.2% for predicted RF≥30mm) R-squ (out-of-sample -639.3% -364.4% -233.6% -242.3% -130.8% for predicted RF≥45mm) R-squ (out-of-sample -863.1% -383.9% -227.6% -361.6% -124.4% for predicted RF≥60mm) R-squ (out-of-sample -1056.5% -350.8% -205.7% -488.4% -171.6% for predicted RF≥75mm) R-squ (out-of-sample -1558.4% -384.5% -212.6% -553.2% -1261.2% for predicted RF≥90mm)

Table 6: R-squared statistics of the regression tree ﬁts with 50,000 splits

22 5.3. Bootstrapped Aggregation of Regression Trees (BAGGING)

A drawback of the regression tree approach is that it can overfit the data and therefore perform poorly out-of-sample. To overcome this, a modified data set can be constructed by sampling with replacement from the original data set, known as bootstrapping, and then a resulting regression tree can be built from this modified data set. As many trees as desired can be built in this fashion, allowing one to calculate a consensus prediction from the built trees. This is known as bootstrapped aggregation of regression trees, due to Breiman [1996], and is commonly called BAGGING.

The BAGGING model is applied to predicting rainfall intensities over durations of one to ﬁve days, with the results using 200 nodes and 40 trees shown in Table 7. The out-of-sample R-squared statistics are higher than those of the OLS and regression tree models.

By increasing the number of nodes to 50,000 and the number of trees to 200, the regression trees are better able to fit the data in-sample, particularly the extreme values, as shown in Table 8. As suggested by the positive R-squared statistic, BAGGING can predict rainfall intensities in excess of 15mm per day for a three-day duration and of 45mm per day for a five-day duration, but mostly its out-of-sample performance is poor. It is likely that many more trees are required to ameliorate this deficiency, which will demand a significant increase in the level of computing power than that used here, and will be investigated in subsequent work on the subject.

23 Statistic 1D 2D 3D 4D 5D R-squ (in-sample) 14.1% 19.1% 23.8% 27.4% 30.2% R-squ (out-of-sample) 8.1% 11.6% 15.1% 18.1% 20.9% R-squ (out-of sample -2.1% -2.6% -0.4% -1.1% -1.6% for predicted RF≥15mm) R-squ (out-of sample -32.2% -12.6% -10.2% N/A -563.1% for predicted RF≥30mm) R-squ (out-of sample -22168.5% N/A N/A N/A N/A for predicted RF≥45mm) R-squ (out-of sample -22168.5% N/A N/A N/A N/A for predicted RF≥60mm) R-squ (out-of sample -22168.5% N/A N/A N/A N/A for predicted RF≥75mm) R-squ (out-of sample N/A N/A N/A N/A N/A for predicted RF≥90mm)

Table 7: R-squared statistics of the bootstrapped aggregation of regression tree ﬁts using 200 splits and 40 trees

Statistic 1D 2D 3D 4D 5D R-squ (in-sample) 65.6% 78.3% 84.3% 88.1% 90.6% R-squ (out-of-sample) 4.9% 8.3% 11.5% 14.8% 17.4% R-squ (out-of sample -2.3% -1.2% 0.1% -0.6% -0.4% for predicted RF≥15mm) R-squ (out-of sample -3.7% -3.3% -4.8% -0.9% -5.3% for predicted RF≥30mm) R-squ (out-of sample -3.9% -4.8% -5.7% -44.6% 52.3% for predicted RF≥45mm) R-squ (out-of sample -15.0% -22.0% N/A N/A N/A for predicted RF≥60mm) R-squ (out-of sample -1454.6% N/A N/A N/A N/A for predicted RF≥75mm) R-squ (out-of sample N/A N/A N/A N/A N/A for predicted RF≥90mm)

Table 8: R-squared statistics of the bootstrapped aggregation of regression tree ﬁts using 50,000 splits and 200 trees

24 6. Conclusions

It has been shown how bootstrapped aggregation of regression trees can provide better prediction of extreme rainfall events out-of-sample than OLS and single regression tree models. Increasing the number of splits in the regression tree models allows adherence of the model to in-sample data and increasing the number of trees beyond 200 in the aggregated regression tree model in Section 5.3, will potentially give reliable out-of-sample predictions of rainfall intensities of 90mm or more and ﬂood events. This will entail the use of greater computing power than used here.

