Flood Studies Update
Technical Research Report
Volume III
Hydrograph Analysis
Kieran O’Connor, Monomoy Goswami and Duncan Faulkner
Derived from Technical Research Reports by NUI Galway and JBA Consulting
Volume I Rainfall Frequency Volume II Flood Frequency Estimation Volume III Hydrograph Analysis Volume IV Physical Catchment Descriptors Volume V River Basin Modelling Volume VI Urbanised and Small Catchments
Volume III Hydrograph Analysis
Abstract
This volume presents methods of constructing the hydrograph to accompany a flood peak of given return period at gauged and ungauged sites in Ireland. The peak flow is typically derived by the methods of flood frequency estimation presented in Volume II.
The preferred route to constructing the hydrograph is based on the analysis of hydrograph widths. Flood hydrographs are made comparable by characterising them by so-called hydrograph widths. For example, W75 represents the duration in hours over which the flow exceeds 75% of the peak value.
Flood hydrographs for many rivers in Ireland typically have a relatively complex shape, with subsidiary peaks and undulations. These reflect the general pattern of successive periods of heavy rainfall leading to the flood but are also moderated by features of the river network.
The aim of hydrograph width analysis is to construct a simplified flood hydrograph shape that is characteristic of the catchment. Two approaches are considered. In one, a particular parametric form is imposed on the hydrograph shape. The shape found most useful is a modified version of the Gamma distribution. In the other approach, the characteristic hydrograph is selected by direct analysis (and averaging) of hydrograph widths.
Regression-based expressions allow hydrograph descriptors at ungauged sites to be estimated from physical catchment descriptors (PCDs). Using the parametric approach, a flood hydrograph with unit peak is constructed as a continuous curve. This is rescaled by the relevant peak flow to obtain the required design flood hydrograph.
A standalone software package, with graphical user interface, called HWA (Hydrograph Width Analysis), was created both as a research tool and as an aid to constructing design flood hydrographs at any site, ideally based on existing or updated flow data.
Although some variation of hydrograph width (and hence hydrograph shape) was noted, no systematic pattern of variation with peak flow magnitude, season of occurrence or pre-event flow value could be established. As expected, arterial drainage was typically found to lead to narrower and peakier hydrographs.
Urbanised catchments are not well represented in the 89 stations subjected to hydrograph width analysis. This is one of several areas of application where the Interactive Bridge Invoking the Design Event Method extends the reach of FSU methods appreciably. The HWA and IBIDEM software packages are available through the FSU Web Portal.
Research on Flood Event Analysis is briefly summarised in Appendix B.
©Office of Public Works 2014
ii Volume III Hydrograph Analysis
Further information about the research
FSU Technical Research Reports (TRRs) are available in their original form for researchers and practitioners who seek additional information about a method. The original TRRs sometimes document exhaustive application of a method to many catchments. In others, additional options are reported.
Inevitably, the relevance of the original TRRs is influenced by OPW decisions on which methods to implement, and how best to arrange and support them. Readers who consult the original TRRs will notice editorial re-arrangements and compressions, and occasional changes in notation and terminology. These were judged necessary to enhance understanding and use of the FSU methods amongst general practitioners. More significant changes are labelled explicitly as editorial notes.
iii Volume III Hydrograph Analysis
Contents i Abstract ii Contents iv Notation xi Symbols xi Subscripts xii Abbreviations and descriptor names xii Glossary of terms xiii 1 Introduction 1 1.1 Overview 1 1.2 The goal and premise of hydrograph width analysis 2 1.3 Catchment selection 3 1.4 Physical catchment descriptors (PCDs) 4 1.5 The characteristic hydrograph 5 1.6 HWA software 5 2 Processing the flow data for HWA 6 2.1 Data screening and checking 6 2.1.1 Data handling 6 2.1.2 Missing flow data 6 2.1.3 Scrutiny of annual maximum flood peaks 6 2.1.4 Stations affected by arterial drainage 7 2.2 Defining the time-window of the flood hydrograph 7 2.3 Selection of flood hydrographs 8 2.4 Numbering of flood hydrographs 9 2.5 Seasonal distribution of flood events 9 2.6 Filtering of selected hydrographs 10 2.6.1 Desire for broadly unimodal hydrographs 10 2.6.2 Decoupling the main flood response within a complex flood event 10 2.6.3 Discarding the complex segments 10 3 Deriving the characteristic hydrograph at gauged sites 12 3.1 Standardising the flood hydrographs 12 3.2 Calculation of hydrograph widths at particular exceedance levels 12 3.3 Procedures for constructing the characteristic hydrograph 13 3.4 Split-sample and whole-sample calibration 15 3.5 Deriving the median hydrograph 16 3.5.1 Basic method 16 3.5.2 Anomalies in the derived median hydrograph 17 3.5.3 Improving the derived median hydrograph 17 4 The parametric approach 19 4.1 Objectives 19 4.2 General approach 19 4.3 UPO-Gamma model for the characteristic hydrograph 20 4.3.1 Gamma distribution 20 4.3.2 Peak of Gamma distribution 20
iv Volume III Hydrograph Analysis
4.3.3 Gamma model with peak at time zero 20 4.3.4 Gamma model with unit peak at time zero 21
4.3.5 Formulation in terms of hydrograph rise time Tr 21 4.3.6 Families of hydrographs constructed using the model 21 4.3.7 Example application of UPO-Gamma model 22 4.4 UPO-ERR-Gamma model for the characteristic hydrograph 23 4.4.1 Formulation 23 4.4.2 Method of fitting 24 4.5 Method of fitting the parametric model 24 4.5.1 Objective function 25 4.5.2 Optimisation scheme 26 4.5.3 Performance evaluation 27 4.6 Reproduction of flood hydrographs of verification events 28 4.7 Other methods 30 5 Performance of methods at gauged sites 31 5.1 Introduction 31 5.2 Relative performance in verification compared to that in calibration 32 5.3 Complexity of hydrographs at Stations 06011 and 34018 33 5.4 Variability in hydrograph widths at some stations 34 5.5 Attenuated response at some stations 38 5.6 General guidance 38 5.7 Results of whole-sample calibration 39 5.7.1 Derived median hydrograph and its descriptors 39 5.7.2 UPO-ERR-Gamma model and its parameters 40 5.7.3 Stations where the flood hydrograph recedes faster than it rises 43 5.8 Characteristic hydrographs on the River Suir 44 5.9 Hydrograph width analysis at gauged sites – a summary 45 5.10 Flood hydrographs having sustained peaks 46 5.10.1 Where the hydrograph shape reflects the temporal pattern of rainfall 46 5.10.2 Where the hydrograph shape is characteristic of the station 46 6 Constructing the characteristic hydrograph at ungauged sites 48 6.1 Introduction 48 6.1.1 Links with other parts of the FSU 48 6.1.2 Assumptions and difficulties 48 6.2 Selection of dependent variables (DVs) 49 6.3 Selection of independent variables (IVs) 50 6.4 Additional notes 51 6.4.1 Treatment of Gamma shape parameter n 51 6.4.2 Software used for the regression analysis 52 6.5 Correlation studies 52 6.5.1 Inter-correlations between PCDs 52 6.5.2 Individual correlations between DVs and initially selected IVs 56 6.5.3 Choosing a subset of PCDs to use as IVs 56 6.5.4 Checking the Normality of the DVs 57 6.5.5 Final selection of the independent variables; a note on the use of BFI 58 6.6 The regression method used 58
6.7 Illustrative results: Estimating W75 when BFI available 58
v Volume III Hydrograph Analysis
6.7.1 Regression models and their performance evaluation 58 6.7.2 Checking the possible influence of collinearity 60 6.7.3 Checking the logical consistency of the model 60 6.7.4 Additional checks 62 6.8 Recommended models for use at ungauged sites 67 6.8.1 The final models 67 6.8.2 Model performance 69 6.8.3 Additional notes on the regression models 70 7 Ancillary investigations 71 7.1 Variation of hydrograph width with peak flow 71 7.2 Variation of hydrograph width with pre-event minimum flow 73 7.3 Variation of hydrograph width with time of year 74 7.4 Effect of arterial drainage on hydrograph widths 75 8 Constructing the characteristic flood hydrograph 79 8.1 Topics covered 79 8.2 Features of the methods 79 8.3 Allowances in design flood hydrographs for pre-event flow 80 8.3.1 Substitution approach 80 8.3.2 Terminology: baseflow or pre-event flow? 81 8.3.3 Choosing the pre-event flow 81 8.4 Estimation of volume of flow 82 8.4.1 Basic method 82 8.4.2 Non-parametric case 82 8.4.3 Parametric case 82 8.5 Deriving the characteristic hydrograph at a gauged site 83 8.6 Estimating the characteristic hydrograph at an ungauged site 84 8.6.1 Using the UPO-ERR-Gamma model 84 8.6.2 Using the parabolic curves method 84 8.6.3 Using IBIDEM 84 8.7 Parabolic curves method 84 8.7.1 Overview 84 8.7.2 Details of method 85 8.7.3 Examples 86 8.7.4 Application at an ungauged site 86 8.8 Constructing the design flood hydrograph 87 8.9 Software 87 8.10 Selection and use of the pivotal catchment 87 8.10.1 Overview 87 8.10.2 Selection of the pivotal catchment 88 8.10.3 Recommended procedure for data transfer 89 8.10.4 Example 89 8.10.5 Further discussion of choice of method and of pivotal catchment 92 8.10.6 Urbanised catchments 93 9 IBIDEM 94 9.1 The idea of IBIDEM 94 9.1.1 Reminder of hydrograph estimation by FSU methods 94 9.1.2 Hydrograph estimation by the FSR design event method 94
vi Volume III Hydrograph Analysis
9.1.3 Basic idea of bridge between the FSR and FSU methods 95 9.2 How IBIDEM fits hydrographs 96 9.3 General approach to the optimisation 97 9.3.1 “First Tp and then SPR” 97 9.3.2 Use of horizontal fitting 98 9.3.3 Deriving Tp by optimising the fit to the FSU flood hydrograph 98 9.3.4 Deriving SPR by matching the required peak flow 100 9.4 Additional IBIDEM options 100 9.4.1 Flood frequency 100 9.4.2 Sensitivity to storm duration 100 9.4.3 Sensitivity to model parameters 100 9.4.4 Sensitivity to changes in urbanisation 101 9.5 Further details of the software 102 9.5.1 Inputs 102 9.5.2 Graphical displays 102 9.5.3 Display options 105 9.5.4 Goodness-of-fit measures 105 9.5.5 Tabular display 107 9.5.6 Export of results 108 9.6 Testing 108 9.6.1 Choice of test sites 108 9.6.2 Estimation of FSU hydrograph shapes 108 9.6.3 Estimation of peak flows 110 9.6.4 Rainfall depth-duration frequency tables 110 9.7 Results 111 9.7.1 Suir at Caher Park 111 9.7.2 Owenboy at Ballea 112 9.7.3 Lagan-Glyde at Aclint 112 9.7.4 Anner at Clonmel 114 9.7.5 Tributary to Tolka at Finglas 115 9.7.6 Illustration of effect of fitting threshold 116 9.7.7 Summary 117 9.8 Additional opportunities provided by IBIDEM 117 9.8.1 Strengths and limitations 117 9.8.2 Urban adjustment to design hydrographs 119 9.8.3 Supplying input hydrographs to river models 119 9.8.4 Allowances for projected land-use change 120 Acknowledgements 121 References 121 Appendices 123 Appendix A Gauges used in Hydrograph Width Analysis 123 Appendix B Précis of UCC research on flood event analysis 128 Appendix C Performance of HWA methods on verification events 130 Appendix D HWA results and their estimates from PCDs 167 Appendix E Application of the HWA software 171
vii Volume III Hydrograph Analysis
E1 Troubleshooting the installation 171 E2 Graphical User Interface (GUI) concepts of HWA software 173 Appendix F Further details of IBIDEM 183 F1 Method of optimising Tp 183 F2 Method of fitting SPR 184 F3 Checks and validation of outputs 185
Maps
Map 1.1: Stations used in hydrograph width analysis 4 Map 5.1: Catchment of Station 25014 Silver at Millbrook 37 Map 9.1: Location of test catchments 109
Figures
Figure 1.1: Screen-shot of start-up window of HWA software 5 Figure 2.1: Hydrographs of flood events displayed within a common window of 275 hours 8 Figure 2.2: Time series of flood events at Station 07009 Boyne at Navan Weir 8 Figure 2.3: Seasonal distribution of flood events at Station 07009 Boyne at Navan Weir 9 Figure 2.4: Decoupling the main components of Events 1, 7 and 24 at Station 07009 11 Figure 3.1: Standardised flood hydrographs for Events 1, 5, 7 and 24 at Station 07009 13 Figure 3.2: Methods of constructing a characteristic hydrograph 15 Figure 3.3: Median hydrograph for Station 07009 Boyne at Navan Weir 16 Figure 3.4: Median hydrographs with irregularities 17 Figure 3.5: (a) Smoothed median hydrograph; (b) Truncated median hydrograph 18 Figure 4.1: UPO-Gamma hydrograph for Tr=50 and different values of shape parameter n 22 Figure 4.2: UPO-Gamma hydrograph for n=3 and different values of scale parameter Tr 22 Figure 4.3: UPO-Gamma curve fitted to median hydrograph at Station 07009 23 Figure 4.4: Exponential replacement recession (ERR) for different values of parameter C 24 Figure 4.5: UPO-ERR-Gamma curve fitted to median hydrograph at Station 07009 25 Figure 4.6: Performance of UPO-ERR-Gamma model on verification events, Station 07009 29 Figure 4.7: Verification performance of UPO-ERR-Gamma calibrated in five versions 30 Figure 5.1: Performance of median hydrograph method across 37 Grade A1 stations 31 Figure 5.2: Performance of UPO-ERR-Gamma method across 37 Grade A1 stations 31 Figure 5.3: Comparison of model performance across 37 Grade A1 stations 32 Figure 5.4: Performance in verification compared to that in calibration 33 Figure 5.5: Varied hydrograph shapes at Station 34018 Turlough at Castlebar 34 Figure 5.6: Wide and narrow-peaked hydrographs at Station 24013 Deel at Rathkeale 35 Figure 5.7: Wide and narrow-peaked hydrographs at Station 25006 Brosna at Ferbane 35 Figure 5.8: Wide and narrow-peaked hydrographs at Station 25014 Silver at Millbrook 36 Figure 5.9: Wide and slanted hydrographs at Station 25025 Ballyfinboy at Ballyhooney 36 Figure 5.10: Rainfall-runoff behaviour in 30 June 1986 flood at Station 25014 37 Figure 5.11: Attenuated hydrographs at Station 25017 Shannon at Banagher 38 Figure 5.12: Median hydrograph for Station 07009 Boyne at Navan Weir (whole sample) 39 Figure 5.13: Summary index, s, of hydrograph skewness (89 Grade A1 + A2 stations) 40 Figure 5.14: Characteristic hydrograph for Station 35002 Owenbeg at Billa Bridge 43 Figure 5.15: Characteristic hydrograph for Station 30005 Robe at Foxhill 44 Figure 5.16: Characteristic hydrographs for four stations on the River Suir 44 Figure 5.17: UPO-ERR-Gamma characteristic hydrographs for four stations on the Suir 45 Figure 5.18: Hydrographs and rescaled characteristic hydrograph for Deel at Rathkeale 47 Figure 6.1: Matrix plot of PCDs that in part represent catchment size (89 stations) 55 Figure 6.2: Normality plots of log-transformed width descriptors and model parameters 57 Figure 6.3: Normality plot of standardised residuals for 5-variable model for ℓnW75 62 Figure 6.4: Plot of standardised residuals for 5-variable model for ℓnW75 63 Figure 6.5: Median hydrographs at: (a) Station 15002 and (b) Station 35071 68
viii Volume III Hydrograph Analysis
Figure 6.6: Derived and modelled values of W75, W50, n, Tr and C (BFI unavailable case) 69 Figure 7.1: Variation of hydrograph width with peak flow at Station 07009 71 Figure 7.2: Variation of hydrograph width with peak flow at Station 07010 72 Figure 7.3: Patterns of variation of hydrograph width with peak flow (at six stations) 72 Figure 7.4: Slope of W75 trend with peak flow magnitude (Grade A1 stations) 73 Figure 7.5: Variation of hydrograph width with pre-event minimum flow (Station 07009) 74 Figure 7.6: Slope of W75 trend with pre-event minimum flow (Grade A1+A2 stations) 74 Figure 7.7: Plot of flood peak against time of year (Station 07009 Boyne at Navan Weir) 75 Figure 7.8: Circular plot of W50, W75 and W90 against time of year (floods at Station 07009) 75 Figure 7.9: Characteristic hydrographs for four sites affected by arterial drainage 78 Figure 8.1: UPO-ERR-Gamma characteristic hydrograph at Stn 07009 by Table 6.7 models 80 Figure 8.2: As Figure 8.1 but with pre-event flow substituting for first part of hydrograph 81 Figure 8.3: Example of parabolic curves method (Station 07009 treated as ungauged) 85 Figure 8.4: Parabolic hydrographs for four stations on the Suir 86 Figure 8.5: Upper hydrographs transferred from Derrycahill to Rookwood 91 Figure 8.6: Derived median and UPO-ERR-Gamma hydrographs for Stns 26002 and 26005 92 Figure 9.1: Design inputs to FSR rainfall-runoff method of flood frequency estimation 95 Figure 9.2: Illustration that FSR T-year peak flow varies with Tp as well as with SPR 97 Figure 9.3: Horizontal fitting by comparing hydrograph widths 98 Figure 9.4: Relationship between peak flow Qpeak and peak rapid response qpeak 98 Figure 9.5: Double-peaked hydrograph 99 Figure 9.6: Display of fitted and imported hydrographs 103 Figure 9.7: Display of how a variable changes with return period 103 Figure 9.8: Display of hydrographs for multiple storm durations 104 Figure 9.9: Display of how a variable changes with storm duration 104 Figure 9.10: Display of sensitivity to an increase in URBEXT 105 Figure 9.11: FSR hydrograph fitted to UPO-ERR-Gamma hydrograph 106 Figure 9.12: As Figure 9.11 but with fitting threshold raised to 60% of peak flow 107 Figure 9.13: Example of IBIDEM tabular display 107 Figure 9.14: Suir at Caher Park 100-year hydrograph fit 111 Figure 9.15: Owenboy at Ballea 100-year hydrograph fit 112 Figure 9.16: Lagan-Glyde at Aclint – derived median hydrograph from HWA software 113 Figure 9.17: Lagan-Glyde at Aclint – 100-year hydrograph fits 113 Figure 9.18: Anner at Clonmel 100-year hydrograph fit 114 Figure 9.19: Tributary of Tolka at Finglas 100-year hydrograph fit 115 Figure 9.20: Urban adjustment to hydrograph for Tolka tributary at Finglas test site 119
Figure B.1: Catchment-average unit hydrographs standardised by area 129
Figure C.1: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 06011 130 Figure C.2: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 06012 131 Figure C.3: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 06013 132 Figure C.4: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 06014 133 Figure C.5: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 06026 134 Figure C.6: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 07007 135 Figure C.7: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 07009 136 Figure C.8: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 07010 137 Figure C.9: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 07012 138 Figure C.10: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 09001 139 Figure C.11: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 14004 140 Figure C.12: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 14006 141 Figure C.13: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 14007 142 Figure C.14: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 14011 143 Figure C.15: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 14018 144 Figure C.16: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 15005 145
ix Volume III Hydrograph Analysis
Figure C.17: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 23002 146 Figure C.18: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 24013 147 Figure C.19: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 25003 148 Figure C.20: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 25006 149 Figure C.21: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 25014 150 Figure C.22: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 25017 151 Figure C.23: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 25025 152 Figure C.24: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 25027 153 Figure C.25: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 25030 154 Figure C.26: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 26007 155 Figure C.27: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 26008 156 Figure C.28: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 26012 157 Figure C.29: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 26019 158 Figure C.30: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 27002 159 Figure C.31: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 29001 160 Figure C.32: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 29011 161 Figure C.33: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 30004 162 Figure C.34: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 30005 163 Figure C.35: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 34018 164 Figure C.36: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 36010 165 Figure C.37: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 36015 166
Figure E.1: Main application window of the HWA program showing a message-box 174 Figure E.2: Data window for entering data 175 Figure E.3: Data windows for displaying graphical and tabular outputs 176 Figure E.4: A dialog window showing a standard windows file-opening dialog-box 176
Boxes
Box 4.1: The choice between horizontal and vertical fitting 26 Box 4.2: Editorial note on performance measures 28 Box 6.1: Collinearity 49 Box 6.2: The role of ALLUV in the hydrograph-width models 61
Tables
Table 3.1: Widths of exceedance for four floods at Station 07009 Boyne at Navan Weir 14 Table 5.1: Outcome of whole-sample calibration at 89 Grade A1 + A2 stations 40 Table 6.1: Summary statistics of dependent variables (DVs) selected for regression analysis 50 Table 6.2: Some summary statistics of the IVs initially selected 51 Table 6.3: Correlation matrix of selected IVs and DVs at (up to) 89 stations 53 Table 6.4: Stepwise regression results for modelling hydrograph width descriptor ℓnW75 59 Table 6.5: Coefficient and collinearity statistics for selected model for ℓnW75 60 Table 6.6: Recommended models – when BFI available 66 Table 6.7: Recommended models – when BFI unavailable 66 Table 7.1: Some leading PCDs of stations on the River Fane 73 Table 7.2: Stations studied for the effect of arterial drainage on hydrograph widths 76 Table 7.3: Pre- and post-drainage values of hydrograph width descriptors/parameters 77 Table 8.1: Selected PCDs for Suck at Rookwood 89 Table 8.2: Data transfers to Suck at Rookwood using parabolic curves method 91 Table 9.1: Some details of the applications to two ungauged test catchments 109 Table 9.2: Design flows (m3s-1) for the five test catchments 110 Table 9.3: IBIDEM input variables for the test catchments 110 Table 9.4: Summary of IBIDEM results for five test catchments (100-year flood case) 116 Table 9.5: Sensitivity to fitting threshold (ungauged site on River Anner) 118
x Volume III Hydrograph Analysis
Table A.1: Stations used in Hydrograph Width Analysis 123 Table A.2: Details of the flow data used (see also Table 7.2) 125
Table B.1: Stations subjected to rainfall-runoff analysis 128
Table D.1: Hydrograph width analysis results for all 89 stations 167
Table E.1: Details of toolbar buttons 182
Table F.1: Example of iteration to find best-fitting value of Tp 184 Table F.2: Checks and outputs 186
Notation Symbols
AF Adjustment factor converting (e.g.) quantile estimates from fixed to sliding duration ANSF Average non-separated flow (m3s-1 per km2) ARF Areal reduction factor (in estimating design depth of catchment rainfall) 3 -1 BF Baseflow in FSR rainfall-runoff method (m s ); equivalent to pre-event flow Q0 C Recession parameter of UPO-ERR-Gamma model (hours) CWI Catchment wetness index (in FSR design event method) D Duration (in hours) of design storm in FSR design event method f(x) Probability density function F(x), F Cumulative distribution function g Ordinate of Gamma distribution I(n,x) Incomplete Gamma function K Scale parameter of Gamma distribution (hours) ℓn Natural logarithm m Number of (% of peak flow) levels at which hydrograph width evaluated when fitting n Shape parameter of Gamma distribution (and of UPO-ERR-Gamma model) N Number of years of record, sample size P Precipitation depth (mm) PR Percentage runoff (in FSR design event method) Q Flow (m3s-1) 3 -1 Q0 Pre-event flow (m s ) 3 -1 QT T-year peak flow (m s ) QMED Median annual flood (m3s-1) r2 Coefficient of determination s Hydrograph skewness or eccentricity parameter, e.g. in parabolic curves method s(t) S-curve (i.e. cumulative response curve) SPR Standard percentage runoff (in FSR design event method) T Return period (years) Tflood Return period of flood (in FSR design event method) Tr Rise-time (= translation parameter) of UPO-ERR-Gamma model Train Return period of rainfall (in FSR design event method) Tp Time-to-peak of unit hydrograph (in FSR design event method) Tp(0) Time-to-peak of instantaneous unit hydrograph (in FSR design event method) Vc(p) Semi-dimensionless volume of characteristic hydrograph above p% of peak flow (hours) 3 -1 3 VD(p) Volume of design flood hydrograph above p% of peak flow (m s hours; m ) Var Variance w Weighting function
xi Volume III Hydrograph Analysis
W Hydrograph width (hours) W50 Hydrograph width (hours) at 50% of peak flow W75 Hydrograph width (hours) at 75% of peak flow y Ordinate of Gamma distribution standardised to have a unit peak Γ(n) Gamma function
Subscripts p Value at peak; percentage of peak I Value at point of inflection
Abbreviations and descriptor names
AEP Annual exceedance probability ALLUV Proportion of extent of floodplain alluvial deposit AM Annual maximum AMRE Average value of mean relative error AREA Catchment area (km2) BFI Baseflow index (see Volume IV) CEH Centre for Ecology and Hydrology CFRAM Catchment Flood Risk Assessment and Management CSV Comma-separated values (file format) DDF Depth-duration-frequency DMH Derived median hydrograph DV Dependent variable ERR Exponential replacement recession FEH Flood Estimation Handbook FSE Factorial standard error FSR Flood Studies Report FSSR Flood Studies Supplementary Report FSU Flood Studies Update GIS Geographic information system GUI Graphical user interface HWA Hydrograph Width Analysis; also name of standalone software package for HWA IBIDEM Interactive Bridge Invoking the Design Event Method IH Institute of Hydrology, former name of CEH Wallingford IV Independent variable MA Moving average MRE Mean relative error NERC (UK) Natural Environment Research Council NSE Nash-Sutcliffe efficiency OPW Office of Public Works PCD Physical catchment descriptor (see Volume IV) POT Peaks-over-threshold PR Percentage runoff RMSE Root mean square error SAAR Standard average annual rainfall (mm) – the FSU uses 1961-90 as the standard period SE, se Standard error tsf Tab-separated format UK United Kingdom UPO Unit peak at origin URBEXT Urban extent: fraction of catchment classified as urban WP Work Package (within the FSU research programme)
xii Volume III Hydrograph Analysis
Glossary of terms
Term Meaning Annual Probability of one or more exceedances in a year of a preset rainfall depth (in a exceedance given duration) probability AEP Annual Peak-over-threshold (POT) series comprising the largest N events in N years of exceedance series record Annual maximum Time series containing the largest value in each year (12-month period) of record series for a particular duration Comparison of a model’s predictions with actual data, and adjustment of its Calibration parameters to achieve a better fit with reality Semi-dimensionless hydrograph defined to represent the characteristic shape of Characteristic flood hydrographs. Ordinates of the characteristic hydrograph are standardised hydrograph so that the peak value is 1.0. Abscissae indicate the time (in hours) Coefficient of Proportion of variation accounted for by (e.g.) a regression model determination r2 Confidence Bounds within which a population parameter is estimated to lie with a stated interval (usually %) confidence; used to indicate the reliability of an estimate Easting and Coordinates of a location expressed as distance eastwards and distance Northing northwards from a fixed datum (i.e. reference point) Eccentricity Parameter summarising the skewness of the upper hydrograph Genetic algorithm An optimisation (or calibration) method based on global or heuristic searching nth root of the product of a sample of n values of a positive variable such as Geometric mean rainfall depth Formula specifying the increase of a defined extreme (e.g. peak flow) with return Growth curve period; provides the factor by which the index flood is multiplied to estimate the T-year flood Interpolation Any method of computing new data points from a set of existing data points Parameter representing value subtracted from or added to a variable x to translate Location the graph of its probability distribution along the x-axis. The location of the parameter UPO-ERR-Gamma model is determined by the time of the peak flow. For a given duration (e.g. 24 hours), a time series of independent events Peak-over- (abstracted from the period of record) that exceed a preset threshold; the series threshold (POT) retains the magnitudes (in mm) and dates of the peak exceedances, together with series their times of occurrence; successive POT rainfall events must not overlap Residual Observed value minus the value estimated by a model Average number of years between years with rainfalls exceeding a certain value. T is the inverse of the annual exceedance probability (AEP). Thus, a 50-year Return period T return period corresponds to an AEP of 0.02. The return period is a basic component of the depth-duration-frequency model used to calculate a rainfall depth of the desired frequency.
Parameter controlling the spread of a distribution; e.g. scale parameter Tr Scale parameter controls the width of the characteristic hydrograph in the UPO-ERR-Gamma model
xiii Volume III Hydrograph Analysis
Term Meaning Parameter controlling the shape of a distribution; e.g. shape parameter n controls Shape parameter the shape of the hydrograph in the Gamma part of the UPO-ERR-Gamma model A measure of the departure from symmetry of a distribution; the hydrograph Skewness skewness descriptor s is the mean ratio of the width under the rising limb of the hydrograph to the total hydrograph width at that level. Standard Measure of dispersion (i.e. variation) of values about their mean deviation Estimated standard deviation of a sample statistic such as the mean, i.e. the Standard error standard deviation of the sampling distribution of the statistic Unimodal Having one maximum e.g. on its probability density function or hydrograph Verification (or Assessment or confirmation of a derived model’s performance by reference to validation) additional data (i.e. data not used in calibration of the model)
xiv Volume III Hydrograph Analysis
1 Introduction
1.1 Overview
Need for hydrograph information
Until recent years, practitioners in Ireland have typically used methods of flood frequency estimation based on the Flood Studies Report (FSR). The Flood Studies Update (FSU) builds on the methodologies of the FSR (NERC, 1975) by using updated databases of Irish hydrometric data and by applying GIS tools for the computation of Physical Catchment Descriptors (PCDs). While the estimation of peak flows (see Volume II) is of general importance, in a proportion of cases it is also necessary to construct the flood hydrograph associated with the T-year peak flow. The requirement is clearest where a flooding problem or a flood alleviation scheme is sensitive to prolonged high flows.
Because of “the relative richness of hydrograph data in Ireland and the relative paucity of rainfall data” (Reed, pers. comm., 2006), it was not envisaged that rainfall records would be used in the FSU for flood hydrograph estimation. The hydrological analyses reported here therefore have the objective of establishing methodologies for estimating design flood hydrographs from recorded flow data only. PCDs are used to develop regression-based estimates of such hydrographs so that they can be constructed at ungauged sites.
In many applications, the design flood hydrograph corresponding to a specified return period is required at a particular site. The site of interest – referred to as the subject site – may be gauged or ungauged. Whereas the design peak flow of a given return period is obtained in the case of a gauged site by statistical frequency analysis of flood peak data, complementary methods are required to produce the characteristic hydrograph to be associated with that peak flow. In the FSU, the requirement is met by Hydrograph Width Analysis.
Some earlier methods
Reed and Marshall (1999) list three approaches to defining the design hydrograph: adjusting the FSR rainfall-runoff model parameters, borrowing a standard hydrograph shape from the FSR rainfall-runoff method, and applying a simplified model of hydrograph shape. The first two approaches are taken forward in the FSU by development of IBIDEM (see Chapter 9). In the third approach, the upper part of the hydrograph – beneath a flood peak of the required return period – is synthesised by a quadratic function of W50 defined as the width of the hydrograph (measured in hours) at 50% of the peak flow. If required, the lower part of the hydrograph is sketched subjectively. At gauged sites, W50 is estimated from the analysis of observed flood hydrographs. At ungauged sites, a regression-based estimate relating W50 to the unit hydrograph time-to-peak is used. In the FEH method (Reed and Marshall, 1999), the upper part of the flood hydrograph is constrained to be symmetric about its peak. This approach is taken forward in the parabolic curves method (see Section 8.7). This exploits additional hydrograph width information and permits the upper hydrograph to be asymmetric.
Archer et al. (2000) develop a non-parametric method for the synthesis of design flood hydrographs. This is based on direct analysis of the shape of flood hydrographs observed at a site. Archer et al. analyse hydrographs for flood events drawn from the annual maximum series. They note the durations of exceedance of selected percentages of the peak flow,
1 Volume III Hydrograph Analysis distinguishing the elements before and after the peak flow. For each exceedance percentage point in turn, the median duration is noted across the N annual maximum events. A hydrograph is thereby derived which is non-dimensional with respect to discharge. The resulting characteristic hydrograph is then applied to the peak flow of given return period to synthesise the required design flood hydrograph. By distinguishing periods before and after the peak flow, the characteristic hydrograph is not constrained to be symmetric.
