OFFICE OF PUBLIC WORKS

FLOOD STUDIES UPDATE PROGRAMME

WORK-PACKAGE WP-2.2 “FREQUENCY ANALYSIS”

Final Report

Department of Engineering Hydrology & The Environmental Change Institute

National University of Ireland, Galway

September 2009

Table of Contents

List of Figures...... iv List of Tables ...... vii List of Equations...... viii List of Symbols and Abbreviations ...... xi Abstract ...... xiv

1 Introduction and data...... 1 2 Trend and Randomness ...... 5 3 Descriptive Statistics and Inferences from them ...... 8 3.1 Descriptive Statistics ...... 8 3.2 Preliminary distribution choice with help of Skewness versus Record Length Plots ...... 25 3.3 Preliminary distribution choice with the help of L-Moment Ratio diagrams. 29 3.4 Largest recorded values and outliers...... 34 4 Probability Plots and Inferences from them ...... 36 4.1 Probability Plots...... 36 4.2 Linear Patterns...... 40 4.3 Non Linear Patterns...... 41 4.4 Variation of shape of probability plot with Skew coefficient...... 45 4.5 Flood volumes associated with largest peaks on convex probability plots .... 49 5 Flood Seasonality...... 57 6 m-year maxima ...... 63 7 Flood statistics on some rivers with multiple gauges...... 66 8 Determining the T-year Flood Magnitude Q T by the Index Flood Method ...... 69 8.1 Introduction...... 69 8.2 Subject, Donor and Analog Sites ...... 71 9 Determining Q med ...... 72 9.1 Subject site is Gauged...... 72 9.2 Subject size is Ungauged...... 73 9.2.1 Use of Regression equation...... 74 9.2.1.1 Regression equation on its own...... 74 9.2.1.2 Regression equation used with donor site ...... 75 9.2.1.3 Regression equation used with analog site...... 76 9.2.2 Region of Influence Method...... 76 9.3 Assessment of Q med estimation using donor or analog sites...... 78 9.3.1 Tests on Qmed estimates via donor and analog catchments...... 81 9.3.2 Results...... 83 9.3.2.1 Results at individual subject sites ...... 83 9.3.2.2 Results from all donor sites collectively ...... 86 9.3.3 Conclusions:...... 92 9.4 Summary of Q med estimation and associated error...... 92 10 Determining the Growth Factor X T...... 94 10.1 Introduction...... 94 10.2 At-site and Pooling Group Estimates of X T...... 94 10.3 Pooling Groups...... 95 10.4 Growth curve estimation...... 97 11 Effect of catchment type and period of record on X T and Q T ...... 101 11.1 Data and Data Screening...... 101

ii 11.2 Effect of catchment type on pooled growth curve estimates...... 104 11.3 Temporal effect on pooled growth curve estimates ...... 111 11.4 Arterial drainage effect on pooled growth curve estimates ...... 118 11.5 Implications for Flood Frequency Estimation ...... 121 12 Selection of distance measure variables...... 123 12.1 Selecting variables for pooling...... 123 12.2 Pooled uncertainty measure, PUM...... 124 12.3 Selecting pooling variables for Irish data ...... 125 13 Standard errors of Q med , Q T and X T...... 130 13.1 Standard Error of Qmed...... 131 13.2 Standard Error of At-site estimate of Q T ...... 131 13.3 Standard Error of pooled estimate of X T and of Q T ...... 135 13.4 Standard Error of Q T based on ungauged catchment estimate of Qmed and pooled estimate of X T...... 140 14 Guidelines for determining Q T...... 142 14.1 Introduction...... 142 14.2 Determining Q T from at-site data...... 143 14.3 Determining Q T from Pooled data...... 145 14.3.1 Data available at the subject site...... 145 14.3.1.1 Determining Qmed...... 145 14.3.1.2 Determining X T...... 146 14.3.2 Subject site is ungauged...... 146 14.3.2.1 Determining Qmed...... 146 14.3.2.2 Determing X T...... 147 14.4 Some examples...... 147 14.5 Flood growth curves with an upper bound or without an upper bound?...... 156 14.6 Choice between at-site and pooled estimates and between 2 and 3 parameter pooled estimates...... 157 15 Summary and Conclusions...... 159 15.1 Data...... 159 15.2 Descriptive Statistics ...... 159 15.3 Seasonal Analysis...... 159 15.4 m-year Maxima ...... 160 15.5 Estimation of Design Flood ...... 160

References ...... 163 Appendix 1...... 165 Appendix 2...... 166 Appendix 3...... 177 Appendix 4...... 194

iii List of Figures

Figure 3. 1 A map showing Specific Qmed (m 3/s/km 2) for A1 , A2 and B category stations...... 19 Figure 3.2 A map showing estimated Cv at 110 station locations (A1 & A2 category stations) ...... 20 Figure 3.3 A map showing estimated Hazen skewness at 110 station locations (A1 & A2 category stations)...... 21 Figure 3.4 Histograms of Cv and L-Cv values...... 22 Figure 3.5 Histograms of Hazen skewness and L-skewness values...... 23 Figure 3.6 L-Cv Vs Cv for A1 and A2 stations. Note if intercept is included the relevant trendlines are y = 0.4426x + 0.0268...... 24 Figure 3.7 L-Skewness Vs Hazen Skewness for A1 and A2 stations. Note if intercept is included the relevant trendlines are y = 0.1225x + 0.0145...... 24 Figure 3.8 Observed Skewness calculated at 110 no. A1 and A2 gauging stations plotted versus record length. Also shown are 67% and 95% Confidence Intervals for skewness calculated from (a) Normal distribution samples and (b) EV1 distribution samples. .... 26 Figure 3.9 Observed L- Skewness calculated at 110 no. A1 and A2 gauging stations plotted versus record length. Also shown are 67% and 95% Confidence Intervals for L- Skewness calculated from (a) Normal distribution samples and (b) EV1 distribution samples...... 28 Figure 3.10 L-moment ratio diagram for annual maximum flow for 110 stations of Ireland ...... 30 Figure 3.11 L-moment ratio diagram of a) observed data b) data simulated from EV1 distribution c) data simulated from LN2 distribution and d) data simulated from LO distribution...... 30 Figure 3.12 First set of two realizations of four realizations of L-moment ratio diagrams of data simulated from EV1, LN2 and LO distribution ...... 32 Figure 3.13 Second set of two realizations of four realizations of L-moment ratio diagrams of data simulated from EV1, LN2 and LO distribution...... 33

Figure 4.1 (a, b) Random Samples of (a) size 25 (b) size 50 from an EV1(100,30) population. (Cv = 0.33)...... 39 Figure 4.2 Hydrograph volumes of different durations during the 4 largest annual maximum flood peaks at 24082 River Maigue at Islandmore...... 52 Figure 4.3 Hydrograph volumes of different durations during the 3 largest annual maximum flood peaks at 7006 River Moynalty at Fyanstown...... 53 Figure 4.4 Hydrograph volumes of different durations during the 3 largest annual maximum flood peaks at 15003 River Dinan at Dinan Bridge...... 55 Figure 4.5 A plot between FAI vs Hazen Skewness...... 56

Figure 5.1 Seasonal occurrence of annual maximum flood values in 202 stations...... 57 Figure 5.2 Flood seasonality showing actual magnitude in 202 Irish stations...... 58 Figure 5.3 Flood seasonality with specific flood magnitude in 201 Irish stations ...... 59 Figure 5.4 Month corresponding to maximum flow in each of 202 AM series...... 60

Figure 6.1 Probability plots of 1, 2 and 3 year maxima showing contrasting behaviours...... 65

iv Figure 7.1 (a) Probability plot summary for stations on Barrow River (b) a scatter plot of Qmed vs Catchment area for these stations. Stations are ordered from upstream. .... 66 Figure 7.2 (a) Probability plot summary for stations on Suir River (b) a scatter plot of Qmed vs Catchment area for these stations. Stations are ordered from upstream...... 67 Figure 7.3 (a) Probability plot summary for stations on Suck River (b) a scatter plot of Qmed vs Catchment area for these stations. Stations are ordered from upstream...... 68

Figure 9.1Increase in R 2 value with number of catchment descriptors included in regression equation in log domain, (i) as in FSR (1975) and (ii) in NUI Maynooth Draft WP2.3 Report (2008)...... 75 Figure 9.2 Relationships between different methods of estimating Q med or of enhancing a Q med estimate for an ungauged catchment...... 77 Figure 9.3 Location of 38 potential subject sites...... 80 Figure 9.4 predicted Qmed by regression vs observed Qmed . Each point represents a catchment , 35 in total as all catchment descriptors were unavailable for 3 of the 38 sites ...... 83 Figure 9.5 Performance of data transfer method of estimating Qmed at Station 14006, Barrow at Pass Bridge, using regression based adjustment. The figures above the boxes are catchment areas. The final figure labelled Regression refers to estimate obtained by eq (10)...... 85 Figure 9.6 Performance of data transfer method of estimating Qmed at Station 14006, Barrow at Pass Bridge, using area based adjustment...... 86 Figure 9.7 Adjusted Qmed (Regression method) vs observed Qmed. Each point represents a catchment and panels relate to different transfer method ...... 90 Figure 9.8 Adjusted Qmed (Area method) vs observed Qmed (m 3/s). Each point represents a catchment and each plot relate to different area based data transfer methods ...... 91

Figure 10.1 Simplified qualitative outline of simulation results about flood quantile estimates. (a) and (b) show the effect of choice of wrong distribution having fixed but incorrect skewness. (c) and (d) show the effect of using a flexible distribution i.e. low bias and large standard error when used in at-site mode but with much reduced se as well as low bias when us in at-site +pooling group mode. (Adapted from Figure 5.1 in WMO ,1989)...... 100

Figure 11.1EV1 probability plot for the discordant stations...... 104 Figure 11.2 Location of the 85 stations and the four regions...... 105 Figure 11.3 Box plots of X100 values from region of influence pooling estimates of X100 obtained by (a) GEV, (b) GLO estimating equations showing effect of %peat and FARL. The numbers in brackets (85, 44, 30…) indicate the number of X100 values contributing to each box plot...... 106 Figure 11.4 Values of (a) H1 (b) H2 for the analyses summarised in Figure 11.3 ...... 107 Figure 11.5 Box plots of (a)X 100 (GEV) (b) X 100 (GLO) showing effect of geographical area and catchment size...... 109 Figure 11.6 Values of (a) H1, (b) H2 for the analyses summarised in Figure 11.5...... 110 Figure 11.7 Variation of X100 with period of record (first half versus second half of each record) and variation of X50 with decades, as estimated by (a) GEV, (b) GLO from each individual stations pooling group using the 5T rule...... 112 Figure 11.8 Values of (a) H1, (b) H2 for the analyses summarised in Figure 11.7...... 113

v Figure 11.9 (a)Column chart of X 50 (GEV) values (b) Column chart of Q med values (c) Column chart of Q 50 values for the decadal data (‘70, ‘80 and ‘90) of 50 stations...... 115 Figure 11.10(a)Column chart of X 50 (GEV) values (b)Column chart of Q med values (c) Column chart of Q 50 values for the decadal data (’50, ’60 ‘70, ‘80 and ‘90) of 26 stations...... 116 Figure 11.11 (a) Column chart of X 50 (GEV) values (b) Column chart of Q med values (c) Column chart of Q 50 values for the decadal data (’57-66, ’67-76 ’77-86, ’87-96 and ’97- 06) of 34 stations ...... 117 Figure 11.12 (a) Column chart of X 50 (GEV) values (b) Column chart of Q med values (c)Column chart of Q 50 values for the pre- and post-drainage data of 16 stations...... 120

Figure 12.1 Box-plot of 100-year PUM values for different sets of catchment descriptors used in defining the distance measure dij . (a) subject site is not included in its own pooling-group when PUM is evaluated (b) subject site is included in its own pooling-group when PUM is evaluated...... 128 Figure 12.2 Box-plot of (a) H1 values, (b) H2 values for different sets of catchment descriptors used in defining the distance measure dij ...... 129

Figure 13.1 (a) Box-plot of percentage standard error of flood quantile estimates for EV1 distribution using simulation method for 85 stations for the 5, 10, 25, 50, 100, 500- year return period. (b) The theoretical EV1 percentage standard errors for the average sample size of 37 years and for a range of L-Cv values ...... 133 Figure 13.2 (a) Box-plot of percentage standard error of flood quantile estimates for GEV distribution using simulation method for 85 stations for the 5, 10, 25, 50, 100, 500-year return period. (b)The theoretical GEV (k=-0.1) percentage standard errors for the average sample size of 37 years and for a range of L-Cv values...... 134 Figure 13.3 Box-plot of percentage standard error of growth factors estimates for EV1 distribution using simulation method for 85 stations for the 5, 10, 25, 50, 100,500-year return period...... 138 Figure 13.4 Box-plot of percentage standard error of quantile estimates for EV1 distribution using simulation method for 85 stations for the 5, 10, 25, 50, 100,500-year return period...... 138 Figure 13.5 Box-plot of percentage standard error of growth factors estimates for GEV distribution using simulation method for 85 stations for the 5, 10, 25, 50, 100,500-year return period...... 139 Figure 13.6 Box-plot of percentage standard error of quantile estimates for GEV distribution using simulation method for 85 stations for the 5, 10, 25, 50, 100,500-year return period...... 139

Figure 14.1 At-site and pooled quantiles for station 25016...... 147 Figure 14.2 At-site and pooled quantiles for station 9001...... 149 Figure 14.3 At-site and pooled quantiles for station 9010...... 151 Figure 14.4 At-site and pooled quantiles for station 25002...... 152 Figure 14.5 At-site and pooled quantiles for station 8009...... 154 Figure 14.6 At-site and pooled quantiles for station 9002...... 155 Figure 14.7 At-site and pooled quantiles for station 15001...... 155 Figure 14.8 At-site and pooled quantiles for station 26008...... 156

vi List of Tables

Table 2.1 Number of cases in which the null hypothesis of zero trend or randomness is rejected at stated level out of 91 no. AM cases ...... 6 Table 2.2 Number of cases in which the null hypothesis of zero trend or randomness is rejected at stated level out of 117 no. 5-year median cases ...... 7 Table 3.1 Calculation of moments and their dimensionless ratios...... 10 Table 3.2 Summary statistics for 45 No. A1 stations ...... 11 Table 3.3 Summary statistics for 70 No. A2 stations ...... 12 Table 3.4 Summary statistics for 67 No. B stations...... 14 Table 3.5 Average values of some statistics at 43 No. A1 and 67 No. A2 gauging stations...... 17 Table 3.6 High outliers from 110 Irish stations...... 35 Table 3.7 Low outliers from 110 Irish stations ...... 35 Table 4.1 Linear pattern statistics for A1 and A2 stations combined. Scores at individual stations are tabulated in Table 4.3 and Table 4.4...... 40 Table 4.2 Non- linear pattern statistics for A1 and A2 stations. Scores at individual stations are tabulated in Table 4.3 and Table 4.4...... 42 Table 4.3 Probability plot linear and non-linear scores for A1 category Stations...... 43 Table 4.4 Probability plot linear and non-linear scores forA2 category Stations ...... 44 Table 4.5 A1 stations ranked by skewness...... 46 Table 4.6 A2 stations ranked by skewness...... 47 Table 4.7 Sample of 8 stations showing convex upward appearance on probability plots...... 50 Table 5.1 Station Number with percent of peak floods in the October-March (percentage)...... 61 Table 5.2 Station number with month corresponding to maximum flow in an AM data series...... 62 Table 6.1Examples of m-year maxima ...... 64

Table 9.1 List of Rivers that have 3 or more gauging stations on them: ...... 78 Table 9.2 38 Gauging Stations used in the Q med Assessment...... 79 Table 9.3 Estimates from donor catchments – summary of minimum, maximum and average percentage difference from observed Qmed...... 84 Table 9.4 Qmed estimates from donor catchments – minimum, maximum and average % difference form observed Qmed...... 87

Table 11.1 Stations with site characteristics, L-moment ratios and discordancy measure ...... 102 Table 11.2 Four stations out of 94 showing large discordancy values...... 104 Table 11.3 List of 16 no. Pre and post- drainage stations...... 119 Table 12.1Summary of AMF data sets used in dij study...... 126 Table 12.2 Variation in the mean PUM100, H1 and H2 values for different sets of pooling variables...... 127 Table13.1 Ranges of percentage se for growth factors, X T, and quantile estimates, Q T...... 137

vii List of Equations

-1 QT = F (1-1/T) --- (8. 1)...... 70 QT = Q I X T --- (8. 2) ...... 70

36.0 se (Qmed ) = Qmed --- (9. 1)...... 72 N  s   Q by regression  Q s = Qd med --- (9. 2)...... 76 med med  d   Qmed by regression 

s d As Qmed = Qmed --- (9. 3)...... 76 Ad  Q s by regression  Q s = Q a  med  --- (9. 4) ...... 76 med med  a   Qmed by regression   m  Q i  s 1  med  Qmed = As  ∑  --- (9. 5) ...... 76 m i=1 Ai  0.829 0.898 -1.539 Q med = 0.000302 × AREA × SAAR × BFI --- (9. 6) ...... 82

ln AREA −ln AREA 2 ln SAAR −ln SAAR 2  BFIHOST − BFIHOST 2 1 i j   i j   i j  dij =   +  +  --- (10. 1) 2 σln AREA   σln SAAR   σ BFIHOST  ...... 96 2 2 2  ln AREA i − ln AREA j   ln SAAR i − ln SAAR j   ln BFI i − ln BFI j  dij =   +   +   --- (10. 2)  σ ln AREA   σ ln SAAR   σ ln BFI  ...... 96 m m (i) t2 = ∑ wi t 2 / ∑ wi --- (10. 3)...... 97 i=1 i=1 m m (i) t3 = ∑ wit3 / ∑ wi --- (10. 4)...... 97 i=1 i=1 β  T  x = 1+ (ln )2 k − (ln ) k  , k ≠ 0 --- (10. 5)...... 98 T k  T −1  2 ln 2 k = 7.8590c + 2.9554c 2 where c = - --- (10. 6)...... 98 3 + t3 ln 3

kt 2 β = k −k --- (10. 7)...... 98 t2 ()Γ 1( + k) − (ln )2 + Γ 1( + k)( 1− 2 )

xT = 1+ β (ln(ln )2 − ln( −ln( 1− /1 T)) ) --- (10. 8) ...... 99 t β = 2 --- (10. 9) ...... 99 ln 2 − t2 []γ + ln(ln )2

β k β −k xT = 1+ (1−{}()1− F / F ) = 1+ (1−{}T −1 ), k ≠ 0 --- (10. 10)...... 99 k k

viii kt 2 sin( πk) k = - t 3 and β = --- (10. 11)...... 99 kπ (k + t 2 ) − t 2 sin( πk)

xT = 1+ β ln( T − )1 --- (10. 12)...... 99

β = t2 --- (10. 13) ...... 99

1 D = N()u − u T A −1 ()u − u --- (11. 1)...... 101 i 3 i i

n 2 d ij = ∑Wk ()X k,i − X k, j --- (12. 1)...... 123 k =1

2 2 2 ln Ai −ln Aj  ln SAAR i −ln SAAR j   FARL i −FARL j   FPEXTi −FPEXT j  dij = 2.3  + 5.0   + 1.0   + 2.0   --- (12. 2)  28.1   37.0   05.0   04.0  ...... 124 2 M long p ∑ ni ()ln xT − ln x Ti i i=1 PUM T = --- (12. 3) ...... 124 M long

∑ ni i=1

2 2 2  ln AREA i − ln AREA j   ln SAAR i − ln SAAR j   BFI i − BFI j  dij = 7.1   +   + 2.0   --- (12. 4)  σ ln AREA   σ SAAR   σ BFI  ...... 127

ˆ α 2 EV1: SE [QT ]= [ .1 1128 + .0 4574 y + .0 8046 y ] --- (13. 1) ...... 131 n ˆ α 2 3 2/1 GEV: se [QT ]= [exp {a0 ()()()()()T + a1 T exp − k + a2 T k + a3 T k }] --- (13. 2) 131 n

2 N M  Xˆ T − Xˆ T  1 1  i, m i  SE (X T )[]% = ∑ ∑ ×100 --- (13. 3)...... 135 N M  X T  i=1m = 1  i 

2  T T  N M Qˆ − Qˆ 1 1  i, m i  SE (QT )[]% = ∑ ∑ ×100 --- (13. 4)...... 135 N M  QT  i=1m = 1  i  ˆ ˆ QT = Qmed × xˆT --- (13. 5)...... 140 2 ˆ ˆ 2 ˆ ˆ ≈ E(xˆT ) Var (Qmed ) + E(Qmed ) Var (xˆT ) + 2E(Qmed ) ⋅ E(xˆT ) ⋅ cov( Qmed ⋅ xˆT ) --- (13. 6)140 ˆ 2 ˆ 2 Var (QT ) ≈ (xT ) |m Var (Qmed ) + (Qmed ) |m Var (xˆT ) --- (13. 7)...... 140 36.0 se (QT ) QT ≈ --- (13. 8)...... 140 N

i − 44.0 y = − ln( −ln( F )) where F = , i = 2,1 ,...... N --- (14. 1)...... 144 i i i N + 0.12

ix i − 8/3 y = Φ −1 (F )) where F = , i = 2,1 ,...... N --- (14. 2) ...... 144 i i i N +1/ 4 2M − M L α = 110 100 = 2 --- (14. 3)...... 144 ln( )2 ln( )2 u = M 100 − .0 5772 α = L1 − .0 5772 α --- (14. 4) ...... 145

QT = u +αyT --- (14. 5)...... 145 1 n µ z = ∑ zi where z i = log 10 (Q i) --- (14. 6) ...... 145 n i=1 2/1  1 n  z z 2 σ z =  ∑( i − )  --- (14. 7) ...... 145 n −1 i=1 

ZT QT =10 --- (14. 8) ...... 145 Qmed = .1 083 ×10 −5 AREA .0 938 BFIsoils − 88.0 SAAR .1 326 FARL .2 233 DRAIND .0 334 rural (14. × S1085 .0 188 1( + ARTDRAIN )2 .0 051 9) ...... 146

x List of Symbols and Abbreviations

Symbols

T return period in years N, n record length Q flood peak discharge

QI index flood

Qmed median of AM flood series

Qmean, , Q , Qbar mean of AM flood series

Qmax maximum flood on record

Qmin minimum flood on record

QT the T year flood

XT the T year growth factor dij similarity distance D discordancy measure Cv coefficient of variation H-skew Hazen corrected skewness L-Cv coefficient of L-Cv L-Skew coefficient of L-skewness L-kur coefficient of L-kurtosis t2 sample L-Cv value t3 sample L-Skewness value t4 sample L-kurtosis value ln natural logarithm log 10 10 base logarithm E expected value Var variance F(Q), F cumulative distribution function f(Q) probability density function w weighting term R2 coefficient of determination A1 gauging station grade A1

xi A2 gauging station grade A2 B gauging station grade B H1 heterogeneity (using L-Cv) H2 heterogeneity (using L-Cv and L-skewness) Q s Q at the subject site med med Q d Q at the donor site med med Q a Q at the analog site med med

As catchment area of the subject site

Ad catchment area of the donor site u location parameter in extreme value distributions α scale parameter in extreme value distributions β scale parameter (growth curve) k shape parameter of Generalized Extreme Value distribution k shape parameter of Generalized Logistic distribution γ Euler’s constant Γ Gamma function st M100 1 PWM nd M110 2 PWM rd M120 3 PWM st λ1 1 L-moment nd λ2 2 L-moment rd λ3 3 L-moment µ mean σ standard deviation rd µ3 3 central moment g coefficient of skewness y reduced variate z variate used in statistical exposition, z = ln X

σ z standard deviation of z = ln X Φ Standardised Normal Distribution Function

xii Abbreviations

A, AREA catchment area (km2) AEP annual exceedance probability AM annual maximum ARTDRAIN2 % of the catchment river network included in the Drainage Schemes BFI baseflow index BFIHOST baseflow index derived from HOST soils data (U.K) BFIsoils baseflow index derived from soils data DRAIND drainage density EPA Environment Protection Agency ESB Electricity Supply Board EV1 Extreme Value Type 1 (distribution), (= Gumbel distribution) FAI flow attenuation index FARL flood attenuation by reservoir and lake FEH Flood Studies Handbook fse factorial standard error FSR Flood Studies Report FSU Flood Studies Update GEV Generalized Extreme Value (distribution) GLO Generalized Logistic (distribution) iid independently identically distributed LN2, LN 2 parameter Log Normal (distribution) LN3 3 parameter Log Normal (distribution) LO 2 parameter Logistic (distribution) OPW Office of Public Works PEAT % of land area covered by peat bogs PUM Pooled Uncertainty Measure (as defined in FEH) PWM Probability Weighted Moment S1085 10-85% stream slope (m/km) SAAR standard annual average rainfall (mm) se standard error

xiii Abstract

In 2005 the Office of Public Works announced a Flood Studies Update Programme arising out of the report of the Flood Policy Review Group in 2004. The work described in this report deals with the topic of flood frequency analysis and estimation. Flood frequency analysis addresses the issue of the rarity of large and medium floods for flood risk assessment and the specification of flood flow values for the design flood of alleviation and control works and for flood zoning and spatial planning.

The report provides a statistical summary of flood flow records in Ireland which have been taken from the data archives of the OPW, EPA and ESB. The summary is based on analysis of annual maximum flow records, ranging from 10 to 55 years long, at approximately 200 river flow gauging sites. More detailed research has been carried out on a subset of approximately 85 gauging sites, at which the quality of the gauged flows is the most reliable.

Most floods occur during the winter half of the year with some notable floods occurring during summer, especially in August, also. Because of Ireland’s humid climate the range of variation of flow values from year to year, as measured by coefficient of variation, is quite small by international standards. Likewise the skewness of flow series is modest. While no single statistical distribution can be considered to be “best” at all locations it has been found that both the Extreme Value Type 1 (Gumbel) and the lognormal distributions provide a reasonable model for the majority of stations.

Guidance is provided on the estimation of the design flood of required annual exceedance probability at both gauged and ungauged locations and on how to express the uncertainty in the resulting estimates. The use of growth curves or growth factors based on regional pooling of data with the use of suitable 3 parameter distributions is recommended for most applications. A number of examples are presented in Chapter 14 which discusses a range of difficult cases that can arise in practice. In some of these cases the user may be recommended to consider using an at-site based estimate in preference to the generally recommended regional pooling based method. What this indicates is that flood estimation cannot be reduced to a single strict formula based procedure but that individual analysts must make choices which depend on the problem circumstances and which take into account their own knowledge and experience.

xiv 1 Introduction and data

Flood frequency analysis is concerned with the assessment of flood magnitudes of stated frequency or stated degree of rarity for use as input into the process of flood risk assessment and management. Flood risk assessment is needed in the design of flood relief and protection works and in the assessment of the safety of existing and planned elements of infrastructure whether they be domestic houses, commercial or industrial buildings, bridges and roads, railways or other vital pieces of infrastructure such as hospitals, electrical stations, gas stations or water and waste-water works. No project can be guaranteed to be immune from flooding during its projected life. Flood proofing every project would be prohibitively costly. As a result projects for which the consequence and cost of occasional flooding is modest may be required to tolerate occasional flooding. Calculation of the flood risk associated with such projects requires that the flood magnitude which needs to be exceeded to cause flooding has to have its associated probability of occurrence calculated. It is these probabilities that are calculated with the help of flood frequency analysis.

This report arises in the context of the Government's multifaceted approach to flood risk assessment and management which was first outlined in the Report of the Flood Policy Review Group (2004) prepared by an expert group appointed by the Minister of State at the office of Public Works. This approach ranges from hard engineering solutions such as constructed flood relief schemes to improved flood awareness, education and preparedness and the implementation of planning and development control. One key element identified was the need for improved hydrological information for input into decision making processes and a proposal was included for a Flood Studies Update (FSU) programme to achieve this. This programme has many components relating to rainfall frequency analysis, flood frequency analysis, hydrograph analysis, flood plain effects on floods as well as the application of Geographic Information Systems to automate the measurement of catchment descriptors from digital mapping data and the use of Information Technology techniques to deliver much of the FSU's output to potential users via the world wide web. These components were divided into Work Packages, one of which is WP2.2 Flood Frequency Analysis. This report describes the work carried out in this Work Package at NUI Galway between 2006 and 2009.

1 It also makes recommendations about guidelines for future practice in the area of flood frequency analysis and design flood determination. The work programme undertaken is as proposed in the NUI Galway Department of Engineering Hydrology’s Tender Document of October 2005.

The Work Package 2.2 study is based on the annual maximum flow data for 203 hydrometric stations which were provided by OPW in stages between June 2006 and February 2007. A final set of 3 data sets from ESB stations were provided in April 2008. Attention was focused on those stations whose quality grading, based on rating curve and water level measurement reliability as indicated by OPW, fall into the categories A1, A2 and B. There are • 45 Grade A1 stations • 70 Grade A2 stations • 67 Grade B stations and a further • 17 stations that have pre- and post-drainage records giving a total of 199 gauging stations.

The main recommendations of the study are based solely on the results obtained from A1 and A2 graded stations. The B grade stations are analysed and displayed but no general deductions are made from them. However the median values obtained from both A and B grade stations have been used in determining relationships between Qmed and catchment descriptors, both in part of this study (i.e. equation (9. 6)) and in the separate work Package WP2.3.

Some of the studies described in this report are based on subsets of the entire A1 and A2 dataset. This is due to a number of reasons – some studies needed data that did not have gaps in the record, some studies needed multiple gauges on each river to be included, some needed Baseflow Index (BFI) which was not available for all catchments, some studies were conducted before some of the data series became available . In each part of the report a note is included in the relevant introduction about the selection of data used in that part of the study.

2 A number of remarkable floods which have occurred in Ireland do not appear in these systematically measured data series because gauging stations did not exist on some such rivers. The River Dargle at Bray is an example where Hurricane Charlie caused a notable flood in August 1986. Earlier ungauged large floods also are known to have occurred in the Dargle in 1904, 1932 and 1967. Another such example is the Newcastlewest flood of 1st August 2008. Some catchments which are known to have experienced significant floods, eg Deel at Crossmolina, have gauging stations but data have not been supplied for them because the gauging stations are not up to the required standard for inclusion in category A1, A2 or B. Further examination of these floods is merited because some of the floods occurred on small, relatively steep catchments which are not well represented in the main database of A1 and A2 grade stations although this further examination is not reported on here.

This report first describes the results of statistical analyses of the annual maximum series under the headings; • Trend and randomness • Descriptive statistics • Probability plots • Flood seasonality • m-year maxima • Statistics of some rivers with multiple gauges.

It then considers methods of determining the T year return period flood magnitude, including • Introduction to index flood method • Determination of the index Flood Qmed in both
gauged and
ungauged catchments using regression and/or donor or analog catchments • Assessment of donor and analog catchment techniques • Determining the Growth Factor from at-site and from pooling group data

• Effect of catchment type and period of record on pooled estimates of Q T • Choice of catchment descriptors for pooling group selection

3 • Standard errors of Qmed, X T and Q T.

Finally summary guidelines for determining the design flood QT are given, followed by some examples where the unchallenged application of a single prescribed strategy may provide QT estimates that are counter intuitive and this leads on to a discussion of such topics as • Flood growth curves with or without an upper bound

• Choice between at-site and pooled estimates of QT and • Choice between 2 and 3 parameter distributions.

This discussion shows that there can be no blind application of a single rule based flood frequency estimation method but that a number of factors such as existing at-site flood experience, the trade-off between at-site and regional estimates and the plausibility or otherwise of an upper bounded distribution must be considered in any determination of

QT. The most suitable measure of rarity of any flood magnitude for general communication with the public is the annual exceedance probability, AEP, of that flood. For example a large flood may have an AEP of 0.01 or 1% meaning that it has only one chance out of 100 of being exceeded in any year. However as in most technical reports the measure of rarity of any flood magnitude used in this report is return period, T, which is defined as the average number of years elapsing between successive exceedances of that flood magnitude. The word average in the definition needs to be stressed as any magnitude can be exceeded at unequal intervals of time. The two measures are consistent with one another since AEP = 1/T.

4 2 Trend and Randomness

It is assumed in flood frequency analysis that successive values in a flood series have all emerged randomly and independently from a common population of flood values. In particular this means that there should be no upward or downward trend in the series of data nor should there be any clustering together of groups of larger than average or smaller than average values during the passage of time. To determine how well these assumptions are met recognised tests of trend and randomness are conducted.

This testing work is based on analysis of data from 117 Grade A1 and A2 gauging stations, i ncluding some post-drainage records of catchments which had earlier experienced arterial drainage works. A number of stations from among those currently available were not available when the trend and randomness analyses were carried out in 2007, hence the number 117 rather than 120.

A total of 6 tests were applied. These took into account the tests which were reported in UK Flood Studies Report (NERC, 1975), WMO Report on Trend (Kundzewicz, 2004) and also software called TREND developed by University of Melbourne for hydrological purposes. These tests are

A) Tests for serial persistence or trend: i) Mann-Kendall (non-parametric test for trend) ii) Spearman’s Rho (non-parametric test for trend) iii) Mean-weighted Linear Regression test (parametric test for trend)

B) Tests for progressive change in the mean and median with time : iv) Mann-Whitney U test (non-parametric test for step jump in median)

C) Tests for serial dependency of time series v) Turning Points (Kendall non-parametric test for randomness) vi) Rank Difference (Meachem non-parametric test for randomness)

5 The results are summarised below and are presented in a separate 50 page report by Mr. U. Mandal, which is attached at the end of this volume.

Initial exploratory data analysis (EDA) found that 11 stations showed, on a visual basis on a chronological plot, an upward or downward trend with time. In this section also EDA of pre- and post-drainage records was conducted. Surprisingly three of 14 such stations showed a decrease in flood magnitudes during the post-drainage period.

Statistical trend analyses were carried out on 91 No. annual maximum series 117 No. 5-year median series.

The median series, from non-overlapping 5 year segments of each record, are examined in order to give a macro view of the data which is unaffected by occasional larger than usual or smaller than usual values in the annual maximum series. Only 91 stations contributed to tests carried out on annual maximum series since 26 of the 117 data series available had some gaps that prevented application of the tests.

When 6 tests are applied to data of approximately 100 no. stations it is to be expected that the null hypothesis (of zero trend or of independence etc.) would be rejected on a small number of occasions even if no trend were present or if all data sets were truly random. The results are summarised below in Table 2.1 and Table 2.2 (which are Tables 7 and 8 of Mr. Mandal's report, Appendix 1).

Table 2.1 Number of cases in which the null hypothesis of zero trend or randomness is rejected at stated level out of 91 no. AM cases Split sample Trend and persistence Randomness test Sing. test Level Mann- Spearman Mean-weighted Mann Whitney Turning Rank Kendall Rho Linear Regressn. U test point difference

11 12 7 22 4 8 1%

25 25 14 36 8 14 5%

6 Table 2.2 Number of cases in which the null hypothesis of zero trend or randomness is rejected at stated level out of 117 no. 5-year median cases Split sample Trend and persistence Randomness test Sing. test Level Mann- Spearman Mean-weighted Mann Whitney Turning Rank Kendall Rho Linear Regressn. U test point difference

9 14 7 47 2 10 1%

22 35 10 73 4 18 5%

These results show an unexpected degree of non-randomness where , across all tests on annual maximum series, approximately 10% of stations reject the null hypothesis at the 1% level whereas in a truly random situation only 1 station from approximately 100 stations would be expected to reject the null hypothesis. Similar remarks can be made about the 5 year median tests . Thus a big question mark hangs over the traditional "independently identically distributed" (iid) assumption on which all flood frequency analysis is based. However use of the iid assumption still allows progress to be made in estimating flood probabilities and the strengths of the method, which has been widely used and developed throughout the world, mean that it is the most suitable form of assumption on which to base a national flood frequency methodology. The iid assumption also allows data series which are incomplete due to gaps in the record, due to malfunctioning recorders or subsequent loss of data, to be treated in the same way as continuous records. For instance if the annual maximum values from 5 years of a 40 year record are unavailable the record is considered to be equivalent to a record of 35 years regardless of whether the 5 unavailable years are contiguous or not.

7 3 Descriptive Statistics and Inferences from them

3.1 Descriptive Statistics

The steps carried out here involved calculating basic statistics and corresponding dimensionless statistics for each station, the preparation of probability plots on Extreme Value Type I ( EV1) and logistic bases as well as log-normal probability plots and studies which attempt to infer from the dimensionless statistics collectively what the most appropriate form of 2 parameter distribution might be best suited as a universal choice for annual maximum flood data.

While data are available for 115 stations and descriptive statistics are calculated for them (Tables 3.2 and 3.3) and their data are displayed on probability plots (Chapter 4 and Appendix 4) the results for eight of these stations are not included in the statistical summary in Table 3.5 nor in the inferences about distribution suitability based on skewness and L-Moment ratios in Sections 3.2 and 3.3 below. These so omitted include • 27070 River L. Inchiquinn at Baunkyle because of suspected extreme lake effects • 36037 River Woodford (Ballyconnell Canal East) at Ballyheady because of suspected man made effects • Stations 25001, 25002, 25003, 25004, 25005 and 25158 in the Mulkear catchment because of embankments which contain 4 out of 5 annual maximum flows but which are overtopped by the larger flows causing damping of the larger flow magnitudes which leads to extremely concave downwards probability plots. The annual maximum flood values used in the present study provide the basic information on which the entire study depends. The entire data sets can be viewed as histograms or as probability plots but for the purposes of applying statistical models, such as probability distributions based on the identically independently distributed assumption, the data need to be summarised by a number of statistical measures for use as input into the calibration of such models. Such summary statistics, especially those expressed in dimensionless form can also be used for comparison of flood frequency behaviour on different catchments and for drawing inferences about the suitability of

8 probability distributions for describing flood data. This Chapter of the report describes these measures and the results of calculating them from the available data series.

In flood hydrology the most useful statistics relate to 1. overall magnitude of the flood data as conveyed by their mean µ or median Qmed 2. the scale or spread of the data as measured by standard deviation σ or 2 nd L-

moment λ2 3. the skewness or the lack of symmetry of the data, as measured by the 3 rd central rd moment µ3 or by the 3 L-moment, λ3.

Dimensionless forms of (ii) and (iii), expressed as • Coefficient of variation, Cv • L-Cv • Skewness, g • L-Skewness are defined in Table 3.1 in terms of the basic population quantities. Formulae for calculation of these from sample data are also provided in Table 3.1.

For comparative studies these dimensionless measures of scale and asymmetry are useful for detection of common behaviour among the several series in the data set. Dimensionless measures of magnitude are not feasible although specific flow value Q/(catchment area) is sometimes used for comparison purposes. Other dimensionless quantities such as Q max /Q med , Q max /Q mean or Q mean /Q med can also be compared between catchments although it has to be realised that Q max /Q mean and Q max /Q med are very much dependent on the period of record available and have high sampling variability.

