and Labrador

Prepared by

© Shabnam Mostofi Zadeh

A Thesis submitted to the Schoo l of Graduate Stu dies in parti al fulfi ll ment of the requiremen ts for the degree of

M aster of E ng ineerin g

Faculty of E ngineerin g a nd A pplied Scienc e Memorial University of Newfoundland

St. J ohn ' s

November 2012

Newfoundl and

ABSTRACT

The objectives of thi s study were to quantify the characteristics of low fl ows in rivers of the province of Newfoundland and Labrador, and to develop equations which could be used to estimate the magnitude, frequency, duration, and spells of low flo w events. These

different aspects of low flows were analyzed by applying methods of flow frequency,

flow duration, and flow spell analysis, respectively. Sixty hydrometric stations in the

Island of Newfoundland which have more than 20 years of complete data were selected

for the current low flow study. Becau se of the sparseness and shortness of h ydrometric

data in Labrador, sites with more than 15 years of data were chosen with a total of 12

stations. An L-moment based approach was applied for regional frequency analysis of

annual minimum 1-day and 7-day flows for two separate homogeneous regions, Island of

Newfoundland, and Labrador and it yielded prediction equations for low flo ws of

different durations and return periods. The performance of these regional models was

verified using n ew sets of data, and showed re liab le results. Therefore, one can use th ese

prediction models for ungauged sites in Newfoundland and Labrador. To perform

regional flow duration a na lysis, physiographic parameters of the regions under study were

regressed against quantiles of flow duration curves obtained for each h ydrometric station

to produce a regional mode l for predicting flow duration curves at any ungauged sites .

R egional mode l of flow durations were validated su ccessfully using a new set of data , and

the results were promising. Different hydrological methodologies were applied to define

flow spells for rivers in Newfoundland and Labrador, and regional models were defined

to predict the annual maximum flow spell variables in Newfoundland and Labrador.

ACKNOWLEDGEMENTS

I would like to express my sincere gratitude to my supervisor Dr. Leonard M. Lye for his continual guidance , teaching, financial support and encouragement for pursu ing the Master of E ngineering Program at Memorial University of Newfoundland (MUN) . Specia l thank goes to Dr. Amir Ali Khan, Department of Water Resources, Government of Newfoundland and Labrador for providing help . Funding for the study was prov ided by the Government of Newfoundland and Labrador through the Institute for Biodiversity Ecosyste m Science and Sustainabi lity (IBES). I wish a lso to thank the School of Graduate Studies for the financia l support.

My heart-felt appreciation goes to my dear husband, Vandad, for his support and sharing my aspiration during the study pe riod a nd thesis p reparation. I appreciate my beloved parents, for being understanding, and helping me to endure the ir distance from me during the period of my study.

A number of people have helped in data gathering for this study. I wish to thank Mr.

Michael Colbert, Hydrological Mode ling Section, D epartment of Environment and Con servation, Government of Newfoundland and Labrador for he lp ing with the hydrologica l records, and deli vering the requested data for the province of Newfoundland and Labrador.

Lastly, I wish to thank the academic staff , the support staff , and friends who have

helped me in many ways during the period of my study.

Table of Contents

ABSTRACT ... ... .. ... ... .. .... ... ... ... ... ... .. ... ... ... ... ii

ACKNOWLEDGEMENTS ... ... ... ... ... ... ... ... .... ... .. ... .. ... iii

Table of Contents ... .. .... ... ... ... ... ... ... ... ... ... ... ... ... ... .. ... ... ... ... iv

1 Introduction .. ... ... ... .. .. ... ... ... .... ... ... ... ... ... ... ... ... ... 1

1.1 General ... ... ... ... ... ... ... .... .. ... ... ... 1

1. 2 Low Flow Analysis for the Island ofNewfoundland ... ... ... ... ... ... 5

1.3 Research Objectives .... ... ... ... ... ... ... ... ... ... .. ... ... 6

1.4 Outline ofThes is ... ... ... ... .... ... ... .... ... ... ... ... .. ... .. .... ... ... 7

2 Literature Review .. .. .... ... ... ... ... ... ... ... ... ... ... 8

2 .1 Low Flow Frequency Analysis .. ... ... ... .. .. .... ... ... ... 9

2.1. 1 General ... ... ... ... ... ... ... ... ... .. ... ... ... .. ... .. ... 9

2 .1.2 Regional Flow Frequency Analysis .. .. ... ... ... .. .. ... ... .. ... ... .... ... 10

2.1.2. 1 Data Sc reening .. .. ... ... .... ... ... .. ... .. .. .. ... ... .. .. ... 11

2.1.2 .2 Delineation of Homo geneo us Regions .. .. .. .. ... ... ... .... .. 12

2. 1.2 .3 Regional Homogeneity T ests ... .... .. ... .. .. .. ... ... 14

2.1 .2.4 Selection and Es timation of Regional Distribution ... .. .. ... 16

2.1 .2 .5 Estimation of F low Magnitudes ... ... ... ... ... ... 18

2.2 Flow Duration Analysis .... ... ... ... ... ... ... .. . 2 1

2.2 . 1 Flow Duration Curve Construction ... ... .... ... o ···o····o ·· ·· ·· ·· ·o··· ·· ··· · ·· ···· ···-21

2 .2 . 1.1 Period of Record FDCs .. ... o ... o .... ... . ... .. .. .. . o·· ··· · ·· ····oo·o·o ·o·ooo o .. .. . 22

2.2.1 .2 Monthly or Seasonal FDCs ^{· ·· ·· ···· o ··· ···· o ··· ·· ···· ··· · ·· ·· o · ···· ··· oo ·o · o·· oo o .. .. .. 23}

2.2. 1.3 Annual FDCs ... ... o··o··· ··· o··· ···· ··· ·o···· · ·o ·· ··· ·· ··· ···· o··· · ··· o ·· ·· · ··oo· •o · ·· ·-23

2.2.2 Application ... .. ... .. . ·o·· ····o··· · ·o ·o· .. . .. .. .. ... .. . ··· ··· · o· · ···· o ·· · ···· ·· · .. o . ... . ···o ·o ··· ·· o · o · o · ·· .. 23

2.2 .3 Interpretation and Indices ... o .. .. ... . o · ··· ···· ··· oo ··· ·oo · ·oooo o.oo oo o·· · ··o ·o··· · ··ooo ... .. 24

2.2.4 Flow Duration Curve Estimation for Ungauged Catchments ·· ·· ·ooo ooo o oo oo o··· ··· 25

2.2.4. 1 Regional Regression Approach · ···o··o ·o oo oo ooo o oo o· ··o·oo·· ·o·o·· ··· ·oo ... .. . ooo o o···· ··25

2.2.4.2 Regional Prediction Curve ... . ooo· · ···o ·o·· ··o oooo ooo o···oo o .. .. ooo o·· ·· ·· ·· ·· ·· ·· ·· -27 2. 3 Flow Spell Analysis ·· ·· ···· ··· o··· ·· · ··· ·· ·o··· ·o ·oo ·· ·oo · oo· ·· ·ooo oo·oo oo · ·· o·o· ·· ···o o ···· · ·o oo .... .. .. . 29

2.3.1 Environmental Instream Flow Requirements .... ... o·· ··· ···· · o · ·· ·· ··o· ·· ··· · ·· ·· ···· ·· ·· ·29 2.3 .2 Continuous Low F low Events and Deficit Volumes ... ... .. ... ... .... ... ... ... ... 30

2 .3 .2. 1 Theory of Runs ... ... o·o·· .. .. .

