CFCAS ProjectMeeting #3

@ University of Waterloo

Presentation Progress on work at UWaterloo:

Quantifying the uncertainty in modelled estimates of future extreme precipitation events:

Uncertainty of Estimates of IDF Curves

By: Teklu T. HailegeorgisJuly 17, 2009Waterloo

TopicsTopics

• Identification of the main sources of uncertainites• Data, tools and Methodology• Delineation of homogeneous pooling groups• Parameter estimation & Choice of frequency distribution• Uncertainty assessment: pooling group delineation• Averaging Quantiles estimates• IDF Curves• Density estimation for quantiles: Non-parameteric (Kernel)• Conclusions and remarks

Identification: Main uncertainites in FA of extreme precipitationevents

Homogeneity of pooling groups

Intersite correlation

At-site

Quality

Pooled

AMS

TS

PDS

Trend free & Randomness

Stationary

IID

Data series

Sampling

Models: GCMs

Downscaling methods

Dynamic Statistical

Emission scenarios

Data Parameter estimation

Distribution (model)

Frequency analysis

Sources of Uncertainites

Impacts of climate change and projections Statistical frequency analysis

Data, tools and MethodologyData, tools and Methodology

• Data series: Annual maximum series precipitation intensity (AMS)• Data source: Environment Canada• Total stations considered at first = 16• No. of homogeneous sites = 13• Data range 1926 to 2003 (most of the data are from 1963 to 2003)• Record lengths 23 to 74 years• Durations: 1-, 2-, 6 -, 12- & 24 hr• Return periods: 2, 5, 10, 20, 50 & 100 years• Frequency analysis: At-site and pooled FA• Parameter estimation methods: L-moment (Hoskings and Wallis, 1988) and Maximum likelihood

& Probability weighted moments (Rao and Hamed, 2002)• Regionalization: Region of influence + LM homogenity tests• Goodness-of-fit tests: LM (Z-values and LM diagram) and ML & PWM (Chi-square and

Kolmogorov-Smirnov-KS)• Uncertainity assessment: pooling groups delineation• Uncertainity assessment method: Non-parameteric balanced bootstrap resampling• Uncertainity representations: Box plots, quantile estimates with 95% Confidence intervals.• Quantile averaging: Bayesian rule for discrete case• Construction of IDF Curves• Density estimation for quantle estimates: Non-parameteric Kernel (Gaussian)

LL--moments: Delineation of homogeneous pooling groupsmoments: Delineation of homogeneous pooling groups

• Region of influence method: Burn (1990; 2000)

( )( )1

2 2

1

1

i

N i R

iN

ii

n t tV

n

=

=

⎧ ⎫−⎪ ⎪⎪ ⎪= ⎨ ⎬

⎪ ⎪⎪ ⎪⎩ ⎭

∑

∑

The similarity among the sites is measured by a weighted and scaled Euclidian distance metric in P-dimensional space defined by sets of:

i. Geographical locations (X and Y co-ordinates) of the stations;i. Elevations of the stations above mean sea level; andii. Mean annual rainfall for the stations considered.

2

,1

P

i jk k

jik kd w

k

x xs=

=⎛ ⎞−

∑ ⎜ ⎟⎝ ⎠

• Discordancy measure (D) & Heterogeneity statistics (H): Hosking & Wallis (1988;1990;1993;1997)- D: Distance of vector ui = [t (i) t3

(i) t4(i) ]T from regional average

- H: fitting Kappa dist. to the regional Avg. L-moment ratios 1, tR, t3R & t4

R for Nsim MC simulations

, 1 , 2 , 3i ii

i

V

V

o b sVH i

μσ

−= =

( )( ) ( )( )1

12 2 2

3 31

2 N

ii

N i iR Ri

in t t t t

Vn

=

=

⎧ ⎫− + −⎨ ⎬

⎩ ⎭=∑

∑

( )( ) ( )( )1

2 2 2

3 3 4 41

3

1

N i iR Ri

iN

ii

n t t t tV

n

=

=

⎧ ⎫− + −⎨ ⎬

⎩ ⎭=∑

∑

( ) ( )( ) ( )1

1

13

T T

i

N

ii i i iD N u u u u u u u u

−

=

⎛ ⎞= − − − −⎜ ⎟

⎝ ⎠∑

Delineation of homogeneous pooling groups Delineation of homogeneous pooling groups Precipitation stations: on use (left) & for possible future use Precipitation stations: on use (left) & for possible future use (right)(right)

Delineation of homogeneous pooling groups Delineation of homogeneous pooling groups Target site (atTarget site (at--site): 55 observationssite): 55 observations

Delineation of homogeneous pooling groups Delineation of homogeneous pooling groups Homogeneous Pooling group Homogeneous Pooling group -- 2 sites: 80 observations2 sites: 80 observations

Delineation of homogeneous pooling groups Delineation of homogeneous pooling groups Homogeneous Pooling group Homogeneous Pooling group -- 3 sites: 103 observations3 sites: 103 observations

Delineation of homogeneous pooling groups Delineation of homogeneous pooling groups Homogeneous Pooling group Homogeneous Pooling group -- 4 sites: 142 observations4 sites: 142 observations

