1. Statistical modelling of precipitation time series including probability assessments of extreme events Silke Trömel and Christian-D. Schönwiese Institute for Atmosphere and Environment J. W. Goethe University Frankfurt/M., Germany

2. Gaussian assumptions

3. Statistical modelling of climate time series Parameter P1(t): Trends Annual cycle Episodic component Modell: Gaussian distribution

4. Statistical modelling of climate time series Parameter P1(t): Trends Annual cycle Episodic component Parameter P2(t): Trends Constant annual cycle Modell: Gaussian distribution

5. Statistical modelling of climate time series Parameter P1(t): Trends Annual cycle Episodic component Parameter P2(t): Trends Constant annual cycle Modell: Gumbel distribution

6. Statistical modelling of climate time series Parameter P1(t): Trends Annual cycle Episodic component Parameter P2(t): Trends Constant annual cycle Modell: Gumbel distribution

7. Statistical modelling of climate time series Parameter P1(t): Trends Annual cycle Episodic component Parameter P2(t): Trends Constant annual cycle Modell: Weibull distribution

8. Statistical modelling of climate time series Parameter P1(t): Trends Annual cycle Episodic component Parameter P2(t): Trends Constant annual cycle Modell: Weibull distribution

9. The distance function Gaussian distribution PDF Least Squares ML Distance function ML

10. Different distributions and their distance functions Gaussian distribution: Least-squares : Random number Pdf Random number Distance function

11. Different distributions and their distance functions Weibull distribution: Frequency Precipitation [mm] Distance function Precipitation [mm] Gumbel distribution: Precipitation [mm] Pdf

12. Analyses of a German station network 132 time series of monthly precipitation totals, 1901-2000 Realization of a Gumbel distributed random variable Eisenbach-Bubenbach

13. Example: Eisenbach-Bubenbach [47.97 o N, 8.3 o E]

15. The expected value

16. …of a Gumbel distributed random variable with time-dependent location parameter a G (t) and time-dependent scale parameter b G (t) Precipitation [mm] Pdf [1/mm]

17. The expected value …of a Gumbel distributed random variable with time-dependent location parameter a G (t) and time-dependent scale parameter b G (t) Precipitation [mm] Pdf [1/mm]

18. Germany: Changing probability of extreme events > 95th percentile January < 5th percentile January

19. Germany: Changing probability of extreme events < 5th percentile August > 9 5th percentile August

20. Trend estimates by comparison LS January Gumbel January

21. Conclusions The introduced generalized time series decomposition technique allows a free choice of the underlying PDF The signal is detected in two instead of one parameter of the PDF Statistical modeling of precipitation time series can be achieved The analytical description of the time series 1.allows probability assessments of extreme values for every time step during the observation period 2. provides trend estimates taking into account the statistical characteristics (of precipitation)