1. Advanced delta change method for
time series transformation
Jules Beersma
Adri Buishand & Saskia van Pelt
Workshop “Non-stationary extreme value
modelling in climatology”
Technical University of Liberec
February 15-17, 2012

2. Outline
• Introduction
• Delta methods
• Study area: Rhine basin
• Results
• Conclusions
• Future work
• Natural variability…
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3. Introduction
Climate
model
Delta method
Direct method
or
? Time series
transformation
Impact model
e.g. change in river discharge
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4. Delta method
Temperature: additive change
T* = T + (TF –TC)
Precipitation: factorial change
P* = PF / PC × P
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5. Delta method
● Linear: P* = aP (classical delta method)
Relative change in std. deviation and all
quantiles is the same as that in the mean
● Non-linear: P* = aPb
Changes in the quantiles different from the
change in the mean if b ≠ 1
May however give unrealistic changes
in the extremes if b > 1
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6. Advanced delta method
P* = aPb for P ≤ Q
P* = aQb + EF/EC (P - Q) for P > Q
where:
Q is a large quantile
EC is the mean excess over the quantile Q
in the Control climate
EF the same for the Future climate
Coefficients a and b follow from future changes in e.g.
P0.60 and P0.90
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7. Advanced delta method
P* = aPb for P ≤ Q
P* = aQb + EF/EC (P - Q) for P > Q
log{g 2 × P0.90 /(g1 × P0.60 )}
F F
b=
log{g 2 × P0.90 /(g1 × P0.60 )}
C C
1− b
a = P0.60 (P0.60 ) b × g1
F C
g1 = P0.60 P0.60
O C
g 2 = P0.90 P0.90
O C
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8. Advanced delta method
P* = aPb for P ≤ Q
P* = aQb + EF/EC (P - Q) for P > Q
This transformation is obtained if:
● Excesses follow a Generalized Pareto Distribution (GPD)
● The shape parameter of the GPD does not change
May be robust against the GPD, but it is essential that
the shape of the upper tail does not change
 difficult to check
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9. Advanced delta method
Generalized Pareto Distribution:
−1 / κ
G ( x) = 1 − (1 + κ x / σ ) , x≥0
Quantile function (inverse):
xG =
[
σ (1 − G )
−κ
−1 ]
κ
Assume GC and GF are the distributions of the excesses in
the Current and the Future climate with respectively
σC , κC and σF , κF then:
x = G [ GC ( x ) ]
∗ −1
F
∗
x =
[
σ F (1 + κ C x / σ C )
κ F /κC
−1 ]
κ F Lib e re c, 1 5-1 7 F e b ru ary 201 2
TU of 9

10. Advanced delta method
x = ∗ σF [ (1 + κ C x /σC )
κ F /κC
]
−1
κF
If κ F = κ C then:
x = (σ F / σ C ) x
∗
And the mean of the excesses:
σ
E= and thus x ∗ = ( E F / EC ) x
1− κ
Similarly for the Weibull distribution:
x ∗ = (σ F / σ C ) x of andc, 1 E 7=eσ ary 201 2+ 1 ν )
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Γ (1 10

11. Points of attention (1)
Choice of Q
Change in mean excess EF / EC
(Empirical estimates based on order statistics)
Default Median
Q=
(SPLUS, R) unbiased
P0.90 1.25 1.23
P0.95 1.29 1.12
P0.95, overlapping 5d 1.25 1.21
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12. Points of attention (2a)
Bias correction factors
are needed to correct coefficients a and b
because of systematic climate model biases in
PC0.60 and PC0.90:
g1 = PO0.60 / PC0.60
g2 = PO0.90 / PC0.90
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13. Points of attention (2b)
Effect of bias correction factors
Relative
change in the
mean annual
maxima of
10-day basin-
average
precipitation
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14. Points of attention (3)
Smoothing
Smoothing of coefficients and quantiles in space
and/or time
P0.60 and P0.90: varies over the year (3-month moving
average)
EF / EC and b: varies over the year but smoothed spatially
a: varies over the year and over space
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15. Study area:
Rhine basin
13 GCMs & 5 RCMs(A1B)
134 sub catchments
(for hydrological modelling)
Extreme river discharges
 Extreme multi-day
precipitation amounts
5 RCMs; bias corrected,
direct method
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16. Study area:
Rhine basin
● P ≡ 5-day precipitation sums at the grid cell scale
● Quantiles P0.60 and P0.90 , coefficients a and b and
excesses E are calculated for each grid cell and each
calendar month:
● a calendar month is six 5-day periods (= 30 days) or
● zeven 5-day periods (= 35 days) for December
● Temporal smoothing (3-month moving averages) of
quantiles and excesses
● Spatial smoothing (median of grid cells) of b
and EF / EC
 similar effect as regional frequency analysis
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17. Schematic
representation
of the
procedure
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18. Schematic
representation of
the procedure
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19. Schematic
representation of
the procedure
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20. Schematic
representation of the
procedure
● Each sub basin gets the same R as the corresponding
grid cell
● Daily amounts get the same R as the 5-day amounts
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21. Results (1a)
● 13 GCMs (A1B)
● 5 RCMs (A1B)
● 5 RCMs (bias corrected; direct method)
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22. Results (1b)
GCM RCM GCM References RCM References
CGCM3.1T63 (Flato, 2005)
CNRM-CM3 (Salas-Mélia et al., 2005)
CSIRO-Mk (Gordon et al., 2002)
ECHAM5r1 REMO_10 (Roeckner et al., 2003) (Jacob, 2001)
ECHAM5r3 RACMO (Lenderink, 2003)
REMO (Jacob, 2001)
GFDL-CM2.0 (Delworth et al., 2006)
GFDL-CM2.1
HADCM3Q0 CLM (Gordon et al., 2000) (Steppeler et al., 2003)
HADCM3Q3 HADRM3Q3 (Jones, 2004)
IPSL-CM4 (Marti et al., 2005)
MIROC3.2 hires (Hasumi and Emori, 2004)
MIUB (Min et al., 2005)
MRI-CGCM2.3.2 (Yukimoto et al., 2006)
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23. Results (1c)
● 13 GCMs (A1B)
● 5 RCMs (A1B)
● 5 RCMs (bias corrected; direct method)
 Quantiles of 10-day precipitation
● Future (2081 – 2100) w.r.t. Current (1961-1995) climate
● basin-average
● winter half year (Oct – Mar)
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24. Results (2)
13 GCMs
5 RCMs
10-day precipitation (mm)
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25. Results (3)
10-day precipitation (mm)
Delta
method
Bias
correction
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26. Conclusions
● Extreme quantiles of 10-day basin-average
precipitation in winter increase in the future climate in
all (18) climate model simulations
● 13 GCMs and 5 RCMs have similar spread in extreme
quantiles of 10-day basin-average precipitation
● Similar changes and spread of changes between the
5 RCMs based on the (advanced) delta method and
on a (non-linear) bias correction method.
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27. Future work
● Large ensemble of GCMs ~50 from CMIP5
● Coupling to hydrological model (HBV) of the Rhine
● Test performance under dry conditions (left tail)
● Application to different river basins / areas?
● Advanced delta change method for daily precipitation
rather than 5-day amounts
 problem of changing wet/dry day frequency
● Use of a similar transformation to remove the
precipitation bias in RCM output (bias correction method)
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28. Natural variability…
13 GCMs
10-day precipitation (mm)
Essence
Natural variability dominates
uncertainty range
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29. Natural variability…
How good can we
determine the real climate
change signal in
extremes?
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