Van thanh van nguyen and students endowed brace professor chair in civil engineering l.jpg
This presentation is the property of its rightful owner.
Sponsored Links
1 / 74

Van- Thanh -Van Nguyen (and Students) Endowed Brace Professor Chair in Civil Engineering PowerPoint PPT Presentation


  • 235 Views
  • Uploaded on
  • Presentation posted in: General

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. OUTLINE. INTRODUCTION

Download Presentation

Van- Thanh -Van Nguyen (and Students) Endowed Brace Professor Chair in Civil Engineering

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Van thanh van nguyen and students endowed brace professor chair in civil engineering l.jpg

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


Outline l.jpg

OUTLINE

  • INTRODUCTION

    • Design Rainfall and Design Storm Concept – Current Practices

    • Extreme Rainfall Estimation Issues?

    • Climate Variability and Climate Change Impacts?

  • OBJECTIVES

  • DOWNSCALING METHODS

    • Spatial Downscaling Issues

    • Temporal Downscaling Issues

    • Spatial-Temporal Downscaling Method

  • APPLICATIONS

  • CONCLUSIONS


Introduction l.jpg

INTRODUCTION

  • 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.

    • Average annual flood damages in the U.S. are US$2.1 billion dollars. (US NRC)

  • Information on extreme rainfalls is essential for planning, design, and management of various water-resource systems.

  • Design Rainfall = maximum amount of precipitation at a given site for a specified duration and return period.


Design rainfall estimation methods l.jpg

Design Rainfall Estimation Methods

  • The choice of an estimation method depends on the availability of historical data:

    • Gaged Sites Sufficient long historical records (> 20 years?) At-site Methods.

    • Partially-Gaged Sites Limited data records Regionalization Methods.

    • Ungaged Sites Data are not available Regionalization Methods.


Design rainfall and design storm estimation l.jpg

Design Rainfall and Design Storm Estimation

  • At-site Frequency Analysis of Precipitation

  • Regional Frequency Analysis of Precipitation

    ⇒ Intensity-Duration-Frequency (IDF) Relations

    ⇒ DESIGN STORM CONCEPT for design of hydraulic structures

    (WMO Guides to Hydrological Practices: 1st Edition 1965 → 6th Edition: Section 5.7, in press)


Extreme rainfall estimation issues 1 l.jpg

Extreme Rainfall Estimation Issues (1)

  • 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


Extreme rainfall estimation issues 2 l.jpg

1

2

3

4

Geographically contiguous fixed regions

Geographically non contiguous fixed regions

Hydrologic neighborhood type regions

Extreme Rainfall Estimation Issues (2)

Regionalization methods

  • GEV/Index-flood method.

    • Index-Flood Method (Dalrymple, 1960):

    • Similarity (or homogeneity) of point rainfalls?

    • How to define groups of homogeneous gages? What are the classification criteria?

Proposed Regional Homogeneity:

PCA of rainfall amounts at different sites for different time scales.

PCA of rainfall occurrencesat different sites.

(WMO Guides to Hydrological Practices: 1st Edition 1965 → 6th Edition: Section 5.7, in press)


Extreme rainfall estimation issues 3 l.jpg

Extreme Rainfall Estimation Issues (3)

  • The “scale” problem

    • 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?

    • 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.


Slide9 l.jpg

Extreme Rainfall Estimation Issues (4)

  • Climate Variability and Change will have important impacts on the hydrologic cycle, and in particular the precipitation process!

  • 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 km2

      • Temporal resolution ~ (daily), month, seasonal

      • Reliability of some GCM output variables (such as cloudiness  precipitation)?


Slide10 l.jpg

  • How to develop Climate Change scenarios for impacts studies in hydrology?

    • Spatial scale ~ a few km2 to several 1000 km2

    • 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


Idf relations l.jpg

IDF Relations

  • At-site Frequency Analysis of Precipitation

  • Regional Frequency Analysis of Precipitation

    ⇒ Intensity-Duration-Frequency (IDF) Relations

    ⇒ DESIGN STORM for design of hydraulic structures.

  • Traditional IDF estimation methods:

    • Time scaling problem:no consideration of rainfall properties at different time scales;

    • Spatial scaling problem:results limited to data availability at a local site;

    • Climate change: no consideration.