The basic statistical descriptors mean, median, Qmax, Cv, Hazen Skew, L-Cv, L-Skew and L-Kurtosis as well as ratios Qmax/mean, Qmax/median, Qmax/Area and Qmean/Qmedian are shown below in tabular form in Table 3.2, Table 3.3 and Table 3.4 for A1, A2 and B grade stations respectively. The quantity specific median flood, Qmed/Area, for A1, A2 and B stations is mapped in Figure 3. 1.

9 Table 3.1 Calculation of moments and their dimensionless ratios

Definitions based on ordinary moments

Population quantities Sample estimates 1 N µµµ = Mean Q = ∑Qi N i=1 2 N (Q − Q) σσσ = Standard deviation σˆ = ∑ i i=1 N −1 N rd N 3 µµµ3 = 3 Central Moment µˆ3 = ∑(Qi − Q ) (N − )(1 N − )2 i=1 σ σˆ Cv = = Coefficient of variation Cˆv = µ Q µ µˆ g = 3 = Coefficient of Skewness gˆ = 3 σ 3 σˆ 3  5.8  gˆ = gˆ 1+  = Hazen's Unbiassed H  N  Skewness

Definitions based on Probability Weighted Moments (PWMs)

Population quantities Sample estimates

st ˆ M100 = 1 PWM M100 = Q = sample mean N nd ˆ 1 (i − )1 M110 = 2 PWM M 110 = ∑ Q(i) N i=1 (N − )1 N rd ˆ 1 (i − )(1 i − )2 M120 = 3 PWM M 120 = ∑ Q(i) N i=1 (N − )(1 N − )2 th where Q (i) is the i smallest value st λλλ1 = M 100 = 1 L-Moment nd λλλ2 = 2M 110 - M 100 = 2 L-Moment rd λλλ3 = 6M 120 - 6M 110 + M 100 = 3 L-Moment λ τ = 2 = L-Cv λ1

λ3 τ 2 = = L-Skewness λ2 Sample values of M 100 , M 110 and M 120 are inserted in these expressions to give the samples estimates of L-Cv and L-Skewness.

10

Table 3.2 Summary statistics for 45 No. A1 stations

L- Qmax/ Qmax/ Qmax/ Qmed/

Station No., River & Station Name Area N Qmean Qmed Qmax H.Skew CV L-CV Skew L-Kur Qmean Qmed Area Qmean 06011 RIVER FANE @ MOYLES MILL 234 48 15.86 15.39 26.36 0.81 0.20 0.11 0.09 0.07 1.66 1.71 0.11 0.97 06013 RIVER DEE @ CHARLEVILLE WEIR 307 30 27.81 27.37 41.84 0.10 0.27 0.16 0.02 0.01 1.50 1.53 0.14 0.98 06014 RIVER GLYDE @ TALLANSTOWN 270 30 22.56 21.46 39.40 1.31 0.27 0.15 0.22 0.12 1.75 1.84 0.15 0.95 06025 RIVER DEE @ BURLEY BRIDGE 176 30 18.32 18.69 23.57 -0.33 0.16 0.09 -0.07 0.19 1.29 1.26 0.13 1.02 06026 RIVER GLYDE @ ACLINT BRIDGE 144 46 13.87 12.30 24.12 1.05 0.32 0.18 0.24 0.09 1.74 1.96 0.17 0.89 06070 RIVER MUCKNO L. @ MUCKNO 153.5 24 13.32 13.19 20.93 0.78 0.25 0.14 0.14 0.14 1.57 1.59 0.14 0.99 07009 RIVER BOYNE @ NAVAN WEIR 1610 29 162.64 134.80 297.60 0.93 0.38 0.21 0.21 0.11 1.83 2.21 0.18 0.83 08002 RIVER DELVIN @ NAUL 37 20 5.62 5.32 8.96 1.98 0.21 0.11 0.27 0.17 1.59 1.68 0.24 0.95 08009 RIVER WARD @ BALHEARY 62 11 10.38 6.59 53.60 5.43 1.41 0.56 0.68 0.65 5.17 8.13 0.86 0.64 09001 RIVER RYEWATER @ LEIXLIP 215 48 38.71 35.46 91.50 1.17 0.44 0.24 0.19 0.15 2.36 2.58 0.43 0.92 09002 RIVER GRIFFEEN @ LUCAN 37 24 7.24 5.40 23.70 2.40 0.83 0.42 0.39 0.25 3.28 4.39 0.64 0.75 09010 RIVER DODDER @ WALDRON'S BRIDGE 95.2 19 70.15 48.00 269.00 3.24 0.86 0.42 0.42 0.30 3.83 5.60 2.83 0.68 10021 RIVER SHANGANAGH @ COMMON'S ROAD 30.9 24 7.87 7.36 14.30 0.95 0.37 0.21 0.19 0.07 1.82 1.94 0.46 0.94 10022 RIVER CABINTEELY @ CARRICKMINES 10.4 18 3.84 3.85 6.89 0.36 0.40 0.23 0.06 0.06 1.79 1.79 0.66 1.00 14006 RIVER BARROW @ PASS BRIDGE 1096 51 83.76 80.52 137.38 1.46 0.20 0.11 0.24 0.21 1.64 1.71 0.13 0.96 14007 RIVER STRADBALLY @ DERRYBROCK 115 25 16.94 16.20 29.30 1.12 0.30 0.17 0.22 0.07 1.73 1.81 0.25 0.96 14011 RIVER SLATE @ RATHANGAN 163 26 12.07 12.30 18.70 0.19 0.25 0.14 0.02 0.16 1.55 1.52 0.11 1.02 14018 RIVER BARROW @ ROYAL OAK 2415 51 141.83 147.98 216.07 0.22 0.24 0.14 0.04 0.06 1.52 1.46 0.09 1.04 14019 RIVER BARROW @ LEVITSTOWN 1660 51 103.46 102.41 162.86 0.61 0.24 0.14 0.09 0.13 1.57 1.59 0.10 0.99 16011 RIVER SUIR @ CLONMEL 2173 52 234.52 223 389 0.42 0.3 0.17 0.09 0.05 1.66 1.74 0.18 0.95 25003 RIVER MULKEAR @ ABINGTON 397 51 69.45 68.98 92.79 0.05 0.15 0.08 0.00 0.11 1.34 1.35 0.23 0.99 25006 RIVER BROSNA @ FERBANE 1207 52 86.77 81.91 147.21 0.71 0.25 0.14 0.14 0.18 1.70 1.80 0.12 0.94 25014 RIVER SILVER @ MILLBROOK BRIDGE 165 54 17.67 17.25 27.03 0.57 0.23 0.13 0.10 0.13 1.53 1.57 0.16 0.98 25017 RIVER SHANNON @ BANAGHER 7980 55 413.25 407.68 596.51 0.18 0.20 0.12 0.04 0.08 1.44 1.46 0.07 0.99 25023 RIVER LITTLE BROSNA @ MILLTOWN 116 52 12.14 11.22 20.05 0.57 0.29 0.16 0.14 0.06 1.65 1.79 0.17 0.92 25025 RIVER BALLYFINBOY @ BALLYHOONEY 160 31 10.15 10.18 17.40 0.57 0.29 0.17 0.08 0.15 1.71 1.71 0.11 1.00 25027 RIVER OLLATRIM @ GOURDEEN BRIDGE 118 43 23.32 22.10 40.46 0.27 0.28 0.16 0.05 0.13 1.73 1.83 0.34 0.95 25030 RIVER GRANEY @ SCARRIFF BRIDGE 279 48 43.80 40.64 87.04 1.01 0.32 0.18 0.18 0.13 1.99 2.14 0.31 0.93

11 25158 RIVER BILBOA @ CAPPAMORE 116 18 47.66 43.88 75.09 0.17 0.29 0.17 0.05 0.13 1.58 1.71 0.65 0.92 26006 RIVER SUCK @ WILLSBROOK 182 53 26.57 24.23 70.06 3.87 0.37 0.15 0.40 0.41 2.64 2.89 0.38 0.91 26007 RIVER SUCK @ BELLAGILL BRIDGE 1184 53 91.75 88.15 147.84 1.05 0.19 0.11 0.16 0.16 1.61 1.68 0.12 0.96 26008 RIVER RINN @ JOHNSTON'S BRIDGE 292 49 23.68 22.94 41.02 1.49 0.19 0.10 0.19 0.19 1.73 1.79 0.14 0.97 26019 RIVER CAMLIN @ MULLAGH 260 51 22.34 21.18 37.03 1.17 0.25 0.14 0.23 0.12 1.66 1.75 0.14 0.95 26020 RIVER CAMLIN @ ARGAR BRIDGE 128 32 11.21 11.27 15.59 0.16 0.19 0.11 0.03 0.07 1.39 1.38 0.12 1.01 26059 RIVER INNY @ FINNEA BRIDGE 249 17 12.98 12.20 16.70 0.31 0.18 0.10 0.11 0.17 1.29 1.37 0.07 0.94 27002 RIVER FERGUS @ BALLYCOREY 562 51 34.22 32.60 59.76 1.37 0.23 0.12 0.18 0.22 1.75 1.83 0.11 0.95 29001 RIVER RAFORD @ RATHGORGIN 119 40 14.17 13.46 19.66 0.34 0.19 0.11 0.09 0.10 1.39 1.46 0.17 0.95 29011 RIVER DUNKELLIN @ KILCOLGAN BRIDGE 354 22 31.94 28.89 66.52 3.37 0.30 0.14 0.41 0.32 2.08 2.30 0.19 0.90 31002 RIVER CASHLA @ CASHLA 72 26 12.89 12.16 21.10 1.92 0.24 0.13 0.31 0.18 1.64 1.74 0.29 0.94 33070 RIVER CARROWMORE L. @ CARROWMORE 90 28 7.90 7.67 11.97 1.19 0.16 0.09 0.10 0.18 1.52 1.56 0.13 0.97 34018 RIVER CASTLEBAR @ TURLOUGH 93 27 11.50 11.28 17.33 0.93 0.20 0.11 0.18 0.03 1.51 1.54 0.19 0.98 36010 RIVER ANNALEE @ BUTLERS BRIDGE 774 50 66.56 66.80 106.62 1.05 0.22 0.12 0.15 0.22 1.60 1.60 0.14 1.00 36012 RIVER ERNE @ SALLAGHAN 263 47 14.22 14.12 21.63 0.21 0.22 0.13 0.03 0.10 1.52 1.53 0.08 0.99 36015 RIVER FINN @ ANLORE 175 33 23.14 22.08 49.99 2.62 0.32 0.16 0.32 0.30 2.16 2.26 0.29 0.95 36018 RIVER DRONMORE @ ASHFIELD BRIDGE 233 50 15.84 16.25 24.43 0.43 0.18 0.10 0.04 0.10 1.54 1.50 0.10 1.03

Table 3.3 Summary statistics for 70 No. A2 stations L- L- L- Qmax/ Qmax/ Qmax/ Qmed/

Station No., River & Station Name Area N Qmean Qmed Qmax H.Skew CV CV Skew Kur Qmean Qmed Area Qmean 06031 RIVER FLURRY @ CURRALHIR 45.3 18 13.58 11.70 35.80 3.12 0.52 0.26 0.39 0.32 2.64 3.06 0.79 0.86 07006 RIVER MOYNALTY @ FYANSTOWN 176 19 26.73 27.93 34.05 -1.09 0.21 0.12 -0.20 0.07 1.27 1.22 0.19 1.05 07033 RIVER BLACKWATER @ VIRGINIA HATCHERY 129 25 14.93 14.62 26.58 1.74 0.24 0.13 0.16 0.27 1.78 1.82 0.21 0.98 08005 RIVER SLUICE @ KINSALEY HALL 10.1 18 3.04 2.50 7.81 1.54 0.68 0.38 0.23 0.19 2.57 3.13 0.77 0.82 08008 RIVER BROADMEADOW @ BROADMEADOW 110 25 44.55 40.90 123.69 1.74 0.63 0.34 0.28 0.16 2.78 3.02 1.12 0.92 12001 RIVER SLANEY @ SCARAWALSH 1036 50 169.50 157.00 399.00 1.59 0.36 0.19 0.18 0.17 2.35 2.54 0.39 0.93 14005 RIVER BARROW @ PORTARLINGTON 398 48 40.81 38.27 80.42 2.01 0.29 0.15 0.29 0.22 1.97 2.10 0.20 0.94 14009 RIVER CUSHINA @ CUSHINA 68 25 6.69 6.79 11.19 1.42 0.23 0.13 0.14 0.24 1.67 1.65 0.16 1.02 14013 RIVER BURRIN @ BALLINACARRIG 154 50 16.54 16.05 26.32 0.37 0.26 0.15 0.07 0.06 1.59 1.64 0.17 0.97 14029 RIVER BARROW @ GRAIGUENAMANAGH 2762 47 162.54 160.74 206.21 0.18 0.14 0.08 0.06 0.07 1.27 1.28 0.07 0.99 14034 RIVER BARROW @ BESTFIELD 2060 14 137.30 125.00 247.00 2.23 0.32 0.17 0.33 0.17 1.80 1.98 0.12 0.91

12 15001 RIVER KINGS @ ANNAMULT 443 42 89.39 88.75 151.00 0.14 0.28 0.16 0.00 0.08 1.69 1.70 0.34 0.99 15002 RIVER NORE @ JOHN'S BRIDGE 1605 35 211.98 197.00 393.00 0.59 0.31 0.18 0.08 0.09 1.85 1.99 0.24 0.93 15003 RIVER DINAN @ DINAN BRIDGE 298 50 143.58 150.76 187.52 -0.78 0.20 0.11 -0.15 0.09 1.31 1.24 0.63 1.05 15004 RIVER NORE @ McMAHONS BRIDGE 491 51 38.96 37.28 74.96 0.96 0.31 0.17 0.13 0.15 1.92 2.01 0.15 0.96 16001 RIVER DRISH @ ATHLUMMON 140 33 15.65 15.66 24.49 0.30 0.22 0.12 0.01 0.12 1.56 1.56 0.17 1.00 16002 RIVER SUIR @ BEAKSTOWN 512 51 55.40 52.66 123.88 1.79 0.30 0.16 0.17 0.19 2.24 2.35 0.24 0.95 16003 RIVER CLODIAGH @ RATHKENNAN 246 51 31.17 29.98 45.72 0.85 0.18 0.10 0.21 0.05 1.47 1.53 0.19 0.96 16004 RIVER SUIR @ THURLES 236 48 22.17 21.37 34.89 0.53 0.20 0.11 0.07 0.11 1.57 1.63 0.15 0.96 16005 RIVER MULTEEN @ AUGHNAGROSS 87 30 23.11 21.79 34.31 1.33 0.18 0.10 0.20 0.13 1.48 1.57 0.39 0.94 16008 RIVER SUIR @ NEW BRIDGE 1120 51 90.66 92.32 110.91 -0.32 0.13 0.07 -0.06 0.05 1.22 1.20 0.10 1.02 16009 RIVER SUIR @ CAHIR PARK 1602 52 159.29 162.21 206.00 -0.41 0.17 0.10 -0.10 0.05 1.29 1.27 0.13 1.02 18004 RIVER BALLYNAMONA @ AWBEG 324 46 30.96 31.20 52.70 1.25 0.17 0.09 0.04 0.33 1.70 1.69 0.16 1.01 18005 RIVER FUNSHION @ DOWNING BRIDGE 363 50 56.69 53.05 109.73 1.63 0.27 0.14 0.22 0.18 1.94 2.07 0.30 0.94 19001 RIVER OWENBOY @ BALLEA UPPER 106 48 15.87 15.42 22.03 0.48 0.17 0.09 0.09 0.12 1.39 1.43 0.21 0.97 19020 RIVER OWENNACURRA @ BALLYEDMOND 75 28 24.63 22.40 38.70 0.12 0.34 0.20 0.03 0.04 1.57 1.73 0.52 0.91 23001 RIVER GALEY @ INCH BRIDGE 196 45 97.39 99.05 210.07 1.22 0.33 0.18 0.13 0.18 2.16 2.12 1.07 1.02 23012 RIVER LEE @ BALLYMULLEN 60 18 16.87 15.66 31.74 2.98 0.29 0.15 0.39 0.33 1.88 2.03 0.53 0.93 24002 RIVER CAMOGUE @ GRAY'S BRIDGE 231 27 24.06 23.49 35.21 0.32 0.19 0.11 0.06 0.17 1.46 1.50 0.15 0.98 24008 RIVER MAIGUE @ CASTLEROBERTS 805 30 120.96 119.13 194.86 0.28 0.26 0.15 0.05 0.09 1.61 1.64 0.24 0.98 24022 RIVER MAHORE @ HOSPITAL 39.7 20 9.83 9.80 20.50 1.12 0.41 0.23 0.12 0.16 2.09 2.09 0.52 1.00 24082 RIVER MAIGUE @ ISLANDMORE 764 28 135.47 140.01 206.35 -0.22 0.26 0.15 -0.04 0.13 1.52 1.47 0.27 1.03 25001 RIVER MULKEAR @ ANNACOTTY 646 49 133.95 132.88 178.58 -0.19 0.16 0.09 -0.02 0.16 1.33 1.34 0.28 0.99 25002 RIVER NEWPORT @ BARRINGTONS BRIDGE 223 51 61.15 62.64 74.64 -0.65 0.16 0.09 -0.14 0.06 1.22 1.19 0.33 1.02 25005 RIVER DEAD @ SUNVILLE 190 46 28.73 29.63 33.42 -0.98 0.11 0.06 -0.19 0.10 1.16 1.13 0.18 1.03 25016 RIVER CLODIAGH @ RAHAN 274 42 23.04 22.57 36.14 0.62 0.22 0.13 0.08 0.16 1.57 1.60 0.13 0.98 25021 RIVER LITTLE BROSNA @ CROGHAN 493 44 28.03 28.58 35.80 -0.13 0.14 0.08 -0.03 0.07 1.28 1.25 0.07 1.02 - 25029 RIVER NENAGH @ CLARIANNA 301 33 54.12 56.48 74.06 -0.09 0.24 0.14 -0.02 0.02 1.37 1.31 0.25 1.04 25034 RIVER L. ENNELL TRIB @ ROCHFORT 12 24 1.50 1.48 2.21 -0.48 0.29 0.17 -0.08 0.12 1.47 1.49 0.18 0.99 25040 RIVER BUNOW @ ROSCREA 30 20 3.78 3.59 6.29 1.32 0.27 0.15 0.20 0.18 1.67 1.75 0.21 0.95 25044 RIVER KILMASTULLA @ COOLE 98.9 33 25.38 22.70 45.75 1.23 0.34 0.19 0.25 0.15 1.80 2.02 0.46 0.89 25124 RIVER BROSNA @ BALLYNAGORE 254 18 12.79 13.65 22.50 -0.31 0.36 0.20 -0.07 0.23 1.76 1.65 0.09 1.07 26002 RIVER SUCK @ ROOKWOOD 626 53 56.98 56.56 104.87 2.29 0.22 0.11 0.22 0.29 1.84 1.85 0.17 0.99 26005 RIVER SUCK @ DERRYCAHILL 1050 51 92.80 93.21 135.94 0.27 0.18 0.10 0.03 0.14 1.46 1.46 0.13 1.00 26009 RIVER BLACK @ BELLANTRA BRIDGE 97 35 13.66 13.22 18.76 0.93 0.16 0.09 0.20 0.11 1.37 1.42 0.19 0.97

13 26018 RIVER OWENURE @ BELLAVAHAN 118 49 9.19 8.95 13.98 0.77 0.20 0.11 0.13 0.11 1.52 1.56 0.12 0.97 26021 RIVER INNY @ BALLYMAHON 1071 30 65.88 66.34 92.51 -0.86 0.25 0.14 -0.11 0.19 1.40 1.39 0.09 1.01 26022 RIVER FALLAN @ KILMORE 950 33 6.64 6.49 11.06 0.47 0.30 0.17 0.09 0.05 1.67 1.71 0.01 0.98 27001 RIVER CLAUREEN @ INCH BRIDGE 48 30 20.65 20.10 31.70 1.55 0.20 0.11 0.19 0.25 1.53 1.58 0.66 0.97 27003 RIVER FERGUS @ COROFIN 168 48 24.01 22.92 40.50 0.71 0.24 0.13 0.09 0.22 1.69 1.77 0.24 0.95 29004 RIVER CLARINBRIDGE @ CLARINBRIDGE 123 32 11.39 11.30 14.77 0.77 0.15 0.08 0.14 0.08 1.30 1.31 0.12 0.99 29071 RIVER L.CUTRA @ CUTRA 123.5 26 16.00 15.70 24.30 0.74 0.23 0.13 0.11 0.19 1.52 1.55 0.20 0.98 30007 RIVER CLARE @ BALLYGADDY 458 31 61.93 62.98 95.98 1.30 0.20 0.11 0.11 0.20 1.55 1.52 0.21 1.02 30061 RIVER CORRIB ESTUARY @ WOLFE TONE BRIDGE 3111 33 274.97 247.97 601.59 3.04 0.32 0.15 0.41 0.38 2.19 2.43 0.19 0.90 32012 RIVER NEWPORT @ NEWPORT WEIR 138.3 24 30.06 29.85 36.60 -0.12 0.12 0.07 0.00 0.18 1.22 1.23 0.26 0.99 34001 RIVER MOY @ RAHANS 1911 36 174.76 174.61 286.56 1.08 0.19 0.10 0.08 0.21 1.64 1.64 0.15 1.00 34003 RIVER MOY @ FOXFORD 1737 29 180.42 178.00 282.00 1.26 0.17 0.09 0.11 0.25 1.56 1.58 0.16 0.99 34009 RIVER OWENGARVE @ CURRAGHBONAUN 113 33 28.37 27.48 38.58 0.43 0.17 0.10 0.08 0.16 1.36 1.40 0.34 0.97 34011 RIVER MANULLA @ GNEEVE BRIDGE 144 30 18.80 18.73 26.05 0.78 0.16 0.09 0.10 0.23 1.39 1.39 0.18 1.00 34024 RIVER POLLAGH @ KILTIMAGH 128 28 20.70 20.80 24.70 -0.39 0.12 0.07 -0.05 0.14 1.19 1.19 0.19 1.00 35001 RIVER OWENMORE @ BALLYNACARROW 299 29 30.52 31.16 46.04 0.10 0.21 0.12 -0.01 0.19 1.51 1.48 0.15 1.02 35002 RIVER OWENBEG @ BILLA BRIDGE 90 34 51.78 50.48 69.37 0.07 0.17 0.10 0.03 0.08 1.34 1.37 0.77 0.97 35005 RIVER BALLYSADARE @ BALLYSADARE 642 55 77.78 75.42 132.71 1.04 0.26 0.14 0.20 0.14 1.71 1.76 0.21 0.97 35071 RIVER L.MELVIN @ LAREEN 247.2 30 26.95 26.29 37.91 0.76 0.18 0.10 0.12 0.20 1.41 1.44 0.15 0.98 35073 RIVER L.GILL @ LOUGH GILL 384 30 54.81 54.05 78.40 0.34 0.22 0.12 0.08 0.14 1.43 1.45 0.20 0.99 36019 RIVER ERNE @ BELTURBET 1501 47 89.60 89.95 119.43 -0.16 0.18 0.10 -0.03 0.06 1.33 1.33 0.08 1.00 36021 RIVER YELLOW @ KILTYBARDEN 23 27 24.96 23.37 43.57 1.85 0.22 0.12 0.20 0.22 1.75 1.86 1.89 0.94 36031 RIVER CAVAN @ LISDARN 52 30 6.85 6.45 13.70 4.19 0.22 0.10 0.37 0.39 2.00 2.12 0.26 0.94 39008 RIVER LEANNAN @ GARTAN BRIDGE 78 33 28.34 28.18 43.88 0.73 0.26 0.15 0.13 0.11 1.55 1.56 0.56 0.99 39009 RIVER FERN O/L @ AGHAWONEY 207 33 45.91 45.72 76.67 1.03 0.26 0.14 0.16 0.15 1.67 1.68 0.37 1.00

Table 3.4 Summary statistics for 67 No. B stations L- L- Qmax/ Qmax/ Qmax/ Qmed/

Station No., River & Station Name Area N Qmean Qmed Qmax H.Skew CV CV Skew L-Kur Qmean Qmed Area Qmean 1041 RIVER DEELE @ SANDY MILLS 45.3 32 85.08 82.61 147.16 0.22 0.29 0.17 0.02 0.13 1.73 1.78 3.25 0.97 1055 RIVER MOURNE BEG @ MOURNE BEG WEIR 10.8 9 2.92 2.70 4.72 1.03 0.38 0.23 0.16 0.02 1.61 1.75 0.44 0.92 6021 RIVER Glyde @ Mansfieldstown 321 50 21.54 21.50 33.00 0.45 0.24 0.13 0.09 0.09 1.53 1.53 0.10 1.00 6030 RIVER BIG @ BALLYGOLY 10.2 30 20.58 10.03 122.00 3.42 1.61 0.61 0.68 0.48 5.93 12.17 11.96 0.49 6033 RIVER WHITE(DEE) @ CONEYBURROW BR. 57.4 25 27.88 18.60 92.30 2.49 0.84 0.41 0.46 0.27 3.31 4.96 1.61 0.67

14 8003 RIVER BROADMEADOW @ FIELDSTOWN 76.2 18 26.88 22.55 110.00 4.21 0.87 0.39 0.36 0.34 4.09 4.88 1.44 0.84 8007 RIVER BROADMEADOW @ ASHBOURNE 34 15 9.88 8.24 18.79 0.89 0.49 0.29 0.18 -0.01 1.90 2.28 0.55 0.83 8011 RIVER NANNY @ DULEEK ROAD BRIDGE 181 23 31.00 32.22 45.41 -0.93 0.25 0.14 -0.15 0.20 1.46 1.41 0.25 1.04 8012 RIVER STREAM @ BALLYBOGHIL 22.1 19 4.21 4.35 8.15 -0.54 0.58 0.33 -0.09 0.11 1.93 1.87 0.37 1.03 9035 RIVER CAMMOCK @ KILLEEN ROAD 54.7 9 12.04 11.70 27.80 2.75 0.60 0.33 0.33 0.23 2.31 2.38 0.51 0.97 10002 RIVER AVONMORE @ RATHDRUM 233 47 88.19 83.49 266.64 2.85 0.43 0.21 0.27 0.31 3.02 3.19 1.14 0.95 10028 RIVER AUGHRIM @ KNOCKNAMOHILL 204.1 16 56.69 46.95 102.00 1.57 0.38 0.21 0.30 0.08 1.80 2.17 0.50 0.83 11001 RIVER OWENAVORRAGH @ BOLEANY 148 33 49.85 47.17 120.7 3.03 0.33 0.16 0.24 0.3 2.42 2.56 0.82 0.95 12013 RIVER SLANEY @ RATHVILLY 185 30 45.16 43.55 72.30 -0.01 0.27 0.15 0.03 0.12 1.60 1.66 0.39 0.96 14033 RIVER OWENASS @ MOUNTMELLICK 185 22 22.59 19.50 33.00 0.53 0.28 0.16 0.14 -0.07 1.46 1.69 0.18 0.86 15005 RIVER ERKINA @ DURROW FOOT BRIDGE 387 50 28.47 27.44 61.85 1.78 0.34 0.18 0.19 0.21 2.17 2.25 0.16 0.96 15012 RIVER NORE @ BALLYRAGGET 945 16 77.16 77.11 133.00 0.90 0.30 0.17 0.08 0.21 1.72 1.72 0.14 1.00 16006 RIVER MULTEEN @ BALLINCLOGH BRIDGE 75 33 30.37 27.87 58.07 0.34 0.39 0.23 0.06 0.03 1.91 2.08 0.77 0.92 16007 RIVER AHERLOW @ KILLARDRY 273 51 79.18 75.84 138.03 0.46 0.32 0.18 0.09 0.06 1.74 1.82 0.51 0.96 16011 RIVER SUIR @ CLONMEL 2173 52 234.52 223.00 389.00 0.42 0.30 0.17 0.09 0.05 1.66 1.74 0.18 0.95 16012 RIVER TAR @ TAR BRIDGE 228 36 55.20 54.57 92.20 0.39 0.28 0.16 0.06 0.08 1.67 1.69 0.40 0.99 16013 RIVER NIRE @ FOURMILEWATER 91 33 101.69 93.21 207.02 0.86 0.42 0.24 0.16 0.08 2.04 2.22 2.27 0.92 16051 RIVER ROSSESTOWN @ CLOBANNA 34.18 13 2.95 2.85 5.67 2.53 0.36 0.19 0.33 0.26 1.92 1.99 0.17 0.96 18001 RIVER BRIDE @ MOGEELY BRIDGE 335 48 71.07 71.49 96.93 -0.16 0.19 0.11 -0.03 0.10 1.36 1.36 0.29 1.01 18002 RIVER BALLYDUFF @ BALCKWATER(MUNSTER) 2338 49 353.65 344.00 479.00 0.24 0.16 0.09 0.06 0.10 1.35 1.39 0.20 0.97 18003 RIVER BLACKWATER @ KILLAVULLEN 1258 49 282.76 266.15 502.74 1.10 0.23 0.13 0.16 0.10 1.78 1.89 0.40 0.94 18006 RIVER BLACKWATER @ CSET MALLOW 1058 27 291.30 286.00 397.00 1.10 0.14 0.08 0.19 0.12 1.36 1.39 0.38 0.98 18016 RIVER BLACKWATER @ DUNCANNON 113 24 80.99 79.65 114.82 0.49 0.23 0.14 0.12 0.00 1.42 1.44 1.02 0.98 18048 RIVER BLACKWATER @ DROMCUMMER 881 23 222.77 220.00 269.00 1.11 0.08 0.05 0.15 0.22 1.21 1.22 0.31 0.99 18050 RIVER BLACKWATER @ DURRIGLE 244.6 24 121.96 124.50 175.00 0.12 0.20 0.11 -0.01 0.09 1.43 1.41 0.72 1.02 19014 RIVER LEE @ DROMCARRA (ESB) N/A 20 79.69 71.89 157.21 1.67 0.38 0.21 0.30 0.12 1.97 1.40 0.69 0.90 19016 RIVER BRIDE @ OVENS BRIDGE (ESB) N/A 8 28.74 29.58 34.87 -1.35 0.16 0.09 -0.14 0.22 1.21 1.65 0.60 1.03 19031 RIVER SULLANE @ MACROOM (ESB) N/A 9 131.09 135.90 201.18 1.46 0.27 0.16 0.16 0.17 1.53 2.27 0.67 1.04 19046 RIVER MARTIN @ STATION ROAD 60.4 9 31.09 29.95 41.95 -0.34 0.26 0.16 -0.04 0.06 1.35 2.01 0.70 0.96 20002 RIVER BANDON @ CURRANURE 431 31 140.60 126.28 287.11 2.13 0.37 0.19 0.37 0.28 2.04 1.43 0.44 0.90 20006 RIVER ARGIDEEN @ CLONAKILTY W.W. 79.3 25 30.25 27.70 55.60 1.67 0.30 0.16 0.27 0.18 1.84 1.22 0.25 0.92 22006 RIVER FLESK @ FLESK BRIDGE 325 51 165.89 169.09 282.83 0.66 0.25 0.14 0.07 0.12 1.70 2.07 0.43 1.02 22009 RIVER DREENAGH @ WHITE BRIDGE 37 24 11.91 11.47 16.35 1.17 0.16 0.09 0.18 0.19 1.37 1.64 0.63 0.96 22035 RIVER LAUNE @ LAUNE BRIDGE 559.65 14 112.81 116.40 141.48 -0.86 0.20 0.12 -0.15 0.03 1.25 1.29 0.40 1.03

15 24004 RIVER MAIGUE @ BRUREE 246 52 54.86 50.63 104.55 0.91 0.39 0.22 0.18 0.10 1.91 1.39 0.29 0.92 24011 RIVER DEEL @ DEEL BRIDGE 273 33 103.01 104.55 171.74 0.42 0.22 0.12 -0.02 0.24 1.67 1.41 0.48 1.01 24012 RIVER DEEL @ GRANGE BRIDGE 359 41 110.45 109.99 141.95 -0.02 0.16 0.09 0.01 0.13 1.29 2.37 0.16 1.00 24030 RIVER DEEL @ DANGANBEG 248 25 52.89 52.00 72.20 0.88 0.15 0.08 0.11 0.10 1.37 2.05 0.45 0.98 25004 RIVER BILBOA @ NEWBRIDGE 125 30 41.70 42.30 59.50 -0.16 0.23 0.13 -0.04 0.12 1.43 1.73 0.49 1.01 25011 RIVER BROSNA @ MOYSTOWN 1227 51 85.64 82.02 194.25 1.31 0.34 0.18 0.14 0.23 2.27 2.38 0.41 0.96 25020 RIVER KILLIMOR @ KILLEEN 197 35 46.60 43.65 89.55 0.93 0.33 0.19 0.16 0.11 1.92 1.66 0.32 0.94 25038 RIVER NENAGH @ TYONE 139 17 42.08 39.30 67.90 1.30 0.27 0.15 0.17 0.22 1.61 2.28 0.21 0.93 26010 RIVER CLOONE @ RIVERSTOWN 100 35 20.03 17.17 40.80 1.64 0.41 0.22 0.35 0.18 2.04 1.27 0.14 0.86 26014 RIVER LUNG @ BANADA BRIDGE 222 16 44.10 42.82 70.90 1.80 0.23 0.12 0.19 0.26 1.61 2.72 0.77 0.97 26058 RIVER INNY UPPER @ BALLINRINK BRIDGE 59 24 5.98 5.35 12.20 1.60 0.38 0.21 0.28 0.18 2.04 1.62 0.15 0.89 26108 RIVER OWENURE @ BOYLE ABBEY BRIDGE 533 15 56.29 57.32 73.07 0.28 0.18 0.11 0.06 -0.06 1.30 1.23 0.14 1.02 28001 RIVER INAGH @ ENNISTIMON 168 17 52.69 47.58 129.28 4.89 0.40 0.16 0.47 0.61 2.45 2.23 0.44 0.90 29007 RIVER L.CULLAUN @ CRAUGHWELL 278 22 27.83 26.49 42.93 1.03 0.22 0.12 0.15 0.20 1.54 1.30 0.14 0.95 30012 RIVER CLARE @ CLAREGALWAY 1075.4 9 126.89 126.00 155.00 1.00 0.12 0.07 0.13 0.15 1.22 1.79 0.02 0.99 30021 RIVER ROBE @ CHRISTINA'S BR. 138 26 28.17 27.20 60.70 2.17 0.34 0.18 0.21 0.29 2.15 2.38 0.12 0.97 30031 RIVER CONG @ CONG WEIR 891 24 94.35 93.88 122.28 -0.62 0.18 0.10 -0.02 0.13 1.30 1.93 1.82 0.99 30037 RIVER ROBE @ CLOONCORMICK 210 21 1.80 1.79 3.19 -0.42 0.37 0.21 -0.09 0.18 1.77 1.96 1.59 0.99 31072 RIVER CONG @ CONG WEIR 891 26 49.08 43.20 103.00 2.04 0.38 0.20 0.31 0.20 2.10 1.94 0.57 0.88 32011 RIVER BUNOWEN @ LOUISBERG WEIR 68.6 26 74.88 64.87 125.00 0.75 0.30 0.17 0.17 0.06 1.67 2.35 1.28 0.87 33001 RIVER GLENAMOY @ GLENAMOY 73 25 62.11 59.30 116.00 1.65 0.28 0.15 0.20 0.19 1.87 1.07 0.53 0.95 34007 RIVER AT DEEL @ BALLYCARROON 156 53 90.37 84.48 198.91 0.96 0.36 0.20 0.14 0.10 2.20 1.25 0.12 0.93 34010 RIVER MOY @ CLOONACANNANA 471 12 123.29 113.72 193.07 1.31 0.3 0.17 0.21 0.11 1.57 1.7 0.41 0.92 35011 RIVER BONET @ DROMAHAIR 294 36 116.02 115.36 188.01 0.04 0.30 0.17 0.01 0.08 1.62 1.39 1.26 0.99 36011 RIVER ERNE @ BELLAHILLAN 318 49 17.91 18.23 23.6 -0.51 0.18 0.1 -0.1 0.11 1.32 1.29 0.07 1.02 36071 RIVER L.SCUR @ GOWLY 66 20 6.36 6.49 8.13 -0.06 0.15 0.09 -0.04 0.01 1.28 1.78 3.25 1.02 38001 RIVER OWENEA @ CLONCONWAL 109 33 70.02 70.63 113.38 1.78 0.16 0.09 0.08 0.26 1.62 1.75 0.44 1.01 39001 RIVER NEW MILLS @ SWILLY 49 30 44.88 44.25 61.50 0.06 0.20 0.12 0.01 0.08 1.37 1.53 0.10 0.99

16 Hazen skewness g h is ordinary skewness g multiplied by the factor {1 + 8.5/N}. This correction factor described by Hazen (1932) was shown by Wallis et al. (1974) to provide an almost unbiased estimator of skewness for lognormal and EV1 distributions and for [mildly] skewed distributions in general.

The values relating to Cv, Skewness and Qmax are of particular interest because they convey information about the flood regime of a region which may be used for inter- comparison with other regions. The average values of these for A1 and A2 stations are shown in Table 3.5 below. The estimated Cv and Skewness values are shown in map format in Figure 3.2 and Figure 3.3.

The average values of Qmed /Q mean are 0.94 for A1 stations and 0.98 for A2 stations. These are just below the values expected in data from normal distributions whereas if data were from an EV1 distribution this ratio would have a value of 0.95 for population Cv values in the range 0.2 to 04, values which are relevant for Ireland. The inverse of

0.95 ≈ 1.06 which is similar to the average value of Q mean /Q med report in NERC (1975), Volume 1, p-132.

Table 3.5 Average values of some statistics at 43 No. A1 and 67 No. A2 gauging stations.

Station Category Qmed/Qmean Cv L-Cv H.Skew L-skew Qmax/Qmean Qmax/Area A1 0.94 0.31 0.16 1.15 0.17 1.85 0.28 A2 0.98 0.25 0.14 0.84 0.11 1.65 0.31 A1& A2 0.96 0.28 0.15 1.00 0.14 1.75 0.29

The table shows that average Cv lies between 0.25 and 0.31 which is a little lower than the average value reported in the FSR for Ireland (0.30 based on 63 stations of record lengths > 15 years). The average Hazen corrected skewness lies between 0.84 and 1.15 which is considerably less that the average value of 1.63 reported in FSR, without Hazen's correction, which was also based on 63 stations with record lengths > 15 years. However the average record length of these 63 records used in FSR was < 20 years whereas the average used in this part of the present study is 36 years. Thus Ireland has a low Cv and low Skewness flood hydrology regime, typical of very humid conditions where between year variation is relatively small. Furthermore the degree of "lowness"

17 has decreased from the 1960s, whether due to real hydrological change or due to change in available record length.

The Cv and skewness statistics are shown in graphical form also, in Figure 3.4 and Figure 3.5 while Figure 3.6 and Figure 3.7 show the relationships between Cv and L- CV and between L-Skewness and Hazen Skewness.

