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1 A SPATIAL-TEMPORAL DOWNSCALING APPROACH TO CONSTRUCTION OF INTENSITY-DURATION-FREQUENCY RELATIONS IN CONSIDERATION OF GCM-BASED CLIMATE CHANGE SCENARIOS Van-Thanh-Van Nguyen (and Students) Endowed Brace Professor Chair in Civil Engineering McGill University Montreal, Quebec, Canada Brace Centre for Water Resources Management Global Environmental and Climate Change Centre Department of Civil Engineering and Applied Mechanics School of Environment

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2 December 19, 2007, Climate Change Symposium, Singapore OUTLINE n INTRODUCTION Design Rainfall and Design Storm Concept Current Practices Design Rainfall and Design Storm Concept Current Practices Extreme Rainfall Estimation Issues? Extreme Rainfall Estimation Issues? Climate Variability and Climate Change Impacts? Climate Variability and Climate Change Impacts? n OBJECTIVES n DOWNSCALING METHODS Spatial Downscaling Issues Spatial Downscaling Issues Temporal Downscaling Issues Temporal Downscaling Issues Spatial-Temporal Downscaling Method Spatial-Temporal Downscaling Method n APPLICATIONS n CONCLUSIONS

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3 December 19, 2007, Climate Change Symposium, Singapore INTRODUCTION n Extreme storms (and floods) account for more losses than any other natural disaster (both in terms of loss of lives and economic costs). Damages due to Saguenay flood in Quebec (Canada) in 1996: $800 million dollars. Damages due to Saguenay flood in Quebec (Canada) in 1996: $800 million dollars. Average annual flood damages in the U.S. are US$2.1 billion dollars. (US NRC) Average annual flood damages in the U.S. are US$2.1 billion dollars. (US NRC) n Information on extreme rainfalls is essential for planning, design, and management of various water- resource systems. n Design Rainfall = maximum amount of precipitation at a given site for a specified duration and return period.

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4 December 19, 2007, Climate Change Symposium, Singapore n The choice of an estimation method depends on the availability of historical data: Gaged Sites Sufficient long historical records (> 20 years?) At-site Methods. Gaged Sites Sufficient long historical records (> 20 years?) At-site Methods. Partially-Gaged Sites Limited data records Regionalization Methods. Partially-Gaged Sites Limited data records Regionalization Methods. Ungaged Sites Data are not available Regionalization Methods. Ungaged Sites Data are not available Regionalization Methods. Design Rainfall Estimation Methods

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5 December 19, 2007, Climate Change Symposium, Singapore Design Rainfall and Design Storm Estimation n At-site Frequency Analysis of Precipitation n Regional Frequency Analysis of Precipitation Intensity-Duration-Frequency (IDF) Relations DESIGN STORM CONCEPT for design of hydraulic structures (WMO Guides to Hydrological Practices: 1 st Edition 1965 6 th Edition: Section 5.7, in press)

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6 December 19, 2007, Climate Change Symposium, Singapore Extreme Rainfall Estimation Issues (1) n Current practices: At-site Estimation Methods (for gaged sites): Annual maximum series (AMS) using 2- parameter Gumbel/Ordinary moments method, or using 3-parameter GEV/ L- moments method. Which probability distribution? Which estimation method? How to assess model adequacy? Best-fit distribution? Problems: Uncertainties in Data, Model and Estimation Method

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7 December 19, 2007, Climate Change Symposium, Singapore Extreme Rainfall Estimation Issues (2) Regionalization methods n GEV/Index-flood method. Index-Flood Method (Dalrymple, 1960): Index-Flood Method (Dalrymple, 1960): Similarity (or homogeneity) of point rainfalls? How to define groups of homogeneous gages? What are the classification criteria? 4 3 2 1 Geographically contiguous fixed regions Geographically non contiguous fixed regions Hydrologic neighborhood type regions (WMO Guides to Hydrological Practices: 1st Edition 1965 6th Edition: Section 5.7, in press) Proposed Regional Homogeneity: 1.PCA of rainfall amounts at different sites for different time scales. 2.PCA of rainfall occurrences at different sites.