Summary l.jpg

Summary

  • Recent developments:

    • 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.

    • Various spatial downscaling methods have been developed to provide the linkage between (GCM) large-scale data and local scale data.

  • 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 at daily temporal scale, while many applications require data at sub-daily scales (hourly, 15 minutes, …).


Objectives l.jpg

OBJECTIVES

  • To review recent progress in downscaling methods from both theoretical and practical viewpoints.

  • To assess the performance of statistical downscaling methods to find the “best” method in the simulation of dailyprecipitation time series for climate change impact studies.

  • 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).

  • To assess the climate change impacts on the extreme rainfall processes at a regional or local scale.


Downscaling methods l.jpg

DOWNSCALING METHODS

Scenarios


Spatial dynamic downscaling methods l.jpg

(SPATIAL) DYNAMIC DOWNSCALING METHODS

  • Coarse GCM + High resolution AGCM

  • Variable resolution GCM (high resolution over the area of interest)

  • GCM + RCM or LAM (Nested Modeling Approach)

    • 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.

    • Considerable computing resource requirement.


Spatial statistical downscaling methods l.jpg

(SPATIAL) STATISTICAL DOWNSCALING METHODS

  • Weather Typing or Classification

    • Generation daily weather series at a local site.

    • Classification schemes are somewhat subjective.

  • Stochastic Weather Generators

    • Generation of realistic statistical properties of daily weather series at a local site.

    • Inexpensive computing resources

    • Climate change scenarios based on results predicted by GCM (unreliable for precipitation)

  • Regression-Based Approaches

    • Generation daily weather series at a local site.

    • Results limited to local climatic conditions.

    • Long series of historical data needed.

    • Large-scale and local-scale parameter relations remain valid for future climate conditions.

    • Simple computational requirements.


Applications l.jpg

APPLICATIONS

  • 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)

    • 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.


Slide18 l.jpg

  • Statistical Downscaling Model (SDSM)

    (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)

    • 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.


Slide19 l.jpg

Geographical locations of sites under study.

Geographical coordinates of the stations


Slide20 l.jpg

  • DATA:

    • 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.

    • Calibration: 1961-1975; validation: 1976-1990.


Slide21 l.jpg

Evaluation indices and statistics


Slide22 l.jpg

Themean of daily precipitationfor the period of1961-1975

BIAS = Mean (Obs.) – Mean (Est.)


Slide23 l.jpg

The mean of daily precipitationfor the period of1976-1990

BIAS = Mean (Obs.) – Mean (Est.)


Slide24 l.jpg

The90th percentile of daily precipitationfor the period of1976-1990

BIAS = Mean (Obs.) – Mean (Est.)


Slide25 l.jpg

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)


Summary26 l.jpg

SUMMARY

  • Downscaling is necessary!!!

  • 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

Is it feasible?

Daily and Sub-Daily Extreme Precipitations


The scaling concept l.jpg

The Scaling Concept


The scaling generalized extreme value gev distribution l.jpg

The Scaling Generalized Extreme-Value (GEV) Distribution.

  • The scaling concept

  • The cumulative distribution function:

  • The quantile:


The scaling gev distribution l.jpg

The Scaling GEV Distribution


Slide30 l.jpg

  • The first three moments of GEV distribution:


Application estimation of extreme rainfalls for gaged sites l.jpg

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.


Slide32 l.jpg

Scaling of NCMs of extreme rainfalls with durations: 5-min to 1-hour and 1-hour to 1-day.

red: 1st NCM; blue: 2nd NCM; black: 3rd NCM; markers: observed values; lines: fitted regression


Results on scaling regimes l.jpg

Resultson 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-dayrainfalls.


Slide34 l.jpg

5-min Extreme Rainfalls estimated from 1-hour Extreme Rainfalls

markers: observed values – lines: values estimated by scaling method

markers: observed values – lines: values estimated by scaling method


Slide35 l.jpg

1-hour Extreme Rainfalls estimated from 1-day Extreme Rainfalls

markers: observed values – lines: values estimated by scaling method


The spatial temporal downscaling approach l.jpg

The Spatial-Temporal Downscaling Approach

  • GCMs: HadCM3 and CGCM2.

  • NCEP Re-analysis data.

  • Spatial downscaling method: the statistical downscaling model SDSM (Wilby et al., 2002).