While Cv and skewness vary between gauging stations some of this variation is due to sampling variability while the remaining may be due to true differences between the gauging stations themselves.

From Table 3.5 it can be seen, using the combined average A1 and A2 values, that the ratio L-CV / Cv = 0.535 and the ratio L-skewness / Skewness = 0.14. These compare very well with the corresponding theoretical values of 0.540 and 0.150 respectively in an EV1 distribution. Alternative values of these ratios, namely 0.52 and 0.13, can be obtained from the slopes of the lines on Figure 3.6 and Figure 3.7, where individual values of L-Cv are plotted against individual values of Cv and likewise for L-Skewness and H-Skewness.

18 Legend

Sp.Qmed !( !( !( 0.006829 - 0.151544 (! !( 0.151545 - 0.270321 (! !( !( !( 0.270322 - 0.418750 !( 0.418751 - 0.812329

!( (! 0.812330 - 1.823686

!( !( !( !( !( !( !( !( !( !( !( !( !( !( !( !!( !(( !( !( !( (! !( !( !( !( !( !( !( !( !( !( !( !( !( !( !( !( !( !( !( !( !( !( !( (! !( !( !( !( !( !( !( !( !( !( !( !( !( !( !( !( !( !( !( !(!(!( !( !( !( !(!( !( !( !( !( !( !( !( !( !( !(!( !( !( !( !( !( !( !( !( !( !( !( !( !(

!( !( !( !( !( !(!( !(!( !( !( !( !( !( !( !( !( !( !( !( !( !( !( !(

!( !( !( !( !( !( !( !( !(!( !( !( !( !( !( !( !( !( !( !(!( !( !( !( !( !( !( !( (! !( !( !( !( !( !( !( !( !( !( !( !( !( !( !( !( !( !( !(

!( !(!( !( ´

Figure 3. 1 A map showing Specific Qmed (m 3/s/km 2) for A1 , A2 and B category stations.

19 Legend CV ´ !( 0.110000 - 0.190000

!( 0.190001 - 0.270000 39009 39008!( !( !( 0.270001 - 0.440000 !( 0.440001 - 0.860000 (! 0.860001 - 1.410000

35071 !(

35073 33070 35005!( !( 36015 35002!( !( !( 34001 36019 6070 6031 !( 36021 !( 36018 !( 36010 !( !( !(!( 36031 6011 35001 34010!( !( !( !( !( 36011 34003!( !( 6026 32012 34018 36012 !( !( !( !( 6013 !( 34024 26009 !( 6025 !( 26018 !( 7033 !( !( 34011 26017!( !(26008 !( !( 26059 !( 26006 26020 26022 7006 !( !(!( !( !( 26019 7009 !( 8002 26002 26021 !( 30007 !( !( !( 8008 8009 !( 8005 26005 25124 (! !( 9001 9002 !( 26007 !( !( !( 9010 31002 !( 30061 25016 10022 !( 25006!( 10021!( !(29004 29011!( !( 25014 14011 !(!( !( 14009 !( 25017!( 14005 !( !( !( 14006 !( !( 25021 !( 14007 29071 25025 !( !( !( 27003 25040 25044 14019 !( 25030 !( !( 25029 25023!( !( 27002 !( !( 15004 14034 27001!( !(25027 !( 14013!( !( !( 15003 !( 14018 16004 !( 11001 25002!( !( !( 25003!( 25158 16003 16001 !( !( !( !( !(25005 16002 12001 24008 !( 15001 14029 !( 24082 16005 !( !( !( 23001 !( !( 24022 !( 24002 16008 !( !( !( 25034 !( 16009 !( 23012 !( 18004 !( 18005 !(

19020 !(

19001 !(

Figure 3.2 A map showing estimated Cv at 110 station locations (A1 & A2 category stations)

20 Legend H_SKEW

!( -1.090000 - -0.090000 !( ´ -0.089999 - 0.620000 !( 0.620001 - 1.460000 39009 !( 1.460001 - 2.620000 39008!( !( (! 2.620001 - 5.430000

35071 !(

35073 33070 35005 !( !( 36015 !(!(35002 34001 !( 6070 36019 6031 !( 36021 !( 36018 36010 !( !( 34009 !( 36031 6011 (! 35001 !( !( !( !( !( (! 6026 32012 34018 36012 !(!( !( 34011 26009 !( 6013 !( 34024 26018 7033 6014 6025 !( !( !( !(26008 !( !( !( 26059 !( 26006 !( 26020 7006 !(26022 !( (! !( 26019 7009 !( 8002 26002 26021 !( 30007 !( !( !( 8008 8009 26005 !( 8005 !( 25124 !( 26007 !( 9001 9002 !( 9010 31002 !( !( 30061 25006 25016 !( !( 10021(! 10022 (!29004 29011!( !( 25014 14011 !(!( !( 29001 14009!( (! 25017!( 14005 !( !(!( 14006 25021 !( !( 29071 14007 25025 !( !( !( 27003 25040!( 25044 14019 !( 25030 25023!( !( 27002 25029 14034 !( 25027 !( 15004 27001!( !( !( !( 14013 !( !( 15003 14018 16004 !( !( 15002 !( 16003!(!( !( !( 16002 12001 24008 16005 15001 14029 !( 24082 !( !( !( 23001 !( !( 24022 !( 24002 16008 !( !( !( 25034 !( 16009 16011 !( !( 23012 (! 18004 !( 18005 !(

19020 !(

19001 !(

Figure 3.3 A map showing estimated Hazen skewness at 110 station locations (A1 & A2 category stations)

21

60 60 Cv of 43 A1 Stations L-Cv of 43 A1 Stations 50 50

40 40

30 30

20 20 No. ofOccurances No. ofOccurances No.

10 10

0 0 3 1 4 6 0.1 15 25 0. 0. 0. 0. 0. 0.2 0. >0.3 0.2 0.3 0.5 >0.6 to to to to to to 0.05 to to 1 2 4 0 05 15 2 0.0 to 0. 0. 0.3 to 0. 0.5 to Cv Range 0. 0. 0.1 0. L-Cv Range0. 0.25 to

60 60 Cv of 67 A2 Stations L-Cv of 67 A2 Stations

50 50

40 40

30 30

No. ofOccurances 20

No. of Occurances 20

10 10

0 0

2 3 3 4 6 15 25 0. 0. 0.1 0. 0. >0. o 0. 0. o >0.3 to 0.2 to to to 0.6 t o o t 2 3 to 0.05 t t .5 0 05 .1 2 0.0 to 0.1 0.1 0. 0. Cv Range 0.4 to 0.5 0 0. 0. 0 0.15 to 0. 0.25 L-Cv Range

60 60 Cv of 110 A1 and A2 Stations L-Cv of 110 A1 and A2 Stations

50 50

40 40

30 30

20 20 No. ofOccurances No. No. ofOccurances

10 10

0 0 1 2 3 2 3 6 05 0. 0. 0. 0. 0. >0.6 0. o >0. to 0.1 to o to 0.4 to 0.5 o o 0 3 4 .0 t .05 to .15 t .2 to 0.25 .25 to 0.3 0. 0.1 0.2 t 0. 0. 0.5 t 0 0 0.1 to 0.15 0 0 0 Cv Range L-Cv Range

Figure 3.4 Histograms of Cv and L-Cv values

22 40 40 H.Skewness of 43 A1 Stations L-Skewness of 43 A1 Stations 35 35

30 30

25 25

20 20 15 15 No. ofOccurances No.

No. of Occurances 10 10 5 5 0 0 1 3 4 0. 0. 0. 4> 1 5 0 5 1 5 2 5 3 5 0.2 0. . > <-0.1 o o - . . . . 5 t < 0 o 0 o 1 o 2 o 3 . - t t t t 3 -0.1 to 0 o o o o 0.0 to 0.1 t 0.2 to 0.3 o 5 t 5 t 5 t 5 t t . . . . 0 0 0 1 1 2 2 3 1 - - Skewness Range L-Skewness Range

40 40 H.Skewness of 67 A2 Stations L-Skewness of 67 A2 Stations 35 35

30 30

25 25

20 20

15 15 No. ofOccurances No. No. ofOccurances No.

10 10

5 5

0 0

1 0 5 1 5 2 5 3 5 5 . . . . > - . 5 0 4 0 o 0 o 1 o 2 o 3 . < - t t t t 0.1 o 0. 4> o o o o 3 - t 0. 5 t 5 t 5 t 5 t o to . . . . < 1 to 0.2 t 0 0 0 1 1 2 2 3 1 - -0. 1 3 - Skewness Range 0.0 to 0.1 L-Skewness0. Range0.2 to 0.3 0.

40 40 H.Skewness of 110 A1 & A2 Stations L-Skewness of 110 A1 & A2 Stations 35 35

30 30

25 25

20 20

15 15 No. ofOccurances No. No. ofOccurances No. 10 10

5 5

0 0

0 1 2 3 4 1 5 0 5 1 5 2 5 3 5 > .1 . . . . > - . . . . . 4 0 0 1 2 3 5 0 o 0 0 0 0 . - o o o o . - < t t t t t 0 o o o o 3 < o o o o o .5 t 5 t 5 t 5 t 1 t t t t t . . . . 0 0 0 1 1 2 2 3 0 0 1 2 3 1 - - . . . . - Skewness Range 0 L-Skewness0 Range0 0

Figure 3.5 Histograms of Hazen skewness and L-skewness values

23 0.80 y = 0.5152x 0.70

0.60

0.50

0.40 L-Cv

0.30

0.20

0.10

0.00 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 Cv

Figure 3.6 L-Cv Vs Cv for A1 and A2 stations. Note if intercept is included the relevant trendlines are y = 0.4426x + 0.0268.

0.80 y = 0.1293x 0.70

0.60

0.50

0.40

0.30

0.20 L-skew 0.10

0.00

-0.10

-0.20

-0.30 -2.00 -1.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 H-skew

Figure 3.7 L-Skewness Vs Hazen Skewness for A1 and A2 stations. Note if intercept is included the relevant trendlines are y = 0.1225x + 0.0145.

24 3.2 Preliminary distribution choice with help of Skewness versus Record Length Plots

Another way of viewing skewness values is shown on Figure 3.8 and Figure 3.9 which show skewness or L-Skewness plotted versus length of record along with confidence intervals based on Normal distribution and EV1 distribution assumptions.

In the interest of model parsimony it is always desirable to consider the adequacy of 2 – parameter models and consequently only Normal and EV1 distributions are considered in this sub-section . The normal distribution confidence intervals for skewness are calculated using the standard expression

6N(N − )1 se (skewness ) = (N − )(1 N + )(2 N + )3 while all other confidence intervals are obtained by simulation. At the lower end of skewness range it is seen that there is a sizeable number, 14, of stations which have negative skewness which fall below the EV1 confidence interval band. At the upper end of the skewness range there are 46 stations which fall above the upper confidence band for the Normal distribution. There is a group of 49 stations which fall within both sets of confidence intervals.

Because of the relatively low values of observed skewness at many gauging stations it is appropriate to ask whether the data as a whole could be considered to have come from a Normal distribution. All but 1 of the lower values fall above the 95% lower Normal CI while 46 of the larger values lie above the upper 95% Normal CI. If the data as a whole were Normal then only 2.5% of 110 or say 3 values would lie above the upper 95% Normal CI, so that as a whole the Normal hypothesis cannot be accepted. The same question is considered in Figure 3.8(b) in relation to the EV1 distribution. Only 5 values lie above the upper 95% EV1 CI ( ≈ 3) while 14 values ( >> 3) lie below the lower 95% EV1 CI. Thus the hypothesis that all the samples have come from the EV1 family cannot be accepted as a whole.

25 Skewness vs RL (Normal Dist.) 4

3

2

1 Skewness

0

-1

-2 0 10 20 30 40 50 60 Record Length Observed Theoretical Skew. Value for N.D. 95% C.I. 67% C.I.

Skewness vs RL (EV1) 4

3

2

1 Skewness

0

-1

-2 0 10 20 30 40 50 60 Record_Length Observed Theoretical Skew. value for EV1 67% C.I. 95% C.I.

Figure 3.8 Observed Skewness calculated at 110 no. A1 and A2 gauging stations plotted versus record length. Also shown are 67% and 95% Confidence Intervals for skewness calculated from (a) Normal distribution samples and (b) EV1 distribution samples.

26 Very similar findings are obtained when analogous plots of L-Skewness are examined, Figure 3.9 below.

The overall result is that, on the basis of skewness values, the stations with the larger values of skewness could be modelled by an EV1 distribution but with the reservation that there is a serious number ( 19 stations or 17% of all stations) which are not in keeping with this choice. Thus while 83 % of station skewness values are consistent with EV1 there is a large proportion, 53%, which is consistent with the normal distribution and 45% of all stations are consistent with both Normal and EV1 assumptions.

Thus it would seem that the choice of a single distribution for all stations in Ireland is of doubtful validity. However skewness in the usual sample sizes found in hydrology is not renowned for its precision. L-Skewness offers some advantages but the same conclusions can be drawn from Figure 3.9 as from Figure 3.8. An alternative to the Skewness versus Record Length analysis is the use of L-Moment Ratio diagrams and these are considered in the next section.

27 0.8 L-Skewness vs RL (Normal Dist.)

0.6

0.4

0.2 L-Skewness

0.0

-0.2

-0.4 0 10 20 30 40 50 60 Record_Length Observed Theoretical L-Skew.value for N.D. 95% C.I. 67% C.I.

L-Skewness vs RL (EV1) 0.8

0.6

0.4

0.2 L-Skewness 0.0

-0.2

-0.4 0 10 20 30 40 50 60 Record_Length Observed Theoretical L-Skew value for EV1 67% C.I. 95% C.I.

Figure 3.9 Observed L- Skewness calculated at 110 no. A1 and A2 gauging stations plotted versus record length. Also shown are 67% and 95% Confidence Intervals for L-Skewness calculated from (a) Normal distribution samples and (b) EV1 distribution samples .

28 3.3 Preliminary distribution choice with the help of L-Moment Ratio diagrams

Another way to explore the suitable of different probability distributions is to use moment ratio diagrams. Moment ratio diagrams based on conventional moment based skewness and kurtosis are regarded as having very poor discriminating capability (WMO, 1989, Appendix 3) but L-moment ratio diagrams can be more reliable as a diagnostic tool (Hosking and Wallis, 1997 p-40). Nevertheless even L-moment ratio diagrams cannot be regarded as entirely reliable in the context of the available sample sizes.

The L-moment ratio diagram is a graph between L-kurtosis and L-skewness. Usually a two-parameter distribution with a location and a scale parameter plots as a single point while a three parameter distribution with location, scale and shape plots as a line or curve on the diagram. Generally the distribution selection process involves plotting the sample L-moment ratios as a scatter plot and comparing them with theoretical L- moment ratio points or curves of candidate distributions.

Plots of dimensionless L moment ratios are shown in Figure 3.10. It compares the observed and the theoretical relations between L-skewness and L-kurtosis for the annual maximum flood flows at the 110 A1 and A2 sites.

In the context of 2-parameter distributions alone the data are on average closer to the population L-moments of an EV1 distribution rather than to those of the LO and LN distributions. Figure 3.11 shows sample L-moments of data simulated from EV1, LO and LN distributions each with the same record lengths and parameters as the actual records. In the case of EV1 simulated data the scatter of the points is narrower than in Figure 3.10 especially on the left hand side. There are many more negative historical values of L-skewness than there are among the simulated data. The LO and LN simulated L-moments cover the range of observed values in the negative L-skewness domain more adequately than the EV1 values but do not extend as far to the right as the EV1 simulated and observed L-skewness values. Four further realizations of this kind of simulation are shown in Figure 3.12 and Figure 3.13.

29 0.70 L-Moment Ratio Diagram for 110 A1 & A2 Stations 0.60

0.50

0.40

0.30

0.20 L-Kurtosis

sample_data 0.10 sample_avg EV1 LO 0.00 LN GEV -0.10 LN3 GLO Normal -0.20 -0.30 -0.20 -0.10 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 L-Skewness

Figure 3.10 L-moment ratio diagram for annual maximum flow for 110 stations of Ireland

0.7 0.7 L-Moment Ratio( Observed data) L-Moment Ratio (EV1 simulated data) 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 L-Kurtosis 0.2 L-Kurtosis 0.1

0.1 0.0

0.0 -0.1

-0.1 -0.2 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 L-Skewness L-Skewness

0.8 0.7 L-Moment Ratio (LO simulated data) 0.7 L-Moment Ratio( LN simulated data) 0.6

0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 L-Kurtosis L-Kurtosis 0.2 0.1 0.1 0.0 0.0

-0.1 -0.1

-0.2 -0.2 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 L-Skewness L-Skewness

Figure 3.11 L-moment ratio diagram of a) observed data b) data simulated from EV1 distribution c) data simulated from LN2 distribution and d) data simulated from LO distribution

30 While on the basis of average values EV1 looks a more suitable candidate than either of LO or LN the occurrence of so many negative values of L-skewness among the observed data throws doubt on the suitability of EV1 as a universal 2 parameter model for Ireland . Note also that the average of the data points falls roughly half way between the GEV and GLO curves.

31

0.7 0.7 L-Moment Ratio( Observed data) L-Moment Ratio (EV1 simulated data) 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 L-Kurtosis 0.2 L-Kurtosis 0.1

0.1 0.0

0.0 -0.1

-0.1 -0.2 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 L-Skewness L-Skewness

0.7 0.7 L-Moment Ratio (LO simulated data) L-Moment Ratio( LN simulated data) 0.6 0.6

0.5 0.5

0.4 0.4

0.3 0.3

0.2 0.2 L-Kurtosis L-Kurtosis

0.1 0.1

0.0 0.0

-0.1 -0.1

-0.2 -0.2 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 L-Skewness L-Skewness

0.7 0.7 L-Moment Ratio( Observed data) L-Moment Ratio (EV1 simulated data) 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 L-Kurtosis 0.2 L-Kurtosis 0.1

0.1 0.0

0.0 -0.1

-0.1 -0.2 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 L-Skewness L-Skewness

0.7 0.7 L-Moment Ratio (LO simulated data) L-Moment Ratio( LN simulated data) 0.6 0.6

0.5 0.5

0.4 0.4

0.3 0.3

0.2 0.2 L-Kurtosis L-Kurtosis

0.1 0.1

0.0 0.0

-0.1 -0.1

-0.2 -0.2 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 L-Skewness L-Skewness

Figure 3.12 First set of two realizations of four realizations of L-moment ratio diagrams of data simulated from EV1, LN2 and LO distribution

32 0.7 0.7 L-Moment Ratio( Observed data) L-Moment Ratio (EV1 simulated data) 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 L-Kurtosis 0.2 L-Kurtosis 0.1 0.1 0.0

0.0 -0.1

-0.1 -0.2 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 L-Skewness L-Skewness

0.7 0.7 L-Moment Ratio( LN simulated data) L-Moment Ratio (LO simulated data) 0.6 0.6

0.5 0.5

0.4 0.4

0.3 0.3

0.2 0.2 L-Kurtosis L-Kurtosis

0.1 0.1

0.0 0.0

-0.1 -0.1

-0.2 -0.2 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 L-Skewness L-Skewness

0.7 0.7 L-Moment Ratio( Observed data) L-Moment Ratio (EV1 simulated data) 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 L-Kurtosis 0.2 L-Kurtosis 0.1

0.1 0.0

0.0 -0.1

-0.1 -0.2 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 L-Skewness L-Skewness

0.7 0.7 L-Moment Ratio( LN simulated data) L-Moment Ratio (LO simulated data) 0.6 0.6

0.5 0.5

0.4 0.4

0.3 0.3

0.2 0.2 L-Kurtosis L-Kurtosis

0.1 0.1

0.0 0.0

-0.1 -0.1

-0.2 -0.2 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 L-Skewness L-Skewness

Figure 3.13 Second set of two realizations of four realizations of L-moment ratio diagrams of data simulated from EV1, LN2 and LO distribution

33 3.4 Largest recorded values and outliers

The largest flow values, Qmax , in each record are also listed in Tables 3.2 to 3.4 as are dimensionless values Qmax/Qbar and Qmax/Qmed and the specific value Qmax/A. The largest values of Qmax/Qbar and Qmax/Qmed among the 45 Grade A1 stations are 6.1 and 10.7 respectively, at station 08009 on the River Ward at Balheary. These resemble those found in UK data but there are only two other stations where these statistics exceed 3 and 4 respectively, namely the 09002 River Griffeen at Lucan and the 09010 River Dodder at Waldron’s Bridge. At Balheary the record is 17 years long but there are 7 incomplete years, leaving only 10 valid annual maxima which are 10.30, 6.59, 6.77, 3.62, 6.39, 2.72, 6.95, 2.08, 3.31, 53.6

Regardless of whether the record is short or long the 1993 (04/02/1994) value of 53.6, even allowing for difficulties of measurement and extrapolation of the rating curve, is truly a large outlier. This station's data are not included in the inferences and summaries of the preceding sub-sections. The large value at Waldron's Bridge occurred in August 1986 as a result of Hurricane Charlie and of all the exceptional flows which occurred in South Dublin and Wicklow at that time it is the only flow to have been measured. Again, even allowing for difficulties of measurement and extrapolation of the rating curve, it is truly a very large flood in the Irish context. At all other 42 A1 stations the values of these ratios are less than 3 which are extremely low by European standards. It should be noted that these quantities depend on record length and not all records in the data set are of equal length, varying from 15 to 55 years. The corresponding largest values of Qmax/Qbar and Qmax/Qmed among 70 Grade A2 stations are 2.8 and 3.1 respectively.

The largest sp ecific values Qmax/A in cumec/km 2 are also listed as are the quantities Qmax/A . The last quantity is of interest because as is well known flood values do not increase linearly with A but rather with A a where the power a varies from 0.75 to 0.85 depending on circumstances.

34 An outlier is an observation that lies an abnormal distance from other values in a random sample from a population. In other words an outlier lies significantly away from the trend of the remaining data. The presence of outliers causes problem with linearity of pattern on probability plots, see next Chapter. The 17 stations with the most obvious outliers have been selected from the probability plots and are listed in Table 3.6. This shows only three stations with a Qmax/Q med value in excess of 3 and only 8 stations with values in excess of 2. These numbers are extremely low by comparison with UK data for instance. There are other instances of notably large floods which have occurred on ungauged catchments or at Grade B stations and these are not included in Table 3.6 eg Dargle at Bray in 1986 (ungauged) and Deel at Crossmolina ( no grade). Table 3.7 lists two stations with notable low outliers. Low outliers can have a negative effect on skewness values and on the appearance of probability plots.

Table 3.6 High outliers from 110 Irish stations

Station Station Record largest Q 2nd 3rd Qmed Qmax/ L.Q/ 2ndL.Q/ No Class Length or Qmax Largest Q Largest Q Qmed 2ndL.Q 3rdL.Q 6011 A1 48 26.36 21.07 19.99 15.39 1.71 1.25 1.05 6031 A2 18 35.8 24.4 18.1 11.7 3.06 1.47 1.35 7033 A2 25 26.58 20.56 18.71 14.62 1.82 1.29 1.1 8009 A1 11 53.6 11.8 10.3 6.59 8.13 4.54 1.15 9010 A1 19 269 156 112 48 5.60 1.72 1.39 14005 A2 48 80.42 80.42 59.99 38.27 2.10 1 1.34 15004 A2 51 74.96 73.61 56.88 37.28 2.01 1.02 1.29 16002 A2 51 123.88 84.42 81.86 52.66 2.35 1.47 1.03 23012 A2 18 31.74 26.27 19.12 15.66 2.03 1.21 1.37 24022 A2 20 20.5 15 14.2 23.49 0.87 1.37 1.06 26006 A1 53 70.06 68.66 41.81 29.23 2.40 1.02 1.64 29011 A1 22 66.52 44.34 40.61 28.89 2.30 1.5 1.09 33070 A1 28 11.97 9.44 9.35 7.67 1.56 1.27 1.01 34001 A2 36 286.56 224.44 219.12 174.61 1.64 1.28 1.02 36015 A1 33 49.99 43.3 31.77 22.08 2.26 1.15 1.36 36021 A2 27 43.57 32.16 31.79 23.37 1.86 1.35 1.01 36031 A2 30 13.7 8.87 8.32 6.45 2.12 1.54 1.07

Table 3.7 Low outliers from 110 Irish stations

Lowest 2nd rd nd rd Station Station Record Q Lowest 3 Lowest Qmed Qmed 2 L.Q 3 L.Q No Class Length or Qmin Q Q /Qmin /L.Q /2 nd L.Q 15003 A2 50 61.13 96.72 98.02 150.76 2.47 1.58 1.01 26059 A1 17 8.23 10.7 11.4 12.2 1.48 1.3 1.07

35 4 Probability Plots and Inferences from them

4.1 Probability Plots

Probability plots are useful techniques for the graphical display and analysis of flood data, particularly to determine whether or not a particular sample is consistent with from a particular population distribution. Probability plots employ an inverse distribution scale so that a simple, usually 2 parameter, cumulative distribution function (CDF) plots as a straight line, i.e the points (x(i), yi); i = 1, 2, ..., n expect to be close to line y =a + b*x, where a and b are location and scale parameters respectively; conversely, strong deviation from this line is evidence that the distribution did not produce the data. It is acknowledged that what constitutes "strong deviation" from such a line is usually judged subjectively. Nevertheless they are widely used in flood hydrology for data display if not for determining a final distribution choice.

A small selection of probability plots of samples, of size 25 and 50, drawn randomly from an EV1 population are shown in Figure 4.1(a) and (b). These show, in some cases, departures from linear patterns that could cause an EV1 hypothesis to be rejected, on the basis of visual inspection, in an isolated case that might occur in practice. Hence deductions from probability plots have to be considered carefully.

Three probability plots namely Gumbel (EV1), logistic (2 parameter or LO) and Lognormal (2 parameter or LN) are plotted for the annual maximum discharge data of 187 Irish stations. The annual flood data from these 186 stations vary in record length from 8 to 55 years and the associated catchment areas range from 10.1 to 2780 km 2. Data from an additional 17 stations which have pre- and post-drainage records are also plotted. The probability plots are shown in Appendix 4 which is subdivided into A1, A2, B gauging station categories and which contains two further sections for ESB stations and pre- and post-drainage stations.

Plotted data for 110 A1 and A2 stations were investigated from the standpoint of linearity and non-linearity. In the linearity investigation a best-fit least squares straight line is drawn through the data and is given a score from 1 to 5 where, 1 stands for a very poor fit and 5 stands for a very good fit. On the other hand in the non-linearity

36 investigation an eye-guided assessment of the plotted data is given a curve pattern like linear, concave, convex or S-curve by visual judgment. It is acknowledged that deductions drawn from probability plots have low statistical power i.e. there is a high probability that a conclusion could be adopted even though it is not true. Nevertheless probability plots have played a prominent role in hydrological frequency analysis in the past and it is considered worthwhile to use them in this study as a way of viewing the available data.

37

300 250 300

250 200 250 200 200 150 Q Q 150 Q 150 100 100 100 50 50 50 0 0 0 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 EV1 y EV1 y EV1 y

300 250 350 250 300 200 200 250 150 200 Q

150 Q Q 100 150 100 100 50 50 50 0 0 0 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 EV1 y EV1 y EV1 y

300 350 350 250 300 300 200 250 250 200 200

Q 150 Q 150 Q 150 100 100 100 50 50 50 0 0 0 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 EV1 y EV1 y EV1 y Figure 4.1(a)

38 300 350 350 250 300 300 250 250 200 200 200

Q 150 Q Q 150 150 100 100 100 50 50 50 0 0 0 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 EV1 y EV1 y EV1 y

450 300 350 400 300 350 250 250 300 200 250 200 Q Q

Q 150 200 150 150 100 100 100 50 50 50 0 0 0 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 EV1 y EV1 y EV1 y

300 300 300 250 250 250 200 200 200

Q 150 Q Q 150 150 100 100 100 50 50 50 0 0 0 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 -2 -1 0 1 2 3 4 5 EV1 y EV1 y EV1 y

Figure 4.1 (a, b) Random Samples of (a) size 25 (b) size 50 from an EV1(100,30) population. (Cv = 0.33)

39 4.2 Linear Patterns

Each data series was examined from the point of view of linearity and subjectively scored from 1 to 5 by visual judgment, where 1= very poor; 2= poor; 3= medium; 4= good; and 5= very good fit.

These scores are listed individually for each station in Table 4.3 and Table 4.4 for A1 and A2 stations respectively and a summary of these scores is given in Table 4.1, as follows:

Table 4.1 Linear pattern statistics for A1 and A2 stations combined. Scores at individual stations are tabulated in Table 4.3 and Table 4.4.

Linear pattern counts for A1 stations Type of plot EV1 LO LN Modal Score 4 2 3 Mean Score 3.3 2.6 3.3 Linear pattern counts for A2 stations Type of plot EV1 LO LN Modal Score 4 3 4 Mean Score 3.4 3.1 3.4 Linear pattern counts for A1& A2 stations Type of plot EV1 LO LN Modal Score 4 3 4 Mean Score 3.3 2.9 3.3

The above statistics show that EV1 and LN give almost identical results and they provide a better fit than LO. It can be deduced from the linear patterns that Irish flood data are more likely to be distributed as EV1 or LN rather than LO among 2-parameter distributions.

40 4.3 Non Linear Patterns

Each data series was investigated from the point of view of non-linearity and was assigned a curve pattern by visual judgment. Four curve patterns linear, concave, convex and elongated S are introduced and, again each has been classified as degree one and degree two for greater discrimination. However, there is also an ‘X’ introduced if the patterns of the end points of the plot do not match the main body pattern well. In summary, non-linear patterns are classified as:

L1= perfectly straight line L2= little deviation from straight line L2X= body pattern is quite straight but end points disturb the linearity U1= mild concave upwards U2= severe concave upwards U1X= mild concave with end disturbance U2X= severe concave with end disturbance D1= mild convex upwards D2= severe convex upwards D1X= mild convex upwards with end disturbance D2X= severe convex upwards with end disturbance S1= mild S-curve S2= severe S-curve S1X = mild S-curve with end disturbance S2X= severe S-curve with end disturbance

These scores are listed individually for each station in Table 4.3 and Table 4.4 for A1 and A2 stations respectively and a summary of these scores is given in Table 4.2 as follows:

In EV1 probability plots for A1 stations, 22 out of 43 show a linear trend and a convex trend comes out as second dominant pattern with 8 stations; while for A2 which contains 67 stations, the corresponding numbers are 29 and 16 respectively. Interestingly very few stations exhibit concave pattern on EV1 probability plots. This is

41 because in many cases the largest floods on record are close together in magnitude and in very few cases there are notably large floods with very few cases of large outliers.

Table 4.2 Non- linear pattern statistics for A1 and A2 stations. Scores at individual stations are tabulated in Table 4.3 and Table 4.4.

Non linear score counts Non linear score counts Non linear score counts Curve : A1 : A2 : A1& A2 Patterns Type of plot Type of plot Type of plot EV1 LO LN EV1 LO LN EV1 LO LN L1 5 0 5 5 0 7 10 0 12 L2 17 9 17 24 19 21 41 28 38 D1 8 1 1 9 2 7 17 3 8 D2 0 0 0 7 1 2 7 1 2 U1 2 11 5 0 17 3 2 28 8 U2 3 4 0 3 5 2 6 9 2 S1 1 8 8 9 13 11 10 21 19 S2 1 3 1 3 5 5 4 8 6 D1x 0 0 0 0 0 1 0 0 1 D2x 0 1 1 1 1 0 1 2 1 U1x 0 1 0 0 1 0 0 2 0 U2x 1 1 1 0 0 1 1 1 2 S1x 2 1 2 0 0 1 2 1 3 S2x 2 1 1 1 1 0 3 2 1 L2x 1 2 1 5 2 6 6 4 7

In Logistic probability plots concave and linear patterns emerge as the dominant patterns and a significant number of S shapes was also found. Hardly any convex pattern was found in LO plots. There was no station found with a perfect straight line.

In the case of lognormal plots a linear trend was found as the dominant one. A considerable number of S and convex patterns was also observed. In Table 4.3 and Table 4.4 it was seen that in many cases EV1 and Lognormal complement one another.

In many cases it was observed that those stations which fit quite straight on EV1 plots show a concave fit on Logistic paper. The reason might be due to the range of the probability axis, because the probability axis of LO is relatively compressed compared to EV1. That is why it was also noticed that those which fit as convex on EV1 plots are found to be quite linear on LO.

42 Table 4.3 Probability plot linear and non-linear scores for A1 category Stations

Record Linear Score Non linear Score Station No, Location Length EV1 LO LN EV1 LO LN 6011 FANE 48 4 2 3 L2 U1 L2 6013 DEE 30 2 3 4 D1 S1 L2 6014 GLYDE 30 4 2 3 L1 U1 L2 6025 DEE 30 2 4 3 D1 L2 S1 6026 GLYDE 46 3 1 2 S1 S2 S2 6070 MUCKNO 24 4 4 5 L2 L2 L1 7009 BOYNE 29 4 1 3 L2 D2X L2 8002 DELVIN 20 4 1 2 L2 U1 U1 8009 BALHEARY 11 1 1 2 U2 U2 U1 9001 RYEWATER 48 5 2 5 L1 U1 L1 9002 GRIFFEEN 24 1 1 4 S2X U2 L1 9010 DODDER 19 2 2 4 U2 U2 L2 10021 SHANGANAGH 24 4 2 3 L2 S1 L2 10022 CARRICKMINES 18 4 3 4 L2 L2X L2 14006 BARROW 51 5 2 3 L1 U1 U1 14007 DERRYBROCK 25 4 2 3 L2 U1 S1 14011RATHANGAN 26 3 3 3 L2 L2 L2 14018 BARROW 51 3 3 4 L2 S1 S1 14019 BARROW 51 3 3 4 D1 S2 S1 16011 SUIR 52 3 2 3 U1 S1 S1 25006 BROSNA 52 4 3 4 L2 L2 S1 25014 SILVER 54 4 3 4 L2 U1X L2 25017 SHANNON 55 3 4 5 D1 L2 L1 25023 LITTLE BROSNA 52 2 2 2 S2 S2 S1X 25025 BALLYFINBOY 31 5 4 5 L1 L2 L2 25027 OLLATRIM 43 4 5 3 D1 L2 D1 25030 GRANEY 48 4 2 4 S1X U1 L2 26006 SUCK 53 1 1 1 S2X S2X S2X 26007 SUCK 53 5 3 5 L1 U1 L1 26008 RINN 49 4 3 3 L2 U1 L2 26019 CAMLIN 51 4 3 3 L2 S1 L2 26020 CAMLIN 32 3 3 2 L2 S1 L2 26059 INNY 17 2 2 1 D1 D1 D2X 27002 FERGUS 51 4 2 4 L2 U1 L2 29001 RAFORD 40 3 4 4 D1 S1 S1 29011 DUNKELLIN 22 2 2 2 U1 S1 S1 31002 CASHLA 26 2 1 1 U2X U2X U2X 33070 CARROWMORE 28 4 3 4 L2X L2X L2X 34018 CASTLEBAR 27 4 3 3 L2 U1 U1 36010 ANNALEE 50 4 3 4 S1X S1X S1X 36012 ERNE 47 3 4 3 D1 L2 L2 36015 FINN 33 2 2 2 U2 U2 U1 36018 DRONMORE 50 4 4 4 L2 L2 L2

43 Table 4.4 Probability plot linear and non-linear scores forA2 category Stations

Record Linear Score Non Linear Score Station No,Location Length Ev1 LO LN EV1 LO LN 6031 FLURRY 18 2 2 3 U2 U2 U2 7006 MOYNALTY 19 2 3 2 D2 D1 D2 7033 BLACKWATER 25 4 3 3 S1 U1 S1 8005 SLUICE 18 4 3 5 L2 U1 L1 8008 BROADMEADOW 25 4 2 4 L2 U1 L2 12001 RIVER SLANEY 50 4 2 3 L2 U1 L2 14005 BARROW 48 3 2 2 L2 U1 U1 14009 CUSHINA 25 4 3 4 S1 U1 S1 14013 BURRIN 50 4 3 4 S1 S2 S1 14029 BARROW 47 3 4 5 D1 S1 L1 14034 BARROW 14 4 2 2 L2X U2X U2X 15001 RIVER KINGS 42 2 2 3 D2 S2 S1 15002 NORE 35 4 3 4 L2 U1 L2 15003 DINAN 50 2 3 2 D2 S1 D2 15004 NORE 51 5 3 5 L2 U1 L2 16001 DRISH 33 5 4 5 L1 U1 L1 16002 SUIR 51 3 2 4 L2X U1X L2X 16003 CLODIAGH 51 4 3 3 L2 S1 S1 16004 SUIR 48 5 4 5 L1 U1 L1 16005 MULTEEN 30 5 2 2 L2 U1 U1 16008 SUIR 51 2 4 3 D2 L2 D1 16009 SUIR 52 2 3 3 D2X D2X D1X 18004 BALLYNAMONA 46 2 2 2 S2 S2 S2 18005 FUNSHION 50 4 2 3 L2X U2 L2X 19001 OWENBOY 48 4 4 4 L2 L2 S1X 19020 OWENNACURRA 28 3 4 3 D2 S1 D1 23001 GALEY 45 4 2 3 L2 U1 L2 23012 LEE 18 2 1 1 U2 U2 U2 24002 CAMOGUE 27 4 5 4 L2 L2 L2X 24008 MAIGUE 30 3 5 4 L2 L2 L2X 24022 MAHORE 20 5 3 5 L1 U1 L2 24082 MAIGUE 28 2 5 3 D1 L2 D1 25016 CLODIAGH 42 4 4 5 L2 L2 L2 25021 BROSNA 44 3 4 5 D1 L2 L2 25029 NENAGH 33 3 3 4 S2 S2 S1 25034 L. ENNELLTRIB 24 2 4 3 D2 L2 D1 25040 BUNOW 20 4 3 4 L2 U1 L2 25044 KILMASTULLA 33 3 3 3 S1 S1 S2 25124 BROSNA 18 3 4 2 D1 L2 D1 26002 SUCK 53 2 1 2 S2X S2X L2X 26005 SUCK 51 4 4 5 L1 L2 L1 26009 BLACK 35 4 3 2 S1 S1 S2 26018 OWENURE 49 4 3 4 L2 S1 L2 26021 INNY 30 1 2 1 D2 D1 D1 26022 FALLAN 33 4 3 4 L2 S1 S1 27001 CLAUREEN 30 4 3 3 L2 U1 L2 27003 FERGUS 48 4 4 4 L2 L2 L2 29004 CLARINBRIDGE 32 4 3 3 L2 L2 S1

44 29071 L.CUTRA 26 4 3 4 L2 S1 L2 30007 CLARE 31 4 3 4 L2X L2X L2X 30061 CORRIB ESTUARY 33 1 1 2 U2 U2 U1 32012 NEWPORT 24 3 5 3 D1 L2 L2 34001 MOY 36 4 3 4 L2X L2X L2 34003 MOY 29 3 3 3 S1 S1 S2 34009 OWENGARVE 33 4 4 5 L2 L2 L1 34011 MANULLA 30 4 4 4 S1 L2 L2 34024 POLLAGH 28 3 4 3 D1 L2 L2 35001 OWENMORE 29 4 5 4 L2 L2 L2 35002 OWENBEG 34 3 5 4 D1 L2 L2 35005 BALLYSADARE 55 4 2 4 S1 S1 L2 35071 L.MELVIN 30 4 3 4 L2 U1 L2 35073 L.GILL 30 3 4 4 D1 L2 D1 36019 ERNE 47 2 3 3 D1 S1 S1 36021 YELLOW 27 3 3 3 L2 U2 S1 36031 CAVAN 30 2 1 1 S2 S2 S2 39008 GARTAN 33 3 2 3 S1 S1 S1 39009 FERN O/L 33 5 3 5 L1 U1 L1

4.4 Variation of shape of probability plot with Skew coefficient

Having noted that skewness is negative or a small positive value in many AM series the estimated skewness values have been ranked in increasing order and the corresponding curve pattern of the data series was noted alongside for the case of EV1 probability plots. This summary is presented in Table 4.5 and Table 4.6 for A1 stations and A2 stations respectively.