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8 December 19, 2007, Climate Change Symposium, Singapore n The scale problem The properties of a variable depend on the scale of measurement or observation. The properties of a variable depend on the scale of measurement or observation. Are there scale-invariance properties? And how to determine these scaling properties? Are there scale-invariance properties? And how to determine these scaling properties? Existing methods are limited to the specific time scale associated with the data used. Existing methods are limited to the specific time scale associated with the data used. Existing methods cannot take into account the properties of the physical process over different scales. Existing methods cannot take into account the properties of the physical process over different scales. Extreme Rainfall Estimation Issues (3)

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9 December 19, 2007, Climate Change Symposium, Singapore n Climate Variability and Change will have important impacts on the hydrologic cycle, and in particular the precipitation process! n How to quantify Climate Change? General Circulation Models (GCMs): A credible simulation of the average large-scale seasonal distribution of atmospheric pressure, temperature, and circulation. (AMIP 1 Project, 31 modeling groups) Climate change simulations from GCMs are inadequate for impact studies on regional scales: Spatial resolution ~ 50,000 km 2 Temporal resolution ~ (daily), month, seasonal Reliability of some GCM output variables (such as cloudiness precipitation)? Extreme Rainfall Estimation Issues (4)

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10 December 19, 2007, Climate Change Symposium, Singapore How to develop Climate Change scenarios for impacts studies in hydrology? How to develop Climate Change scenarios for impacts studies in hydrology? Spatial scale ~ a few km 2 to several 1000 km 2 Temporal scale ~ minutes to years A scale mismatch between the information that GCM can confidently provide and scales required by impacts studies. Downscaling methods are necessary!!! GCM Climate Simulations Precipitation (Extremes) at a Local Site

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11 December 19, 2007, Climate Change Symposium, Singapore IDF Relations n At-site Frequency Analysis of Precipitation n Regional Frequency Analysis of Precipitation Intensity-Duration-Frequency (IDF) Relations DESIGN STORM for design of hydraulic structures. n Traditional IDF estimation methods: Time scaling problem: no consideration of rainfall properties at different time scales; Time scaling problem: no consideration of rainfall properties at different time scales; Spatial scaling problem: results limited to data availability at a local site; Spatial scaling problem: results limited to data availability at a local site; Climate change: no consideration. Climate change: no consideration.

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12 December 19, 2007, Climate Change Symposium, Singapore Summary n Recent developments: Successful applications of the scale invariant concept in precipitation modeling to permit statistical inference of precipitation properties between various durations. Successful applications of the scale invariant concept in precipitation modeling to permit statistical inference of precipitation properties between various durations. Global climate models (GCMs) could reasonably simulate some climate variables for current period and could provide various climate change scenarios for future periods. Global climate models (GCMs) could reasonably simulate some climate variables for current period and could provide various climate change scenarios for future periods. Various spatial downscaling methods have been developed to provide the linkage between (GCM) large-scale data and local scale data. Various spatial downscaling methods have been developed to provide the linkage between (GCM) large-scale data and local scale data. n Scale Issues: GCMs produce data over global spatial scales (hundreds of kilometres) which are very coarse for water resources and hydrology applications at point or local scale. GCMs produce data over global spatial scales (hundreds of kilometres) which are very coarse for water resources and hydrology applications at point or local scale. GCMs produce data at daily temporal scale, while many applications require data at sub-daily scales (hourly, 15 minutes, ). GCMs produce data at daily temporal scale, while many applications require data at sub-daily scales (hourly, 15 minutes, ).

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13 December 19, 2007, Climate Change Symposium, Singapore OBJECTIVES n To review recent progress in downscaling methods from both theoretical and practical viewpoints. n To assess the performance of statistical downscaling methods to find the best method climate change impact studies. n To assess the performance of statistical downscaling methods to find the best method in the simulation of daily precipitation time series for climate change impact studies. n To develop an approach that could link daily simulated climate variables from GCMs to sub-daily precipitation characteristics at a regional or local scale (a spatial-temporal downscaling method). n To assess the climate change impacts on the extreme rainfall processes at a regional or local scale.

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14 December 19, 2007, Climate Change Symposium, Singapore DOWNSCALING METHODS Scenarios

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15 December 19, 2007, Climate Change Symposium, Singapore (SPATIAL) DYNAMIC DOWNSCALING METHODS n Coarse GCM + High resolution AGCM n Variable resolution GCM (high resolution over the area of interest) n GCM + RCM or LAM (Nested Modeling Approach) More accurate downscaled results as compared to the use of GCM outputs alone. More accurate downscaled results as compared to the use of GCM outputs alone. Spatial scales for RCM results ~ 20 to 50 km still larges for many hydrologic models. Spatial scales for RCM results ~ 20 to 50 km still larges for many hydrologic models. Considerable computing resource requirement. Considerable computing resource requirement.