.o ·o·oo.

o o·.

.o ·oo o· .. · ·ooo· ... .. oo o . .... .. ... 3 1 2.3 . 2 .2 Flow Spells ... o . .. .. . o ... ... . .. .... .. o· ··· ·· ··· · ·· ·· ·o· ·· ··· · ··· · ···· ··· ·· ··o···· ··· ··- 32 2.4 Previous Low F low Studies in the Province · ··· · ·· ··· · ·· ·· ·o·· · ··· · ···o··· · ···o···· ··· ·· 34

2 . 5 Rationale ofthe The sis · · ··· · ··· ···· ·· ··· · ·o·· ·· ··o··oo · ··· oooo· ·· ··o· ·· ·o·o · ·ooooo· · ··oo o·o· ·· ·o ·o··· ·oo . . .. 38

3 Methodology ... ... .. ... ... .. o ... . ... .... .. .. ... o .... . ... . ... .... .... o .. .... o . .. .. .. ooo·o ·· ·· oo o ... . 39

3. 1 General ... ... o···· ··· ···· ··· ·· ···· ·· · ··· ·· · ··· o·· · ··· o· ···o· ·· ·· ·o· ·· ··· o ··· ··· · oo oo .... 39

3.2 Regional Low F low Frequency Ana lysis .... ... oo · · ···· ··· · ·· o ·· ·· ·· ··· · ··· · ·o · ···· ·· ···o39

3 .2.1 Probability Distribution ... ... ... .... ... ... ... ... ... .... ... ... 40

3.2.2 Moments .... .... .... ... ... ... ... ... ... ... .... ... .... .... ... ... ... 41

3.2.3 L-Moments ... .. ... ... .... .... ... ... ... .... .. ... ... ... 42

3.2.4 Sample L-moments .... ... ... ... .... ... ... ... ... .. ... ... ... ... 44

3.2.5 Steps in Regional Frequency Analysis ... ... ... ... .. .... ... ... .. .. ... ... ... 4 5 3.2.5 .1 Data Screening ... ... ... ... ... ... ... ... .45

3.2 .5.2 Delineation ofHomogeneous Regions .. ... .... ... .... ... ... ... .. 46

3 .2.5.3 Regional Homogeneity Test ... ... .... ... .. .. ... ... ... ... ... .47

3.2.5.4 Selection and Estimation ofRegional Di stributi on ... ... .... .. .. ... ... .49

3.2.5.5 Flow Quantile Estimation ... ... ... .... .. ... .... ... ... ... ... ... . 52

3.3 Reg ional Flow Duration Analysis ... ... ... ... ... ... ... ... ... ... 55

3. 3 .1 Constructing Flow Duration Curves .... ... .... .... .... ... .... ... ... .. ... 55

3.3. 1.1 Period ofRecord FDC ... .... ... ... ... ... ... ... ... .... ... ... ... ... ... .. 56

3.3. 1.2 Annual FDC ... .... ... ... ... ... ... ... ... ... 5 6 3.3.2 Flow Duration Curves Region al Regression ... .. ... .. ... ... .... ... ... ... .. ... ... .. 57

3.3.2. 1 Regress ion Mode l ... ... ... .... ... .... ... ... ... ... ... ... 58

3.3 .2.2 FDC Quanti les .... ... ... .... .... .... ... ... ... 59

3.3 .2.3 Physiograph ic Parameters ... .. ... ... ... ... ... ... ... ... 59

3.4 Regional F low Spell Analysis ... .. ... ... .. .. .... ... ... .. .. 60

3.4.1 Defining Flow Spell ... .... .. ... ... .. ... ... ... ... ... ... 60

3.4.2 Environmental Instream Requirement as Threshold .. ... ... ... ... ... ... .. 61

3.4.2. 1 Percentiles ofFDC and AFDC ... .... .. ... ... ... ... ... 61

3.4.2.2 Percent of Mean Annual F low ... ... ... ... . 62

3.4.2.3 7Q10 .. ... ... .... .... ... ... ... .. ... .. ... ... ... . 63

3.4.3 Predicting Flow Spells ... ... ... ... ... . 64

3.5 Study Area and Data ... ... ... ... .. ... ... ... ... ... ... ... 65

4 Low Flow Frequency Analysis and Results ... .. ... .. ... 7 1 4.1 Data Screening: Discordancy measure ... .. ... ... ... ... ... ... .. ... ... ... . 71

4.2 Identifying Homogeneous Regions ... ... ... ... ... ... ... ... 72

4.3 Ide ntification of Regional Frequency Di stribution .. ... ... ... . 79

4.4 Reg ional Estimation using Index- flow Procedure (Regional Growth Curve) ... 81

4.5 Low Flow Estimation for Un gau ged Sites .... ... ... ... ... ... ... ... ... . 86

4.6 Verification ofResults ... .... ... ... ... ... ... ... ... .... 88

5 Flow Duration Analysis and Results ... ... ... ... .... ... ... ... ... ... . 92

5.1 Percentiles ofFDCs and AFDCs ... ... ... ... .. ... ... 93

5.2 Physiographic Parameters .... ... ... ... .... ... ... ... ... .... ... ... ... . 93

5.3 Sets of Regression Models ... ... .. ... ... ... ... ... .. ... .... ... 99

5.4 Verification ofResults ... ... ... .. ... ... .. ... .. ... .... 102

6 Flow Sp ell An alysis and Results .. ... .... .... ... ... .. .. ... ... ... ... ... I 07 6.1 Ins tream Flow Thresho ld Values .. ... ... ... ... .... ... ... ... .... l07 6.2 Comparison of Estimated Flows at Different Thresho lds .. .... ... .... ... 1 L 0

6.3 Regiona lization of Flow Spell s ... ... .... ... ... ... ... ... ... 114

6. 3 . 1 R egiona l Prediction of Thresho ld Va lues ... .. .... ... ... ... .. ... 114