Archer et al. suggest that the characteristic hydrograph shape thus derived provides a more realistic basis for generating a design flood hydrograph. The method is claimed to be simpler and quicker, and not to require the separate assessment of baseflow and storm runoff (Archer et al., 2000). However, their method is applicable only at gauged sites.
Approach adopted
The Hydrograph Width Analysis (HWA) reported below takes the Archer et al. method as its starting point. The method presented in Chapters 2 and 3 is applied to flood hydrographs from 89 gauging stations. Using physical catchment descriptors (PCDs) developed in Volume IV, the method is generalised in Chapter 6 to allow synthesis of a characteristic hydrograph at ungauged sites.
Some notes on the structure of volume
Volume III is largely based on HWA research undertaken at NUI Galway as Work Package 3.1 of the Flood Studies Update. Later chapters discuss the application of methods to design flood hydrograph construction at gauged and ungauged sites. The IBIDEM software package (see Chapter 9) developed by JBA Consulting extends both the applicability of the HWA methods and the case-by-case interpretation of T-year flood estimates developed using Volume II methods.
Station 07009 Boyne at Navan Weir is used as the primary example in illustrating the HWA procedures. There was no special reason for choosing this station for demonstration purposes. Other gauged catchments are used where appropriate to illustrate particular features of HWA. In addition, testing of IBIDEM considers three gauged and two ungauged sites.
An overarching requirement was that the HWA methods developed needed to be simple enough to give scope to generalise their use at ungauged (as well as gauged) sites.
1.2 The goal and premise of hydrograph width analysis
The primary goal of the hydrograph width analysis (HWA) research was to devise a methodology to “flesh out” the hydrograph beneath a given peak value. The aim was to define the shape of the design flood hydrograph based on typical flood hydrographs which have occurred. A subsidiary objective was to investigate the extent to which specific factors influence the typical shape of the flood hydrograph. Ancillary studies reported in Chapter 7 examine the influence on flood hydrograph shape of the flood magnitude, its season of occurrence, the pre-event flow and effects arising from arterial drainage works.
The premise of HWA is that hydrographs of floods occurring at a particular station are broadly similar in character and that the typical shape of the hydrograph of a future flood –
2 Volume III Hydrograph Analysis embodied in the design flood hydrograph – can be expected to reflect the general features of those which already occurred. Assessment of similarity is subjective, being largely based on visual judgement.
It is sometimes found that, because of prolonged rainfall at the peak-inducing intensity, a few flood hydrographs at a station can exhibit a highly flattened and prolonged peak segment. Use of the median hydrograph width – as in Archer et al. (2000) – ensures that such atypical flood hydrographs do not receive undue weight in the analysis. Nevertheless, a statistical measure is required to summarise the degree of similarity between synthesised and observed flood hydrographs.
Whereas the shapes of the upper parts of observed flood hydrographs (e.g. the parts with flows above 50% of the respective peak flows) are often found to be generally similar, those of the lower parts tend to vary widely. The variation reflects a number of factors including the occurrence of preceding and/or following floods subsidiary to the main event.
If only the upper part of the hydrograph of the design flood is deemed important, a procedure to model the lower part is not required. However, in a proportion of applications, the complete hydrograph is required.
1.3 Catchment selection
Detailed flow data are required for hydrograph width analysis, and flow data at 15-minute interval were obtained for 90 gauging stations operated by the OPW. Physical catchment descriptors (PCDs) were not initially available for Station 39008. The study therefore considered a network of 89 stations.
The stations chosen were selected with regard to the quality of flow data expected. Grade A1 and Grade A2 are the highest categories of rating curve and water level measurement reliability (see Appendix A1.1 of Volume II). The selected stations comprise 37 stations graded A1 and 52 graded A2. Their catchment areas range in size from 23 to 7980 km2, with a median value of 285 km2. The 89 catchments are identified in Map 1.1 and in Table A.1 of Appendix A.
Hydrograph data were supplied as 15-minute data in a time-series format. Each *.tsf datafile held the date and time of occurrence, the flow (in m3s-1) and a quality code against the measurement. Table A.2 indicates the period of flow data abstracted, the completeness of 15-minute data across that period, the number of annual maximum (AM) values represented, and the median of the annual maxima (i.e. QMED) across that period. Also shown is the period during which any arterial drainage works were carried out on the catchment. For stations affected by drainage works, the main HWA used the post-drainage record only.
3 Legend
! Stations considered in the "Hydrograph Width Analysis"
Volume III Hydrograph• Analysis
! Station location ! 39009 (The boundaries shown mark the Hydrometric Areas of Ireland)
! 35071
35005 35002! 36015 ! ! 34001 ! 36027 36019 6012 ! 35001 36021 ! ! ! 34009 ! ! 36010 6011 ! 36011 ! ! 26012 ! 6014 34018 ! ! ! 26009 7033 6026 6013 ! ! 26008! ! ! 7004 26022 26019 ! 7006 7012 30005 ! ! 7011! ! ! ! ! 7009 26002 26021 7010 30007 ! ! ! ! 7002 ! 7007 26005 ! ! ! 9001 30004 26007 ! ! 30061 29001 2500625016 14004 ! ! ! ! ! 14011 ! 29011 ! ! 29004 ! ! 14006 25017 14009! 14007 25025 ! ! 25030 25029 27002 ! 27001! ! ! 15005 ! 25027 ! 15003 16001 25001 16004 ! ! ! ! 11001 ! ! 14018 ! ! 25005 ! ! 2400825003! 16003 15002 ! ! 23001 ! ! 24082 ! 16008 ! ! 15006 ! 23002 24013 16005! 16009 ! 23012 ! 18004 ! 18005 ! 22071 !
19001 !
0 20 40 80 Kilometers
Map 1.1: Stations used in hydrograph width analysis
1.4 Physical catchment descriptors (PCDs)
Physical catchment descriptors (PCDs) at 216 gauged locations on rivers and lakes in Ireland were supplied by the OPW. Details are given in Volume IV. That volume includes three special PCDs developed in the FSU research: FAI, BFIsoil and FLATWET.
The special descriptor FLATWET was available and, in certain circumstances, plays a role in constructing the characteristic hydrograph at an ungauged site. However, the FAI and BFIsoil descriptors had not been developed at the time of the hydrograph width research. To allow
4 Volume III Hydrograph Analysis consideration of soil permeability and other storage effects on hydrograph widths, median values of the baseflow index (BFI) were supplied for 198 stations. BFI values were unavailable for ten of the 89 stations selected for HWA. Consequently, Stations 07007, 07011, 14004, 15002, 25017, 26009, 29001, 36010, 36027 and 39009 had to be omitted from those analyses requiring BFI.
1.5 The characteristic hydrograph
The aim of the research is to allow the user to construct the design hydrograph of a given return period. The hydrograph represents flows in m3s-1. Following Archer et al. (2000), the need is met by defining a semi-dimensionless flood hydrograph. This has time coordinates in hours but the peak flow is standardised to be 1.0. The terminology adopted in the FSU is to call this the characteristic hydrograph.
At gauged sites, the characteristic hydrograph is constructed by direct analysis of hydrograph data observed in large floods, i.e. by hydrograph width analysis. At ungauged sites, the characteristic hydrograph is synthesised from PCDs by methods presented in Chapter 6. Typically, the required design hydrograph in m3s-1 is obtained by scaling up the characteristic hydrograph by the T-year flood peak estimated using Volume II methods.
1.6 HWA software
Written in Visual Basic and Fortran, the standalone software package HWA provides an interactive tool for Hydrograph Width Analysis, displaying data and results in tabular and graphical forms. Outputs can be saved under user-defined filenames for future access and use. A screen-shot of the HWA start-up window is shown in Figure 1.1. Technical details appear later in Appendix E. The software is available to practitioners through the data and software download module of the FSU Web Portal.
Figure 1.1: Screen-shot of start-up window of HWA software
5 Volume III Hydrograph Analysis
2 Processing the flow data for HWA
This chapter describes the processing procedures by which flow data were screened and flood events selected and filtered for hydrograph width analysis.
2.1 Data screening and checking
2.1.1 Data handling
For gauging stations in the OPW network, river-flow data at 15-minute interval are held in datafiles having the .tsf filename extension. For many stations, these files contain more than a million items of data.
Scanning such long data series for the purpose of identifying and selecting flood events is both tedious and restrictive within commonly used software applications. Prior to introduction of the 2007 version, Microsoft Excel had a limit of 65,536 rows. Even in the 2007 version, the row limit is too small to accommodate 15-minute data series of 30 years or longer.
Application-specific programs were therefore written in Fortran to accept data in the .tsf format and to search the data series for flood events. Visual Basic was used to create a standalone user-interface for displaying flood hydrographs with interactive graphics. Additional programs were written to analyse the identified flood hydrographs and to select and implement generally suitable methods for final adoption. The development resulted in the HWA software package (see Appendix E).
2.1.2 Missing flow data
The HWA research considered flow data from 89 Grade A1 or A2 stations. There are occasional data gaps in the 15-minute flow records. A typical reason given is that the chart was missing. Percentages of missing data are listed in Table A.2 for each station. Eight stations had more than 10% of data missing but only Stations 25005 and 26021 had more than 20% of data missing. The median proportion of missing data was 2.9%.
Complete hydrograph data were therefore not always available for all flood events. In a few cases it proved possible to analyse the rising or receding limb of the flood event where the gap in flow data affected only the opposite limb of the hydrograph.
2.1.3 Scrutiny of annual maximum flood peaks
Annual Maximum (AM) flood peaks were extracted and scrutinised for anomalies. In some cases, the existence of a very large value in the AM series called for special investigations. This occasionally led to the discard of doubtful data.
The index flood adopted in the FSU is QMED, the median of the AM floods. The QMED value corresponding to the period of record used in HWA is given in the penultimate column of Table A.2 (see Appendix A).
6 Volume III Hydrograph Analysis
2.1.4 Stations affected by arterial drainage
For information, the final column of Table A.2 identifies those of the 89 stations which have experienced arterial drainage within or prior to their overall period of record. The hydrological and spatial descriptors of the FSU catchments (see Volume IV) were assembled relatively recently, and generally reflect the post-drainage characteristics of such catchments. Consequently, the main HWA for these catchments was carried out using hydrographs from the post-drainage period only.
Stations 07007, 07010, 23002, 26012 and 30004 had 15-minute flow data available for both pre and post-drainage periods. These five catchments are used specifically in Section 7.4 to examine the effect of arterial drainage on the typical shape of flood hydrographs.
2.2 Defining the time-window of the flood hydrograph
The convention adopted was to position the time origin at the peak of the hydrograph. Thus, each flood hydrograph is presented with its peak occurring at t = 0, with its rising and receding limbs shown on the left and right-hand sides of the peak respectively.
In some instances, due to prolonged rainfall of peak-inducing intensity, an observed flood hydrograph exhibits a flattened (i.e. sustained or persistent) peak. In such cases, the first occurrence of the peak flow is considered as the origin of the time axis for the purpose of hydrograph width analysis. Thus, in the case of sustained peak flow, the receding limb of the flood hydrograph includes the peak flow over the period during which it persists after its first occurrence.
The user of the HWA software specifies the start and end of the flood hydrograph by indicating the time intervals from the peak back to the start of the hydrograph and from the peak forward to the end of the hydrograph. The sum of these two time intervals defines the time-base (or window) that embraces the flood peak and the flood hydrograph about it.
It is convenient to adopt a common time-base when displaying a number of flood hydrographs for a particular station. The user of the HWA software guesses values for the time intervals (before and after the flood peak). Each of the observed flood hydrographs is then displayed within a window of common size. Finalising values of the time intervals requires a number of trials, the objective being to contain all the selected flood hydrographs within the final common window.
Figure 2.1 provides an example of flood hydrographs for three events displayed within a common window. For this station, the 275-hour window comprises 50 hourly time-steps on the rising limb of the hydrograph and 225 hourly time-steps on the receding side. This reflects that flood hydrographs for Station 07009 Boyne at Navan Weir are generally observed to rise (from the level of the pre-event flow to the peak) over about two days and to recede (from the peak to the level of the baseflow) over about nine days.
7 Volume III Hydrograph Analysis
Figure 2.1: Hydrographs of flood events displayed within a common window of 275 hours
2.3 Selection of flood hydrographs
The final selection of flood hydrographs was made by applying a peaks-over-threshold (POT) criterion. This ensures that the hydrograph width analysis gives due weight to the largest floods recorded. Where the annual maximum series comprises N floods (i.e. representing N water-years of record), the criterion applied was to select the N largest flood events from this period. This special case of a POT series is known as the annual exceedance series.
Figure 2.2 illustrates the annual exceedance series for Station 07009 and shows how it relates to the annual maximum (AM) series. The AM events are marked by small green circles, with the broken green line drawn to indicate the QMED of 134.8 m3s-1. Events in the annual exceedance series are marked by blue circles.
There are 29 annual maxima at this station. Some 19 of these 29 flood events are in the annual exceedance series. The annual exceedance series includes ten further floods (from flood-rich years) in place of the ten smallest annual maxima (in flood-poor years).
The blue line marks the level of the 30th largest flood in the annual exceedance series and defines the threshold above which the annual exceedance series has been extracted to yield 29 events in 29 years. [Editorial note: For reasons that are not entirely clear, the 30th largest flood has been included in the annual exceedance series at this station. The dates shown along the x-axis of Figure 2.2 are correct but are not very helpful. The tick-marks are spaced at an interval of 50675 15-minute periods i.e. about every 528 days.]
Figure 2.2: Time series of flood events at Station 07009 Boyne at Navan Weir
8 Volume III Hydrograph Analysis
An exception was made in two cases where the number of flood events would otherwise have been very small. Stations 14011 and 25003 had hydrograph data for only five and seven water-years respectively. The number of events selected in these cases was arbitrarily doubled: i.e. using ten events at Station 14011 and 14 at Station 25003.
2.4 Numbering of flood hydrographs
As part of the selection process, flood events are numbered according to their ranking in the annual exceedance series. Thus, Event 1 is the largest flood event and Event 10 is the tenth largest flood event. The event numbering carries through to the flood hydrographs.
[Editorial note: This is thoroughly effective in ensuring that hydrographs of the largest floods attract particular attention in analysis. It remains to be seen whether the device leads to confusion when flow records are updated by the extraction of hydrographs for later events.]
2.5 Seasonal distribution of flood events
The circular plots of Figure 2.3 illustrate the seasonal distribution of flood events at Station 07009. The angular position denotes the calendar date. Again, the green circles mark the AM events and the blue circles denote the annual exceedance events. The plots in Figure 2.3 are identical except that the radial axis is marked in m3s-1 in the left-hand diagram and in multiples of QMED in the right-hand diagram.
The mean time-of-year of floods is found by plotting the centroid (analogous to the centre of mass) of the data points. The centroids in Figure 2.3 are marked by filled circles: green for the AM series and blue for the annual exceedance series. The mean time-of-year of flood events at Station 07009 is seen to be early January.
Figure 2.3: Seasonal distribution of flood events at Station 07009 Boyne at Navan Weir
[Editorial note: Radial positions in Figure 2.3 indicate the magnitudes of the flood events. Calculation of the mean time-of-year of flooding has been weighted by flood magnitude.]
9 Volume III Hydrograph Analysis
2.6 Filtering of selected hydrographs
2.6.1 Desire for broadly unimodal hydrographs
Essentially unimodal (i.e. single-peaked) flood hydrographs are to be favoured when deriving the characteristic hydrograph. [Editorial note: The characteristic hydrograph is to be applied to design flood peaks derived by Volume II methods to yield the T-year flood hydrographs required in many flood risk assessments. Such design events are hypothetical floods. It is therefore reasonable to allow the characteristic hydrograph itself to be stylised. In essence, the aim is to construct a design flood hydrograph that represents the typical catchment flood response to one heavy rainfall event rather than to a succession of events.]
It was observed that many high-ranking flood events across the 89 stations had complex (i.e. multi-peaked) hydrographs. At some stations, very few single-peaked floods could be identified. This precluded the option of deriving the characteristic hydrograph from only those flood events with broadly one-peaked hydrographs. Techniques for decoupling complex floods were devised to overcome this difficulty.
2.6.2 Decoupling the main flood response within a complex flood event
A complex flood event is one having multiple peaks. Typically, it represents the catchment flood response to more than one period of heavy rainfall. Several approaches were investigated for decoupling (i.e. isolating) the main flood response within a complex event.
One approach constructed a master recession curve from segments of the receding limbs of single-peaked hydrographs observed for the station. An analogous master rising curve was similarly constructed from segments of the rising limbs of single-peaked hydrographs at the station. Whilst dealing with some individual stations adequately, the approach failed to generalise for use across all stations. There was a specific concern that the approach led to bias in the characteristic hydrograph.
The approach ultimately adopted was simply to discard the complex segments of the flood hydrograph: retaining only the broadly unimodal part.
2.6.3 Discarding the complex segments
Only that part of the observed complex hydrograph embracing the largest peak is considered relevant for deriving a generalised shape of the design flood hydrograph. Parts of the hydrograph that can be visually associated with flood responses before/after the largest- peaked flood response are discarded. The decoupling is done subjectively, using interactive features of the HWA software.
The decoupling is illustrated for three flood events at Station 07009 Boyne at Navan Weir. Figure 2.4 shows the hydrographs for the largest, 7th largest and 24th largest floods analysed. The cyan-coloured sections of the plotted hydrographs indicate the parts discarded. The grey- coloured section indicates the decoupled component of the main flood response. This is the hydrograph used in the subsequent Hydrograph Width Analysis.
10 Volume III Hydrograph Analysis
Figure 2.4: Decoupling the main components of Events 1, 7 and 24 at Station 07009
11 Volume III Hydrograph Analysis
3 Deriving the characteristic hydrograph at gauged sites
Methods are described for deriving the characteristic hydrograph at a gauged site. Two groups of methods are distinguished: parametric and non-parametric. The non-parametric approach derives the characteristic hydrograph by a statistical averaging of hydrographs. It is non-parametric because it does not make any particular assumptions about the shape of the hydrograph.
In contrast, the parametric approach specifies a formulaic shape for the characteristic hydrograph. Its basic aim is to represent the hydrograph by a smooth curve expressible in terms of a small number of parameters. Although other strategies are possible, it was found most effective to fit the parametric form to the characteristic hydrograph first obtained using the non-parametric method. The parametric approach is therefore taken up in Chapter 4.
3.1 Standardising the flood hydrographs
The flood hydrographs comprise isolated single-peaked floods and decoupled (unimodal) segments of more complex flood events (see Section 2.6). Collectively, these might be termed filtered hydrographs. Hereafter they are chiefly just called flood hydrographs.
The flood hydrographs are standardised by dividing the flow ordinates by the magnitude of the peak flow. Each standardised flood hydrograph thus has a peak value of 1.0. This corresponds to the 100th percentile of the peak flow, in that 100% of the hydrograph is less than or equal to this flow. [Editorial note: This terminology is correct when applied at the peak but not otherwise. The authors refer to percentiles of the peak flow when they ought to refer to percentages of the peak flow. It was not practical for editing to correct the mislabelling in every case. Users of the HWA package should therefore interpret percentile of peak flow as meaning percentage of peak flow.]
The ordinate scale of the standardised hydrograph is percentages of the peak flow. The time scale of the hydrograph is unaltered, with the abscissa in hours. Thus, the standardised flood hydrograph is semi-dimensionless. As noted previously, the time origin is taken at the time of the peak flow.
Flow levels at increments of 5% of the peak flow are identified to the extent that a particular hydrograph allows. Figure 3.1illustrates this for four hydrographs at Station 07009 Boyne at Navan Weir. Events 1, 7 and 24 are decoupled from complex flood events; Event 5 is an isolated unimodal hydrograph.
3.2 Calculation of hydrograph widths at particular exceedance levels
The hydrograph width at a given percentage of the peak flow is defined as the time during which the flow at a station exceeds the flow corresponding to that percentage of the peak flow. The total width of exceedance is a time in hours. Following Archer et al. (2000), the width is divided into two components: one on the rising limb and another on the receding limb. As seen in Figure 3.1, the width component at a particular percentage of the peak flow is sometimes available on one limb but not on the other.
12 Volume III Hydrograph Analysis
Figure 3.1: Standardised flood hydrographs for Events 1, 5, 7 and 24 at Station 07009
In characterising the hydrographs, the first step is to choose a series of percentages. If the objective is to derive only the upper part of the characteristic hydrograph, percentages above and including 50% of the peak flow are considered sufficient. For other purposes, a fuller description of the hydrograph is required. The research chose to use percentages at 5% intervals from 95% down to 5%. In addition, some use was made of the width at 98% of the peak flow.
The precise value of the flow that corresponds to a selected percentage of the peak will not generally appear in the 15-minute record of flow data for the event. Linear interpolation is used to compute the time at which a particular percentage of the peak flow occurs. The required hydrograph widths can then be found.
For many selected events, hydrograph widths at the lower percentages are generally unavailable. The widths of exceedance at selected percentages are shown in Table 3.1 for the sample events of Figure 3.1. For Event 24, the width of exceedance at 65% of the peak flow is 15.49 hours on the rising side but is undefined on the receding side. [Editorial note: The terms hydrograph width and width of exceedance are used interchangeably. There is an unexplained discrepancy between Table 3.1 and Figure 3.1 in the hydrograph widths on the receding limb of Event 1.]
In certain applications it can be helpful to focus on hydrograph widths at one or two fixed percentages of the peak flow. There is particular interest here in W75 and W50, which denote the hydrograph widths at 75% and 50% of the peak flow.
3.3 Procedures for constructing the characteristic hydrograph
A major element of the research was to develop procedures for constructing the characteristic hydrograph. The principal methods considered in the research are summarised in Figure 3.2. The applicability of a particular procedure depends on whether the subject site is gauged or
13 Volume III Hydrograph Analysis ungauged and whether the complete characteristic flood hydrograph or only its upper half is of interest. The methods ultimately recommended are highlighted.
Table 3.1: Widths of exceedance for four floods at Station 07009 Boyne at Navan Weir
Width of exceedance (hours)
Event 1 Event 5 Event 7 Event 24
Percentage
Rising Receding Total Rising Receding Total Rising Receding Total Rising Receding Total
98 2.84 2.44 5.29 2.81 3.65 6.45 4.24 2.25 6.49 2.96 7.01 9.97 95 5.23 4.25 9.48 4.36 6.54 10.91 7.77 5.32 13.09 5.39 10.22 15.61 90 8.32 6.60 14.92 6.45 9.60 16.05 9.64 8.44 18.07 8.24 14.41 22.65 85 14.11 8.75 22.86 7.84 12.16 20.00 10.78 11.55 22.33 10.56 18.17 28.73 80 16.63 10.34 26.97 9.02 14.49 23.51 10.94 14.23 25.17 12.41 21.98 34.39 75 18.02 10.04 17.18 27.22 11.60 17.34 28.94 13.89 25.83 39.72 70 19.29 10.79 21.06 31.85 12.67 20.68 33.35 15.49 65 20.37 12.20 26.81 39.01 14.27 23.44 37.72 17.41 60 21.07 12.97 33.67 46.64 15.14 27.12 42.26 55 21.79 13.64 40.54 54.17 15.82 30.84 46.66 50 22.54 14.41 47.18 61.59 17.00 35.80 52.79 45 23.51 15.16 53.70 68.86 18.71 43.59 62.31 40 24.42 15.85 61.55 77.39 22.45 56.43 78.88 35 25.33 16.61 73.71 90.32 72.50 30 26.44 17.50 110.40 127.90 97.82 25 28.50 19.11 136.67 155.78 151.27 20 44.84 22.69 200.38 223.07 15 10 5
The present chapter discusses the non-parametric approach in which a semi-dimensionless flood hydrograph is derived broadly following Archer et al. (2000). Observed (isolated or de- coupled) flood hydrographs are selected. The characteristic hydrograph is constructed to be unimodal and to have exceedance widths before and after the peak that take the average of the corresponding exceedance widths in the sample hydrographs. In the FSU hydrograph width research, the recommended approach was to use the median value. The characteristic hydrograph obtained in this way is referred to as the derived median hydrograph or just the median hydrograph.
14 Volume III Hydrograph Analysis
Subject site
Gauged site Ungauged site
Non-parametric method based Parametric method on widths of exceedance by curve fitting If complete hydrograph required For upper hydrograph alone
Apply Gamma curve with Estimate width parameters * Derived Derived exponential replacement W75 and W50 from PCDs . median mean recession, estimating parameters Adopt suitable value of * hydrograph hydrograph n, Tr and C from PCDs eccentricity parameter (s)
Gamma curve with Gamma curve Hayashi et Negative Inverse Construct Apply modified Gamma algebraic with exponential al. (1986) Binomial Gaussian parabolic Gamma model curve replacement replacement curve curve curve curves that with parameters recession recession Sketch subjectively respect n, Tr and C that required yield required
values of values of W75, Fit to event Fit to event Fit to Fit to W75, W50 & s W50 and s derived derived hydrographs hydrographs Fit to event median mean individually; individually; hydrographs hydrograph hydrograph adopt median adopt mean collectively Alternate formulations are given. parameter values parameter values One version requires BFI. BFIsoil might be used but was not available at the time of the HWA research.
Figure 3.2: Methods of constructing a characteristic hydrograph
In the parametric approach, the characteristic hydrograph is obtained by fitting an algebraic formula to the observed hydrographs or to the characteristic hydrograph previously obtained by the non-parametric approach. The coefficients and constants in the model form the parameters. The choice of model structure is a matter of judgement but is aided by an objective measure of how well the modelled hydrograph fits the characteristic hydrograph derived by the non-parametric approach. The recommended parametric model is the Gamma curve with exponential replacement recession. This UPO-ERR-Gamma model is introduced in Section 4.4.
3.4 Split-sample and whole-sample calibration
The 37 Grade A1 stations were used for testing and comparison of the developed methods. At each station, three flood hydrographs were reserved for verification, with the remaining events used in calibration. This procedure is referred to as split-sample calibration. To avoid bias, one large, one medium and one small event were enrolled as the verification events. [Editorial note: Other parts of the Technical Research Report refer to validation rather than verification. The terms are interchangeable.]
As detailed in Section 2.4, the M hydrographs at a station are numbered so that the one with the largest flood peak is Event 1 and the one with the smallest flood peak is Event M. To select the three verification events, the M events were divided into three equal or near-equal groups characterising the ranges of large, medium and small flows. In the case of Station 07009 Boyne at Navan Weir, there are 30 flood events in the annual exceedance series. Events 1 to 10 are deemed large flood events, Events 11 to 20 are deemed medium events and Events 21 to 30 are deemed small events. A middle-ranking flood event in each group was adopted as a verification event. For Station 07009, Events 5, 15 and 24 were thus selected as verification events. The remaining 27 events were used in calibration.
15 Volume III Hydrograph Analysis
In the first phase of analysis, the most promising approaches were identified based on how well the derived characteristic hydrograph fitted the three verification events. The test was made across all 37 Grade A1 stations.
Having identified the most suitable non-parametric and parametric methods, these were applied across the whole set of flood events at all 89 stations in a second phase of analysis. This procedure is referred to as whole-sample calibration. Ultimately, the final estimates of the characteristic hydrograph were based on all selected flood events at the particular station. Two best methods were carried forward: one based on the non-parametric approach and the other based on the parametric approach.
3.5 Deriving the median hydrograph
3.5.1 Basic method
The preferred method in the non-parametric approach is to take median values of the various component widths of the standardised hydrographs: the averaging being taken across the group of events being analysed. As noted earlier, widths are defined at various percentages of the peak flow, and components before and after the time of peak flow are distinguished. Component hydrograph widths are not available at all percentages. Thus, the median is taken across those hydrographs that provide a component hydrograph width at the relevant percentage of the peak flow.
The characteristic hydrograph is constructed so that its component widths at all percentages correspond to the relevant median value. The resulting hydrograph is referred to as the derived median hydrograph. The example in Figure 3.1 is for the split-sample calibration at Station 07009. Thus the hydrograph is the median of 27 standardised hydrographs. Three of the 30 flood events at this station have been withheld for use in verification. The embedded table notes the values of the hydrograph widths at 75%, 50% and 25% of the peak.
100 Percentage Hydrograph width (hours)
90 of peak flow On rising limb On receding limb Total 80 70 75% 10.49 17.32 27.81
60 50% 15.15 38.75 53.90
50 25% 24.11 135.43 159.54 40 30
20
Percentage of peak flow 10 0 -50 -37.5 -25 -12.5 0 56.25 112.5 168.75 225 Time in hours (relative to time of peak flow)
Figure 3.3: Median hydrograph for Station 07009 Boyne at Navan Weir
16 Volume III Hydrograph Analysis
3.5.2 Anomalies in the derived median hydrograph
Unrealistic kinks appear at some lower percentages in the rising/receding limb of the derived median hydrograph. These appear because of the changing (i.e. reducing) number of events across which the median is drawn at lower percentages of the peak. In the case of Station 07009, the kinks are barely discernible. But, for many stations, the anomalies are unrealistic and rather distracting e.g. Station 07007 Boyne at Aqueduct in Figure 3.4a. The kinks reflect both the non-availability of widths at lower percentages in some events and the wide variation in those widths that are available at lower percentages.
(a) Station 07007 Boyne at Aqueduct (b) Station 14006 Barrow at Pass Bridge
Figure 3.4: Median hydrographs with irregularities
3.5.3 Improving the derived median hydrograph
Two techniques were applied to moderate irregularities in the derived hydrographs. These were implemented interactively using options developed within the HWA software.
In some cases, a central 3-term moving-average filter was applied one or more times to smooth the pattern of variation down the offending limb of the hydrograph. Where smoothing failed to produce an acceptable hydrograph, the lower parts were simply discarded. Sometimes the lower parts of the derived median hydrograph were discarded; sometimes the lower parts were discarded after smoothing.
Three applications of the moving-average filter led to a satisfactory outcome for Station 07007 (see Figure 3.5a). At Station 14006, the best that could be achieved was to truncate the median hydrograph by discarding parts below 20% of the peak flow (see Figure 3.5b).
There was concern that repeated use of a moving-average filter was unduly arbitrary and subjective. The primary technique recommended for removing unacceptable features of the derived median hydrograph is therefore to truncate the hydrograph by simply discarding the lower parts that exhibit inconsistent widths.
Derived median hydrographs for the 37 Grade A1 stations can be glimpsed in Appendix C. These are based on the initial split-sample calibration. Final results for the whole-sample calibration at all 89 Grade A1 + A2 stations are summarised later in Table 5.1.
17 Volume III Hydrograph Analysis
(a) Station 07007 Boyne at Aqueduct (b) Station 14006 Barrow at Pass Bridge
Figure 3.5: (a) Smoothed median hydrograph; (b) Truncated median hydrograph
18 Volume III Hydrograph Analysis
4 The parametric approach
The parametric approach specifies a formulaic shape for the characteristic hydrograph. The favoured model has three parameters.
4.1 Objectives
Generation of the median hydrograph from observed flood events at a gauged site is relatively straightforward using the HWA software to implement the non-parametric approach described in Section 3.5. That the median hydrograph is regularly spaced on the standardised flow axis rather than on the time axis is a minor inconvenience. This can be overcome by linear or other interpolation. Where the non-parametric approach is truly limiting is in the difficulty of generalising the method to allow the characteristic hydrograph to be constructed at an ungauged site.
In the parametric approach, the characteristic hydrograph is represented by an algebraic form involving two or three parameters only. This leads to design hydrographs that are smoothly varying and therefore more likely to be intuitively acceptable. However, the chief prize is that a parametric form for the characteristic hydrograph gives much greater scope for generalisation of the model for application at ungauged sites.
The two or three parameters of the algebraic form can be related to physical catchment descriptors (PCDs) available for all sites: gauged and ungauged. [Editorial note: Generalisation and automation of procedures can, for example, assist applications in Catchment Flood Risk Assessment and Management (CFRAM).] A further benefit is that the functional form makes it possible to define the flood hydrograph in its entirety, including the low parts at the beginning and end. This aids the calculation of flood volumes and supports other applications in which the whole hydrograph is required.