These tables show that:

1. There are 1 A1 station and 13 A2 stations with negative skewness and a further 22 A1 stations and 27 A2 stations with skewness between 0 and 1. The average skewness of 110 Irish stations is 1.0. 2. Stations with negative or left skewness predominantly have convex upwards curve patterns. 3. Stations with higher or right skewness mostly have concave upward shape. 4. In addition, most of the high outliers are found in the high right skewness zone and conversely most of the low outliers are found in left skewness zone.

45 Table 4.5 A1 stations ranked by skewness Record Curve patterns on Hazenskew Station No, Location length EV1 Remarks -0.33 6025 DEE 30 convex 0.10 6013 DEE 30 convex 0.16 26020 CAMLIN 32 linear 0.18 25017 SHANNON 55 convex 0.19 14011RATHANGAN 26 linear 0.21 36012 ERNE 47 convex 0.22 14018 BARROW 51 linear 0.27 25027 OLLATRIM 43 convex 0.31 26059 INNY 17 convex Low outlier 0.34 29001 RAFORD 40 convex 0.36 10022 CARRICKMINES 18 linear 0.43 36018 DRONMORE 50 linear 0.42 16011 SUIR 52 convex 0.57 25014 SILVER 54 linear 25023 LITTLE 0.57 BROSNA 52 linear 0.57 25025 BALLYFINBOY 31 linear 0.61 14019 BARROW 51 convex 0.71 25006 BROSNA 52 linear 0.78 6070 MUCKNO 24 linear 0.81 6011 FANE 48 linear High Outlier 0.93 34018 CASTLEBAR 27 linear 0.93 7009 BOYNE 29 linear 0.95 10021 SHANGANAGH 24 linear 1.01 25030 GRANEY 48 s-curve 1.05 26007 SUCK 53 linear 1.05 6026 GLYDE 46 s-curve 1.05 36010 ANNALEE 50 s-curve 1.12 14007 DERRYBROCK 25 linear 1.17 26019 CAMLIN 51 linear 1.17 9001 RYEWATER 48 linear 1.19 33070 CARROWMORE 28 linear High Outlier 1.31 6014 GLYDE 30 linear 1.37 27002 FERGUS 51 linear 1.46 14006 BARROW 51 linear 1.49 26008 RINN 49 linear 1.92 31002 CASHLA 26 concave 1.98 8002 DELVIN 20 linear 2.40 9002 GRIFFEEN 24 s-curve 2.62 36015 FINN 33 concave High Outlier 3.24 9010 DODDER 19 concave High Outlier 3.37 29011 DUNKELLIN 22 concave High Outlier 3.87 26006 SUCK 53 s-curve High Outlier 5.43 8009 BALHEARY 11 concave High Outlier

46 Table 4.6 A2 stations ranked by skewness Record Curve patterns on Hazenskew Station No, Location length EV1 Remarks -1.09 7006 MOYNALTY 19 convex -0.86 26021 INNY 30 convex -0.78 15003 DINAN 50 convex Low outlier -0.48 25034 L. ENNELLTRIB 24 convex -0.41 16009 SUIR 52 convex -0.39 34024 POLLAGH 28 convex -0.32 16008 SUIR 51 convex -0.31 25124 BROSNA 18 convex -0.22 24082 MAIGUE 28 convex -0.16 36019 ERNE 47 convex -0.13 25021 BROSNA 44 convex -0.12 32012 NEWPORT 24 convex -0.09 25029 NENAGH 33 s-curve 0.07 35002 OWENBEG 34 convex 0.10 35001 OWENMORE 29 linear 0.12 19020 OWENNACURRA 28 convex 0.14 15001 RIVER KINGS 42 convex 0.18 14029 BARROW 47 convex 0.27 26005 SUCK 51 linear 0.28 24008 MAIGUE 30 linear 0.30 16001 DRISH 33 linear 0.32 24002 CAMOGUE 27 linear 0.34 35073 L.GILL 30 convex 0.37 14013 BURRIN 50 s-curve 0.43 34009 OWENGARVE 33 linear 0.47 26022 FALLAN 33 linear 0.48 19001 OWENBOY 48 linear 0.53 16004 SUIR 48 linear 0.59 15002 NORE 35 linear 0.62 25016 CLODIAGH 42 linear 0.71 27003 FERGUS 48 linear 0.73 39008 GARTAN 33 s-curve 0.74 29071 L.CUTRA 26 linear 0.76 35071 L.MELVIN 30 linear 0.77 26018 OWENURE 49 linear 0.77 29004 CLARINBRIDGE 32 linear 0.78 34011 MANULLA 30 s-curve 0.85 16003 CLODIAGH 51 linear 0.93 26009 BLACK 35 s-curve 0.96 15004 NORE 51 linear High outlier 1.03 39009 FERN O/L 33 linear 1.04 35005 BALLYSADARE 55 s-curve 1.08 34001 MOY 36 linear High outlier 1.12 24022 MAHORE 20 linear High outlier 1.22 23001 GALEY 45 linear 1.23 25044 KILMASTULLA 33 s-curve 1.25 18004 BALLYNAMONA 46 s-curve 1.26 34003 MOY 29 s-curve 1.30 30007 CLARE 31 linear

47 1.32 25040 BUNOW 20 linear 1.33 16005 MULTEEN 30 linear 1.42 14009 CUSHINA 25 s-curve 1.54 8005 SLUICE 18 linear 1.55 27001 CLAUREEN 30 linear 1.59 12001 RIVER SLANEY 50 linear 1.63 18005 FUNSHION 50 linear 1.74 8008 BROADMEADOW 25 linear 1.74 7033 BLACKWATER 25 s-curve High outlier 1.79 16002 SUIR 51 linear High outlier 1.85 36021 YELLOW 27 linear High outlier 2.01 14005 BARROW 48 linear High outlier 2.23 14034 BARROW 14 linear 2.29 26002 SUCK 53 s-curve 2.98 23012 LEE 18 concave High outlier 30061 CORRIB 3.04 ESTUARY 33 concave 3.12 6031 FLURRY 18 concave High outlier 4.19 36031 CAVAN 30 s-curve High outlier

48 4.5 Flood volumes associated with largest peaks on convex probability plots

For many of the stations which display a convex upwards appearance on probability plots it is noted that the largest and the 2nd largest floods are not appreciably larger than the 3rd largest flood. This contributes to the flattening out of the probability plot and the calculated skewness tends to be small or negative. Such stations are identified as D1, D2 or DX in Table 4.3 and Table 4.4 where there are 8 and 21 such occurrences respectively. These stations are identified as "convex" in Table 4.5 and Table 4.6.

There are other stations, not classified as D1, D2 or D2X in the subjective classification, which also display such behaviour to a lesser degree but such stations are not considered further here.

It could be asked if this behaviour can be attributed to catchment slope or tendency to show over-bank flow at high flow values. In other words is there any storage available which would dampen out the flood peaks while the flood volumes are different between the largest and second/third larges floods i.e. could volume be increasing when peak discharge is stagnant?

To investigate this question hydrograph volumes have been calculated for each of the three or four largest flood peak values in a selection of 8 series from the 29 series identified as having convex upwards series. Having examined the results, see discussion below, it was decided not to proceed with assembly of hydrograph data for the remaining 21 stations. The stations examined are listed in Table 4.7.

49 Table 4.7 Sample of 8 stations showing convex upward appearance on probability plots.

Station Hazen Duration of Grade Skewness hydrograph data Station No. River and Station Name available 7006 R. Moynalty at Fyanstown A2 -1.088 1956-2005* 15003 R. Dinan at Dinan Bridge A2 -0.996 1972-2005 16008 R. Suir at New Bridge A2 -0.324 1954-2005 16009 R. Suir at Cahir Park A2 -0.409 1940-2005 24013 R. Deel at Rathkeale A1 -0.908 1972-2003 24082 R. Maigue at Islandmore A2 -0.216 1975-2001 25017 R. Shannon at Banagher A1 0.018 1989-2003 25021 Little Brosna at Croghan A2 -0.134 1961-2003 * (1987-2006 post-drainage period alone was used in this part of the study )

Volume calculation:

On examination of hydrographs it is seen that some of the larger annual maximum flows occur as a result of a single rainfall event but many of them occur as a result of longer rainfalls of varying intensities which lead to multi peak hydrographs around the time of the annual maximum flood. As a result no single time duration for calculation of hydrograph volume can be specified. Hence volumes were calculated for a number of time durations, or windows, symmetrically located around the time of the peak. The windows corresponded to ±12hours, ± 24 hours, ±84 hours, ±1week and ±2 weeks leading to window sizes of 1, 2, 7,14 and 28 days respectively.

The results for stations 24082 River Maigue at Islandmore and 7006 River Moynalty at Fyanstown are summarised in Figure 4.2 and Figure 4.3 below. The former example is based on 46 years observations, 1977 – 2004 while the later one is based on 19 years of post drainage observations, 1986 – 2004.

50

250 24082 RIVER MAIGUE @ ISLANDMORE EV1 200 '89 '98 '88 '00 150 w inter peak summer peak 100 AMF(m3/s)

50 2 5 10 25 50 100 500

0 -2-1 0 1 2EV1 y 3 4 5 6 7

250 Hydrograph during Maximum Flood in the year 1989 Red zone= Vol. of 1day Yellow zone= Vol. of 2day Blue zone= Vol. of 1week 200

150

100 Discharge (m3/s) Discharge

50

0 01/02/1990 00:00 03/02/1990 00:00 05/02/1990 00:00 07/02/1990 00:00 09/02/1990 00:00 11/02/1990 00:00 Time 200 Hydrograph during Maximum Flood in the year 2000 Red zone= Vol. of 1day 180 Yellow zone= Vol. of 2day Blue zone= Vol. of 1week 160

140

120

100

80 Discharge Discharge (m3/s) 60

40

20

0 31/10/2000 00:00 02/11/2000 00:00 04/11/2000 00:00 06/11/2000 00:00 08/11/2000 00:00 10/11/2000 00:00 12/11/2000 00:00 Time

51 200 Hydrograph during Maximum Flood in the year 1988 Red zone= Vol. of 1day 180 Yellow zone= Vol. of 2day Blue zone= Vol. of 1week 160

140

120

100

80 Discharge (m3/s) Discharge 60

40

20

0 14/10/1988 16/10/1988 18/10/1988 20/10/1988 22/10/1988 24/10/1988 26/10/1988 28/10/1988 00:00 00:00 00:00 00:00 00:00 00:00 00:00 00:00 Time 200 Hydrograph during Maximum Flood in the year 1998 Red zone= Vol. of 1day 180 Yellow zone= Vol. of 2day Blue zone= Vol. of 1week 160

140

120

100

80 Discharge (m3/s) Discharge 60

40

20

0 23/12/1998 00:00 25/12/1998 00:00 27/12/1998 00:00 29/12/1998 00:00 31/12/1998 00:00 02/01/1999 00:00 04/01/1999 00:00 Time 160 Volume of hydrographs of different year during max peak 140 1998 1988 120 2000 100 1989

80

60 Mill cu. meter 40

20

0 1 2Days 7 14 30

Figure 4.2 Hydrograph volumes of different durations during the 4 largest annual maximum flood peaks at 24082 River Maigue at Islandmore.

In Figure 4.2 the flood hydrographs are shown in decreasing order of peak magnitudes from 1989 (largest) to 1998 (smallest) while in the final part of Figure 4.2 the flood volumes are shown in increasing order of magnitude 1998, corresponding to the

52 smallest of the 4 floods, 1988 (2 nd smallest), 2000 (2 nd largest) and 1989 (largest of the record). One day and two day flood volumes are noticeably similar for these four events. For 7 and 14 and 30 days the 1989 event, the largest peak, gives the largest volume among the four.

40 7006 RIVER MOYNALTY @ FYANSTOWN 35 '93 30 '00 '04 '95 w inter peak summer peak 25

20

AMF(m3/s) 15

10

5 2 5 10 25 50 100 500 0 -2-10 1 2EV1 y 3 4 5 6 7

25 Volume of hydrographs of different year during max peak

2000 20 1995 1993

15

10 Mill. cu. meter

5

0 1 2Days 7 14 30

Figure 4.3 Hydrograph volumes of different durations during the 3 largest annual maximum flood peaks at 7006 River Moynalty at Fyanstown.

In the Figure 4.3 the individual hydrographs are not shown and the flood volumes are shown in the order 2000 (corresponding to the smallest of the 3 floods), 1995, and 1993 (largest flow on record) . The flood peak magnitudes increase slightly in the order shown, i.e. 2000 peak is the smallest while 1993 peak is the largest. One day flood volumes are noticeably similar for these three events. For 2, 7 and 14 and 30 days the 1995 event shows a slightly

53 higher volume than the 2000 event but the 1993 event, that with the largest peak gives the least volume among the three; in fact it is 50% of the 1995 event volume for 7, 14 and 30-day. The 2000 and 1995 flood events display multiple peaks which probably explain why they have greater volumes than the 1993 event.

Of the remaining 6 stations so considered, 5 of them show the same behaviour as that shown above, namely 1-day and 2-day volumes which are almost identical across the 3 or 4 flood events analysed. As a result it may be concluded that flood volumes do not reveal any noticeable growth with rank among the highest ranking floods in those series which display convex upwards patterns on probability plots. That is, no clue about the flattening out of the plots can be found among the flood volumes which would seem to rule out local attenuation effects, meaning that attenuation effects if any are attributable to the effects of the upstream catchment.

The exception is 15003 River Dinan at Dinan Bridge where there is noticeable growth of volume with flood peak magnitude, at 1 and 2 day durations and slight growth of volume at 7 days see Figure 4.4 below. However this is reversed for the larger durations where the largest flood peak (1985) is associated with the smallest volume.

54 15003 RIVER DINAN @ DINAN BRIDGE 200

'98 '68 '97 '85 150

w inter peak 100 summer peak AMF(m3/s)

50 2 5 10 25 50 100 500

0 -2-1 0 1 2EV1 y 3 4 5 6 7

50 Volume of hydrographs of different years during max peak

45 1998 40 1997 1985 35

30

25 20

Mill. cu. meter 15

10 5

0 1 2Days 7 14 30

Figure 4.4 Hydrograph volumes of different durations during the 3 largest annual maximum flood peaks at 15003 River Dinan at Dinan Bridge.

The entire sample of 8 stations is presented in Appendix 3. The conclusion remains that, except in the case of the Dinan, where the largest flood peaks show very little variation the flood volumes also remain more or less constant so that no explanation for the flattening out of the probability plots can be found among the flood volumes.

As part of Work Package 5.3, a catchment descriptor called Floodplain Attenuation Index, FAI, was developed for each gauged catchment in the study. The Index falls in the range 0 to 1 and it might be hoped that it could explain some of the behaviours described above. A plot of skewness , as an indicator of convex upwards behavour , against FAI for those catchments whose skewness is less than 1 does not reveal any repentance between these variables ( Figure 4.5) leading to the conclusion that the concave downwards behaviour cannot be explained by the Flood Attenuation Index.

55

FAI vs H.skewness Data point: 60 ( stations whose skewness are less than 1.0)

0.35

0.30

0.25

0.20 FAI 0.15

0.10

0.05

0.00 -1.50 -1.00 -0.50 0.00 0.50 1.00 1.50 Hazen Skewness

Figure 4.5 A plot between FAI vs Hazen Skewness

56 5 Flood Seasonality

Annual maximum floods were examined, with respect to seasonality, on 202 stations (A1, A2 and B categories). After the seasonality study had been completed some of the 202 stations were considered unsuitable for further statistical analysis because data quality issues. These issues did not affect the seasonality studies. A total of 6969 station years of data were involved. Data for 203 stations are available for categories A1, A2 and B but one of these does not have date of occurrence listed.

1. Floods are found to occur at all times of year, but most rivers register in between 65% and 100% in the winter time i.e. October-March half-year. In all, 6094 of the 6969 annual maxima occurred in winter. Table 5.1 lists all stations with the corresponding percentage of peak floods in the October-March period. There are 11 stations where no peak floods were observed in the summer time i.e. April-September.

2. Figure 5.1 shows the frequency of flood peaks in each of the 12 months. The months December and January are associated with the greatest number of AM flood events followed by November and February. In the summer time considerable numbers of flood peaks were observed in August while July has the least number of AM floods.

Seasonal Flood Frequency 1800 1600 Total no. of station years: 6969 1400 1200 1000 800

600

No. of Occurances 400 200 0 Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Month

Figure 5.1 Seasonal occurrence of annual maximum flood values in 202 stations

57 3. Figure 5.2 illustrates the seasonal distribution of AM flood peaks for the 202 gauging stations. The radial position indicates season and the distance from the center shows magnitude of the flood peak. As stated above most of the big floods were found in the months between October and February. There were six occasions when more than 600 cumec discharges were observed, one of which occurred in August. However, all these discharges occurred at a single station, 23002 Feale at Listowel with a drainage area 646 km2.

Apr 800

6005 8 0

400

200

1 2 0

Jul -800 -34 0 1 2 0 5 8 0 Jan

-3 4 0

Oct-8 00

Figure 5.2 Flood seasonality showing actual magnitude in 202 Irish stations

4. As drainage area exercises a substantial control over flood peaks, specific AM (flood magnitude divided by catchment area) values are plotted on Figure 5.3, which is similar in appearance to Figure 5.2. A number of large specific floods were observed in August

58 and September. In fact among the biggest five, four were from a single station Sandy Mills (1041) on the river Deele with drainage area 45.3 km2, while the other one was found in August during Hurricane Charlie in August 1986 at the station Waldron's Bridge (9010) of the river Dodder. It is also known that very large floods occurred in other rivers along the east coast as a result of Hurricane Charlie in August 1986, the river Dargle being a notable one where Q/A may have exceeded 3 cumec/ km2. However this and others are not included in Figures 13 and 14 as no gauged record exists.

Apr 3.253 .2 5

2

1 0.5

0 Jul -3.25 0 3 .25 Jan

Oct-3 .2 5

Figure 5.3 Flood seasonality with specific flood magnitude in 201 Irish stations

59 5. Finally the months corresponding to occurrence of maximum flow in each AMF data series are listed in Table 5.2 and are shown graphically in Figure 5.4. Among the 202 stations examined, Figure 5.4 shows that the largest value in the series occurred during the summer months at 20 of them. Of those 20, 8 occurred in August. The month of December provided the series maximum in 91 stations (45%).

Month corresponding to Max. flow in an AM series 100

90

80

70

60

50

40

No. of occurences 30

20

10

0 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Month

Figure 5.4 Month corresponding to maximum flow in each of 202 AM series

60 Table 5.1 Station Number with percent of peak floods in the October-March (percentage)

Station %of Station %of Station %of Station %of Station %of No events(oct- No events(oct- No events(oct- No events(oct- No events(oct- mar) mar) mar) mar) mar) 1041 91 14006 88 20006 84 26009 89 34018 100 1055 89 14007 76 22006 86 26010 91 34024 90 3051 100 14009 80 22009 79 26012 98 34029 86 6011 100 14011 81 22035 100 26014 100 35001 90 6013 90 14013 82 23001 87 26017 94 35005 93 6014 97 14018 96 23002 80 26018 96 35011 89 6021 94 14019 92 23012 83 26019 88 35071 90 6025 80 14029 98 24001 88 26020 88 35073 87 6026 91 14033 82 24002 81 26021 90 36010 96 6030 63 14034 82 24004 87 26022 88 36011 96 6031 94 15001 88 24008 87 26058 83 36012 98 6033 80 15003 84 24011 85 26059 96 36015 94 6070 85 15004 92 24012 88 26108 100 36018 92 7002 96 15005 90 24013 87 27001 73 36019 96 7003 83 15012 75 24022 100 27002 96 36021 73 7004 98 16001 88 24030 88 27003 88 36027 93 7005 91 16002 90 24082 82 27070 79 36031 90 7006 84 16003 82 25001 73 28001 82 36071 75 7007 91 16004 90 25002 76 29001 85 38001 73 7009 90 16005 87 25003 78 29004 81 39001 83 7010 93 16006 82 25004 87 29007 82 39008 82 7011 98 16007 84 25005 87 29011 86 39009 88 7012 94 16008 90 25006 92 29071 83 7033 84 16009 90 25011 78 30001 94 7041 100 16011 90 25014 81 30004 97 8002 76 16012 92 25016 93 30005 86 8003 78 16013 88 25017 98 30007 94 8005 67 16051 85 25020 91 30012 100 8007 73 18001 90 25021 91 30021 92 8008 84 18002 94 25023 71 30031 100 8009 67 18003 96 25025 90 30037 57 8011 87 18004 85 25027 88 30061 91 8012 68 18005 84 25029 85 31002 81 9001 77 18006 93 25030 92 31072 69 9002 68 18016 83 25034 83 32011 65 9010 63 18048 83 25038 82 32012 88 9035 44 18050 92 25040 75 33001 68 10002 83 19001 92 25044 83 33070 89 10021 63 19014 90 25124 94 34001 97 10022 60 19016 100 25158 72 34003 97 10028 69 19020 89 26002 91 34004 92 11001 91 19031 89 26005 94 34007 85 12001 90 19046 67 26006 94 34009 91 12013 73 20001 93 26007 94 34010 58 14005 90 20002 94 26008 92 34011 97

61 Table 5.2 Station number with month corresponding to maximum flow in an AM data series Month Month Month Month Month corres- corres- corres- corres- corres- ponding ponding ponding ponding ponding max Station max flow Station max flow Station max flow Station max flow Station flow in a No. in a series No. in a series No. in a series No. in a series No. series 1041 12 12001 11 19014 12 25034 12 30031 12 1055 12 12013 11 19016 1 25038 12 30037 11 3051 12 14005 12 19020 12 25038 2 30061 1 6011 12 14006 12 19031 2 25040 10 31002 10 6013 12 14007 2 19046 3 25044 12 31072 7 6014 12 14009 2 20001 10 25124 2 32011 9 6021 12 14011 2 20002 12 25158 1 32012 12 6025 11 14013 12 20006 12 26002 10 33001 9 6026 12 14018 2 22006 12 26005 12 33070 10 6030 11 14019 2 22009 12 26006 12 34001 10 6031 12 14029 12 22035 1 26007 11 34003 10 6033 12 14033 12 23001 12 26008 11 34004 10 7002 11 14034 2 23002 8 26009 10 34007 10 7003 12 15001 12 23012 8 26010 11 34009 11 7004 11 15003 8 24001 10 26012 2 34010 6 7005 12 15004 2 24002 2 26014 11 34011 10 7006 10 15005 12 24004 12 26017 12 34018 12 7007 12 15012 2 24008 12 26018 12 34024 11 7009 12 16001 12 24011 12 26019 10 34029 1 7010 11 16002 12 24012 12 26020 12 35001 10 7011 11 16003 12 24013 12 26021 12 35005 10 7012 12 16004 12 24022 12 26022 12 35011 10 7033 1 16005 9 24030 8 26058 1 35071 1 7041 1 16006 12 24082 12 26059 1 35073 12 8002 11 16007 9 25001 12 26108 1 36010 12 8003 8 16008 12 25002 10 27001 1 36011 12 8005 11 16009 12 25003 12 27002 12 36012 12 8007 8 16011 11 25004 12 27003 12 36015 10 8008 11 16012 11 25005 11 27070 12 36018 12 8009 6 16013 11 25006 12 28001 12 36019 12 8011 12 16051 1 25011 12 29001 1 36021 10 8012 8 18001 10 25014 12 29004 1 36027 1 9001 12 18002 10 25016 12 29007 1 36031 10 9002 12 18003 12 25017 12 29011 1 36071 4 9010 12 18004 1 25020 1 29071 12 38001 9 9035 10 18005 12 25021 12 30001 12 39001 9 10002 11 18006 11 25023 11 30004 11 39008 12 10021 5 18016 1 25025 1 30005 12 39009 12 10022 5 18048 12 25027 12 30007 2 10028 10 18050 12 25029 2 30012 12 11001 8 19001 1 25030 12 30021 1

62 6 m-year maxima

The theory of extreme values states that if Z is the maximum of a number of other independent random variables X 1, X 2, ……X N then the distribution of Z converges towards one of the 3 types of Extreme Value distribution as N becomes infinitely large. In practice it is assumed that the limiting case is reached or almost reached for quite modest values of N, c.f. the analogous case of the Central Limit Theorem, although it is unlikely that the CL findings would easily transfer to the EV case. In the context of extreme value theory Gumbel (1940) suggested that annual maximum flows ought to have an extreme value distribution because each one is the maximum of a sufficiently large number of individual flows during the year and he used some examples where the EV1 distribution looked to be a good descriptor of AM flow values. However what constitutes "large" in the context of N was never really discussed. Some writers felt that N=365 based on the number of days in the year but this is hardly realistic since successive daily flows are not independent. In most Irish rivers there are no more that 10 or 12 separate flood peaks in any one year (leaving aside consideration of what constitutes independent peaks) and this number may not be sufficient to support the large N assumption. If N in individual years is insufficient to produce EV behaviour then one would hope that decade maxima might display such behaviour and analyses of such values might lead to identification of which type of EV distribution would be appropriate. However with the length of record available a realistic assessment cannot be made on the basis of decade maxima. Instead a more modest approach base on m-year maxima where m=1,2 and 3 is explored here in order to see what information might be unearthed. Probability plots of 1-year , 2-year and 3-year maxima have been prepared thus far for the 41 No. A1 grade stations. The results obtained are not uniform. In some cases the three lines are parallel, in some case the three lines diverge at high return periods and in some cases they converge at high return periods. Examples of each type of case are shown overleaf inTable 6.1 and Figure 6.1. It is disappointing that no clear single behaviour exists. The behaviour is further compared with the behaviours observed in previous plots and tables. The parallel line case, Station 9001, shows the classical EV1 behaviour with a maximum score of 5 for linearity in Table 4.3 and a skewness almost identical to the theoretical EV1 skewness of 1.14. The diverging lines case, Station 36015, scores very

63 low in linearity on any type of plot and is classified a severe concave upwards in Table 4.5, with a large skewness and presence of outliers. In the converging case, Station 6013, while Table 4.3 shows medium scores for linearity Table 4.5 shows convexity and accompanying low skewness. The pattern seems to be that the behaviour of m-year maxima is very much predictable from the annual maximum behaviour. It appears that here is no relief to be gained, from m-year maxima analysis, from the uncertainty of identifying a single underlying parent distribution for Irish AM floods.

Table 6.1Examples of m-year maxima Station and AM Prob. Plot Behaviour Skewness Outliers? pattern of Table 8 Table 10 Tables 3 and m-year maxima EV1,LO,LN EV1,LO, LN 12 9001 parallel 5, 2, 5 L1, U1, L1 1.17 No 36015 diverging 2, 2, 2 U2,U2, U1 2.616 Yes 6013 converging 3, 3, 4 D2, S1, L2 0.10 No, but convex

64 Station 9001

100 y = 14.032x + 40.423 80

60 y = 13.351x + 47.107 y = 13.641x + 30.97 40 20

m-year maxima,m-year cumec 0 -2 -1 0 1 2 3 4 5 EV1 Reduced Variate

Station 36015

60 50 y = 7.1632x + 22.326 40 30 y = 8.285x + 24.198 20 y = 5.7818x + 19.878 10

m-year maxima,m-year cumec 0 -2 -1 0 1 2 3 4 5 EV1 Reduced Variate

Station 6013 60

50 y = 4.798x + 29.494 40 y = 4.4222x + 31.09 y = 5.8484x + 24.516 30

20

10 m-year maxima, cumec maxima, m-year

0 -2 -1 0 1 2 3 4 5 EV1 Reduced Variate

Figure 6.1 Probability plots of 1, 2 and 3 year maxima showing contrasting behaviours.

65 7 Flood statistics on some rivers with multiple gauges

As part of exploratory data analysis it is interesting to view flow data from several gauging stations on the same river, where such multiple gauging stations exist. This gives an opportunity to see how flood magnitude increases in the downstream direction. The data for catchments (Barrow, Suir, and Suck) are illustrated below in the form of probability plots and plots of median annual flood vs catchment area. Catchment area acts as a surrogate for distance along main channel. It can be seen that median annual flood increases in a regular, mostly linear, manner with area.

250 Probability Plot Summary for stations on Barrow River

200

14005 150 14006 14018 14019 AMF(m3/s) 100 14029

50

2 5 10 25 50 100 500 0 -2-1 0 1 2EV1 y 3 4 5 6 7

Qmed vs Catchment Area relationship for stations on Barrow River 180

160

140

120

100

80 Qmed (m3/s) 60

40

20

0 0 500 1000 1500 2000 2500 3000 Catchment Area (km2)

Figure 7.1 (a) Probability plot summary for stations on Barrow River (b) a scatter plot of Qmed vs Catchment area for these stations. Stations are ordered from upstream.

66 500 Probability plot summary for stations on Suir River 450 2 5 10 25 50 100 500 400

350 16004

300 16002

250 16008

AMF(m3/s) 200 16009 150 16011 100

50

0 -2-1 0 1 2 3 4 5 6 7 EV1 y

Qmed vs Catchment Area Relationship for stations on Suir River 250

200

150

100 Qmed (m3/s)

50

0 0 500 1000 1500 2000 2500 Catchment Area (km2)

Figure 7.2 (a) Probability plot summary for stations on Suir River (b) a scatter plot of Qmed vs Catchment area for these stations. Stations are ordered from upstream.

67

160 Probability Plot Summary for stations on Suck River

140

120

100 26006 26002 80 26005

AMF(m3/s) 26007 60

40

20 2 5 10 25 50 100 500 0 -2-1 0 1 2 3 4 5 6 7 EV1 y

Qmed vs Catchment Area relationship for stations on Suck River 100 90 80 70 60 50 40 Qmed (m3/s) Qmed 30 20 10 0 0 200 400 600 800 1000 1200 1400 Catchment Area (km2)

Figure 7.3 (a) Probability plot summary for stations on Suck River (b) a scatter plot of Qmed vs Catchment area for these stations. Stations are ordered from upstream.

68 8 Determining the T-year Flood Magnitude Q T by the Index Flood Method

8.1 Introduction

Since the time of Fuller (1914) other developments were made by Foster (1924) and Hazen (1932). Gumbel (1940) brought the basis of analysis to a new level by applying to extreme value theory and the findings of Fisher and Tippet (1928) Gumbel introduced the Extreme Value Type I distribution to flood frequency analysis. This distribution is often known as the Gumbel distribution but is denoted EV1 in this report. Powell (1943) introduced extreme value probability paper , which in one form or another (ruled in terms of probability or reduced variate) is still widely in use today. Then Langbein (1949) provided the link between the T-year flood estimated by both annual maximum and partial duration (or peaks over a threshold) models.

The concept of standard error and bias of statistical estimates of Q T was introduced by Kimball (1948) for the EV1 distribution and by Kaczmarek(1957) for EV1 and other distributions although this information filtered only slowly into hydrological practice. The first simulation based studies of standard errors were reported by Nash and Amorocho(1965) and by Lowery and Nash (1969) while by the mid-1970s extensive simulation studies on bias and standard error were reported by Mathalas and wallis(197); Mathalas, Wallis and Slack (197) and other analogous studies up to including that by Lu and Stedinger(1992) were reported in the intervening years.

Since the time of Fuller (1914) flood magnitudes have been estimated by fitting distributions, either graphically or numerically, to series of annual maximum floods treated as if they are random samples from known distributions.

The T-year return period flood is the flood magnitude which is exceeded on average once in every T years and which has a probability of being exceeded in any one year of p =1/T. Let F(Q) be the distribution function of annual maximum flood magnitudes, then

69 F(Q) = Non-exceedance probability of magnitude Q in any one year = 1 – 1/T and

-1 QT = F (1-1/T) --- (8. 1) as illustrated in Figure 8. 1

Probability Density Function Distribution Function

1 1 - F(q) = 1/T 0.4 0.8 0.3 area = 0.6 1 - 1/T area = 1/T 0.2 0.4 F(q) = 1 - 1/T 0.1 0.2 0 Probability Density Function Q T Non Exceedance0 Probability, F(q) QT

Figure 8. 1 probability density function and distribution function

Regional or geographical pooling of dimensionless flow values was introduced by

Dalrymple (1960) so as to provide regional growth factors X T for use at ungauged sites, in the form

QT = Q I X T --- (8. 2)

where Q I = Index Flood = Mean or Median annual flood

X T = QT / Q I = Growth factor.

FSR (1975) introduced the idea that equation (8. 2) also be used for Q T estimation at gauged sites if T > 2N where N = period of observed or gauged record while FEH (1999), more stringently, recommended that equation (8. 2) be used at gauged sites if T > N/2. In between publication of FSR and FEH considerable research was carried out on the efficiency of the index flood approach (Greis & Woods (1981), Hosking, Wallis & Woods (1985), Lettenmaier, Wallis and Wood (1987), WMO (1989), Hosking and Wallis (1997)) and the index flood approach is now widely accepted as a valid and well understood method (both in its implementation and its properties) . While Dalrymple (1960) and FSR (1975) used the Index Flood a method with geographically defined regions the Region of Influence method of selecting catchments for regionalisation or pooling developed by Burn (1990) was favoured and found to be more effective by FEH (1999).

70 The index flood approach involves selection of a suitable index flood such as Q or Q med or possibly Q 5. FSR choose Q , FEH choose Q med while FSR Volume II (Rainfall

Studies) choose M 5 the 5 year return period rainfall. In the present study Q med is chosen in line with FEH.

Use of the index flood approach involves (i) estimation of the index flood for the site in question – the subject site.

(ii) estimation of the growth factor X T or the growth curve, the X-T relation.

These two steps are more or less independent and will be dealt with separately in the following sections. First some terminology relating to subject, donor and analog sites is explained.

8.2 Subject, Donor and Analog Sites

Any river location at which flood estimate is required for any purpose or project is called a subject site or alternatively a project location. In some cases a subject site coincides with or nearly coincides with a gauging station location but the majority of subject sites do not. If not, the subject site is then an ungauged site.

In keeping with the terminology adopted by FEH (1999) a donor site is considered to be a gauging station on the same river as a subject site and either upstream or downstream from it.

Likewise in keeping with FEH (1999) an analog site is considered to be a gauging station which is not on the same river as the subject site but may be on another tributary within the same parent catchment as the subject site or it may be on a neighbouring catchment area. It is assumed that an analog site is chosen so that its catchment’s main characteristics such as area, soil, slope etc. are similar to those of the subject site’s catchment area. While traditionally an analog site was presumed to be "close" to the subject site that restriction may be relaxed if an objective measure of similarity of catchments, such as Euclidean distance measure in catchment descriptor space, e.g. equation (10. 2) below, is used.

71 9 Determining Q med

9.1 Subject site is Gauged

At a gauged site Qmed is obtained as the median of the annual maximum flow series

Q med = Median (Q 1, Q2 ….. Q N)

The Q i values are ranked from smallest to largest

Q (1) < Q (2) < ……. Q (N)

If N is odd there is a single mid value which can be taken as Q med. If N is even the two middle most values are averaged to give Q med.

The standard error of the median in a random sample from a normal distribution is se(Q med ) ≈ 1.253 σ / N . Since Irish flood data are slightly more skewed than the normal distribution then se(Q med ) for flood data will be slightly greater, the multiplier 1.25 being increased to approximately 1.28 to 1.30 when N = 5, a result obtained by simulation. Adopting the

_ larger value of 1.30 and taking average values of Cv and of the ratio Qmed/ Q for Irish A1 and A2 stations into account an approximate value of the multiplier of Qmed is obtained as follows.

36.0 se (Qmed ) = Qmed --- (9. 1) N

If N = 5, se(Q med ) ≈ 0.161 Q med ( = 0.22 Q med in FEH,3, Table 2.2)

If N = 10, se(Q med ) ≈ 0.114 Q med ( = 0.14 Q med in FEH,3, Table 2.2)

If N = 20, se(Q med ) ≈ 0.084 Q med ( = 0.08 Q med in FEH,3, Table 2.2) If N = 50, se(Qmed) ≈ 0.051Qmed ( not included in FEH,3,Table 2.2) The multipliers given for FEH are slightly larger possibly because of the large Cv and skewness of UK data.

In FEH it is recommended that if N < 14 then Q med can be estimated with slightly smaller standard error from a peak over a threshold series (POT). The reduction in

72 standard error in the UK case is from 22% Q med to 18% Q med when N = 5, with similar kinds of reduction when N < 5 and with lesser reductions when 5 < N < 14 (FEH, 3, Table 2.2).

The standard error of the median is 25% large than that of the mean but this is partially balanced by the fact that Qmed is not affected by low or high outliers in the data sample and by the fact that its return period of 2 is easier to interpret.

Another method of reducing the standard error of estimate is by “record extension” whereby a short at-site record at the subject site is “extended” by exploiting the correlation between the AM series at the subject site and the AM series of one or more neighbouring sites (Fiering, 1963). This has not been considered further here. However another form of data transfer is described in the following section.

9.2 Subject size is Ungauged

The standard method is to use an equation, calibrated by multiple regression analysis, which gives Q med in terms of catchment descriptors. α s r Qmed,cds = cA S R ………….. This can be used for a quick preliminary estimate for a pre-feasibility study but such an estimate should not be relied on its own, because of its low inherent precision, until all other possibilities have been investigated. Every effort should be made to find suitable donor or analog sites that may be able to enhance the precision of the Qmed estimate.

This enhancement results from what is called "data transfer" from one or more donor or analog sites where such sites are available or have been identified. Selection of analog sites could be made by human judgement as to similarity of catchments or more formally by using a region of influence method where a Euclidean distance measure in catchment descriptor space is used to express similarity.

When data transfer is employed either the regression based estimate is adjusted so that it conforms with reality at a donor station on the same river as the subject site or at an analog site if there is evidence to support confidence in the use of a particular analog site.