  • Temporal downscaling method: the scaling GEV model (Nguyen et al. 2002).


The spatial temporal downscaling approach37 l.jpg

The Spatial-Temporal Downscaling Approach

  • 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);

    • 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.

  • Temporal downscaling:

    • investigating the scale invariant property of observed AMPs at a local site; and

    • determining the linkage between daily AMPs with sub-daily AMPs.


Application l.jpg

Application

  • Study Region

    • Precipitation records from a network of 15 raingages in Quebec (Canada).

  • Data

    • GCM outputs:

      • HadCM3A2, HadCM3B2,

      • CGMC2A2, CGCM2B2,

      • Periods: 1961-1990, 2020s, 2050s, 2080s.

    • Observed data:

      • Daily precipitation data,

      • AMP for 5 min., 15 min., 30 min., 1hr., 2 hrs., 6 hrs., 12 hrs.

      • Periods: 1961-1990.


Slide39 l.jpg

Daily AMPs estimated from GCMs versus observed daily AMPs at Dorval.

Calibration period: 1961-1975

CGCMA2

HadCM3A2


Slide40 l.jpg

Calibration period: 1961-1975

Residual = Daily AMP (GCM) - Observed daily AMP (local)

CGCMA2

HadCM3A2


Slide41 l.jpg

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


Slide42 l.jpg

CGCMA2

HadCM3A2


Conclusions 1 l.jpg

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 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 observed statistical properties of the daily precipitation process at a local site.


Conclusions 2 l.jpg

CONCLUSIONS (2)

  • 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.

  • Differences between quantile estimates from observed daily AMPs and from GCM-based daily AMPs could be described by a second-order non-linear function.

  • 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.

  • The proposed scaling GEV method could provide accurate AMP quantiles for sub-daily durations from daily AMPs.

  • 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.


Slide45 l.jpg

Thank you

for

your attention!


Slides required for presentations l.jpg

Slides required for presentations


Design storm concept l.jpg

DESIGN STORM CONCEPT

  • Watershed as a linear system

    • Stormwater removal Qpeak Rational Method: Qpeak = CIA Uniform Design Rainfall

  • Watershed as a nonlinear system.

    • Environmental control Entire Hydrograph Q(t)  More realistic temporal rainfall pattern(or Design Storm) formore realistic rainfall-runoff simulation.

  • A design storm describes completely the distribution of rainfall intensity during thestorm durationfora givenreturn period.


Design storm concept50 l.jpg

DESIGN STORM CONCEPT

  • Two main types of “synthetic” design storms:

    • Design Storms derived from the IDF relationships.

    • Design Storms resulted from analysing and synthesising the characteristics of historical storm data.

  • A typical design storm:

    • Maximum Intensity: IMAX

    • Time to peak: Tb

    • Duration: T

    • Temporal pattern


Design storm estimation issues l.jpg

Design Storm Estimation Issues

  • Different synthetic design storm models available in various countries:

    • US Chicago storm model (Keifer and Chu, 1957)

    • US Normalized storm pattern by Huff (1967)

    • Czechoslovakianstorm pattern by Sifalda (1973)

    • Australian design storm by Pilgrim and Cordery (1975)

    • UK Mean symmetric pattern (Flood Studies Report, 1975)

    • French storm model by Desbordes (1978)

    • US storm pattern by Yen and Chow (1980)

    • Canadian Atmospheric Environment Service (1980)

    • US balanced storm model (Army Corps of Engineer, 1982)

    • Canadian temporal rainfall patterns (Nguyen, 1981,1984)

    • 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?


Slide52 l.jpg

Intensity-Duration-Frequency curves for Montreal area.


Slide53 l.jpg

Chicago

IDF

Design Storm


Design storm patterns for southern quebec canada l.jpg

Design Storm Patterns for southern Quebec (Canada)


Design storm patterns for southern quebec canada55 l.jpg

Design Storm Patterns for southern Quebec (Canada)


Summary56 l.jpg

SUMMARY

  • Results indicated:

    • For runoff peak flows:

      • the Canadian AES design storm

      • the Desbordes model (with a peak intensity duration of 30 minutes)

    • 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.


The 1 hr optimal storm pattern for southern quebec canada l.jpg

The 1-hr optimal storm pattern for southern Quebec (Canada)


Assessment of the proposed optimal storm pattern l.jpg

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.