4.2 General approach
The general approach taken was to fit parametric curves to the median hydrographs derived in Section 3.5. The following models were considered in the research:
Cubic polynomials (fitted separately before and after the peak); The Negative Binomial distribution curve; The Inverse Gaussian distribution curve; The Hayashi et al. (1986) curve; Various formulations and extensions of the Gamma distribution curve.
These models were subject to extensive exploration, although not all methods were applied to all catchments in the 89-station dataset. In general, the parametric curves were fitted to the median hydrograph in its native semi-dimensionless form i.e. with flows in percentages of the peak and times in hours before and after the peak.
In the case of statistical distributions such as the Inverse Gaussian and Gamma, the model is customarily standardised to have unit volume under the curve rather than a unit peak. It is a matter of algebraic manipulation to come up with a variant that meets the unit peak criterion.
19 Volume III Hydrograph Analysis
This is now illustrated for the UPO-Gamma curve, which denotes a Gamma curve reformulated to have a Unit Peak at the Origin (abbreviated to UPO). The parametric model ultimately recommended is a modification of the UPO-Gamma curve.
4.3 UPO-Gamma model for the characteristic hydrograph
4.3.1 Gamma distribution
The Gamma distribution rises from the origin at x = 0 and encloses a unit volume under the curve. It has the functional form:
n1 1 x x g f(x) exp where x ≥ 0 4.1 K Γn K K
The model has two parameters: the shape parameter n and the scale parameter K. Γ(n) is the Gamma function, defined by standard formulae or tables.
The Gamma distribution has a long history of application in rainfall-runoff modelling as the Nash-cascade model for the so-called instantaneous unit hydrograph (Nash, 1957), and in flood routing as the Kalinin-Milyukov routing method (Kalinin and Milyukov, 1957). Examples appearing later (see Figure 4.1 and Figure 4.2) confirm the hydrograph-like curves generated by the distribution.
4.3.2 Peak of Gamma distribution
It can be shown that the peak of the Gamma distribution occurs at:
xp Kn 1 4.2 and, when n > 1, takes the value: 1 n1 4.3 g n 1 exp n 1 p K Γn The Gamma curve is not hydrograph-like if n ≤ 1. For example, when n = 1, it degenerates to an exponential recession. Thus, its application here is limited to the case n > 1.
4.3.3 Gamma model with peak at time zero
For the peak to occur at time zero, the distribution needs to be shifted left by xp units, i.e. by K(n-1). Thus, the Gamma distribution with peak at time zero is given by:
n1 1 x Kn 1 x Kn 1 g exp where x + K(n-1) ≥ 0 4.4 K Γn K K
This simplifies to:
20 Volume III Hydrograph Analysis
n1 1 x x g n 1 exp n 1 where x + K(n-1) ≥ 0 4.5 K Γn K K
4.3.4 Gamma model with unit peak at time zero
The requirement is for a model that can be fitted to the characteristic hydrograph. Typically, this will be the median hydrograph derived in Section 3.5. The required model therefore has a peak of 1.0 at time zero.
Dividing Equation 4.3 by Equation 4.5, we obtain the Gamma distribution with unit peak at time zero: n1 x x 4.6 y g gp 1 exp Kn 1 K
This provides the UPO-Gamma model for the characteristic hydrograph, where UPO denotes Unit Peak at Origin. y denotes the flow as a proportion of the peak flow. The hydrograph rises from y = 0 at time x = -K(n-1) to y = 1 at time x = 0, and recedes thereafter.
4.3.5 Formulation in terms of hydrograph rise time Tr
The formulation preferred here replaces K with Tr/(n-1), where Tr denotes the rise time of the Gamma distribution (see Equation 4.2). The UPO-Gamma model is then:
n1 x xn 1 y 1 exp 4.7 Tr Tr
The model rises from 0 at time x = -Tr to 1 at time x = 0. [Editorial note: The Tr parameter is sometimes referred to as the hydrograph rise time and sometimes as the translation parameter. This is because moving the time origin from the beginning of the hydrograph to the peak of the hydrograph has translated the time axis by Tr units.]
4.3.6 Families of hydrographs constructed using the model
Figure 4.1 shows hydrographs constructed using the UPO-Gamma model that all have a rise time of Tr = 50 hours. The effect of varying the parameter n is seen and its role as a shape parameter confirmed.
Figure 4.2 shows hydrographs that all have n = 3. The effect of varying the parameter Tr is seen and its role as a scale parameter confirmed. The hydrograph rises over time Tr, measured in hours.
[Editorial note: Some notational changes have been made. HWA uses q to denote the standardised flow, i.e. the hydrograph with unit peak. This has been changed to y to avoid clashing with the use of q in IBIDEM to denote the rapid response element of the total hydrograph in m3s-1.]
21
Volume III Hydrograph Analysis
Percentage of peak flow peak of Percentage
Time in hours (relative to time of peak flow) Figure 4.1: UPO-Gamma hydrograph for Tr=50 and different values of shape parameter n
Percentage of peak flow peak of Percentage
Time in hours (relative to time of peak flow)
Figure 4.2: UPO-Gamma hydrograph for n=3 and different values of scale parameter Tr
4.3.7 Example application of UPO-Gamma model
Figure 4.3 shows the outcome of fitting the UPO-Gamma model to the median hydrograph for Station 07009 Boyne at Navan Weir derived in Section 3.5. The model has two parameters: the shape parameter n and the rise-time parameter Tr. The use of only two parameters limits how well the model can fit a derived hydrograph.
It can be seen in Figure 4.3 that the model fits the rising limb of the median hydrograph at Station 07009 rather well, and also respects the general shape of the upper hydrograph. However, it provides a poor representation of the lower part of the receding limb.
22 Volume III Hydrograph Analysis
100 Median hydrograph 90 UPO-Gamma model 80 70 60 50 40 30
20 Percentage of peak flow 10 0 -50 -37.5 -25 -12.5 0 56.25 112.5 168.75 225 Time in hours (relative to time of peak Figure 4.3: UPO-Gamma flowcurve) fitted to median hydrograph at Station 07009
Experience in applying the model across 89 stations found the behaviour in Figure 4.3 to be rather typical, with the modelled hydrograph receding too quickly. This limitation was addressed by developing a version of the model offering an extended recession. This is the UPO-ERR-Gamma model.
4.4 UPO-ERR-Gamma model for the characteristic hydrograph
The UPO-ERR-Gamma model replaces the lower part of the recession of the UPO-Gamma model by an exponential recession. ERR denotes Exponential Replacement Recession. An example is shown in Figure 4.5 below.
4.4.1 Formulation
The UPO-ERR-Gamma model follows the UPO-Gamma model of Section 4.3 until its point of inflection xI on the receding limb of the hydrograph. It can be shown that the inflection occurs at:
xI Tr n 1 where n > 1 4.8
Here, x again denotes time relative to the time of peak flow. From Equation 4.7, the hydrograph ordinate at the inflection point is given by:
n1 1 yI 1 exp n 1 4.9 n 1
Thereafter, the UPO-ERR-Gamma model replaces the Gamma curve by an exponential recession. The recession curve is defined by:
23 Volume III Hydrograph Analysis
x x I y yIexp 4.10 C where parameter C controls the rate of recession. It is the time over which the recession descends to a proportion e-1/C of its value. The recession decays to 0.5 of its value over the time 0.693C, the so-called recession “half-life”.
Figure 4.4 provides an impression of the family of shapes supported by the exponential replacement recession. The case illustrated is where the point of inflection on the Gamma curve occurs 15 hours after the peak flow and at 60% of its value.
Shapes of the exponential recession curve for different values of the parameter C 70 C = 25 xo = 15, yo = 60 C = 50 60 C = 75 50 C = 100 C = 125 40 C = 150 30 C = 175 C = 200 20 C = 225 curvepercentile) (in C = 250 Ordinate of the recession the of Ordinate 10 Percentage of peak flow peak of Percentage C = 275 0 C = 300 0 50 100 150 200 250 300 Time (hr) Figure 4.4: Exponential replacement recession (ERR) for different values of parameter C
The coupling of the exponential recession replacement to the Gamma curve means that hydrographs constructed using the UPO-ERR-Gamma model exhibit a kink at the join. This reflects a discontinuity in the derivative of the modelled hydrograph. This undesirable feature was considered acceptable given the improved performance achieved by the UPO-ERR- Gamma model in comparison to the UPO-Gamma model. If required, the kink could be moderated by local smoothing.
4.4.2 Method of fitting
Various methods of fitting the UPO-ERR-Gamma model were explored. That ultimately favoured was to fit the model to the median hydrograph by optimising the three parameters (n, Tr and C) simultaneously using a genetic algorithm (see next section). The method is implemented within the HWA software package. Figure 4.5 shows the outcome for Station 07009 Boyne at Navan Weir.
4.5 Method of fitting the parametric model
Fitting the UPO-ERR-Gamma model to the characteristic hydrograph requires a criterion of what is considered a good fit and a means of seeking the parameter values that provide the best fit by this measure. The criterion is the objective function and the means of seeking the best parameters is the optimisation scheme.
24 Volume III Hydrograph Analysis
4.5.1 Objective function
The objective function adopted is the so-called least-squares criterion. The best fit is judged the one that minimises the sum of squared differences between modelled and observed values of the feature of interest.
100 Median hydrograph 90 UPO-ERR-Gamma model 80 70 Kink at join of Gamma and exponential replacement recession 60 50 40 30
Percentage of peak flow ofpeak Percentage 20 10 0 -50 -37.5 -25 -12.5 0 56.25 112.5 168.75 225 Time relative to time of peak flow (hours) Figure 4.5: UPO-ERR-Gamma curve fitted to median hydrograph at Station 07009
For applications in rainfall-runoff modelling, the feature of interest is typically ordinates of the flow hydrograph along its length. In the HWA research, the feature of interest is the hydrograph width at various percentages of the peak value. In either application, a weight can be applied to prioritise a good fit in the part of the hydrograph of most interest. The model fitting is then said to have been done by weighted least-squares.
In fitting the UPO-ERR-Gamma curve to a standardised hydrograph, the objective is to optimise its three parameters (n, Tr and C) so as to match the known widths of exceedance as closely as possible in the least-squares sense. The standardised hydrograph being fitted may be that of a single flood event or a characteristic hydrograph such as the median hydrograph derived in Section 3.5.
The objective function used for fitting the UPO-ERR-Gamma curve is the weighted sum-of- squares: N ˆ 2 S w i yi yi 4.11 i1 th Here, yi is the i ordinate of the standardised hydrograph, i is the corresponding ordinate modelled by the UPO-ERR-Gamma, wi is a weighting factor, and S is the sum of squares of the differences between yi and i across ordinates 1 to N. The summation takes place across all reference ordinates for which the hydrograph is defined, the reference ordinates corresponding to end-points to which the hydrograph width (or a component of the hydrograph width) at 98%, 95%, 90%, 85%, … 5% of the peak flow is defined. If, for example, the widths of exceedance on the rising side are available at percentiles 98, 95, 90, 85, 80, 75, 70, 65, 60, 55, 50 and 45 and the widths of exceedance on the receding side are available at percentiles 98, 95, 90, 85, 80, 75, 70, 65, 60, 55, 50, 45, 40, 35 and 30 then N in
25 Volume III Hydrograph Analysis
Equation 4.11 will be 12 + 1 + 15 = 28. Box 4.1 explains why some users may be puzzled by this choice of objective function.
Equation 4.11 includes the weight wi. The HWA software supports three weighting systems:
Greater weight to the fitting of the widths at higher percentages of the peak flow, by 2 setting: wi = (yi) ; Greater weight to the fitting of the widths at lower percentages of the peak flow, by 2 setting: wi = (1/yi) ;
Even weighting, by setting: wi = 1.
The first option was adopted in the research, i.e. assigning greater weight to the fitting of widths at higher percentages of the peak flow.
Box 4.1: The choice between horizontal and vertical fitting
Editorial note: Models such as the UPO-ERR-Gamma have a defined functional form, albeit a moderately complicated one. It ought to have been possible to fit the hydrograph width model using an objective function defined in terms of horizontal departures of the modelled hydrograph from the observed hydrograph. Instead, the HWA authors chose an objective function based on vertical fitting of the modelled hydrograph to the observed hydrograph.
Although the difference may not have changed results very appreciably, it is confusing to have evaluated the comparative performance of different models (in Section 4.5.3 below) using a measure based on horizontal departures and yet fitted the hydrograph width models by minimising vertical departures.
Unsurprisingly, the developers of IBIDEM (see Chapter 9) chose a different course. They chose an objective function that fitted hydrographs from the FSR rainfall-runoff method to FSU design hydrographs by minimising horizontal departures.
Practitioners are encouraged to make full use of the HWA and IBIDEM software. Visual display of hydrographs is integral to both packages and should allow users to obtain effective results. Researchers are to be encouraged to focus on horizontal fitting in any further development of hydrograph width analysis.
4.5.2 Optimisation scheme
The Genetic Algorithm (GA) was chosen as the optimisation scheme for fitting the UPO- ERR-Gamma curve. GAs are search algorithms based on the mechanics of natural selection and genetics. The method was developed by John Holland, colleagues and students at the University of Michigan (Holland, 1975). The method combines “survival of the fittest among string structures with a structured yet randomized information exchange to form a search algorithm with some of the innovative flair of human search” (Goldberg, 1989). The University of Michigan research aimed to: Abstract, and rigorously express, the adaptive processes of natural systems; Design artificial systems that retain the important mechanisms of natural systems.
26 Volume III Hydrograph Analysis
Genetic algorithms aim to be robust and to balance the efficiency and efficacy necessary for survival in many different environments. In each generation, a new set of artificial creatures (strings) is created using bits and pieces of the fittest of the old strings. However, the occasional new part is tried for good measure (Goldberg, 1989). [Editorial note: A few details of the procedure implemented are given in the original research report but these are not specific to the particular application. The method is said to be based on Duan (2003).]
In hydrological applications, the chief advantage of the genetic algorithm over more conventional methods of optimisation is that the search is globally oriented. This can help when fitting rather complex models to hydrological problems. [Editorial note: If the “surface” being searched (to find the minimum of the objective function) is irregular, gradient methods of optimisation may lead the user to parameter values corresponding to a local minimum of the objective function. In contrast, genetic algorithms are equipped to leap clear and are more likely to find the parameter values corresponding to the global minimum of the objective function parameters.] Wang (1991) finds the GA to be a robust and efficient method for calibrating conceptual rainfall-runoff models.
4.5.3 Performance evaluation
Several measures were considered for judging the relative performance of one method (of modelling the characteristic hydrograph) over another, including the root mean square error (RMSE) and the Nash-Sutcliffe efficiency coefficient (Nash and Sutcliffe, 1970). The measure ultimately adopted was the Mean Relative Error (Elshorbagy et al., 2000). Whereas the Nash-Sutcliffe efficiency and the RMSE are inflated by squared terms, the range of variation of the values of the MRE was generally found to be small.
In the application here, the MRE at a given percentage pj of the peak flow is defined as:
iN j ˆ 1 Wi,j Wj MRE 4.12 p j N j i1 Wi,j
th where Wi,j is the exceedance width of the i observed hydrograph, j is the exceedance width given by the model and Nj is the number of observed hydrographs for which the width is defined. In each case, the subscript j denotes the value relevant at percentage pj of the peak flow. The measure was evaluated at (up to) 20 percentages of the peak flow, with p1 = 98, p2 = 95, p3 = 90,…, p19 = 10 and p20 = 5. For reasons explained in Section 3.2, not all events have hydrograph widths defined at all percentages of the peak flow. The lower the value of MRE, the more efficient is the parametric model at reproducing the observed hydrograph widths (e.g. the widths of the characteristic hydrograph derived by the Section 3.5 method).
In practice, the quality of model fit at and above a given percentage of the peak flow is of interest. The Average MRE was therefore defined as: 1 kM AMRE Average MRE MRE 4.13 p j p j pk,j M k1 where k counts over different percentages of the peak flow at and above pj%. For example, when the quality of fit at and above 50% of the peak flow is of interest:
27 Volume III Hydrograph Analysis
1 AMRE MRE MRE MRE MRE MRE 4.14 50 11 50 55 90 95 98 To simplify matters a little, assessments focused on three reference percentiles corresponding to the fit above 75%, 50% and 25% of the peak flow. The lower the value of AMRE75, the better is the fit of the model to the upper part of the hydrograph above 75% of the peak flow.
Box 4.2: Editorial note on performance measures
An array of measures is used to assess performance. Mention of the Nash-Sutcliffe efficiency (NSE) is perhaps justified by its widespread use in hydrological modelling. However, it is poorly adapted to the case of modelling hydrograph widths. More aptly, the root mean square error (RMSE) in hydrograph width estimation is used to good effect in IBIDEM (see Chapter 9, in particular Section 9.5.4).
The authors’ preference for the Equation 4.12 measure (MRE) is uncomfortable. When defining the mean relative absolute error, it is more usual for the denominator to be taken as the minimum absolute value of the terms being differenced in the numerator. The disadvantage of the Equation 4.12 measure is that it does not treat modelled and observed values even-handedly. n consequence, large modelled values of the hydrograph width (i.e. j) degrade the evaluated performance much more than small values do. ndeed, large values of can lead to values of MRE (and AMRE) that exceed 1.0. j
4.6 Reproduction of flood hydrographs of verification events
An independent test of the relative merit of competing parametric models was possible by exploiting the three events reserved for verification at each station (see Section 3.4). Figure 4.6 illustrates the performance achieved with the UPO-ERR-Gamma model for Station 07009 Boyne at Navan Weir. The model reproduces the observed flood hydrographs quite well for Events 5 and 15, and the fit is especially good above 50% of the peak flow. In the case of Event 24, only the upper part of the hydrograph was available, this being the main flood response decoupled from a complex event. The fit of the UPO-ERR-Gamma model is poorer for this event but considered acceptable given that the flood is not a major one, being only the 24th largest of the 30 events at Station 07009.
When evaluated across all 37 Grade A1 station, the UPO-ERR-Gamma model was judged to provide the best overall performance (amongst all the parametric models considered) in modelling the upper parts of the hydrographs reserved for verification. Modelled and observed hydrographs for these 111 verification events are shown in Figure C.1 to Figure C.37 of Appendix C. Fits achieved by the non-parametric approach are also shown.
28 Volume III Hydrograph Analysis
100 Median hydrograph
90 UPO-ERR-Gamma model Event 5 hydrograph 80 70 60 50 40 30 20
Percentage 10 of peak flow 0 -50 -25 0 56.25 112.5 168.75 225 Time relative to time of peak flow (hours)
100 Median hydrograph
90 UPO-ERR-Gamma model Event 15 hydrograph 80
flow 70
60
50 40 30
20
Percentage 10 of peak 0 -50 -25 0 56.25 112.5 168.75 225
Time relative to time of peak flow (hours)
100 Median hydrograph
90 UPO-ERR-Gamma model Event 24 hydrograph 80 70 60 50
40 30 20
Percentage 10 of peak flow
0 -50 -25 0 56.25 112.5 168.75 225 Time relative to time of peak flow (hours) Figure 4.6: Performance of UPO-ERR-Gamma model on verification events, Station 07009
29 Volume III Hydrograph Analysis
4.7 Other methods
The research considered many models, and multiple ways of fitting them. The Gamma-UPO- ERR parametric model was calibrated extensively. The primary versions assessed were:
v1 Fitting to the derived median hydrograph (as recommended and illustrated above); v2 Fitting to the derived mean hydrograph; v3 Taking the median of parameter values obtained when fitting to flood hydrographs for individual events; v4 Taking the mean of parameter values obtained when fitting to flood hydrographs for individual events; v5 Fitting to the event hydrographs collectively.
Further stratification of the study took the objective function as minimising Average MRE values calculated above 50% of the peak flow, above 75% of the peak flow and above other percentages. The reader is referred to the original research report and its many appendices.
[Editorial note: It is the practice in applied research to prune back the number of methods, and to consider in detail only the most promising sub-variations. Some investigators are, however, reluctant to judge one method better than another without exhaustive analyses.]
Figure 4.7 provides an example of the richness of material generated in the research. A good method constructs the upper hydrographs of the verification events with low error. The performance measure AMRE is the average value of the Mean Relative Error in reproducing upper hydrograph widths (see Section 4.5.3). The recommended version (v1) is seen to perform best in verification across the 37 Grade A1 stations; v3 performs next best.
2.0 Gamma I median Gamma I mean Gamma II median Gamma II mean Gamma III 1.8
1.6 v1 v2 v3 v4 v5
50 1.4 1.2 1.0 MRE 0.8
AMRE 0.6 0.4 0.2
0.0
6011 6012 6013 6014 6026 7007 7009 7010 7012 9001
14004 14006 14007 14011 14018 15005 23002 24013 25003 25006 25014 25017 25025 25027 25030 26007 26008 26012 26019 27002 29001 29011 30004 30005 34018 36010 36015
2.0 Gamma I median Gamma I mean Gamma II median2.0 Gamma II Gammamean I medianGamma III Gamma I mean Gamma IIStation median no. Gamma II mean Gamma III 1.8 1.8 1.6 1.6 v1 v2 v3 v4 v5
1.4 1.4 1.2 50 1.2
1.0 1.0 MRE 0.8 MRE 0.8 0.6 0.6 0.4 AMRE 0.4 0.2 0.2
0.0 0.0
6011 6012 6013 6014 6026 7007 7009 7010 7012 9001
6011 6012 6013 6014 6026 7007 7009 7010 7012 9001
14004 14006 14007 14011 14018 15005 23002 24013 25003 25006 25014 25017 25025 25027 25030 26007 26008 26012 26019 27002 29001 29011 30004 30005 34018 36010 36015
14006 14007 14011 14018 15005 23002 24013 25003 25006 25014 25017 25025 25027 25030 26007 26008 26012 26019 27002 29001 29011 30004 30005 34018 36010 36015 14004 Station no. Station no. Figure 4.7: Verification performance of UPO-ERR-Gamma calibrated in five versions
30 Volume III Hydrograph Analysis
5 Performance of methods at gauged sites
5.1 Introduction
The performance of the recommended methods is now examined. The recommended method in the non-parametric approach is the median hydrograph of Section 3.5. The recommended method in the parametric approach is the UPO-ERR-Gamma model developed in Section 4.4. The initial assessment is made across the 37 Grade A1 stations.
It is helpful to assess the methods jointly. Where poor performance is identified, it is then possible to distinguish whether this arises more from the choice of method or more from the variable nature of hydrographs at a particular station.
Figure 5.1 and Figure 5.2 summarise the performance of the median hydrograph and UPO- ERR-Gamma methods in calibration and in verification across the 37 Grade A1 stations. The measure shown is AMRE50.
AMRE denotes the average MRE. This is a double averaging. The mean relative error (MRE) in modelling hydrograph widths is averaged across available segments of the hydrograph above 50% of the peak flow, and is also averaged across the available flood hydrographs. For the verification results, the averaging is across the three events withheld for the purpose. For the calibration, the averaging is across the remaining flood hydrographs.
1.2 Calibration Verification 1.0
0.8 Calibration Verification 50
0.6 MRE 0.4
AMRE 0.2
0.0
6011 6012 6013 6014 6026 7007 7009 7010 7012 9001
14006 14007 14011 14018 15005 23002 24013 25003 25006 25014 25017 25025 25027 25030 26007 26008 26012 26019 27002 29001 29011 30004 30005 34018 36010 36015 14004 Figure 5.1: Performance of median hydrographStation no. method across 37 Grade A1 stations
1.2 Calibration Verification 1
Calibration Verification
50 0.8 0.6
0.4 MRE
AMRE 0.2
0
6011 6012 6013 6014 6026 7007 7009 7010 7012 9001
14006 14007 14011 14018 15005 23002 24013 25003 25006 25014 25017 25025 25027 25030 26007 26008 26012 26019 27002 29001 29011 30004 30005 34018 36010 36015 14004 Figure 5.2: Performance of UPO-ERR-GammaStation method no. across 37 Grade A1 stations
A small value of AMRE50 indicates that the method typically represents the upper hydrograph well. An arbitrary reference line AMRE50=0.4 is drawn across the figures to aid discrimination of unusual cases. It is seen that AMRE50 lies below 0.4 at most stations.
Although only just discernible in Figure 5.1 and Figure 5.2, the median hydrograph method outperforms the UPO-ERR-Gamma method. This is more clearly seen in Figure 5.3.
31 Volume III Hydrograph Analysis
1.2
Station 34018 h
p 1.0
a
r
g o
r Station 06011
d 0.8
y
h
n
a
i d
e 0.6
m
r
o
f
0 0.4
5
E R
M 0.2 A
0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 AMRE50 for UPO-ERR-Gamma Figure 5.3: Comparison of model performance across 37 Grade A1 stations
In calibration, the median AMRE50 (across the 37 stations) is 0.227 for the median hydrograph method and 0.246 for the Gamma model. The difference is more noticeable in verification; the median AMRE50 is 0.200 for the median hydrograph method (better than in calibration!) and 0.252 for the Gamma model. The superior performance of the median hydrograph method is to be expected given its greater flexibility to accommodate different shapes of characteristic hydrographs.
A number of factors influence these results. Some stations exhibit wide variation in hydrograph shapes across events. These are characterised by relatively high AMRE in calibration. Prime examples are Stations 06011 and 34018. A further factor is that AMRE is less well defined at stations for which the flood hydrograph is typically complex. AMRE50 nominally measures the average error in estimating hydrograph widths across the upper half of the hydrograph. However, in practice, AMRE is evaluated across such widths as are available after filtering of the hydrograph (see Section 2.6).
5.2 Relative performance in verification compared to that in calibration
The median hydrograph method performs better in verification than in calibration at 23 stations and worse at 13 stations, with Station 07010 too close to call. The UPO-ERR- Gamma method performs better in verification than in calibration at 21 stations and worse at 15 stations, with Station 34018 too close to call.
As illustrated in Figure 5.4, the improvement in performance is most evident at Stations 06011, 06012, 14006, 14007, 14018, 26007 and 36015 (for both methods) and at Station 26012 (for the median hydrograph method). A deterioration in performance in verification is most evident at Stations 07012 and 25003 (for both methods) and at Station 25017 (for the median hydrograph method).
32 Volume III Hydrograph Analysis
4
25017 25003 2
07012 Calib
50 25014 22550202405016143004 36 0016014 3207000042 26008 1 3401 807010 21550 0300075 00067026
250297001
AMRE
/ 23002 1 400610113 26007 092060201910911
Verif Verif 07009
50 0.5 14006 30005 36015 1410400718
06011 26012
median hydrograph methodmedianhydrograph
AMRE
for 0.25 06012
0.125 0.125 0.25 0.5 1 2 4 Verif Calib AMRE50 / AMRE50 for UPO-ERR-Gamma model
Figure 5.4: Performance in verification compared to that in calibration
[Editorial note: With many events typically used in calibration, there is considerable scope for wide variation in hydrographs at a particular station to degrade model performance. With only three events used for verification at each station, improved performance may in part be a matter of chance. Three samples may show broad conformity to the calibrated model or wide departures from it. Nevertheless, these are good results, since one would normally expect performance in verification to be inferior to performance in calibration.
The unexpectedly strong performance in verification evident in Figure 5.4 may reflect that the selection of verification events (see Section 3.4) was not fully automated. With over 2800 flood events across 89 catchments, the selection was a very demanding task in itself. Stations occasionally exhibit individual hydrographs of a highly unusual shape. There may have been reluctance – instinctive rather than conspiring – to allow patently unusual events to join the select few allocated to the verification set.]
5.3 Complexity of hydrographs at Stations 06011 and 34018
The recommended methods perform poorly at Stations 06011 Fane at Moyles Mill and 34018 Castlebar at Turlough. Other methods considered in the research (see Sections 4.2 and 4.7) also performed poorly on these catchments. A particular difficulty is that flood hydrographs at these stations are generally complex. At Station 06011, the total width of exceedance is undefined below 65% of the peak flow in all the hydrographs studied. The restriction is even more severe at Station 34018, where none of the hydrographs defines the total width of exceedance below 80% of the peak flow. The hydrographs of four of the eight largest floods at this station are illustrated in Figure 5.5.
33 Volume III Hydrograph Analysis
Event 1 Event 2
)
)
1
1
-
-
s
s
3
3
Flow (m Flow (m
Event 6 Event 8
)
)
1
1
-
-
s
s
3
3
Flow (m Flow (m
Figure 5.5: Varied hydrograph shapes at Station 34018 Turlough at Castlebar
A further difficulty is that upper segments of the hydrographs at Stations 06011 and 34018 have a mix of sharp-peaked and rounded shapes. The characteristic hydrographs derived by the median and UPO-ERR-Gamma methods are gently rounded (see Figure C.1 and Figure C.35) yet some individual hydrographs have upper segments that are sharp-peaked.
The variability in upper-hydrograph shape is particularly marked at Station 34018 Castlebar at Turlough (see Figure 5.5). This may reflect special features of the catchment that provide particular scope for temporal and spatial patterns in rainfall to lead to varied and complex hydrograph shapes. [Editorial note: The name Castlebar at Turlough suggests that geolog- ical features may in part account for the complexity and variability of hydrographs at Station 34018. The authors note the large loughs in the catchment and suggest that the layout increases the sensitivity of response timings to the spatial pattern of rainfall.]
5.4 Variability in hydrograph widths at some stations
Hydrographs at stations such as 24013 Deel at Rathkeale, 25006 Brosna at Ferbane, 25014 Siler at Millbrook and 25025 Ballyfinboy at Ballyhooney exhibit wide variability in widths. As examples, two relatively wide and two relatively narrow hydrographs at each of these stations are reproduced in Figure 5.6 to Figure 5.9. In the case of Station 25025, the two narrower hydrographs are rather curiously slanted.
In the context of such wide variations in the shapes of flood hydrographs, it is desirable that the reasons be investigated and explanations sought in terms of catchment features and/or the meteorological conditions leading to the observed floods at these stations. Station 24013 is investigated further in Section 5.10.2.
34 Volume III Hydrograph Analysis
Event 2 Event 11
)
)
1
1
-
-
s
s
3
3
Flow (m Flow (m Flow
Event 20 Event 27
)
)
1
1
-
-
s
s
3
3
Flow (m Flow (m Flow
Figure 5.6: Wide and narrow-peaked hydrographs at Station 24013 Deel at Rathkeale
Event 1 Event 4
)
)
1
1
-
-
s
s
3
3
Flow (m Flow (m Flow
Event 3 Event 31
)
)
1
1
-
-
s
s
3
3
Flow (m Flow (m Flow
Figure 5.7: Wide and narrow-peaked hydrographs at Station 25006 Brosna at Ferbane
35 Volume III Hydrograph Analysis
Event 7 Event 23
)
)
1
1
-
-
s
s
3
3
Flow (m Flow (m Flow
Event 9 Event 21
)
)
1
1
-
-
s
s
3
3
Flow (m Flow (m Flow
Figure 5.8: Wide and narrow-peaked hydrographs at Station 25014 Silver at Millbrook
Event 5 Event 13
)
)
1
1
-
-
s
s
3
3
Flow (m Flow (m Flow
Event 2 Event 31
) )
1 1
- -
s s
3 3
Flow (m Flow Flow (m
Figure 5.9: Wide and slanted hydrographs at Station 25025 Ballyfinboy at Ballyhooney
36 Volume III Hydrograph Analysis
Twelve flood events at Station 25014 Silver at Millbrook were studied as part of FSU work on Flood Event Analysis (see 0). Six events were used to derive an average unit hydrograph, which was tested on six further events. One of these was the 30 June 1986 flood, which corresponds to the second part of Event 9 (see lower left diagram in Figure 5.8).
The University College Cork analysis in Figure 5.10 suggests that the catchment responded particularly quickly to rainfall in this event. DR denotes the direct runoff from rainfall. A possible explanation for the unusually narrow hydrograph is that the flood arose from heavy rainfall on part of the lower catchment only. The catchment configuration is somewhat unusual (see Map 5.1 based on UCC research).