6 .3.2 Regiona l Pred iction of Annual M aximum Spe ll Variab les ... ... .. ... .. . 117

6 .3.3 R egiona l Prediction ofProbabi lity Distribution ... .. ... .. ... ... ... 123

7 S ummary of th e R esults .. ... ... ... ... ... .. .. ... ... .... ... ... ... ... ... .. 124

7. 1 Ge neral ... ... .. ... ... ... ... .. ... ... .... .. ... ... .... ... .... ... ... ... 124

7.2 C onclusion s: R egio nal Low F low Fr equ ency Ana lys is .... ... ... .. ... ... ... ... l 25 7.3 Co nclusions : R egiona l Flow Duration An a lysis ... ... .. ... .. ... ... .. ... ... 126

7.4 C onclusions : Reg iona l F low Spell An alysis ... .... ... ... .. ... ... ... l 27 7. 5 R ecommendations ... ... ... ... ... .. ... ... ... ... .. ... ... .. 128

Bibliography ... ... .... ... ... ... ... ... ... ... .... ... ... .... .... .. ... ... ... ... .... 129

Appendi x .... ... ... ... ... ... .. ... ... ... ... ... .... ... ... ... ... ... ... ... .... 138

List of Tables

Table 3-1 Critical values for discordancy measure, Di (after Hosking and Wallis, 1997)46

Table 3-2 Tennant's Method (adopted from McMahon et al., 2004) ... ... ... 63

Tab le 3-3 Selected Hydrometric Stations in Newfoundland (HYDAT database) .... .. ... .... 66

Table 3-4 Selected Hydrometric Stations in Labrador (HYDAT database) .... ... . 68

Table 4-1 Statistics summary of gauging stations in Labrador.. ... ... ... .. . 73

Table 4-2 Statistics summary of gauging stations in Newfoundland . .. ... .. ... .. ... 74

Tab le 4-3 Weighted regional average L-statistics and weighted regional standard deviation ... .. .. ... ... ... ... ... ... ... ... ... ... ... ... 78

Table 4-4 Kappa distribution parameters and heterogeneity measures ... .... ... .. .... 79

Table 4-5 L-Kurtosis based goodness-of-fit measure .... ... ... ... ... ... .... .. ... ... 80

Tab le 4-6 lognonnal distribution parameters ... ... ... ... .. .. .... .... ... ... ... 83

Table 4-7 mean annua l minimum flow prediction equations ... ... .. ... ... . 86

Table 4-8 minimum low flow prediction equations .... .. .... .... ... ... .... ... ... ... . 88

Table 4-9 Selected sites for verification of Newfoundland regiona l models ... ... 89

Tab le 4-10 Se lected sites for verification of Labrador regional m odel s .. ... .. .. ... ... .... .. 89

Ta ble 5- l Physiographi c database ... ... ... ... ... ... .... ... ... ... ... ... ... .. 94

Tab le 5-2 Sets of regression equations for FDC quantiles in Newfoundland and Labrador ... ... ... ... ... .... .... ... ... .. ... .. .. ... ... ... .. ... .. ... 100

Tab le 5-3 Se ts of regress ion equations for AFDC quantiles in N ewfoundland and

La brador ... ... .... ... ... ... .. ... ... ... ... ... ... ... ... ... 101

Table 5-4 NSE values for FDC and AFDC predictions in validation sites ... ... ... .. 1 04

Table 6-1 Results of thresho lds (m ³ /s) obtained for rivers in Labrador.. ... ... .. ... ... 107

Tab le 6-2 Results of thresho lds (m ³ /s) o btained for rivers in Newfoundland ... 1 08 Tab le 6-3 Probability of exceedance by fl ow duration analysis for Labrador ... ... 112

Table 6-4 Probability of exceedance by flow duration analys is for Newfoundland ... 113

Table 6-5 Re lat ionship between thresho lds and drainage areas in Newfoundland .... ... . 117

Table 6-6 Relationship between thresho lds and drainage areas in Labrador.. ... 117

Tab le 6-7 Relationship between thresholds and mean annual maximum volume in Newfoundland ... ... ... ... ... ... .... ... ... ... .... ... ... .... ... ... ... ... ... ... 122

Table 6-8 Re lationship between thresho lds and mean annua l maximum volume in Labrador ... ... ... ... ... .. .. ... ... ... ... .. ... .... ... ... ... 122

Table 6-9 Re lationship between thresho lds and mean annual maximum intensity in Newfound land ... ... .... ... ... ... .. .. .... ... .... .. .. ... ... 122

Table 6- 10 Re lationship between thresho lds and mean annual maximum intensity in

Labrador ... ... .. ... ... ... ... ... ... ... ... ... ... ... ... ... .... ... 122

List of Figures

Figure 2-1 Low flow frequency curve ... .. .... ... .... ... .... .... ... ... ... ... ... .. .... ... 9

Figure 2-2 Daily flow duration curve ... ... .. ... ... ... ... .. ... .... ... ... 21

Figure 3- 1 L-moment ratio diagram, key to distributions: E-exponentia1 , G-gumbel, N- nonnal, U-uniform, GP A, generalized pareto, GEV -generalized extreme value, GLO- generalized logistic, LN3-lognormal, PE3- Pearson type III . .... ... .... .... ... ... .... .... 50

Figure 3-2 Typical regional growth curve ... ... .. .. .. ... ... ... ... .... ... ... ... .... .. ... 55

Figure 3-3 General diagram of defining flow spells .... ... .... ... ... ... ... .... .... 60

Figure 3-4 Location of hydrometric stations in Newfoundland ... ... .... .... .... 69

Figure 3-5 Location of hydrometric stations in Labrador ... .... .... ... ... ... ... ... ... ... .. . 70

Figure 4-1 L-moment ratios in Newfoundland (a: 1-day; b : 7 -day) and in Labrador (c: 1- day; d: 7-day) .. ... .. ... ... .. .. ... ... ... ... ... ... ... ... ... .... ... 77

Figure 4-2 L-moment ratio diagram and regional averages ... ... ... ... ... ... 82

Figure 4-3 Regional comparison between fitted lo gnormal distributions 1-day ... .. 84

Figure 4 -4 Regional comparison between at-site and fitted lognormal distribution, Newfoundland 1-day ... ... .. ... .... .. ... ... ... ... ... ... 84

Figure 4 -5 Regional comp arison between at-site and fitted lognormal distributi on , Labrador 1-day .. .... ... ... ... ... ... .... ... ... ... .. ... ... ... ... .. ... ... ... .... ... ... 84

Figure 4-6 Regional comparison between fitt ed lognonna l di stribution s 7 -day .... ... .. 85

Figure 4-7 Reg ional comparison between at-site and fitted lognonnal distribution,