73

Use of a regression equation on its own could also be considered to be a "data transfer" method so that these procedures are not entirely independent. Their dependencies will be considered further below.

The alternative to use of the regression method as the starting point is to calculate the average specific median annual flood at a group of analog sites, selected by a region of influence method, and then to transfer this to the subject site and scale it up by As , the catchment area at the subject site. This is a less tried and tested method of estimation although it has been used in the area of low flow estimation.

The following sections give a more formal description of the possible methods as follows (i) Use of Regression model. a. Regression on its own b. Regression equation used with donor site or with analog site (ii) Region of Influence Method as an alternative to Regression model.

A comparison of the similarities and differences between these approaches can be made with the help of Figure 9.2 below.

9.2.1 Use of Regression equation The regression model can be used on its own, section 9.2.1.1, or it can be used in conjunction with a donor station, section 9.2.1.2 below.

9.2.1.1 Regression equation on its own

In this case a regression based estimate is used namely

Log 10 Q med = c + a log 10 AREA+ r log 10 SAAR + s log 10 S1085 + ….. c a r s or Q med = 10 AREA SAAR S1085 ……….. where AREA, SAAR, S1085 …. are numerically expressed catchment descriptors and c, a, r, s, … are parameters of the linear model estimated by regression analysis in log space.

74 In such equations AREA plays a dominant role and on its own explains 2 R = 55% - 60% of the variance of log 10 Q med . When the 3 most significant variables are included about 80% of the variance is explained. Generally it is useful to include no more than a further 4 variables in which case R 2 may increase to c. 90%. The relationship between R 2 and number of variables included is demonstrated in Figure 9.1. The data from FSR (1975) is largely based on UK data, with some Irish data included. It is included here to show that the nature of the relationship is almost the same for the two eras and two countries.

FSR(1975) NUI Mayn'th(2008)

1 0.8 0.6 0.4

R R Squared 0.2 0 0 2 4 6 8 10 Number of Variables

Figure 9.1Increase in R 2 value with number of catchment descriptors included in regression equation in log domain, (i) as in FSR (1975) and (ii) in NUI Maynooth Draft WP2.3 Report (2008).

As pointed out by Nash & Shaw (1966), the use of such a regression equation is only as good as having an at-site record length of 1 i.e. Qmed ( Q in their case) estimated from a single year of gauged record would have the same precision, as expressed by standard error, as the regression estimate. This was confirmed in a simulation experiment by Hebson & Cunnane (1988). However, other authors claim that the regression estimate is as useful, from a standard error point of view, as up to 5 years of flow data.

9.2.1.2 Regression equation used with donor site Because of the large standard error associated with the regression method it is advantageous to use it in conjunction with the donor site approach. The regression

d estimate at the subject site can be modified by the ratio of observed Qmed to the regression estimate of Q med at the donor site e.g.

75  s   Q by regression  Qs = Qd med --- (9. 2) med med  d   Qmed by regression 

This method of adjustment is recommended in FEH 3, Chapter 4. It was also hinted at in FSR I, pp 342-3. If the regression model were based on the single dominant variable Area (A) alone then equation (9. 2) would be reduce to a specific flood form of adjustment

s d As Qmed = Qmed --- (9. 3) Ad

9.2.1.3 Regression equation used with analog site

In the absence of any suitable donor site and if a suitable analog site can be identified then the regression estimate can be adjusted by

 Q s by regression  Q s = Q a  med  --- (9. 4) med med  a   Qmed by regression 

Qa Q Q Where med and med by regression are the observed and calculated values of med at the analog site.

9.2.2 Region of Influence Method

Another method of obtaining Q med at ungauged site is based on a region of influence approach in which the similarity of all possible analog sites to the subject site is expressed by a Euclidean distance measure in catchment descriptor space and the 1, 2, 3, 4, 5, … closest catchments chosen on this basis as is done in regional frequency analysis. The average of the specific median flood values is then transferred to the subject site and scaled up by its catchment area As,

 m  Qi  s 1  med  Qmed = As  ∑  --- (9. 5) m i=1 Ai  where m is the number of sites included. The relationships between the ungauged catchment estimators can be seen in Figure 9.2 below.

76 Regression Equation on its own α s r Qmed,cds = cA S R …………..

Use of Regression Equation Use of Regression Equation with donor site with analog site  Q s   Q s  Q s = Q d  med ,cds  Q s = Q a  med ,cds  med med  d  med med  a   Qmed ,cds   Qmed ,cds 

Special Special case case

Use of 1 variable regression Use of 1 variable regression Equation with donor site Equation with analog site

 s α  s α s d  A  s a  A  Qmed = Qmed  d  Qmed = Qmed  d   A   A 

If α = 1 and If α = 1 m = 1 and m = 1 Analog case  m  Q j  Donor case s s 1  med  Qmed = A  ∑ j  m j=1  A 

Transfer of pooled average specific median annual flood

Figure 9.2 Relationships between different methods of estimating Q med or of enhancing a Qmed estimate for an ungauged catchment

77 9.3 Assessment of Q med estimation using donor or analog sites A total of 184 hydrometric gauging stations were identified as being potentially suitable candidates for this assessment. It was decided to limit the assessment to subject sites on rivers that have at least three gauging stations on the same river and to stations having at least 20 years of data so that se (Q med ) < 10% Q med approximately. Therefore on each river so selected there is a choice of 3 subject sites, each of which have 2 possible donor sites. There were 38 such gauging stations available located in 10 different catchments. These stations are displayed in Figure 9.3, and the stations are identified further in Table 9.1and Table 9.2.

Table 9.1 List of Rivers that have 3 or more gauging stations on them: River Name Basin Name Hydrometric Code No. of Gauging Stations Suir Suir 16 5 Barrow Barrow 14 6 Blackwater Blackwater 18 5 Erne Erne 36 3 Suck Upper Shannon 26 4 Inny Upper Shannon 26 3 Brosna Lower Shannon 25 3 Deel Shannon Estuary 24 3 Maigue Shannon Estuary 24 3 Glyde Glyde 6 3

78 Table 9.2 38 Gauging Stations used in the Q med Assessment

Station Record No Station Name River_Name Area, km2 length Obs Qmed 6014 Tallanstown Weir Glyde 270 30 21.46 6021 Mansfieldstown Glyde 321 50 21.50 6026 Aclint Lagan (Glyde) 144 46 12.30 14005 Portarlington Barrow 398 48 38.27 14006 Pass Bridge Barrow 1096 51 80.52 14018 Royal Oak Barrow 2415 51 147.98 14019 Levitstown Barrow 1660 51 102.41 14029 Graiguenamanagh Barrow 2762 47 160.74 14034 Bestfield Barrow 2060 17 117.00 16002 Beakstown Suir 512 51 52.66 16004 Thurles Suir 236 48 21.37 16008 New Bridge Suir 1120 51 92.32 16009 Caher Park Suir 1602 52 162.21 16011 Clonmel Suir 2173 52 223.00 18003 Killavullen Blackwater 1258 49 266.15 18006 Cset mallow Blackwater 1058 27 286.00 18016 Duncannon Blackwater 113 24 79.65 18048 Dromcummer Blackwater 881 23 220.00 18050 Duarrigle Blackwater 245 24 124.50 24004 Bruree Maigue 246 52 50.63 24008 Castleroberts Maigue 805 30 119.13 24011 Deel Bridge Deel 273 33 104.55 24012 Grange Bridge Deel 359 41 109.99 24030 Danganbeg Deel 248 25 52.00 24082 Islandmore Weir Maigue 764 28 140.01 25006 Ferbane Brosna 1207 52 81.91 25011 Moystown Brosna 1227 51 82.02 25124 Ballynagore Brosna 254 18 13.65 26002 Rookwood Suck 626 53 56.56 26005 Derrycahill Suck 1050 51 93.21 26006 Willsbrook Suck 182 53 24.23 26007 Bellagill Suck 1184 53 88.15 26021 Ballymahon Inny 1071 30 66.34 26058 Ballinrink br. Inny upper 59 24 5.35 26059 Finnea br. Inny 249 23 12.20 36011 Bellahillan Erne 318 49 18.23 36012 Sallaghan Erne 263 47 14.12 36019 Belturbet Erne 1501 47 89.95

79

´

36019 (! 36011 (! 06026 36012 (!(! (!06021 (! 06014 26059 26006 (! (!26058 (!

26002 26021 (! (! 26005 ! 25124 ( 26007 (! (! 25011(! (! 25006 14005 14006 (! (!

14019 (! 14034 (!

14018 16004 16002 (! (!(! 24008 14029 (! 24082 (! 24012 (! 16008 (! (! 24030 24004 (! 24011(! (! 16009 16011 (! (!

18006 18050 ! 18016 (! (! (18003 (! (! 18048

Legend

(! subjectsite areas

Figure 9.3 Location of 38 potential subject sites

80 9.3.1 Tests on Qmed estimates via donor and analog catchments

The approach used here to estimate Q med at ungauged sites consists of three steps: • selection of appropriate donors and analogues for each subject site,

• determination of Q med and catchment descriptors at the donor or analog site

• calculation of adjusted Q med for the subject site using data of each donor and analog site. Simultaneous use of multiple donor sites has not been considered.

The donor site selection technique uses the nearest gauge on the same stream as the subject site. The 2 nd nearest, 3 rd nearest etc. in the upstream and downstream directions are also examined where such sites exist. The donor site selection methods examined are the • up-stream site(s), • down-stream site(s) and

Analog sites, analogs are selected according to the similarity of physiographic catchment characteristics. For this component of the study Catchment area, SAAR and BFI have been used to find the similarity between subject sites and possible analog sites. For the Area based adjustment catchment area alone is used for identifying an analog station.

For each subject site and donor/analog site combination two methods for determining

Qmed are examined here.

The first method, adopted from FEH, adjusts the Qmed at the donor / analog site by the ratio of the regression estimates of Qmed at the subject site and donor/analog sites that is use of equation (9. 2).

In order to apply this method a regression model has been developed between log Qmed and log catchment descriptor values, independently of any work being carried out in Project WP2.3 , as WP2.3 Work was being carried out in parallel when this work was being done.

81 Data from 164 sites are used for the calibration of the regression model. Three catchment descriptors namely Catchment Area, SAAR and BFI were used in this model to find the regression equation for use in this part of the study and these descriptors are also used in the pooling analysis to pool the data when selecting catchments for the analog with regression based adjustment. The resulting regression model is found to be:

0.829 0.898 -1.539 Q med = 0.000302 × AREA × SAAR × BFI --- (9. 6)

which explains 82% of the total variance in lnQ med with RMSE = 0.4725 in the ln 0.205 domain and RMSE = 0.205 in the log 10 domain and hence FSE = 10 = 1.60. (In WP2.3 the same 3 variables are reported to account for 83% of the variance), the difference being due to the different basic data set used in each case.

Figure 9.4 shows the scatter plot between predicted Qmed using this regression equation vs observed Qmed for the 35 stations used in the current tests (the other 3 stations are not used due to unavailability of BFI catchment descriptors) with a coefficient of determination of 80% (which, as expected is similar to the value of 82% found when the equation was derived).

The second method adjusts the Q med obtained at the donor/analog site by scaling it by the ratio of catchment areas of the subject site and the donor/analogue site that is use of equation (9. 3). The second method is a subset of first method and can be expected to be less effective.

82 300 Regression y = 0.8169x R2 = 0.8033 250

200

150 Qmed_pred 100

50

0 0 50 100 150 200 250 300 350 400 Qm ed_obs

Figure 9.4 predicted Qmed by regression vs observed Qmed . Each point represents a catchment , 35 in total as all catchment descriptors were unavailable for 3 of the 38 sites

9.3.2 Results

The results are considered in two subsections, one deal with details at individual subject sites and one which amalgamates the results and examines them collectively.

9.3.2.1 Results at individual subject sites

The results are provided in

Appendix 2 and the details for Site 14006, River Barrow at Pass Bridge are demonstrated here. This station has one donor station located upstream of it, namely 14005 River Barrow at Portarlington, and 4 potential donor stations downstream of it, namely 14019 (Levitstown), 14018 (Royal Oak), 14034 (Bestfield) and 14029 (Graiguenamanagh) in order of increasing catchment area. Station 14034 is not used in this case – as all of the catchment descriptors for it are not available. Figure 9.5 and Figure 9.6 show in tabular and graphical form the estimates of Qmed at station 14006 for both area based adjustment and regression based adjustment, where Donor_up means station 14005, Donor_down_1 means Station 14019, Donor_down_2

83 means Station 14018 and so on. The estimated Qmed at 14006 for each case is shown in the 2 nd last row of the each table and this value expressed as a percentage of observed Qmed at 14006 is shown in the last row. These latter percentages are shown in graphical form also where the red line represents 100%. Summaries of these percentages for the 4 donor catchment based estimates are shown in Table 9.3. These show that the regression based adjustment provides more reliable estimates than the area based adjustment although the average performance is comparable in magnitude between the two, 21% and 18%. Both donor methods are better, in this case, than the estimate obtained from the catchment descriptor based regression equation alone (9. 6).

Table 9.3 Estimates from donor catchments – summary of minimum, maximum and average percentage difference from observed Qmed.

Regression based adjust'nt Area based adjustment Minimum difference 3% -16% Maximum difference 23% 31% Average abs. difference 18% 21% Difference using catchment descriptors +31% (Absolute +31% (Algebraic Regression only value) value)

Use of analog catchments is much more subjective and the choice of catchment greatly affects the performance measure of the Qmed estimate . For the Area based adjustment analog sites have been chosen by a Euclidean distance which is based on Area alone. On the other hand for the Regression based adjustment analog sites have been chosen by a Euclidean distance which is based on Area, SAAR and BFI.

84

Regression Adjustment Subject_site Donor_up Donor_down1 Donor_down2 Donor_down3 Analog Regression St.No. 14006 14005 14019 14018 14029 24008 Obs Qmed 80.5 38.3 102.4 148 160.7 119.1 Area,km2 1096 398 1660 2415 2762 806.04 Adj.Qmed 80.5 62.4 82.7 98.7 98.7 130.3 105.2 %obs Qmed 100 78 103 123 123 162 131

Minimum difference via donors +3% Maximum difference via donors +23% Average abs. difference via donors 18% Difference from regression estimate +31%

14006 Barrow @ Pass Bridge, Area=1096 km2 (Regression Adjustment)

180 806.04 160

140 2415 2762 120 1660 100 398 80

%obsQmed 60

40

20

0 Donor_up Donor_down1 Donor_down2 Donor_down3 Analog Regression Data Transfer Category

Figure 9.5 Performance of data transfer method of estimating Qmed at Station 14006, Barrow at Pass Bridge, using regression based adjustment. The figures above the boxes are catchment areas. The final figure labelled Regression refers to estimate obtained by eq (10).

85 Area Adjustment Subject_site Donor_up Donor_down1 Donor_down2 Donor_down3 Analog Regression St.No. 14006 14005 14019 14018 14029 30012 Obs Qmed 80.5 38.3 102.4 148 160.7 126 Area km2 1096 398 1660 2415 2762 1075.4 Adj.Qmed 80.5 105.4 67.6 67.2 63.8 128.4 105.2 %obs Qmed 100 131 84 83 79 159 131

Minimum difference via donors -16% Maximum difference via donors +31% Average abs. difference via donors 21% Difference from regression estimate +31%

14006 Barrow @ Pass Bridge, Area=1096 km2 (Area Adjustment) 180 1075.4 km2 160

140 398 km2

120

100 1660 km2 2415 km2 2762 km2 80 %obs Qmed 60

40

20

0 Donor_up Donor_down1 Donor_down2 Donor_down3 Analog Regression Data Transfer Category

Figure 9.6 Performance of data transfer method of estimating Qmed at Station 14006, Barrow at Pass Bridge, using area based adjustment.

9.3.2.2 Results from all donor sites collectively

Results are assessed in 2 ways, one being as in Table 9.3 above, see Table 9.4 below, and one in which Qmed estimated by each method is plotted against observed Qmed, see Figure 9.7 and Figure 9.8 below.

86

Table 9.4 Qmed estimates from donor catchments – minimum, maximum and average % difference form observed Qmed

Diff. from Subject Min diff% Max diff% Avr. abs. diff% Regression Site Reg Area Reg Area Reg Area Actual Abs ST14005 29 -24 58 -41 44 35 +69 69 ST14006 3 -16 23 31 18 21 +31 31 ST14019 -3 -1 -25 56 16 18 +27 27 ST14034 2 69 24 N/A ST14018 0 1 -37 57 18 18 +7 7 ST14029 0 -2 -37 65 18 21 +7 7 ST18016 2 -28 -17 -79 7 58 -32 32 ST18050 0 38 -19 -71 7 52 -33 33 ST18048 20 8 29 182 24 84 -18 18 ST18006 -4 -8 -22 161 10 76 -36 36 ST18002 0 69 -19 378 7 194 -33 33 ST25124 10 24 23 26 17 25 +13 13 ST25006 -9 -1 12 -21 11 11 +3 3 ST25011 -10 2 -19 -20 15 11 -8 8 ST24030 12 46 38 83 25 64 -18 18 ST24011 -19 -20 -28 -45 23 33 -41 41 ST24012 -11 25 23 -32 17 28 -27 27 ST36012 7 12 9 N/A ST36011 39 5 39 -6 39 5 +41 41 ST36019 -28 -4 -28 -10 28 7 +1 1 ST6026 2 -7 2 -22 2 14 +40 40 ST6014 0 7 0 -16 0 12 +38 38 ST6021 19 27 23 N/A ST26058 -12 -32 27 -46 20 39 +22 22 ST26059 14 26 45 85 29 56 +39 39 ST26021 -21 -21 -31 46 26 34 -4 4 ST24004 5 -11 23 -28 14 20 -15 15 ST24008 -5 24 17 39 11 31 -19 19 ST24082 -14 12 -19 -19 17 16 -31 31 ST26006 -1 -32 9 -44 5 37 +35 35 ST26002 -3 -2 5 47 4 22 +30 30 ST26005 1 2 11 50 6 23 +37 37 ST26007 5 21 10 79 8 40 +24 24 ST16004 26 -9 74 14 52 12 +41 41 ST16002 1 0 -35 -20 17 8 -8 8 ST16008 -21 10 38 25 26 20 +12 12 ST16009 -1 1 -35 -19 17 8 -9 9 ST16011 -11 0 -42 -20 23 8 -19 19

Average 0 3 3 26 18 32 5 25 St dev'n 14 21 31 82 12 33 29 15

The values in Table 9.4 show that the regression based adjustment is more reliable. This is to be expected since the 3 variable regression equation accounts for 82% of the

87 variance of lnQ med while area accounts for only 55%-57% of that variance. While the average minimum differences of 3 and 0 are quite similar the standard deviations of 21 and 14 respectively show that the regression based method is more reliable in terms of the minimum differences. In terms of maximum differences the regression based method is clearly superior both in average value and standard deviation and the same applies to average absolute differences. These performances can be judged relative to the result of using an estimate based on regression on catchment descriptors alone, equation (9. 6), where the mean absolute difference is 25(15) in comparison to 32(33) and 18(11) for area based and regression based adjustment respectively, where the figures in brackets are the relevant standard deviations.

This one measure alone confirms that the catchment descriptor based regression formula estimate is better than the area based adjustment donor estimate. This happens probably because the variance in Qmed is better explained by the 3 catchment descriptors (Catchment Area, SAAR and BFI) rather than the by Catchment Area alone, see Figure 9.1. Area based adjustment can be considered as a special case of regression based adjustment (see Figure 9.2) where only Catchment area has been used in the regression equation and hence would be expected to be less effective than the regression on catchment descriptors.

Qmed values obtained by each method have also been compared to the observed Qmed values in Figure 9.7 which shows the scatter plot between predicted Qmed vs observed Qmed when the regression based method has been used for the adjustment of Qmed at the subject site. Each point represents a catchment and the 3 plots in Figure 9.7 relate to different data transfer methods. The downstream donor selection method performs best with a very good coefficient of determination of 95% followed by upstream donor selection method (91%) and analogue selection method (79%).

Figure 9.8 shows the corresponding results as Figure 9.7 for the case where the Area based method is used in the data transfer procedure. These results show that the downstream donor selection method performs best with the coefficient of determination of 78% while in case of analogue selection method it performs worst with a low coefficient of determination of 3%. The other method, upstream donor selection also

88 performed well with coefficients of determination of 77%. It is quite noticeable that with this adjustment technique the coefficient of determination for analogue selection method is smaller compared to the case of regression based transfer.

It should be noted that fewer data points i.e. subject sites are used in Figure 9.7 than in Figure 9.8 due to unavailability of catchment descriptor values for some subject sites at the time that this part of the study was conducted. If the same donor sites were used in Figure 9.8 as in Figure 9.7 the corresponding coefficients of determination would then be 76%, 77% and 3% for upstream selection, downstream selection and analogue selection method respectively.

89 700 Regression based adjustment Upstream site 600 y = 0.9551x R2 = 0.9169 500

400

300 Qmed_pred

200

100

0 0 100 200 300 400 500 600 700 Qmed_obs

700 Downstream site 600

500 y = 1.0608x R2 = 0.9515 400

300 Qmed_pred

200

100

0 0 100 200 300 400 500 600 700 Qmed_obs

700 Analog site 600 y = 0.957x R2 = 0.7908 500

400

300 Qmed_pred

200

100

0 0 100 200 300 400 500 600 700 Qmed_obs

Figure 9.7 Adjusted Qmed (Regression method) vs observed Qmed. Each point represents a catchment and panels relate to different transfer method

90 700 Area based adjustment Upstream site 600

500 y = 1.2819x R2 = 0.7719 400

300 Qmed_pred

200

100

0 0 100 200 300 400 500 600 700 Qmed_obs

700 Downstream site 600

500 y = 0.8568x R2 = 0.7816 400

300 Qmed_pred

200

100

0 0 100 200 300 400 500 600 700 Qmed_obs

700 Analog site 600

500 y = 0.8098x R2 = 0.0291 400

300 Qmed_pred

200

100

0 0 100 200 300 400 500 600 700 Qmed_obs

Figure 9.8 Adjusted Qmed (Area method) vs observed Qmed (m 3/s). Each point represents a catchment and each plot relate to different area based data transfer methods

91 9.3.3 Conclusions:

1. The comparisons show that the downstream selection method performs best in both adjustment techniques. 2. Upstream selection method also performs well and is better than analogue site selection method. 3. Table 9.4 shows a substantial improvement in using regression based adjustment over using area based adjustment in terms of average minimum difference between observed Qmed and adjusted Qmed. Likewise in terms of average maximum difference and in terms of average of the average difference measured at the 38 sites. These differences are also apparent in Figures 8 and 9. In the case of analogue sites the regression adjustment transfer method performs much better than the Area adjustment method. This happens probably because the variance in Qmed is better explained by the 3 catchment descriptors (Catchment Area, SAAR and BFI) rather than the Catchment Area alone, see Figure 9.1. Area based adjustment can be considered as a special case of regression based adjustment (see Figure 9.2 and equation (9. 2)) where only catchment area has been used with exponent α =1 in the regression equation and hence would be expected to be no more effective than the regression on catchment descriptors alone.

9.4 Summary of Q med estimation and associated error

• At site data available: • Qmed can be estimated from the at-site data and the corresponding standard error is approximately 0.363 Qmed/ √N. • At site data not available (ungauged catchment case): • Qmed estimated from catchment descriptor based formula. i. If this estimate is as good as only one year of data, as described

by Nash & Shaw (1965), then se(Qmed) = σQ, the standard deviation of the flood population ≈ 0.28 µ ≈ 0.29 Qmed.

92 ii. If this estimate is as good as, say, 4 year's data then

se(Qmed) = σQ/√4 = 0.5 σQ = 0.5 x 0.28 µ = 0.14 µ ≈ 0.145 Qmed.

iii. The factorial standard error, FSE, of such estimates varies from 1.47 (in FSR) to 1.55 (in FEH) with WP2.3 giving 1.36. These

give lower 67% confidence limits of between 65% Q med and

70%Q med and upper 67% confidence limits of between 1.44 Qmed and 1.55 Qmed.

(i) and (iii) should be compatible since they are derived from the same basic information, namely the rmse ( see ) of the regression residuals. If se(Qmed) = 0.29Qmed then Qmed – 1 se(Qmed) = 0.71Qmed which could taken as equalling the 70% Qmed in (iii).

• If a donor catchment is used the analysis described above gives some indication of the average estimation error as follows from Table 9.4: i. Using regression based data transfer: Abs(Av. abs diff) = 18% ii. Using area based data transfer: Abs(Av. abs diff) = 32% iii. The corresponding values using regression formula alone = 25% iv. The corresponding values for transfer from a single analog catchment are 30 % and 69% respectively using regression based and area based data transfer. Note that these are average values of differences from a finite number of cases and are not directly comparable to standard error values which are expected root mean square values rather than average values.

93 10 Determining the Growth Factor X T

10.1 Introduction

The growth factor X T is the factor which when multiplied by the index flood Q I gives the flood magnitude of return period T, Q T, as in equation (8. 2) : QT = Q I. X T

The relationship between X T and T is often referred to as the growth curve. In this study Q med is used as the index flood.

In this Section two methods of determining the growth factor X T and the resulting Q T are outlined. The effects of catchment type (Area, Storage amount and Peat content),

Geographical Location and Period of Record on X T and on Q T are examined. Also the effectiveness of different combinations of catchment descriptors in defining the distance measure on which region of influence pooling groups are formed is examined.

10.2 At-site and Pooling Group Estimates of X T

If a subject site coincides with a gauging station and if a sufficiently long record of AM floods exists at the station then a suitable distribution can be fitted to the data of that site for direct estimation of Q T. Here the word “direct” means that Q T can be obtained without going through the intermediate steps of determining Q med and X T.

What constitutes a “sufficiently long record” is arguable. In FSR (1975) the criterion N > 0.5T was adopted, where N is available record length and T is required return period whereas FEH (1999) adopted the much more stringent criterion of N > 2T. If the record is not sufficiently long the standard error of estimate of Q T, se (Q T) is large.

Adopting a pooling group estimate of X T, and hence of Q T, reduces the value of se (Q T). Obviously the choice of using a single site estimate or pooling group estimate involves a trade-off between the convenience of a single site estimate and the extra work involved in assembling and analysing data for a pooling group estimate as well as adoption of the assumption of pooling group homogeneity.

Another question arises as to what constitutes a suitable distribution for at-site estimation. Generally it is better to adopt a 2-parameter distribution than a 3-parameter

94 distribution because se(Q T) is much larger in the latter case even though bias may be reduced. Some guidance on choice of 2-parameter distributions for Irish data is available from Chapter 2, where EVI and lognormal distributions are seen overall to be better descriptors of gauged AM data than the 2 parameter logistic distribution. However there are many samples of data which seem not to be able to be described by any 2 parameter distribution either through extreme concave or convex upward curvature or an S-shape on a probability plot. In the case of these latter samples perhaps this is a genuine permanent departure from 2 parameter behaviour or it may merely be a period of record effect i.e. the next N years may not exhibit this behaviour.

One approach might be to avoid using an at-site estimate at a station where the sample shows a strong departure from 2 parameter behaviour and to adopt a pooling group estimate instead, on the basis that this reflects an average type of Q-T behaviour for the type of catchment in question. Note also that probability plots of random samples drawn from a 2-parameter distribution can display non-linear behaviour in a small proportion of cases (see Figure 4.1).

10.3 Pooling Groups

While X T can be estimated at a gauged site from the at-site record by conventional flood frequency analysis methods it is generally accepted that an advantage can be gained, in terms of reduction of standard error of estimate of Q T, if the assumption is made that X T does not vary between gauging stations that belong to a homogeneous pooling group. A pooling group of gauging stations is said to be homogeneous if the same growth curve applies to all stations in the group (or even more strictly if the same value of X T, for stated T, can be applied to all stations in the group). With this assumption a space – time trade-off is achieved e.g. n years of record at each of m stations is assumed to be equivalent to nm years of record at a single site. Hence if X T is estimated from the nm values (after they have been suitably standardised e.g. by division by their respective

Qmed values) it will have smaller standard error of estimate than an X T value obtained from an at-site estimate, the reduction being of the order of m .

95 Pooling groups can be formed by using geographical regions but FEH (1999, III, Fig. 16.5) found that such pooling groups were less homogeneous than those formed by a region of influence approach of the type proposed by Burn (1990). The region of influence approach selects stations, which are nearest to the subject site in catchment descriptor space, to form the pooling group for that subject site. For instance in FEH (1999) the distance between subject site i and any other site j is taken to be

ln AREA −ln AREA 2 ln SAAR −ln SAAR 2  BFIHOST − BFIHOST 2 1 i j   i j   i j  dij =   +  +  --- (10. 1) 2 σln AREA   σln SAAR   σ BFIHOST 

In choosing a pooling group and distance measure d ij a decision has to be made about how many and which catchment descriptors are to be included in the distance measure (3 in the above case) and what weightings are to be applied to them and whether logarithms or other transformations are to be used. In addition a decision has to be taken about how many stations are to be included in the pooling group. FEH (1999) investigated a range of pooling group sizes and decided on adoption of the 5T rule, namely that the total number of station years of data to be included when estimating the T year flood should be at least 5T. Adoption of such a rule is a compromise. If too few stations are included precision of the Q T estimate is sacrificed whereas if far too many stations are included the assumption of homogeneity may be compromised. Hosking & Wallis (1997) however show that a small departure from homogeneity can be tolerated so that having too few stations included may be less desirable than having slightly too many.

In this study the distance measure

2 2 2  ln AREA i − ln AREA j   ln SAAR i − ln SAAR j   ln BFI i − ln BFI j  dij =   +   +   --- (10. 2)  σ ln AREA   σ ln SAAR   σ ln BFI  has been adopted. Choice of AREA and SAAR follows from the FEH case, combined with the knowledge that both Area and SAAR play a major role in regression studies of relations between Q med and catchment descriptors. Measured BFI is used as the third variable because it has been found to be useful in regression studies of relations between

Qmed and catchment descriptors and also between some hydrograph shape parameters and catchment descriptors.

96

In this study also the 5T rule, for determining which stations to include in the X T estimation, has been used. It is recognised that in some practical situations the 5T rule cannot always be implemented.

10.4 Growth curve estimation

Growth curves are expressed as algebraic equations for X T such as equations (10. 5) and

(10. 8) below. Suppose there are m gauging stations in a particular group. Let Q i , i =

1, 2 … n i be the recorded annual maximum floods at the ith of m sites.

Probability weighted moments M 1jo are calculated at each site and the dimensionless L- moment ratios L-CV = t 2 and L-Skewness = t 3 are calculated for each site.

The regional weighted average value of each of these is calculated as

m m (i) t2 = ∑ wi t 2 / ∑ wi --- (10. 3) i=1 i=1

m m (i) t3 = ∑ wit3 / ∑ wi --- (10. 4) i=1 i=1

(i) (i) where t2 and t3 are the values of t 2 and t 3 at the ith site and w i is a weighting factor.

If equal weights are given to all stations then w i = 1 for all i and Σ w i = m. If t values are weighted according to length of record then w i = n i and Σ w i = total number of station years of data in the pooling group.

Note that the dimensionless 1 st L-Moment is unity.

A distribution's parameters can be estimated from the dimensionless L –Moments,

 from 1 and L-Cv in the case of a 2 parameter distribution, such as EV1 or LO

97  from 1, L – Cv and in the case of a 3 parameter distribution, such as L-skewness GEV or GLO

 from 1, L-CV, L-Skewness in the case of a 4 parameter distribution, such as the & L-Kurtosis kappa.

In pooled regional flood frequency estimation the values t2 , t3 , and t 4 if required, are equated to expressions for these quantities written in terms of the distribution's unknown parameters (when the distributions growth curve is expressed in dimensionless form) and the resulting equations are solved for the unknown parameters.

A three parameter distribution can be used for a pooling group because the extra data involved in the estimation ensures that the resulting standard error is smaller than in the at-site or single sample case. Further the three parameter distribution avoids any possible bias resulting from using a fixed shape 2 parameter distribution when in fact a 3 parameter distribution is more appropriate --- see (d) in Figure 10.1.

In this study the GEV and its special case EV1 and GLO and its special case LO forms of growth curve are used. The dimensionless GEV growth curve is defined by a shape factor k and a scale parameter β as follows:

β  T  x = 1+ (ln )2 k −(ln ) k  , k ≠ 0 --- (10. 5) T k  T −1 

The parameters k and β are estimated from sample t 2 = L-CV and sample t 3 = L- skewness, as follows Hosking & Wallis (1997, p 196)

2 ln 2 k = 7.8590c + 2.9554c 2 where c = - --- (10. 6) 3 + t3 ln 3

kt 2 β = k −k --- (10. 7) t2 ()Γ 1( + k) − (ln )2 + Γ 1( + k)( 1− 2 )

98 The EV1 growth curve is defined as:

xT = 1+ β (ln(ln )2 − ln( − ln( 1− /1 T )) ) --- (10. 8) t β = 2 --- (10. 9) ln 2 − t2 []γ + ln(ln )2 where, t2 is the L-coefficient of variation (L-CV) and γ is Euler’s constant = 0.5772.

The dimensionless GLO growth curve is defined by a shape factor k and a scale parameter β as follows:

β k β −k xT = 1+ (1−{}()1− F / F ) = 1+ (1−{}T −1 ), k ≠ 0 --- (10. 10) k k

The parameters k and β are estimated from sample t 2 = L-CV and sample t 3 = L- skewness, as follows Hosking & Wallis (1997, p 197)

kt 2 sin( πk) k = - t 3 and β = --- (10. 11) kπ (k + t 2 ) − t 2 sin( πk)

The LO growth curve is defined as :

xT = 1+ β ln( T − )1 --- (10. 12)

β = t2 --- (10. 13) where, t2 is the L-coefficient of variation (L-CV)

99

FIXED SKEWNESS MODELS (usually with two parameters)

(a) Model Skewness < Parent (b) Model Skewness > Parent Skewness Skewness Hence negative bias i.e. Hence positive bias i.e. overestimation at large underestimation at large T T

1 1

0 T 0 T

- - 1 1

VARIABLE SKEWNESS MODELS (usually with > two parameters)

(c) At-site use (d) Pooled X T estimation + At-site Qmed Small bias, Small bias large se and small se

1 1

0 T 0 T

- - 1 1

T

10 100 10 100 T

Bias/Q T%

[ Bias ± 1.96 se ]/Q T %

Figure 10.1 Simplified qualitative outline of simulation results about flood quantile estimates. (a) and (b) show the effect of choice of wrong distribution having fixed but incorrect skewness. (c) and (d) show the effect of using a flexible distribution i.e. low bias and large standard error when used in at-site mode but with much reduced se as well as low bias when us in at-site +pooling group mode. (Adapted from Figure 5.1 in WMO ,1989).

100 11 Effect of catchment type and period of record on X T and

QT

11.1 Data and Data Screening

Data of 88 A1 and A2 category stations which have BFI values available were used in this part of the study.

Very unusual or discordant data sets have been excluded using t he Hosking Wallis discordancy measure:

1 D = N()u − u T A −1 ()u − u --- (11. 1) i 3 i i where ui is a vector containing the L-moment ratios for station i, namely the L-Cv (t),

L-Skewness (t 2), and L-Kurtosis (t 3), u is the unweighted regional average and A is the matrix of sums of square and cross products.

The D values are shown in Table 11.1 where 3 values are seen to be greater than the discordant cut-off value of 3. The data of these stations have been examined to and possible reasons for their discordancy are listed in Table 11.2. The EV1 probability plots of these data are shown in Figure 11.1. These stations were excluded from further analysis for this part of the study even though it can be argued that this reduces the natural variability within the data set being studied.