Assessment of the proposed optimal storm pattern59 l.jpg

Assessment of the Proposed Optimal Storm Pattern


Slide60 l.jpg

Climate Trends and Variability1950-1998

  • Maximum and minimum temperatures have increased at similar rate

  • Warming in the south and west, and cooling in the northeast (winter & spring)

Trends in

Winter

Mean Temp

(°C / 49 years)

Trends in

Spring

Mean Temp

(°C / 49 years)

Trends in

Summer

Mean Temp

(°C / 49 years)

Trends in

Fall

Mean Temp

(°C / 49 years)

From X. Zhang, L. Vincent, B. Hogg and A. Niitsoo, Atmosphere-Ocean, 2000


Slide61 l.jpg

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]


Slide62 l.jpg

300km

50km

10km

1m

Point

Climate Scenario development need: from coarse to high resolution

A mismatch of scales between what climate models can supply and what environmental impact models require.

Impact models require ...

GCMs or RCMs supply...

P. Gachon


Choice of distribution model for fitting annual extreme rainfalls l.jpg

Choice of distribution model for fitting annual extreme rainfalls

  • Common probability distributions:

    • Two-parameter distribution:

      • Gumbel distribution

      • Normal

      • Log-normal (2 parameters)

    • 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


Choice of distribution model for fitting annual extreme rainfalls64 l.jpg

Choice of distribution model for fitting annual extreme rainfalls

  • Generalized Gamma distribution

  • Generalized Normal distribution

  • Generalized Pareto distribution

  • Four-parameter distribution

    • Two-component extreme value distribution

  • Five-parameter distribution:

    • Wakeby distribution

  • No general agreement on the choice of distribution for extreme rainfalls!!!


  • Choice of distribution model for fitting annual extreme rainfalls65 l.jpg

    Choice of distribution model for fitting annual extreme rainfalls

    • A three-parameter distribution can provide sufficient flexibility for describing extreme hydrologic data.

    • A two-parameter distribution could be adequate for prediction.

    • 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).


    Estimation of model parameters l.jpg

    Estimation of model parameters

    • Graphical method (Probability plots)

      • Different plotting-position formulas

    • Frequency factor method

    • Method of moments

      • Sample mean, variance, and skewness.

      • 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)

    • Should we use higher-order moments?


    Slide67 l.jpg

    Estimation of model parameters

    • Method of maximum likelihood

      • Optimal estimators (unbiased, minimum variance) of the parameters.

      • Iterative numerical methods.

      • It could give bad estimators for small samples.

    • Method of L-moments

      • Linear combination of order statistics

      • Sample L-moments are found less biased than traditional moment estimators  better suited for use with small samples?

    • Other methods

      • Maximum entropy method

      • Etc.


    Model assessment l.jpg

    MODEL ASSESSMENT

    • Descriptive Ability

      • Graphical Display: Quantile-Quantile Plots

      • Numerical Comparison Criteria

    • Predictive Ability

      • Bootstrap Method


    Numerical comparison criteria l.jpg

    Numerical Comparison Criteria

    • Root Mean Square Error

    • Relative Root Mean Square Error

    • Maximum Absolute Error

    • Correlation Coefficient


    Bootstrap method l.jpg

    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.


    Location of the 20 climatological stations l.jpg

    Location of the 20 Climatological Stations

    Record Length

    Max: 52 yrs

    Min: 24 yrs


    Goodness of fit on the right tail l.jpg

    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)


    Extrapolated right tail quantiles l.jpg

    Extrapolated Right-Tail Quantiles

    Box Plots of Extrapolated Right-Tail Bootstrap Data for

    5-Minute Annual Precipitation Maxima at McGill Station


    Results for at site frequency analysis of extreme rainfalls in quebec l.jpg

    Results for At-site Frequency Analysis of Extreme Rainfalls in Quebec

    • Comparable performance for all distributions in terms of Descriptive and Predictive abilities.

    • Top three distributions – WAK,GEV and GNO

    • Computational simplicity

      • GUM>GPA>BEP>BEK>GEV>GNO>PE3>WAK>LP3

    • Theoretical basis of GEV

      ⇒GEV is recommended as the most suitable for representing annual maximum precipitation in Southern Quebec


  • Login