Newfound land 7-day .... .... ... ... .. .. ... ... ... ... ... .... ... ... ... .... 85

Figure 4-8 Regional comparison between at-site and fitted lognormal distributi on , Labrador 7 -day ... ... .... ... .... ... ... .. ... ... ... ... ... ... ... ... ... ... .. 85 Figure 4-9 Regress ion of index flow with basin areas in Newfoundland and Labrador ... 87 Figure 4-10 Obs erved and regional estimated growth factor, Newfoundland 1-day AM verification sites .. ... ... ... .... .... ... ... .... ... ... .. ... ... ... ... . 90 Figure 4-11 Observed and regional estimated growth factor, Labrador 1-day AM verification sites ... ... ... .... ... .... ... ... .. .. .. ... ... ... .... .. ... ... .... .. . 90 Figure 4-12 Observed and regional estimated growth factor, Newfoundland 7-day AM verification sites ... ... ... .. .. ... ... .. .... ... ... .... ... ... .. ... ... ... ... 91 Figure 4- 13 Observed a nd reg ional estimate d growth factor, Labrador 7-day AM verification sites ... ... ... ... ... .... ... ... ... ... ... ... 91 Figure 5-1 Comparison of observed and estimated FDCs for validation sites ... ... .. .... 1 03 Figure 5-2 Comparison of observed and estimated AFDCs for validation s ite ... . 1 05 Figure 6- 1 Comparison of the estimated flows for different threshold methods with 25%

MAF for Newfoundland .. .... ... ... ... ... ... ... ... ... .... ... .. .. ... .. ... ... 111 Figure 6-2 Comparison of the estimated flow for different thresho ld methods with 25%

MAF for Labrador ... ... ... .. ... ... .. ... .. ... ... ... ... ... 111 Figure 6-3 Threshold values as a function of drainage areas of Newfoundland stations 115 Figure 6-4 Threshold values as a function of drain age areas of Labrador stations ... ... 11 6

F igure 6-5 Relationship between threshold va lue and mean of annua l maximum volumes

for Newfoundland ... .. ... .. .... .. .... ... ... ... .. ... .. ... ... .... ... ... .. 11 8

Figure 6-6 Relationship between threshold value and mean of annual maximum intensity

for Newfoundland ... ... ... ... ... ... .. .. ... .... ... ... .. ... 119

Figure 6-7 Relationship between threshold value and mean of annual maxim um vo lume

for Labrador .. ... ... ... ... .... .... ... ... ... ... .. ... .... ... ... .. ... 120

Figure 6-8 Relationship between threshold value and mean of annual m aximum intensity

for Labrador ... .. ... ... ... ... ... ... .... ... .. ... ... ... ... .... .. .. ... .. ... 121

List of Symbols and Abbreviation s

a Scale parameter of the distribution ar Population probability weighted moment B ₄ Bias of sample regional L-kurtosis f3r Population probability weighted moment Cv Coefficient of variation

Di Discordancy measure

E(x) Expectation ofrandom variab le x F Non-exceedance probability

<D Standard normal CDF

F ( x) Cumul ative distribution function

<D - ¹ Inverse of standard nonnal CDF

h 4th parameter of the distribution H Heterogeneity measure

k Shape parameter of the distribution

ln natural log

Ar Population L- moments lr Sample L-moments

11 mean llv mean ofVs

N sim Number of s imulated regions

Q Flow rate

CJ Sta nda rd deviati on

CJ 2 Vari ance

CJ ₄ Standa rd deviation of sampl e regional L -kurtosis

CJv sta nda rd deviatio n ofV s T Re tum period

T L-CV

t Sa mple L-C V T3 L-Skewness

t 3 Sampl e L-S kewness

t/ ^{Regiona l a} verage sampl e L-Sk

T 4 L-Kurtos is

t 4 Sa mple L-Kurtosis

T 4 ^Dist Distributio n L-kurtosis

t 4 R Regiona l a verage s ampl e L-Ku t R Reg io na l average sampl e L-C V

u L- mome nt ratio vector

V weig hted standard deviatio n of sample L-CVs ( Loca tio n pa ra mete r o f di stributio n

X( F) Qua ntile fun ction of frequ ency dis tribution

Z Di st Goodn es s- of-fi t measure of the candida te di stributi o n

7Q 10 7-day consecutive low flow with ten year return period AFDC Annua l F low Duration Curve

AM Annual Minimum Eff-P Effective Precipitation

FDC F low Duration Curve L-CV Coefficient of L-variation

L-ku Coefficient of L-Kurtosis NSE Nash-Sutcliffe Efficiency OLS Ordinary Least Squares

PWM Probability Weighted Moments

List of Appendices

A-1 Discordancy Measure ... .

138

A-2 Kapp a Distribution .. ....

140

A-3 Heterogeneity Test. ... . . .. ... ..

148

A-4 Goodness-of-fit Test. .. . . . ..

150

A-5 Parameters of Lognormal Di stribution . ..

^{· ·o.. . .. . .. . .. . ..} 154

A-6 Quantile of Lognorma l Distribution . ....

o. 156

1.1 General

Stream flows naturally vary both during a year and from year to year. In the face of these variabilities, water management decisions can only be made with predicted estimates of stream flows. More importantly, des ign and planning of water resources projects requires the assessment of the probability of extreme hydrological events such as low or high flows . A low flow condition can be defined as a period during which the average stream flow is a minimum for the year. The characteristics and est imation of low flows are important issues in hydrologic studies such as the detennination of minimum downstream flow requirement of hydropower station , estimation of available water supply for municipal and industrial uses, water quality management, detenninatio n of potential capacity for effluent dilution, assessing the impact of low flows on aquatic ecosystem, and in genera l for environment impact assessment stud ies (Govt. of Newfound land and Labrador, 1991).

The low flow reg ime of a ri ver can be analyzed in a variety of ways depending on the type of data initially availab le and the type of output infonnation required (Smakhtin , 2001 ). Low flow studies often reqmre that the hydrolo gists estimate the magnitude , frequency, duration , and spells of low flow events as different aspects of low flow analys is by applying me thods of flow frequency, flow duration, and flow spell analysis.

Flow frequency, flow duration, and flow spe ll anal ysis are the three main objectives of

the c urre nt study.