What is apparent from the R-squared statistics is that the prediction powers of the models are higher for rainfall intensities of ﬁve days’ duration than for shorter durations. Clearly, it is the high intensity rainfall events over short durations which cause most ﬂoods and therefore there is much improvement to be done on the analysis here.

Another point is that no matter how good the model, if there is little information content in the data then the predictive power of the model will be low. Thus, additional improvements to the modelling can be made by incorporating additional data such as temperature data, air pressure data, river gauge readings, hydrological maps and topographical maps. Subsequent work will incorporate such additional data as well as spatial models such as those described in Gschlössl [2006] and Lehmann et al. [2015].

25 References

R. A. Berk. Statistical Learning from a Regression Perspective. Springer, 2008.

O. Boucher, J. Haigh, D. Hauglustaine, J. Haywood, G. Myhre, T. Nakajima, G.Y. Shi, and S. Solomon. Radiative forcing of climate change. In Climate Change 2001: The Scientiﬁc Basis, pages 351–416. Intergovernmental Panel on Climate Change, 2001.

L. Breiman. Bagging predictors. Machine Learning, 24(2):123–140, 1996.

S. Davis. Record-breaking La Nina events. http://www.bom.gov.au/climate/enso/ history/La-Nina-2010-12.pdf, July 2012.

D. Day. The Weather Watchers: 100 Years of the Bureau of Meteorology. Melbourne University Publishing, 2007.

A. Donald. Australian rainfall and the MJO. The Australian Cotton Grower, pages 40–42, April-May 2004.

O. Gal and R. Chen-Morris. The archaeology of the inverse square law: Metaphysical images and mathematical practices. History of Science, 43: 391–414, 2005.

S. Gschlössl. Hierarchical bayesian spatial regression models with applications to non-life insurance. Doctoral dissertation, Technische Universität München, 2006.

E.A. Lehmann, A. Phatak, A. Stephenson, and R. Lau. Spatial modelling framework for the characterisation of rainfall extremes at different durations and under climate change. Environmetrics, 27(4):239–251, 2015.

R. A. Madden and P.R. Julian. Description of global-scale circulation cells in the tropics with a 40-50 day period. Journal of Atmospheric Science, 29:1109– 1123, 1972.

M. Milankovic. Theorie Mathematique des Phenomenes Thermiques produits par la Radiation Solaire. Gauthier-Villars Paris, 1920.

L. F. Richardson. Weather Prediction by Numerical Process. Cambridge, The University press, 1922.

J.S. Risbey, M.J. Pook, P.C. Mclntosh, M.C. Wheeler, and H.H. Hendon. On the remote drivers of rainfall variability in Australia. American Meteorological Society Monthly Weather Review, 137(10):3233–3253, 2009.

A. Savitzky and M.J.E. Golay. Smoothing and differentiation of data by simpliﬁed least squares procedures. Analytical Chemistry, 36(8):1627–1639, 1964.

26 J. Solheim, K. Stordahl, and O. Humlum. The long sunspot cycle 23 predicts a signiﬁcant temperature decrease in cycle 24. Journal of Atmospheric and Solar-Terrestrial Physics, 80:267–284, 2012.

M.C. Wheeler and H.H. Hendon. An all-season real-time multivariate MJO index: Development of an index for monitoring and prediction. American Meteorological Society Monthly Weather Review, 132(8):1917–1932, 2004.

