The document assesses projected changes in seasonal precipitation extremes in the Czech Republic using a non-stationary index-flood statistical model. The model is applied to precipitation data from multiple regional climate models to evaluate extremes in the present climate and projected changes. Key results include an evaluation of biases in quantiles and distribution parameters for the present climate, as well as relative changes in quantiles projected between 1961-1990 and 2070-2099. Homogeneous precipitation regions in the Czech Republic are identified for application of the statistical modeling approach.

1. Assessment of projected changes in seasonal
precipitation extremes in the Czech Republic
with a non-stationary index-flood model
Martin Hanel, Adri Buishand
Technical University in Liberec (TUL)
Royal Netherlands Meteorological Institute (KNMI), De Bilt
Non-stationary extreme value modelling in climatology
2012/02/16 Liberec

2. OUTLINE
1. Introduction, study area & data
2. Statistical model
3. Evaluation for present climate & projected changes
4. Model diagnostics
5. Conclusions

3. INTRODUCTION
Work iniciated within the ENSEMBLES project - WP5.4
Evaluation of extreme events in observational and RCM data
Questions:
- are the precipitation extremes properly represented in the RCM
simulations?
- what are the projected changes in precipitation extremes?
- how large is the uncertainty associated with the estimation of
precipitation extremes and their changes in the climate model
data?
statistical model developed to answer these questions

4. CASE STUDIES
Statistical model applied to assess
1-day summer and 5-day winter precipitation extremes in the
Rhine basin
54
52
5
4
50
2
3
48
1
46
4 6 8 10 12

5. CASE STUDIES
Statistical model applied to assess
1-day summer and 5-day winter precipitation extremes in the
Rhine basin
1-day and 1-hour annual precipitation extremes in the
Netherlands

6. CASE STUDIES
Statistical model applied to assess
1-day summer and 5-day winter precipitation extremes in the
Rhine basin
1-day and 1-hour annual precipitation extremes in the
Netherlands
1-, 3-, 5-, 7-, 10-, 15-, 20- and 30-day precipitation extremes for
all seasons in the Czech Republic

7. DATA
RCM DATA
model acronym source period
--- ECHAM5 driven ---
RACMO RACMO_EH5 KNMI 1950-2100
REMO REMO_EH5 MPI 1951-2100
RCA RCA_EH5 SMHI 1951-2100
RegCM RegCM_EH5 ICTP 1951-2100
HIRHAM HIR_EH5 DMI 1951-2100
--- HadCM3Q0, HadCM3Q3, HadCM3Q16 driven ---
HadRM HadRM_Q0 Hadley Centre 1951-2099
CLM CLM_Q0 ETHZ 1951-2099 ≈ 25 km × 25 km
HadRM HadRM_Q3 Hadley Centre 1951-2099
RCA RCA_Q3 SMHI 1951-2099
HadRM HadRM_Q16 Hadley Centre 1951-2099
SRES A1B
RCA RCA_Q16 C4I 1951-2099
transient
--- ARPEGE driven ---
HIRHAM HIR_ARP DMI 1951-2100 simulations
CNRM-RM CNRM_ARP CNRM 1951-2100
ALADIN-CLIMATE/CZ ALA_ARP CHMI 1961-2100
--- BCM driven ---
RCA RCA_BCM SMHI 1961-2100
OBSERVATIONAL DATA (≈ 25 km x 25 km)
acronym source period
CHMI_OBS CHMI 1950-2007

8. GEV DISTRIBUTION FUNCTION
−1 
 

 x−ξ κ
F(x) = exp − 1 + κ , κ 0
 

 α 

x−ξ
F(x) = exp − exp − , κ=0
α
PROBABILITY DENSITY FUNCTION
ξ ... location parameter
0.3
dgev(seq(−3, 10, 0.1), 0, 1, 0)
0.2
The GEV parameters can
vary over the region (spatial heterogeneity) 0.1
vary with time (climate change)
0.0
−2 0 2 4 6 8 10

