Statistical distribution of climate time series 2.5
2.5.2 Extreme-value analysis of the observed hydro-climate series
Regardless of whether the empirical cumulative density functions (ECDF) can be fitted by a normal or other theoretical distribution, the salient statistical attributes of the ECDF, such as the mean, the median and the variance or standard deviation provide some valuable identification parameters of the climate variable under question. These statistical parameters are summarized in Table 2.9, and this not only for the annual-, but also for the 4-season series of the various hydro-climate variables.
An even more detailed picture is provided by the quantiles of the ECDF at certain percentile levels (Ferro et al. 2005, Schlünzen et al. 2010). These are listed in
Table 2.10 and they serve as the reference values of the hydro-climate for 1971-2006 time period. Based on these distributions of the 20th-century – hydro-climate variables, the possible changes of the climate in the future will be assessed and further discussed in Chapter 6.
The goodness of fit is computed by means of the deviance parameter, , defined as:
where y is the data, ̂ is the vector of the fitted parameters of the null model, i.e. the model to be fitted and ̂ is the fitted vector of the saturated model, i.e. the model with as many unknown parameters as observations, which theoretically should result in the best fit. Obviously, the smaller the deviance, the better is the fit of the model distribution to the data.
The above fitting processes are realized with the package “evd”, as implemented in the R- programming environment (Stephenson 2012).
In Table 2.11, the deviances of the fits of the various climate parameters by the four theoretical extreme values distributions are listed. The results of the table indicate that the GEV distribution provides the best fit to the empirical distributions of all hydro-climate time series, while the Gumbel distribution works second best.
Table 2.11. Average deviance of the fits of the various climate parameters in the study area by the four theoretical extreme values distributions.
station average deviance of the distribution model
GEV Gumbel Fréchet Weibull
PCP 188 192 >1x106 >1x106
WetDay 209 242 >1x106 >1x106
Tmax 90 95 >1x106 >1x106
Tmin 76 81 >1x106 >1x106
Streamflow 98 102 >1x106 >1x106
From the quantiles q of these fitted four distributions, i.e. the inverse of distribution function for a given probability p or its inverse, the return period T=1/p, exceedance values are then computed for each climate- as well as the streamflow series. The corresponding return-period level plots are shown, up to the 100-year recurrence period, in Figure 2.22 and Figure 2.23 for the precipitation and streamflow, respectively. One may notice that when the return period is above 10 years, the differences in the exceedance levels for the different fitted distributions are becoming significantly large which, in turn, reflects the differences in the distributional fits.
In Figure 2.24, the fits of the streamflow data at station z15 by the GEV- and Gumbel models are analyzed.in more detail. The estimated extreme values from both models are nearly similar up to the 10-year return period. However, at larger return periods, up to the maximal observed period of about 30 years, GEV underestimates the observed streamflow, while Gumbel overestimates them. As shown in the corresponding panels for the density functions, this is due to the differences in the fits of the empirical density function by the two theoretical extreme value distributions, namely, in the upper tail section of the observed density function.
In Table 2.12 and Table 2.13 the exceedance values for various return periods, up to 100 years, as retrieved from the results of the extreme value analysis with the Gumbel- and the GEV-model, are listed for the average climate data series in the study area and the three streamflow series, respectively. Note that in these two tables, in addition to the regular annual series already analyzed, results of the data series separated into the 4-season scheme, as discussed in the previous section, is also presented. The exceedance values in the two tables indicate that for the
( ( | ̂ ) ( | ̂ )) (2.15)
Figure 2.22. Exceedance/return period plots of 1971-2006 monthly precipitation at 24 stations fitted with four extreme value distributions: GEV, Gumbel, Fréchet and Weibulls.
Figure 2.23. Similar to Figure 2.22, but for the monthly streamflow at the three gauge stations.
Figure 2.24. Probability-, quantile- density function- and return level plots of the extreme monthly 1971-2006 runoff at gauge-station z15 using the generalized extreme value (GEV)- (top four panels) and the Gumbel distribution (bottom four panels).
