Presentation at conference, NRC International Workshop on Flood-Resistant Buildings, Ottawa, 27 Feb 2020.Canada First Research Excellence FundNon-Peer ReviewedTo better use climate information available in the historical record, a recommended approach is 
 composing precipitation as the product of precipitable water and precipitation efficiency

Ben Alaya, Mohamed Ali

Zhang, Xuebin

Zwiers, Francis

Presentation at conference, NRC International Workshop on Flood-Resistant Buildings, Ottawa, 27 Feb 2020.Canada First Research Excellence FundNon-Peer ReviewedTo better use climate information available in the historical record, a recommended approach is 
 composing precipitation as the product of precipitable water and precipitation efficiency

University of Saskatchewan Research Archive

University of SaskatchewanHARVEST, https://harvest.usask.ca/Global Institute for Water Security Global Water Futures2020-02-27Long period return level estimates ofextreme precipitationZwiers, FrancisUniversity of Victoria Pacific Climate Impacts Consortiumhttps://hdl.handle.net/10388/14876Presentation at conference, NRC International Workshop on Flood-Resistant Buildings, Ottawa,27 Feb 2020.Downloaded from HARVEST, University of Saskatchewan's Repository for ResearchLong period return level estimates of extreme precipitationFrancis Zwiers and M.A. Ben Alaya, PCIC, University of VictoriaXuebin Zhang, Climate Research Division, Environment and Climate Change CanadaNRC International Workshop on Flood-Resistant Buildings, Ottawa, 27 Feb 2020Photo: F. Zwiers (Zhenyuan)Outline• Introduction• Problem definition• Possible solution• Conclusions23Introduction:The simple & elegant theoretical basisPhoto: F. Zwiers (Xiaoqikong)Introduction• Extreme value theory provides two general approaches• Block maximum• Set a fixed ”block” length (typically a year)• Analyze a time series of block maxima• Leads to the Generalized Extreme Value (GEV) distribution• Peaks over threshold• Set a high threshold• Analyze exceedances above the threshold, usually after de-clustering• Leads to the Generalized Pareto distribution (GPD)4Block maximum approach• Approach most widely used in engineering design problems• Natural block length is a year è annual maxima• Seeks to estimate a point in the upper tail of the distribution (e.g., an n-year ”return level”)• Most work uses the Generalized Extreme Value (GEV) distribution𝐹(𝑌 = 𝑦|𝜇, 𝜎, 𝜉) =exp − 1 + ! "#$%# ⁄' !, 𝜉 < 0, 𝑦 < 𝜇 − ⁄𝜎 𝜉exp − 1 + ! "#$%# ⁄' !, 𝜉 > 0, 𝑦 > 𝜇 − ⁄𝜎 𝜉exp −exp − "#$% , 𝜉 = 05Reversed Weibull, or Type IIIFréchet, or Type IIGumbel, or Type I6R D Gill - Created by R D Gill, 4 January 2013, using R scriptReversed Weibull, or Type IIIFréchet, or Type IIGumbel, or Type I2 5 10 20 50 100 200 500 10000246810Return PeriodReturn Level𝜉 = 0.5𝜉 = −0.52 5 10 20 50 100 200 500 10000246810Return PeriodReturn Level𝜉 = 0.1𝜉 = −0.1Block maximum approach• The ”Extremal Types Theorem” provides some justification• This theorem has been generalized to some types of stationary processes• This is a limit theorem, like the Central Limit Theorem7Let 𝑀! = max{𝑋", 𝑋#, … , 𝑋!} where 𝑋$ are iid random variables.  If for some constants 𝑎! > 0, 𝑏!, 𝑃{𝑎! 𝑀! − 𝑏! ≤ 𝑥}→%𝐺(𝑥)for some nondegenerate 𝐺, then 𝐺 is one of the three extreme value types that comprise the GEV distributionReal world applications• This is a limit theorem, like the Central Limit Theorem• Working assumption is that 1-year blocks are large enough• But … observed processes are generally not iid or even stationary• There can be strong serial dependence and a strong annual cycle (e.g., rainfall and streamflow) à effective block lengths << 1-year• There might be “surprises” in the upper tail (e.g., hurricanes or atmospheric rivers) that are not consistently observed in each block8It is safe to use the GEV (or the GPD) for rainfall and wind?