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16 December 19, 2007, Climate Change Symposium, Singapore (SPATIAL) STATISTICAL DOWNSCALING METHODS n Weather Typing or Classification Generation daily weather series at a local site. Generation daily weather series at a local site. Classification schemes are somewhat subjective. Classification schemes are somewhat subjective. n Stochastic Weather Generators Generation of realistic statistical properties of daily weather series at a local site. Generation of realistic statistical properties of daily weather series at a local site. Inexpensive computing resources Inexpensive computing resources Climate change scenarios based on results predicted by GCM (unreliable for precipitation) Climate change scenarios based on results predicted by GCM (unreliable for precipitation) n Regression-Based Approaches Generation daily weather series at a local site. Generation daily weather series at a local site. Results limited to local climatic conditions. Results limited to local climatic conditions. Long series of historical data needed. Long series of historical data needed. Large-scale and local-scale parameter relations remain valid for future climate conditions. Large-scale and local-scale parameter relations remain valid for future climate conditions. Simple computational requirements. Simple computational requirements.

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17 December 19, 2007, Climate Change Symposium, Singapore APPLICATIONS n LARS-WG Stochastic Weather Generator (Semenov et al., 1998) Generation of synthetic series of daily weather data at a local site (daily precipitation, maximum and minimum temperature, and daily solar radiation) Generation of synthetic series of daily weather data at a local site (daily precipitation, maximum and minimum temperature, and daily solar radiation) Procedure: Procedure: Use semi-empirical probability distributions to describe the state of a day (wet or dry). Use semi-empirical distributions for precipitation amounts (parameters estimated for each month). Use normal distributions for daily minimum and maximum temperatures. These distributions are conditioned on the wet/dry status of the day. Constant Lag-1 autocorrelation and cross-correlation are assumed. Use semi-empirical distribution for daily solar radiation. This distribution is conditioned on the wet/dry status of the day. Constant Lag-1 autocorrelation is assumed.

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18 December 19, 2007, Climate Change Symposium, Singapore n Statistical Downscaling Model (SDSM) (Wilby et al., 2001) (Wilby et al., 2001) Generation of synthetic series of daily weather data at a local site based on empirical relationships between local-scale predictands (daily temperature and precipitation) and large- scale predictors (atmospheric variables) Generation of synthetic series of daily weather data at a local site based on empirical relationships between local-scale predictands (daily temperature and precipitation) and large- scale predictors (atmospheric variables) Procedure: Procedure: Identify large-scale predictors (X) that could control the local parameters (Y). Find a statistical relationship between X and Y. Validate the relationship with independent data. Generate Y using values of X from GCM data.

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19 December 19, 2007, Climate Change Symposium, Singapore Geographical locations of sites under study. Geographical coordinates of the stations

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20 December 19, 2007, Climate Change Symposium, Singapore n DATA: Observed daily precipitation and temperature extremes at four sites in the Greater Montreal Region (Quebec, Canada) for the 1961-1990 period. Observed daily precipitation and temperature extremes at four sites in the Greater Montreal Region (Quebec, Canada) for the 1961-1990 period. NCEP re-analysis daily data for the 1961-1990 period. NCEP re-analysis daily data for the 1961-1990 period. Calibration: 1961-1975; validation: 1976-1990. Calibration: 1961-1975; validation: 1976-1990.

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21 December 19, 2007, Climate Change Symposium, Singapore NoCodeUnitTime scaleDescription 1Prcp1%Season Percentage of wet days (daily precipitation 1 mm) 2SDIImm/r.daySeasonDaily Mean: sum of daily precipitations / number of wet days 3CDDdaysSeasonMaximum number of consecutive dry days (daily precipitation < 1 mm) 4R3daysmmSeasonMaximum 3-day precipitation total 5Prec90pmmSeason90 th percentile of daily precipitation amount 6Precip_meanmm/dayMonthSum of daily precipitation in a month / number of days in that month 7Precip_sdmmMonthStandard deviation of daily precipitation in a month Evaluation indices and statistics

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22 December 19, 2007, Climate Change Symposium, Singapore The mean of daily precipitation for the period of 1961-1975 BIAS = Mean (Obs.) Mean (Est.)

