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 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. 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 50 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 = p (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.