Table 2.12. Average extreme exceedance values for monthly climate variables by using 4 temperature stations and 24 precipitation gauges during years 1971-1999 at return periods 2,5,10, 20, 50 and 100-years, as computed from a fitted GEV- and a Gumbel- extreme value distribution, and using the full annual climate data or separating the latter into four seasons: dry, pre-monsoon (pm), monsoon1 (ms1) and monsoon2 (ms2) (see Table 2.4).
variable season exceedance climate values for different return periods
GEV Gumbel
2-yr 5-yr 10-yr 20-yr 50-yr 100-yr 2-yr 5-yr 10-yr 20-yr 50-yr 100-yr Tmax
(°C)
annual 33.8 34.6 35.1 35.5 36.1 36.6 33.7 34.6 35.2 35.7 36.4 37.0 dry 32.7 33.5 33.9 34.4 34.9 35.3 32.6 33.5 34.0 34.6 35.3 35.8 pm 33.8 34.5 35.0 35.4 35.8 36.2 33.7 34.6 35.1 35.7 36.4 37.0 ms1 32.2 32.9 33.4 33.8 34.3 34.8 32.2 32.9 33.4 33.8 34.4 34.8 ms2 32.1 32.8 33.2 33.6 34.1 34.4 32.0 32.8 33.3 33.8 34.4 34.9 Tmin
(°C)
annual 26.9 27.5 27.8 28.1 28.5 28.7 26.8 27.5 28.0 28.5 29.1 29.6 dry 25.5 26.2 26.5 26.7 27.0 27.1 25.3 26.3 27.0 27.7 28.5 29.1 pm 26.8 27.4 27.7 28.0 28.2 28.4 26.7 27.6 28.1 28.6 29.3 29.8 ms1 26.0 26.5 26.9 27.2 27.7 28.1 26.0 26.5 26.9 27.3 27.7 28.1 ms2 24.1 24.5 24.7 24.8 25.0 25.1 24.0 24.6 25.0 25.4 25.8 26.2 PCP
(mm /day)
annual 9.9 12.7 14.6 16.5 19.1 21.4 9.8 12.8 14.7 16.6 19.1 20.9 dry 1.6 2.8 3.7 4.7 6.2 7.6 1.7 2.9 3.6 4.3 5.3 6.0 pm 6.6 8.8 10.2 11.4 13.0 14.1 6.4 8.9 10.5 12.1 14.1 15.6 ms1 7.9 10.8 12.9 15.3 18.8 21.9 8.1 10.8 12.6 14.3 16.5 18.2 ms2 6.7 9.8 11.8 13.6 16.0 17.8 6.6 9.9 12.0 14.1 16.7 18.7
Table 2.13. Similar to Table 2.12, but for the monthly 1974-2006 streamflow at the three gauge stations.
station season exceedance streamflow (m3/s) for different return periods
GEV Gumbel
2-yr 5-yr 10-yr 20-yr 50-yr 100-yr 2-yr 5-yr 10-yr 20-yr 50-yr 100-yr
Z4
annual 10.9 14.5 15.8 16.5 17.0 17.2 9.0 14.4 18.0 21.5 26.0 29.4 dry 1.5 2.1 2.5 2.9 3.4 3.7 1.5 2.1 2.5 2.9 3.4 3.8 pm 5.9 7.5 8.1 8.5 8.8 9.0 5.2 7.3 8.8 10.1 11.9 13.2 ms1 5.5 9.5 13.6 18.9 28.9 39.6 6.2 9.6 11.9 14.0 16.8 18.9 ms2 11.1 14.3 15.7 16.6 17.5 17.9 10.1 14.3 17.1 19.8 23.3 25.9
Z15
annual 6.0 8.8 10.5 11.9 13.4 14.5 5.8 8.9 11.0 13.0 15.6 17.6 dry 0.2 0.5 0.9 1.5 2.9 4.5 0.4 0.8 1.1 1.4 1.7 2.0 pm 1.8 3.2 4.3 5.4 7.0 8.3 1.9 3.3 4.1 5.0 6.0 6.8 ms1 3.1 6.3 9.9 15.2 26.2 39.3 3.9 6.5 8.2 9.9 12.0 13.6 ms2 5.0 7.7 9.7 11.9 15.1 17.9 5.2 7.7 9.4 11.0 13.1 14.6
Z38
annual 7.2 8.8 9.2 9.5 9.7 9.7 6.1 9.1 11.1 13.0 15.4 17.3 dry 0.8 1.3 1.7 2.0 2.5 2.9 0.8 1.3 1.7 2.0 2.4 2.8 pm 2.7 4.5 5.7 6.8 8.3 9.3 2.7 4.5 5.7 6.8 8.3 9.4 ms1 4.3 7.0 8.9 10.9 13.6 15.8 4.5 7.0 8.7 10.4 12.5 14.1 ms2 6.7 8.4 9.1 9.6 10.0 10.2 6.2 8.4 9.8 11.3 13.1 14.4
extreme temperatures, the precipitation and the streamflow the corresponding estimates are higher for the Gumbel- than for the GEV model.
The results of this extreme-value analysis of the past observed (20th–century) hydro-climate data in the study region will serve as a reference, when analyzing possible extremes of the future climate and its effect on extreme streamflow (floods) in Chapter 6.