Flow frequency analysis is traditionally b ased on fitting a probability distribution to the availabl e data at a sp ecifi c site of interest. This p robability only gives us an idea how like ly a flow is to happen in future. It is generall y assumed that flow magnitude can be reli ab ly estimated from a long perio d of data records . However, the ava ilab le hi storica l fl ow data at the s ite of interest are o ften too short to give these reliab le estimates of critical flow (low or high ). This conditi on has led h ydro lo gis ts to wonder w heth er the estimation from one sample can be more ac curate by not jus t using infonnation fro m one sample but also from other related sa mples . Therefore approach es have been developed to augment the limite d flow record available at a specific location by invo lving data from neighboring locations, the so called homogeneous hydro logical region. Thi s tech nique wo uld not only improve the estimates at the site of interest with short data record s, but would also provide a basis for flow estimation at a ny ungauged locations within that ho mogeneous region . The process of using data from several sites to estimate the fre quency dis tribution is known as regional freq uency ana lysis. This procedure can be used for estimating any fl ow statistics such as mean , low or hi gh fl ows (Hosking and Wa llis, 1997) . In thi s study th e interest is in the m inimum low flow estimation, thus the outcome of th e regional frequen cy analys is would b e the low flow min imums with associated frequency of flow being equ al or below this a mount.

T he genera l procedure of conduc ting regional fre q uency analys is involves the fo llowing b asic step s: collecting low fl ow d ata at the gauged rivers ; screenmg the collected da ta for any gross errors or any oth er causes that makes the data un usable;

ide nti fying ho mogeneous regions and test ing their ho mogeneity; determin ing the regional

prediction equations (growth curves or regression relations) for the homogeneous regions;

and establi shing the flow quantiles of interest. Estimating flow magnitudes u sing the regional approach has been documented for the last four decades (Ho sking an d Wallis, 1997).

The index flow method suggested b y the USGS (Dalrymple, 1960) is the earliest and still most popular approach for regional estima tion w hich is sti ll in use with s ligh t modifications over time. Regression on quantiles was suggested as an a lternative approach to overcome th e apparent problems associated with the orig inal index fl ow method regarding its assumption about th e distribution characteristics of flow data within a region. With the introducti on of the L -moments approach in statistics and its app li cation in hydrolo gy the index flow method has been firmly re-established as a general procedure of flow frequ ency analysis. This approach h as been used for conducting th e regional low flow frequency analys is in this s tudy .

To find out what percentage of time of a year fl ow in a river will be below a c ertain amount, it is necessary to conduct flow duration ana lysis using flow-duration curves.

Flow-duration curves simply prov ide the relationship between streamflow and the

percentage of time it is exceed ed (Vogel and Fennessy, 1994) . Flow-duration curves, as a

comparison to flo od or low flow frequency ana lys is , are derived from a ll the historic data

availab le for a stream rath er than just the annua l lowest flow . Simil ar to flow frequenc y

analysis, regiona l fl ow-duration analysis can be conducted for a region. There ar e

different m ethods for regionali zation of fl ow-duration c urves, the most common being the

multiple-regression approach. Wh ere no flow measu rements exists at a site, a common

approach for estimating streamflow is to develop a relationship between fl ow measurements from a gauged river and physiographic parameters of its basin . Some of th e factors which have been considered include catchment area, ma in chan ne l slope, drai nage dens ity, difference in elevation, and percentage of the area covere d by forest, swamps, lakes and impermeable rock. Regress ion equations can be develo ped between these factors and any streamflow indices d eriv ed from the flo w-duration curves. And finally for any ungauged location within the defined region, stream flow indi ces can be estimated from the multiple regress ion equation and estimates of these physiographic factors (McMahon et al., 2 004).

Environmental instream flow requirm ent are flows in a river that are deemed as a minimum to maintain the river ecosystem (Karim, 1995) . Therefore, it is critical to have an estimate of these required instream flows, and planning for th e time that streamflow goes below this amo unt. There are several ways that instr eam flow can be estimated . Methods such as percentil es of the flow duration curves , p ercentages of the mean an nual flow, and consecutive seven-day averaged low fl ow with an estimated te n year return period . Selecting a method to estimate the environmental instream flow depends on the particular requirement that is be ing con sidered for the ecosystem (McMah on et a!, 2 004).

In order to estimate how long strea mflow will be below a certain amount (instream

requirement), and how large the deficit volume is, it is necessary to condu ct flow spell

ana lysis. This may be found by us ing the aforementioned instream minimum flows as a

thres hold on the sequ ential daily flows. Flows below these thresholds are considered as

spe lls (IH, 1980) that may be quantified in tenns of duration (in days), volume (in m ³ )

and intensity of flow spell (volume divided by the duration). Therefore, flow spell analysis takes into account the sequencing of flows. Flow duration curves, in contrast, give no infonnation on how the low flow days are distributed.

1.2 Low Flow Analysis for the Island of Newfoundland

The history of low flow estimation in the is land of Newfoundland dates back to 1991 , when the Government of Newfoundland and Labrador (Govt. of Newfoundl and and Labrador, 1991) conducted the first study to quantify the characteristics of low stream flows and came up with set of equations to estimate low flows of various durations and return periods on ungauged streams. However, at the time of that study the pres ent state- of-the-art regional frequency analysis techniques were not available, and the recorded data period was short.

A hydrological study of the Island of Newfoundland was perforn1ed in 1995 (Richter and Lye, 1 995) to identify the key basin characteristics associated wit h flow measures and assessed several methods of regional subdivision for improving flow estimates at ungauged sites. A data set of 40 stations with more than 10 years of record was used in this study.

In 1997 a research study was performed on duration, volumes, and intensities of flow

spell s for a few rivers in Newfoundland and Labrador (Shaughne ssy, 1997) . Different

methods of estimating environmental instream flow requirements were used as the

threshold values. Aga in, the number of suitable gauged rivers and their record period was

s hort in this study. A more detailed review of the aforem entioned studi es is given in

Section 2 .4.

1.3 Research Objectives

The first objective of this research is to app ly the popular L-moments based index flow approach to conduct a regional frequency analysis for low flows for rivers of Newfoundland and Labrador, Canada. The L-moments and regional frequency ana lysis based on L-moments were introduced in the early 1990's (Hosking, 1990, Hosking and Wallis, 1993) . The 1991 low flow study for the Island of Newfoundland was based on 'regression on quantiles' approach, and data records were short at that time. The Island of Newfoundland was the only region used in the 1991 study, and no research was perfonned on rivers of Labrador. In the present study the more efficient 'L-moments' approach will be used to conduct a regional analysis for rivers in both the Island of Newfo undland, and the Labrador region where the records are now of sufficient lengt h for frequency analysis.

The next objective of the proposed research is the development of regiona l flow duration estimation equations for Newfoundl and and Labrador. A regiona l regression approach will be used between flow indices of flow duration curves and related phys iographic parameters of river basins, to produce a set of prediction equations for ungauged sites.

The final goal of this thesis is to provide a means of estimating flow spells for rivers of

Newfoundland and Labrador, and to quantif y duration, volume and inten sity of those

spe lls based on different instream flow requirements. This part of the study is revis iting

the previous study und ertaken in 1997 by us ing longer availab le record data, and

additiona l gau ged rivers for the analysis.