Figure 5.10: Rainfall-runoff behaviour in 30 June 1986 flood at Station 25014
Map 5.1: Catchment of Station 25014 Silver at Millbrook
37 Volume III Hydrograph Analysis
5.5 Attenuated response at some stations
In the case of (e.g.) Station 25017 Shannon at Banagher, the flood hydrographs are exceedingly attenuated (i.e. dampened or “flat”), with only very slow rises to and recessions from the peak. The peak segment is maintained for several days, and sometimes represents a sustained high flow with no outstanding peak. Four such hydrographs are reproduced in Figure 5.11.
Event 1
)
) Event 4
1
1
-
-
s
s
3
3
Flow (m Flow (m Flow
Event 10
)
)
1
1
-
-
s
s 3
Event 7 3
Flow (m Flow (m Flow
Figure 5.11: Attenuated hydrographs at Station 25017 Shannon at Banagher
For a station characterised by such very flat flood hydrographs, only a very small number of percentile widths are available for calibrating a model of the characteristic hydrograph. It is therefore unsurprising that a derived hydrograph produced from such a small number of ill- defined events may not reproduce the flood hydrograph of a verification event very well.
5.6 General guidance
On the basis of the above, it is concluded that, for reliability in producing a design hydrograph:
Standardised hydrographs should be derived for as many flood events as possible; Where these differ widely from event to event, the reasons should be explored on the basis of the physical characteristics of the catchment and of the flood-producing rainfall;
38 Volume III Hydrograph Analysis
A characteristic hydrograph fitted to very varied hydrographs, or one developed from flood hydrographs that are highly attenuated (i.e. dampened or “flat”), should be applied with caution when producing design flood hydrographs.
5.7 Results of whole-sample calibration
Having identified “best” non-parametric and parametric methods – i.e. the derived median hydrograph and the UPO-ERR-Gamma model – based on 37 Grade A1 stations, the methods were applied to the entire set of 89 stations in the study. All available events were now used: hence, the term whole-sample calibration.
5.7.1 Derived median hydrograph and its descriptors
Figure 5.12 shows the outcome of the whole-sample calibration for Station 07009 Boyne at Navan Weir.
100 Percentage Hydrograph width (hours)
90 of peak flow On rising limb On receding limb Total 80 70 75% 10.46 17.29 27.75
60 50% 14.94 39.20 54.14
50 25% 23.38 136.67 160.05 40 30
flow ofpeak Percentage 20 10 0 -50 -37.5 -25 -12.5 0 56.25 112.5 168.75 225 Time relative to time of peak flow (hours)
Figure 5.12: Median hydrograph for Station 07009 Boyne at Navan Weir (whole sample)
The table embedded in Figure 5.12 notes the values of the hydrograph widths at 75%, 50% and 25% of the peak. Comparison with Figure 3.3 reveals that, for this station, the median hydrograph is little changed after inclusion of the three events previously reserved for verification.
Table 5.1 summarises the outcome of hydrograph width analysis applied to the full set of 89 stations. Three width descriptors are presented in the central columns. Two are the widths of exceedance W75 and W50 at 75% and 50% of the peak flow. Hydrographs are insufficiently defined to establish these width descriptors for Stations 30061 and 34018; additionally, W50 is undefined for Stations 06011, 25017 and 34001.
The third width descriptor is the mean ratio, s, of the width on the rising side to the total width. This average value is calculated across all available widths of all the available flood hydrographs. The ratio s summarises the skew of the hydrograph, with a value of 0.5 indicating broad symmetry about the peak. Figure 5.13 shows the histogram of values of s
39 Volume III Hydrograph Analysis across the 89 stations analysed. It is seen that s is appreciably less than 0.5 at most stations. This reflects that floods typically rise more quickly than they fall. For Station 07009, s = 0.345.
7 <-- s = 0.5 (symmetric) 6
5
y c
n 4
e
u q
e 3 r F 2 1 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 s = mean ratio of width before peak to total hydrograph width
Figure 5.13: Summary index, s, of hydrograph skewness (89 Grade A1 + A2 stations)
[Editorial note: Ratios (such as s) should be averaged by taking the geometric mean not by taking the arithmetic mean. Although not of great significance here, the difference can be important in particular hydrological applications.]
5.7.2 UPO-ERR-Gamma model and its parameters
The UPO-ERR-Gamma was applied to the entire set of 89 stations, fitting the model by optimisation using the genetic algorithm (see Section 4.5.2) to the derived median hydrograph for the particular station. The optimised parameter values appear in the final three columns of Table 5.1. These are values of the shape parameter n, the rise-time or translation parameter Tr and parameter C of the exponential replacement recession.
The C parameter could not be calibrated for Station 30061. The Gamma curve fitted to the derived median hydrograph for this Grade A2 station implies a point of inflection on the receding limb at 71% of the peak flow, some 30 hours after the peak. For this station, the median hydrograph does not define after-the-peak hydrograph widths below 90% of the peak flow. Thus, it was not possible to fit the exponential replacement recession.
Table 5.1: Outcome of whole-sample calibration at 89 Grade A1 + A2 stations Station Station Median hydrograph method UPO-ERR-Gamma model
number grade W75 W50 s n Tr C 06011 A1 114.35 N/A 0.374 1.300 29.690 360.834 06012 A1 136.25 243.02 0.325 2.457 93.311 280.948 06013 A1 41.99 70.87 0.354 7.782 59.989 102.855 06014 A1 95.87 154.30 0.394 3.014 89.910 113.849 06026 A1 82.47 146.16 0.390 4.998 102.176 140.966 07001 A2 18.36 34.20 0.377 5.006 23.309 38.360 07002 A2 46.74 93.14 0.351 2.770 32.736 151.959 07004 A2 110.14 171.87 0.382 3.014 101.979 116.780
40 Volume III Hydrograph Analysis
Station Station Median hydrograph method UPO-ERR-Gamma model
number grade W75 W50 s n Tr C 07006 A2 15.42 30.38 0.331 7.782 20.874 58.149 07007 A1 26.22 52.53 0.355 3.745 26.837 69.875 07009 A1 27.75 54.14 0.345 5.276 33.040 95.526 07010 A1 30.42 119.06 0.348 3.458 25.784 222.317 07011 A2 108.21 170.42 0.344 2.988 92.964 116.780 07012 A1 27.75 49.10 0.375 6.111 36.407 88.197 07033 A2 46.41 84.10 0.372 6.111 60.818 105.054 09001 A1 11.21 22.64 0.394 8.399 21.751 22.970 11001 B 7.54 10.84 0.456 26.356 25.241 5.380 14004 A1 48.17 74.35 0.447 5.406 66.484 53.751 14006 A1 57.46 83.02 0.487 4.710 75.828 41.292 14007 A1 9.34 17.14 0.487 21.412 31.787 11.243 14009 A2 26.27 46.20 0.378 3.884 27.746 46.422 14011 A1 45.96 89.57 0.361 5.276 54.734 116.780 14018 A1 65.08 121.78 0.350 2.944 57.728 116.780 15001 A2 15.74 28.49 0.434 9.417 30.458 23.703 15002 A2 8.12 19.23 0.428 17.635 24.966 21.504 15003 A2 5.77 8.79 0.489 10.541 12.422 6.113 15005 B 60.06 114.00 0.417 5.267 83.234 106.520 15006 A2 19.95 35.64 0.517 13.829 53.250 22.970 16001 A2 30.61 46.41 0.359 5.554 40.156 39.826 16002 A2 58.04 102.32 0.411 8.338 91.266 114.581 16003 A2 39.53 77.32 0.216 2.527 16.148 188.604 16004 A2 63.79 102.80 0.429 6.390 90.244 99.191 16005 A2 13.76 21.02 0.541 28.192 51.750 5.380 16008 A2 88.08 146.96 0.243 3.153 44.158 410.670 16009 A2 39.50 92.10 0.360 6.111 49.988 163.685 18004 A2 38.18 73.27 0.327 4.998 43.894 105.054 18005 A2 15.63 26.35 0.427 24.604 52.750 15.641 19001 A2 26.34 44.94 0.245 2.788 12.675 112.383 22071 A2 96.29 190.22 0.348 1.883 50.563 243.571 23001 A2 6.98 11.84 0.442 12.536 16.654 5.380 23002 A1 5.81 9.27 0.528 30.265 22.999 3.182 23012 A2 11.67 18.29 0.474 18.226 32.131 11.243 24001 A2 18.24 27.88 0.360 4.441 19.161 35.429 24008 A2 19.08 30.54 0.363 3.884 19.404 28.100 24013 A1 19.66 28.77 0.570 12.333 48.750 6.846 24082 A2 18.66 27.51 0.316 5.137 21.420 38.360 25001 A2 15.51 29.51 0.460 9.452 31.941 35.429 25003 A1 15.38 29.57 0.343 4.302 16.506 50.087 25005 A2 14.00 33.17 0.314 4.998 13.136 69.875
41 Volume III Hydrograph Analysis
Station Station Median hydrograph method UPO-ERR-Gamma model
number grade W75 W50 s n Tr C 25006 A1 35.86 63.91 0.396 5.276 42.419 94.060 25014 A1 17.63 30.05 0.393 8.965 30.830 36.162 25016 A2 20.26 36.19 0.332 3.884 18.404 62.546 25017 A1 231.91 N/A 0.442 1.500 109.750 374.759 25025 A1 85.33 187.22 0.285 4.136 111.286 222.317 25027 A1 7.66 14.25 0.438 19.409 23.416 19.305 25029 A2 15.87 29.50 0.416 8.886 28.700 50.087 25030 A1 70.22 122.12 0.395 4.223 81.857 94.060 26002 A2 118.67 163.98 0.462 6.581 186.500 39.826 26005 A2 136.04 209.83 0.418 3.597 148.511 84.533 26007 A1 156.97 257.02 0.445 3.040 159.806 107.985 26008 A1 106.69 206.81 0.419 3.875 124.500 177.610 26009 A2 26.52 38.47 0.486 8.469 49.000 14.175 26012 A1 177.27 285.94 0.400 3.849 195.500 257.496 26019 A1 72.81 114.01 0.432 3.362 71.390 70.608 26021 A2 52.68 179.79 0.275 3.484 24.799 527.933 26022 A2 41.75 74.82 0.404 4.998 53.640 66.943 27001 A2 14.87 20.26 0.385 4.998 17.722 11.243 27002 A1 270.72 493.96 0.430 5.276 429.250 187.871 29001 A1 49.94 86.19 0.429 5.328 74.750 46.422 29004 A2 76.87 130.05 0.526 2.770 79.500 76.471 29011 A1 138.65 242.46 0.458 4.014 182.750 114.581 30004 A1 44.33 68.49 0.453 5.302 59.439 46.422 30005 A1 39.87 56.46 0.562 7.399 77.000 22.970 30007 A2 38.52 59.31 0.549 6.111 67.750 20.771 30061 A2 N/A N/A 0.447 2.910 41.901 N/A 34001 A2 95.52 N/A 0.416 2.788 58.347 829.752 34009 A2 19.67 26.03 0.559 9.095 39.699 8.312 34018 A1 N/A N/A 0.315 1.274 35.584 187.871 35001 A2 103.62 154.99 0.435 3.875 116.937 105.787 35002 A2 7.62 11.39 0.577 17.765 24.011 0.25 35005 A2 38.18 85.88 0.340 4.998 36.804 187.871 35071 A2 123.67 210.68 0.321 4.223 99.464 456.843 36010 A1 75.15 155.52 0.440 3.040 73.102 152.692 36011 B 316.02 613.41 0.371 2.214 295.250 229.646 36015 A1 40.08 67.19 0.485 6.189 68.000 35.429 36019 A2 295.51 491.91 0.421 2.483 254.500 302.202 36021 A2 3.11 4.90 0.462 14.698 7.655 2.449 36027 A2 332.71 539.54 0.376 2.605 300.500 213.522 39009 A2 45.10 86.66 0.348 9.104 74.098 115.314
42 Volume III Hydrograph Analysis
[Editorial note: The parameter values of Table 5.1 reveal some unusual features of the optimisation. Excessive decimal places are shown to confirm that parameter n takes certain preferential values, with n = 4.998 at six stations and n = 6.111 at a further four stations. Parameter C also has preferential values, with C = 116.780 at four stations and the values 5.380, 11.243, 27.970, 35.429, 46.422 and 187.871 each appearing at three stations. While the fits achieved are fully satisfactory, the preferential values suggest that results obtained with the genetic algorithm are not always fully optimised.]
5.7.3 Stations where the flood hydrograph recedes faster than it rises
Flood hydrographs at Station 35002 Owenbeg at Billa Bridge typically fall more steeply than they rise. In consequence, the derived median hydrograph in Figure 5.14 also has this feature. The station has the most skewed hydrograph (s = 0.577) of any in the 89-station dataset. Reference to Figure 4.1 indicates that this is not a shape that the Gamma family of curves can accommodate. When forced to do so, the exponential replacement recession in the adopted UPO-ERR-Gamma is unreasonably steep, with C = 0.25 hours (see Figure 5.14). [Editorial note: In essence, the ERR part of the model is attempting to compensate for overestimation (by the Gamma curve) of the hydrograph width component that occurs after the peak.]
100 Median hydrograph
90 UPO-ERR-Gamma model 80
70 60 50
40
Percentage of peak flow ofpeak Percentage 30 20
10- 25 -18.75 -12.5 -6.25 0 18.75 37.5 56.25 75 Time in hours (relative to time of peak flow)
Figure 5.14: Characteristic hydrograph for Station 35002 Owenbeg at Billa Bridge
Station 30005 Robe at Foxhill also exhibits hydrographs that typically fall more steeply than they rise. However, the feature is less pronounced and the UPO-ERR-Gamma model might be considered just about acceptable (see Figure 5.15).
Overall, the UPO-ERR-Gamma curve was found to model the characteristic hydrograph adequately at most of the 89 stations. One station having a somewhat unusual characteristic hydrograph shape is Station 16008 Suir at New Bridge, investigated in the next section.
43 Volume III Hydrograph Analysis
100 Median hydrograph
90 UPO-ERR-Gamma model
80 70 60 50 40 30 Percentage of peak flow ofpeak Percentage 20 10-75 -56.25 -37.5 -18.75 0 43.75 87.5 131.25 175 Time relative to time of peak flow (hours)
Figure 5.15: Characteristic hydrograph for Station 30005 Robe at Foxhill
5.8 Characteristic hydrographs on the River Suir
Multiple gauging stations allow one to see how the characteristic hydrograph changes down the river system. Figure 5.17 presents the outcome of applying the recommended methods to four stations on the Suir. The timescale is the same in each graph. Of particular interest is that the width of hydrographs does not appear to widen appreciably as floods pass down the river system, except around New Bridge. The steepening of the rising limb is particularly noticeable at New Bridge and Caher Park.
Median hydrograph Median hydrograph UPO-ERR-Gamma model UPO-ERR-Gamma model 16004 Thurles 16002 Beakstown 229 km2 486 km2
Median hydrograph Median hydrograph UPO-ERR-Gamma model UPO-ERR-Gamma model 16008 New Bridge 16009 Caher Park 1090 km2 1583 km2
Figure 5.16: Characteristic hydrographs for four stations on the River Suir
The period of record used in the HWA is substantial. The analysis spans 1954 to 2000 at all four sites, and somewhat longer periods at Stations 16008 and 16009 (see Table A.2). The effects evident in Figure 5.17 are therefore not thought to be an artefact of the data samples.
44 Volume III Hydrograph Analysis
The UPO-ERR-Gamma model represents well the steepening of the rising limb (see Figure 5.17). However, the model has difficulty with the unusual shape of the median hydrograph at Station 16008 Suir at New Bridge. Small values of n and Tr are needed to represent the rising limb and the UPO-ERR-Gamma model cannot accommodate the appreciable “hunch” on the receding limb.
The Suir example provides a strong reminder of the very substantial hydrograph records held for many Irish rivers, and the scope for hydrograph width analysis to complement the statistical analysis of flood peaks in Volume II.
1.00 16004 Suir at Thurles 16002 Suir at Beakstown
h 16008 Suir at New Bridge
p 0.75 a
r 16009 Suir at Caher Park
g
o
r
d
y
h
c
i 0.50
t
s
i
r
e
t
c
a
r a
h 0.25 C
0.00 -96 -72 -48 -24 0 24 48 72 96 120 144 168 192 216 Time relative to time of peak flow (hours) Figure 5.17: UPO-ERR-Gamma characteristic hydrographs for four stations on the Suir
5.9 Hydrograph width analysis at gauged sites – a summary
Table 5.1 has summarised the hydrograph width analysis (HWA) undertaken for the 89 Grade A1 + A2 stations. The results presented comprise:
Three hydrograph-width descriptors (W75, W50 and s) summarising the upper part of the median hydrograph derived using the recommended non-parametric method;
Three parameters (n, Tr and C) of the UPO-ERR-Gamma model, which is the recommended parametric model for the characteristic hydrograph.
The design flood hydrograph of required return period can be produced by scaling up the ordinates of the characteristic hydrograph by the relevant peak flow derived by Volume II methods.
It is desirable that the HWA results are updated and extended as new data become available. Hydrograph width analysis is assisted by provision of the HWA software (see Section 1.6 and Appendix E). It is anticipated that users will upload their HWA results to the FSU Web Portal: both to facilitate their own applications and to share the hydrograph-width descriptors and parameters values obtained at gauged sites with the wider community.
45 Volume III Hydrograph Analysis
5.10 Flood hydrographs having sustained peaks
Sometimes flood hydrographs at a station are observed to have a sustained crest segment in which the flow varies little around the peak. Such flood hydrographs may or may not be representative of the typical flood hydrographs occurring at a station. Two cases are discussed.
5.10.1 Where the hydrograph shape reflects the temporal pattern of rainfall
During a storm event, the continuation of heavy rainfall at the peak-inducing intensity for a considerable period of time may cause the resulting flood hydrograph to exhibit a sustained peak. [Editorial note: This behaviour may arise when a rain-producing weather system becomes relatively fixed, e.g. with moist air continually fed into a particular zone where topography induces heavy rainfall. The weather system is sometimes termed a “seeder-feeder mechanism”. The topographic effect is referred to as “orographic enhancement”.] Although these situations can give rise to a flood of unusually large volume, such events are unlikely to be a regular occurrence at any particular station.
The premise of hydrograph width analysis is to represent only the characteristic shape of the hydrograph. Where only one or two events exhibit the sustained hydrograph peak, it is proper that they do not unduly influence the characteristic hydrograph. This is the case for the recommended non-parametric method, which takes the median (rather than the mean) of hydrograph widths.
5.10.2 Where the hydrograph shape is characteristic of the station
At some stations, relatively flat-peaked flood hydrographs are a characteristic feature attributable to floodplain storage or other features upstream. A prime example is Station 24013 Deel at Rathkeale. A floodplain effect is inherently threshold-sensitive, so it is prudent to study actual hydrographs (in m3s-1) as well as standardised hydrographs.
Filtered hydrographs for all 31 events at Station 24013 are overlain in Figure 5.18. The hydrographs have been translated to synchronise their peaks but not otherwise adjusted. It is seen that a slowly rising crest segment is indeed typical of flood hydrographs at this site. The characteristic hydrographs shown in the figure have been scaled up by multiplying the median standardised hydrograph and the UPO-Gamma standardised hydrograph by the median hydrograph peak of 116.75 m3s-1.
The non-parametric method is seen to perform well, with the thick blue line in Figure 5.18 representing the tendency for the crest segment of Deel at Rathkeale flood hydrographs to rise slowly to a peak before dropping away. The UPO-Gamma model is incapable of representing this effect.
[Editorial note: Confusingly, the thick black line in Figure 5.18 shows the UPO-Gamma model rather than the recommended UPO-ERR-Gamma. As explained in Section 5.7.3, the exponential replacement recession can compensate a little for the intrinsic unsuitability of the
46 Volume III Hydrograph Analysis
Gamma curve in cases where the characteristic hydrograph falls more rapidly than it rises. The relevant recession is marked by the black broken line superposed on Figure 5.18.]
Figure 5.18: Hydrographs and rescaled characteristic hydrograph for Deel at Rathkeale
47 Volume III Hydrograph Analysis
6 Constructing the characteristic hydrograph at ungauged sites
6.1 Introduction
A major part of research was to generalise Hydrograph Width Analysis (HWA) to allow the characteristic hydrograph to be constructed at ungauged sites. The original research report provides a very detailed description of the work. A précis is presented here. Holder (1985) provides a primer on the use of multiple regression in hydrology.
6.1.1 Links with other parts of the FSU
The generalisations are based on the physical catchment descriptors (PCDs) presented in Volume IV. There is some commonality with the Volume II work on generalising a model of the index flood QMED. Both studies use multiple regression analysis, and develop linear models between logarithmic transforms of the variable of interest and logarithmic transforms of the PCDs. Individual researchers favour particular notations, terminology and refinements of the basic techniques. Editing has sought to unify the descriptions.
Whereas the flood frequency research generalises one variable only (QMED), the HWA research considers three descriptors (W75, W50 and s) and three model parameters (Tr, n and C). A generic term is needed and these six are referred to below as the dependent variables (DVs) of the regression study. Having adopted this terminology, it is natural to refer to the PCDs as the independent variables (IVs).
6.1.2 Assumptions and difficulties
The following standard assumptions of multiple linear regression are noted:
The relationship between the DV and the IVs is linear; The expected value of the arithmetic mean of the errors is zero (i.e. unbiased); The error variance is constant across the range of values of the IVs (i.e. homo- scedasticity); The errors associated with one set of IV values are uncorrelated with those of another (i.e. independence); The errors are normally distributed with zero mean and constant variance (i.e. normality) – while normality is necessary for the t-tests to be valid, estimation of the model coefficients requires only that errors be identically and independently distributed; The model is properly specified (i.e. all relevant IVs are included and all irrelevant ones discarded).
Other issues to be considered are:
The avoidance of over-determination. An over-determined or “over-parameterised” model is one which fits too many free parameters in relation to the number of data samples available. An adequately determined model is needed if the derived model is to generalise sensibly to estimate the DV at ungauged sites.
48 Volume III Hydrograph Analysis
Checking that individual observations do not exert undue influence on the model. Checking for collinearity amongst the IVs (see Box 6.1). If IVs are highly correlated, the regression coefficients are unlikely to be robust; values are likely to be unduly sensitive to changes such as the addition of a station to the calibration dataset.
It can readily be appreciated that, in hydrological applications, complete satisfaction of the assumptions and full resolution of the above issues cannot be achieved. Indeed, some of the assumptions are expressly contradicted:
Some subsets of the PCDs are known to exhibit high collinearity; The sample size of stations is unlikely to be large enough to be able to generalise a model that includes all factors thought physically relevant; The homogeneity and representativeness of samples is often difficult to ascertain.
Box 6.1: Collinearity
Collinearity refers in a strict sense to the presence of exact linear relationships within a set of variables. Typically, these are a set of candidate explanatory (i.e. predictor) variables in a regression-type model. In statistical usage, collinearity also refers to near-collinearity, i.e. when variables are close to linearly related.
In a multiple regression with collinearity, least-squares regression coefficients are highly sensitive to very minor changes in the input data. The least-squares problem or the dataset is said to be ill-conditioned. Some or all of the regression coefficients are likely to be meaningless.
A typical approach to overcoming collinearity is to simplify the problem, e.g. by retaining only one of the subset of variables that are highly correlated. Relatively arbitrary decisions – as to which variables to retain and which to remove – are sometimes unavoidable and inevitably influence the final model achieved.
6.2 Selection of dependent variables (DVs)
Regression models are developed for three hydrograph-width descriptors (W75, W50 and s) and three model parameters (Tr, n and C). These respectively characterise the recommended non-parametric and parametric approaches to HWA. These six hydrograph descriptors are the dependent variables (DVs) of the regression analyses.
Four of the variables W75, W50, Tr and C are times in hours. The other two are dimensionless, with the mean ratio s taking values between 0 and 1 and n being a real number > 1.0. Values of the six DVs obtained at 89 gauging stations are given in Table 5.1 and form the basic HWA data used. These data are further summarised in Table 6.1.
For reasons noted in Section 5.7, values of W75, W50 and C could not be derived for a few stations. The six variables were ℓn-transformed prior to the main model-building, where ℓn denotes the natural logarithm.
49 Volume III Hydrograph Analysis
Table 6.1: Summary statistics of dependent variables (DVs) selected for regression analysis Dependent No. of Geom. Arith. Unit Minimum Median Maximum CV variable stations mean mean
W75 h 87 3.11 39.87 332.71 38.70 63.31 1.09
W50 h 84 4.90 72.07 613.41 65.07 106.36 1.11 s – 89 0.22 0.40 0.58 0.40 0.41 0.18 n – 89 1.27 5.00 30.27 5.44 7.04 0.86
Tr h 89 7.66 49.99 429.25 49.38 69.16 1.00 C h 87 2.45 84.53 829.75 63.46 116.04 1.14
6.3 Selection of independent variables (IVs)
In principle, some 36 PCDs might have been considered as independent variables (IVs) in the regression analysis. Those available included such items as the Eastings and Northings of the station location and of the catchment centroid. It was judged prudent to use only a selection of them in order to minimise the expected effects of collinearity (see Box 6.1).
The number of stations for which the dependent variable is available (i.e. 84 to 89 in this research) is a limiting factor on the number of IVs that can be effectively supported in a model. Tabachnick and Fidell (2001) and Brace et al. (2003) are amongst those putting forward rules of thumb for the maximum number of IVs that should be considered when searching for a “best” model. The limit adopted here was to allow no more than 12 Vs in the main model-building.
Table 6.2 presents a brief description and some summary statistics for the 19 PCDs considered initially. Eighteen of these are explicitly discussed in Volume IV and need no further introduction here. The additional PCD was CGDIST: the distance in km from catchment centroid to catchment outlet (i.e. the station location). When used in conjunction with other measures of catchment size, CGDIST can help to represent catchment shape. Some aspects of hydrograph shape are expected to reflect catchment shape.
[Editorial note: The flood attenuation index FAI and the permeability descriptor BFIsoil were unavailable at the time of study. The omission of FAI is unfortunate, given that upper hydrograph shape is expected to be influenced by floodplain effects. Non-availability of BFIsoil was dealt with here by developing model variants according to whether a gauged BFI value is available. It will be noted from Volume IV that BFIsoil coincides with gauged BFI at the 166 stations used in its calibration. Of the two indices of arterial drainage, ARTDRAIN was favoured over ARTDRAIN2. The opposite choice was made in modelling QMED in Volume II, so it is important for FSU users to distinguish the two descriptors of arterial drainage. Both S1085 and TAYSLO definitions of mainstream slope were retained but S1085 ultimately proved the more useful.]
The 19 PCDs were ℓn-transformed prior to the main model-building, where ℓn denotes the natural logarithm. Where the lower range of a particular descriptor can take a value of zero, 1.0 is added to the value prior to the ℓn-transformation. This applies to the fractions ALLUV, ARTDRAIN, FOREST, PASTURE, PEAT and URBEXT.
50 Volume III Hydrograph Analysis
Table 6.2: Some summary statistics of the IVs initially selected # of Notation Brief description Unit Min Median Max Mean CV stns ALLUV Alluvial fraction – 89 0.0029 0.0349 0.0977 0.0370 0.53 AREA Catchment area km2 89 23.41 292.67 7980.41 624.76 1.60 Fraction of area mapped as ARTDRAIN benefiting from arterial – 89 0.00 0.0015 0.3669 0.0576 1.45 drainage BFI Baseflow index – 79 0.27 0.61 0.83 0.60 0.20 Distance from catchment CGDIST km 89 4.83 11.72 54.19 13.86 0.58 centroid to gauging station km/ DRAIND NETLEN/AREA 89 0.39 0.96 1.81 0.99 0.29 km2 Index of flood attenuation by FARL – 89 0.66 1.00 1.00 0.94 0.09 reservoirs and lakes FOREST Forested fraction – 89 0.0161 0.0751 0.5619 0.0934 0.87 FLATWET Wetness index – 89 0.54 0.62 0.73 0.63 0.07 MSL Mainstream length km 89 13.92 38.14 214.61 47.65 0.64 NETLEN Length of upstream network km 89 23.99 293.54 6428.92 580.01 1.47 PASTURE Pasture fraction 89 0.2319 0.8134 0.9738 0.7737 0.22 PEAT Peat fraction 89 0.00 0.0527 0.4826 0.0994 1.09 Average annual potential SAAPE mm 89 449.60 503.96 546.98 498.22 0.05 evapotranspiration SAAR Average annual rainfall mm 89 783.26 1023.31 2101.83 1068.66 0.19 Number of segments in STMFRQ – 89 7.00 288.00 5490.00 576.66 1.48 upstream river network Mainstream slope (excluding S1085 m/km 89 0.00021 0.00206 0.01867 0.00277 1.02 top 10% and bottom 15%) TAYSLO Taylor-Schwarz stream slope m/km 89 0.00001 0.00106 0.01308 0.00175 1.25 URBEXT Urban fraction – 89 0.00 0.0064 0.0553 0.0081 0.99
6.4 Additional notes
6.4.1 Treatment of Gamma shape parameter n
One of the six DVs – the shape parameter n – is required to take a value >1 if the Gamma distribution is to provide a hydrograph that rises to a peak rather than immediately decays. This case did not arise in fitting the UPO-ERR-Gamma at any of the 89 stations analysed. The minimum value found was 1.29. Most of the modelling used the DV in the form ℓn(n). However, it was prudent in the final modelling to use the DV in the form ℓn(n-1). This guaranteed that hydrographs generated by the model at ungauged sites always rise to a peak before decaying.
51 Volume III Hydrograph Analysis
6.4.2 Software used for the regression analysis
A one-sentence summary of the goal of multiple regression is “to identify the fewest IVs necessary to predict a DV where each IV predicts a substantial and independent segment of the variability in the DV” (Tabachnick and Fidell, 2001). The SPSS statistical software package was used for: exploring inter-correlations, developing regression equations, assumption-checking and making additional diagnostic checks. Details and examples are given in books such as Pallant (2001), Tabachnick and Fidell (2001) and Brace et al. (2003).
6.5 Correlation studies
Correlations amongst the PCDs were examined, both in their native form and after logarithmic transformation. Those most strongly correlated were examined further. The correlation matrix of the 19 IVs and six DVs is given in Table 6.3. This shows the ordinary (Pearson) correlation coefficient for the ℓn-transformed variables.
6.5.1 Inter-correlations between PCDs
Matrix plots showing pairwise scatter-plots were extensively examined. Figure 6.1 illustrates the extent to which AREA, MSL, NETLEN, STMFRQ and CGDIST compete to represent catchment size. These strong correlations are understandable. [Editorial note: STMFRQ denotes the number of streams in the catchment. This is one greater than the number of stream junctions. Larger catchments tend to have more stream junctions. Thus, STMFRQ in the FSU is correlated with catchment size. In contrast, the FSR (NERC, 1975) defined STMFRQ as the number of junctions per unit area.]
Using a Pearson correlation with absolute value greater than 0.6 as an arbitrary guide, other strong correlations evident in Table 6.3 are briefly discussed.
The strong inverse correlation between mainstream slope (represented by ℓnS1085) and variables indicative of catchment size (e.g. ℓnAREA, ℓnMSL and ℓnCGDIST) is principally a function of topography: steep catchments are inevitably rather small. [Editorial note: Such correlations reflect both physical properties and the available network of stations. When all ≈134,000 FSU ungauged catchments are considered, the inverse correlation between ℓnS1085 and ℓnAREA is weaker, with r = -0.41 (as opposed to r=-0.66 in Table 6.3). Many small catchments in Ireland have mild stream-slopes but the majority are not sufficiently important to be gauged.]
Another grouping of PCDs is PASTURE, PEAT and SAAR. The strong inverse correlation (r = -0.84) between ℓn(1+PASTURE) and ℓn(1+PEAT) appears straightforward. Catchments dominated by more peaty formations tend to have fewer managed pastures. Also, the land- cover classifications are mutually exclusive; in cases where most of a catchment is classed as pasture, there is little available to be classed as peatlands. The evolution of peatlands in the form of blanket and raised bogs is typically attributed to high rainfall combined with poor drainage. In the Irish context, it is therefore understandable that a high value of ℓn(1+PEAT) is often associated with a high value of ℓnSAAR (r = 0.54), as well as with a low value of ℓn(1+PASTURE). The strong inverse correlation between ℓn(1+PASTURE) and ℓnSAAR (r = -0.70) is consistent with drier areas being more readily put to pasture.
