Bias correction with CDF-Transform · QQ-mapping !CDF observations (1980 2016) = CDF observations...

Introduction Quantile-mapping CDF-Transform Application Ongoing Summary

Bias correction with CDF-Transform

Komlan A. KPOGO-NUWOKLO, Henning W. RUST, Uwe ULBRICH, ChristosVAGENAS and Edmund MEREDITH

Institut fur Meteorologie, Freie Universitat Berlin, Berlin, Germany

Berlin-Workshop on Bias Correction in Climate Studies

Berlin-Dahlem, 4-6 October 2016

K. A. KPOGO-NUWOKLO et al. Bias correction with CDF-t 1 / 22

Introduction Quantile-mapping CDF-Transform Application Ongoing Summary

• Systematic biases in the Regional Climate Models

6.8 7.0 7.2 7.4 7.6

50.9

51.0

51.1

51.2

51.3

51.4

Observation

lon

lat

4.0

4.5

5.0

5.5

6.0

6.5

6.8 7.0 7.2 7.4 7.6

50.9

51.0

51.1

51.2

51.3

51.4

Model

lon

lat

4.0

4.5

5.0

5.5

6.0

6.5

Figure: Mean of daily precipitation (mm/day) at Wupper catchment (case of July)

K. A. KPOGO-NUWOKLO et al. Bias correction with CDF-t 2 / 22

Introduction Quantile-mapping CDF-Transform Application Ongoing Summary

• Difference in the distribution

0 10 20 30 40 50 60

0.0

0.2

0.4

0.6

0.8

1.0

lon= 7.09 lat= 51.18

Precipitation [mm/day]

CD

F

ObsModel

Need to correct the whole distribution and not only the mean

K. A. KPOGO-NUWOKLO et al. Bias correction with CDF-t 3 / 22

Introduction Quantile-mapping CDF-Transform Application Ongoing Summary

Existing bias correction methods

Mean-based approaches

Linear scaling [Lenderink et al., 2007]

Local intensity scaling [Schmidli et al., 2006]

Variance scaling [Leander and Buishand, 2007]

Power transformation (for precipitation) [Chen et al., 2013]

Distribution-based approaches

Quantile-mapping [Panofsky and Brier, 1968]

CDF-Transform [Michelangeli et al., 2009]

K. A. KPOGO-NUWOKLO et al. Bias correction with CDF-t 4 / 22

Introduction Quantile-mapping CDF-Transform Application Ongoing Summary

Existing bias correction methods

Mean-based approaches

Linear scaling [Lenderink et al., 2007]

Local intensity scaling [Schmidli et al., 2006]

Variance scaling [Leander and Buishand, 2007]

Power transformation (for precipitation) [Chen et al., 2013]

Distribution-based approaches

Quantile-mapping [Panofsky and Brier, 1968]

CDF-Transform [Michelangeli et al., 2009]

K. A. KPOGO-NUWOKLO et al. Bias correction with CDF-t 4 / 22

Introduction Quantile-mapping CDF-Transform Application Ongoing Summary

Existing bias correction methods

Mean-based approaches

Linear scaling [Lenderink et al., 2007]

Local intensity scaling [Schmidli et al., 2006]

Variance scaling [Leander and Buishand, 2007]

Power transformation (for precipitation) [Chen et al., 2013]

Distribution-based approaches

Quantile-mapping [Panofsky and Brier, 1968]

CDF-Transform [Michelangeli et al., 2009]

K. A. KPOGO-NUWOKLO et al. Bias correction with CDF-t 4 / 22

Introduction Quantile-mapping CDF-Transform Application Ongoing Summary

Table of contents

1 Quantile-mapping

2 CDF-Transform

3 Application

4 Ongoing work

5 Summary

K. A. KPOGO-NUWOKLO et al. Bias correction with CDF-t 5 / 22

Introduction Quantile-mapping CDF-Transform Application Ongoing Summary

Quantile-mapping

• Fo → CDF of observations

• Fm → CDF of model output

Fo(xo) = Fm(xm) =⇒ xbc = F−1o [Fm(xm)]

