Temperature

The data used in the development of a 2030 data set were based on the TMY data for Adelaide. This was constructed using the methodology of TMY2 as in Marion and Urban (1995). It has 8,760 hourly values. The year runs from 1 December as the year is split into seasons, with summer being December to February. The first step is to identify and model the seasonality which can be represented using discrete Fourier transforms which reveal periodicities in the data as well as the relative strengths of those periodic components (Boland 2008). Several significant cycles were identified using spectral analysis. The power spectrum in Figure 2.1 illustrates that the significant peaks are located at 1,365 and 730 cycles/year, these being the annual, daily and twice-daily cycles. From this analysis, the Fourier series for the seasonality in Adelaide is of the form given in Equation 2.1, with time, ‘t’, in hours.

(2.1)

Figure 2.1: Power spectrum for temperature – the X-axis is in cycles/year

Here T (t) is the seasonal component which can be seen overlaid on the actual data in Figure 2.2. Subtracting the seasonal component from the original hourly time series gives what is called the de-seasoned data. From Figure 2.3, it can be seen that there is a higher variance in summer than winter. However, on examining only the summer data (Figure 2.4), it is apparent that the data are now homoscedastic (common variance) and thus the variance does not need to be considered in the context of

A Framework for Adaptation of Australian Households to Heat Waves 11 climate change adaption. To enable the adjustment of the residuals to create a higher frequency of extreme events in 2030, an appropriate probability distribution is fitted to the frequency distribution of the summer residuals. One that is defined on a closed interval with no assumption of symmetry is the beta distribution, as given in Equation 2.2 where B(a,f3) is the beta function, a and f3 are the shape parameters and the range is a x b.

The model fit (Figure 2.5) is not as good as one would hope. One issue was identified that ultimately led to an improvement. The essential assumption about the residuals from the Fourier series model is that they should be independent of the original series.

However, in plotting these residuals against the original data (Figure 2.6), there is an upward trend in the residuals with an increase in temperature. The dependence can be accounted for using simple linear regression. The frequency distribution of these residuals (data minus model) from this regression analysis is then fitted with a beta distribution. With these new residuals, the beta distribution fit is far more successful as seen in Figure 2.7. This is supported by Figure 2.8 for the first week in December. This is a comparison between the real data after the seasonality has been removed and the corresponding values from the beta model correlated to the frequency distribution of these residuals as depicted in Figure 2.7. Note that when going through this process, a time stamp is added to each data value so that after all adjustments are made to a specific data value, it can be put back into the final model at precisely the same time from which it was derived.

(2.2)

Figure 2.2: Adelaide seasonal model for temperature during the summer

12 A Framework for Adaptation of Australian Households to Heat Waves

Figure 2.3: De-seasoned temperature

Figure 2.4: De-seasoned summer temperature

A Framework for Adaptation of Australian Households to Heat Waves 13 Figure 2.5: Summer standardised residuals and beta distribution

Figure 2.6: Summer temperature residuals against original data

14 A Framework for Adaptation of Australian Households to Heat Waves

Figure 2.7: Summer standardised residuals and beta distribution

Figure 2.8: Summer residuals with beta model

The ‘Climate Change in Australia’ report (Watterson et al. 2007) states that during summer, the maximum temperature increases by up to 5% more than the mean in coastal locations. At the same time, the average increase in Australian minimum temperature is predicted to be approximately 10% higher than the average. This indicates that there needs to be a differential change in the temperature, the minimum is changing at a different rate to that of the maximum. Previous work in this area uses

A Framework for Adaptation of Australian Households to Heat Waves 15 the technique of morphing whereby two transformations are applied to the present data sets (Belcher et al. 2005; Jentsch et al. 2008). With reference to the climate change projections, a climate variable, for example, temperature, is both translated and stretched, thus changing both the mean and the variance. There is no allowance for the differential change of minimum and maximum temperatures. In the present work, the Fourier series daily profile is altered more in the minimum than the maximum.

The process for developing the adjusted data set for temperature is detailed in the following steps:

1. Using the current TMY data, the year is rearranged to flow from December to November to assist with seasonal flow.

2. A Fourier series for the annual, daily and twice-daily components is identified as per Equation 2.1. This analysis is performed for the whole year’s data.

3. The data for the season of interest is extracted and the residuals from the Fourier series are regressed against the temperature data to obtain a linear model for the trend in the residuals. This trend is then subtracted to form the second stage residuals.

4. A beta model is fitted to the frequency distribution of the residuals.

5. The ‘Climate Change in Australia’ report’s (Watterson et al. 2007) projections for climate change mean there will be an alteration in the distributional specifics of the temperature. The procedure in this example entails taking the beta model of the standardised residuals and, keeping the 10^th percentile static, increasing the 50^th percentile by a small amount and the 90^th percentile by a greater amount to account for the increased number of days over 35^oC. This was done by solving a simple optimisation problem wherein we perturbed the α and β, parameter estimates so that the distribution has these altered percentile values.

The effects of the changes to the residuals are evident in the comparison between present and projected cumulative distribution functions as shown in Figure 2.9.

6. The Fourier series in Step 1 is now refitted to accommodate the amount of the seasonal increase in temperature as identified for the location less the amount already accounted for in Step 5.

7. The new Fourier series, the model for the residuals trend (Step 3) and the beta residuals from Step 4 are added together to form the new adjusted data set for the season. The effects of the alterations are shown in Figures 2.10 and 2.11.

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Figure 2.9: Comparison of cumulative distribution functions (CDFs) of original and altered residuals

Figure 2.10: Comparison of maximum and minimum temperatures during the Adelaide summer

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