•We study this question with a climate model9Problem definition:     … using the CanRCM4 large ensemblePhoto: F. Zwiers (Squall on Juan de Fuca Strait)CanRCM4 large ensemble• 50-members, 50 km resolution, driven by the CanESM2 large ensemble• historical + RCP8.5 forcing• hourly precipitation archived for 35-members• considering 1951-2000 only, we have 35x50=1750 annual maximaGEV fitting method• assume stationarity over 1951-2000• fit via maximum likelihood• results are similar if using probability weighted momentsPhoto: F. Zwiers (American Bittern, Delta)Bias in return level estimates12Relative bias of extreme quantile estimatesRelative bias in extreme quantiles of CanRCM4 simulated 1-hour precipitation accumulations for 1951-2000 based on fitting a GEV distribution to 1750 annual extremes for 1951-200033   644  645  646 Figure 7. Maps of the bias in return levels estimated using GEV-1 distributions (GEV fitted to 647 annual maximum hourly precipitation) and GEV-10 distributions (GEV fitted to 10-year maxima 648 of hourly precipitation) using the 35 CanRCM4 historical simulations of the period 1951-2000. 649 Bias is expressed in % relative to the corresponding empirical estimates. 650  651  652  653   654 100-year return level 1000-year return level33   644  645  646 Figure 7. Maps of the bias in return levels estimated using GEV-1 distributions (GEV fitted to 647 annual maximum hourly precipitation) and GEV-10 distributions (GEV fitted to 10-year maxima 648 of hourly precipitation) using the 35 CanRCM4 historical simulations of the period 1951-2000. 649 Bias is expressed in % relative to the corresponding empirical estimates. 650  651  652  653   654 The fit of the GEV and tail stabilityPhoto: F. Zwiers (California Quail, Emlyn Cove)24   480 Figure 4. Return level estimates based on fitting the GEV distribution to annual maxima at four 481 different locations A (in (b)), B (in (d)), C (in (f)) and D (in (h)) using one CanRCM4 simulation 482 of 1951-2000 (50 annual maxima, in blue) and the 35 simulations (1750 annual maxima, in red). 483 Geographical positions of the four locations are shown in (a). Black dots in panels (b), (d), (f) 484 and (h) show empirical quantile estimates obtained using the 1750 annual maxima. Estimates of 485 the shape parameter versus block length based on 1750 years of CanRCM4 simulations are 486 shown by the black line for the four locations A (in (c)), B (in (e)) , C (in (g)) and D (in (i)). 487 These panels also show estimated shape parameters based on annual maxima from a single 488 CanRCM4 simulation (in blue) and the 35 ensemble members (in red), with the extension to 489 longer blocks reflecting the max-stability assumption. Shading indicates 80% confidence 490 intervals obtained by bootstrapping. 491 GEV fits to block maxima at 2 locations14VancouverEmpirical distribution from 1750 annual maximaGEV from 50 annual maxima GEV from 1750 annual maxima GEV from 1 to 20 year block maximaExtreme quantiles based on 1750-years of CanRCM4 simulated 1-hour precipitation accumulations for 1951-2000Return Period (yr) Block length (yr)Return Level (mm)Shape parameter24   480 Figure 4. Return level estimates based on fitting the GEV distribution to annual maxima at four 481 different locations A (in (b)), B (in (d)), C (in (f)) and D (in (h)) using one CanRCM4 simulation 482 of 1951-2000 (50 annual maxima, in blue) and the 35 simulations (1750 annual maxima, in red). 483 Geographical positions of the four locations are shown in (a). Black dots in panels (b), (d), (f) 484 and (h) show empirical quantile estimates obtained using the 1750 annual maxima. Estimates of 485 the shape parameter versus block length based on 1750 years of CanRCM4 simulations are 486 shown by the black line for the four locations A (in (c)), B (in (e)) , C (in (g)) and D (in (i)). 487 These panels also show estimated shape parameters based on annual maxima from a single 488 CanRCM4 simulation (in blue) and the 35 ensemble members (in red), with the extension to 489 longer blocks reflecting the max-stability assumption. Shading indicates 80% confidence 490 intervals obtained by bootstrapping. 