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

CDF

Observation

Model

xbc

xm

Quantile-mapping can be used with both empirical and parametric CDF

K. A. KPOGO-NUWOKLO et al. Bias correction with CDF-t 6 / 22

Introduction Quantile-mapping CDF-Transform Application Ongoing Summary

Limit of the Quantile-mapping

Future simulation

Observations do not cover the same time period as the model output

Example:

• Model output→ 1980− 2026 (historical+future)

• Observations→ 1980− 2016 (historical)

QQ-mapping→ CDF observations (1980− 2016) = CDF observations (1980− 2026)

Quantile mapping does not take into account any information on the distribution of thefuture modelled dataset

K. A. KPOGO-NUWOKLO et al. Bias correction with CDF-t 7 / 22

Introduction Quantile-mapping CDF-Transform Application Ongoing Summary

Limit of the Quantile-mapping

Future simulation

Observations do not cover the same time period as the model output

Example:

• Model output→ 1980− 2026 (historical+future)

• Observations→ 1980− 2016 (historical)

QQ-mapping→ CDF observations (1980− 2016) = CDF observations (1980− 2026)

Quantile mapping does not take into account any information on the distribution of thefuture modelled dataset

K. A. KPOGO-NUWOKLO et al. Bias correction with CDF-t 7 / 22

Introduction Quantile-mapping CDF-Transform Application Ongoing Summary

Limit of the Quantile-mapping

Future simulation

Observations do not cover the same time period as the model output

Example:

• Model output→ 1980− 2026 (historical+future)

• Observations→ 1980− 2016 (historical)

QQ-mapping→ CDF observations (1980− 2016) = CDF observations (1980− 2026)

Quantile mapping does not take into account any information on the distribution of thefuture modelled dataset

K. A. KPOGO-NUWOKLO et al. Bias correction with CDF-t 7 / 22

Introduction Quantile-mapping CDF-Transform Application Ongoing Summary

Limit of the Quantile-mapping

Future simulation

Observations do not cover the same time period as the model output

Example:

• Model output→ 1980− 2026 (historical+future)

• Observations→ 1980− 2016 (historical)

QQ-mapping→ CDF observations (1980− 2016) = CDF observations (1980− 2026)

Quantile mapping does not take into account any information on the distribution of thefuture modelled dataset

K. A. KPOGO-NUWOKLO et al. Bias correction with CDF-t 7 / 22

Introduction Quantile-mapping CDF-Transform Application Ongoing Summary

CDF-Transform

Historical period Future periodObservation Fo,h(x) Fo,f (x)Model Fm,h(x) Fm,f (x)

• If it is possible to estimate Fo,f , then future model output can be corrected throughquantile-mapping:

xbc = F−1o,f [Fm,f (x)]

How can we approximate Fo,f ?

K. A. KPOGO-NUWOKLO et al. Bias correction with CDF-t 8 / 22

Introduction Quantile-mapping CDF-Transform Application Ongoing Summary

CDF-Transform

Historical period Future periodObservation Fo,h(x) Fo,f (x)Model Fm,h(x) Fm,f (x)

• If it is possible to estimate Fo,f , then future model output can be corrected throughquantile-mapping:

xbc = F−1o,f [Fm,f (x)]

How can we approximate Fo,f ?

K. A. KPOGO-NUWOKLO et al. Bias correction with CDF-t 8 / 22

Introduction Quantile-mapping CDF-Transform Application Ongoing Summary

CDF-Transform

Historical period Future periodObservation Fo,h(x) Fo,f (x)Model Fm,h(x) Fm,f (x)

• If it is possible to estimate Fo,f , then future model output can be corrected throughquantile-mapping:

xbc = F−1o,f [Fm,f (x)]

How can we approximate Fo,f ?