101 Table 11.1 Stations with site characteristics, L-moment ratios and discordancy measure

StationNo RL AREA SAAR BFI FARL L-Cv L-Skew L-Kur Discordancy 6011 48 229.19 1028.98 0.71 0.87 0.113 0.090 0.074 0.438 6013 30 309.15 873.08 0.62 0.97 0.157 0.019 0.012 0.971 6014 30 270.38 927.45 0.63 0.93 0.149 0.224 0.121 0.572 6026 46 148.48 940.87 0.66 0.92 0.177 0.240 0.092 1.019 6031 18 46.17 930.66 0.56 1.00 0.258 0.392 0.317 1.978 6070 24 162.02 1046.28 0.73 0.83 0.145 0.135 0.143 0.019 7006 19 177.45 936.67 0.55 0.99 0.118 -0.205 0.074 2.462 7009 29 1658.19 868.55 0.71 0.99 0.211 0.206 0.106 0.738 7033 25 124.94 1032.22 0.44 0.89 0.129 0.162 0.272 0.721 8002 20 33.43 791.12 0.60 1.00 0.112 0.271 0.175 0.859 8005 18 9.17 710.76 0.52 1.00 0.378 0.228 0.189 4.715 9001 48 209.63 783.26 0.51 1.00 0.243 0.191 0.146 0.826 9002 24 34.95 754.75 0.67 1.00 0.417 0.390 0.253 6.261 9010 19 94.26 955.04 0.56 0.96 0.423 0.416 0.305 6.855 10021 24 32.51 799.07 0.65 1.00 0.213 0.186 0.068 1.100 10022 18 12.94 821.92 0.60 1.00 0.233 0.057 0.061 1.275 12001 50 1030.75 1167.31 0.72 1.00 0.194 0.176 0.171 0.193 14005 48 405.48 1014.90 0.50 1.00 0.149 0.291 0.223 0.535 14006 51 1063.59 899.07 0.57 1.00 0.106 0.238 0.212 0.549 14007 25 118.59 814.07 0.64 1.00 0.170 0.219 0.072 1.125 14009 25 68.35 831.24 0.67 1.00 0.125 0.136 0.242 0.473 14011 26 162.30 806.97 0.60 1.00 0.143 0.020 0.157 0.423 14018 51 2419.40 857.46 0.67 1.00 0.138 0.036 0.064 0.389 14019 51 1697.28 861.46 0.62 1.00 0.136 0.092 0.126 0.052 14029 47 2778.15 876.50 0.69 1.00 0.078 0.058 0.067 0.733 15001 42 444.35 935.24 0.51 1.00 0.159 0.003 0.075 0.488 15003 50 299.17 933.86 0.38 1.00 0.111 -0.146 0.087 1.618 16001 33 135.06 916.42 0.61 1.00 0.124 0.012 0.125 0.294 16002 51 485.70 932.15 0.63 1.00 0.156 0.172 0.191 0.050 16003 51 243.20 1192.01 0.55 1.00 0.099 0.207 0.054 1.851 16004 48 228.74 941.36 0.58 1.00 0.111 0.069 0.108 0.190 16005 30 84.00 1153.57 0.56 1.00 0.098 0.204 0.133 0.731 16008 51 1090.25 1029.63 0.64 1.00 0.072 -0.062 0.055 0.939 16009 52 1582.69 1078.57 0.63 1.00 0.099 -0.096 0.052 1.014 16011 52 2143.67 1124.95 0.67 1.00 0.173 0.089 0.053 0.618 18004 46 310.30 985.41 0.68 1.00 0.087 0.043 0.333 2.891 18005 50 378.47 1190.37 0.71 1.00 0.142 0.224 0.182 0.213 19001 48 103.28 1175.68 0.64 1.00 0.095 0.095 0.119 0.284 19020 28 73.95 1179.07 0.66 1.00 0.196 0.029 0.037 0.987 23001 45 191.74 1084.01 0.32 1.00 0.180 0.128 0.181 0.181 23012 18 61.63 1264.00 0.46 1.00 0.146 0.386 0.326 1.650 24008 30 806.04 939.47 0.54 1.00 0.151 0.055 0.092 0.223 24022 20 41.21 942.31 0.53 1.00 0.233 0.123 0.164 0.809 24082 28 762.84 941.70 0.52 1.00 0.152 -0.036 0.130 0.804 25006 52 1162.76 931.99 0.71 0.96 0.137 0.140 0.183 0.040 25014 54 164.42 1007.65 0.67 1.00 0.128 0.103 0.125 0.071 25016 42 275.17 947.09 0.61 1.00 0.125 0.082 0.155 0.080 25023 52 113.86 922.49 0.65 1.00 0.163 0.142 0.062 0.679

102 25025 31 161.20 904.54 0.73 1.00 0.167 0.083 0.146 0.143 25027 43 118.87 1021.15 0.65 1.00 0.158 0.046 0.126 0.219 25029 33 292.67 1108.68 0.58 1.00 0.139 -0.020 -0.020 1.367 25030 48 280.02 1183.81 0.54 0.85 0.178 0.185 0.132 0.235 25034 24 10.77 968.60 0.76 1.00 0.166 -0.080 0.115 1.358 25040 20 28.02 989.64 0.64 1.00 0.154 0.204 0.181 0.100 25044 33 92.55 1186.86 0.58 1.00 0.188 0.248 0.149 0.484 25124 18 215.45 954.84 0.87 0.78 0.200 -0.069 0.231 2.954 26002 53 641.45 1067.03 0.61 0.98 0.111 0.224 0.293 0.968 26005 51 1085.38 1054.40 0.56 0.98 0.101 0.035 0.142 0.282 26006 53 184.76 1120.64 0.54 0.97 0.152 0.397 0.412 2.833 26007 53 1207.22 1045.62 0.65 0.98 0.107 0.164 0.157 0.244 26008 49 280.31 1035.47 0.61 0.86 0.104 0.193 0.190 0.347 26018 49 119.48 1043.90 0.72 0.76 0.114 0.126 0.113 0.248 26019 51 252.96 979.62 0.54 0.99 0.139 0.234 0.121 0.715 26021 30 1098.78 945.25 0.83 0.81 0.140 -0.112 0.187 2.475 26022 33 61.88 915.82 0.58 1.00 0.173 0.091 0.047 0.695 26059 17 256.64 976.44 0.91 0.73 0.099 0.111 0.175 0.218 27001 30 46.70 1476.89 0.28 0.99 0.105 0.193 0.248 0.557 27002 51 564.27 1336.35 0.70 0.84 0.122 0.181 0.216 0.223 29004 32 121.44 1107.47 0.52 0.99 0.083 0.144 0.085 0.942 29011 22 354.14 1079.37 0.63 0.98 0.145 0.410 0.318 1.811 30007 31 469.90 1115.06 0.65 0.99 0.107 0.112 0.203 0.281 30061 33 3136.08 1422.43 0.78 0.66 0.147 0.410 0.379 2.383 31002 26 71.35 1530.25 0.53 0.63 0.129 0.308 0.178 1.030 32012 24 146.16 1784.36 0.59 0.84 0.071 0.001 0.181 0.902 34001 36 1974.76 1322.66 0.78 0.83 0.104 0.080 0.211 0.481 34003 29 1802.38 1339.66 0.80 0.82 0.093 0.114 0.253 0.854 34009 33 117.11 1256.71 0.40 1.00 0.097 0.081 0.158 0.220 34018 27 95.40 1554.59 0.66 0.73 0.114 0.177 0.033 1.725 34024 28 127.23 1177.46 0.52 0.92 0.066 -0.051 0.140 1.013 35001 29 299.45 1172.84 0.60 0.92 0.119 -0.009 0.190 0.923 35002 34 88.82 1380.56 0.42 0.99 0.098 0.026 0.081 0.405 35005 55 639.66 1198.32 0.61 0.90 0.144 0.205 0.143 0.271 35071 30 247.21 1364.46 0.77 1.00 0.100 0.118 0.197 0.285 36015 33 153.06 1090.72 0.42 0.96 0.159 0.325 0.299 0.987 36018 50 234.40 950.12 0.69 0.85 0.104 0.045 0.100 0.264 36019 47 1491.76 971.21 0.79 0.76 0.104 -0.029 0.064 0.579 36021 27 23.41 1569.64 0.27 1.00 0.119 0.201 0.222 0.300 36031 30 63.77 910.43 0.48 0.96 0.098 0.365 0.391 2.885

103

Table 11.2 Four stations out of 94 showing large discordancy values River Station Name D Value Observation on discordancy 8005 Sluice 6.16 Large Cv of 0.6 9002 Griffeen 6.26 Large Cv of 0.8 9010 Dodder 8.94 Outlier due to Hurricane Charlie

9010 RIVER DODDER @ W ALDRON'S BRIDGE 8005 RIVER SLUICE @ KINSALEY HALL 300 10 250 EV1 8 200 w inter peak 6 w inter peak summer peak 150 summer peak 4 100 AMF(m3/s) AMF(m3/s) 2 50 2 5 10 25 50 100 500 0 0 2 5 10 25 50 100 500 -2-10 1 2 3 4 5 6 7 EV1 y -2-10 1 2 3 4 5 6 7 9002 RIVER GRIFFEEN @ LUCAN 30 EV1 y 25 20 w inter peak 15 summer peak 10 AMF(m3/s) 5 0 2 5 10 25 50 100 500 -2-10 1 2 3 4 5 6 7 EV1 y

Figure 11.1EV1 probability plot for the discordant stations

11.2 Effect of catchment type on pooled growth curve estimates.

Catchment have been divided into subgroups on the basis of • catchment storage as measured by FARL values • peat content of catchment as measured by % Peat value • size of catchment as measure by area • location of catchments in one of 4 regions (see in Figure 11.2)

In all cases the effect on X 100 has been assessed for simplicity rather than try to compare entire growth curves simultaneously. The exception is that X 50 is used in the case of estimates from decadal data because of the obvious smaller amount of data available.

104 The growth curve ordinate X 100 has been estimated for each of the 85 gauging stations from its own pooling group based on the distance measure defined in equation (10. 2) and using the 5T rule for number of data included in each. Estimates were obtained by both GEV and GLO estimating equations.

Legend (! Gauging station ´

(! SHANNON

(! (! (! (! (! (! (! (! (! ! (! (! (! ( (! (! (! (! (! (! (! (! (! (! (! (! (!(! (! (! (! ! ! WEST (! ( ( (! (! (! (! (! (! (! (! (!! (! ! ( (! (! (! ( (! (! ! (! ( ! EAST ( (! (! (! (! (!(! (! (! (!(! (! (!(! (! (! (! (! (! (! (! (! (! (! (! (! (! (! (!

SOUTH -WEST (! (!

Figure 11.2 Location of the 85 stations and the four regions

105

The effects of catchment storage and of percentage peat are shown in Figure 11.3 and

Figure 11.4 in box plot form. Each box plot in Figure 11.3 summarises values of X 100 while Figure 11.4 summarise the corresponding Hosking- Wallis homogeneity statistics H1 and H2. The leftmost box plot represents all 85 stations in the study.

Pooling Growth Curve(GEV) 2.4

2.2

2

1.8

X100 1.6

1.4

1.2

1 All st(85) ≥5% ≥10% ≥15% ≥20% farl farl farl farl peat(44) peat(30) peat(22) peat(14) ≤.98(30) ≤.95(23) ≤.90(19) ≤.85(15)

Figure 11.3(a)

Pooling Growth Curve(GLO) 2.4

2.2

2

1.8

X100 1.6

1.4

1.2

1 All st(85) ≥5% ≥10% ≥15% ≥20% farl farl farl farl peat(44) peat(30) peat(22) peat(14) ≤.98(30) ≤.95(23) ≤.90(19) ≤.85(15)

Figure 11.3 Box plots of X100 values from region of influence pooling estimates of X100 obtained by (a) GEV, (b) GLO estimating equations showing effect of %peat and FARL. The numbers in brackets (85, 44, 30…) indicate the number of X100 values contributing to each box plot.

106 Homogeneity Statistics(H1) 14

12

10

8

6 H1

4

2

0

-2 All st(85) ≥5% ≥10% ≥15% ≥20% farl farl farl farl peat(44) peat(30) peat(22) peat(14) ≤.98(30) ≤.95(23) ≤.90(19) ≤.85(15)

Figure 11.4(a)

Homogeneity Statistics(H2) 14

12

10

8

6 H2

4

2

0

-2 All st(85) ≥5% ≥10% ≥15% ≥20% farl farl farl farl peat(44) peat(30) peat(22) peat(14) ≤.98(30) ≤.95(23) ≤.90(19) ≤.85(15)

Figure 11.4 Values of (a) H1 (b) H2 for the analyses summarised in Figure 11.3

The median X 100 values differ only slightly with variation in peat percentage and without a definite pattern. The median X 100 values show a slight monotonically decreasing trend with increasing amount of storage i.e. smaller FARL values. The results are substantially the same for both GEV and GLO estimation. In nearly all cases

107 the median H1 values are greater than 2 while nearly all the median H2 values are less than 2 and those few that do exceed 2 do so by a very small amount.

The effects of geographical region and of catchment area are shown in Figure 11.5 and

Figure 11.6. In terms of geographical region South-West has the lowest median X 100 of approximately 1.52 while the East has the highest value of approximately 1.9. However catchments of all sizes, degree of FARL and of percentage peat are mixed together in these subjectively selected geographical groupings. The median X 100 values shows a monotonic decrease with increase in catchment area. Figure 11.6 display H1 and H2 values that are slightly more dispersed than those in Figure 11.4, but with similar overall results, showing median values of H2 less than 2.

108

Pooling Growth Curve(GEV) 2.4

2.2

2

1.8

X100 1.6

1.4

1.2

1 All st(85) West(17) South- East(24) Shannon A≤100(19) A~101- A~201- A≥500(22) West(18) (26) 200(20) 500(24)

Figure 11.5(a)

Pooling Growth Curve(GLO) 2.4

2.2

2

1.8

X100 1.6

1.4

1.2

1 All st(85) West(17) South- East(24) Shannon A≤100(19) A~101- A~201- A≥500(22) West(18) (26) 200(20) 500(24)

Figure 11.5 Box plots of (a)X 100 (GEV) (b) X 100 (GLO) showing effect of geographical area and catchment size

109

Homogeneity Statistics(H1) 12

10

8

6 H1 4

2

0

-2 All st(85) West(17) South- East(24) Shannon (26) A≤100(19) A~101- A~201- A≥500(22) W est(18) 200(20) 500(24)

Figure 11.6(a)

Homogeneity Statistics(H2) 12

10

8

6 H2 4

2

0

-2 All st(85) West(17) South- East(24) Shannon A≤100(19) A~101- A~201- A≥500(22) West(18) (26) 200(20) 500(24)

Figure 11.6 Values of (a) H1, (b) H2 for the analyses summarised in Figure 11.5

110 11.3 Temporal effect on pooled growth curve estimates

Pooled growth curves have been obtained from different periods of record as follows: • distinguishing between early and late halves of each record and • distinguishing also by decades, 1972 - 81, 1982 – 91 and 1992 – 2001 (50 stations) and • distinguishing by decades, 1952 - 61, 1962 - 71, 1972 - 81, 1982 – 91 and 1992 – 2001 (for which data are available for 26 stations). • distinguishing by decades 1957-66, 1967-76, 1977-86, 1987-96 and 1997-2006 where 2006 is the most recent year of record in the data set. There are 34 records available for all 5 decades and 68 records available for the last 3 such decades. This part of the study was completed using a total of 90 gauging stations before the data provider had indicated some of these are unsuitable for detailed statistical analysis. Hence the number 90 is greater than 85 mentioned above in Section11.1. It is believed that the overall nature of the findings would not be greatly altered if the study had been repeated with just 85 stations. This would be specially true if the five omitted stations had reasonable water level records and were rejected on the basis of doubts about the reliability of rating curves.

The effects of period of record (1 st half of each record compared with 2 nd half) and of successive decades are shown in

Figure 11.7 and Figure 11.8. H1 an H2 values are shown in Figure 11.8. Median X 100 shows a decrease between the earlier and later halves of the records while the decade rd based values of X 50 show no particular pattern. Indeed the 3 decade value of median st X50 is not much different to the 1 decade median X 50 value.

111

Pooling Growth Curve(GEV)

2.4

2.2 2

1.8 1.6

1.4 1.2

1 X100 (X50 in case of decade analysis) All record(90) 1st half 2nd half Decade70(52) Decade80(83) Decade90(82) record(90) record(90)

Figure 11.7(a)

Pooling Growth Curve(GLO)

2.4

2.2

2 1.8

1.6 1.4

1.2 1 X100(X50 in case of decade analysis) All record(90) 1st half 2nd half Decade70(52) Decade80(83) Decade90(82) record(90) record(90)

Figure 11.7 Variation of X100 with period of record (first half versus second half of each record) and variation of X50 with decades, as estimated by (a) GEV, (b) GLO from each individual stations pooling group using the 5T rule

112

Homogeneity Statistics(H1)

12

10

8

6 H1 4

2

0 -2 All record(90) 1st half 2nd half Decade70(52) Decade80(83) Decade90(82) record(90) record(90)

Figure 11.8(a)

Homogeneity Statistics(H2)

12

10

8

6 H2 4

2

0

-2 All record(90) 1st half 2nd half Decade70(52) Decade80(83) Decade90(82) record(90) record(90)

Figure 11.8 Values of (a) H1, (b) H2 for the analyses summarised in Figure 11.7

113 There are 50 stations for which decade estimates are available for all 3 decades. The values of X 50 for each decade for each station are displayed in Figure 11.9(a). From this it can be seen that the largest X 50 values occur in the middle decade, 1982-91, in every case. The corresponding Qmed values, standardized at each station by the 1972-81 value of Qmed, are shown in Figure 11.9(b). In most cases the Qmed values for the 2 nd and 3 rd decade exceed the 1 st decade value and while the 3 rd decade displays a number of remarkable Qmed values, only about one third of the stations have their largest Qmed rd in the 3 decade. Figure 11.9(c) shows the estimated Q 50 values, standardized at each nd rd station, by the 1972-81 value. The majority of Q 50 values in the 2 and 3 decades exceed those of the 1 st decade, with more that half of these exceedances occurring during the 2 nd decade.

There are 26 stations for which decade estimates are available for 5 decades. At some stations there are not a full 10 years of data available for the 1 st decade. Only stations which have a minimum of 7 years of data available for the 1 st decade are included. The results of the analyses are examined in the same way as the 3 decades result above and nd presented in Figure 11.10. The 2 decade (1960s) value of X 50 is larger than those of the other decades at every station with an average value of approximately 1.7. This contrasts strongly with the results of the last decade (1990s) where the average value is approximately 1.4. Figure 11.10(b) shows the decadal standardized Qmed values. This shows that the largest values tend to be from the 2 nd (1960s) and the last (1990s) decades. The Q50 values in Figure 11.10(c) also conveys the impression that the largest Q50 values occur during the 2 nd (19960s) and last (1990s) decades.

The results for the decades ending in 1966, 1976, 1986, 1996 and 2006 show results that are similar to those described in the last paragraph. Results are shown in Figure 11.11 which is analogous to Figure 11.10.

114 1.9 Pooling Growth Curve (GEV) Decade70 Decade80 Decade90 1.8

1.7

1.6

1.5 X 5 0 1.4

1.3

1.2

1.1

1 60 11 60 26 90 01 110 01 120 01 140 05 140 06 140 18 140 19 150 03 160 01 160 02 160 03 160 04 160 08 160 09 180 04 180 05 190 01 230 01 250 01 250 03 250 05 250 06 250 14 250 16 250 23 250 27 250 29 250 30 260 02 260 05 260 06 260 07 260 08 260 18 260 19 260 22 270 01 270 02 290 04 300 61 340 01 340 09 350 01 350 02 360 11 360 15 360 18 360 19 Station No. Figure 11.9(a)

1.8 Decade70 Decade80 Decade90 Decadal Qmed value 1.6

1.4

1.2

1

0.8 decade70) 0.6

0.4 Qm ed (standardised by Qm ed of 0.2

0 6 0 1 1 6 0 2 6 9 0 0 1 1 1 0 0 1 1 2 0 0 1 1 4 0 0 5 1 4 0 0 6 1 4 0 1 8 1 4 0 1 9 1 5 0 0 3 1 6 0 0 1 1 6 0 0 2 1 6 0 0 3 1 6 0 0 4 1 6 0 0 8 1 6 0 0 9 1 8 0 0 4 1 8 0 0 5 1 9 0 0 1 2 3 0 0 1 2 5 0 0 1 2 5 0 0 3 2 5 0 0 5 2 5 0 0 6 2 5 0 1 4 2 5 0 1 6 2 5 0 2 3 2 5 0 2 7 2 5 0 2 9 2 5 0 3 0 2 6 0 0 2 2 6 0 0 5 2 6 0 0 6 2 6 0 0 7 2 6 0 0 8 2 6 0 1 8 2 6 0 1 9 2 6 0 2 2 2 7 0 0 1 2 7 0 0 2 2 9 0 0 4 3 0 0 6 1 3 4 0 0 1 3 4 0 0 9 3 5 0 0 1 3 5 0 0 2 3 6 0 1 1 3 6 0 1 5 3 6 0 1 8 3 6 0 1 9 Station No.

Figure 11.9(b) 2 Decadal Q50 value 1.8 Decade70 Decade80 Decade90

1.6

1.4

1.2

1

decade70)0.8

0.6

Q50 (standardised0.4 by Q50 of

0.2

0 6 0 1 1 6 0 2 6 9 0 0 1 11 00 1 12 00 1 14 00 5 14 00 6 14 01 8 14 01 9 15 00 3 16 00 1 16 00 2 16 00 3 16 00 4 16 00 8 16 00 9 18 00 4 18 00 5 19 00 1 23 00 1 25 00 1 25 00 3 25 00 5 25 00 6 25 01 4 25 01 6 25 02 3 25 02 7 25 02 9 25 03 0 26 00 2 26 00 5 26 00 6 26 00 7 26 00 8 26 01 8 26 01 9 26 02 2 27 00 1 27 00 2 29 00 4 30 06 1 34 00 1 34 00 9 35 00 1 35 00 2 36 01 1 36 01 5 36 01 8 36 01 9 Station No.

Figure 11.9 (a)Column chart of X 50 (GEV) values (b) Column chart of Q med values (c) Column chart of Q 50 values for the decadal data (‘70, ‘80 and ‘90) of 50 stations

115 2.00 Pooling Growth Factor

1.80

1.60

1.40 Decade50 1.20 Decade60 Decade70 1.00 X 5 0 Decade80 0.80 Decade90 0.60

0.40

0.20

0.00 1 4 0 0 5 1 4 0 0 6 1 4 0 1 8 1 4 0 1 9 1 5 0 0 3 1 6 0 0 2 1 6 0 0 3 1 6 0 0 4 1 6 0 0 8 1 6 0 0 9 1 8 0 0 5 2 5 0 0 1 2 5 0 0 3 2 5 0 0 5 2 5 0 0 6 2 5 0 1 4 2 5 0 2 3 2 6 0 0 2 2 6 0 0 5 2 6 0 0 6 2 6 0 0 7 2 6 0 0 8 2 6 0 1 9 2 7 0 0 2 3 6 0 1 1 3 6 0 1 8 Station No.

Figure 11.10(a) 1.8 Qmed Value 1.6 1.4

1.2 Decade50 1 Decade60 Decade70 0.8

decade50) Decade80 0.6 Decade90

0.4

Qmed( standardised0.2 by Qm ed of

0 14005 14006 14018 14019 15003 16002 16003 16004 16008 16009 18005 25001 25003 25005 25006 25014 25023 26002 26005 26006 26007 26008 26019 27002 36011 36018 Station No.

Figure 11.10(b)

1.80 Q50 Value 1.60

1.40

1.20 Decade50 1.00 Decade60 Decade70 0.80 Decade80 decade50) Decade90 0.60

0.40 Q50( standardised by Q50 of

0.20

0.00 1 4 0 0 5 1 4 0 0 6 1 4 0 1 8 1 4 0 1 9 1 5 0 0 3 1 6 0 0 2 1 6 0 0 3 1 6 0 0 4 1 6 0 0 8 1 6 0 0 9 1 8 0 0 5 2 5 0 0 1 2 5 0 0 3 2 5 0 0 5 2 5 0 0 6 2 5 0 1 4 2 5 0 2 3 2 6 0 0 2 2 6 0 0 5 2 6 0 0 6 2 6 0 0 7 2 6 0 0 8 2 6 0 1 9 2 7 0 0 2 3 6 0 1 1 3 6 0 1 8 Station No.

Figure 11.10(a)Column chart of X 50 (GEV) values (b)Column chart of Q med values (c) Column chart of Q 50 values for the decadal data (’50, ’60 ‘70, ‘80 and ‘90) of 26 stations

116 3 Decade 57-66 Growth Factor X50 Decade 67-76 2.5 Decade 77-86 Decade 87-96 Decade 97-06 2

1.5 X50

1

0.5

0 6011 6026 9001 12001 14005 14006 14018 14019 15003 16002 16003 16008 16009 18004 18005 19001 25001 25003 25006 25014 25023 25030 26002 26005 26006 26007 26008 26018 26019 27002 35005 36011 36018 36019 Station No.

Figure 11.11(a) 2 Decadal Qmed Value Decade 57-66 1.8 Decade 67-76 1.6 Decade 77-86 Decade 87-96 1.4 Decade 97-06 1.2

1 66)

0.8

0.6

0.4

Qmed(standardised byQmed decade57- of 0.2

0 6011 6026 9001 12001 14005 14006 14018 14019 15003 16002 16003 16008 16009 18004 18005 19001 25001 25003 25006 25014 25023 25030 26002 26005 26006 26007 26008 26018 26019 27002 35005 36011 36018 36019 Station No.

Figure 11.11(b) 2 Decadal Q50 Value Decade 57-66 1.8 Decade 67-76 1.6 Decade 77-86 Decade 87-96 1.4 Decade 97-06 1.2

1

0.8

0.6

0.4

Q50(standardised by0.2Q50 of decade 57-66)

0 6011 6026 9001 12001 14005 14006 14018 14019 15003 16002 16003 16008 16009 18004 18005 19001 25001 25003 25006 25014 25023 25030 26002 26005 26006 26007 26008 26018 26019 27002 35005 36011 36018 36019 Station No.

Figure 11.11 (a) Column chart of X 50 (GEV) values (b) Column chart of Q med values (c) Column chart of Q 50 values for the decadal data (’57-66, ’67-76 ’77-86, ’87-96 and ’97-06) of 34 stations

117 11.4 Arterial drainage effect on pooled growth curve estimates

Pooled growth curves have been obtained from the pre and post drainage records of 16 gauging stations, 8 of which are located in the Boyne catchment. These 16 stations are those that have their pre- and post-drainage annual maximum series listed in Table 11.3 and displayed on probability plots in Appendix 4D. The descriptor BFI is needed in the pooling procedure but 3 of these stations have not had a BFI value provided. Two of these are on the Boyne catchment (7007 and 7011) so the values at neighboring gauging stations on the same river have been used for these instead – 7003 for 7007 and 7004 for 7011. The third station lacking a BFI value, namely 30004, was assigned the BFI value for gauging station 30005 even though this is not on the same stream. A visual check was made on the appearance of yearly hydrographs of these two gauging stations to check if it is reasonable to make this assignment. The agreement is reasonable but not perfect. The probability plots in Appendix 4D show a clear increase in magnitude at all return periods for only 9 gauging stations. These are 3051, 7003, 7004, 7010, 7011, 7012, 24001, 24013 and 30004. One gauging station's plot, 30001, shows quite the opposite effect while the remaining 6 show a variety of cross-over effects namely that the pre-drainage large floods exceed the post drainage large floods or else lines fitted to each period of record might cross over.

The results of the pooled estimation of X50, Qmed and Q50 for these stations, from both pre-drainage and post-drainage data are shown in Figure 11.12. These show that the pre-drainage growth factor exceeds the post-drainage growth factor in every case, whereas the Qmed values are greater in the post-drainage period than in the pre- drainage period in every case except station for the above mentioned 30001. The estimated Q50 values for the post-drainage period exceed the pre-drainage values in 10 cases, 9 of them being those noted above in relation to their probability plot behaviour, the 10th one being 26012 which shows just a small increase.

It should be noted that the extensive studies have already been carried out by OPW (1981) on the effect of catchment drainage on flood magnitude return period relationships as well as on unit hydrograph characteristics

118

Table 11.3 List of 16 no. Pre and post- drainage stations

Station Record Drainage Missing Number Station Name River Name Catchment length Timespan works year 1975- 3051 FAULKLAND BLACKWATER Blackwater 26 2004 86-'89 1986-89 1953- 3/74- 7002 Killyon Deel Boyne 46 2004 4/79 1973-78 Blackwater 1953- 1970- 7003 Castlerickard (Enfield) Boyne 46 2004 1975 1956,70-74 Blackwater 1956- 1973- 7004 Stramatt (Kells) Boyne 48 2004 1981 1973-81 1953- 1971- 7005 Trim Boyne Boyne 47 2004 1974 1971-74 Boyne 1953- 7007 Aqueduct Boyne Boyne 45 2004 73-78 1973-78 Blackwater 1953- 1982- 1970,71,82- 7010 Liscartan (Kells) Boyne 46 2004 1986 85 O'Dalys Blackwater 1958- 7011 Bridge (Kells) Boyne 44 2004 80- 83 1980-82 1940- 7012 Slane Castle Boyne Boyne 65 2004 69-86 None 1946- 23002 Listowel Feale Feale 59 2004 51-59 None 1953- 24001 Croom Maigue Maigue 51 2003 75- 76 None 1953- 1963- 24013 Rathkeale Deel Maigue 46 2004 1968 1963-68 1957- 26012 Tinacurra Boyle Boyle 48 2004 82-91 None 1952- 30001 Cartronbower Aille Cara 48 2001 70-71 1970-71 1952- 1958- 30004 Corrofin Clare Cara 35 2003 58-64 73,79,86 1953- 30005 Foxhill Robe Cara 45 2004 73-78 1961,73-78

119 2.5 Pre-Drainage Pooling Growth Factor Post-Drainage 2

1.5 X50 1

0.5

0 3051 7002 7003 7004 7005 7007 7010 7011 7012 23002 24001 24013 26012 30001 30004 30005 Station No.

3.5 Pre-Drainage Qmed value Post-Drainage 3

2.5

2

1.5 drainage series) drainage 1

0.5 Qmed (standardised by Qmed of pre- of Qmed by (standardised Qmed 0 3051 7002 7003 7004 7005 7007 7010 7011 7012 23002 24001 24013 26012 30001 30004 30005 Station No.

3 Q50 value Pre-Drainage Post-Drainage 2.5

2

1.5

1 pre-drainage series) pre-drainage 0.5 Q50 (standardised by Q50 of of Q50 by (standardised Q50

0 3051 7002 7003 7004 7005 7007 7010 7011 7012 23002 24001 24013 26012 30001 30004 30005 Station No.

Figure 11.12 (a) Column chart of X 50 (GEV) values (b) Column chart of Q med values (c)Column chart of Q 50 values for the pre- and post-drainage data of 16 stations

120 11.5 Implications for Flood Frequency Estimation

The FARL value, catchment area and geographical location appear to have an effect on the X 100 growth factor with percentage peat having virtually no noticeable effect. In temporal terms the middle of the 3 decades between 1972 and 2001 produces the largest

X50 values as a whole, while the same decade displays many, but not all, of the largest

Q50 values. The 1972-81 decade displays the smaller values of X 50 , Q 50 and Q 50 values with just a small number of exceptions. When the smaller number of cases for which 5 decades of data are available the impression is that the 2 nd (1960s) and last (1990s) decades produce the largest floods.

In view of the effect of FARL, catchment area and geographical location on X T the question arises as to whether the catchments which are allowed into a particular pooling group should be screened or filtered in advance of pooling for any particular subject site. The disadvantage of this is to reduce the pooling group constituency from which the pooling group members are to be drawn. Given that the region of influence based pooling method takes catchment area into account in the distance measure d ij this variable's effect on X T is partially taken into account. The geographical location may be partially taken into account by SAAR but not uniquely so while FARL is not taken into account in any manner.

It may be best to draw users' attention to these issues and to leave it open to users to decide for themselves whether to screen the data or not. It is likely that users will want to take geographical location explicitly into account in some cases, especially for east coast locations. Users will have to advised that there is a trade-off involved between tailoring the pooling group constituency on the one hand and loss of station years and lack of precision on the other.

It should also be noted that the effects refereed to above have not been statistically tested to determine whether these relatively small observed effects would in fact reject the null hypothesis of no effect.

With regard to temporal effects and whether only recent records should be used in flood estimation the evidence is not quite as clear cut that only recent data should be included.

121 This may conflict with other published views that the past is not a good guide to the future, cf "The past is longer the key to the future, and the future is uncertain" in Ireland at Risk, published by Irish Academy of Engineering, 2007, p8. However this latter view conflicts with the well established practice of using design flood estimates based on historical records.

122 12 Selection of distance measure variables

As stated earlier "The region of influence approach selects stations, which are nearest to the subject site in catchment descriptor space, to form the pooling group for that subject site." Equation (10. 2) is one definition of a distance measure used for this purpose which has been used in the investigations described in the last subsection, 1. However other catchment descriptors could also be used in the definition, fewer or more of them could be used and each component could have different weights.

12.1 Selecting variables for pooling

In this study four catchment descriptors, AREA, SAAR, BFI and FARL have been considered in eight different combinations as described later below:

The objective is to find which of these combinations leads to pooling groups which are most homogeneous and therefore most effective at exploiting the information about the flood distribution contained in the pooling groups. This is examined with the help of two statistics

• H1, the Hosking- Wallis homogeneity statistic based on similarity of L-CV values in the pooling group, and

• PUM 100 , the FEH defined Pooled Uncertainty Measure of the 100 year quantile:

The general form of the similarity measure used for selecting members of a pooling group, is

n 2 d ij = ∑Wk ()X k,i − X k , j --- (12. 1) k =1

th where, n is the number of catchment descriptors, X i,k is the value of k catchment

th descriptor at the i site and Wk is the weight applied to descriptor k reflecting the relative importance of that catchment descriptor. The subscript j applies to the ‘subject site’ and the subscript i applies to the i th ‘donor site’.

123

In choosing a distance measure dij a decision has to be made about which catchment descriptors are to be included in the distance measure and what weightings are to be applied to them and whether logarithms or other transformations are to be used.

The Flood Estimation Handbook (1999) gives a number of useful maxims for choosing a distance measure. It uses pooled uncertainty measure (PUM) for selecting the final set of pooling variables. For UK data the distance measure is taken as equation (10. 1), recalled here again :

2 2 2 1  ln Ai − ln Aj   ln SAAR i − ln SAAR j   BFIHOST i − BFIHOST j  dij =   +   +   2  σ ln A   σ ln SAAR   σ BFIHOST 

The weight of 0.5 is given to AREA only to allow SAAR and BFIHOST to play a slightly more significant role in forming the pooling groups in the distance measure.

Recently a revised distance measure has been developed for UK data (Kjeldsen et al., 2008) where two new variables FARL and FPEXT (the extent of flood plains in a catchment) has been included and the variable BFIHOST has been omitted:

ln A −ln A  ln SAAR −ln SAAR 2  FARL −FARL 2  FPEXT −FPEXT 2 i j i j i j i j --- (12. 2) dij = 2.3  + 5.0   + 1.0   + 2.0    28.1   37.0   05.0   04.0 

12.2 Pooled uncertainty measure, PUM

The PUM is a weighted average of the differences between each site growth factor and the pooled growth factor measured on a logarithmic scale. The pooled uncertainty measure for return period T, PUM T is defined by (FEH, 1999) as

2 M long p ∑ ni ()ln xT − ln x Ti i i=1 PUM T = --- (12. 3) M long

∑ ni i=1

124 where M is the number of long-record sites, n is the record length of the i th site, x is long i Ti

th p the T-year site growth factor for the i site, and ln x Ti is the T-year pooled growth factor for the i th site. PUM is a measure of how effective a pooling method is at identifying a homogeneous region. A good pooling method will yield low values of PUM.

12.3 Selecting pooling variables for Irish data

For Irish conditions, four catchment descriptors have been selected for the potential pooling variables. They are AREA, SAAR, BFI and FARL. These variables have been found to be effective in explaining the observed variation in Q med values in regression studies.

In FEH pooling variables were chosen based on the pooled uncertainty measure (PUM). In this study PUM as well as heterogeneity measure (H) is used to select the final set of pooling variables.

In this study PUM has been evaluated at 100-year return period for catchments whose record length is 20 or more years of data. 90 sites of category A have been considered for the pooling analysis. Among them 85 sites have 20 or more years of data. For each of these catchments, the pooling group was selected from 90 sites. For determining the size of the pooling group the 5T rule has been used here.

Generalised Extreme Value Distribution (GEV) has been used to calculate the pooled and single site growth factors. In FEH the subject site was not included in its own pooling group when PUM was evaluated. In this study PUM is evaluated in both the presence and absence of the subject site.

The following combinations of the four variables have been tested based on PUM and H lnAREA lnAREA,lnSAAR lnAREA,lnSAAR,BFI lnAREA,lnSAAR,BFI,FARL lnSAAR

125 BFI lnAREA+BFI lnAREA,lnSAAR,lnBFI

The data sets that have been used in the study are summarised Table 12.1.

Table 12.1Summary of AMF data sets used in dij study Number of stations 90 Shortest record length 18 Longest record length 55 Mean record length 37.47 Number of AMF events 3372

Initially all weights W k in equation (12. 1) were set to unity. Figure 12.1 show the variation in 100-year PUM value for different sets of catchment descriptors used in equation (12. 1) in box-plot form. In Figure 12.1(a) the PUM value is evaluated without inclusion of subject site in its own pooling group while in Figure 12.1(b) the subject site is included. Figure 12.2(a, b) summarise the corresponding heterogeneity statistics H1 and H2.

Further Table 12.2 summarises the corresponding mean variation of PUM100, H1 and H2 values for different sets of pooling variables which shows that the numerical measures of effectiveness vary by very little between rows. As the standard errors of these measures are unknown it is not possible to state whether the values in any one column are significantly different from one another in the statistical sense.

From Table 12.2 it is shown that the set of two variables lnAREA and lnSAAR and the set of one variable lnSAAR alone performed best in terms of providing the lowest PUM value. In terms of H1 heterogeneity the set consisting of lnAREA and lnSAAR comes second best to the set consisting of lnAREA, lnSAAR, BFI and FARL and in terms of H2 heterogeneity it comes third behind lnAREA on its own and behind lnSAAR o n its own.

Overall the set of variables comprised of lnAREA and lnSAAR can be considered as being the most suitable set of pooling variables for Irish condition. However if there is also a desire to incorporate another physical catchment effect then BFI could be

126 included with these two, as was done in equation (10. 2) above. While inclusion of just one or two catchment descriptors may be indeed be best there is an attraction in representing some descriptor of catchment response also even at a small apparent loss in effectiveness.

Table 12.2 Variation in the mean PUM100, H1 and H2 values for different sets of pooling variables

PUM(without taking PUM( taking subject site into subject site into Variables used in model account) account) H1 H2 lnAREA 0.1956 0.1951 6.1670 2.7697 lnAREA,lnSAAR 0.1929 0.1924 5.2276 2.7684 lnAREA,lnSAAR,BFI 0.1958 0.1951 5.4381 2.8232 lnAREA,lnSAAR,BFI,FARL 0.1966 0.1959 5.0160 2.9584 lnSAAR 0.1913 0.1929 5.5279 2.7258 BFI 0.2053 0.2048 7.1408 3.2138 lnAREA+BFI 0.2044 0.2023 6.1348 2.9501 lnAREA,lnSAAR,lnBFI 0.1958 0.1953 5.5582 2.8771

An extension to this investigation was done with varying values of weights W k in equation (10. 2). The results of all variations examined are not reported in detail here. It was found that the weights 1.7, 1.0 and 0.2 for ln AREA, ln SAAR and BFI respectively gave PUM100 = 0.1889 and 0.189, H1 = 5.20 and H2 = 2.62 which offer small improvements on the Wk = 1.0 values used in the calculations on which Table 12.2 are based.

Thus either equation (10. 2) or the following one is considered suitable for calculating the distance measure.

2 2 2  ln AREA i − ln AREA j   ln SAAR i − ln SAAR j   BFI i − BFI j  dij = 7.1   +   + 2.0   --- (12. 4)  σ ln AREA   σ SAAR   σ BFI 

127

Pooled Uncertainty Measure(PUM) 0.35

0.30

0.25

0.20

P UM0.15 100

0.10

0.05

0.00 lnA lnA+lnS lnA+lnS+B lnA+lnS+B+F lnS B lnA+B lnA+lnS+lnB

Figure 12.1(a)

Pooled Uncertainty Measure (PUM) 0.35

0.30

0.25

0.20

P UM 100 0.15

0.10

0.05

0.00 lnA lnA+lnS lnA+lnS+B lnA+lnS+B+F lnS B lnA+B lnA+lnS+lnB

Figure 12.1 Box-plot of 100-year PUM values for different sets of catchment descriptors used in defining the distance measure d ij . (a) subject site is not included in its own pooling- group when PUM is evaluated (b) subject site is included in its own pooling-group when PUM is evaluated.

128 Homogeneity Statistics(H1) 14

12

10

8

6 H1

4

2

0 lnA lnA+lnS lnA+lnS+B lnA+lnS+B+F lnS B lnA+B lnA+lnS+lnB -2

Figure 12.2(a)

Homogeneity Statistics (H2) 14

12

10

8

6 H2

4

2

0

-2 lnA lnA+lnS lnA+lnS+B lnA+lnS+B+F lnS B lnA+B lnA+lnS+lnB

Figure 12.2 Box-plot of (a) H1 values, (b) H2 values for different sets of catchment descriptors used in defining the distance measure dij .

129 13 Standard errors of Q med , Q T and X T

The standard error se of an estimate of Q T is an indication of how reliable that estimate is. It is based on the assumption that the data upon which the estimate is based are randomly drawn from a single population – an assumption that cannot easily be proven. If an infinite number of similarly sized data sets were to be drawn from the same population and the value of Q T obtained from each set by the same procedure then the se is defined as

se (QT ) = Standard deviation of all the possible set of Q T values.