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23 December 19, 2007, Climate Change Symposium, Singapore BIAS = Mean (Obs.) Mean (Est.) The mean of daily precipitation for the period of 1976-1990

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24 December 19, 2007, Climate Change Symposium, Singapore The 90 th percentile of daily precipitation for the period of 1976-1990 BIAS = Mean (Obs.) Mean (Est.)

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25 December 19, 2007, Climate Change Symposium, Singapore GCM and Downscaling Results (Precipitation Extremes ) 1- Observed 2- SDSM [CGCM1] 3- SDSM [HADCM3] 4- CGCM1-Raw data 5- HADCM3-Raw data From CCAF Project Report by Gachon et al. (2005)

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26 December 19, 2007, Climate Change Symposium, Singapore SUMMARY Downscaling is necessary!!! LARS-WG and SDSM models could provide good but generally biased estimates of LARS-WG and SDSM models could provide good but generally biased estimates of the observed statistics of daily precipitation at a local site. GCM-Simulated Daily Precipitation Series Daily and Sub-Daily Extreme Precipitations Is it feasible?

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27 December 19, 2007, Climate Change Symposium, Singapore The Scaling Concept

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28 December 19, 2007, Climate Change Symposium, Singapore The Scaling Generalized Extreme-Value (GEV) Distribution. n The scaling concept n The cumulative distribution function: The quantile: The quantile:

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29 December 19, 2007, Climate Change Symposium, Singapore The Scaling GEV Distribution

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30 December 19, 2007, Climate Change Symposium, Singapore n The first three moments of GEV distribution:

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31 December 19, 2007, Climate Change Symposium, Singapore APPLICATION: Estimation of Extreme Rainfalls for Gaged Sites Data used: Raingage network: 88 stations in Quebec (Canada). Rainfall durations: from 5 minutes to 1 day. Record lengths: from 15 yrs. to 48 yrs.

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32 December 19, 2007, Climate Change Symposium, Singapore red : 1 st NCM; blue : 2 nd NCM; black : 3 rd NCM; markers : observed values; lines : fitted regression Scaling of NCMs of extreme rainfalls with durations: 5-min to 1-hour and 1-hour to 1-day.

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33 December 19, 2007, Climate Change Symposium, Singapore Results on scaling regimes: Non-central moments are scaling. Two scaling regimes: 5-min. to 1-hour interval. 1-hour to 1-day interval. Based on these results, two estimations were made: 5-min. extreme rainfalls from 1-hr rainfalls. 1-hr. extreme rainfalls from 1-day rainfalls.

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34 December 19, 2007, Climate Change Symposium, Singapore 5-min Extreme Rainfalls estimated from 1-hour Extreme Rainfalls markers: observed values lines: values estimated by scaling method

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35 December 19, 2007, Climate Change Symposium, Singapore 1-hour Extreme Rainfalls estimated from 1-day Extreme Rainfalls markers : observed values lines : values estimated by scaling method

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36 December 19, 2007, Climate Change Symposium, Singapore The Spatial-Temporal Downscaling Approach n GCMs: HadCM3 and CGCM2. n NCEP Re-analysis data. n Spatial downscaling method: the statistical downscaling model SDSM (Wilby et al., 2002). n Temporal downscaling method: the scaling GEV model (Nguyen et al. 2002).

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37 December 19, 2007, Climate Change Symposium, Singapore The Spatial-Temporal Downscaling Approach n Spatial downscaling: calibrating and validating the SDSM in order to link the atmospheric variables (predictors) at daily scale (GCM outputs) with observed daily precipitations at a local site (predictand); calibrating and validating the SDSM in order to link the atmospheric variables (predictors) at daily scale (GCM outputs) with observed daily precipitations at a local site (predictand); extracting AMP from the SDSM-generated daily precipitation time series; and extracting AMP from the SDSM-generated daily precipitation time series; and making a bias-correction adjustment to reduce the difference in quantile estimates from SDSM- generated AMPs and from observed AMPs at a local site using a second-order nonlinear function. making a bias-correction adjustment to reduce the difference in quantile estimates from SDSM- generated AMPs and from observed AMPs at a local site using a second-order nonlinear function. n Temporal downscaling: investigating the scale invariant property of observed AMPs at a local site; and investigating the scale invariant property of observed AMPs at a local site; and determining the linkage between daily AMPs with sub-daily AMPs. determining the linkage between daily AMPs with sub-daily AMPs.