52 Volume III Hydrograph Analysis
Table 6.3: Correlation matrix of selected IVs and DVs at (up to) 89 stations
PCD
or variable
AREA MSL NETLEN STMFRQ DRAIND CGDIST FARL ARTDRAIN S1085 TAYSLO FLATWET SAAR SAAPE URBEXT FOREST PEAT PASTURE ALLUV BFI ℓn AREA 1 ℓn MSL 0.94 1 Row above uses abbreviated names; full name is in first column ℓn NETLEN 0.96 0.93 1 ℓn STMFRQ 0.88 0.85 0.97 1 Red denotes Pearson correlation ≤ -0.6 or ≥ 0.6 ℓn DRAIND -0.14 -0.04 0.15 0.32 1 Orange indicates other correlations significant at 1% ℓn CGDIST 0.83 0.89 0.80 0.71 -0.11 1 ℓn FARL -0.24 -0.26 -0.28 -0.33 -0.14 -0.15 1 Green indicates correlations significant at 5% ℓn(1+ARTDRAIN) 0.22 0.22 0.21 0.16 -0.02 0.26 0.01 1
ℓn S1085 -0.66 -0.65 -0.56 -0.44 0.36 -0.60 0.21 -0.18 1 ℓn TAYSLO -0.25 -0.23 -0.22 -0.18 0.11 -0.23 0.07 -0.11 0.44 1 ℓn FLATWET -0.03 0.05 0.03 0.10 0.23 0.03 -0.48 0.00 -0.13 -0.10 1
ℓn SAAR -0.15 -0.06 0.02 0.19 0.59 -0.12 -0.41 -0.20 0.32 0.12 0.59 1 ℓn SAAPE -0.02 -0.09 -0.01 0.01 0.03 -0.09 0.39 -0.02 0.28 0.10 -0.83 -0.24 1 ℓn(1+URBEXT) -0.05 -0.10 -0.08 -0.08 -0.11 -0.02 -0.04 0.18 0.06 -0.01 -0.13 -0.08 0.15 1 ℓn(1+FOREST) -0.25 -0.18 -0.14 -0.07 0.36 -0.21 0.04 -0.26 0.37 0.21 0.09 0.49 -0.05 -0.26 1 ℓn(1+PEAT) -0.03 -0.02 0.01 0.06 0.14 -0.12 -0.22 -0.05 -0.04 -0.05 0.52 0.54 -0.43 -0.02 0.42 1 ℓn(1+PASTURE) 0.11 0.10 0.04 -0.07 -0.23 0.18 0.27 0.16 -0.16 -0.12 -0.49 -0.70 0.30 0.06 -0.58 -0.84 1 ℓn(1+ALLUV) 0.07 0.04 0.12 0.08 0.16 0.03 0.38 0.13 0.28 0.08 -0.58 -0.26 0.56 0.03 0.06 -0.41 0.40 1 ℓn BFI 0.42 0.38 0.29 0.23 -0.45 0.38 -0.39 0.22 -0.49 -0.24 0.08 -0.27 -0.16 0.08 -0.48 -0.14 0.15 -0.19 1
ℓn W75 0.33 0.32 0.24 0.22 -0.31 0.23 -0.64 -0.16 -0.49 -0.31 0.41 0.04 -0.40 -0.09 -0.31 0.07 -0.09 -0.48 0.63 ℓn W50 0.32 0.31 0.22 0.20 -0.32 0.22 -0.63 -0.15 -0.48 -0.36 0.36 0.00 -0.35 -0.10 -0.32 0.01 -0.04 -0.44 0.67 ℓn s -0.05 0.00 -0.04 -0.03 0.01 -0.06 0.14 -0.13 0.04 0.05 0.16 0.09 -0.22 -0.12 0.19 0.22 -0.15 -0.17 -0.36 ℓn n -0.26 -0.22 -0.19 -0.16 0.22 -0.23 0.53 -0.09 0.35 0.26 -0.32 -0.02 0.29 -0.14 0.30 -0.04 0.03 0.32 -0.48 ℓn Tr 0.24 0.26 0.16 0.16 -0.27 0.12 -0.43 -0.27 -0.42 -0.25 0.36 0.04 -0.39 -0.13 -0.20 0.10 -0.11 -0.47 0.47 ℓn C 0.37 0.32 0.30 0.30 -0.22 0.32 -0.56 0.04 -0.40 -0.22 0.28 -0.01 -0.26 0.04 -0.32 -0.02 -0.04 -0.27 0.74
53 Volume III Hydrograph Analysis
(This page is intentionally blank.)
54 Volume III Hydrograph Analysis
AREA AREA
ℓn AREA
MSL
MSL AREA AREA
ℓn MSL MSL
MSL NETLEN NETLEN
ℓn NETLEN
NETLEN
NETLEN
STRMFRQ STRMFRQ
ℓn STMFRQ
DRAIND
DRAIND STRMFRQ STRMFRQ
ℓn CGD ST
DRAIND
CGDIST
DRAIND CGDIST
ℓn ℓn ℓn ℓn ℓn AREA MSL NETLEN STMFRQ CGDIST
W75
W75
Width Width Width Width
Figure 6.1: MatrixCGDIST plot of PCDs that in part represent catchment size (89 stations) CGDIST
AREA AREAMSL NETLENMSL STRMFRQNETLEN STRMFRQDRAIND DRAINDCGDIST CGDISTWidth Width [Editorial note: From Table 6.2, PASTURE is seen to be the dominant classification,W75 with a W75
mean of 77.4%W75 of catchment land-cover as opposed to 9.9% for PEAT and 12.7% to other
W75 Width Width Width Width classifications. When all ≈134,000 FSU ungauged catchments are considered, the mean values of PASTURE, PEAT and “other” are 57.3%, 23.8% and 18.9% respectively. The AREA AREAMSL NETLENMSL STRMFRQNETLEN STRMFRQDRAIND DRAINDCGDIST CGDISTWidth Width corresponding mean values for SAAR are 1069 mm for the 89 catchments studied andW75 1293 W75 mm for the ≈134,000 FSU ungauged catchments. Thus there appears to be a bias towards gauging drier and more agriculturally productive catchments than is the Irish norm. This emphasises the importance of the research undertaken to generalise methods of estimating the characteristic hydrograph that take full account of catchment properties.]
The strong inverse correlation (r = -0.83) between ℓnSAAPE and ℓnFLATWET reflects that these descriptors derive respectively from standardised estimates – based on climate data – of potential evaporation and soil moisture. FLATWET is the proportion of the time for which soils can be expected to be typically quite wet. FLATWET is expected to be greater in areas where rainfall is high but the potential for evaporation (indexed by SAAPE) is relatively low.
[Editorial note: SAAR is the pre-eminent climatological descriptor in many hydrological applications. The lack of correlation between ℓnSAAR and the hydrograph-width variables (see bottom six rows of Table 6.3) is striking.]
55 Volume III Hydrograph Analysis
The strong inverse correlation between ℓnS1085 and ℓnCGD ST (r = -0.60) is more inscrutable. It may reflect that streams tend to be steeper on catchments where the drainage pattern is fan-shaped (with a more compact catchment) than when the drainage pattern is elongated (with a relatively large CGDIST).
Interestingly, the correlation between ℓnS1085 and ℓnTAYSLO is not especially high (r = 0.44), indicating that these are rather different measures of mainstream channel slope.
6.5.2 Individual correlations between DVs and initially selected IVs
The correlations between each of the six DVs and the 19 initially selected IVs are shown in the bottom rows of Table 6.3. There are two stand-out sets of associations:
ℓnW75 and ℓnW50 are highly negatively correlated with ℓnFARL;
ℓnW75, ℓnW50 and ℓnC are highly correlated with ℓnBFI.
These are consistent with physical interpretations that:
Lakes and other water bodies (consistent with a smaller value of ℓnFARL) attenuate the flood passing down the river system, tending to lead to hydrographs that are more prolonged than otherwise (consistent with larger values of ℓnW75 and ℓnW50); Catchments that are relatively permeable (consistent with a larger value of ℓnBFI) tend to lead to hydrographs that are more prolonged than otherwise (consistent with larger values of ℓnW75 and ℓnW50).
6.5.3 Choosing a subset of PCDs to use as IVs
In view of the strong correlations (see Section 6.5.1) amongst various groups of PCDs, it was decided that a subset would suffice for the main modelling. For reasons discussed in Section 6.3, the main regression study would use no more than about 12 IVs.
The most suitable IVs in each of three competitive groups were chosen by favouring those most strongly correlated with the six DVs under study. The relevant correlations are shown in the bottom six rows of Table 6.3.
From inspection, it can be seen that:
The magnitudes of the correlations associated with ℓnAREA are generally a little larger than those associated with ℓnMSL, ℓnNETLEN, ℓnSTMFRQ or ℓnCGDIST; The magnitudes of the correlations associated with ℓn(1+PASTURE) and ℓn(1+PEAT) are larger than those associated with ℓnSAAR; The magnitudes of the correlations associated with ℓnFLATWET are evenly balanced with those associated with ℓnSAAPE.
Accordingly, ℓnAREA, ℓn(1+PASTURE) and ℓnFLATWET were selected as the representative variables, and the seven other PCDs discarded. This achieved the target of allowing no more than 12 IVs in the regression modelling.
56 Volume III Hydrograph Analysis
6.5.4 Checking the Normality of the DVs
The log-transforms yield DVs that are acceptably Normal, satisfying one of the requirements for linear regression (see Section 6.1.2). The Gamma shape parameter n is somewhat less convincing than the others in this respect (see Figure 6.2).
)
ℓnW75 ℓnW50
ed ed value
Expect
(if Normally distributed
Observed value Observed value
)
ℓn s ℓn n
ed ed value
Expect
Normally distributed
(if
Observed value Observed value
)
ℓnTr ℓn C
ed ed value
Expect
(if Normally distributed
Observed value Observed value
Figure 6.2: Normality plots of log-transformed width descriptors and model parameters
Reading across the 4th-last row of Table 6.3, the hydrograph width descriptor ℓn s is seen to be no more than weakly correlated with the available IVs. Although an attempt was made to model this index of hydrograph skewness, the best that could ultimately be achieved was to adopt a fixed value of s = 0.40. From Table 6.1 it is seen that this is very close to both the arithmetic and geometric means of s, i.e. to the arithmetic means of s and ℓn s. It proved possible to develop useful regression models for the other five DVs.
57 Volume III Hydrograph Analysis
6.5.5 Final selection of the independent variables; a note on the use of BFI
The 12 variables finally selected as IVs were (in no particular order):
ℓnAREA, ℓnDRAIND, ℓnFARL and ℓn(1+ARTDRAIN); ℓnS1085, ℓnTAYSLO, ℓnFLATWET and ℓn(1+URBEXT); ℓn(1+FOREST), ℓn(1+PASTURE), ℓn(1+ALLUV) and ℓnBFI.
BFI is not strictly a PCD. Rather, it is a hydrological index derived by the analysis of daily mean flow data. At the time of the hydrograph width research, the mechanism by which BFI would be modelled at ungauged sites was unclear. Accordingly, two sets of generalisations – of the hydrograph width variables W75 and W50 and the UPO-ERR-Gamma model parameters n, Tr and C – were developed according to whether a BFI value is/isn’t available.
6.6 The regression method used
Various forms of multiple regression are possible. That adopted here was stepwise linear regression. In stepwise regression, IVs are introduced into the regression model one variable at a time. At each step, the IV offering the greatest reduction in the objective function (least- squares or weighted least-squares) is added into the model. But IVs already included in the model “may also be deleted at any step where they no longer contribute significantly to regression” (Tabachnick and Fidell, 2001). The stepwise method results in a parsimonious model i.e. one frugal in its use of parameters. The default settings in SPSS were used in respect of the statistical criteria for including or removing an IV. Accordingly, the thresholds used were 0.05 (5% significance) for inclusion of an IV and 0.10 (10% significance) for removal of an IV, the F-test statistic being used to assess the significance of the departure from the null hypothesis (that there is no linear relationship between the IV and the DV).
[Editorial note: For datasets of the moderate size considered here, computer power is typically such that “best subsets” regression rather than stepwise regression can be used. Indeed, the best subsets approach is adopted for the Volume II modelling of QMED. Many researchers feel more comfortable with the stepwise approach, believing it to provide more defined safeguards against over-determination of models. An over-determined model is one which fits too many free parameters in relation to the number of data samples available. In many cases, the different techniques lead to the same final model.]
6.7 Illustrative results: Estimating W75 when BFI available
Generalisation of the hydrograph width descriptor W75 provides an insight into the methods. The regression is based on linear least-squares, with ℓnW75 taken as the DV. The stepwise regression analysis is summarised in Table 6.4, which reports intermediate results for 1, 2, 3, and 4-variable models as well as the final 5-variable model.
6.7.1 Regression models and their performance evaluation r2 indicates the proportion of variation explained by the regression model (i.e. by the IVs). It is often termed the coefficient of determination. There is progressive improvement in r2 as additional IVs are allowed into the model. The final 5-variable model explains about 72% of the variability in ℓnW75. Explaining this much of the variability is regarded as quite good.
58 Volume III Hydrograph Analysis
It is inevitable that r2 increases as additional IVs are allowed into the model. By this criterion, a model using all 12 available IVs would be judged best.
To avoid over-determination, it is necessary to examine the adjusted r2. The adjustment takes into account the number of “degrees of freedom” consumed by having to fit additional parameters in the linear least-squares regression. The adjusted r2 provides a reasonable estimate of how well the regression model might be expected to fit another dataset drawn from the same population. The adjusted r2 value of 0.701 indicates that the 5-variable model provides a reasonably good fit.
Table 6.4: Stepwise regression results for modelling hydrograph width descriptor ℓnW75 Factorial Standard No. standard No. of IV Adjusted error of of IV added r2 error of catchments removed r2 estimate IVs estimate (SEE) (FSE) 1 77 ℓnBF .399 .391 .788 2.20 2 77 ℓn(1+ALLUV) – .575 .564 .668 1.95 3 77 ℓnFARL – .643 .629 .616 1.85 4 77 ℓn(1+ARTDRA N) – .686 .669 .581 1.79 5 77 ℓnS1085 – .721 .701 .552 1.74
The standard error of estimate (SEE) is the standard deviation of the residuals or errors in predictions by the model: in this case, in the estimates of ℓnW75. This means that – on the assumption that the model residuals are normally distributed – the actual value of ℓnW75 is expected to lie within 0.552 of the predicted value in about 68% of cases.
Because our principal interest is in estimating W75 rather than ℓnW75, the factorial standard error (FSE) is generally more relevant. The FSE is just the exponential of the SEE. For the final (5-variable) model, the FSE is e0.552 = 1.74. This means that (in about 68% of cases) the actual value of W75 is expected to lie within the factorial range 0.57 75 to 1.74 75 where 75 is the estimated value of the width descriptor W75 obtained from the regression model and 0.57 is the reciprocal of 1.74. This confidence interval is considered reasonably good.
For the scenario when BFI is available, the recommended model is:
ℓnW = 3.548 + 1.861 ℓnBFI – 12.199 ℓn(1+ALLUV) – 3.946 ℓnFARL 75 6.1 – 3.324 ℓn(1+ARTDRA N) – 0.246 ℓnS1085
Exponentiating both sides of the equation, the model can be written:
1.86 -12.20 -3.95 -3.32 -0.25 W75 = 34.74 BFI (1+ALLUV) FARL (1+ARTDRAIN) S1085 6.2
Table 6.5 confirms that all coefficients are significant at the 0.05 level (│t statistic│> 1.96). The standardised coefficients (β in the table) highlight the relative contribution of each descriptor to explaining the variation in ℓnW75. ℓnBF is seen to be the most important predictor. This vindicates the decision to develop models for the case when this variable is available.
59 Volume III Hydrograph Analysis
Table 6.5: Coefficient and collinearity statistics for selected model for ℓnW75 Variance Standard β t inflation Term/regressor Coefficient error of Tolerance value statistic factor coefficient (VIF) Constant 3.548 0.68 5.20 ℓnBFI 1.861 0.34 0.43 5.53 0.660 1.52 ℓn(1+ALLUV) -12.199 3.82 -0.23 -3.19 0.767 1.30 ℓnFARL -3.946 0.94 -0.30 -4.20 0.751 1.33 ℓn(1+ARTDRAIN) -3.324 0.92 -0.24 -3.62 0.903 1.11 ℓnS1085 -0.246 0.08 -0.22 -2.95 0.708 1.41
6.7.2 Checking the possible influence of collinearity
In order to assess the possible impact of collinearity (see Box 6.1), the variance inflation factor (VIF) was also studied. This statistic is the reciprocal of the tolerance, which in turn denotes the proportion of the variance in a given catchment descriptor that cannot be explained by the other regressors.
High VIF values (i.e. small tolerances) indicate that a large amount of the variance in one regressor can be explained by the other regressors. VIF thus indexes the impact of collinearity (amongst the regressors) on the stability of the multiple regression model. VIF values are (by definition) greater than or equal to 1. Whilst only a guide, VIF values greater than 10 are often regarded as indicating serious problems of collinearity. In weaker models, values above 2.5 may sometimes be a cause for concern.
It is seen from the final column of Table 2.7 that VIF is less than 1.6 for all regressors. Collinearity is therefore judged not to be a problem with the selected model.
6.7.3 Checking the logical consistency of the model
Logical consistency is often the overriding factor in the final choice of a regression model. Is the model consistent with what we know about catchment flood behaviour? The 5-variable model for ℓnW75 appears credible in this respect:
W75 increases with permeability and storage (indicated by a larger value of BFI);
W75 decreases with larger values of FARL (a larger value of FARL is associated with reduced attenuation of flood water by storage elements such as lakes and reservoirs; flood hydrographs in such a catchment can be expected to have a higher peak and a narrower width);
W75 decreases with larger values of ARTDRAIN, consistent with a flashier response after arterial drainage works (i.e. a flood hydrograph with a higher peak and a narrower width);
W75 decreases with larger values of S1085, indicating a flashier response from steeper catchments;
W75 decreases when the proportion of land-cover classed as alluvial (ALLUV) is larger (more alluvium in a catchment indicates a lower potential of the catchment for
60 Volume III Hydrograph Analysis
accepting rainfall; the resulting flood response is therefore expected to be faster, with the flood hydrograph expected to be narrower than that in a catchment having less alluvium).
The last feature is not wholly convincing, because of the high magnitude of the exponent of 1+ALLUV in the Equation 6.2 model for W75. The role of ALLUV in the hydrograph-width models is discussed further in Box 6.2.
Box 6.2: The role of ALLUV in the hydrograph-width models
Editorial note: Discussion is warranted of an uncomfortable feature of the Equation 6.2 model for W75. This is the high magnitude (-12.20) of the exponent of 1+ALLUV. ALLUV denotes the proportion of the catchment classed as alluvium in the Teagasc classification of soils (see Volume IV).
The effect might conceivably reflect that alluvial areas lie close to the river network and tend to contribute runoff quickly. However, the attribution of physical effects to results obtained by regression is often hazardous.
From Table 6.2 it is seen that alluvial fractions for the 89 catchments studied range from 0.00 to 0.10. Thus, the modelled factorial effect of the alluvium on W75 ranges from 1.00-12.20 = 1.000 (when ALLUV=0.00) to 1.10-12.20 = 0.313 (when ALLUV = 0.10). This may just be reasonable.
PCDs for ungauged sites were not available at the time of the hydrograph width research. 98.3% of the ≈134,000 ungauged catchments supported by the FSU have a value of ALLUV within the range 0.00 to 0.10. However, 0.25% of catchments are classified as having ALLUV≥0.20. For these cases, the W75 model (Equation 6.2) implies that classification of land as alluvium reduces W75 by a factor of more than 9, since (1+0.20)-12.20 = 0.108 ≈1/9. This does not seem reasonable.
The urban fraction (URBEXT) is the only classification of land cover to play a major role in the Volume II procedure for estimating the T-year flood peak at an ungauged site. It is notable that ALLUV plays no role. Thus, an ungauged catchment classified as having an unusually large alluvial fraction will be modelled as generating hydrographs that are exceedingly narrow but not especially high-peaked. This appears physically unrealistic. Should the catchment under study be mapped as having an unusually large alluvial fraction, it will be prudent for the user to make special checks.
The Equation 6.2 model for W75 has not in fact been implemented. The models implemented through the FSU Web Portal are shown later in Table 6.7. Only one of these – the model for estimating Tr – includes the 1+ALLUV term. Moreover, the exponent of 1+ALLUV in that model is somewhat less severe (-8.83) than for the Equation 6.2 model. Users are nevertheless to be encouraged to make special checks should they be using the model to estimate Tr on a catchment for which ALLUV≥0.20.
The role of ALLUV in the hydrograph-width models may warrant further exploration.
61 Volume III Hydrograph Analysis
6.7.4 Additional checks
Additional checks were made on the sensitivity of the 5-variable model to the inclusion of particular catchments in the calibration. The statistics examined include a leverage statistic (to identify catchments whose observed values of the selected DV influence the regression model more than others) and Cook’s distance (which measures how much the model coefficients change when a particular catchment is dropped from the analysis). Some weak evidence was found that Station 26021 Inny at Ballymahon might have undue influence. However, rules of thumb indicated that this was not so marked as to warrant its exclusion.
The model residuals (i.e. prediction errors) were tested for normality. The probability- probability (P-P) plot shown in Figure 6.3 shows some departure from a perfect 1:1 line but not enough to judge the model inadequate.
Expected value value Expected (if Normally distributed) Normally (if
Observed cumulative probability Figure 6.3: Normality plot of standardised residuals for 5-variable model for ℓnW75
Model residuals were also tested for homoscedasticity. This requires that the standard deviations of errors of prediction are approximately equal for all predicted values of the DV. Homescedastic derives from the Greek for equal scatter. Homoscedasticity is exhibited when the plot of residuals displays a cloud of dots and the band enclosing the residuals is approximately equal in width at all values of the predicted DV.
A lack of homoscedasticity (i.e. heteroscedasticity) is characterised by a pattern such as a funnel shape, indicating greater errors associated with larger predicted values. This may arise when there is an interaction between an IV and a variable not in the regression model, or when some IVs are skewed while others are not. No serious violation of the assumption of homoscedasticity is evident in the Figure 6.4 plot of standardised residuals against the standardised predicted values. The 5-variable model for ℓnW75 is therefore considered acceptable.
62
Volume III Hydrograph Analysis
predicted value predicted Regression standardised standardised Regression
Regression standardised residual
Figure 6.4: Plot of standardised residuals for 5-variable model for ℓnW75
63 Volume III Hydrograph Analysis
(This page is intentionally blank.)
64 Volume III Hydrograph Analysis
(This page is intentionally blank.)
65 Volume III Hydrograph Analysis
Table 6.6: Recommended models – when BFI available Hydrograph # of Adj. Stations identified as possible Eqn 2 FSE Recommended model descriptor stations r outliers* # 1.86 -3.95 -12.20 W75 = 34.74 BFI FARL (1+ALLUV) W75 77 0.701 1.74 26021 6.3 (1+ARTDRAIN)-3.32 (S1085/1000)-0.25 2.11 -4.55 -10.24 W50 = 63.05 BFI FARL (1+ALLUV) W50 75 0.735 1.70 None 6.4 (1+ARTDRAIN)-3.17 (S1085/1000)-0.25 06011, 11001, 14007, 16005, n 79 0.377 2.02 n = 1 + 2.90 BFI-1.12 FARL4.37 18005, 25027, 34018 6.5
1.32 -13.08 -3.70 -0.20 Tr 79 0.493 1.77 26021 Tr = 54.98 BFI (1+ALLUV) (1+ARTDRAIN) (S1085/1000) 6.6
3.44 -4.88 C 77 0.630 2.16 16005 C = 310.75 BFI FARL 6.7 *Assignment based on a test of
standardised deleted residuals
Table 6.7: Recommended models – when BFI unavailable Hydrograph # of Adj. Stations identified as possible 2 FSE Recommended model Eqn # descriptor stations r outliers* -0.88 -5.85 -2.86 W75 = 31.28 DRAIND FARL (1+FOREST) W75 87 0.675 1.79 15002, 35071 6.8 (1+ARTDRAIN)-2.92 FLATWET3.12 (S1085/1000)-0.27 -0.95 -6.59 -2.83 W50 = 34.02 DRAIND FARL (1+FOREST) W50 84 0.675 1.80 35071 6.9 (1+ARTDRAIN)-2.60 FLATWET2.44 (S1085/1000)-0.29 0.63 5.46 2.46 n 89 0.412 1.99 06011, 11001, 14007, 34018 n = 1 + 4.78 DRAIND FARL (1 + FOREST) 6.10
-0.48 -2.54 -3.16 Tr = 13.85 DRAIND FARL (1+ARTDRAIN) Tr 89 0.213 2.03 26021, 30061 6.11 (S1085/1000)-0.25 (1+ALLUV)-8.83 -0.97 -7.65 -3.70 0.26 C 87 0.485 2.42 16008, 23002, 35071 C = 11.78 DRAIND FARL (1+FOREST) AREA 6.12 *Assignment based on a test of
standardised deleted residuals
66
6.8 Recommended models for use at ungauged sites
6.8.1 The final models
The techniques of Sections 6.6 and 6.7 were applied to develop models for the hydrograph width measures/parameters: W75, W50, n, Tr and C. The recommended models are presented in Table 6.6 (for the case when BFI is available) and Table 6.7 (for the case when BFI is unavailable). These models (see tables on previous page) are now jointly discussed:
When judged by the factorial standard error (FSE), the models using BFI (Table 6.6) generally outperform those that do not (Table 6.7). This shows the value of BFI in helping to model hydrograph shape. (Precise comparisons are inhibited because of the somewhat different datasets used in Table 6.6 and Table 6.7. BFI values were available for only 79 of the 89 stations.) The likely importance of storage attenuation and the effectiveness of FARL in its characterisation are highlighted by the appearance of the descriptor in all but one of the ten models. Mainstream slope S1085 appears in the models of the three main characteristic times W75, W50 and Tr. The exponent of S1085 is reassuringly similar in all cases. [Editorial note: For a reason known only to the research contractor, values of S1085 were divided by 1000 prior to the regression modelling of hydrograph widths. This explains why the term S1085/1000 appears in the models for W75, W50 and Tr shown in Table 6.6 and Table 6.7. This led to difficulties for the contractor testing IBIDEM at two ungauged sites (see Sections 9.7.4 and 9.7.5) but should have no impact for the user.] When BFI is unavailable, the importance of FARL is heightened and DRAIND also becomes indispensable. Both descriptors appear in all five models. FOREST also then proves helpful (appearing in four of the models). The exponents of DRAIND and FOREST indicate that hydrographs are narrower when drainage and forest cover are more extensive. This likely reflects the faster conveyance of water, noting that afforestation is often accompanied by drainage works such as ditching.
There is reassuring consistency in the models derived for W75 and W50. The same terms appear in both models. Moreover, the exponents are broadly comparable. Except for parameter C, the FSEs are in the range 1.7 to 2.0. This is a moderately good performance by hydrological standards. [Editorial note: The excellent performance achieved in estimating QMED on rural catchments (FSE = 1.37, see Volume II) suggests that the task of modelling typical peak flows on an ungauged catchment is rather easier than that of modelling typical hydrograph widths. This likely reflects the highly attenuated nature of many flood hydrographs in Ireland. It may also reflect the varied duration and often complex pattern of rainfalls that give rise to flooding.] The term least well-estimated is parameter C of the UPO-ERR-Gamma model. This determines the rate of decay of the exponential replacement recession. The poor performance in modelling C is neither surprising nor especially worrying. The parameter has no influence on the peak segment of the hydrograph. The model for C in the BFI unavailable case is the only occasion on which a PCD indicative of catchment size appears in the hydrograph width modelling. It is also notable that general wetness indexed by SAAR does not appear in any model. 67
However, the wetness descriptor FLATWET proves useful in modelling W75 and W50 in the BFI unavailable case. The analyses identify a number of stations that are possible outliers. Those implicated in estimation of the parameters of the UPO-ERR-Gamma model may not warrant much concern. It is inevitable that the limited family of hydrograph shapes supported by the UPO-ERR-Gamma model (see Figure 4.1, Figure 4.2 and Figure 4.4) will not suit all stations.
The sites implicated as possible outliers in the modelling of W75 and W50 are Stations 15002, 26021and 35071. These are all Grade A2 stations. Station 26021 had by far the largest proportion of missing flow data of any of the 89 stations analysed (see Table A.2). The poor performance of the PCD-based models at Stations 15002 and 35071 may warrant further investigation. The characteristic hydrographs at these stations (see Figure 6.5) are relatively well defined and, in the case of Station 15002 Nore at John’s Bridge, rather finely shaped.
100 Median hydrograph 90 (a) Station 15002 Nore at John’s Bridge
80 70 60 50
40 30
Percentageof flowpeak 20 10
0 -50 -37.5 -25 -12.5 0 50 100 150 200 Time in hours (relative to time of peak flow)
100 Median hydrograph 90 (b) Station 35071 Lareen at L. Melvin 80 70
60
50 40 30
Percentage of peak flow ofpeak Percentage 20
10 0 -150 -112.5 -75 -37.5 0 75 150 225 300 Time in hours (relative to time of peak flow)
Figure 6.5: Median hydrographs at: (a) Station 15002 and (b) Station 35071
68
6.8.2 Model performance
The quality of performance achieved by the regression models in the BFI unavailable case is encapsulated in Figure 6.6. The plots show “observed” and predicted values of the width descriptors (W75 and W50) and the UPO-ERR-Gamma parameters (n, Tr and C) for the 89- station dataset. The supporting data are presented in Table D.1 of Appendix D.
W75 (hours) W50 (hours)
FSE = 1.79 FSE = 1.80
n
FSE = 1.99
C (hours) Tr (hours)
FSE = 2.03 FSE = 2.42
Figure 6.6: Derived and modelled values of W75, W50, n, Tr and C (BFI unavailable case)
Note the logarithmic scale of the plots. The factorial standard error (FSE) provides a one- number summary of the performance achieved. While many values are predicted within a factor of two, some for the C parameter are seen to be in error by up to a factor of ten.
69
It is confirmed that the generalisation for application at ungauged sites is more successful for the width descriptors W75 and W50 than for parameters of the UPO-ERR-Gamma model.
It is concluded that, for the BFI unavailable case, the UPO-ERR-Gamma model may not be sufficiently reliable to construct the characteristic flood hydrograph at ungauged sites. An alternative is to use the parabolic curves method (see Section 8.7) with the width descriptors W75 and W50 estimated by Equations 6.8 and 6.9. This synthesises the upper hydrograph only.
In view of the better performance of the regression models derived for the BFI available case, consideration might be given to using any or all of the models from Table 6.6 at ungauged sites by substituting BFIsoil for BFI. This possibility is discussed in Section 8.6.
6.8.3 Additional notes on the regression models
The Gamma shape parameter n is dimensionless. Both FARL and BFI are themselves dimensionless. So it is encouraging that the regression model is dimensionally correct in the BFI available case (Equation 6.5).
The other four hydrograph descriptors (W75, W50, Tr and C) are all characteristic times in hours. None of the PCDs available to the generalisation research had a dimension involving time. Thus, it is not possible for the regression models to be strictly dimensionally correct.
70
7 Ancillary investigations
The research checked for systematic variations in the shape of the flood hydrograph with:
The magnitude of the peak flow; The pre-event minimum flow; Season; Arterial drainage in the catchment.
7.1 Variation of hydrograph width with peak flow
The variation of hydrograph width with peak flow was checked for all 89 stations studied. Figure 7.1 presents a typical outcome for Station 07009 Boyne at Navan Weir. There is no particular trend evident towards wider or narrower hydrographs according to the magnitude of the flood peak.
[Editorial note: The broken lines connect widths for different events. This is to help visual detection of any underlying trend. Formal tests for linear trend were also undertaken using ordinary least-squares regression.].