R. Whitaker. The Complete Book of Australian Weather. Allen and Unwin, 2010.

27 A. List of Predictive Factors

# Predictive Factor 1 Sunspot number 2 SG Filter of sunspot number 3 Proportion of time in current sunspot cycle 4 Latitude at which Sun is overhead 5 Proportion of time in current tilt cycle 6 Distance from Sun 7 Proportion of time in current orbital period 8 Southern Oscillation Index 9 Indian Ocean Dipole 10 Southern Annular Mode 11 Madden-Julian Oscillation: RMM1 12 Madden-Julian Oscillation: RMM2 13 Madden-Julian Oscillation: r 14 Madden-Julian Oscillation: theta 15 Madden-Julian Oscillation: phase 16 Longitude of rainfall station 17 Latitude of rainfall station 18 Rainfall station’s height above sea level 19 Distance rainfall station from nearest major river 20 Distance rainfall station from coast 21 Intensity of Sun at rainfall station 22 Average daily rainfall over previous years in that month at location 23 Standard deviation of daily rainfall over previous years in that month at location

Table 9: List of predictive factors for rainfall levels

28 B. List of Rainfall Stations

# Station Name State Station # Long. Lat. Alt. 1 NORTH ADELAIDE SA 23011 138.6 -34.92 48 2 ADELAIDE AIRPORT SA 23034 138.52 -34.95 2 3 ADELAIDE (GLEN OSMOND) SA 23005 138.65 -34.95 128 4 ADELAIDE (MORPHETTVILLE RACECOURSE) SA 23098 138.54 -34.97 11 5 NORTH ADELAIDE SA 23011 138.6 -34.92 48 6 ALBANY WA 9500 117.88 -35.03 3 7 ALICE SPRINGS AIRPORT NT 15590 133.89 -23.8 546 8 BOND SPRINGS HOMESTEAD NT 15631 133.92 -23.54 764 9 CAPE LEEUWIN WA 9518 115.14 -34.37 13 10 FOREST GROVE WA 9547 115.08 -34.07 60 11 BALLARAT AERODROME VIC 89002 143.79 -37.51 435 12 BUNGAREE (KIRKS RESERVOIR) VIC 87014 143.93 -37.55 511 13 RAYWOOD VIC 81041 144.21 -36.53 127 14 CLIFTON HILLS SA 17016 138.89 -27.02 32 15 BIRDSVILLE POLICE STATION QLD 38002 139.35 -25.9 37 16 ROSEBERTH STATION QLD 38020 139.59 -25.79 47 17 BORROLOOLA NT 14710 136.3 -16.07 17 18 BORROLOOLA AIRPORT NT 14723 136.3 -16.08 16 19 CENTRE ISLAND NT 14703 136.82 -15.74 12 20 MCARTHUR RIVER MINE AIRPORT NT 14704 136.08 -16.44 40 21 BOULIA AIRPORT QLD 38003 139.9 -22.91 162 22 ELROSE STATION QLD 37103 140.11 -22.82 108 23 MARION DOWNS QLD 38014 139.66 -23.37 124 24 COLLERINA (KENEBREE) NSW 48052 146.52 -29.77 107 25 FORDS BRIDGE (DELTA) NSW 48194 145.29 -30.08 111 26 TOOWONG BOWLS CLUB QLD 40245 152.99 -27.49 10 27 ALDERLEY QLD 40224 153 -27.42 39 28 BROKEN HILL (PATTON STREET) NSW 47007 141.47 -31.98 315 29 BROKEN HILL AIRPORT AWS NSW 47048 141.47 -32 281 30 BROKEN HILL (STEPHENS CREEK RESERVOIR) NSW 47031 141.59 -31.88 238 31 BROOME AIRPORT WA 3003 122.24 -17.95 7 32 BIDYADANGA WA 3030 121.78 -18.68 11 33 ROEBUCK PLAINS WA 3023 122.47 -17.93 15 34 FAIRYMEAD SUGAR MILL QLD 39037 152.36 -24.79 5 35 DAYBORO POST OFFICE QLD 40063 152.82 -27.2 68 36 MT MEE QLD 40145 152.78 -27.06 465 37 WAMURAN QLD 40343 152.87 -27.04 33 38 CAIRNS AERO QLD 31011 145.75 -16.87 2 39 KURANDA RAILWAY STATION QLD 31036 145.64 -16.82 340 40 MULGRAVE MILL QLD 31049 145.79 -17.09 15 41 AINSLIE TYSON ST NSW 70000 149.14 -35.26 585 42 CARNARVON AIRPORT WA 6011 113.67 -24.89 4 43 BOOLATHANA WA 6003 113.69 -24.65 4 44 BRICKHOUSE WA 6005 113.79 -24.82 15 45 AUGUSTUS DOWNS STATION QLD 29001 139.87 -18.54 75