This measure only represents the degree of scatter of the several estimates and does not refer to whether the mean of these equals the true value in the population. If this equality holds then the procedure whereby Q T is calculated is said to be unbiased – otherwise it is said to be biased.

In flood hydrology randomness of annual maximum floods and lack of trend with time is generally assumed. Likewise it is assumed that a single form of statistical distribution describes all the AM flood series in a region or country. Such assumptions cannot be proven and there is some evidence to suggest that Irish flood data are not entirely trend free. The presence of low or high outliers in some data sets also makes interpretation difficult. If the data are drawn are not truly from a unique homogeneous parent population then the se cannot strictly be defined. However the se concept is widely used in flood hydrology. Its value is determined on the assumption of a unique homogeneous parent and the value of se so obtained may be considered as a lower bound on what the true value, if it could be defined, actually is. Hence it provides a useful guide to the precision of the Q T value obtained in any situation.

The aim in this section is to provide expressions or graphs which give an indication of the order of magnitude of se (Q T) in at-site estimation and in pooling group based estimation of Q T. This will be restricted to the EV1 and GEV cases on the assumption

130 that the GLO/LO and LN3/LN2 standard errors would be of the same order of magnitude.

13.1 Standard Error of Qmed

This has been investigated in section 9.1 above and the result, adapted to Irish AM flood condiditons, is reproduced here as follows:

36.0 se (Q ) = Qmed med N

If N = 5, se(Q med ) ≈ 0.161 Q med ( = 0.22 Q med in FEH,3, Table 2.2)

If N = 10, se(Q med ) ≈ 0.114 Q med ( = 0.14 Q med in FEH,3, Table 2.2)

If N = 20, se(Q med ) ≈ 0.084 Q med ( = 0.08 Q med in FEH,3, Table 2.2)

If N = 50, se(Q med ) ≈ 0.051 Q med ( not tabulated in FEH,3, Table 2.2)

13.2 Standard Error of At-site estimate of Q T

Theoretical expressions for L-Moment based estimates of Q T in both EV1 and GEV cases have been given by Lu & Stedinger (1992) as follows.

ˆ α 2 EV1: SE [QT ]= [ .1 1128 + .0 4574 y + .0 8046 y ] --- (13. 1) n

ˆ where QT is the estimate for the T-year flow event ; α is the EV1 scale parameter;

y = yT = −ln (− ln (1− /1 T )); and n is the number of observations in the sample.

ˆ α 2 3 2/1 GEV: se [QT ]= [exp {a0 ()()()()()T + a1 T exp − k + a2 T k + a3 T k }] --- (13. 2) n

ˆ where QT is the estimate for the T-year flow event; a0 (T ), a1(T ), a2 (T ), and a3 (T ) are coefficients that depend on the return period, T ; α and k are scale and shape parameters respectively; and n is the number of observations in the sample. The values

131 for constant coefficients for different return periods are tabulated in Lu and Stedinger (1992). For example, for T=10: a0= -2.667, a1= 4.491, a2= -2.207, a3= 1.802 for T=100: a0= -4.147, a1= 8.216, a2= -2.033, a3= 4.780

The percentage standard error, se (Q T)/Q T % is, in particular, focused on here. In the

EV1 case this percentage can be expressed as a function of the ratio α/u and y T or equivalently as a function of Cv and y T or of L-Cv and y T. Note that L-Cv ≈ 0.5 Cv in the EV1 case.

Equation (13. 1) has been applied to the data of 85 A1 and A2 stations in which the values of u, α and Q T were estimated from each site's data separately and the ratio se (Q T)/Q T % formed. By using estimated values of parameters in these expressions the true value of se (Q T) is not obtained but it is considered useful to examine the range of values so obtained because part of their scatter is caused by inter-site heterogeneity, assuming such exists, as well as by random sampling effects and unequal record lengths (with an average of 37 years). A box plot of these 85 values for 6 different return periods is shown in Figure 13.1. It is suggested that the upper and lower values indicated by the box plot i.e. the middle 50% of values, ought to give a good indication of the range of values in which the true se values fall. The extreme high and low values can be considered to be unrepresentative of the whole and hence the suggestion of focusing on the boxes.

In Figure 13.1(b) the same results are re-presented along with the theoretical EV1 percentage standard errors of equation (13. 1) for the average sample size of 37 years and for a range of L-Cv values that covers the range of L-Cv values experienced among Irish AM flood data. [The difference in appearance of the boxes in Figure 13.1(b) from those shown in Figure 13.1(a) arises from a technical difficulty in graphing two different types of series.]

What these show, under the EV1 assumption, is that the se can be assumed to be between 5% and 7% for the 10 year quantile and between 6.5% and 10% for the 100 year quantile.

132 Corresponding results for the GEV case are presented in Figure 13.2. The theoretical figures in Figure 13.2(b) are now for a single value of shape parameter k = -0.1 which it is felt ought to cover the most extreme underlying population case that would arise in Irish conditions. From the box plots we can conclude, using the same arguments as above, that under the GEV assumption, that the se can be assumed to be between 6% and 8% for the 10 year quantile and between 8% and 15% for the 100 year quantile. However it can be seen that the most extreme values are extremely large but these are not representative of the true se value.

Relative Standard Error (At site/EV1) 18

16

14

12 % T 10 )/Q T ^ 8 se(Q 6

4

2

0 T=5 T=10 T=25 T=50 T=100 T=500

Figure 13.1(a) 18 Relative Standard Error (Atsite/EV1) 16

14

LCV=0.3 12 LCV=0.25 % T 10 LCV=0.2 )/Q

T LCV=0.15 8 se(Q LCV=0.1 6

4

2

0 T=5 T=10 T=25 T=50 T=100 T=500

Figure 13.1 (a) Box-plot of percentage standard error of flood quantile estimates for EV1 distribution using simulation method for 85 stations for the 5, 10, 25, 50, 100, 500-year return period. (b) The theoretical EV1 percentage standard errors for the average sample size of 37 years and for a range of L-Cv values

133

Relative Standard Error (At site/GEV) 100

90

80

70

% 60 T )/Q

T 50 ^

40 se(Q

30

20

10

0 T=5 T=10 T=25 T=50 T=100 T=500

Figure 13.2(a)

Relative Standard Error(At-site/GEV) 50 K=-0.1

40 T 30 )/Q T se(Q 20

LCV=0.3 10 LCV=0.25 LCV=0.2 LCV=0.15 LCV=0.1 0 T=5 T=10 T=25 T=50 T=100 T=500

Figure 13.2 (a) Box-plot of percentage standard error of flood quantile estimates for GEV distribution using simulation method for 85 stations for the 5, 10, 25, 50, 100, 500-year return period. (b)The theoretical GEV (k=-0.1) percentage standard errors for the average sample size of 37 years and for a range of L-Cv values

134 13.3 Standard Error of pooled estimate of X T and of Q T

A simulation based estimate of the order of magnitude of the pooled estimate of X T, se(X T), for selected values of T, was obtained by simulation for each of the 85 stations. Each gauging station was selected in turn as the subject site and the following procedure applied.

1. Identify the gauging stations in the subject site's pooling group using dij values of equation (10. 2) and with a minimum of 500 station years of data in the pooling group, which satisfies the 5T rule for the 100 year quantile. 2. Random samples are drawn from EV1 populations for the subject site and for each site in the pooling group. For each site the sample size is taken as equal to the length of the observed historical record at the site and the parameter values u and α are those estimated from the observed record by L-Moments. 3. The sample Qmed is obtained for the subject site. 4. The L-Cv value is obtained for each sample in the pooling group and the average of these is calculated to represent the pooled average L-Cv. 5. The pooled average L-Cv is used to determine the pooling group's EV1 growth curve parameter β

6. The subject site's xT = 1+ β (ln(ln )2 − ln( − ln( 1− /1 T )) ) is calculated for T = 5, 10, 25,………500 years

7. The subject site's Q T = Qmed X T is calculated for T = 5, 10, 25,………500 years

8. Steps 2 to 7 are repeated 1000 times to provide 1000 values of XT and QT at the

subject site and the se(X T) and se(Q T) are calculated for the subject site by the following equations:

2 N M  Xˆ T − Xˆ T  1 1  i, m i  SE (X T )[]% = ∑ ∑ ×100 --- (13. 3) N M  X T  i=1m = 1  i 

2  T T  N M Qˆ − Qˆ 1 1  i, m i  SE (QT )[]% = ∑ ∑ ×100 --- (13. 4) N M  QT  i=1m = 1  i 

135 where Xˆ T and Qˆ T are the estimated T-year growth factor and T-year flood quantile i, m i, m

ˆ T ˆ T respectively at a site i at mth repetition; X i and Qi are the mean of those estimated

T T measures; X i and Qi are the assumed true T-year growth factor and T-year quantile at site i; N is the number of sites in the pooling group and M is the number of repetitions.

These expressions average the standard error over all the sites in the pooling group to take heterogeneity in the pooling group into account. It should be noted that the actual standard error for any individual site could be either smaller or larger than the calculated value depending on the at site value of L-Cv and L-skewness. The extreme values in the Box plots illustrate the range that could occur.

Steps 1 to 8 were repeated for each of the 85 stations and the percentage standard errors for X T and Q T are presented in box plot form in Figure 13.3 and Figure 13.4. The same remarks apply to the interpretation of these plots as for those of the at-site standard errors above.

The values of percentage se for X T vary from 1.1 to 1.2 at T = 10 to 1.8 to 2.0 for T =

100 while the corresponding percentage se values for Q T remain relatively constant over the whole range of T value, varying between from 3.7 to 8.5 at T = 10 and from 4.0 to 9.0 at T = 100.

The simulation procedure described above was also applied with the GEV distribution used instead of EV1 in steps 2 to 6. Because the shape parameter k can be very variable and hence unreasonably large or small when estimated from individual flow records a single value of k = -0.1 was used in all simulations. This is a conservative (worst case from the hydrological point of view) choice in the context of Irish flood data. In step 4 L-Skewness and average L-Skewness is calculated as well as L-Cv. In step 5 the GEV growth curve parameters β and k are calculated by equations (10. 6) and (10. 7); and in step 6 the expression for X T given in equation (10. 5) is used.

The results are presented, in box plot form, in Figure 13.5 and Figure 13.6. The values of percentage se for X T vary from 1.4 to 1.8 at T = 10 to 3.7 to 5.0 for T = 100 while the

136 corresponding percentage se values for Q T show a increasing trend, varying between 3.8 to 9.0 to at T = 10 and from 4.6 to 10.6 at T = 100. These ranges for all return periods and both distributions, EV1 and GEV, are summarized in tabular form in Table13.1. Since the 100 year flood is used for design purposes far many major projects in Ireland, such as major bridges, it is worth noting that its estimate's standard error can be considered to be approximately 10% of Q 100 .

Table13.1 Ranges of percentage se for growth factors, X T, and quantile estimates, Q T.

T Se(X T)%,EV1 Se(Q T)%,EV1 Se(XT)%,GEV Se(QT)%,GEV 5 0.7 to 0.8 3.7 to 8.4 0.8 to 1.0 3.7 to 9.0 10 1.1 to 1.2 3.7 to 8.5 1.4 to 1.8 3.8 to 9.0 25 1.4 to 1.6 3.8 to 8.7 2.3 to 3.0 4.1 to 8.9 50 1.6 to 1.8 3.9 to 8.9 3.0 to 3.9 4.3 to 9.6 100 1.8 to 2.0 4.0 to 9.0 3.7 to 5.0 4.6 to 10.6 500 2.1 to 2.3 4.1 to 9.3 5.4 to 7.3 5.7 to 12.2

137

Relative Standard Error of Growth Curve (Pooled/EV1-Simulation) 3.0

2.5

2.0 % T )/X

T 1.5 ^ se(X 1.0

0.5

0.0 T=5 T=10 T=25 T=50 T=100 T=500

Figure 13.3 Box-plot of percentage standard error of growth factors estimates for EV1 distribution using simulation method for 85 stations for the 5, 10, 25, 50, 100,500-year return period.

Relative Standard Error of Quantile Estimate (Pooled/EV1-Simulation) 35

30

25 %

T 20 )/Q T ^ 15 se(Q

10

5

0 T=5 T=10 T=25 T=50 T=100 T=500

Figure 13.4 Box-plot of percentage standard error of quantile estimates for EV1 distribution using simulation method for 85 stations for the 5, 10, 25, 50, 100,500-year return period.

138 Relative Standard Error of Growth Curve (Pooled/GEV-Simulation) 12

10

8 % T )/X

T 6 ^ se(X 4

2

0 T=5 T=10 T=25 T=50 T=100 T=500

Figure 13.5 Box-plot of percentage standard error of growth factors estimates for GEV distribution using simulation method for 85 stations for the 5, 10, 25, 50, 100,500-year return period.

Relative Standard Error of Quantile estimates(Pooled/GEV-Simulation) 45

40

35

30 % T 25 )/Q T ^ 20 se(Q 15

10

5

0 T=5 T=10 T=25 T=50 T=100 T=500

Figure 13.6 Box-plot of percentage standard error of quantile estimates for GEV distribution using simulation method for 85 stations for the 5, 10, 25, 50, 100,500-year return period.

139 13.4 Standard Error of Q T based on ungauged catchment estimate of

Qmed and pooled estimate of X T

Using the index flood method ˆ ˆ QT = Qmed × xˆT --- (13. 5)

The variance of the T-year event is approximated using Taylor series expansion as:

ˆ ˆ Var (QT ) = Var (Qmed × xˆT )

2 ˆ ˆ 2 ˆ ˆ ≈ E(xˆT ) Var (Qmed ) + E(Qmed ) Var (xˆT ) + 2E(Qmed ) ⋅ E(xˆT ) ⋅ cov( Qmed ⋅ xˆT ) --- (13. 6)

Ignoring effects arising from - bias in the sample median - bias in the sample L-moment ratio - covariance between the sample median and the estimated regional growth curve leads to

ˆ 2 ˆ 2 Var (QT ) ≈ (xT ) |m Var (Qmed ) + (Qmed ) |m Var (xˆT ) --- (13. 7)

The expression for var(Q T) is dominated by the var(Qmed) term as var(X T) affects only the 3 rd or 4 th decimal point in the value of the expression. Hence the expression effectively reduces to

se (QT ) ≈ X T ⋅ se (Qmed ) se (QT ) QT ≈ X T ⋅ se (Qmed ) (X T ⋅Qmed ) se (QT ) QT ≈ se (Qmed ) Qmed

36.0 Qmed se (QT ) QT ≈ ⋅ N Qmed

36.0 se (QT ) QT ≈ --- (13. 8) N

140 Therefore, se (QT ) QT is independent of T but dependent on value of N associated with the worth of a catchment descriptor based estimate of Qmed which is considered to be equivalent to either 1 year of AM data or perhaps a little more (e.g. Nash and Shaw, 1975; FSR, 1975, I, eq 4.15; Hebson and Cunnane, 1987). 36.0 Hence for ungauged catchments, with N=1, se (QT ) QT = = 0.36= 36%. 1 This result applies regardless of whether the growth curve is obtained using EV1 or GEV. This may seem counter intuitive but it is caused by the fact that the uncertainty in

QT is dominated by the uncertainty in Qmed. The 36% value of percentage standard error in this derivation is consistent with FSE =1.36 in equation (14. 9).

141 14 Guidelines for determining Q T

14.1 Introduction

QT may be determined from at-site data if sufficient data exist and the required T value is relatively small. In some instances although adequate at-site data may exist interpretation may be problematical e.g. an irregular shaped probability plot, in which case it may be necessary to use a combined at-site/pooling group approach.

For large return periods a combined at-site/pooling group approach is recommended. When no data exist at the subject site then the site's Qmed must be estimated from the catchment descriptor equation and enhanced where possible by use of suitable donor site data.

While most hydrologists are greatly influenced by the appearance of at-site data on probability plots when making choices about which procedures or estimates to adopt it must be borne in mind that random samples from any particular statistical population show tremendous inter-sample variation when displayed on probability plots. Some such samples do not always display convincing straight line behaviour even when the parent distribution is represented by a straight line on such a plot. Figure 4.1 show nine random samples from an EV1 population with Cv = 0.33, a value which is slightly higher than the average Cv for Irish A1 station data. Sample sizes are 25 and 50. It can be seen that in the smaller samples the departures from straight line behaviour is more striking than in the larger samples. Some of the sample of 25 would have been given a low score in the assessment of probability plots described earlier in ( Chapter 4). However most of the size 50 samples do not depart too markedly from straight line behaviour.

A series of factors must be borne in mind when selecting a design flood magnitude including

142 (i) The at-site or pivotal station's annual maxima probability plot on ordinary and on log scales (ii) The range of variation that can occur between random samples drawn from a single population – which emphasizes that a single at-site estimate may differ from the true value. (iii) One or more pooled growth curves (iv) Preference for 2 parameter over 3 parameter distributions in at-site mode (v) The overall statistical superiority of the at-site Qmed plus the pooled growth curve estimation method in homogenous pooling groups (c.f.Figure 10.1 (d)) (vi) The robustness of the at-site Qmed plus pooled growth curve estimation method to small departures from pooling group homogeneity (vii) Consideration of straight line versus concave downwards relations including the issue of upper bounds on flood magnitudes

(viii) Comparison of estimated Q T and corresponding water levels with ground truth i.e. does the proposed estimate make sense in the context of what has been previously observed or not observed in the vicinity? (ix) Careful consideration of possible underestimation when the at-site Qmed

plus the pooled growth curve estimate of Q T is smaller than some observed at-site floods (x) The credibility of large outliers and how they might be viewed, assessed or accommodated.

14.2 Determining Q T from at-site data

The required flood quantile may be estimated from a single record of at-site data providing • the record length N is at least 10 years long • the required quantile return period T is less than N, or not appreciably larger than N and certainly not more than 2N

Ordinarily a 2 parameter distribution should be used, either the EV1 or LN2. In the case of EV1 the parameters should be estimated by the method of probability weighted moments (or the L-moment equivalent). In the case of the LN2 the parameters should

143 be estimated from the mean and standard deviation of the logarithms of the data. Use of L-moments, as in the EV1 case, is statistically efficient as well as being simple to use. Use of ordinary of the logarithms of the data, as in the LN2 case, is also statistically efficient and simple to use and indeed simpler than use of L-moments in this case.

It is assumed that the basic statistics of the at-site data will have been calculated and that dimensionless statistics have been examined to see where they fall in the range of observed values among all gauged catchments. In other words it is assumed that the user has carried out a thorough check on the at-site data in the context of the overall data set for the country and especially in the context of data for similar type catchments. (We may define similarity using the distance measure dij in pooling group nomenclature, e.g. equation (10. 2) above).

The at-site data should be displayed on an EV1 based probability plot, using Gringorten Plotting Positions, and also on a lognormal probability plot, using Blom Plotting Positions as follows:

Gringorten Plotting Positions:

i − 44.0 y = − ln( − ln( F )) where F = , i = 2,1 ,...... N --- (14. 1) i i i N + 0.12

Blom Plotting Positions:

i − 8/3 y =Φ −1 (F )) where F = , i = 2,1 ,...... N --- (14. 2) i i i N +1/ 4 and Φ(y) is the Standardised Normal Distribution Function.

EV1 parameters u and α are estimated as

2M − M L α = 110 100 = 2 --- (14. 3) ln( )2 ln( )2

144 u = M 100 − .0 5772 α = L1 − .0 5772 α --- (14. 4)

Then

QT = u +αyT --- (14. 5) 1 where y = − ln( −ln( 1− )) T T

LN2 parameters are estimated as

1 n µ z = ∑ zi where z i = log 10 (Q i) --- (14. 6) n i=1

2/1  1 n  2 --- (14. 7) σ z =  ∑(zi − z )  n −1 i=1 

Then

ZT QT =10 --- (14. 8) 1 where Z =Φ −1 1( − ) and Φ −1 (F) is the standardised inverse Normal variate. T T

14.3 Determining Q T from Pooled data

14.3.1 Data available at the subject site

14.3.1.1 Determining Qmed

If a gauging station exists at the subject site then Qmed is obtained from the AM series of floods at that site. If the record is short Qmed so obtained is subject to larger standard error than if the record is long. However since gauged data are so much better that information obtained from a catchment descriptor based formula it is recommended to base the Qmed value on the observed data even if the record is short. The data of upstream and downstream gauging stations on the same river as well as on other neighbouring stations should be checked to see if the Qmed values obtained from the same period of record as those at

145 the subject site are smaller or larger than the long term Qmed at these sites. The ratio of long term Qmed to short term Qmed could be applied to the subject site's Qmed as a correction factor.

14.3.1.2 Determining X T

The growth factor should be determined from that the data of a pooling group selected by means of the distance measure dij of equation (10. 2) or (12. 4) as desired. If the 5T rule is applied to the 100 year flood estimation then a minimum of 500 station years of data should be included in the pooling group. It is suggested that such a pooling group be used for all return periods rather than constructing a separate group for every return period of interest. If the hydrologist wishes to depart from this advice and to use geographical regions or exclude certain types of catchment then the reasons for doing so will be expected to be documented. Consideration must also be given to choice of a 2 parameter or a 3 parameter distribution in line with the discussion below in Section 14.6 below. In the event that the at-site estimate of Q-T relation is steeper that the pooled one then consideration will have to be given to using a combination of the at-site estimate and the pooled estimate for design flood estimation. While there is a possibility that this could lead to over design it avoids the less desirable outcome of underdesign.

14.3.2 Subject site is ungauged

14.3.2.1 Determining Qmed

The value of Qmed will have to initially be obtained from the catchment descriptor data using either the WP2.3 Rural Equation

Qmed = .1 083 ×10 −5 AREA .0 938 BFIsoils − 88.0 SAAR .1 326 FARL .2 233 DRAIND .0 334 rural (14. 9) × S1085 .0 188 1( + ARTDRAIN )2 .0 051 or a modified version which takes urbanization into account through an urban adjustment factor, UAF. The Factorial Standard Error (FSE) of the above equation is 1.36. Use of a donor catchment, as described above in Section 9.3, is recommended or in the absence of a donor catchment on the same river a suitably chosen analog catchment

146 might be used. However an unsuitable analog catchment may add very little useful information. The equation for Qmed is applied with the catchment descriptors of the donor catchment and the adjusted Qmed is obtained at the subject site by use of equation (9. 2).

As shown in Section 9.3 use of equation (9. 2) introduces a noticeable improvement in the estimate of Qmed.

14.3.2.2 Determing X T

The same remarks apply as in the Section 14.3.1.2 above .

14.4 Some examples

Example 1 – Data series shows good straight line behaviour on a probability plot

60 25016 RIVER CLODIAGH @ RAHAN

50 AMF data At-site EV1 fit 40 Pooled EV1 fit EV1 pooled GEV fit 30 AMF(m3/s) 20

10

0 2 5 10 25 50 100 500 -2-10 1 2EV1 3 y 4 5 6 7 8

Figure 14.1 At-site and pooled quantiles for station 25016

A good example of such behaviour is shown by the River Clodiagh at Rahan (Stn No 25016) with 42 years of record. In Table 4.3 this station was assigned scores of 4 and L2 in the linear and non-linear pattern assessments where 4 means “good agreement with straight line” and L2 means “little deviation from straight line”. The Cv = 0.22 and

147 Hazen Skewness = 0.63 which is less than the theoretical value of 1.14 in the EV1 distribution. It seems reasonable to suggest that the 50 year or even 100 year return period flood could be estimated by the EV1 distribution fitted to the at-site data in this case. The standard error of Q 100 in such a case would be approximately 6% to 10%, indicated in Figure 13.1 providing the underlying parent distribution is EV1. The at-site Cv is lower than the national average. The pooled growth curves shown in Figure 14.2 are in general agreement with the at-site data and in this case the pooled estimate should be used for design flood determination.

Example 2 – Data series shows good straight line behaviour on a probability plot but lack of agreement between at-site and pooled estimates

A very good example of such behaviour is shown by the River Ryewater at Leixlip (Stn No 9001) with 48 years of record. The Cv = 0.44 and Hazen Skewness = 1.17 which is practically equal to the theoretical value of 1.14 in the EV1 distribution. It seems reasonable to suggest that the 50 year or even 100 year return period flood could be estimated by the EV1 distribution fitted to the at-site data in this case. The standard error of Q 100 in such a case would be approximately 4% to 8%, indicated in Table13.1 providing the underlying parent distribution is EV1. The at-site Cv is higher than the national average and higher than that of other stations in the hinterland. Hence the pooled growth curves are lower than the at-site curve which is shown in Figure 14.2.

However much faith is place in the superiority of the pooled estimate in general, it is a fact in this case that the pooled estimate of Q 50 has been exceeded 5 times in 48 years already. This evidence cannot be overlooked when a final design flood is being selected even for long return periods. It could be that the observed sample displays a steepness on the probability plot that is at the upper end of the scale of steepness and that the sample actually has come from a population with a less steep growth curve i.e. with lower Cv. However no practical design project would ignore the at-site data in such a case.

148 140 9001 RIVER RYEWATER @ LEIXLIP 120 AMF data At-site EV1 fit 100 Pooled EV1 fit 80 Pooled GEV fit

60 AMF(m3/s)

40

20 2 5 10 25 50 100 500 0 -2-10 1 2EV1 y 3 4 5 6 7 8

Figure 14.2 At-site and pooled quantiles for station 9001

Other very good examples of straight line behaviour are shown by 14006 Barrow at Pass Bridge (N = 51, Cv = 0.20 and Hazen Skewness = 1.46) and 26007 Suck at Bellagill (N = 53, Cv = 0.19, Hazen Skewness = 1.05). Because these have low Cv values the pooled growth curves are lower than the at-site one.

Caveat: Random samples drawn from an EV1 distribution do not always display straight line behaviour on probability plots. The likelihood of this happening decreases as the sample size gets larger as seen in Figure 4.1(a, b). Conversely some random samples drawn from non-EV1 distributions could display straight line behaviour on probability plots. The proportion which does so decreases of course as the departure of the parent distribution from EV1 increases.

Example 3 – Data series shows concave upward behaviour on a probability plot or displays one or more high outliers.

A very good example of such behaviour is shown by the River Dodder at Waldron's Bridge (Stn No 9010) even though it has a relatively short 19 years of record. Even if the underlying parent growth curve is a straight line the short record makes it much more likely to observe departure from linearity on the probability plot. The Cv = 0.86 and Hazen Skewness = 3.24, both of which are among the highest values recorded among the entire FSU data set. While a three parameter GEV distribution provides a

149 good graphical fit (see in Figure 14.3) in this case it has to be borne in mind, from a theoretical point of view, that GEV at-site estimates have high standard error especially with short data sets. That means that other similar sized later records might not show the same pattern or be so steep. Hence, notwithstanding the good at-site fit, it would normally be recommended that a regional pooling estimate be used for flood estimation at this site except for low, T < 25 years. Because most other stations included in the pooling group will more than likely have smaller Cv and skewness than that of the at- site data the pooling group based growth curve is lower than the at-site growth curve. However the pooled growth curve looks extremely low in comparison with the observed data.

The hydrologist then has to balance the weight of evidence provided by several publications relating to the advantages of the regional pooling method over the at-site method and his/her own beliefs about the representative nature of the observed data and whether the behaviour shown could be repeated in another similar length of record. For instance the at-site GEV estimate of Q 100 ≈ 350 cumec. One has to ask if this is credible. The largest flood on record (269 cumec) was caused by Hurricane Charlie in 1986 and its counterpart on the neighbouring Dargle catchment had earlier been equalled twice in the previous 80 years. Undoubtedly therefore the 269 cumec on the

Dodder could occur again but this does not necessarily support the figure of Q 100 = 350 cumec. On the other hand the pooling group estimate of Q 100 is seen on the plot to have been exceeded 4 times in 19 years which indicates that in this case the pooling based curve is definitely too low.

This sort of dilemma always presents itself when the subject site record has a Cv and skewness which is considerably larger than the regional average, as has occurred above in Example 2. In such cases a prudent approach has to be adopted where "Ground Truth" cannot be overlooked. In such cases it would be wise not to place too much faith in estimates of Q T for large T.

150 400 9010 RIVER DODDER @ WALDRON'S BRIDGE 350 AMF data

300 Atsite GEV fit Pooled GEV fit 250 Pooled EV1 fit 200

AMF(m3/s) 150

100

50 2 5 10 25 50 100 500 0 -2-10 1 2EV1 y 3 4 5 6 7 8

Figure 14.3 At-site and pooled quantiles for station 9010

Other very good examples of this behaviour are 29011 River Dunkellin at Kilcolgan Bridge (N = 22, Cv = 0.30, Hazen Skewness = 3.37) and 36015 Finn at Anlore (N = 33, Cv = 0.32, Hazen Skewness = 2.62).

It is noticeable that these three examples are based on records which considerable shorter that the longest records available in the study and also that the Dodder Catchment is more urbanized than many other Irish catchments.

Example 4 – Data series shows concave downward behaviour on a probability plot or displays a number of high values which are almost equal and possibly some very low values among the smallest values in the series.

A very good example of such behaviour is shown by the River Newport at Barrington's Bridge (Stn No 25002) with 51 years of record. The Cv = 0.16 and Hazen Skewness = - 0.65 both of which are among the lowest values recorded among the entire FSU data set. While a three parameter GEV distribution provides a very good graphical fit (see in Figure 14.4) in this case it has to be borne in mind, from a theoretical point of view, that GEV at-site estimates have high standard error especially with short data sets although 51 years is not considered short. In addition the k value is positive which implies an upper bound on the fitted distribution. This is not reasonable from a hydrological point of view and while it might be satisfactory to use the fitted distribution up to return periods of 25 years it is recommended that a regional pooling estimate be used for flood

151 estimation at this site. Because most other stations included in the pooling group will more than likely have larger skewnesses than that of the at-site data the pooling group based growth curve is steeper than the at-site growth curve. This may require that the pooling based estimate for high return periods be interrogated further in the context of "Ground Truth" considerations referred to above. It is also known that this river as well as the neighbouring Malkear river has been embanked since the 1920’s and these embankments have an effect on flood flows as they are overtopped in one out of every 5 years. See also further discussion below on the issue upper bounds.

160 25002 RIVER NEWPORT @ BARRINGTONS BRIDGE 140 AMF data 120 Atsite GEV fit Pooled GEV fit 100 Pooled EV1 fit 80

AMF(m3/s) 60

40

20 2 5 10 25 50 100 500 0 -2-10 1 2EV1 y 3 4 5 6 7 8

Figure 14.4 At-site and pooled quantiles for station 25002

Other very good examples of this behaviour are provided by 25021 River Little Brosna at Croghan (N = 44, Cv = 0.14, Hazen Skewness = -0.13) and 34024 Pollagh at Kiltimagh (N = 29, Cv = 0.0.16, Hazen Skewness = -2.095). It is noticeable that two of these three examples are based on records which are considered to be among the longest in the study.

Example 5 – Data series contains one large outlier or two large outliers which are noticeably out of line from the other data points in a probability plot.

A very extreme example of such behaviour is provided by 8009, River Ward at Balheary where 10 of 11 values available are less than 12 cumec and the largest is

152 recorded as 53.6 cumec which was a summer occurrence in the year 1992-93. The at- site and pooled quantile curves are given in Figure 14.5. The small sample size exacerbates the problem of trying to estimate a design flood for this location. This station has by far the highest recorded skewness, 5.58, of the whole data set. It has to be acknowledged that skewness measured from such a short record is unreliable. The median value, which is not affected by the largest flood, is 5 cumec and even assuming a conservative growth factor of 2.5, the Q 100 estimate would be 12.5 cumec. The probability of obtaining a largest value in 15 years which is more than 4 times the estimated Q 100 is virtually zero.

As a result of this statistical incompatibility there can be no unique recommendation made in this case. The following points should be considered • The meteorological conditions leading to the 1993 summer storm should be examined and the likelihood of these being repeated anywhere in the region (including the subject site) should be assessed • Even if the rating curve or other features of the measurement are less than satisfactory the water level achieved in the locality will be known. This could be used as the basis of design of important works and floor levels of dwellings. • The hydrologist has to balance between the belief that the 1992 flood was so large that it could never happen again or the belief that if it happened once it could happen again. For instance the Hurricane Charlie 1986 flooding in Bray was so large that many believed it was unrepeatable until it was discovered that almost exactly similar events had occurred in 1904 and 1932.

Hence for small return period events a standard at-site Qmed and a pooled estimate of

XT could be used, but for long return periods the above points have to be taken into account.

153 90 8009 RIVER WARD @ BALHEARY 80 AMF data Atsite EV1 fit 2 5 10 25 50 100 500 70 Pooled EV1 fit 60 Pooled GEV fit 50

40 AMF(m3/s) 30

20

10

0 -2-10 1 2EV1 y 3 4 5 6 7 8

Figure 14.5 At-site and pooled quantiles for station 8009

Further examples of outliers but not of such extreme nature are provided by 36021 River Yellow at Kiltybarden , by 36031 River Cavan at Lisdarn and by 26006, River Suck at Willsbrook. The same points as made above apply to these cases also.

Example 6 – Data series shows irregular behaviour of one sort or another e.g an elongated S shape or a noticeable change of slope between lower and upper segments of the graph.

Examples are 9002 River Griffeen at Lucan (N = 25 years), 15001 Kings River at Annamult ( N = 42 years) and 26008 River Rinn at Johnston's Bridge (N = 50 years). The at-site and pooled quantile curves for these stations are displayed in Figure 14.6, Figure 14.7 and Figure 14.8 respectively.

The River Griffeen is particularly problematical as there is a distinctive increase in slope caused by the 4 largest floods in the record. It should be noted also that it has experienced an increase in its urban fraction over the past 2 decades especially. Qmed is 5.25 and again taking a conservatively large value of X 100 = 2.5 gives Q 100 = 13 cumec, a value which has been well exceeded 3 times in 25 years. Additional local knowledge must be gained about the meteorological and physical conditions leading to these floods. The observed largest water levels will also have to act as a guide as to what design conditions to adopt.

154

The 15001 case is one of a probability plot with reverse curvature or elongated S shape. However the overall appearance is too far from a straight line, in the context of Figure 4.1(a, b) and hence application of a pooled growth factor should provide a satisfactory

QT estimate.

The 26008 case is a more extreme version of 15001. While no single distribution could describe the at-site probability plot the upper end is not too dissimilar to some of the samples in Figure 4.1(a, b). Hence application of a pooled growth factor should provide a satisfactory estimate of Q T.

40 9002 RIVER GRIFFEEN @ LUCAN AMF data 35 Atsite EV1 fit 30 Pooled EV1 fit

25 Pooled GEV fit

20

AMF(m3/s) 15

10

5 2 5 10 25 50 100 500 0 -2-10 1 2EV1 y 3 4 5 6 7 8

Figure 14.6 At-site and pooled quantiles for station 9002

250 15001 RIVER Kings @ Annamult AMF data 200 At-site EV1 fit Pooled EV1 fit 150 Pooled GEV fit

100 AMF(m3/s)

50

2 5 10 25 50 100 500 0 -2-10 1 2EV1 3 y 4 5 6 7 8

Figure 14.7 At-site and pooled quantiles for station 15001

155

60 26008 RIVER RINN @ JOHNSTON'S BRIDGE

50 AMF data At-site EV1 fit 40 Pooled EV1 fit Pooled GEV fit 30 AMF(m3/s) 20

10

2 5 10 25 50 100 500 0 -2-10 1 2EV1 y 3 4 5 6 7 8

Figure 14.8 At-site and pooled quantiles for station 26008

14.5 Flood growth curves with an upper bound or without an upper bound?

Three parameter distributions, such as GEV and GLO, have an upper bound if the shape parameter k >0. This does not seem hydrologically realistic regardless of whether fitted to at-site or pooled data because some unprecedented rainfall could occur in the future which could have a runoff or routing mechanism different to all previous floods and which could therefore produce a flood much in excess of the largest flood on record. When k>0 a two parameter distribution growth curve (i.e. a straight line growth curve) should be estimated instead. In such a case it is nevertheless necessary to use judgement, following assessment of all relevant physical factors and water levels reached during previous floods, before a design flood can be specified. There is a possibility that a straight line two parameter distribution, when extrapolated to estimate very rare floods, may produce flood estimates that may seem implausibly large in the context of all relevant physical factors and the hydrologist's experience i.e. positive bias as in Figure 10.1(b). Hence the hydrologist will have to apply judgement to the estimation process in such a case.

156 14.6 Choice between at-site and pooled estimates and between 2 and 3 parameter pooled estimates

In ordinary circumstances a three parameter distribution should not be used with at-site data. An exception could be made if the data series is very long, say > 50 years, and the required return period is small, say ≤ 25 years. A three parameter distribution is more flexible and may give a better fit visually on a probability plot but the estimation of a 3 rd parameter has the effect of increasing the standard error of estimate of the estimated quantile, see Figure 10.1(c). The contrary can also occur where a three parameter distribution, fitted to individual at-site data, can give a Q-T relation that is not intuitively acceptable because of extreme upwards or downwards curvature.

In the event that the probability plot of available at-site data shows a marked departure from straight line behaviour, either by way of upward or downward curvature or because of some form of an elongated S shape, then consideration has to be given to using a pooling group estimate of the growth factor X T for use with the at-site value of Qmed. However even this may not remove all doubts about the suitability of the method chosen as the pooled growth curve may disagree significantly from whatever general pattern is shown by the at-site data as in Examples 1 and 2 above. In such a case a decision has to be made whether to trust the pooled growth curve as a matter of good practice or principle, based on the well documented reduced standard error of estimate and robustness of the pooling method, e.g. Figure 10.1(d). However when the at-site data display Cv and skewness greatly in excess of the pooling group average it is not always easy to trust the pooled growth curve.

If the available at-site record is long it may be worthwhile displaying the maximum of non-overlapping pairs or triplets of years to see what kind of pattern is displayed on a probability plot

If a very large flood is observed during the period of record the question arises as to whether it should it over-ride any more modest estimate of Q T obtained by a pooling group approach or whether a weighted combination of the pooling group estimate and the at-site estimate should be adopted. If a combination is used the weights to be given

157 to the two components of the combination cannot be specified by any rule based on scientific evidence but must be chosen in an arbitrary, however one would hope a reasonable, way.

In the case of a very large observed flood it is also possible, under certain assumptions, to calculate the probability that such a large flood could occur. For instance if a sample of 50 floods are drawn randomly from an EV1 distribution with Cv = 0.3, then the probability that the largest flood would exceed 1.25Q 100 , where Q 100 is the population value of the 100 year return period flood, is less that 7% and the probability that it would exceed 1.5Q 100 is less than 1%. In practice the true value of Q100 is unknown and has to be replaced by an estimate which makes these percentage probabilities less reliable. Nevertheless such calculations could be adapted to provide an informal test of the hypothesis that the observed outlier is consistent with the assumed parent population. [The quoted probabilities are based on the fact that if Q is EV1(u, α) then Qmax in N years is EV1(u+ αlnN, α) ].