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38 December 19, 2007, Climate Change Symposium, Singapore Application n Study Region Precipitation records from a network of 15 raingages in Quebec (Canada). Precipitation records from a network of 15 raingages in Quebec (Canada). n Data GCM outputs: GCM outputs: HadCM3A2, HadCM3B2, CGMC2A2, CGCM2B2, Periods: 1961-1990, 2020s, 2050s, 2080s. Observed data: Observed data: Daily precipitation data, AMP for 5 min., 15 min., 30 min., 1hr., 2 hrs., 6 hrs., 12 hrs. Periods: 1961-1990.

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39 December 19, 2007, Climate Change Symposium, Singapore Daily AMPs estimated from GCMs versus observed daily AMPs at Dorval. Calibration period: 1961-1975 CGCMA2 HadCM3A2

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40 December 19, 2007, Climate Change Symposium, Singapore Residual = Daily AMP (GCM) - Observed daily AMP (local) Calibration period: 1961-1975 CGCMA2 HadCM3A2

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41 December 19, 2007, Climate Change Symposium, Singapore Daily AMPs estimated from GCMs versus observed daily AMPs at Dorval. Validation period: 1976-1990 CGCMA2 HadCM3A2 Adjusted Daily AMP (GCM) = Daily AMP (GCM) + Residual

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42 December 19, 2007, Climate Change Symposium, Singapore CGCMA2 HadCM3A2

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43 December 19, 2007, Climate Change Symposium, Singapore CONCLUSIONS (1) Significant advances have been achieved regarding the global climate modeling. However, GCM outputs are still not appropriate for assessing climate change impacts on the hydrologic cycle. Downscaling methods provide useful tools for this assessment. Calibration Calibration of the SDSM suggested that precipitation was mainly related to zonal velocities, meridional velocities, specific humidities, geopotential height, and vorticity. In general, LARS-WG and SDSM models could provide good but biased estimates of the In general, LARS-WG and SDSM models could provide good but biased estimates of the observed statistical properties of the daily precipitation process at a local site.

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44 December 19, 2007, Climate Change Symposium, Singapore CONCLUSIONS (2) n It is feasible to link daily GCM-simulated climate variables with sub-daily AMPs based on the proposed spatial- temporal downscaling method. IDF relations for different climate change scenarios could be constructed. n Differences between quantile estimates from observed daily AMPs and from GCM-based daily AMPs could be described by a second-order non-linear function. n Observed AMPs in Quebec exhibit two different scaling regimes for time scales ranging from 1 day to 1 hour, and from 1 hour to 5 minutes. n The proposed scaling GEV method could provide accurate AMP quantiles for sub-daily durations from daily AMPs. n AMPs derived from CGCM2A2 outputs show a large increasing trend for future periods, while those given by HadCM3A2 did NOT exhibit a large (increasing or decreasing) trend.

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45 December 19, 2007, Climate Change Symposium, Singapore Thank you for your attention!

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46 December 19, 2007, Climate Change Symposium, Singapore Slides required for presentations

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47 December 19, 2007, Climate Change Symposium, Singapore

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48 December 19, 2007, Climate Change Symposium, Singapore

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49 December 19, 2007, Climate Change Symposium, Singapore DESIGN STORM CONCEPT n Watershed as a linear system Stormwater removal Q peak Rational Method: Q peak = CIA Uniform Design Rainfall Stormwater removal Q peak Rational Method: Q peak = CIA Uniform Design Rainfall n Watershed as a nonlinear system. Environmental control Entire Hydrograph Q(t) More realistic temporal rainfall pattern (or Design Storm) for more realistic rainfall-runoff simulation. Environmental control Entire Hydrograph Q(t) More realistic temporal rainfall pattern (or Design Storm) for more realistic rainfall-runoff simulation. n A design storm describes completely the distribution of rainfall intensity during the storm duration for a given return period.