Delineation of homogeneous pooling groups Delineation of homogeneous pooling groups Homogeneous Pooling group Homogeneous Pooling group -- 5 sites: 173 observations5 sites: 173 observations

Delineation of homogeneous pooling groups Delineation of homogeneous pooling groups Homogeneous Pooling group Homogeneous Pooling group -- 6 sites: 206 observations6 sites: 206 observations

Delineation of homogeneous pooling groups Delineation of homogeneous pooling groups Homogeneous Pooling group Homogeneous Pooling group -- 7 sites: 242 observations7 sites: 242 observations

Delineation of homogeneous pooling groups Delineation of homogeneous pooling groups Homogeneous Pooling group Homogeneous Pooling group -- 8 sites: 268 observations8 sites: 268 observations

Delineation of homogeneous pooling groups Delineation of homogeneous pooling groups Homogeneous Pooling group Homogeneous Pooling group -- 9 sites: 306 observations9 sites: 306 observations

Delineation of homogeneous pooling groups Delineation of homogeneous pooling groups Homogeneous Pooling group Homogeneous Pooling group -- 10 sites: 343 observations10 sites: 343 observations

Delineation of homogeneous pooling groups Delineation of homogeneous pooling groups Homogeneous Pooling group Homogeneous Pooling group -- 11 sites: 417 observations11 sites: 417 observations

Delineation of homogeneous pooling groups Delineation of homogeneous pooling groups Homogeneous Pooling group Homogeneous Pooling group -- 12 sites: 474 observations12 sites: 474 observations

Delineation of homogeneous pooling groups Delineation of homogeneous pooling groups Homogeneous Pooling group Homogeneous Pooling group -- 13 sites: 514 observations13 sites: 514 observations

Results: HResults: H--values, Zvalues, Z--values & Choice of frequency distribution: 1 hrvalues & Choice of frequency distribution: 1 hr

1H <

1H <

Best-fit distributions

ML PWM

Z-values H-values

GLOG GEV GNOR PE3 GPAR H1 H2 H3

At site GEV GEV GEV GEV

2 0.75 0.12 -0.18 -0.72 -1.44 GEV -0.07 -0.55 -0.38

3 1.9 1.11 0.74 0.08 -0.85 PE3 -0.52 -1.05 0.25

4 2.82 1.76 1.36 0.62 -0.79 PE3 0.47 -0.74 0.18

5 2.31 1.36 0.86 -0.02 -1.05 PE3 1.38* -0.18 0.69

6 1.74 0.81 0.3 -0.6 -1.54 GNOR 0.74 -0.66 0.35

7 1.47 0.41 -0.09 -0.97 -2.23 GNOR 0.95 -0.32 0.33

8 1.46 0.36 -0.14 -1.04 -2.34 GNOR 0.67 -0.8 -0.16

9 1.7 0.37 -0.14 -1.07 -2.82 GNOR 0.56 -0.51 -0.01

10 2.05 0.52 0.02 -0.95 -3.09 GNOR 0.35 -0.33 0.15

11 2.56 0.73 0.25 -0.73 -3.48 GNOR 0.31 0.03 0.39

12 2.75 0.79 0.27 -0.78 -3.74 GNOR 0.11 -0.39 -0.13

13 2.9 0.74 0.22 -0.85 -4.22 GNOR 1.39* -0.32 -0.27

“Best-fit”distributio

nχ2 KS χ2 KS

LMPooling

groups: No.

of sites

44 4

41 .64

D ISTD IST

RtZ

τ βσ− +

= ≤1; 1 H < 2

H 2H < ≤

≥

Z: fitting Kappa dist. to the regional Avg. L-moment ratios 1, tR, t3R & t4R for Nsim MC simulations-Bias & Stdev t4R

Results: HResults: H--values, Zvalues, Z--values & Choice of frequency distribution: 12 hrvalues & Choice of frequency distribution: 12 hr

Best-fit distributions

ML PWM

Z-values H-values

GLOG GEV GNOR PE3 GPAR H1 H2 H3

At site EV1EV1/GLOG

GEV/PE3 EV1

2 1.97 0.85 0.76 0.44 -1.55 PE3 1.98* 0.32 0.91

3 2.71 1.53 1.36 0.93 -1.09 PE3 1.30* 0.31 0.71

4 3.2 1.97 1.66 1.03 -0.87 PE3 1.26* 0.5 0.51

5 2.78 1.55 1.19 0.48 -1.33 PE3 0.85 0.08 0.35

6 2.13 1.01 0.58 -0.2 -1.69 PE3 0.16 -0.05 0.47

7 2.14 0.87 0.44 -0.36 -2.16 PE3 0.23 0.05 0.28

8 2.41 1.04 0.63 -0.16 -2.17 PE3 0.14 0.03 0.28

9 2.92 1.3 0.88 0.02 -2.42 PE3 -0.05 -0.03 0.15

10 3.46 1.57 1.18 0.33 -2.71 PE3 -0.22 0.45 0.45

11 3.41 1.43 0.97 0.01 -3.08 PE3 -0.5 0.11 0.21

12 3.77 1.66 1.15 0.11 -3.18 PE3 -0.6 -0.22 -0.18

13 3.58 1.45 0.85 -0.33 -3.51 PE3 -0.9 -0.06 0.1

Best-fit

distributio

nχ2 KS χ2 KS

LM

Pooling groups: No. of sites

Results: Parameter estimation & Choice of frequency distribution: L-M diagram & Distribution comparison for 1 hr duration