1.4 Outline of Thesis

The th esis is organi ze d into three major groups of chapters :

- Introdu ction to the problem, overv iew and a pproaches: C hapter I and 2 - Main me thod o logies: Chapter 4 , 5 and 6

Summary and conclusions : Chapte r 7

Chapter 1 covers the introduction of th e topic in which the gen eral concepts of low

fl ow estimati on including regiona l low fl ow frequency, fl ow durat ion, a nd fl ow spell

analysis and the ir appli cation in Newfoundland and Labrador a re briefl y discussed .

Chapter 2 su rveys the ex is ting litera ture review on flow estimatio n me thod s w ith the

particular emph asis o n regiona li zation tec hniques. The main methodo logies proposed,

regional low flow , fl ow duration , and flow s pe ll ar e briefl y introduce d in Chapter 3, and

then followed up by their a pplicati on fo r se lected rive rs w ithin New foundl and a nd

Labrad or in Chapte rs 4, 5 a nd 6, res pec tive ly. Summary an d conclusions of this study can

be found in Chapter 7. F in all y, the comp uter programs that were deve loped for various

processing of th e da ta are presented in the appendi ces.

2 Literature Review

Low flows have been investigated on ly in the recent past few decades. This includes low flow frequency analysis, base flow separation, recession analysis, flow spell analysis , and low flow estimation at ungauged sites. Although there is a high interest in low flow studies, the mass of literature has still been relatively less compared with flood or precipitation studies. It could be a result of that low flows are viewed less destructi ve as floods . The characteristics and estimation of low fl ows are important issues in many hydrologic studies and in general for environmental impact assessment studies. Such studies often require that the hydro logists estimate the magn itude, frequency, durati on , and spells of low flow events as different aspects of low flow analysis (Smakhtin, 2001).

Proposing any solutions to a problem is only justifiable after a complete know ledge and

understanding of the existing solution(s) to th e problem at hand or problems with some

sim ilar characteri stics . For this reason, this chapter reviews the deve lopments and existing

theories and methods that are relevant to low flow analysis in general. At the end, the

earlier report of the Provincial Gov ernment of Newfoundland and Labrador on th e low

flow characteristics of the rivers in the Is land , the study on relationship between flow and

basin variables on the I sland by Richter and Lye (1 995), and also flow spell analys is

research by Shaughnessy ( 1997) for rivers in th e province are reviewed .

2.1 Low Flow Freq uency Ana lys is

2.1.1 General

Unlike the flow duration curve wh ich shows the proportion of time during which a flow is exceeded, a low flow frequency curve shows the proportion of years when a flow is exceeded or equiva lently the average interval in years (return period or recurrence interva l) that the streamflow falls be low a give n discharge . F igure 2-1 illustrates a typica l lo w flow frequency curve.

2 . - - -- - - -- - - , - Flow Freouencv Curve

"' ..._

"' E ¹

_Q

3:

u..

0.5

0

1 10 100

Return Period {years)

Figure 2-l Low flow frequency curve

Low flow frequency analys is forn1 a part of the frequency analys is of extreme events and as s uch has been covered in ma ny classical hydro logy text books (e.g. McMahon e t a!., 2004). Some authors note that the literature on low flow frequency ana lys is remains to be limited compared, for example, with the litera ture on flood frequency (e.g. Voge l and Wilson, 1996) .

The ex ist ing approaches in flow estim atio n can broadly be divided into two sections:

(1) statisticall y based ; (2) physically based. Statistically based approaches refer to the

analysis of raw d ata co ll ected from a s ite or a region us ing state-of-the-art statistical tool s

in deriving probabilistic functions or frequency distributions pertaining to flow (low flow)

quantiles. Physically ba sed approaches essentially model the actua l flow conditions in a river channe l based on all the available phys ical theories and d ata. Two distincti ve components can be delineated based on the theories used: hydrologic and hydraulic. S ince the physically based approac h is not within the scope of this study, only the statistical approach is reviewed in the following sections.

Traditional methods that dominate the statistical approach are single station flow frequency ana lys is and regional versions of this analysis. Flow frequency analysis is a standard procedure for the planning and design of water resources projects and other civil engineering works. It provides the probabilistic assessment of the magnitude of (flood or low) flows associated w ith a certain risk tolerance level. It was discussed before that one of the objectives of this study is to develop regional lo w flow frequency models, and to apply the mature method of L-moments for this reg ional analysis. Therefore, the res t of this section wi ll be confined to some ofthe history of regional flow frequency analys is .

2.1.2 Regional Flow F requency Analysis

This section will review the literature on regional flow frequency analysis under the fo llowing subheadings th at constitute the general procedure of the analysis:

- Data screening;

- Delineation of homogeneous regions;

- Region al homogenei ty test;

Selection and estimation of regional frequency distribution;

- Es timation of flow magnih1des; and

Quanti le estimat ion accuracy assessment.

2.1.2.1 Data Screening

The first essential step of any statistical data analysis is to check that the data are appropriate for the analysis. For frequency analysis of any hydrological event, the data collected at a site must be a true representation of the quantity being measured and must be drawn from the same frequency distribution. It is also based on the assumption that the data are random, independent and homogeneous. For hydrological data errors could be due to incorrect recording, systematic changes over time (type or location of the measuring instrument), human-induced flow regulation, or any combination of these . These errors may cause data to have outliers, non-homogeneity, serial correlation, and trends which subsequently reduce the reliability of the frequency analysis based on these data.

Statistical tests for outliers and trends can be found in the literature (e.g. K endall, 1990;

Barnett and Lewis, 1994). Double-mass plots and quantile-quantil e plots are some of the

techniques that can be used for between-site comparisons. In addition, there are many

computer software packages that can perform tests for outliers, trends, and serial

correlation (e .g. Environment Canada CF A 3 .1 ). In th e context of regiona l frequency

analysis usin g L -moments, Hosking and Wa llis (1997) found that comparing sample L-

moment ratios of different sites provide usef ul infonnation. They noted L-moments of the

data can reflect the incorrect data values, outliers, trends, and shift in the mean of a

samp le. They introduced a composite statistic based on L-moment ratios , a measure of

discordancy between the L-moment ratios of a site and the average L-moment ratios of a

group of s imilar site s, called the discordancy measure (Di). The details on computations and interpretation ofDi statis tics are given in Section 3.2 .5.1.

2.1.2.2 Delineation of Homogeneous Regions

The identifi cati on of homogeneous regions is usually the most difficult stage in a regional frequency analys is, and requires the greatest amount of subj ecti ve judgme nt. The aim is to form groups of sites such that their frequency distributions are identical except for a site-spec ifi c scale factor (Hosking and Wallis, 1997).