K. A. KPOGO-NUWOKLO et al. Bias correction with CDF-t 8 / 22

Introduction Quantile-mapping CDF-Transform Application Ongoing Summary

A solution to approximate Fo,f

Historical period Future periodObservation Fo,h(x) Fo,f (x)Model Fm,h(x) Fm,f (x)

• T : [0, 1] −→ [0, 1],

T (Fm,h(x)) = Fo,h(x) (1)

K. A. KPOGO-NUWOKLO et al. Bias correction with CDF-t 9 / 22

Introduction Quantile-mapping CDF-Transform Application Ongoing Summary

A solution to approximate Fo,f

Historical period Future periodObservation Fo,h(x) Fo,f (x)Model Fm,h(x) Fm,f (x)

• T : [0, 1] −→ [0, 1],

T (Fm,h(x)) = Fo,h(x) (2)

T (Fm,f (x)) = Fo,f (x) (3)

K. A. KPOGO-NUWOKLO et al. Bias correction with CDF-t 10 / 22

Introduction Quantile-mapping CDF-Transform Application Ongoing Summary

A solution to approximate Fo,f

Historical period Future periodObservation Fo,h(x) Fo,f (x)Model Fm,h(x) Fm,f (x)

• T : [0, 1] −→ [0, 1],

T (Fm,h(x)) = Fo,h(x) (4)

T (Fm,f (x)) = Fo,f (x) (5)

T is modelled by replacing x by F−1m,h(u),

T (u) = Fo,h(F−1m,h(u)) (6)

ans thus,

Fo,f (x) = Fo,h(F−1m,h(Fm,f (x))) (7)

K. A. KPOGO-NUWOKLO et al. Bias correction with CDF-t 11 / 22

Introduction Quantile-mapping CDF-Transform Application Ongoing Summary

Application

Wupper catchment (Germany)

Model output from COSMO-CLM (1979-2015)

WFDEI used as reference data (1979-2013)

K. A. KPOGO-NUWOKLO et al. Bias correction with CDF-t 12 / 22

Introduction Quantile-mapping CDF-Transform Application Ongoing Summary

QQplot of daily precipitation (mm/day): January (left) - July (right)

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

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0 10 20 30 40 50

010

2030

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WFDEI

CO

SM

O−

CLM

lon= 7.46 lat= 51.11

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Model without correctionModel with CDFt correction

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0 10 20 30 40

010

2030

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WFDEI

CO

SM

O−

CLM

lon= 7.46 lat= 51.11

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K. A. KPOGO-NUWOKLO et al. Bias correction with CDF-t 13 / 22

Introduction Quantile-mapping CDF-Transform Application Ongoing Summary

Monthly mean (left) and standard deviation (right) of daily precipitation

lon= 7.46 lat= 51.11

Mea

n of

pre

cipi

tatio

n [m

m/d

ay]

23

45

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WFDEICOSMO−CLM without correctionCOSMO−CLM with CDFt correction

lon= 7.46 lat= 51.11

Std

of p

reci

pita

tion

[mm

/day

]

24

68

10

J F M A M J J A S O N D

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WFDEICOSMO−CLM without correctionCOSMO−CLM with CDFt correction

K. A. KPOGO-NUWOKLO et al. Bias correction with CDF-t 14 / 22

Introduction Quantile-mapping CDF-Transform Application Ongoing Summary

Deviation in the mean [Mean(reference)−Mean(model)] for January and July

6.8 7.0 7.2 7.4 7.6

50.9

51.1

51.3

Model without correction

lon

lat

−1.0

−0.5

0.0

0.5

1.0jan

6.8 7.0 7.2 7.4 7.6

50.9

51.1

51.3

Model with CDFt correction

lon

lat

−1.0

−0.5

0.0

0.5

1.0jan

6.8 7.0 7.2 7.4 7.6

50.9

51.1

51.3

Model without correction

lon

lat

−2

−1

0

1

2jul

6.8 7.0 7.2 7.4 7.6

50.9

51.1

51.3

Model with CDFt correction

lon

lat

−2

−1

0

1

2jul

K. A. KPOGO-NUWOKLO et al. Bias correction with CDF-t 15 / 22

Introduction Quantile-mapping CDF-Transform Application Ongoing Summary

Bias correction for decadal climate predictions with CDF-t

• Goal: Bias correction of daily MiKlip decadal predictions ( spatial resolution = 0.11◦ )