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50 December 19, 2007, Climate Change Symposium, Singapore DESIGN STORM CONCEPT n Two main types of synthetic design storms: Design Storms derived from the IDF relationships. Design Storms derived from the IDF relationships. Design Storms resulted from analysing and synthesising the characteristics of historical storm data. Design Storms resulted from analysing and synthesising the characteristics of historical storm data. n A typical design storm: Maximum Intensity: I MAX Maximum Intensity: I MAX Time to peak: T b Time to peak: T b Duration: T Duration: T Temporal pattern Temporal pattern

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51 December 19, 2007, Climate Change Symposium, Singapore n Different synthetic design storm models available in various countries: US Chicago storm model (Keifer and Chu, 1957) US Chicago storm model (Keifer and Chu, 1957) US Normalized storm pattern by Huff (1967) US Normalized storm pattern by Huff (1967) Czechoslovakian storm pattern by Sifalda (1973) Czechoslovakian storm pattern by Sifalda (1973) Australian design storm by Pilgrim and Cordery (1975) Australian design storm by Pilgrim and Cordery (1975) UK Mean symmetric pattern (Flood Studies Report, 1975) UK Mean symmetric pattern (Flood Studies Report, 1975) French storm model by Desbordes (1978) French storm model by Desbordes (1978) US storm pattern by Yen and Chow (1980) US storm pattern by Yen and Chow (1980) Canadian Atmospheric Environment Service (1980) Canadian Atmospheric Environment Service (1980) US balanced storm model (Army Corps of Engineer, 1982) US balanced storm model (Army Corps of Engineer, 1982) Canadian temporal rainfall patterns (Nguyen, 1981,1984) Canadian temporal rainfall patterns (Nguyen, 1981,1984) Canadian storm model by Watt et al. (1986) Canadian storm model by Watt et al. (1986) No general agreement as to which temporal storm pattern should be used for a particular site How to choose? How to compare? No general agreement as to which temporal storm pattern should be used for a particular site How to choose? How to compare? Design Storm Estimation Issues

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52 December 19, 2007, Climate Change Symposium, Singapore Intensity-Duration-Frequency curves for Montreal area.

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53 December 19, 2007, Climate Change Symposium, Singapore Chicago IDF Design Storm

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54 December 19, 2007, Climate Change Symposium, Singapore Design Storm Patterns for southern Quebec (Canada)

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55 December 19, 2007, Climate Change Symposium, Singapore Design Storm Patterns for southern Quebec (Canada)

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56 December 19, 2007, Climate Change Symposium, Singapore SUMMARY n Results indicated: For runoff peak flows: For runoff peak flows: the Canadian AES design storm the Desbordes model (with a peak intensity duration of 30 minutes) For runoff volumes: For runoff volumes: the Canadian pattern proposed by Watt et al. None of the eight design storms was able to provide accurate estimation of both runoff parameters. None of the eight design storms was able to provide accurate estimation of both runoff parameters.

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57 December 19, 2007, Climate Change Symposium, Singapore The 1-hr optimal storm pattern for southern Quebec (Canada)

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58 December 19, 2007, Climate Change Symposium, Singapore Assessment of the Proposed Optimal Storm Pattern Probability distributions of runoff peak flows and volumes for a square basin of 1 ha Similar results of probability distributions for all tested basins.

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59 December 19, 2007, Climate Change Symposium, Singapore Assessment of the Proposed Optimal Storm Pattern

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60 December 19, 2007, Climate Change Symposium, Singapore Climate Trends and Variability 1950-1998 Climate Trends and Variability 1950-1998 Maximum and minimum temperatures have increased at similar rate Warming in the south and west, and cooling in the northeast (winter & spring) Trends in Fall Mean Temp (C / 49 years) Trends in Spring Mean Temp (C / 49 years) Trends in Winter Mean Temp (C / 49 years) Trends in Summer Mean Temp (C / 49 years) From X. Zhang, L. Vincent, B. Hogg and A. Niitsoo, Atmosphere-Ocean, 2000

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61 December 19, 2007, Climate Change Symposium, Singapore Validation of GCMs for Current Period (1961-1990) Winter Temperature (C) Model mean =all flux & non-flux corrected results (vs NCEP/NCAR dataset) [Source: IPCC TAR, 2001, chap. 8]

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62 December 19, 2007, Climate Change Symposium, Singapore 300km 50km 10km 1m Point GCMs or RCMs supply... Impact models require... A mismatch of scales between what climate models can supply and what environmental impact models require. Climate Scenario development need: from coarse to high resolution P. Gachon

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63 December 19, 2007, Climate Change Symposium, Singapore Choice of distribution model for fitting annual extreme rainfalls n Common probability distributions: Two-parameter distribution: Two-parameter distribution: Gumbel distribution Normal Log-normal (2 parameters) Three-parameter distributions: Three-parameter distributions: Beta-K distribution Beta-P distribution Generalized Extreme Value distribution Pearson Type 3 distribution Log-Normal (3 parameters) Log-Pearson Type 3 distribution