491 DenverReturn Period (yr) Block length (yr)Return Level (mm)Shape parameter22   459 Figure 2. Estimated shape parameter of the GEV distribution for CanRCM4 simulated 1, 6, 12 460 and 24-hour precipitation accumulations (rows) for the historical period 1951-2000 over North 461 America. The first column shows the shape parameter estimates when the GEV distribution is 462 fitted to a sample of 50 annual maximum values from only one of the 35 CanRCM4 simulation. 463 The remaining columns shows shape parameter estimates using block maxima from the 464 ensemble of 35 CanRCM4 simulations for 1-year (GEV-1; 1750 blocks), 5-year (GEV-5; 350 465 blocks), 10-year (GEV-10; 175 blocks) and 20-year blocks (GEV_20; 87 blocks).  466  467   468   469 22   459 Figure 2. Estimated shape parameter of the GEV distribution for CanRCM4 simulated 1, 6, 12 460 and 24-hour precipitation accumulations (rows) for the historical period 1951-2000 over North 461 America. The first column shows the shape parameter estimates when the GEV distribution is 462 fitted to a sample of 50 annual maximum values from only one of the 35 CanRCM4 simulation. 463 The remaining columns shows shape parameter estimates using block maxima from the 464 ensemble of 35 CanRCM4 simulations for 1-year (GEV-1; 1750 blocks), 5-year (GEV-5; 350 465 blocks), 10-year (GEV-10; 175 blocks) and 20-year blocks (GEV_20; 87 blocks).  466  467   468   469 !𝜉Shape parameters of extreme 1-hour precipitation16Relative bias of extreme quantile estimatesRelative bias in extreme quantiles of CanRCM4 simulated 1-hour precipitation accumulations for 1951-2000 based on fitting a GEV distribution to 175 decadal extremes for 1951-2000100-year return level 1000-year return level33   644  645  646 Figure 7. Maps of the bias in return levels estimated using GEV-1 distributions (GEV fitted to 647 annual maximum hourly precipitation) and GEV-10 distributions (GEV fitted to 10-year maxima 648 of hourly precipitation) using the 35 CanRCM4 historical simulations of the period 1951-2000. 649 Bias is expressed in % relative to the corresponding empirical estimates. 650  651  652  653   654 33   644  645  646 Figure 7. Maps of the bias in return levels estimated using GEV-1 distributions (GEV fitted to 647 annual maximum hourly precipitation) and GEV-10 distributions (GEV fitted to 10-year maxima 648 of hourly precipitation) using the 35 CanRCM4 historical simulations of the period 1951-2000. 649 Bias is expressed in % relative to the corresponding empirical estimates. 650  651  652  653   654 Discussion• We should worry about tail stability and where we sample• Sampling the annual maximum may leave us ignorant (in relative terms) about surprises deeper in the upper tail• Using a peaks-over-threshold approach does not solve the problem• Extrapolation into the deep tail requires information from somewhere• It is either constructed from basic postulates, assumed, or perhaps can be objectively derived from further information about the underlying physicsA possible SolutionPhoto: F. Zwiers (Huangguoshu Water Falls)Can physical considerations help?• Existing approaches include use of mixtures of distributions, in combination or not with storm classification, or empirical selection of distributions from amongst several heavy-tailed candidate• We use a multivariate approach in which extreme precip is decomposed into two components as PCP=PW×PE, where• PW is the precipitable water in the atmospheric column• PE is the precipitation efficiency (the fraction of PW that is precipitated during the event)• PW is generally bounded, whereas PE can be heavy tailed, with PE >> 1 possible.• The two components represent two different aspects of the physics controlling precipitationProposal …• Model the joint behaviour of extreme PW and PE  and then use Monte Carlo methods to derive the distribution of PCP=PWxPE• Options• Heffernan and Tawn (2004) conditional dependence model• Ben Alaya et al (2018) extreme value copula based model• We opted for Heffernan and Tawn (2004) because the flexibility offered using copula’s is quite limitedSome results for 6-hourly precipitationPhoto: F. Zwiers (Yinlianzhui Water Falls)Relative bias in 1000-year return level estimates34   803 Figure 3. Maps of the relative bias of 1000-year RL estimates, over North America, using the 804 compound approach in (a), the univariate approach in (b), and the univariate approach using the 805 1750 annual maxima in (c). 