Labels mark % of peak flow at which width abstracted
Width of exceedance (hours) ofexceedance Width
Peak flow (m 3s-1)
Figure 7.1: Variation of hydrograph width with peak flow at Station 07009
For a few stations (e.g. Station 07010 Blackwater at Liscartan shown in Figure 7.2), the graphs display generally positive slopes. However, for most other stations, the parts of the graphs corresponding to the two or three highest recorded floods actually show negative slopes, in contrast to the positive slopes for the points corresponding to the floods of lower magnitudes. Some examples of such inconsistently varying slopes are shown in Figure 7.3.
A composite trend-line of the form y = mx + c was fitted to the plots of hydrograph width against the magnitude of the peak flow, y being the total width (in hours), x the peak flow (in m3s-1), m the slope and c the intercept. At each station in turn, trends were examined across all hydrograph widths collectively, and for the reference widths W75 and W50 individually.
71
Labels mark % of peak flow at which width abstracted Width of Width exceedance (hours)
Peak flow (m3s-1) Figure 7.2: Variation of hydrograph width with peak flow at Station 07010
Labels mark % of peak flow at which width abstracted
Station 06026 Lagan-Glyde at Aclint Station 09001 Ryewater at Leixlip Width of exceedance (hours) Peak flow (m3s-1) Peak flow (m3s-1)
Station 14004 Figile at Clonbulloge Station 14011 Slate at Rathangan Width of Width exceedance (hours) Peak flow (m3s-1) Peak flow (m3s-1)
Station 24013 Deel at Rathkeale Station 36015 Finn at Anlore Width of Width exceedance (hours)
3 -1 3 -1 Peak flow (m s ) Peak flow (m s )
Figure 7.3: Patterns of variation of hydrograph width with peak flow (at six stations) 72
Checks for trend were extensive but generally inconclusive. Figure 7.4 shows the slope of the trend in W75 for the 37 Grade A1 stations. The stations are arranged by catchment size, with the smallest (Station 34018) on the left and the largest (Station 25017) on the right. While there is some variation, no marked pattern with catchment size is evident. Slope m of the trend-line Y=mX+c fitting the available total widths at 75 percentile of selected flood events plotted against the respective peak flows for each station 20 90009000 15 2 8000
Catchment area (km ) 8000
line line ) 1 - 10 7000
- 7000 s
3 5 60006000 )
0 50005000 2 m
-5 40004000 (km
6026 6012 9001 6011 6014 6013 7007 7010 7009 7012
14007 25027 29001 25025 14011 25014 36015 30005 26019 14004 25030 26008 29011 15005 25003 24013 26012 27002 23002 30004 36010 14006 26007 25006 14018 25017
-10 34018 30003000 Slope Slope -15 Station # 20002000 Catchment area (hours/m -20 10001000
Slope of trend of Slope -25 00 Station no. Figure 7.4: Slope of W75 trend with peak flow magnitude (Grade A1 stations)
Although the pattern of trend was not generalised, some weak support was found to suggest that hydrographs may tend to become narrower in larger events on steep catchments (e.g. high S1085) with fewer lakes (i.e. high FARL) than otherwise. The opposing slopes of the trends at Stations 06011 (Fane at Moyles Mill) and 06012 (Fane at Clarebane) evident in Figure 7.4 were explored. The reason for the marked difference is unclear, although the mainstream slope (S1085) is indeed much lower at the downstream site (see Table 7.1), suggesting that the intervening area is rather flat.
Table 7.1: Some leading PCDs of stations on the River Fane AREA S1085 FARL Station # Station name Location km2 m/km –
06012 Fane at Clarebane Upstream 162.8 0.0052 0.874 06011 Fane at Moyles Mill Downstream 229.2 0.0027 0.831
7.2 Variation of hydrograph width with pre-event minimum flow
The variation of hydrograph width with peak-event minimum flow was checked for all 89 stations. For the purpose of HWA, the pre-event minimum flow is taken as the minimum flow on the rising side of the flood hydrograph that lies within the time-window defined in Section 2.2.
Figure 7.5 presents the outcome for Station 07009 Boyne at Navan Weir. No particular trend is evident towards wider or narrower hydrographs according to the magnitude of the pre- event flow. No systematic trend or pattern in the variation of hydrograph widths with pre- event minimum flow could be identified across the 89 stations.
Although no pattern of trend could be generalised, some weak support was found to suggest that hydrographs may tend to become wider in sparser river networks (i.e. with lower values of DRAIND) or more permeable catchments (i.e. with higher values of BFI) when the pre- event minimum flow is high than when it is low. A more obscure effect was a possible tendency for hydrographs to become narrower – when the pre-event minimum flow is high – in catchments where the alluvial fraction (ALLUV) is higher (see Figure 7.6). In the figure,
73
the 89 Grade A1+A2 stations are ordered according to the fraction of the catchment mapped as alluvium. See also the discussion of ALLUV in Section 6.7.3.
Labels mark % of peak flow at which width abstracted
Width of exceedance (hours) ofexceedance Width
3 -1 Pre-event minimum flow (m s ) Figure 7.5: Variation of hydrograph width with pre-event minimum flow (Station 07009)
12 ALLUV 0.1616
10 TREND_SLOPEW75_QPreMin )
line 0.1414 1
- 8 ALLUV -
PreMin 0.1212 s
_Q 6
3 75 4 0.1010
2 0.088 ALLUV 0 0.066 -2 4
-4 0.04 TREND_SLOPEW -6 0.022
(hours/m -8 0.000
6012 6011 7002 7004 7033 7011 7001 7010 7012 7007 7009 7006 6014 6026 9001 6013
29004 35001 14004 26022 35071 27002 30005 26009 34009 16001 25025 29011 30007 26021 29001 30004 34001 27001 26002 26007 26005 36011 18005 35005 15003 36015 26019 22071 16002 39009 36019 36010 23012 26008 35002 23002 16004 25030 14006 14011 15005 14009 36027 15002 15006 11001 16008 25014 16009 25016 16005 24013 15001 14018 23001 25006 19001 18004 16003 25027 25029 25001 24008 25003 24001 24082 25005 14007 trend of Slope StationStation No. #
Figure 7.6: Slope of W75 trend with pre-event minimum flow (Grade A1+A2 stations)
7.3 Variation of hydrograph width with time of year
The variation of hydrograph width with time of year was checked for all 89 stations using both regular and circular plots. The seasonal distribution of floods at Station 07009 Boyne at Navan Weir is illustrated in Figure 7.7. It can be seen that most floods occur in the winter half-year from October to March, with the three largest in November to January. However, floods sometimes occur in the summer half-year.
The circular plot of Figure 7.8 indicates that wider hydrographs at this station are generally associated with the winter half-year (right half of diagram).
74
) 1
-
s 3
(m flow Peak
Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep
Figure 7.7: Plot of flood peak against time of year (Station 07009 Boyne at Navan Weir)
Radial scale marks Station 07009 Boyne hydrograph width in hours at Navan Weir
Figure 7.8: Circular plot of W50, W75 and W90 against time of year (floods at Station 07009)
However, a systematic pattern could not be established across the 89 stations studied. In essence, the widest flood hydrographs tend to be in winter but this is in any case the season where large floods are most prevalent.
7.4 Effect of arterial drainage on hydrograph widths
According to Table A.2, catchments associated with 19 of the 89 stations have undergone arterial drainage. The drainage schemes were carried out over different periods for different catchments. Of the 19 stations, 13 are Grade A1 and six are Grade A2. Only data for the post-drainage periods have been used in the main HWA reported earlier. An ancillary study investigated the effect of arterial drainage on the shape of the flood hydrographs of the affected stations.
75
Hydrograph data for the pre-drainage periods of five of the 13 Grade A1 stations were used. The stations selected and periods considered are listed in Table 7.2, together with gauged values of the index flood, QMED.
It is seen that the effect of arterial drainage is to increase QMED appreciably. This is to be expected in the post-drainage period because of the improved conveyance characteristics of the channel network. The one exception highlighted in Table 7.2 is that QMED is virtually unchanged for Station 23002.
[Editorial note: Judging from the relevant PCDs, arterial drainage on the Feale is not at all extensive (ARTDRAIN = 0.001 and ARTDRAIN2 = 0.002). The results for this station therefore appear irrelevant to exploring the effect of arterial drainage on hydrograph widths, and further discussion is omitted. As noted in Volume II, Catchment 23002 generates some of the largest floods ever gauged in Ireland. The Feale has been studied by Martin et al. (2000) amongst others.]
Table 7.2: Stations studied for the effect of arterial drainage on hydrograph widths Pre-drainage Post-drainage # QMED # events # QMED # events Station Period Period years (m3s-1) used years (m3s-1) used
01/10/1969 09/04/1979 07007 Boyne to 3 23.72 7* to 25 35.41 25 at Aqueduct 30/09/1972 05/04/2004 07010 14/11/1952 08/12/1986 Blackwater to 29 50.40 29 to 17 69.61 17 at Liscartan 30/09/1981 21/05/2003 18/10/1946 01/10/1959 23002 Feale to 12 373.74 12 to 46 371.84 46 at Listowel 30/09/1958 06/01/2006 01/10/1957 01/01/1991 26012 Boyle to 24 36.95 24 to 11 46.72 11 at Tinacarra 30/09/1981 22/01/2002 04/08/1951 01/10/1964 30004 Clare to 6 43.54 14** to 41 89.83 41 at Corrofin 30/09/1957 01/10/2005 * Event 4 discarded at Station 07007 ** Events 2 and 9 discarded at Station 30004
The procedures recommended in Chapters 3 and 4 were applied to derive characteristic hydrographs for flood hydrographs drawn from the pre-drainage period. Resulting values of the width descriptors (W75 and W50) and UPO-ERR-Gamma model parameters (n, Tr and C) are shown in Table 7.3, together with the corresponding results for the post-drainage period (taken from HWA results and their estimates from PCDs
Table D.1).
76
Table 7.3: Pre- and post-drainage values of hydrograph width descriptors/parameters Pre-drainage Post-drainage
Station W75 W50 n Tr C W75 W50 n Tr C 07007 67.00 N/A 2.20 48.56 70.61 26.22 52.53 3.74 26.84 69.88 07010 76.77 158.23 1.85 44.96 217.82 30.42 119.06 3.46 25.78 222.32 26012 287.76 N/A 1.72 182.25 211.32 177.27 285.94 3.85 195.50 257.50 30004 100.49 213.04 2.76 110.75 140.97 44.33 68.49 5.30 59.44 46.42
The resulting characteristic hydrographs are shown in Figure 7.9. Note that the pre-drainage and post-drainage hydrographs are drawn to the same timescale but that this differs from station to station.
Generally, the width descriptors are smaller in the post-drainage period, indicating narrower flood hydrographs. This corresponds to a quicker passage of flood flow and is consistent with the expected impact of arterial drainage, which Bhattarai and O’Connor (2004) summarise as having the objective of achieving a reduction of the extent and duration of flooding, by inducing faster runoff in the river channel, with higher peaks and shorter recessions of the discharge hydrograph.
Values of the Gamma shape parameter n are found to increase in all cases after arterial drainage. As illustrated in Figure 4.1, higher values of n correspond to a peakier hydrograph. Values of the Gamma rise-time parameter Tr are expected to be shorter after drainage. This is very much the finding at Stations 07007, 07010 and 30004. However, the post-drainage period yields a somewhat larger value of Tr at Station 26012. [Editorial note: Reference to part (c) of Figure 7.9 indicates that the increased value of Tr at Station 26012 is a product of parameter interaction. The pre-drainage value n = 1.72 leads to an abrupt start to the modelled hydrograph whereas the post-drainage value n = 3.85 provides a gentler (and consequently earlier) start. The value of Tr has changed in compensation.]
The research confirms that the effect of arterial drainage is generally to yield peakier hydrographs of reduced width. The swifter response ties in with the basic requirements of drainage schemes.
77
Pre-drainage period Post-drainage period
(a) Station 07007 Boyne at Aqueduct
-50 -25 0 56.25 112.5 168.75 225 -50 -25 0 56.25 112.5 168.75 225
Time in hours relative to time of peak flow Time in hours relative to time of peak flow
(b) Station 07010 Blackwater at Liscartan
-75 -37.5 0 56.25 112.5 168.75 225 -75 -37.5 0 56.25 112.5 168.75 225
Time in hours relative to time of peak flow Time in hours relative to time of peak flow
(c) Station 26012 Boyle at Tinacarra
-175 -87.5 0 125 250 375 500 -175 -87.5 0 125 250 375 500
Time in hours relative to time of peak flow Time in hours relative to time of peak flow
(d) Station 30004 Clare at Corrofin
-125 -62.5 0 87.5 175 -125 -62.5 0 87.5 175
Time in hours relative to time of peak flow Time in hours relative to time of peak flow
Figure 7.9: Characteristic hydrographs for four sites affected by arterial drainage 78
8 Constructing the characteristic flood hydrograph
8.1 Topics covered
This chapter provides guidance in constructing the characteristic flood hydrograph. The material covers a number of topics including:
Recommended methods at gauged sites; Motivation for and description of the parabolic curves method of constructing the upper hydrograph (Section 8.7); Recommendations at ungauged sites; Recommendation to use the pivotal catchment approach (Section 8.10).
The chapter explains the methods chosen for implementation through the FSU Web Portal. The parabolic method was devised by NUI Galway as part of their HWA research. Its description here is simplified.
The characteristic hydrograph is typically needed to construct a design hydrograph to accompany the T-year (peak) flood estimate derived by Volume II methods. The design flood hydrograph is obtained by scaling up the ordinates of the characteristic flood hydrograph by the T-year flood flow, QT. In effect, the methods paint in a hydrograph below the flood peak. Aspects of the synthesis of the T-year design flood hydrograph are therefore also discussed.
8.2 Features of the methods
The non-parametric and parametric approaches to constructing the characteristic hydrograph have strengths and weaknesses. It is helpful to recognise these to understand the methods chosen for implementation. Some difficulties can be circumvented by use of the IBIDEM method presented in Chapter 9.
Non-parametric method
Incompleteness of the lower part of the characteristic hydrograph generated using the non- parametric approach presents a problem in some applications. Many flood analysts use hydraulic modelling techniques that require a complete hydrograph as input. The FSU recommendation is to sketch in the lower part of the hydrograph subjectively or to use IBIDEM (see Chapter 9).
Parametric method
The recommended parametric method is the UPO-ERR-Gamma model of Section 4.4. The characteristic hydrograph generated in this way is continuous and complete. A minor unrealistic feature is that the gradient of the receding limb of the hydrograph changes abruptly at the point where the Exponential Replacement Recession takes over from the Gamma curve (e.g. Figure 8.1). Local smoothing of the hydrograph around the join is recommended should the feature be found troublesome.
79
It must be recognised that the UPO-ERR-Gamma model performs poorly for unusual sites at which the flood hydrograph characteristically falls more steeply than it rises.
100
w 75
o
l
f
k
a
e
p
f o
50
e
g
a
t
n
e
c
r e
P 25
0 -24 0 24 48 72 96 Time in hours (relative to time of peak flow) Figure 8.1: UPO-ERR-Gamma characteristic hydrograph at Stn 07009 by Table 6.7 models
A more general defect is that the UPO-ERR-Gamma portrays the flood hydrograph as rising from a pre-event flow of zero (e.g. Figure 8.1). This is unrealistic for Irish rivers. Ways of incorporating a non-zero pre-event flow are now discussed.
8.3 Allowances in design flood hydrographs for pre-event flow
8.3.1 Substitution approach
The approach recommended by NUI Galway for inclusion of pre-event flow is to:
Step 1 Estimate an appropriate pre-event flow, Q0 (see Section 8.3.3); Step 2 Substitute Q0 for any ordinate of the design flood hydrograph that falls below Q0.
This substitution approach has not been implemented through the FSU Web Portal, where the general recommendation is to use IBIDEM (see Chapter 9). However, the substitution approach may be useful in some applications, especially those focused on a specific site.
Figure 8.2 shows an example where the pre-event flow represents 10% of the peak flow. The substitution approach tends to lead to design hydrographs that rise abruptly from the pre- event flow. However, the approach has the merit that the modelled hydrograph widths are unchanged by the adjustment. Some hydraulic modelling may not cope well with the abrupt change of gradient at the start of the flood hydrograph. Should the feature be found troublesome, the user can deal with this by local smoothing of the hydrograph around the join.
80
8.3.2 Terminology: baseflow or pre-event flow?
The pre-event flow is denoted here by Q0. Many users will refer to this as the baseflow allowance. However, this terminology can lead to misunderstanding. In the absence of heavy rainfall, baseflow decays with time rather than remains constant.
100
w 75
o
l
f
k
a
e
p
f o
50
e
g
a
t
n
e
c
r e
P 25
0 -24 0 24 48 72 96 Time in hours (relative to time of peak flow) Figure 8.2: As Figure 8.1 but with pre-event flow substituting for first part of hydrograph
Additionally, there is scope for the less experienced user to confuse the baseflow allowance with the baseflow index (BFI). BFI is an index of catchment permeability and storage developed from daily mean flow data. It is defined as the proportion of the long-term river flow deriving from subsurface flows or from other delayed responses to rainfall. BFI takes a value between 0 and 1. Further details of BFI are given in Volume IV.
Obscure terminology is nothing new. When formulating a procedure for constructing design flood hydrographs, the FSR (NERC, 1975) coined the term average non-separated flow (ANSF) to represent baseflow. The FSR decision to model ANSF in units of m3s-1 per km2 (i.e. as a standardised baseflow) rather than in m3s-1 added to confusion.
The description in this chapter refers chiefly to the pre-event flow. The FSU advocates use of IBIDEM (see Chapter 9), which builds a bridge between FSU and FSR methods. Inevitably, IBIDEM adopts the FSR terminology of referring to the pre-event flow as the baseflow. The two terms should therefore be treated as synonymous.
8.3.3 Choosing the pre-event flow
The FSU has not researched pre-event flows. An approach possible at gauged sites will be to fix the pre-event flow at the median of pre-event flows noted for past flood events. However, the complexity of hydrographs at some stations may defy definition of a pre-event flow.
An alternative is to estimate pre-event flow using elements of the FSR rainfall-runoff method. IBIDEM (see Chapter 9) specifically accommodates this and is applicable at ungauged as well as gauged sites.
81
When arbitrary choices are made, it is prudent to check sensitivities. The user is therefore urged to test the sensitivity of results to the pre-event flow assumed. Flood risk assessments and the design of flood alleviation works ought not to be greatly impacted by the detail of the lower part of the hydrograph. If the final results are found to be sensitive to the pre-event flow assumed, this may reflect weaknesses in the hydraulic modelling.
8.4 Estimation of volume of flow
Having derived the characteristic hydrograph, the widths of exceedance at given percentages of the peak flow of a design flood are known. The flow volume above a given percentage of the peak of a design flood can be calculated if required.
8.4.1 Basic method
The design flood hydrograph is produced by scaling the ordinates of the characteristic flood hydrograph by the design peak flow QT of return period T. The volume VD of the design flood hydrograph above any given flow Q can be estimated by:
Step 1 Expressing the flow Q as a percentage p of the peak flow;
Step 2 Evaluating the semi-dimensionless volume Vc of the characteristic hydrograph above percentage p of the peak flow; remarkably, Vc is typically measured in hours; Step 3 Multiplying the semi-dimensionless volume Vc by QT to obtain the volume VD of the design flood hydrograph above percentage p of the peak flow; VD is typically measured in m3s-1 hours or in m3.
8.4.2 Non-parametric case
In the non-parametric approach, a numerical method such as Simpson’s rule can be used to integrate the relevant area (i.e. above the flow of interest and below the flood hydrograph) to obtain the required volume of flow. Surprisingly, it can be convenient to work out the volume in horizontal slices, using the widths that define the hydrograph. Where necessary, the hydrograph width is taken to vary linearly between percentages of the peak flow at which the width is expressly tabulated. The HWA software provides this option.
8.4.3 Parametric case
In the parametric approach, theoretical expressions can be derived for the volume represented by the area under the UPO-ERR-Gamma curve and above a given flow. When Equation 4.1 is used as an instantaneous unit hydrograph, the cumulative area under the Gamma defines the so-called S-curve response (Nash, 1957):
s(t) = I(n, t/K) 8.1 where I(n, x) is the Incomplete Gamma function defined by:
1 x In,x ev vn1dv 8.2 Γn 0
82
and Γ(n) is the Gamma function. Numerical algorithms for the Gamma and Incomplete Gamma functions are available in standard packages. The volume under the hydrograph between particular times is obtained as the difference between the S-curve ordinates at those times, appropriately rescaled.
The volume of flow sometimes has to be computed in two parts: before and after the point of inflection at which the exponential replacement recession becomes active in the UPO-ERR- Gamma model. The relevant formulae are incorporated in the HWA software. [Editorial note: Unless the HWA software is being applied, it is likely to be more convenient to evaluate volumes using a numerical method such as Simpson’s rule.]
8.5 Deriving the characteristic hydrograph at a gauged site
The non-parametric approach of Chapter 3 is recommended for producing the characteristic hydrograph at a gauged site with long records of 15-minute flow data. The steps to be followed are now summarised. The HWA software supports many of these steps. Further detail is presented in Chapters 2 and 3 above.
Step 1 A set of single-peaked flood hydrographs is extracted from the entire record at a gauged site. The number of flood hydrographs abstracted is a matter of judgement but use of the annual exceedance series – i.e. a Peaks-Over-Threshold (POT) series yielding an average of one event per year – is recommended. Step 2 At some stations, many of the flood hydrographs will be found to be multi-peaked, and a sufficient number of isolated single-peaked floods cannot be identified. In such cases, some complex floods must be decoupled. In essence, the hydrograph is filtered to retain only the unimodal part at its core. The procedure is described in Section 2.6. Step 3 In some cases, either because of error in the flow data or peculiarity in the flow generation process for the particular event, it may not be possible to identify a usable filtered hydrograph. Such problematic events are best discarded. Step 4 Each extracted hydrograph is standardised by dividing the flow ordinates by the flood peak. The standardised hydrograph thereby attains a peak value of 1.0. Step 5 For each of a number of reference percentages of the peak flow, the widths of exceedance on the rising and receding limbs of the hydrograph are abstracted. The widths are measured in hours. The reference percentages of the peak flow used in the HWA research are: 98, 95, 90, 85, 80, …, 10 and 5%. All such widths are abstracted where available. In some cases, the width at (e.g.) 75% of the peak flow is available on one limb of the hydrograph but not on the other. Step 6 At each reference percentage of the peak flow, the medians of the widths on the rising and receding sides are separately calculated. Step 7 The two median widths (at each reference percentage of the peak flow) are plotted as horizontal lines on a graph on either side of the peak, with width as the abscissa and percentage of peak flow as the ordinate. The time origin of the graph is set at the time of the peak flow. [Editorial note: The occurrence of flat-topped hydrographs sometimes presents a problem. The HWA software adopts the first of exactly equal peak ordinates as the peak of the hydrograph. If required, users can subvert this convention by increasing the value of the most central ordinate (of a number of exactly equal peak ordinates) by a very small amount.] Step 8 The median hydrograph is constructed as a segmented line passing through the left- hand extremities of the horizontal lines, passing through the peak of 1.0 at time 0.0, 83
and passing through the right-hand extremities of the horizontal lines. The median hydrograph thus defined is adopted as the characteristic hydrograph of the station.
8.6 Estimating the characteristic hydrograph at an ungauged site
8.6.1 Using the UPO-ERR-Gamma model
Based on the research reported in Chapters 4 and 6, the UPO-ERR-Gamma model can be used to estimate the characteristic hydrograph at an ungauged site. The steps required are:
Step 1 Abstract the relevant PCDs.
Step 2 Estimate the curve descriptors n, Tr and C using the equations given in Table 6.7. Step 3 Construct the characteristic hydrograph using Equations 4.7 to 4.10. The graph can be drawn in Microsoft Excel or other such application software. It is a unit peak at origin (UPO) model, i.e. the hydrograph is constructed to have a peak value of 1.0 at time 0.0. Thus the time origin is at the time of the hydrograph peak.
Station 07009 Boyne at Navan Weir is again used as the example. The characteristic hydrograph shown in Figure 8.1 is that derived from PCDs using the equations of Table 6.7. [Editorial note: This agrees almost perfectly with that derived earlier by HWA (Figure 4.5). The exceptionally good performance is not typical of that achieved on the 89 catchments as a whole. Estimation from PCDs is generally prone to considerable error, as evidenced by the FSEs shown in Table 6.7.]
Experienced users may wish to consider at Step 2 the alternate models given in Table 6.6 which require a value of the baseflow index. The default at an ungauged site is to adopt BFIsoil as the estimate of BFI.
8.6.2 Using the parabolic curves method
If only the upper hydrograph is required, it is possible to use the parabolic curves method. This is introduced in Section 8.7.
8.6.3 Using IBIDEM
IBIDEM (described in Chapter 9) is capable of constructing the design flood hydrograph in full. The package can be guided to match the characteristic hydrograph derived using the UPO-ERR-Gamma model or that obtained by the parabolic curves method.
8.7 Parabolic curves method
8.7.1 Overview
The parabolic curves method provides a way of estimating the upper hydrograph based on values of W75 and W50 derived in the non-parametric approach. The method is one of a number of techniques developed by NUI Galway and can be applied at ungauged as well as 84
gauged sites. The method is more sophisticated than the equivalent FEH technique (Reed and Marshall, 1999). The parabolic curves method uses hydrograph widths at both 50% and 75% of the peak flow. Moreover, the upper hydrograph is not constrained to be symmetric about the peak flow. An example is shown in Figure 8.3.
1.00
w
o l
f W75
0.75
k
a
e
p
f
o
n
o W50 i
t 0.50
r
o
p
o
r p
Hydrograph descriptors
s a
W = 36.69 hours
0.25 75 w
o W50 = 68.65 hours l
F s = 0.40 (i.e. 40% of width occurs before peak)
0.00 -48 -36 -24 -12 0 12 24 36 48 60 Time in hours (relative to time of peak flow)
Figure 8.3: Example of parabolic curves method (Station 07009 treated as ungauged)
The method is defined by three descriptors: the widths at 75% and 50% of the peak flow (i.e. W75 and W50 measured in hours) and the eccentricity parameter s. The eccentricity (or skewness) parameter is a coefficient defining the proportion of the width that occurs before the time of the peak flow. For the case illustrated in Figure 8.3, s =0.40; this is the default value for applications at ungauged sites.
8.7.2 Details of method
The description here differs from that given by NUI Galway. The upper hydrograph is synthesised in two parts by parabolas passing through the peak flow at (0, 1). The relevant equations can be written:
y = 1 + bx + cx2 on rising limb 8.3 and: y = 1 + Bx + Cx2 on receding limb 8.4
The coefficients b, c, B and C are determined from W75, W50 and s.
At first, it appears that the problem is underspecified, with four unknown coefficients and only three hydrograph descriptors. However, the eccentricity applies at both 75% and 50% of the peak flow. The solution is therefore fully determined.
NUI Galway adopted a root-solving approach. However, an explicit solution is possible:
85
2W 2 W 2 2W W on y 1 75 50 x 75 50 x 2 rising 8.5 4sW W W W 4s2W W W W 75 50 75 50 75 50 75 50 limb
and 2W 2 W 2 2W W on y 1 75 50 x 75 50 x 2 receding 8.6 41 sW W W W 4 1 s 2 W W W W 75 50 75 50 75 50 75 50 limb
8.7.3 Examples
Stations on the Suir provide a convenient example of the range of shapes that the parabolic method supports. The values of W75, W50 and s used are taken from the HWA results summarised in Table D.1 of Appendix D. The upper hydrographs yielded by the parabolic curves method are shown in Figure 8.4. The unusual shape for Station 16009 arises because W75 is less than half W50.
1.00 16004 Suir at Thurles
16002 Suir at Beakstown
w
o l
f 16008 Suir at New Bridge
0.75 k
a 16009 Suir at Caher Park
e
p
f
o
n
o i
t 0.50
r
o
p
o
r
p
s a
0.25
w
o
l F
0.00 -96 -72 -48 -24 0 24 48 72 96 120 144 168 192 216 Time in hours (relative to time of peak flow)
Figure 8.4: Parabolic hydrographs for four stations on the Suir
These characteristic hydrographs can be compared with those given earlier: for the full non- parametric method (in Figure 5.16) and for the UPO-ERR-Gamma model (in Figure 5.17).
8.7.4 Application at an ungauged site
Based on the research reported in Chapters 3 and 6, the parabolic curves method can be used to estimate the upper half of the characteristic hydrograph at an ungauged site. The steps required are:
Step 1 Abstract the relevant PCDs.
86
Step 2 Estimate the hydrograph width descriptors W75 and W50 using the equations given in Table 6.7. Step 3 Construct the upper half of the characteristic hydrograph using Equations 8.5 and 8.6. The graph can be drawn in Microsoft Excel or other such application software. The hydrograph is constructed to have a peak value of 1.0 at time 0.0. Thus the time origin is at the time of the hydrograph peak.
Figure 8.3 (shown earlier) reports the outcome for Station 07009 Boyne at Navan Weir, treating it as an ungauged site. Note that – at an ungauged site – the eccentricity (i.e. skewness parameter) takes the fixed value s = 0.40.
Experienced users may wish to consider at Step 2 the alternate models given in Table 6.6 which require a value of the baseflow index. The default at an ungauged site is to adopt BFIsoil as the estimate of BFI.
8.8 Constructing the design flood hydrograph
In principle, the design flood hydrograph for a given T-year peak flow (derived using Volume II methods) is obtained by scaling up the ordinates of the characteristic flood hydrograph by the appropriate factor. If only the upper part of the design flood hydrograph is required, the procedure is complete.
Where the complete design flood hydrograph is required, the user has a number of options:
To sketch in the lower part of the design hydrograph subjectively; To use IBIDEM (see Chapter 9); To adopt the parametric method of constructing the characteristic hydrograph, i.e. using the UPO-ERR-Gamma model of Section 4.4 (but see notes in Section 8.2).
8.9 Software
Implementation gives the user access to the extensive features of the HWA and IBIDEM packages. These packages are supplied for use offline to the FSU Web Portal. Some technical details of the HWA software are given in Appendix E. IBIDEM is presented in Chapter 9 with further details in Appendix F.
Experienced users will be able to consider and develop additional options for constructing the design flood hydrograph in particular applications.
8.10 Selection and use of the pivotal catchment 8.10.1 Overview
The concept of a pivotal catchment is introduced in Volume II. The pivotal catchment is the gauged catchment judged to be most relevant to the specific flood estimation problem at an ungauged site. The concept is fundamental to application of FSU methods at ungauged sites.
87
The FSU Web Portal is designed to steer all users to select and use a pivotal catchment. This is fundamental to applying Volume II methods for estimating the index flood QMED at an ungauged site.
Less experienced users may wish to become familiar with the pivotal catchment concept by exploring its use in QMED estimation in Volume II. In principle, the pivotal catchment approach is followed whenever the characteristic hydrograph is needed at an ungauged site.
Selection and use of a pivotal catchment promotes the effective use of gauged flood data, even at ungauged sites. The summary description is as follows. First, the performance of the ungauged method of Section 8.6 method is assessed by checking how it performs for the pivotal catchment. Second, the correction factor required to make the ungauged method perform well at the pivotal site is transferred (fully or partially) to adjust the estimate at the ungauged site. The overall procedure is referred to as a data transfer.
8.10.2 Selection of the pivotal catchment
The user assesses the most appropriate data transfer by making a reasoned selection of a pivotal catchment. This is the gauged catchment judged to be most relevant to the specific flood estimation problem.
The pivotal catchment is the user’s assessment of the most relevant catchment on which to base a data transfer. Where flood data are available from a gauge sited upstream or downstream of the subject site, this will often be selected as the pivotal catchment. In other cases, the selection is likely to be more precarious and will hinge on the user’s judgement of catchment similarity.
An algorithmic judgment of catchment similarity is likely to give weight to differences in a few leading factors – e.g. catchment size (represented by AREA), catchment wetness (indexed by SAAR) and catchment permeability (indexed by BFI or BFIsoil) – and to neglect all other factors. This is not a safe approach.
A particular feature present in one catchment and absent from another may lead to strong differences in their flood behaviour. Arterial drainage (indexed by ARTDRAIN or ARTDRAIN2) is perhaps the most notable such feature. Volume II reports evidence that BFIsoil and ARTDRAIN2 are important in characterising the post-drainage flood response of a catchment, whilst the descriptors DRAIND and S1085 are more important in characterising the response of undrained catchments. These findings may help the experienced user to judge which PCDs to examine closely when judging hydrological similarity for the purpose of selecting the pivotal catchment.