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64 December 19, 2007, Climate Change Symposium, Singapore Generalized Gamma distribution Generalized Gamma distribution Generalized Normal distribution Generalized Normal distribution Generalized Pareto distribution Generalized Pareto distribution n Four-parameter distribution Two-component extreme value distribution Two-component extreme value distribution n Five-parameter distribution: Wakeby distribution Wakeby distribution No general agreement on the choice of distribution for extreme rainfalls!!! Choice of distribution model for fitting annual extreme rainfalls

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65 December 19, 2007, Climate Change Symposium, Singapore n A three-parameter distribution can provide sufficient flexibility for describing extreme hydrologic data. n A two-parameter distribution could be adequate for prediction. n The choice of a distribution is not as crucial as an adequate data sample. Discrepancies increase for extrapolation beyond the length of record (model error is more important than sampling error). Choice of distribution model for fitting annual extreme rainfalls

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66 December 19, 2007, Climate Change Symposium, Singapore Estimation of model parameters n Graphical method (Probability plots) Different plotting-position formulas Different plotting-position formulas n Frequency factor method n Method of moments Sample mean, variance, and skewness. Sample mean, variance, and skewness. Sample mean, variance, 1st and/or 2nd moments in log-space (method of mixed moments) Sample mean, variance, 1st and/or 2nd moments in log-space (method of mixed moments) Sample mean, variance, and geometric and/or harmonic mean (generalized method of moments) Sample mean, variance, and geometric and/or harmonic mean (generalized method of moments) Should we use higher-order moments?

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67 December 19, 2007, Climate Change Symposium, Singapore n Method of maximum likelihood Optimal estimators (unbiased, minimum variance) of the parameters. Optimal estimators (unbiased, minimum variance) of the parameters. Iterative numerical methods. Iterative numerical methods. It could give bad estimators for small samples. It could give bad estimators for small samples. n Method of L-moments Linear combination of order statistics Linear combination of order statistics Sample L-moments are found less biased than traditional moment estimators better suited for use with small samples? Sample L-moments are found less biased than traditional moment estimators better suited for use with small samples? n Other methods Maximum entropy method Maximum entropy method Etc. Etc. Estimation of model parameters

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68 December 19, 2007, Climate Change Symposium, Singapore MODEL ASSESSMENT n Descriptive Ability Graphical Display: Quantile-Quantile Plots Graphical Display: Quantile-Quantile Plots Numerical Comparison Criteria Numerical Comparison Criteria n Predictive Ability Bootstrap Method Bootstrap Method

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69 December 19, 2007, Climate Change Symposium, Singapore Numerical Comparison Criteria n Root Mean Square Error Relative Root Mean Square Error Maximum Absolute Error Correlation Coefficient

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70 December 19, 2007, Climate Change Symposium, Singapore BOOTSTRAP METHOD A nonparametric approach that repeatedly draws, with replacement, n observations from the available data set of size N (N >n) and yields multiple synthetic samples of the same sizes as the original observations.

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71 December 19, 2007, Climate Change Symposium, Singapore Location of the 20 Climatological Stations Record Length Max: 52 yrs Min: 24 yrs

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72 December 19, 2007, Climate Change Symposium, Singapore Goodness-of-fit on the Right Tail Quantile-Quantile Plots for the Distributions Fitted to 5-Minute Annual Precipitation Maxima at St-Georges Station Fitted Precipitations (mm) Observed Precipitation (mm)

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73 December 19, 2007, Climate Change Symposium, Singapore Extrapolated Right-Tail Quantiles Box Plots of Extrapolated Right-Tail Bootstrap Data for 5-Minute Annual Precipitation Maxima at McGill Station

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74 December 19, 2007, Climate Change Symposium, Singapore Results for At-site Frequency Analysis of Extreme Rainfalls in Quebec n Comparable performance for all distributions in terms of Descriptive and Predictive abilities. n Top three distributions WAK,GEV and GNO n Computational simplicity GUM>GPA>BEP>BEK>GEV>GNO>PE3>WAK>LP3 GUM>GPA>BEP>BEK>GEV>GNO>PE3>WAK>LP3 n Theoretical basis of GEV GEV is recommended as the most suitable for representing annual maximum precipitation in Southern Quebec