Several methods have been proposed for groupin g sim ilar sites into regions and for use in the regional frequency analys is w hich can be roug hly categorized based on th e following basis:

Geographical convenience: Regions are often defined by sets of contiguo us sites, based on administrative areas (e.g. FEH, 1999; Beable and McKerchar, 1982), or major physiographic sites groupin g (e.g. Mata las et a l, 1975). However, as Wiltshire (1986) and Acreman and Sinclair (1986) discussed, geographica l proximity could not guarantee hydrolog ical homogeneity, as some neighboring basins could be physically very diff erent.

Kachroo et a l (2000) in a more recent study utilized sound j udgment about the hydrological responses of the bas ins based on geographic information and similarity of the statistics of the observed flow data. The geographic regions they delineated were found to be hydrolo gically homogeneous.

Subjective partitioning: Regions can be defined subj ective ly by inspection of the site-

characteristics, especially for small-scale studies. Therefore, the fonn ed regions may or

may not be geographically contiguous. The res ulting regions from this subjective method

can be objectively tested by heterogeneity measure described m Section 3 .2. 5.3 . The Government of Newfoundland and Labrador (1991) study divided the Island of Newfoundland into thre e regions based on site characteristics factors. Gingras et al.

(1994) is an example of subjective partitioning as wel l. They formed regions for annual maximum streamflow data in Ontario and Quebec by grouping the sites according to the time of year at which the largest flood typically occurred . It should be noted that the use of at-site statistics in subj ective partitioning is discouraged as this might affect the validity of test of homogeneity which is usually based on the at-site data itself (Hosking and Wallis, 1997).

Objective partitioning : In this method of partitioning, the sites are assigned to one of two groups depending on whether a chosen site characteristic does or does not exceed some thresho ld value. This threshold value is chosen to minimize a within-group heterogeneity criterion . Wiltshire (1985) used a single measured partitioning value of one or more basin characteristics to group the basins. In an iterative fashion, the optimum size of the region wou ld be defined by minimi zing the within -gro up departure of these statistics. Pearson (1991a) applied similar procedure and used within-group variation of samp le L-moments . Hosking and Wallis ( 1997) described this procedure as an effective 'objective partitioning' approach. They used it in conjunction with an effici ent homogeneity test (heterogeneity measure) as defined in Hosking and Wallis (1993).

Pearson (199lb) successfull y applied this heterogeneity measure along w ith Wiltshire's

( 1985) partitioning criterion for regionalization of streamflow data for small drainage

basins in New Zealand.

Cluster analysis: It is a s tandard method of statistical multivariate analysis and it is used for dividing data into groups. T his method has been successfully used to form regions in regional frequency analysis. In this method, a d ata vector represents the characteristics of a site and the sites are group ed according to the simil arity in their respective data vectors . De Coursey (1973) was the first one wh o applied cluster analys is to fonn groups of sites having similar peak flo w respon se. Acreman and Sinclair ( 1986), Bum (1989), Guttman (1993), and Lim and Lye (2003) are some of examples of using this partitioning method for identi fyi ng homogeneous regions in r egional fre quency analysis.

Hosking and Wallis (1997) regard cluster ana lysis of site characteristics as the most practical method of fonning regio ns from large data sets. However, they noted that the output of this analysis should no t be considered final and it needs subjective decisions at several stages. In addition, they prov ided insigh t into the maximum a nd minimum s ize of the regions to be fonned by thi s procedure for use with the index flow method.

2.1.2.3 Regional Homogeneity Tests

Once re gions are fonned based on the physical characteristics of the s ite, it is required

to assess whether the regions are h ydrologicall y homogeneous , so tha t the inf ormation

obta ined from the region is u seful for flow freq uency analysis. It s hould be tested whether

a region is hom ogeneo us or it needs to be divid ed into more reg io ns, or whether two or

more homogeneous regions are s imilar and so should be combined to fonn one

homogen eous region. The hypothesis of homogeneity is based on the assumpti on that the

at-site freq uency di stribution s of the observed data a t the sites in a hom ogeneous region

are identical except for a s ite-specific scale factor. This test is constructed as a stati stical significance test of the similarity of appropriately chosen s tati sti cs calculated from the di stribution of at-site data. However, selection of which statist ic to use and whi ch distribution to assum e for the at-site data has remained controversial for th e last few decades . Thi s test examines the simi larity between the at-site di stribution and hypothesized regional distribution . Some of the regional homogen eity tests in the literature are reviewed n ext.

D alrymple (1960) apparently is the first published literature on a region al homogeneity

test. He suggested a procedure to test homo geneity of a region for the index flow m ethod

based on the study of 1 0-year flood estimated from the Gumbel frequency distr ibution at

each gau ging station within the region. Wiltshire (1986 a, b) proposed the next two

approaches after Da lrymp le (1960) based on statistica l hypothesis tests. The first

approach involved testing the regional homogeneity based on the coefficient of variat ion

(CV) of standardized annual maximum series, whereas th e second app roach was a

distribution based procedure and used the geometry of the cumulati ve distribution

function of the dimensionless reg ional parent. He concluded that the second approach is

better in tenns of statistical power. In order to evaluate the regional homogeneity,

Wi ltshire u sed a non-parametric jack-knife procedure to estimate the at-site distributio n,

unlike D alrymple who assumed Gumbe l distribution as the parent distribution at each

site. Hosking and Walli s (1 993) proposed the next important statistical test for

homogeneity test based on the sample L-moments ratios . Chowdhury et a !. ( 199 1)

suggested another stati stical test based on L-moments which was more powerf ul than

previous tests ; however, the most rigorous L-moment based test of homogeneity is that of Hosking a nd Wallis (1993). It compares the variability of the L-moment ratios of the sites within a region with the expected variability obtained from simu lation from a co llection of s ites with the same record length as their real world counterparts. A heterogeneity measure is then calculated based on the difference between the weighted standard deviation of the sites' L-CVs in the region and the mean of the same statistics obtained from the simulation . Hosking and Wallis ( 1997) used a 4-parameter Kappa distribution for their simulation. This test has been used as a standard test of homo geneity in recent years (e.g. Castell arin et a!., 2001; Lim and Lye, 2003). Details of this test are discussed in Section 3.2.5.3.

2.1.2.4 Selection and Estimation of Regional Distribution

After confinning the homogeneity of a region in regiona l frequency analysis, a single

frequency di stribution is fitted to the data from several sites within that region . The

candidate distributions are usually eva luated for the accuracy of the quantile estimates for

each site. There are many families of distribution that mi ght be cand idates to be a regiona l

freque ncy distribution. The choice of this distribution can be eva luated by con siderin g its

ability to re produce features of data that are of pmiicular impmiance in modeling. There

may be a pa1iicular ran ge of return periods for which quantile est imates are required, for

example, in analys is of extreme events such as drought, quantiles of one tail of f reque ncy

distribution are of particular interest. Matalas and Wallis (1 973) menti oned th at the

competing dis tributions tha t fit the observed data satisfactorily may differ significantly in

their ta ils. These considerations may affect the choice of a regional fr equency

distribution . Therefore, 'robu stn ess' was reco gnized to b e the m ost important property of a frequency distribution employed for r egional analysis.