Results: Parameter estimation & Choice of frequency distribution: L-M diagram & Distribution comparison for 12 hr duration

Summary of Results : Parameter estimation & Choice of frequency Summary of Results : Parameter estimation & Choice of frequency distributiondistribution

• Z-statistics and L-moment diagram (LM), Chi-square and Kolmogorov-Smirnov (ML & PWM)

“Best-fit” distributions

LM: regional ML: at site PWM: at-site

Z - values LM ratio diagram

χ2 KS χ2 KS

1 GNOR LOGNOR GEV GEV GEV GEV2

GEV GEV GLOG GEV GLOG GEV6

PE3 PE3 EV1/PE3 PE3 PE3 GEV/PE312

PE3 PE3 EV1 EV1/GLOG GEV/PE3 EV124

GPAR PE3 EV1 PE3 EV1 EV1/GEV/PE3

Durations (hrs)

UncertaintyUncertainty assessment: pooling group delineationassessment: pooling group delineation

• Representations: Box plots, quantile estimates and 95 % Confidence intervals (CI) of quantile estimates.

• Method for CI construction: Non-parametric balanced bootstrap resampling: Efron (1982); Davison et al. (1986); Faulkner & Jones (1999); Carpenter (1999); Burn (2003)

Results: Uncertainty assessment: pooling group delineation Boxplots for 1 hr and 12 hr

Results: Uncertainty assessment: pooling group delineation Quantile estimates: 1 hr (GEV) and 24 hr(PE3)

Results: Uncertainty assessment: pooling group delineation Quantile estimates and Confidence Intervals 12 hr (PE3)

Averaging Quantiles estimatesAveraging Quantiles estimates

• Bayesian rule for discrete case

IF evidence (E) is true, THEN hypothesis (H) is true with probability p: if event E occurs, then the probability that event H will occur is p.

E1: Absolute difference from the arithmetic mean of quantiles (AbsDmeanQi)E2: Width of 95 % CI (CIWQi).Hi: quantile estimate from i’s case is the best, where i = 1,2,......13

( ) ( ) ( )

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )( )

1 21 2

1 21

1 2

2

1 21 2

11

F o r E a n d E , a s s u m in g c o n d i t io n a l in d e p e n d e n c e (V ru s ia s , 2 0 0 5 ) :

n i ii n m

n k kk

N

ii

i i

k k kk

ii

p E E . . . E H p Hp H E E . . . E

p E E . . . E H p H

p E H p E H p Hp H E E

p E H p E H p H

M a x A b s D m e a n QA b s D N m e a n Q

=

=

×=

×

× ×=

× ×

=

∑

∑( )

( ) ( )( )

( ) ( )

i

i i

i ii

i i

A b s D m e a n QM a x A b s D m e a n Q M in A b s D m e a n Q

M a x C IW Q C IW QC IW N Q

M a x C IW Q M in C IW Q

−−

−=

−

Averaging Quantiles estimatesAveraging Quantiles estimates

( )

( )

( )

( )( )

( )i

1 2

1 2

1

1

2

= 1

= 1

1

i

/

/

1 a t-s ite an d 1 2 p o o led (2 ,3 ,....,1 3 s ites)1 3

,

1

N

N

i

N

N

kk

i

i i

i

i

kk

i

i

A b sD N m ea n Qp E H

A b sD N m ea n Q

C IW N Qp E H

C IW N Q

P H

W eig h ted Q p H E E Q w h ere

p H E E

=

=

=

=

= →

= ×

=

∑

∑

∑

∑

Quantiles: NonQuantiles: Non--parameteric Density estimationparameteric Density estimation

• Gaussian Kernel

( )1

1ˆ ,

N: Sample size h: Band width (smoothing parameter)

Ni

hi

x xp x K where

Nh h=

−⎛ ⎞= ⎜ ⎟⎝ ⎠

∑

( )( )

( )12

21 exp 0,122

xK x Nπ

⎛ ⎞= − →⎜ ⎟

⎝ ⎠

Results: IDF curves with 95 % CI & sample PDF and CDFResults: IDF curves with 95 % CI & sample PDF and CDF

Conclusions and remarksConclusions and remarks

• Works to date based only on historical observations (i.e. uncertainity pertinent to the impacts of climate change is not included).

• Significant uncertainity observed pertinent to delineation of pooling groups, parameter estimation and choice of frequency distribution

• Progress vs. schedule (work plan) of Dec, 2008:

June/09 - Task C.10 (Presentation of uncertainty from historical observattions in IDF curves- Ok!

The availability of annual time series data of extreme precipitation events for the projected climate change scenarios both for base climate climate and future time slice(s) is crucial for future work.

Thanks