29 # Station Name State Station # Long. Lat. Alt. 46 BURKETOWN POST OFFICE QLD 29004 139.55 -17.74 6 47 COBAR (DOUBLE GATES) NSW 48034 145.48 -31.69 175 48 COBAR MO NSW 48027 145.83 -31.48 260 49 COBAR (TINDAREY) NSW 48075 145.83 -31.12 190 50 COOBER PEDY SA 16007 134.76 -29 215 51 COOBER PEDY (MCDOUALL PEAK) SA 16027 134.9 -29.84 150 52 COOLGARDIE WA 12018 121.17 -30.96 427 53 MALLINA WA 4059 118.03 -20.88 40 54 DAMPIER SALT WA 5061 116.75 -20.73 6 55 DARWIN AIRPORT NT 14015 130.89 -12.42 30 56 DERBY AERO WA 3032 123.66 -17.37 6 57 YEEDA WA 3026 123.65 -17.62 5 58 EDEN (TIMBILLICA) NSW 69029 149.7 -37.36 35 59 ELLIOTT NT 15131 133.54 -17.56 220 60 ESPERANCE DOWNS RESEARCH STN WA 9631 121.78 -33.6 158 61 MYRUP WA 9584 122 -33.74 60 62 EUCLA WA 11003 128.9 -31.68 93 63 MUNDRABILLA STATION WA 11008 127.86 -31.84 20 64 EXMOUTH GULF WA 5004 114.11 -22.38 15 65 LEARMONTH AIRPORT WA 5007 114.1 -22.24 5 66 EXMOUTH TOWN WA 5051 114.13 -21.93 13 67 YANREY WA 5030 114.79 -22.51 25 68 FITZROY CROSSING AERO WA 3093 125.56 -18.18 115 69 FOSSIL DOWNS WA 3027 125.78 -18.14 120 70 GOGO STATION WA 3014 125.59 -18.29 150 71 GERALDTON TOWN WA 8050 114.61 -28.78 3 72 HOWATHARRA WA 8168 114.62 -28.55 70 73 NABAWA WA 8028 114.79 -28.5 145 74 NORTHAMPTON WA 8100 114.64 -28.35 180 75 SANDSPRINGS WA 8116 114.94 -28.79 220 76 BALGO HILLS WA 13007 127.99 -20.14 420 77 GILES METEOROLOGICAL OFFICE WA 13017 128.3 -25.03 598 78 GOVE AIRPORT NT 14508 136.82 -12.27 52 79 ALCAN MINESITE NT 14509 136.84 -12.26 20 80 BINYA POST OFFICE NSW 75006 146.34 -34.23 141 81 WHITTON (CONAPAIRA ST) NSW 74118 146.18 -34.51 132 82 YENDA (HENRY STREET) NSW 75079 146.19 -34.25 129 83 HALLS CREEK AIRPORT WA 2012 127.66 -18.23 422 84 MOOLA BULLA WA 2020 127.5 -18.19 430 85 HOBART BOTANICAL GARDENS TAS 94030 147.33 -42.87 27 86 HOBART (ELLERSLIE ROAD) TAS 94029 147.33 -42.89 51 87 HOWARD SPRINGS NATURE PARK NT 14149 131.05 -12.46 34 88 KOOLPINYAH NT 14032 131.18 -12.39 30 89 MIDDLE POINT RANGERS NT 14090 131.31 -12.58 10 90 NAPPA MERRIE QLD 45012 141.11 -27.6 66