Different regional frequency di stributions were se lected in several regional studies. For example, the Flood Estimation Handbook (1999) recommended an index-flow m ethod employing the GE V distribution for a site with short p eriod of record . Durran s and To mic (1 996) appli ed the log-Pearson III distribution to regiona l low fl ow frequency analysis.

Chen et a l. (2006) analyzed low flow frequenc y in South China, an d selected the three- parameter lognonnal distribution as the most appropriate distribution fo r the region .

As the purp ose of region al frequency analys is is to augm ent the data at one site, it was possible to fit a three or more p arameter di stribu tio n, more reli abl y. Hosking and Wa llis ( 1997), thereby noted that di stributions w ith three to five parameters are appropriate ca ndidates for regional frequ ency analys is, b ecause they yield less biased estimates of quantiles in the tails of the di stributi on. It is possible that more th an one distribution fi ts the data ad equ ately; in th is case, the best cho ice would be one that prov ide s the most robust and e ffic ient quantile estimation . Furthermore, Hosking and W alli s suggest th at the fin al choice of distribution should be m ade b ased o n 'goodn ess -of-fi t' tests of the candidate distribution s. They provided an a pproach that directly involves the regiona l average L-mo me nts. For a three-p arameter di str ibutio n, the goodness-of -fi t is ju dg ed by h ow closely the L-kurtos is of the fitted distrib ution matches its regional averag e counterp art of the observed data.

McC uen ( 1985) introduced the moment ratio d iagram which is a too l to visua lly judg e

th e fit of a partic ular data set to a theoretica l dis tribution. The basic a dvantage of us ing

this diagram is that a single diagra m can vi su ally compare the fit of several distributions for a given set of data . In the regional context, the pos itio n of regional average dimens ionless moments on the diagram would give closer resemblance of the underlying regiona l distribution. Later on, Hosking (1 990) introduc ed the L-mo ment ratio diagram.

Vogel and Fennessy ( 1993) showed that the L-moment ratio diagram s are more accurate than the product moment diagrams in discriminating between th e di stribution and they proposed to replace the product moment diagram with L-mom ent diagram in hydrological investi gation s. Howe ver, Hos king and Wallis (1997) ind icated th at th e L-moment diagra ms is o nly a too l in selecting the candidate distributions and fina l distributio n selection should be made using more objec tive test that re fl ects the rob ustness of the distribution. Details on regional frequency di stribution selection are p rovided in Section 3.2. 5.4

2.1.2.5 Estimation of Flow Magnitudes

The frequency distribution s at the sites within a homogen eou s region are assumed to be identi cal ap art from a scale factor, and a probability distribution wi ll have been chosen for fi tting to each region. Several methods have been proposed for fi ttin g a d istrib ution to data from homogeneous regions, for exampl e, methods based on index-floo d (index- flow), station-year, and maximum likelihood procedures.

The index- flood procedure was first introd uced b y Da lrymple (196 0), in which the

observed annual peaks at each site are first standardized by di viding each data point by its

sa mple mean (th e index-flood), and then a ll the standardi zed observations are used to

est imate an ave rage dimens ion less frequency distribu tion (growth curve). Then, the

quantile for each site within the homogen eou s region is calculated by multiplyi ng the quantile estimate of the regional growth curve by the site ' s sample mean (the index-flo od) of annual records. The procedure is called index -flood because of its first application in flood studies. However, in this study, it has been called the index-flow procedure without loss of ge nera lity. The index flow procedure is very popular among practicing hydrolog ists, and have been adopte d in many regional frequenc y studies with limited modifications.

The well-known station-year method combines the rescaled data (by site-dependent scale factor) from all sites into a sing le sample and fits a distribution by treati ng the combined samples as a single random sample. This method is now rarely in use , as in many cases, it is not appropriate to treat the rescaled data as a single random sample .

An approach based on max imum likelihood estimation treats data as a statistical model that is comp lete l y sp ecifi ed by scale factors and unknown parameters of a regional growth curve. These parameters can be estimated by usin g the method of m aximum like lihood. This method has been used for example by Loaiciga and Marin o (1988).

As disc ussed before, the m ain goal of regional an alysis is to b e able to estimate the

flow variab les at a site where th ere are no records available. In this case, the index flow

variable at the site of interest must be estimated in another way, as th ere is no flow

recorded at this site. Usua lly, the index fl ow is estimated from a regionally calibrated

linear or log-li near relationship between th e mean or median flows and physically

measurable catchment characteristics (Lim and Lye, 2003 ; Mostofi Zadeh et a l. , 20 12) .

T he US Geo logica l Survey (Thomas, 1987; Tasker, 1987) proposed a different approach

from the index flow. They estimated the flow quantile of interest at every station, and regressed these quantiles from a homogeneous region to their respective sets of significant catchment c haracteristics. The quantiles at the site of interest would be obtained by substituting the important catchment characteristics in the respective regional regression relations. This method has been widely used all over the world as well an d it is known as ' regression on quantile ' method of regional an alysis. Two ad vantages of this method over the index flow method is that, firstly, it avoids specif y ing a regiona l average frequency distribution (the growth curve) , and secondly, it uses the regression techniques that are readily understood by hydrolog ists. However, the introduction of L-moments has finnly re -established th e index flow method as a general procedure for regional flow freq uency ana lys is, because the extent of distribution selection and parameter estimation problem in the index flow method have been significantl y reduced by u sing L-moments.

Hosking and Wa llis (1993) provided a general fr amework for carrying out index flow

based regional frequenc y ana lys is using L-moments. As the L-moments approac h gaine d

popul arity among hydro logists, the index flow meth od based on L-moments has been

acce pted as a stand ard method of regional frequency analysis in recent years. Most of the

app lications of this methodology were in fl ood frequency analysis. However, some

researchers have attempted to apply this mature method to low fl ow frequency ana lysis .

Pearson (1995) , Durrans and Tomic ( 1996), Tate et a!. (2000), Kroll and Vo gel (2002),

Chen et a l. (2006), Modarres (2008) , and Shi et a!. (20 1 0) are some of the examples of

reg ion al low fl ow fr equ ency analysis based on L -moments. The suggested L-moments

algorithm by Ho skin g and Wa llis (1997) is summari zed in Section 3.2.5.4.