Jules Beersma: Advanced delta change method for time series transformation

Advanced delta change method for time series transformation

Jules BeersmaAdri Buishand & Saskia van Pelt

Workshop “Non-stationary extreme value modelling in climatology”

Technical University of LiberecFebruary 15-17, 2012

TU of Liberec, 15-17 February 2012 2

Outline

• Introduction

• Delta methods

• Study area: Rhine basin

• Results

• Conclusions

• Future work

• Natural variability…


Introduction

Climate model

Impact modele.g. change in river discharge

Direct methodDelta method

or

Time series transformation

?


Delta method

Temperature: additive change

T* = T + (TF –TC)

Precipitation: factorial change

P* = PF / PC × P


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


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





b11

bC0.60

F0.60 g(PPa )

)}P/(gP{g

)}P/(gP{gb

C0.601

C0.902

F0.601

F0.902

log

log

C0.60

O0.601 PPg C

0.90O

0.902 PPg





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



0 ,)/1(1)( /1 xxxG

11

GxG

Generalized Pareto Distribution:

F

CCFCFx

x 1/1 /

Quantile function (inverse):

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:

xGGx CF1



If then:

And the mean of the excesses:

Similarly for the Weibull distribution:

CF

11 E

1

E

xx CF )/(

xx CF )/( and

and thus xEEx CF )/(

F

CCFCFx

x 1/1 /


Points of attention (1)

Q =Default

(SPLUS, R)Median

unbiased

P0.90 1.25 1.23

P0.95 1.29 1.12

P0.95, overlapping 5d 1.25 1.21

Choice of QChange in mean excess EF / EC

(Empirical estimates based on order statistics)


Points of attention (2a)

Bias correction factors

are needed to correct coefficients a and b because of systematic climate model biases in PC

0.60 and PC0.90:

g1 = PO0.60 / PC

0.60

g2 = PO0.90 / PC

0.90


Points of attention (2b)

Effect of bias correction factors

Relative change in the mean annual maxima of 10-day basin-average precipitation


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


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


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


Schematicrepresentation 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


Results (1a)

● 13 GCMs (A1B)

● 5 RCMs (A1B)

● 5 RCMs (bias corrected; direct method)


Results (1b)

GCM RCM GCM References RCM References

CGCM3.1T63 CNRM-CM3

(Flato, 2005)(Salas-Mélia et al., 2005)

CSIRO-Mk (Gordon et al., 2002)

ECHAM5r1 REMO_10 (Roeckner et al., 2003) (Jacob, 2001)

ECHAM5r3 RACMOREMO

(Lenderink, 2003)(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)


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)


Results (2)

13 GCMs 5 RCMs

10-

da

y p

rec

ipit

ati

on

(m

m)


Results (3)

10-

da

y p

rec

ipit

ati

on

(m

m)

Deltamethod

Biascorrection


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.


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)


Natural variability…

13 GCMsEssence

10-

da

y p

rec

ipit

ati

on

(m

m)

Natural variability dominates uncertainty range


Natural variability…

How good can we determine the real climate

change signal in extremes?

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Jules Beersma: Advanced delta change method for time series transformation

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