30 # Station Name State Station # Long. Lat. Alt. 91 INNAMINCKA STATION SA 17028 140.76 -27.72 53 92 KALGOORLIE-BOULDER AIRPORT WA 12038 121.45 -30.78 365 93 BULONG WA 12013 121.75 -30.75 380 94 KARRATHA STATION WA 5052 116.67 -20.88 30 95 LAUNCESTON (DISTILLERY CREEK) TAS 91181 147.21 -41.43 120 96 LAUNCESTON (KINGS MEADOWS) TAS 91072 147.16 -41.47 67 97 COEN AIRPORT EVAP QLD 27006 143.12 -13.76 161 98 MEIN STATION QLD 27014 142.8 -13.18 122 99 LOCKHART RIVER AIRPORT QLD 28008 143.3 -12.79 19 100 LONGREACH AERO QLD 36031 144.28 -23.44 192 101 BEACONSFIELD QLD 36066 144.6 -23.33 230 102 CAMOOLA PARK QLD 36013 144.52 -23.04 188 103 EVESHAM STATION QLD 36022 143.71 -23.03 190 104 SUMMER HILL QLD 36153 144.81 -23.05 240 105 WHITEHILL QLD 36131 144.05 -23.64 210 106 FARLEIGH CO-OP SUGAR MILL QLD 33023 149.1 -21.1 47 107 MACKAY M.O QLD 33119 149.22 -21.12 30 108 PLEYSTOWE SUGAR MILL QLD 33060 149.04 -21.14 27 109 GABO ISLAND LIGHTHOUSE VIC 84016 149.92 -37.57 15 110 FLEMINGTON RACECOURSE VIC 86039 144.91 -37.79 10 111 CAULFIELD (RACECOURSE) VIC 86018 145.04 -37.88 49 112 FLEMINGTON RACECOURSE VIC 86039 144.91 -37.79 10 113 OAKLEIGH (METROPOLITAN GOLF CLUB) VIC 86088 145.09 -37.91 61 114 CALTON HILLS STATION QLD 29118 139.41 -20.14 350 115 WEST LEICHHARDT STATION QLD 29121 139.69 -20.6 280 116 MOUNT ISA MINE QLD 29126 139.48 -20.74 381 117 OBAN STATION QLD 37035 139.05 -21.23 282 118 NANNUP WA 9585 115.77 -33.98 100 119 NEWCASTLE NOBBYS SIGNAL STATION AWS NSW 61055 151.8 -32.92 33 120 NEWCASTLE WATERS NT 15086 133.41 -17.38 209 121 WILLIAMTOWN RAAF NSW 61078 151.84 -32.79 9 122 NHULUNBUY NT 14512 136.76 -12.19 20 123 NORMANTON AIRPORT QLD 29063 141.07 -17.69 18 124 NORMANTON HOSPITAL QLD 29073 141.08 -17.68 12 125 NORMANTON POST OFFICE QLD 29041 141.07 -17.67 8 126 I DUNNO WA 12077 121.71 -32.86 270 127 SALMON GUMS RES.STN. WA 12071 121.62 -32.99 249 128 OODNADATTA AIRPORT SA 17043 135.45 -27.56 117 129 HAMILTON STATION SA 16083 135.08 -26.71 170 130 PARABURDOO AERO WA 7185 117.75 -23.17 424 131 ASHBURTON DOWNS WA 7119 117.03 -23.39 250 132 PARDOO STATION WA 4028 119.58 -20.11 9 133 PERTH AIRPORT WA 9021 115.98 -31.93 15 134 PERTH AIRPORT WA 9021 115.98 -31.93 15 135 GOSNELLS CITY WA 9106 115.98 -32.05 10