9. GEV DISTRIBUTION FUNCTION
−1 
 

 x−ξ κ
F(x) = exp − 1 + κ , κ 0
 

 α 

x−ξ
F(x) = exp − exp − , κ=0
α
PROBABILITY DENSITY FUNCTION
ξ ... location parameter
0.3
dgev(seq(−3, 10, 0.1), 0, 1, 0)
α ... scale parameter
0.2
The GEV parameters can
vary over the region (spatial heterogeneity) 0.1
vary with time (climate change)
0.0
−2 0 2 4 6 8 10

10. GEV DISTRIBUTION FUNCTION
−1 
 

 x−ξ κ
F(x) = exp − 1 + κ , κ 0
 

 α 

x−ξ
F(x) = exp − exp − , κ=0
α
PROBABILITY DENSITY FUNCTION
ξ ... location parameter
0.3
dgev(seq(−3, 10, 0.1), 0, 1, 0)
α ... scale parameter
κ ... shape parameter
0.2
The GEV parameters can
vary over the region (spatial heterogeneity) 0.1
vary with time (climate change)
+
−
0.0
−2 0 2 4 6 8 10

11. STATISTICAL MODEL
Index flood method assumes that precipitation maxima over a region
are identically distributed after scaling with a site-specific factor
SPATIAL HETEROGENEITY
ξ varies over the region
α
κ, γ = ξ are constant over the region
γ is the dispersion coefficient analogous to the coefficient of variation
T-year quantile at any site s can be represented as
1 −κ
1 − − log 1 − T
QT (s) = µ(s) · qT , qT = 1 − γ ·
κ
where qT is a common dimensionless quantile function (growth curve) and µ(s) is a
site-specific scaling factor ("index flood")
it is convenient to set µ(s) = ξ(s)

12. STATISTICAL MODEL
NON-STATIONARITY
location parameter varies over the region, but with common trend
ξ(s, t) = ξ0 (s) · exp [ξ1 · I(t)]
where I(t) is the time indicator
dispersion coefficient and shape parameter are constant over the
region, but vary with time
γ(t) = exp [γ0 + γ1 · I(t)]
κ(t) = κ0 + κ1 · I(t)
UNCERTAINTY
assessed by a bootstrap procedure (1000 samples for each RCM
simulation, season and duration)

13. RELATIVE CHANGES
T-year quantile in time t and location s:
QT (s, t) = ξ(s, t) · qT (t)
Relative change between t2 and t1 (t2 > t1 ) is:
QT (s, t2 ) ξ(s, t2 ) qT (t2 )
= ·
QT (s, t1 ) ξ(s, t1 ) qT (t1 )
ξ(s, t2 )
ξ(s, t) = ξ0 (s) · exp [ξ1 · I(t)] ⇒ = exp ξ1 · [I(t2 ) − I(t1 )]
ξ(s, t1 )

14. RELATIVE CHANGES
T-year quantile in time t and location s:
QT (s, t) = ξ(s, t) · qT (t)
Relative change between t2 and t1 (t2 > t1 ) is:
QT (s, t2 ) ξ(s, t2 ) qT (t2 )
= ·
QT (s, t1 ) ξ(s, t1 ) qT (t1 )
ξ(s, t2 )
ξ(s, t) = ξ0 (s) · exp [ξ1 · I(t)] ⇒ = exp ξ1 · [I(t2 ) − I(t1 )]
ξ(s, t1 )
⇒ the relative change in quantiles can be written as
QT (s, t2 ) qT (t2 )
= exp ξ1 · [I(t2 ) − I(t1 )] ·
QT (s, t1 ) qT (t1 )
which does not depend on s.