Other notable features to consider when assessing catchment similarity are the extent of urbanisation (indexed by URBEXT) and the presence of large lakes (indexed by FARL). Research reported in Volume II endorses the recommendation to favour geographical closeness – as well as similarity in key PCDs such as FARL – when selecting a pivotal catchment for use in a particular flood estimation problem.
88
8.10.3 Recommended procedure for data transfer
Selection of the pivotal catchment is a demanding task that calls for reasoning and judgement. Some judgements will be unsettling because of the impact that the data transfer has on the final results.
In some cases – not least on small catchments – the pivotal catchment selected may not be wholly convincing. This is not a reason to abandon making a data transfer. However, it may justify making only a partial transfer. An approach to making a partial transfer is included in Step 4 of the procedure now illustrated by worked example.
8.10.4 Example
The methodology for estimating the characteristic hydrograph at an ungauged site is illustrated for the Suck at Rookwood. This corresponds to Station 26002 but is treated here as an ungauged site. The following provides a broad guide to deriving an estimate of the characteristic hydrograph at this location.
The worked example is for use of the parabolic curves method. The same principles apply to use of the UPO-ERR-Gamma model. However, with three parameters to consider, data transfers are likely to be rather complicated to execute for that model.
Step 1 Confirm location: It is important to confirm the location of the subject site and to check the centroid of its catchment. The site location determines the PCDs extracted in Step 2. The centroid is relevant to judging the nearness of the subject catchment to available gauged catchments, when the user is undertaking the important task of selecting the pivotal catchment (see Section 8.10.5). t is the user’s responsibility to check that the FSU digital data provide a fair representation of the real conditions.
Step 2 Derive catchment descriptor information: Table 8.1 lists the PCDs needed in the version of the parabolic curves method illustrated here. The PCDs derive from the digital datasets made available with the FSU.
Table 8.1: Selected PCDs for Suck at Rookwood PCD value unit PCD value unit AREA 641.45 km2 FOREST 0.080 – DRAIND 0.799 km/km2 FARL 0.979 – S1085 0.500 m/km FLATWET 0.690 – ALLUV 0.025 mm ARTDRAIN 0.000 –
Step 3 Apply hydrograph width models: From Table 6.7 we have for the parabolic curves method:
-0.88 -5.85 -2.86 W75 = 31.28 DRAIND FARL (1+FOREST) (1+ARTDRAIN)-2.92 FLATWET3.12 (S1085/1000)-0.27
-0.88 -5.85 -2.86 -2.92 3.12 -0.27 W75 = 31.28 (0.799) (0.979) (1.081) (1.0002) (0.690) (0.500/1000) = 84.42 h
89
-0.95 -6.59 -2.83 W50 = 34.02 DRAIND FARL (1+FOREST) (1+ARTDRAIN)-2.60 FLATWET2.44 (S1085/1000)-0.29
= 34.02 (0.799)-0.95 (0.979)-6.59 (1.081)-2.83 (1.0002)-2.60 (0.690)2.44 (0.500/1000)-0.29 = 142.30 h
[Editorial note: The estimates of W75 and W50 differ very slightly from those shown in the centre-right columns of Table D.1 of Appendix D. This is because, in the recommended models, the exponents have been rounded to 2 decimal places.]
Step 4 Transfer data from gauged locations to improve model prediction at subject site:
The general procedure is to infer an adjustment factor, AdjFac, by reference to the performance of the PCD-based model at a nearby gauged site. Thus, the adjustment factor for W75 would be: W AdjFac 75,gauged W75 8.7 W75,PCD The adjustment is then partially or fully transferred to the subject site:
h 8.8 W AdjFac W 75,adj W75 75,PCD
The typical procedure is to apply a full transfer by setting the exponent h to 1.0. If W75 is found to be 20% greater than the PCD-based estimate, it is assumed that the model will be similarly in error at the subject site. Thus, the estimate of W75 at the subject site is adjusted by multiplying by 1.20.
The exponent h can be thought of as the hardness of the data transfer. h=1 denotes a full (or “hard”) transfer. A partial (or “softer”) transfer might set h=0.5. n this case, if W75 is found (at the donor site) to be 20% greater than given by the PCD model, the 0.5 estimate of W75 at the subject site is adjusted by multiplying by a factor of 1.20 or 1.095.
Much skill attaches to deciding which of several possible donor catchments is pivotal to improving estimation at the subject site. With gauged sites upstream (Station 26006) and downstream (Station 26005) of the subject site, the choice might not be clear-cut for the Suck at Rookwood. However, it transpires that HWA has not been undertaken for Station 26006 Suck at Willsbrook. Thus, the data transfer illustrated here is from Station 26005 Suck at Derrycahill, which drains an area 69% greater than that at Rookwood.
Table 8.2 summarises the data transfer from the Suck at Derrycahill (Station 26005) to the Suck at Rookwood. It is seen that hydrograph widths are appreciably underestimated at the donor site. Use of a hard data transfer assumes that the same relative error will occur when applying the PCD models at Rookwood and adjusts the hydrograph widths accordingly.
When HWA results for the Suck at Rookwood itself are examined (lighter-shaded rows in Table 8.2), it is found that the PCD method does indeed underestimate hydrograph widths there. However, it transpires that, the data transfer from Derrycahill is too strong. In this instance, a soft transfer with h=0.5 would work better (see right-hand column of table). However, the user will not know this for a subject site that is truly ungauged! The upper
90
hydrographs transferred (see Figure 8.5) are somewhat “pointy” in comparison to the notably flat-topped hydrograph (black curve) that the parabolic method yields directly at Rookwood.
Table 8.2: Data transfers to Suck at Rookwood using parabolic curves method Hydrograph width Data Implied HWA PCD Hard Soft Method transfer factorial HWA transfer transfer from adjustment with h = 1 with h = 0.5 hours hours
Width at 75% of hydrograph peak (W75) No data 84.42 transfer Downstream Station 136.04 86.15 1.58 133.4 106.1 donor 26005 Analysis of Station 118.7 gauged data 26002 Width at 50% of hydrograph peak (W50) No data 142.30 transfer Downstream Station 209.83 147.63 1.42 202.1 169.6 donor 26005 Analysis of Station 163.98 gauged data 26002
1.00
w 0.75
o
l
f
k
a
e
p
f o
0.50
n
o
i
t r
o Parabolic curves method p
o HWA at 26002 r
P 0.25 Soft transfer (26005 to 26002) Hard transfer (26005 to 26002)
0.00 -144 -96 -48 0 48 96 144 192 Time in hours (relative to time of peak flow)
Figure 8.5: Upper hydrographs transferred from Derrycahill to Rookwood
91
[Editorial note: Station 26002 Suck at Rookwood provides an example where the parabolas fitted to the rising and receding limbs have turning points just before and just after the nominal peak value of 1.0. This feature is detectable in Figure 8.5 by close scrutiny of the crest segment of the hydrograph shown in black. Where the feature arises, it is reasonable to reduce any ordinate that exceeds 1.0 to the peak value of 1.0, thereby producing a flat-topped hydrograph.]
For completeness, HWA results for the two catchments are shown in Figure 8.6. The characteristic hydrograph at Derrycahill (Station 26005) is relatively smoothly represented by the derived median hydrograph (of Section 3.5), which in turn is especially well modelled by the UPO-ERR-Gamma (of Section 4.4). It is confirmed that hydrographs at Rookwood are rather narrower than at Derrycahill, although the difference is almost entirely in the receding limb.
Figure 8.6: Derived median and UPO-ERR-Gamma hydrographs for Stns 26002 and 26005
8.10.5 Further discussion of choice of method and of pivotal catchment
While the pivotal catchment principle applies to transferring a characteristic hydrograph in much the same as it does to transferring a gauged value of QMED in Volume II, a number of important differences arise in practice:
In the index flood case, there is the one flood measure QMED to be evaluated and transferred. In the characteristic hydrograph case there are several possibilities. There is the choice between use of the UPO-ERR-Gamma model (with three parameters open to adjustment) and use of the parabolic curves method (with two width descriptors open to adjustment). The use of IBIDEM (see Chapter 9) brings additional options. In the index flood case there is essentially the one PCD-based method of estimating QMED, albeit with important adjustments for urbanised catchments. In the characteristic hydrograph case, there is a choice between models that use BFI and models that do not.
The suitability of a donor reflects many factors. Donors on the same river system are likely to be most suitable. However, the typical width and shape of hydrographs will be influenced by any notable feature that intervenes between the subject site and the donor site. The most obvious features are lakes or reservoirs, which are likely to modify the width and shape of hydrographs appreciably. This is confirmed by the appearance of FARL in all ten PCD models presented in Table 6.6 and Table 6.7.
92
Different users will make different judgements. Some will argue that the presence of FARL in the models allows the transfer to proceed. Others may argue that site-specific features mean that the lake has a stronger effect than represented in the necessarily generalised model.
Confidence in the relevance of the data transfer is weakened if a special feature intervenes that is not represented in the model, e.g. if a significant tributary of wholly different character joins the river between the donor and subject sites. If it is the most suitable donor available, the pragmatist will allow the data transfer in part: e.g. setting h to 0.5 (or less).
Finally, it should be noted that it may be appropriate or necessary to choose one gauged site as the pivotal catchment in QMED estimation and another in estimation of the characteristic hydrograph.
8.10.6 Urbanised catchments
Large-scale urbanisation affects flood magnitudes and catchment response times very appreciably. Flood magnitudes are generally increased and flood response times (and hydrograph widths) are compressed, often markedly. Great care is therefore required in making data transfers to a subject catchment that is heavily urbanised. In similar vein, a notably urbanised catchment should not normally be chosen as the pivotal station if the subject catchment itself is largely rural.
Experienced users may find it helpful to apply IBIDEM. This allows the merit of particular data transfers to be interpreted in terms of the parameters of the FSR rainfall-runoff method and the urban adjustments to that method presented in FSSR16 (IH, 1985).
93
9 IBIDEM
9.1 The idea of IBIDEM
IBIDEM stands for Interactive Bridge Invoking the Design Event Method. The idea is to provide a bridge between the FSU method of estimating a design flood hydrograph and the FSR design event method that it replaces. Two parameters of the FSR rainfall-runoff model (time to peak and standard percentage runoff) are chosen by optimisation so that the design hydrograph synthesised by the FSR method matches that produced by the FSU procedures.
This chapter explains how IBIDEM is structured and illustrates its use on five example catchments. The tests indicate that IBIDEM is helpful in assessing design flood hydrographs produced using the FSU procedures and can help the experienced user to judge whether a design hydrograph is consistent with the properties expected of the particular catchment. IBIDEM is supplied as a standalone software package downloadable through the FSU Web Portal. Some technical details are given in Section 9.5 and Appendix F. The package was developed by JBA Consulting.
9.1.1 Reminder of hydrograph estimation by FSU methods
The T-year peak flow is typically estimated as the product of an index flood and a growth curve. The methods are described in Volume II. The required design hydrograph is constructed around the peak flow by applying a hydrograph shape taken from the HWA presented in earlier chapters. There are two main methods:
At gauged sites, the characteristic hydrograph is built up using widths extracted from observed hydrographs at given percentages of the peak flow. This is the non- parametric approach of Chapter 3, and can be executed using the HWA software. At ungauged sites, the parametric approach of Chapter 4 can be applied. Parameters of the UPO-ERR-Gamma model are estimated from PCDs using the equations presented in Chapter 6. Application of the parabolic curves method of Section 8.7 is also supported.
IBIDEM also provides the user with a number of additional options.
9.1.2 Hydrograph estimation by the FSR design event method
Some users may not be very familiar with the FSR design event method. A brief summary is attempted here. The relevant volume of the Flood Estimation Handbook provides further details (Houghton-Carr, 1999).
The T-year design hydrograph is constructed as the output to the “unit hydrograph/losses” rainfall-runoff model. The FSR design event method combines four inputs: the temporal profile, duration, and rarity of the rainfall event and the pre-event catchment wetness. The first three define the rainfall input (to the rainfall-runoff model), whilst the fourth defines the initial condition (of the rainfall-runoff model). These inputs take specific values according to particular rules (see Figure 9.1). The rules reflect some of the general properties of the catchment and its climate. 94
Rainfall profile Rainfall duration Rainfall rarity Pre-event wetness
Profile = constant D = D (SAAR, Tp) T = T ( T ) CWI = CWI (SAAR) rain rain flood
Figure 9.1: Design inputs to FSR rainfall-runoff method of flood frequency estimation
It is necessary to adopt suitable values for the parameters of the rainfall-runoff model itself. For the FSR unit hydrograph/losses model, the parameters are:
The standard percentage runoff, SPR; The unit hydrograph time-to-peak, Tp; The standardised baseflow, known as the “average non-separated flow”, ANSF.
On all but highly permeable catchments, the last parameter tends to be relatively unimportant.
Two other factors play a role in the model: the catchment area (AREA), and the areal reduction factor (ARF). ARF is applied to estimate the design catchment rainfall from the design rainfall depth at a typical point within the catchment. In application here, the rainfall depth-duration-frequency model is taken from Volume I rather than from FSR methods.
9.1.3 Basic idea of bridge between the FSR and FSU methods
The aim of IBIDEM is to link the FSU method of T-year hydrograph estimation to the FSR rainfall-runoff method it replaces. The Tp and SPR parameters of the rainfall-runoff model are chosen so that the design hydrograph synthesised by the FSR method matches that produced by the FSU procedures. The approach offers several gains:
Whereas the HWA methods of Chapter 3 construct only the upper parts of the design hydrograph, IBIDEM synthesises the entire hydrograph. This allows the user to look at runoff volumes (e.g. for assessing flood storage) and to “route” flood hydrographs, as they do when using the FSR rainfall-runoff method. A link with rainfall is made. By noting the percentage runoff (PR) and the rainfall duration (D) implied by IBIDEM, the user is able to check whether the FSU design hydrograph has properties consistent with that expected of the catchment. Those with particular experience of the FSR design event method are able to interpret the Tp and SPR parameters of the rainfall-runoff model to which the FSU design hydrograph is said to be equivalent. Based on experience or further guidance, users can vary these values to test sensitivities and to investigate the possible effects of catchment change on the design flood hydrograph.
95
9.2 How IBIDEM fits hydrographs
IBIDEM fits an FSR rainfall-runoff hydrograph to match the shape and peak flow of an imported FSU design hydrograph. The fitting is achieved by adjusting the time to peak (Tp) and standard percentage runoff (SPR) parameters of the FSR rainfall-runoff method.
IBIDEM implements all parts of the FSSR16 version of the rainfall-runoff method (IH, 1985), other than estimation of Tp and SPR from catchment characteristics. Values of Tp and SPR are instead derived by fitting the implied FSR hydrograph to the imported FSU hydrograph.
In running the rainfall-runoff method, IBIDEM undertakes the following steps:
Step 1 Imports the physical catchment descriptors AREA, SAAR and URBEXT; URBEXT is needed for the urban adjustment to the percentage runoff. Step 2 Finds values of SPR and Tp by optimisation. In some options, adjustments are made by the user. Step 3 Selects an appropriate data interval T based on Tp, adopting a convenient value such as 0.25 hours or 1 hour. Step 4 Calculates the design rainfall duration D from Tp and SAAR, and evaluates the areal reduction factor (ARF) from AREA and D. Step 5 Constructs a triangular unit hydrograph with time-to-peak Tp. Step 6 Evaluates the design rainfall depth from D and a user-supplied flood return period (or set of return periods). The user-supplied flood return period (Tflood) is linked to a rainfall return period (Train) according to the FSR design package used. [Editorial note: The design package is a prescribed a set of rules to be followed when using the FSR rainfall-runoff method. One package corresponds to winter conditions and is typically applied on catchments that are largely rural.] Step 7 IBIDEM requires the rainfall depth for a typical (i.e. average) point in the catchment. The depth is multiplied by ARF (Step 4) to obtain the catchment-average rainfall (P) of required return period. [Editorial note: The user obtains rainfall depths for a set of durations and return periods through the FSU Web Portal. These are supplied to IBIDEM in the form of a table of rainfall depths, with durations (0.25 to 600 hours) in rows and return periods (2 to 200 years) in columns. IBIDEM calculates the required rainfall depth by interpolation, using linear interpolation between durations and logarithmic interpolation between return periods.] Step 8 Distributes the rainfall depth according to a standard temporal profile taken from the FSR. The most commonly used profiles are the so-called 75% winter and 50% summer profiles. To meet the project specification, the default setting in IBIDEM is to use the 75% winter rainfall profile: even on an urbanised catchment. The alternative summer profile can be chosen by the user. Step 9 Calculates the percentage runoff PR from SPR, rainfall depth P and the catchment wetness index CWI, applying an urban adjustment if necessary. The urban adjustment is amended to use URBEXT from the FSU rather than URBAN from the FSR. The relevant formula is:
PR = PRrural (1.0 – 0.47 URBEXT) + 70 (0.47 URBEXT) 9.1 (The factor 0.47 arises as the product of 0.30 and 1.567. The factor 1.567 back- converts URBEXT to be compatible with the URBAN index used in the FSR.) 96
Step 10 Applies the PR to the total rainfall profile to obtain the net rainfall profile, i.e. the portion of rainfall that generates rapid response runoff. Step 11 Convolves the unit hydrograph with the net rainfall profile to give the rapid response hydrograph. Step 12 Adds baseflow to give the total runoff hydrograph. Baseflow is calculated from AREA, SAAR and CWI using a standard equation from FSSR16.
9.3 General approach to the optimisation
9.3.1 “First Tp and then SPR”
The two parameters to be fitted (Tp and SPR) affect the hydrograph in distinct ways. SPR affects the magnitude of the flows but does not have any effect on timings. In cases where baseflow forms only a minor element, the flood magnitude is roughly proportional to SPR (see the example in Figure 9.2).
The Tp parameter affects both the timing and the magnitude of the flows. A shorter Tp alters the T-year flood magnitude in three ways:
It forces an increase in the peak of the unit hydrograph (to maintain the same volume of flow in a shorter time); It shortens the design storm duration, hence increasing the rainfall intensity for a given return period; The resulting change in the design rainfall depth affects the percentage runoff via the DPRRAIN term (see Section F2 of Appendix F).
Figure 9.2 provides an example of the kind of effect that Tp and SPR have on the peak of the design flood hydrograph in the FSR rainfall-runoff method.
) 2525
1
- s
3 2020
SPR = 20% 1515 SPR = 20%
SPRSPR == 40% 40%
eak flow (m flow eak 1010
Peak flow (m3/s) 55 yearp
- T 00 00 22 44 66 8 10 Tp (hours) Tp (hours) Figure 9.2: Illustration that FSR T-year peak flow varies with Tp as well as with SPR
Because SPR has no effect on hydrograph timings, it is convenient to optimise Tp first. SPR is adjusted in a later step to give a peak flow that exactly matches the peak of the imported FSU design hydrograph. For rather intricate reasons – with advantages outweighing disadvantages – this strategy is in fact preferable to that of optimising SPR and Tp jointly.
97
9.3.2 Use of horizontal fitting
IBIDEM fits the FSR rainfall-runoff hydrograph – standardised so that its ordinates are expressed as percentages of the peak flow – to the characteristic hydrograph derived by FSU methods. The process uses horizontal fitting, in which differences in hydrograph widths are minimised between the FSU and FSR representations. This is in contrast to the conventional vertical fitting of hydrographs in terms of differences in flows.
The set-up is illustrated in Figure 9.3. The FSU hydrograph is built up from median hydrograph widths at various percentages of the peak flow, using the non-parametric approach of Chapter 3. [Editorial note: IBIDEM refers to this as the empirical approach.] Fitting is carried out for the portion of the hydrograph above a threshold. This accommodates the feature that FSU hydrograph does not cover the full range of flows down to zero.
FSU hydrograph Sample FSR hydrograph
Threshold for fitting Threshold for fitting
Figure 9.3: Horizontal fitting by comparing hydrograph widths
9.3.3 Deriving Tp by optimising the fit to the FSU flood hydrograph
Tp is obtained by optimising the shape of the FSR rainfall-runoff hydrograph so that it best matches the FSU characteristic hydrograph. These semi-dimensionless hydrographs are expressed as a percentage of the peak flow.
A complication is that baseflow (in the FSR rainfall-runoff method) is defined as a fixed amount in m3s-1 rather than as a proportion of the peak flow. The difficulty is overcome by fitting the widths of the rapid response parts of the hydrographs, i.e. after subtracting the (fixed) baseflow BF from the FSU hydrograph (Figure 9.4).
) 1
- s 3 q peak Qpeak
(m Flow
BF Time (hours) Figure 9.4: Relationship between peak flow Qpeak and peak rapid response qpeak 98
The detailed procedure is:
Step 1 Calculate the baseflow BF (in m3s-1) from the FSSR16 method [Editorial note: FSSR16 provides an equation for ANSF, defined as the baseflow per unit area i.e. in m3s-1 per km2. Thus, BF = ANSFAREA.]
Step 2 Subtract BF from the FSU hydrograph. The peak flow Qpeak is thereby reduced to the response peak qpeak. Step 3 Express the threshold for fitting as a % of qpeak: %qthreshold = %Qthreshold - 100(BF/Qpeak) This meets the IBIDEM requirement that the user specifies the fitting threshold as a percentage of the total peak flow, %Qthreshold. The user is prompted to raise the threshold should the percentage first entered yield a threshold flow that is less than or equal to BF.
Step 4 Calculate a set of m widths WFSU of the FSU response hydrograph for percentage flows between %qthreshold and 100%, at an interval of 1% (see left-hand side of Figure 9.3). Step 5 Run the FSR rainfall-runoff method with an arbitrary (fixed) SPR and an initial guess for Tp.
Step 6 Calculate a set of widths WFSR of the FSR response hydrograph at the same percentages as in Step 4 (see right-hand side of Figure 9.3). 2 Step 7 Evaluate the objective function: Σ[WFSU(i) - WFSR(i)] for i = 1 to m. Step 8 Vary Tp and repeat Steps 5 to 7 until the objective function is minimised (see Section F1 of Appendix F for details of the optimisation method).
The non-parametric method of Chapter 3 may sometimes produce hydrographs with more than one peak (Figure 9.5). Within IBIDEM, these cases are treated by excluding the time when flow is below the relevant percentage of the peak when evaluating the width of the hydrograph. This pragmatic approach ensures that IBIDEM represents the total duration of the hydrograph but without being unduly affected by the separation of the peaks. The output from IBIDEM is always a single-peaked hydrograph.
IBIDEM discounts this portion of the total width of the hydrograph
Figure 9.5: Double-peaked hydrograph
99
9.3.4 Deriving SPR by matching the required peak flow
After Tp has been optimised, the FSR hydrograph is scaled to fit the relevant peak flow, which will typically have been derived by Volume II methods. SPR is calculated by working out the factor by which the hydrograph has to be multiplied to match the FSU peak flow. The details of how IBIDEM does this are given in Section F2 of Appendix F.
9.4 Additional IBIDEM options
The IBIDEM software provides a number of additional options. The principal ones are now summarised. Graphical displays are illustrated later, chiefly in Section 9.5.2.
9.4.1 Flood frequency
The Flood frequency option allows users to optimise a set of hydrographs for different return periods. To allow maximum flexibility of options, users are required to import a separate FSU flood hydrograph for each return period. IBIDEM allows up to seven return periods to be analysed together. A separate pair of Tp and SPR values is fitted at each return period, using the method described in Sections 9.2-9.3.
9.4.2 Sensitivity to storm duration
The terms design rainfall and design storm are used interchangeably. The Sensitivity to storm duration option – and further options – are made available once IBIDEM has performed the hydrograph fit for a single return period. The fitted values of Tp and SPR are retained (i.e. no further optimisation is carried out) and the FSR rainfall-runoff method is re-run for five trial storm durations. A feature of the FSR rainfall-runoff method is that the volume of the flood hydrograph increases as the storm duration increases.
Default durations for the trials are 0.5D, 0.5D, D, 2D and 2D where D is the duration resulting from the Tp value found in the optimisation. The user can change the trial durations if desired.
In addition to testing sensitivities, this option may be helpful in river modelling applications (see Volume V) where a longer-than-normal storm is being applied to a tributary catchment in order to generate a T-year flood further down the river system.
9.4.3 Sensitivity to model parameters
The Sensitivity to model parameters option allows the user to re-run the rainfall-runoff method (after the initial optimisation stage), with altered values of the parameters Tp and/or SPR. If Tp is altered, the storm duration is automatically updated. If SPR is altered, PR is updated. Other settings remain unchanged.
The option allows the user to explore the possible impact of land-use change, by adjusting Tp and/or SPR to represent conditions before and after the change. An implicit assumption is that the user trusts application of the FSR rainfall-runoff method to represent the particular land-use change (such as agricultural drainage or tree planting) adequately.
100
9.4.4 Sensitivity to changes in urbanisation
The Sensitivity to changes in urbanisation option allows the user to re-run the rainfall-runoff method for an altered value of URBEXT. IBIDEM calculates revised values for Tp and PR to reflect the change in URBEXT. Because of the joint use of newer and older technologies, the procedure is rather intricate.
Back-converting URBEXT
The first step is to back-convert the revised URBEXT to the equivalent FSR descriptor of urban extent (URBAN). This is done using: URBAN = 1.567 URBEXT 9.2 In the FSR rainfall-runoff method method, URBAN affects Tp, PR and the choice of design event package (winter or summer).
Choice of design package
Were IBIDEM to choose the design package automatically based on the degree of urbanisation, it would be possible for a small increase in URBEXT to lead to an abrupt change from use of a winter design event to use of a summer design event, with potentially a large change in the design flood hydrograph. To avoid this possibility – and to allow greater flexibility – the choice of design event package is set manually by the user.
Effect on PR
IBIDEM calculates a new PR from SPR using the urban adjustment (Equation 9.1) from Step 9 of the Section 9.2 procedure.
Effect on Tp
Tp for the base condition will have been found during the initial run. IBIDEM updates this for the revised value of URBAN by invoking part of the FSSR16 model for estimating Tp(0). The factor representing the urban effect on response times in the FSSR16 model is the term (1+URBAN)-2.2.
The Tp value obtained in the initial run of IBIDEM is therefore updated for the altered level of urbanisation using: (1 URBAN )2.2 Tp0 = Tp(0) revised 9.3 revised (1 URBAN)2.2
Tp is converted to and from Tp(0) as required, using: Tp = Tp(0) + ΔT/2 where ΔT is the data interval (see Section 9.2). Equation 9.2 is used as required to convert current and projected values of URBEXT to the corresponding values of URBAN.
Summary
Although intricate, this option allows users to investigate the impacts of urban development on design flood hydrographs and peak flows. 101
9.5 Further details of the software
9.5.1 Inputs
The IBIDEM software requires the following inputs in all cases:
Catchment descriptors AREA, SAAR and URBEXT: typed in on the first screen (the FSU descriptor of urbanisation is the fractional urban extent URBEXT, and takes a value between 0 and 1); Design flood hydrograph derived from FSU procedures: supplied as a CSV file (this gives pairs of time and flow values); Rainfall frequency information from FSU procedures: supplied as a CSV file giving a table of design rainfall depths at an average point in the catchment (the tabulated depths are for a standard set of durations and return periods); Return period associated with the FSU hydrograph (default is 2 years); Threshold flow to be used in fitting: expressed as a percentage of the peak flow (default is 50%).
These data are validated during the input process and the user notified of any exceptions (see Section F3 of Appendix F. In particular, the user is warned if:
The FSR baseflow exceeds part of the imported FSU hydrograph. In this case, the user needs to import a hydrograph with a higher minimum flow (e.g. by removing the first or last few values) or to adjust the baseflow. The threshold falls below the lowest flow value in the rising/receding limb of the imported FSU hydrograph. In this case, the user needs to raise the threshold flow used for the fitting or to import a more complete hydrograph.
Additional inputs are required for some of the optional functionality:
For the flood frequency option: Hydrographs for multiple return periods; For the option to vary baseflow: Value for baseflow; For the sensitivity to model parameters option: New value for Tp or SPR or both; For the sensitivity to changes in urbanisation option: New value for URBEXT.
9.5.2 Graphical displays
IBIDEM displays the hydrograph fit, showing the imported FSU hydrograph, the fitted FSR hydrograph and the threshold flow used for fitting. Figure 9.6 provides an example. The two hydrographs are aligned so that they peak at the same time.
When the flood frequency option is selected, the user can choose the return period for which the fit is displayed. Alternatively, it is possible to display a graph showing how a particular variable changes with return period. The user can select any one of peak flow, percentage runoff, SPR, Tp, rainfall depth or runoff volume. The example in Figure 9.7 shows how the
102
runoff volume – expressed as a depth in mm across the catchment – changes with return period.
FSR hydrograph FSU derived fitted by IBIDEM median hydrograph
Figure 9.6: Display of fitted and imported hydrographs
Figure 9.7: Display of how a variable changes with return period
When the sensitivity to storm duration option is selected, the user can plot either a graph showing multiple hydrographs – i.e. one for each duration plus the imported FSU hydrograph (e.g. Figure 9.8) – or a graph showing how a particular variable changes with return period.
The example in Figure 9.9 shows how the peak flow varies with the design storm duration. This provides a way of identifying the critical duration that the bridge to the FSR rainfall- runoff method implies for the catchment. The figure also illustrates the manner in which 103
options are selected on-screen in IBIDEM. Q denotes the peak flow, PR the percentage runoff, P the rainfall depth and V the hydrograph volume.
Figure 9.8: Display of hydrographs for multiple storm durations
Figure 9.9: Display of how a variable changes with storm duration
When the sensitivity to model parameters or the urbanisation option is selected, IBIDEM plots the imported FSU hydrograph, the original fitted FSR hydrograph and the altered FSR hydrograph resulting from the changed parameter(s). Figure 9.10 illustrates this for a case where URBEXT increases from 0.00 to 0.20.
104
Figure 9.10: Display of sensitivity to an increase in URBEXT
9.5.3 Display options
Various options are provided for graph layout and units (the option name is in italics):
Hydrographs can be plotted with the vertical axis showing either % of peak flow (default) or m3s-1. Hydrographs can be plotted with the time origin either at the peak (default) or at the start of the FSR hydrograph. Flow units can be either m3s-1 (default) or mm/hr. Runoff volume units can be either mm equivalent of catchment runoff (default) or m3 or cumec-hours. A cumec-hour is the volume represented by a flow of 1 m3s-1 sustained for one hour, i.e. 3600 m3. Plots of variables against return period can have a horizontal axis showing the Gumbel reduced variate, the Logistic reduced variate or the natural logarithm of return period. In each case, a subsidiary axis shows the return period in years. These are return periods on the annual maximum scale. Thus, the 50-year event corresponds to a value with an annual exceedance probability of 0.02. [Editorial note: Frequency statements should be treated with some degree of caution. Although the peak of the FSU flood hydrograph is nominally of the stated frequency, the frequency assignment does not strictly transfer to quantities (such as the flood volume) derived by invoking the bridge to the FSR rainfall-runoff method.]
9.5.4 Goodness-of-fit measures
IBIDEM summarises the goodness of fit – of the FSR rainfall-runoff hydrograph to the FSU design hydrograph – in two measures: the root mean square error (RMSE) and the Nash- Sutcliffe efficiency (NSE). These are calculated in quite different ways:
105
RMSE is calculated as part of the IBIDEM fitting process. It is the root mean square error in terms of hydrograph width (measured in hours) for the upper portion of the hydrograph over which the fitting is carried out. It indicates how well the FSR and FSU hydrographs match in terms of their widths. A small value of RMSE indicates a good fit. The RMSE obtained using IBIDEM is always the minimum possible given the shape of the imported FSU hydrograph and the family of shapes that the FSR rainfall-runoff hydrograph can take. NSE is a dimensionless measure of hydrograph fit calculated in the conventional (i.e.) vertical direction. It is a measure of the goodness of fit in terms of flow over the duration of the imported FSU hydrograph. Values close to 1.0 indicate an excellent fit. Negative values of NSE indicate that a better fit could be achieved using the mean flow. The statistic is calculated independently of the fitting done by IBIDEM, and will not usually take the minimum possible value.