31 # Station Name State Station # Long. Lat. Alt. 136 MIDLAND WA 9025 116.02 -31.87 15 137 SUBIACO TREATMENT PLANT WA 9151 115.79 -31.96 20 138 QUORN (OLIVE GROVE) SA 19030 138.02 -32.47 420 139 QUORN SA 19038 138.04 -32.35 295 140 PORT HEDLAND AIRPORT WA 4032 118.63 -20.37 6 141 NORTH SHIELDS (PORT LINCOLN AERODROME) SA 18071 135.87 -34.6 11 142 PORT LINCOLN (WOOLGA) SA 18107 135.76 -34.59 228 143 TELEGRAPH POINT (FARRAWELLS ROAD) NSW 60031 152.79 -31.34 10 144 ROCKHAMPTON AERO QLD 39083 150.48 -23.38 10 145 GLENLANDS QLD 39043 150.51 -23.53 42 146 GRACEMERE - LUCAS ST QLD 39049 150.46 -23.46 31 147 MOONMERA QLD 39067 150.4 -23.58 137 148 SOMERSET QLD 27035 142.6 -10.75 NaN 149 BAMAGA QLD 27031 142.38 -10.9 58 150 CAPE YORK POST OFFICE QLD 27004 142.53 -10.7 40 151 FREELING SA 23325 138.81 -34.46 182 152 FREELING RAILWAY SA 23303 138.82 -34.45 191 153 HORN ISLAND AIRPORT QLD 27027 142.29 -10.59 13 154 PUNSAND BAY QLD 27062 142.46 -10.72 11 155 THURSDAY ISLAND TOWNSHIP QLD 27021 142.22 -10.58 8 156 DOONDI QLD 44178 148.46 -28.25 183 157 KATOOTA QLD 43065 148.71 -27.77 204 158 CAPE SORELL TAS 97000 145.17 -42.2 19 159 LAKE MARGARET DAM TAS 97006 145.57 -41.99 665 160 SYDNEY (OBSERVATORY HILL) NSW 66062 151.21 -33.86 39 161 SYDNEY AIRPORT AMO NSW 66037 151.17 -33.95 6 162 SYDNEY BOTANIC GARDENS NSW 66006 151.22 -33.87 15 163 CENTENNIAL PARK NSW 66160 151.23 -33.9 38 164 SYDNEY (OBSERVATORY HILL) NSW 66062 151.21 -33.86 39 165 ATTUNGA (GARTHOWEN) NSW 55000 150.86 -30.91 369 166 MOONBI (BELLEVUE) NSW 55143 151.07 -31.02 462 167 TELFER AERO WA 13030 122.23 -21.71 292 168 LOCHNAVAR WA 4058 121.02 -20.7 NaN 169 RPF 672 MILE WA 4065 121.1 -22.7 NaN 170 RPF 734 MILE WA 4034 121.4 -22.02 NaN 171 RPF 798 MILE WA 4066 121.2 -21.2 NaN 172 WARRAWAGINE WA 4041 120.7 -20.85 159 173 HELIDON POST OFFICE QLD 40096 152.12 -27.55 155 174 WITHCOTT QLD 40672 152.02 -27.55 261 175 WARBURTON AIRFIELD WA 13011 126.58 -26.13 459 176 AURUKUN SHIRE COUNCIL QLD 27000 141.72 -13.35 13 177 WEIPA EASTERN AVE QLD 27042 141.88 -12.63 20 178 MERLUNA QLD 27044 142.45 -13.06 87 179 MORETON TELEGRAPH STATION QLD 27015 142.64 -12.45 40 180 WYNDHAM WA 1013 128.12 -15.49 11 181 CARLTON HILL WA 2005 128.53 -15.49 20

Table 10: List of rainfall stations shown in the map in Figure 8

32 C. List of Major Rivers of Australia

Number River Number River 1 Alice 30 Katherine 2 Ashburton 31 Lachlan 3 Barcoo 32 Latrobe 4 Barwon 33 Leichhardt 5 Blackwood 34 Mackenzie 6 Bogan 35 Macleay 7 Burdekin 36 Macquarie 8 Campaspe 37 Margaret 9 Clarence 38 Mitchell 10 Condamine 39 Murchison 11 Coopers Creek 40 Murray 12 Daly 41 Murrumbidgee 13 Darling 42 Namoi 14 Dawson 43 Nicholson 15 De Grey 44 Ord 16 Derwent 45 Ovens 17 Diamantina 46 Palmer 18 Drysdale 47 Roper 19 Finke 48 Shoalhaven 20 Fitzroy 49 Snowy 21 Fitzroy 50 Staaten 22 Flinders 51 Suttor 23 Fortescue 52 Swan/Avon 24 Gascoyne 53 Tamar 25 Georgina 54 Thompson 26 Goulburn 55 Victoria 27 Hunter 56 Warrego 28 Isaac 57 Wilton 29 Jardine 58 Yarra

Table 11: List of major rivers in Australia