15. TIME INDICATOR
Various choices for I(t) are possible:
year t
CHANGE IN PRECIPITATION
e.g., I(t) = t, but more complicated functions
are needed
temperature/ temperature anomaly
We use:
seasonal global temperature anomaly
of the driving GCM

16. TIME INDICATOR
Various choices for I(t) are possible:
SEASONAL GLOBAL TEMPERATURE ANOMALY
JJA
year t
3
ECHAM5
HadCM3Q0
temperature anomaly
HadCM3Q3
2
HadCM3Q16
e.g., I(t) = t, but more complicated functions ARPEGE
1
BCM
CGCM
are needed
0
−1
temperature/ temperature anomaly
−2
1950 2000 2050 2100
DJF
Index
3
temperature anomaly
2
1
ECHAM5
0
We use: HadCM3Q0
−1
HadCM3Q3
HadCM3Q16
ARPEGE
−2
BCM
seasonal global temperature anomaly CGCM
−3
1950 2000 2050 2100
of the driving GCM

17. IDENTIFICATION OF HOMOGENEOUS REGIONS
For each RCM simulation and season we assessed
the grid box estimates of the GEV parameters for the period
1961-1990 and 2070-2099
their changes between these two periods
focus mainly on the dispersion coefficient and changes in location
parameter and dispersion coefficient
formation of homogeneous regions common for all RCM simulations
and seasons is challenging

18. IDENTIFICATION OF HOMOGENEOUS REGIONS
For each RCM simulation and season we assessed
the grid box estimates of the GEV parameters for the period
1961-1990 and 2070-2099
their changes between these two periods
focus mainly on the dispersion coefficient and changes in location
parameter and dispersion coefficient
formation of homogeneous regions common for all RCM simulations
and seasons is challenging
JJA SON DJF MAM

19. IDENTIFICATION OF HOMOGENEOUS REGIONS
For each RCM simulation and season we assessed
the grid box estimates of the GEV parameters for the period
1961-1990 and 2070-2099
their changes between these two periods
focus mainly on the dispersion coefficient and changes in location
parameter and dispersion coefficient
formation of homogeneous regions common for all RCM simulations
and seasons is challenging

20. IDENTIFICATION OF HOMOGENEOUS REGIONS
obviously it is not possible to identify strictly homogeneous regions
common for all RCM simulations and seasons
- however, regional frequency analysis is more accurate than the
at-site analysis even in not strictly homogeneous regions
- allowing more heterogeneity of precipitation maxima in the
statistical model improves goodness-of-fit, however, the
estimated changes are not much different when compared to
the standard model
it turned out that the regions could be acceptably based on the areas
of the eight river basin districts in the Czech Republic
different groupings of the river basin districts were examined with
respect to the lack-of-fit of the statistical model

21. EVALUATION FOR CONTROL CLIMATE
Relative bias in quantiles
0.4
0.4
0.4
0.4
areal mean JJA areal mean DJF areal mean MAM areal mean SON 50
0.0 0.1 0.2 0.3
0.0 0.1 0.2 0.3
0.0 0.1 0.2 0.3
0.0 0.1 0.2 0.3
40
relative bias [% x 100]
relative bias [% x 100]
relative bias [% x 100]
relative bias [% x 100]
return period [years]
30
20
10
5
−0.2
−0.2
−0.2
−0.2
2
1 5 10 20 30 1 5 10 20 30 1 5 10 20 30 1 5 10 20 30
duration [days] duration [days] duration [days] duration [days]
Relative (ξ, γ) and absolute (κ) bias in parameters
relative [% x 100] or absolute [−] bias
relative [% x 100] or absolute [−] bias
relative [% x 100] or absolute [−] bias
relative [% x 100] or absolute [−] bias
areal mean JJA areal mean DJF areal mean MAM areal mean SON ξ
0.4
0.4
0.4
0.4
γ
κ
0.2
0.2
0.2
0.2
−0.4 −0.2 0.0
−0.4 −0.2 0.0
−0.4 −0.2 0.0
−0.4 −0.2 0.0
1 5 10 20 30 1 5 10 20 30 1 5 10 20 30 1 5 10 20 30
duration [days] duration [days] duration [days] duration [days]

