Statistical modelling of precipitation time series including probability assessments of extreme events Silke Trömel and Christian-D. Schönwiese Institute.

Statistical modelling of precipitation time seriesincluding probability assessments of extreme events

Silke Trömel and Christian-D. Schönwiese

Institute for Atmosphere and Environment J. W. Goethe University

Frankfurt/M., Germany

Gaussian assumptions

Statistical modelling of climate time series

Parameter P1(t):TrendsAnnual cycleEpisodic component

Modell: Gaussian distribution

Statistical modelling of climate time series

Parameter P1(t):TrendsAnnual cycleEpisodic component

Parameter P2(t):TrendsConstant annual cycle

Modell: Gaussian distribution

Statistical modelling of climate time series

Parameter P1(t):TrendsAnnual cycleEpisodic component

Parameter P2(t):TrendsConstant annual cycle

Modell: Gumbel distribution

Statistical modelling of climate time series

Parameter P1(t):TrendsAnnual cycleEpisodic component

Parameter P2(t):TrendsConstant annual cycle

Modell: Gumbel distribution

Statistical modelling of climate time series

Parameter P1(t):TrendsAnnual cycleEpisodic component

Parameter P2(t):TrendsConstant annual cycle

Modell: Weibull distribution

Statistical modelling of climate time series

Parameter P1(t):TrendsAnnual cycleEpisodic component

Parameter P2(t):TrendsConstant annual cycle

Modell: Weibull distribution

The distance function

Gaussian distribution

PDF

Least SquaresML

Distance functionML

Different distributions and their distance functions

Gaussian distribution: Least-squares:

Random number

Pd

f

Random number

Dis

tan

ce

fu

nc

tio

n

Different distributions and their distance functions

Weibull distribution:

Fre

qu

en

cy

Precipitation [mm] Precipitation [mm]

Dis

tan

ce

fu

nc

tio

nD

ista

nc

e f

un

cti

on

Precipitation [mm]

Gumbel distribution:

Precipitation [mm]

Pd

f

Analyses of a German station network

• 132 time series of monthly precipitation totals, 1901-2000

• Realization of a Gumbel distributed random variable

Eisenbach-Bubenbach

Example: Eisenbach-Bubenbach [47.97oN, 8.3oE]

Example: Eisenbach-Bubenbach [47.97oN, 8.3oE]

The expected value…of a Gumbel distributed random variable with time-dependent location parameter aG(t) and time-dependent scale parameter bG(t)

Precipitation [mm]

Pd

f [1

/mm

]

The expected value…of a Gumbel distributed random variablewith time-dependent location parameter aG(t) and time-dependent scale parameter bG(t)

Precipitation [mm]

Pd

f [1

/mm

]

Germany: Changing probability of extreme events> 95th percentile

January

< 5th percentile

January

Germany: Changing probability of extreme events< 5th percentile

August

> 95th percentile

August

Trend estimates by comparisonLS

January

Gumbel

January

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 statisticalcharacteristics (of precipitation)