806  807  808   809 Compound Approach(50-year sample of precip componentsUnivariate Approach(50-year sample of annual maxima)Univariate Approach(1750-year sample of annual maxima)Relative RMSE in 1000-year return level estimatesCompound Approach(50-year sample of precip componentsUnivariate Approach(50-year sample of annual maxima)Univariate Approach(1750-year sample of annual maxima)32   705 Figure 3. Maps of the relative bias (RB) and the relative root mean square errors (RRMSE) of 706 1000-year RL estimates, over North America, using the compound approach in (a) and (d), the 707 univariate approach in (b) and (e), and the univariate approach using the 1750 annual maxima in 708 (c) and (f). 709  710  711   712 Estimated return levels for extreme 6-hour precipitation at two locations32   787 Figure 1. Geographical positions of GB1 and GB2 in North America. 788  789   790 33   791 Figure 2. Estimated return levels using the univariate approach (red curve) and the compound 792 approach (blue curve) at the GB1 (in a) and GB2 (in b). Black dots in (a) and (b) shows 793 empirical estimates obtained using the 1750 annual maxima. Estimated shape parameters against 794 the length of block maxima using the univariate approach (red line) and the compound approach 795 (blue curve) for GB1 and GB2 are shown in (c) and (d) respectively. In (c) and (d) black curve 796 shows the estimated shape parameters using the 1750 annual maxima. The 80% confidence 797 intervals obtained by calculating the 10th and the 90th percentiles obtained by the vector 798 bootstrapping resampling approach are shaded for all the four figures. 799  800  801   802 33   791 Figure 2. Estimated return levels using the univariate approach (red curve) and the compound 792 approach (blue curve) at the GB1 (in a) and GB2 (in b). Black dots in (a) and (b) shows 793 empirical estimates obtained using the 1750 annual maxima. Estimated shape parameters against 794 the length of block maxima using the univariate approach (red line) and the compound approach 795 (blue curve) for GB1 and GB2 are shown in (c) and (d) respectively. In (c) and (d) black curve 796 shows the estimated shape parameters using the 1750 annual maxima. The 80% confidence 797 intervals obtained by calculating the 10th and the 90th percentiles obtained by the vector 798 bootstrapping resampling approach are shaded for all the four figures. 799  800  801   802 GB1 GB2Annual max univariate (50 year sample)Multivariate extremal dependence model (50 year sample)Empirical quantile estimates (1750 year sample)33   791 Figure 2. Estimated return levels using the univariate approach (red curve) and the compound 792 approach (blue curve) at the GB1 (in a) and GB2 (in b). Black dots in (a) and (b) shows 793 empirical estimates obtained using the 1750 annual maxima. Estimated shape parameters against 794 the length of block maxima using the univariate approach (red line) and the compound approach 795 (blue curve) for GB1 and GB2 are shown in (c) and (d) respectively. In (c) and (d) black curve 796 shows the estimated shape parameters using the 1750 annual maxima. The 80% confidence 797 intervals obtained by calculating the 10th and the 90th percentiles obtained by the vector 798 bootstrapping resampling approach are shaded for all the four figures. 799  800  801   802 33   7  i  . ti t  return levels using the univariate ap roach (red curve) and the compound 7  r  ( l  r ) at the B1 (in a) and GB2 (in b). Black dots in (a) and (b) shows 7  e irical esti ates tained using the 1750 an ual maxima. Estimated shape parameters against 794 the length of block axi a using the univariate approach (red line) and the compound approach 795 (blue curve) for 1 and 2 are shown in (c) and (d) respectively. In (c) and (d) black curve 796 shows the esti ated shape para eters using the 1750 annual maxima. The 80% confidence 797 intervals obtained by calculating the 10th and the 90th percentiles obtained by the vector 798 bootstrapping resampling approach are shaded for all the four figures. 