1 McGill University Department of Civil Engineering and Applied Mechanics Montreal, Quebec, Canada

2 STATISTICAL MODELING AND ANALYSIS OF EXTREME PRECIPITATION PROCESSES Van-Thanh-Van Nguyen and Tan-Danh Nguyen Department of Civil Engineering and Applied Mechanics McGill University Montreal, Quebec, Canada and OURANOS, Climate Change Consortium Montreal, Quebec, Canada

October 2003, Victoria, BC OUTLINE n INTRODUCTION n OBJECTIVES n METHODOLOGY n APPLICATIONS n CONCLUSIONS

October 2003, Victoria, BC 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: Saguenay (Quebec) flood damages = CAD $800 million dollars; US average annual flood damages = US$2.1 billion dollars]. n Information on extreme rainfalls and floods is essential for planning, design, and management of water-resource systems. n Design Rainfall = the maximum amount of precipitation falling at a given point (or over a given area) for a specified duration and return period  Frequency analysis of extreme rainfall events. n Climate variability and change will have important impacts on the hydrologic cycle, and in particular extreme storm and flood events  How to quantify these impacts?

October 2003, Victoria, BC DOWNSCALING METHODS

October 2003, Victoria, BC … 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.

October 2003, Victoria, BC Issues Related to Estimation of Extreme Rainfall Events: n At-site methods Current practice: Annual maximum series (AMS) using 2- parameter Gumbel/Ordinary moments method, or using 3- parameter GEV/ L-moments method. Current practice: Annual maximum series (AMS) using 2- parameter Gumbel/Ordinary moments method, or using 3- parameter GEV/ L-moments method. n Regionalization methods Current practice: GEV/Index-flood method. Current practice: GEV/Index-flood method.  Similarity (or homogeneity) of sites?  How to define groups of homogeneous sites? What are the classification criteria?  No general agreement on the choice of a suitable distribution model for extreme rainfalls.  What are the impacts of climate variability and change on annual maximum series?

October 2003, Victoria, BC … 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.

October 2003, Victoria, BC OBJECTIVES n To propose new modelling methods that can take into account the scaling properties of the extreme rainfall process. n To demonstrate the importance of scaling consideration in the estimation of extreme rainfalls. n To propose new regional estimation methods of extreme rainfalls for ungaged sites.

October 2003, Victoria, BC METHODOLOGY n Scaling Methods (for partially-gaged and ungaged sites) The scaling concept: The scaling concept:

October 2003, Victoria, BC Generalized Extreme-Value (GEV) Distribution. n The cumulative distribution function: The quantile: The quantile:

October 2003, Victoria, BC n The first three moments of GEV distribution:

October 2003, Victoria, BC The Scaling GEV Distribution

October 2003, Victoria, BC Estimation of Extreme Rainfalls for Partially-Gaged Sites  Rainfall data are not always available for the time and space scales of interest.  Short time interval rainfall extremes are important for small watersheds, but not always available.  Daily rainfall data are widely available.  Daily scale  shorter time scales ?

October 2003, Victoria, BC Methods of estimation of short-duration extreme rainfalls from long-duration extreme rainfalls n Method 1. Basic equation. where

October 2003, Victoria, BC … n Method 2 Basic equation:  Parameters are estimated by the method of moments.

October 2003, Victoria, BC... Data used:  Rainfall duration: from 5 minutes to 1 day.  Raingage network: 14 stations in Quebec.  Record lengths: from 15 yrs. to 48 yrs.

October 2003, Victoria, BC Observation of scaling regime : 5 min 1 hour1 day 3rd order moment. 2nd order moment. 1st order moment.  k (t) t

October 2003, Victoria, BC Scaling characteristics k  ( k )

October 2003, Victoria, BC Results on scaling regimes:  Non-central moments are scaling.  Two scaling regimes:  5-min. to 1-hour interval.  1-hour to 1-day interval.  The slope of the straight line is the estimate of the scaling exponent b(k).  Relationship between  (k) and k, for k = 1 to 3, are linear.

October 2003, Victoria, BC Based on these results, two estimations were made:  5-min. extreme rainfalls from 1-hr rainfalls.  1-hr. extreme rainfalls from 1-day rainfalls.

October 2003, Victoria, BC 5-min extreme rainfalls from 1-hr extreme rainfalls.

October 2003, Victoria, BC 1-hr extreme rainfalls from 1-day extreme rainfalls.

October 2003, Victoria, BC

October 2003, Victoria, BC Results on the estimation methods:  Extreme rainfalls estimated in two cases by two methods were in good agreement with observations.  Method 2 provided more accurate estimates than method 1, especially at the two extremes.

October 2003, Victoria, BC Regional estimation of daily extreme rainfalls for ungaged sites  Homogeneous sites are defined based on the similarity of rainfall occurrences (e.g., strong correlation of the number of rainy hours).  Regional relations between statistical moments of daily extreme rainfalls and the mean number of rainy hours are developed for the homogeneous group.  Statistical moments of daily extreme rainfalls at an ungaged site are estimated using these regional relations  Distribution of daily extreme rainfalls is estimated for the ungaged site.

October 2003, Victoria, BC

October 2003, Victoria, BC

October 2003, Victoria, BC Results on the regional estimation method  Regional estimates are comparable with corresponding at-site estimates.  Good agreement between the estimates (both at-site and regional) with the observations indicates the feasibility of the proposed regional estimation method.

October 2003, Victoria, BC CONCLUSIONS l Consideration of scaling properties of hydrologic processes could lead to the development of more accurate and more reliable estimation methods. l Consideration of scaling properties of hydrologic processes could provide better understanding of the physical phenomenon studied. l The GEV distribution is suitable for regional estimation of extreme rainfalls and floods.

October 2003, Victoria, BC CONCLUSIONS (Continued) l It is feasible to assess the homogeneity of extreme rainfall conditions at different locations based on the similarity of rainfall occurrences. l Problems related to the estimation of extreme rainfalls are still far from being completely solved.  integrated physical- statistical approaches?

October 2003, Victoria, BC... Thank You!

October 2003, Victoria, BC Slides required for presentations

October 2003, Victoria, BC

October 2003, Victoria, BC

October 2003, Victoria, BC

October 2003, Victoria, BC 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  Generalized Gamma distribution  Generalized Normal distribution  Generalized Pareto distribution Four-parameter distribution Four-parameter distribution  Two-component extreme value distribution Five-parameter distribution: Five-parameter distribution:  Wakeby distribution