statistical modelling of precipitation time series including probability assessments of extreme...
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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
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)