Historical (1979-2014) Future (2015-2004)MiKlip F (M11)

h F (M11)f

WATCH F (W11)h F (W11)

f

CDF-t =⇒ F (W11)f (x) = F (W11)

h

{F

(M11)−1h

[F (M11)

f (x)]}

• Problem: MiKlip historical simulations are only available for spatial resolution of 0.44◦

K. A. KPOGO-NUWOKLO et al. Bias correction with CDF-t 16 / 22

Introduction Quantile-mapping CDF-Transform Application Ongoing Summary

Bias correction for decadal climate predictions with CDF-t

• Goal: Bias correction of daily MiKlip decadal predictions ( spatial resolution = 0.11◦ )

Historical (1979-2014) Future (2015-2004)MiKlip F (M11)

h F (M11)f

WATCH F (W11)h F (W11)

f

CDF-t =⇒ F (W11)f (x) = F (W11)

h

{F

(M11)−1h

[F (M11)

f (x)]}

• Problem: MiKlip historical simulations are only available for spatial resolution of 0.44◦

K. A. KPOGO-NUWOKLO et al. Bias correction with CDF-t 16 / 22

Introduction Quantile-mapping CDF-Transform Application Ongoing Summary

Use CDF-t as statistical downscaling method

K. A. KPOGO-NUWOKLO et al. Bias correction with CDF-t 17 / 22

Introduction Quantile-mapping CDF-Transform Application Ongoing Summary

Summary

Compared to the quantile-mapping, CDF-t has advantage to take into account the changein the distribution for the future period

Satisfactory results are obtained with CDF-t

CDF-t can be used for statistical downscaling

Need of reliable multivariate and spatial bias correction methods

K. A. KPOGO-NUWOKLO et al. Bias correction with CDF-t 18 / 22

Introduction Quantile-mapping CDF-Transform Application Ongoing Summary

References I

Chen, J., Brissette, F. P., Chaumont, D., and Braun, M. (2013).Finding appropriate bias correction methods in downscaling precipitation for hydrologic impact studies over north america.Water Resources Research, 49(7):4187–4205.

Leander, R. and Buishand, T. A. (2007).Resampling of regional climate model output for the simulation of extreme river flows.Journal of Hydrology, 332(3):487–496.

Lenderink, G., Buishand, A., and Deursen, W. v. (2007).Estimates of future discharges of the river rhine using two scenario methodologies: direct versus delta approach.Hydrology and Earth System Sciences, 11(3):1145–1159.

Michelangeli, P.-A., Vrac, M., and Loukos, H. (2009).Probabilistic downscaling approaches: Application to wind cumulative distribution functions.Geophysical Research Letters, 36(11).

Panofsky, H. A. and Brier, G. W. (1968).Some applications of statistics to meteorology.University Park : Penn. State University, College of Earth and Mineral Sciences.

Schmidli, J., Frei, C., and Vidale, P. L. (2006).Downscaling from gcm precipitation: a benchmark for dynamical and statistical downscaling methods.International journal of climatology, 26(5):679–689.

K. A. KPOGO-NUWOKLO et al. Bias correction with CDF-t 19 / 22

Introduction Quantile-mapping CDF-Transform Application Ongoing Summary

Acknowledgements

The BINGO project has received funding from the European Union’sHorizon 2020 Research and Innovation programme, under the GrantAgreement number 641739.

THANK YOU FOR YOUR ATTENTION

K. A. KPOGO-NUWOKLO et al. Bias correction with CDF-t 20 / 22

Introduction Quantile-mapping CDF-Transform Application Ongoing Summary

Future simulation CDFs off the range of the historical ones

Fo,f (x) = Fo,h(F−1m,h(Fm,f (x)))

−2 0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

x

CD

F(x

)

Fo,hFm,hFm,fFo,f without shiftFo,f with shift

K. A. KPOGO-NUWOKLO et al. Bias correction with CDF-t 21 / 22

Introduction Quantile-mapping CDF-Transform Application Ongoing Summary

K. A. KPOGO-NUWOKLO et al. Bias correction with CDF-t 22 / 22