799  800  801   802 Estimated shape parameters for extreme 6-hour precipitation at two locations32   787 Figure 1. Geographical positions of GB1 and GB2 in North America. 788  789   790 GB1 GB2Predicted shape using annual max univariate (50 year sample)Predicted shape using multivariate extremal dependence model (50 year sample)Empirical estimates (1750 year sa ple)ConclusionsPhoto: F. Zwiers (First Bridge, Yangtze River at Nanjing)Conclusions• Traditional univariate analysis assumes a stable upper tail• The underlying process generating extremes, may however, be very complex, implying that stability will only be attained once blocks are large enough to consistently sample extremes from the physical process responsible for the largest events• Large multi-year blocks are infeasible with short historical recordsQuantile−Quantile : OSHAWA WPCP, ON  6155878Gumbel Estimate/Estimation Gumbel (mm)Rainfall/Pluie (mm)Duration/Durée : 5 min0 5 10 15 20 25 30051015202530Annual Maximum1:1Duration/Durée : 10 min0 5 10 15 20 25 30 35 400510152025303540Duration/Durée : 15 min0 5 10 15 20 25 30 35 400510152025303540Maximum annuel1:1Duration/Durée : 30 min0 5 10 15 20 25 30 35 40 45 5005101520253035404550Duration/Durée : 1 h  0 10 20 30 40 50 600102030405060Duration/Durée : 2 h  0 10 20 30 40 50 60 70 8001020304050607080Duration/Durée : 6 h  0 10 20 30 40 50 60 70 80 90 1000102030405060708090100Duration/Durée : 12 h  0 20 40 60 80 100120140160180200020406080100120140160180200Duration/Durée : 24 h  0 20 40 60 80 1001201401601802000204060801001201401601802002014/12/211970 − 200632 years / ansQuantile−Quantile : OSHAWA WPCP, ON  6155878Gumbel Estimate/Estimation Gumbel (mm)Rainfall/Pluie (mm)Duration/Durée : 5 min0 5 10 15 20 25 30051015202530Annual Maximum:1Duration/Durée : 1  min0 5 10 15 20 25 30 35 400510152025303540Duration/Durée : 15 min0 5 10 15 20 25 30 35 400510152025303540Maximum annuel1:1Duration/Durée : 30 min0 5 10 15 20 25 30 35 40 45 5005101520253035404550Duration/Durée : 1 h  0 10 20 30 40 50 600102030405060Duration/Durée : 2 h  0 10 20 30 40 50 60 70 8001020304050607080Duration/Durée : 6 h  0 10 20 30 40 50 60 70 80 90 1000102030405060708090100Duration/Durée : 12 h  0 20 40 60 80 100120140160180200020406080100120140160180200Duration/Durée : 24 h  0 20 40 60 80 1001201401601802000204060801001201401601802002014/12/211970 − 200632 years / ansGumbel quantile (mm)Observed quantile (mm)Conclusions• It is therefore necessary to better use information available in the historical record• One option is to extract information from the constituent variables that produce univariate extremes• We illustrated this approach by decomposing precipitation as the product of precipitable water and precipitation efficiency• The ”compound events” extremal dependence model appears to be able to capture fluctuations in tail shape that result from physical relationships between the component variables.• Bias is, consequently, considerably reduced, even when using a modestly short 50-year sample.• Note that additional information that allows this to happen comes from PWProblem is not limited to extreme precipitation• We see similar issues with extreme wind speed• Fitting GEV distributions to annual maxima of model simulated “instantaneous” wind speed tends to find bounded distributions• Leads to negative bias in long-return period extreme wind speed (and thus extreme wind load) estimatesPhoto: F. Zwiers (Juan de Fuca Strait)Questions?https://www.pacificclimate.org/

Long period return level estimates of extreme precipitation

https://harvest.usask.ca/bitstream/10388/14876/1/2002-feb-27-zwiers.pdf

Long period return level estimates of extreme precipitation

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