If a data series shows negative skewness and 3 or 4 largest floods that differ from one another by only a very small amount then the data series will usually exhibit downwards curvature on a probability plot. Any 3 parameter distribution fitted to the at-site data will usually have an upper bound which is usually not very much larger than the largest recorded flood. However it is unreasonable to expect that there is an upper bound of such modest size and in such a case the advice is to using the pooled growth curve rather than the at-site one. If the pooled growth curve is also concave downwards then a two parameter distribution should be fitted to the pooled data so as to avoid the upper bound. However in that case care has to be taken so as not to specify design floods, for values of T ≈ N, which are greatly in excess of the observed floods and judgement has to be used in such cases. On the other hand some unknown large rainfalls could occur in the future which would cause the catchment to respond differently to those storms which caused the recorded floods and it might be unwise not to accept the two parameter based estimates for higher return periods, T>>N.

158 15 Summary and Conclusions

15.1 Data

Data from approximately 200 gauging stations in the Republic of Ireland, from the arcives of OPW, EPA and ESB were available. Of these 115 were of quality Grade A while 67 were Grade B. Data of 17 stations which had both pre and post drainage records were also available. Summary statistics and probability plots were prepared for all stations. However no further inferences were drawn from the Grade B stations data.

15.2 Descriptive Statistics

Examination of the data for trend and randomness shows that up to 10% of records display some trend or lack of randomness. The usual assumption that annual maximum floods are statistically independent and identically distributed, was nevertheless assumed as in most other national flood studies supported by the fact that 90% of records did not show significant trend or lack of randomness.

The descriptive statistics show that Irish AM data have low C v (average of 0.3) and low skewness (average H-Skewness of 1.0) whether measured by the traditional statistics or by L-moment statistics. Examination of probability plots and moment ratio diagrams suggest that among 2-parameter distributions the Extreme Value Type I (EV1) and lognormal (LN2) are the most appropriate, the exception being for a number of stations that display very low skewness values which in some cases is caused by the 3 or 4 largest values in the record not being appreciably different from another. Explanations for this latter phenomenon were sought by considering flood volumes and by referring to the catchment descriptor, Flood Attenuation Index, but to no avail.

15.3 Seasonal Analysis

Seasonal analysis shows that two thirds of AM flows occur during the winter months of October to March and at 11 stations no AM floods occurred during the summer months. July is the least likely month in which AM floods occur while August and September

159 are the most likely summer months to have AM floods. At 20 out of the 202 stations examined the largest flow on record occurred during summer, 8 of them in August, while 45% of stations had their largest flood on record during December.

15.4 m-year Maxima

As well as annual maxima, maxima of no-overlapping 2 and 3 year periods were also examined on probability plots. The pattern seems to be that the behaviour of m-year maxima i.e. straight line, concave upwards or concave downwards, is very much predictable from the annual maximum behaviour and no new more stable pattern could be gleaned from the m-year maxima than from the annual maxima.

15.5 Estimation of Design Flood

The estimation of the T year flood quantile is considered for each of the full range of possibilities that can arise in practice, from gauged to ungauged catchments and from short to long flow records. The index flood method is recommended whereby Q T is expressed as

QT = Qmed X T

where Q T is the flood of return period T years, Qmed is the median of the annual maximum at the subject site and X T is the growth factor appropriate to the subject site, estimated either from at-site data or from data of an appropriate homogeneous pooling group.

Where at-site data are available Qmed is estimated from those data. If the data record is long the Qmed obtained from it is used on its own. If the available flow record is short or if Qmed is obtained, in the ungauged case, from a catchment descriptor based equation then the Qmed may be adjusted with the assistance of data from a suitable donor or analog site. Tests conducted on the data of 38 gauging stations located in 10 different catchments were used to assess the efficacy of different data transfer methods for improving the Qmed estimate. Data transfer from downstream donor stations was found to be a little superior to data transfer from upstream donor stations and both were

160 found to be superior to use of analog stations. Use of the regression based adjustment technique was found to be superior to use of area based adjustment. This reflects the fact that flood magnitude depends on more than catchment area alone.

Estimation of X T may be based on at-site data if a sufficiently long data record exists at the subject site. Otherwise X T is estimated from the dimensionless L-Cv and L- Skewness values obtained by averaging these quantities obtained over the stations in a homogenous pooling group of stations. The members of the pooling group are chosen with the help of a Euclidean distance or similarity measure, dij. Tests have been carried out and reported on into the effect on the estimated value of X T of catchment area, peat coverage, lakiness (as measured by FARL), geographical location and period of record varying from the 1950s to the 1990s. None of these were judged to be sufficiently influential to make special provision for them. Also, tests were carried out and reported on about the effectiveness of different combinations of catchment descriptors in the definition of dij and the most effective ones were AREA, SAAR and BFI soils.

The standard errors associated with estimates of XT and Q T have also been investigated.

The standard error of Q T estimated by the index flood method is dominated by se(Q med ) and se(Q T), expressed as a percentage, varies only slightly with T. When Qmed is estimated from at-site data and X T is estimated from a pooling group containing approximately 500 station years of data then se(Q T) is of the order of 5% to 10 % Q T regardless of return period. If Qmed is estimated from a catchment descriptor based formula alone and X T is estimated from a pooling group containing approximately 500 station years of data then se(Q T) is of the order of 36% Q T.

Chapter 14 deals with guidelines for the estimation of Q T, from both at-site and pooled data. Several points that need to be taken into account in practical cases are outlined and the point is emphasised that the blind use of a prescribed method can sometimes lead to a Q T estimate which is not always in accordance with “ground truth”. This is especially the case when the at-site data have a higher than average Cv or skewness. A number of examples are discussed, most of which throw up practical problems of the type met in practice where difficult choices have to be considered. Discussion is also provided on the issues of flood distributions with upper bounds and on the choice between two parameter and three parameter distributions. A number of examples are

161 discussed, most of which throw up practical problems of the type met in practice where difficult choices have to be considered. In some of these cases the user may be recommended to consider using an at-site based estimate in preference to the generally recommended regional pooling based method. What this indicates is that flood estimation cannot be reduced to a single strict formula based procedure but that individual analysts must make choices which depend on the problem circumstances and which take into account their own knowledge and experience.

162 References

Burn, D.H.,(1990),Evaluation of regional flood frequency analysis with a region of influence approach. Wat. Res. Res. 26(10), 2257-2265.

Dalrymple,T., (1960), Flood frequency methods. U.S. Geol. Survey, Water Supply Paper 1543 A, pp. 11–51, Washington.

Foster, H. A. (1924), Theoretical frequency curves and their application to engineering problems. Trans. ASCE, 87, 142-173.

Flood Studies Report (1975), Natural Environment Research Council, London, vol. I.

Flood Estimation Handbook (FEH), (1999), Institute of Hydrology, U.K.

Fiering, (1963 ), Use of correlation to improve estimates of the mean and variance. Statistical studies in Hydrology, Geological Survey professional paper 434-C, US Gov. Printing Office.

Greis, N.P. and Wood, E.F., (1981), Regional flood frequency estimation and network design. Water Resour. Res. , 17(4) , 1167 – 1177.

Hazen, A.,(1930), Flood flows, John Wiley, New York. Hebson, C.S. and Cunnane, C., (1988), Assessment of Use of At-site and Regional Flood Data for Flood Frequency Estimation. In V.P. Singh (Editor), Hydrologic Frequency Modelling, D. Reidel, Publ. Co., Dordrecht, pp. 433-448.

Hosking, J.R.M., Wallis, J.R. and Wood, E.F., (1985a), An appraisal of the regional flood frequency procedure in the U.K. Flood Studies Report. Hydrol. Sci. J., 30(1), 85- 109.

Hosking, J.R.M., Wallis, J.R. and Wood, E.F., (1985b), Estimation of the generalised extreme value distribution by the method of probability weighted moments. Technometrics, 27(3), 251-261.

Hosking, J.R.M. and Wallis, J.R., (1997), Regional flood frequency analysis: An approach based on L-Moments, Cambridge Univ. Press.

Irish Academy of Engineering (2007), Ireland at Risk – Impact of Climate Change on the Water Environment. Proc Workshop held at RDS, Ballsbridge, Dublin, May 2007(www.iae.ie/publications).

Kjeldson, T., Jones,D., Bayliss, A., Spencer, P., Surendran, S., Laeger,S., Webster, P. and McDonald, D.,(2008), Improving the FEH Statistical Method . In: Flood & Coastal Management Conference 2008, University of Manchester, 1-3 July 2008 . Environment Agency/Defra. Available at UK Natural Environment Research Council Open Research Archive, http://nora.nerc.ac.uk/3545/ , 10 pages.

163

Lettenmaier, D.P., Wallis, J.R. and Wood, E.F.,(1987), Effect of heterogeneity on Flood frequency estimation. Water Resour. Res., 23(2), 313-323.

Lu & Stedinger (1992), Variance of two-and three-parameter GEV/PWM quantile estimators: formulae, confidence intervals, and a comparision. J. of Hydrol., 138(1-2), 247-267.

Nash, J.E. and Shaw, B.L.,(1965), Flood frequency as a function of catchment characteristics. Institution of Civil Engineers, Symposium on River Flood Hydrology, (published 1966), 115-136.

OPW, (1981), Estimating Flood Magnitude/ Return Period Relationships and the Effect of Catchment Drainage.

WMO, (1989), Operational Hydrology Report no. 33: Statistical Distributions for Flood Frequency Analysis.

164

Appendix 1

TREND ANALYSIS ANNUAL MAXIMUM FLOWS IN IRISH RIVERS

See separate volume

165

Appendix 2

Assessment of effectiveness of Donor and Analog Catchments in adjusting the Catchment Descriptor based estimate of Qmed

166 Area Adjustment Subject_site Donor_up Donor_down1 Donor_down2 Donor_down3 Analog Regression St.No. 16002 16004 16008 16009 16011 25021 Obs Qmed 52.7 21.4 92.3 162.2 223.0 28.6 Area km2 512 236 1120 1602 2173 493 Adj.Qmed 52.7 46.4 42.2 51.8 52.5 29.7 48.3 %obs Qmed 100 88 80 98 100 56 92

Minimum difference via donors 0 Maximum difference via donors 20 Average difference via donors 8 Difference from regression estimate 8

16002 Suir @ Beakstow n, Area=512 km 2 (Area Adjustm ent)

120

1602 2173 100 236 1120 80

493 60

%obsQmed 40

20

0 Donor_up Donor_down1 Donor_down2 Donor_down3 Analog Regression Data Transfer Category

(a)

167 Regression Adjustment Subject_site Donor_up Donor_down1 Donor_down2 Donor_down3 Analog Regression St.No. 16002 16004 16008 16009 16011 6013 Obs Qmed 52.7 21.4 92.3 162.2 223.0 27.3 Area km2 512 236 1120 1602 2173 309 Adj.Qmed 52.7 34.4 43.3 53.0 59.7 40.5 48.3 %obs Qmed 100 65 82 101 113 77 92

Minimum difference via donors 1 Maximum difference via donors 35 Average difference via donors 17 Difference from regression estimate 8

16002 Suir @ Beakstow n, Area=512 km 2 (Regression Adjustment)

120 2173 1602 100 1120 80 309 236

60

%obs Qmed 40

20

0 Donor_up Donor_down1 Donor_down2 Donor_down3 Analog Regression Data Transfer Category

Figure A2. 1 Performance of data transfer method of estimating Qmed at Station 16002, Suir at Beakstown, (a)using area based adjustment (b) using regression based adjustment.

168 Area Adjustment Subject_site Donor_up2 Donor_up1 Donor_down1 Donor_down2 Analog Regression St.No. 16008 16004 16002 16009 16011 14006 Obs Qmed 92.32 21.4 52.7 162.2 223 80.52 Area km2 1120 236 512 1602 2173 1096 Adj.Qmed 92.32 101.43 115.20 113.41 114.94 82.3 103.1 %obs Qmed 100 110 125 123 124 89 112

Minimum difference via donors 10 Maximum difference via donors 25 Average difference via donors 20 Difference from regression estimate 12

16008 Suir @ New Bridge, Area=1120 km2 (Area Adjustment)

140 512 1602 2173 120 236

100 1096

80

60 %obs Qmed 40

20

0 Donor_up2 Donor_up1 Donor_down1 Donor_down2 Analog Regression Data Transfer Category

169 Regression Adjustment Subject_site Donor_up2 Donor_up1 Donor_down1 Donor_down2 Analog Regression St.No. 16008 16004 16002 16009 16011 26007 Obs Qmed 92.32 21.4 52.7 162.2 223 88.1 Area km2 1120 236 512 1602 2173 1207 Adj.Qmed 92.32 73.35 112.34 112.97 127.44 83.4 103.1 %obs Qmed 100 79 122 122 138 90 112

Minimum difference via donors 21 Maximum difference via donors 38 Average difference via donors 26 Difference from regression estimate 12

16008 Suir @ New Bridge, Area=1120 km2 (Regression Adjustment)

160 2173 140 512 1602 120 112

100 1207 236 80

60 %obsQmed

40

20

0 Donor_up2 Donor_up1 Donor_down1 Donor_down2 Analog Regression Data Transfer Category

Figure A2. 2 Performance of data transfer method of estimating Qmed at Station 16008, Suir at Newbridge, (a)using area based adjustment (b) using regression based adjustment.

170

Area Adjustment Subject_site Donor_up Donor_down1 Donor_down2 Donor_down3 Analog Regression St.No. 18050 18016 18048 18006 18002 16003 Obs Qmed 124.5 79.7 220 286 344 29.9 Area km2 244.6 113 881 1058 2333 246 Adj.Qmed 124.5 172.4 61.1 66.1 36.0 29.8 82.9 %obs qmed 100 138 49 53 29 24 67

Minimum difference via donors 38 Maximum difference via donors 71 Average difference via donors 52 Difference from regression estimate 33

18050 Blackwater @ Duarrigle, Area=244.6 km2 (Area Adjustment)

160 113 km2 140

120

100

80

1058 km2 %obs qmed 60 881 km2

40 2333 Km2 246 km2 20

0 Donor_up Donor_down1 Donor_down2 Donor_down3 Analog Regression Data Transfer Category

(a)

171

Regression Adjustment Subject_site Donor_up Donor_down1 Donor_down2 Donor_down3 Analog Regression St.No. 18050 18016 18048 18006 18002 35011 Obs Qmed 124.5 79.7 220 286 344 115.3 Area 244.6 113 881 1058 2333 293 Adj.Qmed 124.5 121.2 100.9 130.1 124.2 76.1 82.9 %obs qmed 100 97 81 104 100 61 67

Minimum difference via donors 0 Maximum difference via donors 19 Average difference via donors 7 Difference from regression estimate 33

18050 Blackwater @ Duarrigle, Area=244.6 km2 (Regression Adjustment)

160

140

120 1058 km 2 113 km 2 2333 Km 2 100 881 km 2 80 67 293

%obsqmed 60

40

20

0 Donor_up Donor_down1 Donor_down2 Donor_down3 Analog Regression Data Transfer Category

Figure A2. 3 Performance of data transfer method of estimating Qmed at Station 18050, Blackwater at Duarrigle, (a)using area based adjustment (b) using regression based adjustment.

172

Area Adjustment Subject_site Donor_up Donor_down1 Analog Regression St.No. 24008 24004 24082 36010 Obs Qmed 119.1 50.6 140 66.8 Area km2 805 246 764 774 Adj.Qmed 119.1 165.7 147.5 69.4 96.2 %obs qmed 100 139 124 58 81

Minimum difference via donors 24 Maximum difference via donors 39 Average difference via donors 31 Difference from regression estimate 19

24008 Maigue @ Castleroberts, Area=805 km 2 (Area Adjustment)

160 246 140 764 120

100

80 774

%obs qmed 60

40

20

0 Donor_up Donor_down1 Analog Regression Data Transfer Category

173

Regression Adjustment Subject_site Donor_up Donor_down1 Analog Regression St.No. 24008 24004 24082 14006 Obs Qmed 119.1 50.6 140 80.5 Area km2 805 246 764 1063 Adj.Qmed 119.1 113.1 139.1 73.6 96.2 %obs qmed 100 95 117 62 81

Minimum difference via donors 5 Maximum difference via donors 17 Average difference via donors 11 Difference from regression estimate 19

24008 Maigue @ Castleroberts, Area=805 km2 (Regression Adjustment)

160

140 764 120 246 100

80 1063

%obsqmed 60

40

20

0 Donor_up Donor_down1 Analog Regression Data Transfer Category

Figure A2. 4 Performance of data transfer method of estimating Qmed at Station 24008, Maigue at Castleroberts, (a) using area based adjustment (b) using regression based adjustment.

174

Area Adjustment Subject_site Donor_up Donor_down1 Donor_down2 Analog Regression St.No. 26002 26006 26005 26007 35005 Obs Qmed 56.6 24.2 93.2 88.2 75.4 Area km2 626 182 1050 1184 642 Adj.Qmed 56.6 83.3 55.6 46.6 73.5 73.8 %obs Qmed 100 147 98 82 130 130

Minimum difference via donors 2 Maximum difference via donors 47 Average difference via donors 22 Difference from regression estimate 30

26002 Suck @ Rockwood, Area=626 km2 (Area Adjustment)

160 182 140 642 642

120

1050 100 1184 80

%obsQmed 60

40

20

0 Donor_up Donor_down1 Donor_down2 Analog Regression Data Transfer Category

175

Regression Adjustment Subject_site Donor_up Donor_down1 Donor_down2 Analog Regression St.No. 26002 26006 26005 26007 30007 Obs Qmed 56.6 24.2 93.2 88.2 62.9 Area km2 626 182 1050 1184 152 Adj.Qmed 56.6 54.7 53.9 59.7 75.1 73.8 %obs Qmed 100 97 95 105 133 130

Minimum difference via donors 3 Maximum difference via donors 5 Average difference via donors 4 Difference from regression estimate 30

26002 Suck @ Rockwood, Area=626 km2 (Regression Adjustment)

160

140 152

120 1184 182 1050 100

80

60 %obsQmed

40

20

0 Donor_up Donor_down1 Donor_down2 Analog Regression Data Transfer Category

Figure A2. 5 Performance of data transfer method of estimating Qmed at Station 26002, suck at Rockwood, (a) using area based adjustment (b) using regression based adjustment.

176 Appendix 3

Flood volumes associated with largest peaks on convex probability

177

7006 RIVER MOYNALTY @ FYANSTOWN 2 Annual Maximum Floods 1986 to 2004.(no missing years) A(km )= 176.00 N= 46 Mean 26.726 Median 27.933 Std.Dev. 5.493 CV 0.206 HazenS. -1.088 40 EV1 35 '93 30 '00 '04 '95 w inter peak summer peak 25

20

AMF(m3/s) 15

10

5 2 5 10 25 50 100 500 0 -2-1 0 1 2 3 4 5 6 7 EV1 y 40 Hydrograph during Maximum Flood in the year 1993 Red zone= Vol. of 1day 35 Yellow zone= Vol. of 2day Blue zone= Vol. of 1week 30

25

20

15 Discharge (m3/s)

10

5

0 02/10/1993 00:00 04/10/1993 00:00 06/10/1993 00:00 08/10/1993 00:00 10/10/1993 00:00 12/10/1993 00:00 Time 35 Hydrograph during Maximum Flood in the year 1995 Red zone= Vol. of 1day 30 Yellow zone= Vol. of 2day Blue zone= Vol. of 1week

25

20

15 Discharge (m3/s)

10

5

0 22/11/1995 00:00 24/11/1995 00:00 26/11/1995 00:00 28/11/1995 00:00 30/11/1995 00:00 02/12/1995 00:00 04/12/1995 00:00 06/12/1995 00:00 Time

178 35 Hydrograph during Maximum Flood in the year 2000 Red zone= Vol. of 1day 30 Yellow zone= Vol. of 2day Blue zone= Vol. of 1week

25

20

15 Discharge (m3/s)

10

5

0 31/10/2000 00:00 02/11/2000 00:00 04/11/2000 00:00 06/11/2000 00:00 08/11/2000 00:00 10/11/2000 00:00 Time 25 Volume of hydrographs of different year during max peak

2000 20 1995 1993

15

10 Mill. cu. meter

5

0 1 2Days 7 14 30

Comments: 1. One day flood volumes are noticeably similar for these three events. For 2, 7 ,14 and 30 days the 1995 event shows slightly higher volume than the 2000 event but the 1993 event, that with the largest peak, gives the least volume among the three; in fact it is 50% of the volume of 1995 event for the 7, 14 and 30-day volume. 2. The 2000 and 1995 flood events display multiple peaks probably explain why they have greater volumes than the 1993 event.

179 15003 RIVER DINAN @ DINAN BRIDGE Annual Maximum Floods 1954 to 2004.(no missing years) A(km 2)= 298.00 N= 46

Mean= 141.820 Median= 150.378 Std.Dev.= 30.597 CV= 0.216 HazenS.= -0.996

EV1 200

'98 '68 '97 '85 150 w inter peak summer peak

100 AMF(m3/s)

50 2 5 10 25 50 100 500

0 -2-1 0 1 2EV1 y 3 4 5 6 7

Hydrograph during Maximum Flood in the year 1985 200 Red zone= Vol. of 1day Yellow zone= Vol. of 2day Blue zone= Vol. of 1w eek

150

100 Discharge (m3/s) Discharge

50

0 22/08/1986 23/08/1986 24/08/1986 25/08/1986 26/08/1986 27/08/1986 28/08/1986 29/08/1986 30/08/1986 00:00 00:00 00:00 00:00 00:00 00:00 00:00 00:00 00:00 Time

Hydrograph during Maximum Flood in the year 1997 200 Red zone= Vol. of 1day Yellow zone= Vol. of 2day Blue zone= Vol. of 1w eek

150

100 Discharge (m3/s)

50

0 14/11/1997 15/11/1997 16/11/1997 17/11/1997 18/11/1997 19/11/1997 20/11/1997 21/11/1997 22/11/1997 00:00 00:00 00:00 00:00 00:00 00:00 00:00 00:00 00:00 Time

180 100 Hydrograph during Maximum Flood in the year 1998 90 Red zone= Vol. of 1day Yellow zone= Vol. of 2day Blue zone= Vol. of 1w eek 80

70

60

50

40 Discharge (m3/s) Discharge

30

20

10

0 15/09/1999 00:00 17/09/1999 00:00 19/09/1999 00:00 21/09/1999 00:00 23/09/1999 00:00 25/09/1999 00:00 27/09/1999 00:00 Time 50 Volume of hydrographs of different years during max peak

45 1998 40 1997 1985 35

30 25 20

Mill. cu. meter 15

10

5 0 1 2Days 7 14 30

Comments: 1. There is a noticeable growth of volume with peak flood magnitude at 1 and 2- day durations and slight growth of volume at 7 days. However this is reversed for the larger durations where the largest flood peak (1985) is associated with the smallest volume.

181 16008 RIVER SUIR @ NEW BRIDGE 2 Annual Maximum Floods 1954 to 2004.(no missing years) A(km )= 1120.00 N= 46 Mean= 90.663 Median= 92.323 Std.Dev.= 11.350 CV= 0.125 HazenS.= -0.324

120 EV1 Plot

'83 '68 '60 100 '87 w inter peak 80 summer peak

60 AMF(m3/s) 40

20 2 5 10 25 50 100 500 0 -2-1 0 1 2EV1 y 3 4 5 6 7

120 Hydrograph during Maximum Flood in the year 1960 Red zone= Vol. of 1day Yellow zone= Vol. of 2day 100 Blue zone= Vol. of 1w eek

80

60

Discharge (m3/s) Discharge 40

20

0 26/11/1960 28/11/1960 30/11/1960 02/12/1960 04/12/1960 06/12/1960 08/12/1960 10/12/1960 12/12/1960 14/12/1960 00:00 00:00 00:00 00:00 00:00 00:00 00:00 00:00 00:00 00:00 Time 140 Hydrograph during Maximum Flood in the year 1968 Red zone= Vol. of 1day Yellow zone= Vol. of 2day 120 Blue zone= Vol. of 1w eek

100

80

60 Discharge (m3/s) Discharge 40

20

0 16/12/1968 18/12/1968 20/12/1968 22/12/1968 24/12/1968 26/12/1968 28/12/1968 30/12/1968 01/01/1969 00:00 00:00 00:00 00:00 00:00 00:00 00:00 00:00 00:00 Time

182 120 Hydrograph during Maximum Flood in the year 1983 Red zone= Vol. of 1day Yellow zone= Vol. of 2day Blue zone= Vol. of 1w eek 100

80

60

Discharge (m3/s) Discharge 40

20

0 29/01/1984 31/01/1984 02/02/1984 04/02/1984 06/02/1984 08/02/1984 10/02/1984 12/02/1984 14/02/1984 16/02/1984 00:00 00:00 00:00 00:00 00:00 00:00 00:00 00:00 00:00 00:00 Time 120 Hydrograph during Maximum Flood in the year 1987 Red zone= Vol. of 1day Yellow zone= Vol. of 2day Blue zone= Vol. of 1w eek 100

80

60

Discharge (m3/s) Discharge 40

20

0 26/01/1988 28/01/1988 30/01/1988 01/02/1988 03/02/1988 05/02/1988 07/02/1988 09/02/1988 11/02/1988 00:00 00:00 00:00 00:00 00:00 00:00 00:00 00:00 00:00 Time 250 Volume of hydrographs of different years during max peak 1987 1983 200 1968 1960 150

100 Mill. cu. meter

50

0 1 2Days 7 14 30

Comments: 1. The flood hydrographs of these four events display very similar shape and that reflect to the calculation of volume as well. There is hardly any difference found among the flood volume of these four events particularly at the 1, 2 and 7-day durations.

183

16009 RIVER SUIR @ CAHIR PARK 2 Annual Maximum Floods 1953 to 2004.(no missing year) A(km )= 1602.00 N= 46 Mean= 159.294 Median= 162.214 Std.Dev.= 27.397 CV= 0.172 HazenS.= -0.409 250 EV1 Plot

200 '68 '00 '60 '89 '04

150

100 AMF(m3/s)

50 2 5 10 25 50 100 500

0 -2-1 0 1 2EV1 y 3 4 5 6 7

250 Hydrograph during Maximum Flood in the year 1989 Red zone= Vol. of 1day Yellow zone= Vol. of 2day Blue zone= Vol. of 1w eek 200

150

100 Discharge (m3/s) Discharge

50

0 30/01/1990 01/02/1990 03/02/1990 05/02/1990 07/02/1990 09/02/1990 11/02/1990 13/02/1990 15/02/1990 00:00 00:00 00:00 00:00 00:00 00:00 00:00 00:00 00:00 Time 250 Hydrograph during Maximum Flood in the year 1960 Red zone= Vol. of 1day Yellow zone= Vol. of 2day Blue zone= Vol. of 1w eek 200

150

100 Discharge (m3/s) Discharge

50

0 25/11/1960 27/11/1960 29/11/1960 01/12/1960 03/12/1960 05/12/1960 07/12/1960 09/12/1960 11/12/1960 13/12/1960 15/12/1960 00:00 00:00 00:00 00:00 00:00 00:00 00:00 00:00 00:00 00:00 00:00 Time

184 250 Hydrograph during Maximum Flood in the year 2000 Red zone= Vol. of 1day Yellow zone= Vol. of 2day Blue zone= Vol. of 1w eek 200

150

100 Discharge (m3/s) Discharge

50

0 29/10/2000 31/10/2000 02/11/2000 04/11/2000 06/11/2000 08/11/2000 10/11/2000 12/11/2000 00:00 00:00 00:00 00:00 00:00 00:00 00:00 00:00 Time 250 Hydrograph during Maximum Flood in the year 2004 Red zone= Vol. of 1day Yellow zone= Vol. of 2day Blue zone= Vol. of 1w eek 200

150

100 Discharge (m3/s) Discharge

50

0 20/10/2004 22/10/2004 24/10/2004 26/10/2004 28/10/2004 30/10/2004 01/11/2004 03/11/2004 05/11/2004 00:00 00:00 00:00 00:00 00:00 00:00 00:00 00:00 00:00 Time 350 Volume of hydrographs of different year during max pe ak 2004 300 2000 250 1960

200 1989

150

Millcu. meter 100

50

0 1 2Days 7 14 30

Comments: 1. This too displays similar kind of hydrographs and the volumes are also found similar in magnitude particularly for the one and two-day volume. 2. At the 30-day duration, the event with the largest peak (1989) gives the largest flood volume which is found more than double than the volume of 2004 event.

185 24013 RIVER DEEL @ RATHKEALE Annual Maximum Floods 1953 to 2004.(missing years: 63-68) A(km 2)= 426.00 N= 46

Mean= 95.213 Median= 101.757 Std.Dev.= 29.944 CV= 0.314 HazenS.= -0.908 160 EV1 Plot 140 '73 '80 '98 120 '94 '88

100

80

AMF(m3/s) 60

40

20 2 5 10 25 50 100 500 0 -2-1 0 1 2EV1 y 3 4 5 6 7

160 Hydrograph during Maxim um Flood in the year 1973 Red zone= Vol. of 1day 140 Yellow zone= Vol. of 2day Blue zone= Vol. of 1week 120

100

80

60 Discharge (m3/s)

40

20

0 26/11/1973 27/11/1973 28/11/1973 29/11/1973 30/11/1973 01/12/1973 02/12/1973 03/12/1973 04/12/1973 05/12/1973 06/12/1973 00:00 00:00 00:00 00:00 00:00 00:00 00:00 00:00 00:00 00:00 00:00 Tim e 160 Hydrograph during Maximum Flood in the year 1998 Red zone= Vol. of 1day 140 Yellow zone= Vol. of 2day Blue zone= Vol. of 1week 120

100

80

Discharge (m3/s) 60

40

20

0 25/12/1998 00:00 27/12/1998 00:00 29/12/1998 00:00 31/12/1998 00:00 02/01/1999 00:00 04/01/1999 00:00 Tim e

186 160 Hydrograph during Maximum Flood in the year 1980 Red zone= Vol. of 1day 140 Yellow zone= Vol. of 2day Blue zone= Vol. of 1week

120

100

80

Discharge (m3/s) Discharge 60

40

20

0 28/10/1980 00:00 30/10/1980 00:00 01/11/1980 00:00 03/11/1980 00:00 05/11/1980 00:00 07/11/1980 00:00 09/11/1980 00:00 Tim e 140 Hydrograph during Maximum Flood in the year 1988 Red zone= Vol. of 1day 120 Yellow zone= Vol. of 2day Blue zone= Vol. of 1week

100

80

60 Discharge (m3/s) Discharge

40

20

0 15/10/1988 00:00 17/10/1988 00:00 19/10/1988 00:00 21/10/1988 00:00 23/10/1988 00:00 25/10/1988 00:00 27/10/1988 00:00 29/10/1988 00:00 Tim e 80 Volume of hydrograph of different years during max peak

70 1988 1980 60 1998

50 1973

40

Million cu.m Million 30

20

10

0 1 2Days 7 14 30 Comments: 1. Not much difference is shown among the volumes of hydrographs of four events for the one and two-day volume. 2. The two largest peak events, 1973 and 1998 are found to be delivered 30% more volume than the volume of 1980 and 1988 events in the case of 7-day volume.

187 24082 RIVER MAIGUE @ ISLANDMORE Annual Maximum Floods 1977 to 2004.( no missing year ) A(km 2)= 764.00 N= 46

Mean= 135.469 Median= 140.010 Std.Dev.= 35.757 CV= 0.264 HazenS.= -0.216 250 EV1 Plot

200 '89 '00 '98 '88 150

100 AMF(m3/s)

50 2 5 10 25 50 100 500

0 -2 -1 0 1 2EV1 y 3 4 5 6 7 250 Hydrograph during Maximum Flood in the year 1989 Red zone= Vol. of 1day Yellow zone= Vol. of 2day Blue zone= Vol. of 1week 200

150

100 Discharge (m3/s) Discharge

50

0 01/02/1990 00:00 03/02/1990 00:00 05/02/1990 00:00 07/02/1990 00:00 09/02/1990 00:00 11/02/1990 00:00 Time 200 Hydrograph during Maximum Flood in the year 2000 Red zone= Vol. of 1day 180 Yellow zone= Vol. of 2day Blue zone= Vol. of 1week 160

140

120

100

80 Discharge (m3/s) Discharge 60

40

20

0 31/10/2000 00:00 02/11/2000 00:00 04/11/2000 00:00 06/11/2000 00:00 08/11/2000 00:00 10/11/2000 00:00 12/11/2000 00:00 Time

188 200 Hydrograph during Maximum Flood in the year 1988 Red zone= Vol. of 1day 180 Yellow zone= Vol. of 2day Blue zone= Vol. of 1week 160

140

120

100

80 Discharge (m3/s) Discharge 60

40

20

0 14/10/1988 16/10/1988 18/10/1988 20/10/1988 22/10/1988 24/10/1988 26/10/1988 28/10/1988 00:00 00:00 00:00 00:00 00:00 00:00 00:00 00:00 Time 200 Hydrograph during Maximum Flood in the year 1998 Red zone= Vol. of 1day 180 Yellow zone= Vol. of 2day Blue zone= Vol. of 1week 160

140

120

100

80 Discharge (m3/s) Discharge 60

40

20

0 23/12/1998 00:00 25/12/1998 00:00 27/12/1998 00:00 29/12/1998 00:00 31/12/1998 00:00 02/01/1999 00:00 04/01/1999 00:00 Time 160 Volume of hydrographs of different year during max peak 140 1998 1988 120 2000 100 1989

80

60 Mill cu. meter cu. Mill 40

20

0 1 2Days 7 14 30

Comments: 1. Not much difference is shown among the volumes of hydrographs of these four events for the one and two-day volume. 2. The highest peak event (1989) is found to be given the largest volume for the 7, 14 and 30-day volume and in the case of 30-day volume it gives 30% more flood volume than the volume of the1998 event.

189 25017 RIVER SHANNON @ BANAGHER 2 Annual Maximum Floods 1950 to 2004.(no missing year) A(km )= N/A N= 46 Mean= 95.213 Median= 101.757 Std.Dev.= 29.944 CV= 0.314 HazenS.= -0.908 700 EV1 Plot 600 '54 '01 '89 '99 500 '94

400

300 AMF(m3/s)

200

100 2 5 10 25 50 100 500

0 -2-1 0 1 2EV1 y 3 4 5 6 7

600 Hydrograph during Maximum Flood in the year 1999 Red zone= Vol. of 1day Yellow zone= Vol. of 2day 500 Blue zone= Vol. of 1week

400

300

Discharge (m3/s) 200

100

0 16/12/1999 00:00 21/12/1999 00:00 26/12/1999 00:00 31/12/1999 00:00 05/01/2000 00:00 10/01/2000 00:00 Time 600 Hydrograph during Maximum Flood in the year 1989 Red zone= Vol. of 1day Yellow zone= Vol. of 2day 500 Blue zone= Vol. of 1week

400

300

Discharge (m3/s) Discharge 200

100

0 22/01/1990 00:00 27/01/1990 00:00 01/02/1990 00:00 06/02/1990 00:00 11/02/1990 00:00 16/02/1990 00:00 21/02/1990 00:00 26/02/1990 00:00 Time

190 600 Hydrograph during Maximum Flood in the year 2001 Red zone= Vol. of 1day Yellow zone= Vol. of 2day 500 Blue zone= Vol. of 1week

400

300

Discharge (m3/s) 200

100

0 03/02/2002 05/02/2002 07/02/2002 09/02/2002 11/02/2002 13/02/2002 15/02/2002 17/02/2002 19/02/2002 21/02/2002 23/02/2002 00:00 00:00 00:00 00:00 00:00 00:00 00:00 00:00 00:00 00:00 00:00 Time 600 Hydrograph during Maximum Flood in the year 1994

Red zone= Vol. of 1day Yellow zone= Vol. of 2day 500 Blue zone= Vol. of 1week

400

300

Discharge (m3/s) 200

100

0 21/01/1995 00:00 26/01/1995 00:00 31/01/1995 00:00 05/02/1995 00:00 10/02/1995 00:00 15/02/1995 00:00 Time 1400 Volume of hydrograph of different years during max peak 1994 1200 2001 1989 1000 1999 800

600 Mill. cu. meter 400

200

0 1 2Days 7 14 30

Comments: 1. All the four events display similar kind of hydrographs and eventually they give similar amount of flood volumes.

191 25021 RIVER LITTLE BROSNA @ CROGHAN 2 Annual Maximum Floods 1961 to 2004.(no missing years) A(km )= 493.00 N= 46 Mean= 28.028 Median= 28.578 Std.Dev.= 3.939 CV= 0.141 HazenS.= -0.134 45 EV1 Plot 40

35 '00 '62 '99 30 '94 w inter peak summer peak 25

20 AMF(m3/s) 15

10

5 2 5 10 25 50 100 500 0 -2-1 0 1 2EV1 y 3 4 5 6 7

40 Hydrograph during Maximum Flood in the year 1962 Red zone= Vol. of 1day 35 Yellow zone= Vol. of 2day Blue zone= Vol. of 1week

30

25

20

15 Discharge (m3/s) Discharge

10

5

0 31/10/1962 00:00 02/11/1962 00:00 04/11/1962 00:00 06/11/1962 00:00 08/11/1962 00:00 10/11/1962 00:00 12/11/1962 00:00 Time 40 Hydrograph during Maximum Flood in the year 2000 Red zone= Vol. of 1day Yellow zone= Vol. of 2day 35 Blue zone= Vol. of 1week

30

25

20

15 Discharge (m3/s) Discharge

10

5

0 29/10/2000 31/10/2000 02/11/2000 04/11/2000 06/11/2000 08/11/2000 10/11/2000 12/11/2000 14/11/2000 00:00 00:00 00:00 00:00 00:00 00:00 00:00 00:00 00:00 Time

192 40 Hydrograph during Maximum Flood in the year 1999 Red zone= Vol. of 1day 35 Yellow zone= Vol. of 2day Blue zone= Vol. of 1week

30

25

20

15 Discharge (m3/s) Discharge

10

5

0 16/12/1999 18/12/1999 20/12/1999 22/12/1999 24/12/1999 26/12/1999 28/12/1999 30/12/1999 01/01/2000 03/01/2000 00:00 00:00 00:00 00:00 00:00 00:00 00:00 00:00 00:00 00:00 Time 40 Hydrograph during Maximum Flood in the year 1994 Red zone= Vol. of 1day 35 Yellow zone= Vol. of 2day Blue zone= Vol. of 1week

30

25

20

15 Discharge (m3/s) Discharge

10

5

0 21/01/1995 23/01/1995 25/01/1995 27/01/1995 29/01/1995 31/01/1995 02/02/1995 04/02/1995 00:00 00:00 00:00 00:00 00:00 00:00 00:00 00:00 Time 70 Volume of hydrographs of different year during max peak 1994 60 1999 50 2000 1962 40

30 Mill cu. meter cu. Mill 20

10

0 1 2Days 7 14 30

Comments: 1. Though the 1962 flood event possesses largest peak discharge, it gives considerably lower flood volume than the volume of other events for the 7, 14 and 30-day durations.

193 Appendix 4

Probability Plots

See separate volume

194