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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
( ) ( ) ( )
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1 21 2
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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
=
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×=
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× ×=
× ×
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∑
∑( )
( ) ( )( )
( ) ( )
i
i i
i ii
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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
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= 1
= 1
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1 a t-s ite an d 1 2 p o o led (2 ,3 ,....,1 3 s ites)1 3
,
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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
=
=
=
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= →
= ×
=
∑
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∑
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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