22. CHANGES BETWEEN 1961-1990 AND 2070-2099
Relative changes in quantiles
0.4
0.4
0.4
0.4
areal mean JJA areal mean DJF areal mean MAM areal mean SON 50
0.0 0.1 0.2 0.3
0.0 0.1 0.2 0.3
0.0 0.1 0.2 0.3
0.0 0.1 0.2 0.3
relative change [% x 100]
relative change [% x 100]
relative change [% x 100]
relative change [% x 100]
40
return period [years]
30
20
10
5
−0.2
−0.2
−0.2
−0.2
2
1 5 10 20 30 1 5 10 20 30 1 5 10 20 30 1 5 10 20 30
duration [days] duration [days] duration [days] duration [days]
Relative (ξ, γ) and absolute (κ) changes in parameters
relative [% x 100] or absolute [−] change
relative [% x 100] or absolute [−] change
relative [% x 100] or absolute [−] change
relative [% x 100] or absolute [−] change
0.3
0.3
0.3
0.3
areal mean JJA areal mean DJF areal mean MAM areal mean SON ξ
γ
κ
0.2
0.2
0.2
0.2
0.1
0.1
0.1
0.1
0.0
0.0
0.0
0.0
−0.1
−0.1
−0.1
−0.1
1 5 10 20 30 1 5 10 20 30 1 5 10 20 30 1 5 10 20 30
duration [days] duration [days] duration [days] duration [days]

23. MODEL DIAGNOSTICS
For each grid box we calculate the residuals:
we transform the seasonal maxima Xt to:
ˆ
 
1  ξ(t) Xt 
Xt = · log 1 + · −1 ,
 
ˆ ˆ(t) ˆ(t)
 
ξ(t)
 γ µ 
which should have a standard Gumbel distribution;
Pr{Xt ≤ x} = exp [− exp(−x)]
if the model is true.
these residuals can be inspected visually and/or e.g. by the
Anderson-Darling statistics:
∞
[FN (x) − F(x)]2
A2 = N dF(x),
−∞ F(x)[1 − F(x)]
where FN (x) is the empirical distribution function of Xt and F(x)
standard Gumble distribution function.
critical values can be derived using simulation
Xt can be further transformed to have a standard normal distribution:
normal residuals: Φ−1 exp[− exp(Xt )] ∼ N(0, 1)

24. MODEL DIAGNOSTICS
2 Switzerland
loess smoohter (span=0.3)
q
q
q
q
average normal residual
q q
q q
q
1
q
q q q q q q q
q q q
q
q q
q q
q
q
q q q q
q q q
q q
qq q q q
q
q
q q q q
q q q q
q q q q q q
q q
q q q
q q q
q q q q q q
0
q q q
qq q q q q q
q q q q q q
q q
q q qq q q
q q q
q q
q q q q q q
qq q q
q q q q
q q q
q q q
q q
q q q q
q q
q q q
qq qq
−1
q q q
q q
q q
q q q
q
q
q
q
−2
1950 2000 2050 2100

25. MODEL DIAGNOSTICS
Switzerland
average autocorrelation of normal residuals
1.0
0.05 critical values
0.8
0.6
0.4
0.2
0.0
0 10 20 30 40 50 60
lag

26. MODEL DIAGNOSTICS
2.5
2.0 Switzerland
Anderson−Darling statistic
global 0.1 critical value ≈ pointwise 0.0017 critical value
1.5
pointwise 0.1 critical value
gridbox value
1.0
0.5
0 10 20 30 40 50 60
gridbox

27. CONCLUSIONS
Regional GEV modelling provides an informative summary of
changes in parameters and various quantiles (rather than a single
quantile only).
Taking ξ and γ constant over a region strongly reduces standard
errors.
A negative ensemble mean bias decreasing (in absolute value) with
duration is found in summer (6–17%)
The bias is mostly small for short duration precipitation extremes in
the other seasons (-5–2%), however, as the duration gets longer, the
bias is increasing, especially in winter (more than 20%)
The average relative changes in the quantiles of the distribution are
positive for all seasons and durations.
The magnitude of the changes depends on duration only for
summer precipitation extremes and large quantiles in autumn

28. Thank you for attention