[Editorial note: The Nash-Sutcliffe efficiency is used to good effect in generalising a model for the baseflow index BFI (see Volume IV). However, NSE is problematic to interpret in some of the cases arising in IBIDEM. Because it gives no special weight to the quality of fit around the peak, NSE is not ideal for evaluating the match to the upper hydrograph. The measure has the minor merit of being independent of the method of fitting used in IBIDEM.]
In the case of Figure 9.11, the fit of the FSR hydrograph is judged to be very poor, with NSE = -0.70. The difficulty arises largely because the receding limb of the FSR hydrograph is much steeper than that of the FSU hydrograph once the inflection point four hours after the peak has been passed. Beyond that time, there is a long period when the FSR hydrograph is much lower than the FSU hydrograph. The recession limb of the imported hydrograph has less influence on the IBIDEM fit if the threshold used is raised from 50% to 60% (see Figure 9.12). The NSE becomes a respectable +0.70.
NSE evaluated across this period NSE = −0.70
FSR hydrograph Characteristic fitted by IBIDEM hydrograph by UPO-ERR-Gamma
Figure 9.11: FSR hydrograph fitted to UPO-ERR-Gamma hydrograph
106
NSE across here NSE = +0.70
Figure 9.12: As Figure 9.11 but with fitting threshold raised to 60% of peak flow
[Editorial note: The take-home messages are: (i) Give greater weight to the RMSE measure than NSE, (ii) Consider the sensitivity of IBIDEM fits to the threshold chosen and (iii) Fitting horizontally rather than vertically has real merit in hydrograph width modelling!]
9.5.5 Tabular display
Below the graph, IBIDEM tabulates the parameter values and other variables (as shown in Figure 9.13). After the hydrograph fitting is carried out, the variables shown are: flow return period, rainfall return period, baseflow (BF), fitted Tp, fitted SPR, PR (based on SPR and other terms including URBEXT), time-step, storm duration (calculated from Tp and SAAR), rainfall depth (calculated from storm duration and return period), peak flow (taken from the imported FSU hydrograph), runoff volume, RMSE and Nash-Sutcliffe efficiency.
Figure 9.13: Example of IBIDEM tabular display
107
Slightly different versions of the table appear in some options. For example, it is not appropriate to show the goodness-of-fit statistics within the Sensitivity to storm duration option. The order of the rows in each version of the table remains the same but the variable that the user has changed (between columns) is highlighted to aid interpretation. For example, the row containing the flow return period is highlighted in Figure 9.13. Another feature is that variables not changing between columns are indicated in grey. Baseflow does not vary with return period, so the BF row is shaded grey in this example.
9.5.6 Export of results
IBIDEM allows export of a summary report in .CSV format, including a record of the FSU and FSR hydrographs, PCDs used in the calculations, fitted parameters and measures of the goodness of fit. A different version of the report is available in the Sensitivity to storm duration option.
9.6 Testing
9.6.1 Choice of test sites
IBIDEM has been tested on a range of catchments and a variety of FSU hydrographs. Design hydrographs on gauged catchments will usually be based on hydrograph width analysis, with the characteristic hydrograph derived by the non-parametric method of Chapter 3. At ungauged sites, the characteristic hydrograph is likely to be based on the UPO-ERR-Gamma model of Section 4.4 or the parabolic curves method of Section 8.7.
The five test catchments (see Map 9.1) are:
Station 16009 Suir at Caher Park – a large rural catchment (1602 km2); Station 19001 Owenboy at Ballea – a small to medium-sized rural catchment (103 km2); Station 06026 Lagan-Glyde at Aclint – a medium-sized rural catchment (144 km2); An ungauged site on a medium to large-sized rural catchment (443 km2) on the Anner (a tributary of the Suir which it joins at Clonmel); An ungauged site on a small urbanised catchment (8 km2) on a tributary of the Tolka at Finglas.
9.6.2 Estimation of FSU hydrograph shapes
For the three test sites with flow data, the HWA software was applied to derive the characteristic hydrograph by the non-parametric method of Chapter 3.
At the two ungauged sites, the characteristic hydrograph was constructed using a version of the Chapter 6 procedure, i.e. estimating parameters of the UPO-ERR-Gamma model from PCDs. The alternative method of constructing the upper hydrograph by the parabolic curves method was also tested. Some details are reported in Table 9.1.
[Editorial note: Testing of IBIDEM was undertaken before HWA recommendations were finalised, and before it was possible to estimate BFI at ungauged sites using the BFIsoil model 108
of Volume IV. It was therefore necessary to borrow BFI values: from Station 15001 for the Anner ungauged site and from Station 08005 for the Tolka tributary. The differences in technique are not thought to compromise the integrity of the testing of IBIDEM.]
Map 9.1: Location of test catchments
Table 9.1: Some details of the applications to two ungauged test catchments Variable Unit River Anner Tributary of Tolka at Finglas Physical catchment descriptors BFI (see editorial note above) – 0.51 0.52 FARL – 0.999 1.000 ALLUV – 0.047 0.000 ARTDRAIN – 0.000 0.014 S1085 m/km 3.4 16.1 Parameters of the UPO-ERR-Gamma model n – 7.35 7.23
Tr hours 9.68 12.62 C hours 30.80 32.77 Width descriptors for use of the parabolic curves method
W75 hours 4.18 4.90
W50 hours 7.03 7.57 s (eccentricity parameter) – 0.40 0.40
109
9.6.3 Estimation of peak flows
For the test sites with flow data, QMED was estimated as the median of the annual maximum flows. At the two ungauged test sites, QMED was estimated from PCDs. [Editorial note: Testing of IBIDEM was undertaken before the Volume II model for estimating QMED from PCDs was finalised. Indeed, feedback from IBIDEM testing on the tributary of the Tolka led to changes in the urban adjustment model finally recommended for QMED estimation.]
Design flows for other return periods were estimated by applying a flood growth curve based on the FSR regional growth curve for Ireland. Due adjustment was made – by dividing the T- year flood growth factor by the 2-year flood growth factor – for the different index variable used in the FSR method. The design flows are shown in Table 9.2. It should be noted that applications of IBIDEM will generally use design flows estimated by Volume II procedures.
Table 9.2: Design flows (m3s-1) for the five test catchments Return period (years) Catchment 2 5 50 100 200 16009 Suir at Caher Park 162 204 303 334 366 19001 Owenboy at Ballea 15.4 19.4 28.8 31.7 34.7 06026 Lagan-Glyde at Aclint 12.9 16.3 24.2 26.6 29.2 Anner at Clonmel 61.5 77.6 115 127 139 Tributary of Tolka at Finglas 1.62 2.05 3.03 3.34 3.66
9.6.4 Rainfall depth-duration frequency tables
IBIDEM requires a table of design rainfall values for a typical point in the catchment. The user will obtain this through the FSU Web Portal. For the purpose of testing, tables of values based on the Volume I rainfall frequency procedure were transferred to IBIDEM in CSV files supplied by Met Éireann. In each case (i.e. for each of the test catchments in turn) the table of design rainfall depths was supplied for a 2-km grid point close to the catchment centroid. Table 9.3 summarises the main input variables used in the IBIDEM testing.
Table 9.3: IBIDEM input variables for the test catchments AREA SAAR Characteristic Approx. centroid Catchment 2 URBEXT (km ) (mm) hydrograph Easting Northing Suir at 1602 1079 0.009 200000 146000 Caher Park Derived median Owenboy 106 1176 0.018 hydrograph of 162000 62000 at Ballea Chapter 3 Lagan-Glyde 144 1072 0.007 310000 224000 at Aclint Anner 443 986 0.003 224000 134000 at Clonmel UPO-ERR-Gamma model (Section 4.4 Tributary of 8 734 0.527 + pilot of Chapter 6) 321000 240000 Tolka at Finglas 110
The heavily urbanised nature of the tributary of the Tolka at Finglas is to be noted.
9.7 Results
The sections below discuss the hydrograph fit obtained with IBIDEM for each of the test catchments in turn. Although testing considered a range of return periods, the examples shown are for the 100-year flood. Results are summarised later in Table 9.4.
9.7.1 Suir at Caher Park
For Station 16009 Suir at Caher Park, the HWA used hydrographs from 54 flood events. For this catchment, the prescribed baseflow by the FSR rainfall-runoff method was 51.0 m3s-1. This was slightly lower than the minimum flow in the imported hydrograph. After IBIDEM provided a warning, the baseflow was reduced to 50 m3s-1 to overcome the difficulty.
The IBIDEM 100-year hydrograph is shown in Figure 9.14, based on the default fitting threshold of 50% of the peak flow. The FSU hydrograph has a rapid rising limb but a much slower recession limb. The FSR rainfall-runoff method results in only a limited range of hydrograph shapes, and cannot capture this feature of the FSU hydrograph. However, the RMSE criterion used by IBIDEM to optimise the fit ensures that the widths of the two hydrographs are broadly similar, on average, for flows above the threshold.
The RMSE for this site is 12.7 hours: the largest for any of the test sites (see Table 9.4). This reflects the propensity of the catchment to produce flood hydrographs of long duration. Those interested in the Suir should also refer to Section 5.8. It is notable that the FSR hydrograph overestimates widths near the peak and underestimates widths at lower flows. The Nash-Sutcliffe efficiency (NSE) is 0.50, indicating a fairly good fit in terms of flow.
The fitted Tp is 52.8 hours and the SPR is 36.7%. These values seem reasonable for this large catchment of moderate permeability (BFI = 0.63).
IBIDEM hydrograph Derived median hydrograph RMSE = 12.7 hours
NSE = 0.50
Figure 9.14: Suir at Caher Park 100-year hydrograph fit
111
9.7.2 Owenboy at Ballea
For Station 19001 Owenboy at Ballea, the HWA used hydrographs from 35 flood events. The IBIDEM 100-year hydrograph is shown in Figure 9.15, based on the default fitting threshold of 50% of the peak flow. As for the Suir at Caher Park, the FSU hydrograph is more skewed than the FSR one, in that it rises rapidly and falls more slowly.
The RMSE for this site is 6.1 hours. The Nash-Sutcliffe efficiency is -0.48. The poor performance in terms of NSE arises because the statistic is calculated as an average over the duration of the FSU hydrograph, i.e. from about 10 hours before the peak to 55 hours after. The statistic could be improved by entering a higher value for baseflow (BF) or by raising the threshold used in fitting.
The fitted Tp is 31.1 hours and the SPR is 29.9%. These values seem reasonable for this moderate-sized catchment of moderate permeability (BFI = 0.64), although the Tp is perhaps rather long, as can be seen from the hydrograph plot. However, a shorter Tp would give a narrower hydrograph and hence a poorer fit in terms of hydrograph widths.
IBIDEM Derived median RMSE = 6.1 hours hydrograph hydrograph NSE = -0.48
Figure 9.15: Owenboy at Ballea 100-year hydrograph fit
9.7.3 Lagan-Glyde at Aclint
Station 06026 Lagan-Glyde at Aclint used hydrographs from 31 flood events and provides an example of a case where basic use of HWA yields a derived median hydrograph with “time reversals”. One of those on the rising limb is at quite a high level, between 70 and 75% of the peak flow (see Figure 9.16). Optionally, these can be removed using interactive features within the HWA software.
The fitting method used in IBIDEM can in fact cope with such time reversals, as shown in the left-hand plot of Figure 9.17. However, users may be reluctant to present a hydrograph in which time appears to run backwards! The anomaly can be avoided by editing the FSU hydrograph within the HWA software or en route to IBIDEM.
112
Time reversal
Figure 9.16: Lagan-Glyde at Aclint – derived median hydrograph from HWA software
IBIDEM Derived median IBIDEM DMH edited to hydrograph hydrograph hydrograph remove time reversal
Figure 9.17: Lagan-Glyde at Aclint – 100-year hydrograph fits
Incompleteness of the imported hydrograph prompts IBIDEM to issue a warning where appropriate. A typical message is: “Threshold flow is less than 100 yr input hydrograph start/end values, please adjust”. Many users would instinctively raise the fitting threshold from the default value of 50% to 75%. However, as shown in the right-hand plot of Figure 9.17, it is permissible to adopt an intermediate threshold at which the hydrograph width (in the imported FSU hydrograph) is defined on one limb of the hydrograph but not the other. This allows the fitting to exploit the width information on the receding limb between 70 and 75% of the peak flow.
IBIDEM gives a good fit to the rising and falling limbs of the FSU hydrograph, above the threshold value used for fitting. The RMSE is 11.6 hours and the Nash-Sutcliffe efficiency is 0.64, indicating a moderately good fit in terms of flow.
The fitted Tp is 97.3 hours. This is a surprisingly long time-to-peak (of the unit hydrograph) given the modest size of the catchment (144 km2). The typically slow flood response may in part reflect the influence of loughs in the catchment (FARL is 0.91).
The fitted SPR is 48.8%. The BFI for this catchment is 0.66, indicating a moderately permeable catchment. The fitted SPR value is surprisingly high for such a catchment. The high value of SPR is partly explained by the long time-to-peak which tends to produce a subdued hydrograph with a relatively low peak. IBIDEM has increased SPR in compensation, in order to ensure that the FSU peak flow is matched. 113
9.7.4 Anner at Clonmel
Two sets of calculations were carried out for this ungauged catchment: one with the characteristic hydrograph constructed using the UPO-ERR-Gamma model and the other using the parabolic curves method. These are shown in the Figure 9.18, along with hydrographs fitted by IBIDEM.
The FSR hydrograph fitted by IBIDEM to the FSU hydrograph in UPO-ERR-Gamma form is shown in the left-hand side of Figure 9.18. The fit is seen to be fairly good, the main defect being that the FSR rainfall-runoff method cannot reproduce the sudden change of gradient on the falling limb when the Gamma curve is replaced by the exponential recession. The RMSE is 1.9 hours. The Nash-Sutcliffe efficiency is a very poor -0.70. This reflects the departure of the IBIDEM and UPO-ERR-Gamma hydrographs in the early and (especially) later part of the period over which the fitting is made. (As explained in Section 9.5.4, NSE is evaluated from vertical differences throughout the period for which the FSU hydrograph exceeds the fitting threshold.)
UPO-ERR-Gamma Parabolic curves IBIDEM hydrograph IBIDEM hydrograph hydrograph hydrograph
Figure 9.18: Anner at Clonmel 100-year hydrograph fit
IBIDEM provides a very good fit to the FSU hydrograph in parabolic form (see right-hand side of Figure 9.18), with RMSE = 1.0 hours and NSE = 0.85.
Interpretation of the Tp and SPR values fitted by IBIDEM is revealing for this catchment. The fitted Tp is 2.8 hours for the parabolic hydrograph and 7.1 hours for the Gamma hydrograph. These times to peak (particularly 2.8 hours) seem unreasonably short for this medium to large rural catchment of 443 km2. Fitted SPR values are 0.4% for the parabolic hydrograph and 2.3% for the Gamma hydrograph. IBIDEM helpfully warns that these SPR values are suspiciously low. An SPR of 0.4% is unreasonably low, because it implies virtually none of the storm rainfall typically becomes rapid response runoff. The very small value is probably due in part to the need to compensate for the excessively small value of Tp. IBIDEM has had to decrease SPR in order to ensure that the FSU peak flow is matched.
[Editorial note: Multipliers in the PCD-based models – for the hydrograph width parameter Tr and the descriptors W75 and W50 – supplied for testing were much too small due to the use of non-standard units for the mainstream slope descriptor S1085. A feature of the FSR rainfall-runoff method is that underestimation of Tp leads to underestimation of SPR also. It is helpful that IBIDEM helpfully warns the user when SPR values are suspiciously low. See also the editorial note in Section 9.7.5.]
114
9.7.5 Tributary to Tolka at Finglas
Two sets of calculations were carried out for this urbanised but ungauged catchment: one with the characteristic hydrograph constructed using the UPO-ERR-Gamma parametric model and the other using the width descriptors W75 and W50 and the parabolic method. These are shown in the Figure 9.18, along with hydrographs fitted by IBIDEM.
Because this small catchment is heavily urbanised, the urban catchment design package was applied within IBIDEM. This means that the 100-year return period rainfall was used to synthesise the 100-year flood hydrograph, and the 50% summer rainfall profile was adopted.
As on the River Anner, the fitted hydrograph matches the FSU shape fairly well for the UPO- ERR-Gamma hydrograph and very closely for the parabolic hydrograph. RMSE and NSE values can be seen in the final columns of Table 9.4, which summarises results for all five test catchments.
IBIDEM Parabolic IBIDEM UPO-ERR-Gamma hydrograph hydrograph hydrograph
Figure 9.19: Tributary of Tolka at Finglas 100-year hydrograph fit
Fitted Tp values are 9.3 hours for the UPO-ERR-Gamma hydrograph and 4.2 hours for the parabolic hydrograph. These values seem suspiciously long for a small heavily urbanised catchment.
For reasons explained in Section 9.3.1, there is interaction between the Tp and SPR parameters that IBIDEM obtains when fitting the FSR rainfall-runoff method to the design flood hydrograph that the user has derived by FSU methods. IBIDEM always respects the peak flow of the imported FSU hydrograph. Whilst the parabolic curves method appears preferable to the UPO-ERR-Gamma model in terms of the Tp values resulting for this catchment, this preference is reversed when the SPR values are considered (see final columns of Table 9.4). The fitted SPR for the parabolic hydrograph is 6.3% which is unusually low.
[Editorial note: As discussed in the editorial note in Section 9.7.4, the developer was supplied with incorrect multipliers in the models for the hydrograph width parameter Tr and the descriptors W75 and W50. Because the PCD-based models had grossly underestimated hydrograph width at both sites used in testing (i.e. the Anner and the Tolka tributary), the IBIDEM developers understandably cast around for a possible explanation. Indeed they expressed surprise that none of the regression models for Tr, W75 or W50 includes a term that reflects catchment size. Such PCDs were considered in the generalisation research of
115
Chapter 6 but found useful only once: AREA appears in the regression model for parameter C of the UPO-ERR-Gamma model in the case where BFI is unavailable (see Table 6.7).
A characteristic feature of flood response times in Ireland and Great Britain is that catchment size often plays a much smaller role than the hydraulically-minded expect. Although mainstream length (MSL) appeared in the FSSR16 model for Tp(0), it did so only to a modest exponent of 0.23. This means that the estimated value of Tp(0) doubles only if the mainstream is 20 times longer. Catchment size is a very poor guide to flood response time!
It is reiterated that the poor results obtained in IBIDEM testing on the Anner and the Tolka tributary arose largely because the developers had been supplied with PCD models for Tr, W75 and W50 that had faulty multipliers.]
Table 9.4: Summary of IBIDEM results for five test catchments (100-year flood case) * * Anner Tolka Variable Suir Owenboy Lagan Gamma Parabolic Gamma Parabolic Rainfall return 140 140 140 140 140 100 100 period (years) BF (m3s-1) 50 3.69 4.56 12.85 12.85 0.13 0.13 Tp (hours) 52.8 31.1 97.3 7.1 2.8 9.3 4.2 SPR 36.7 29.9 48.8 2.3 0.4 25.3 6.3 PR 46.8 40.2 60.7 10.9 6.2 38.1 22.1 Time-step 1 1 1 1 0.25 1 0.25 (hours) Storm duration 111 69 203 15 5.75 17 7.25 (hours) Rainfall depth 122.2 126.9 149.4 102.1 70.8 85.2 64.8 (mm) Peak flow 334.0 31.7 26.6 126.9 126.9 3.3 3.3 (m3s-1) Volume (mm of catcht 84.6 69.9 141.3 15.0 5.9 24.4 10.7 runoff) RMSE (hours) 12.68 6.09 11.58 1.88 1.02 1.69 0.48 NSE 0.50 -0.48 0.64 -0.70 0.85 0.03 0.88 Editorial note: *Test results for the Anner and Tolka catchments are in error due to incorrect multipliers in the PCD-based models originally supplied for Tr, W75 and W50
9.7.6 Illustration of effect of fitting threshold
Tests were carried out using a wide range of thresholds. The effect of varying the threshold is illustrated for the ungauged site on the River Anner, for an FSU hydrograph constructed using the UPO-ERR-Gamma model of hydrograph width. The fitting threshold is marked on each plot in Table 9.5, and a commentary provided.
116
[Editorial note: The option to vary the fitting threshold is an exceedingly valuable feature of IBIDEM. It allows the user to focus on the hydrograph features most relevant to their application. In some cases, only the upper hydrograph may be relevant; in others, the entire hydrograph is important. It is important that the user inspects and interprets values of Tp and SPR thoroughly. Otherwise, a more hydrologically-informed analyst may later judge the choice of fitting threshold to have been made as a matter of convenience.]
9.7.7 Summary
The tests led to improvements in IBIDEM and to timely feedback to other parts of the FSU research. IBIDEM gave broadly sensible results on the three gauged catchments. Tp and SPR values were within expected ranges on the Suir and Owenboy. The inferred Tp was longer than expected for the Lagan-Glyde, with SPR taking on a high value in compensation.
As noted above, results for the two ungauged catchments were undermined by the supply of PCD-based models for Tr, W75 and W50 with incorrect multipliers.
Even when the correct models are used for estimating the characteristic hydrograph from PCDs, users will find cases where the Tp and SPR values – inferred when IBIDEM fits the FSR rainfall-runoff method to the imported FSU hydrograph – are unrealistic with perceived properties of the catchment. There is no simple recipe to deal with such cases. It is largely a matter of experience.
It is further emphasised that the FSU recommendation is to base flood estimates wherever possible on data transfers rather than on PCDs alone. Implementation of the FSU research stresses the importance of choosing a pivotal catchment so that estimates at ungauged sites gain from knowledge of flood behaviour at gauged sites. Section 8.10 discusses the selection and use of the pivotal catchment in the context of estimating the characteristic hydrograph at an ungauged site.
9.8 Additional opportunities provided by IBIDEM
9.8.1 Strengths and limitations
IBIDEM provides a way of assessing FSU outputs using a structured model of hydrograph formation. The test results above illustrate that the software can help in detecting hydrograph shapes or peak flows that appear to be inconsistent with properties of a catchment.
A limitation is that IBIDEM relies on the assumptions made in the FSR design event approach. These may not always be appropriate. The design event method derives a flood hydrograph from a single combination of inputs (rainfall depth, rainfall duration, rainfall profile and catchment wetness index). This combination does not always result in a hydrograph peak of the required return period. Thus, the design event used by IBIDEM may not always be relevant to the supplied FSU design hydrograph. The implication for users is that, while IBIDEM can provide a useful diagnostic test of the FSU design hydrograph, the method is not a cure-all.
117
Table 9.5: Sensitivity to fitting threshold (ungauged site on River Anner)
Threshold Performance used in Hydrograph plot Comment statistics fitting
Threshold
RMSE = 0.31 hr 90% Good fit to widths for small segment above of peak flow NSE = -1.29 threshold
Threshold
RMSE = 0.19 hr 70% Very good fit to widths above threshold (hence of peak flow NSE = -1.42 small RMSE)
RMSE = 1.88 hr Larger RMSE indicates Threshold inferior fit to widths but 50% smaller NSE indicates NSE = -0.70 of peak flow better fit in terms of flow
RMSE = 5.80 hr Continuation of trends 30% noted above ( when of peak flow 50% of peak flow used Threshold NSE = -0.50 as threshold)
118
9.8.2 Urban adjustment to design hydrographs
The options described in Section 9.4 provide additional functionality. A potential further application of IBIDEM is in estimating improved design hydrographs on urban catchments.
One possibility is to use FSU methods to derive the design flood hydrograph as if the catchment were entirely rural, by omitting the urban adjustment to QMED. The hydrograph would then be imported to IBIDEM, with URBEXT set to zero. After carrying out the fitting, the user would select the option to test sensitivity to urbanisation, and enter the correct value of URBEXT. The adjusted FSR hydrograph obtained is this way provides an alternative allowance for urbanisation. A potential advantage of this approach is that it adjusts the hydrograph widths for urbanisation, not just the peak flow. This is possible because the FSR rainfall-runoff method incorporates an urban adjustment to both Tp and PR.
Figure 9.20 illustrates the outcome for the heavily urbanised ungauged tributary of the Tolka at Finglas considered in Section 9.7.5. The adjusted hydrograph has a peak flow over four times larger than the original one, and the time to peak is much shorter. The increase in peak flow is caused by two effects: the increase in percentage runoff and the decrease in time to peak (and consequently storm duration). The increase in peak flow thus synthesised is much greater than that provided by the Volume II urban adjustment to QMED. [Editorial note: Details of this example may have been compromised by the supply of PCD-based models for Tr, W75 and W50 with incorrect multipliers.]
Imported FSU hydrograph Calculated FSR hydrograph Adjusted FSR hydrograph
Figure 9.20: Urban adjustment to hydrograph for Tolka tributary at Finglas test site
9.8.3 Supplying input hydrographs to river models
IBIDEM also has potential in deriving inflows for hydrodynamic river models, which often require hydrographs resulting from design storms with durations different from the critical
119
duration of the particular subcatchment. These hydrographs can be generated using the option to test sensitivity to storm duration.
Hydrographs generated by IBIDEM are likely to be useful for flood storage and flood routing studies because they cover the full range of flows. Although the non-parametric method of Chapter 3 is strongly recommended for use at gauged sites, the characteristic hydrograph it produces does not extend all the way down to zero flow. IBIDEM provides a structured alternative to sketching in the lower part of the hydrograph by hand.
9.8.4 Allowances for projected land-use change
IBIDEM provides a route to assessing the effect of future changes in urbanisation or other land use by varying the parameters of the rainfall-runoff method. The relevant options have been discussed in Sections 9.4.3 and 9.4.4.
120
Acknowledgements
The hydrograph width analysis was undertaken at the Department of Engineering Hydrology in the National University of Ireland Galway, principally by Kieran O’Connor and Monomoy Goswami. Samiran Das undertook some testing of the methods.
IBIDEM was developed by JBA Consulting: principally by Zoë Whiteman and Duncan Faulkner.
The flood event analysis summarised in 0 was undertaken at University College Cork.
The help of organisations and individuals who gather and curate hydrograph data is gratefully acknowledged. Peter Newport of OPW is especially thanked for supplying extensive data.
Volume III was edited by Duncan Reed of DWRconsult.
References
Archer, D., Foster, M., Faulkner, D. and Mawdsley, H. 2000. The synthesis of design flood hydrographs. Proc. Flooding: risks and reactions. CIWEM/ICE Conference, London, October 5, 2000, 45–57.
Bhattarai, K.P. and O’Connor, K.M., 2004. The effects over time of an arterial drainage scheme on the rainfall-runoff transformation in the Brosna catchment. Physics and Chem. of the Earth, 29: 787–794.
Brace, N., Kemp, R. and Snelgar, R., 2003. SPSS for psychologists: a guide to data analysis using SPSS for Windows. Lawrence Erlbaum Associates.
Brune, A. 2007. Comparison of the hydrological behaviour of various Irish catchments. M.Eng.Sc. thesis, Dept of Civil and Environmental Engg University College Cork, 200pp.
Duan, Q. 2003. Global optimization for watershed model calibration. In: Calibration of watershed models (eds Duan, Q, Gupta, H.V., Sorooshian, S., Rousseau, A.N. and Turcotte, R.), Water Science and Application 6, American Geophysical Union, Washington, 89–104.
Elshorbagy, A., Simonovic, S.P. and Panu, U.S. 2000. Performance evaluation of artificial neural networks for runoff prediction. J. Hydrol. Engg, 5: 424–427.
IH, 1979. Short cut to unit hydrograph convolution. Flood Studies Supplementary Report 9, Institute of Hydrology, Wallingford, UK, 6pp.
IH, 1985. The FSR rainfall-runoff model parameter estimation equations updated. Flood Studies Supplementary Report 16, Institute of Hydrology, Wallingford, UK, 5pp.
Goldberg, D.E. 1989. Genetic Algorithms in search, optimization and machine learning. Addison-Wesley, 432pp.
Hayashi T., Nagamine, Y. and Nishida, A., 1986. A vibration model to describe the lactation curve of a dairy cow. Japanese J of Zootechnical Science, 57: 471–478.
Holder, R.L. 1985. Multiple regression in hydrology. Institute of Hydrology, Wallingford, UK, 147pp.
121
Holland, J.H. 1975. Adaptation in natural and artificial systems. Ann Arbor: University of Michigan Press.
Houghton-Carr, H. 1999. Restatement and application of the Flood Studies Report rainfall- runoff method. Volume 4, Flood Estimation Handbook, CEH Wallingford, 288pp.
Kalinin, G.P. and Milyukov, P.I. 1957. O raschete neustanovivshegosya dvizheniya vody v otkrytykh ruslakh (On the computation of unsteady flow in open channels). Met. i Gydrologia Zhurnal, 10: 10–18.
Martin, J., O’Kane, J.P. and Javan, M. 2000. Computer modelling for flood alleviation in the lower Feale catchment. Proc. Water in the Celtic world: managing resources for the 21st century, Aberystwyth, 3-7 July 2000, BHS Occasional Paper No. 11, 163–172.
Nash, J.E. 1957. The form of the instantaneous unit hydrograph. In: Proc. Toronto General Assembly, IASH Publ. No. 45, 114–121.
Nash, J.E. and Sutcliffe, J.V., 1970. River flow forecasting through conceptual models, Part 1: A discussion of principles. J. Hydrol., 10: 282–290.
NERC 1975. Flood Studies Report (5 volumes). Natural Environment Research Council, London, UK.
Pallant, H., 2001. SPSS Survival Manual. A step by step guide to data analysis using SPSS for Windows. Open University Press, Buckingham, the UK.
Reed, D.W. and Marshall, D.C.W. 1999. Defining a design hydrograph. Chapter 10, Volume 3, Flood Estimation Handbook, CEH Wallingford: 59–62.
Tabachnick, B. G. and Fidell, L. S. 2001. Using multivariate statistics. 4th ed., Allyn and Bacon, Needham Heights, Massachusetts, 966pp.
Wang, Q.J., 1991. The Genetic Algorithm and its application to calibrating conceptual rainfall-runoff models. Water Resources Research, 27(9), 2467-2471.
122
Appendices
Appendix A Gauges used in Hydrograph Width Analysis
Table A.1: Stations used in Hydrograph Width Analysis Station Station River Name Area (km2) River Basin District grade 06011 Fane Moyles Mill A1 229.2 Eastern 06012 Fane Clarebane A1 162.80 Eastern 06013 Dee Charleville A1 309.1 Eastern 06014 Glyde Tallanstown A1 270.4 Eastern 06026 Lagan (Glyde) Aclint A1 148.4 Eastern 07001 Tremblestown Tremblestown A2 151.3 Eastern 07002 Deel Killyon A2 285.0 Eastern 07004 Kells Blackwater Stramatt A2 245.7 Eastern 07006 Moynalty Fyanstown A2 177.5 Eastern 07007 Boyne Aqueduct A1 441.2 Eastern 07009 Boyne Navan Weir A1 1658.2 Eastern 07010 Blackwater Liscartan A1 699.7 Eastern 07011 Kells Blackwater O’Daly’s Br A2 281.7 Eastern 07012 Boyne Slane Castle A1 2460.3 Eastern 07033 Kells Blackwater Virginia Hatchy A2 124.9 Eastern 09001 Ryewater Leixlip A1 209.6 Eastern 11001 Owenavorragh Boleany B 155.1 South-Eastern 14004 Figile Clonbulloge A1 268.9 South-Eastern 14006 Barrow Pass Bridge A1 1063.6 South-Eastern 14007 Stradbally Derrybrock A1 118.6 South-Eastern 14009 Cushina Cushina A2 68.4 South-Eastern 14011 Slate Rathangan A1 162.3 South-Eastern 14018 Barrow Royal Oak A1 2419.4 South-Eastern 15001 Kings Annamult A2 444.3 South-Eastern 15002 Nore John’s Br A2 1644.1 South-Eastern 15003 Dinin Dinin Br A2 299.2 South-Eastern 15005 Erkina Durrow Foot Br B 379.4 South-Eastern 15006 Nore Brownbarn A2 2418.3 South-Eastern 16001 Drish Athlummon A2 135.1 South-Eastern 16002 Suir Beakstown A2 485.7 South-Eastern 16003 Clodiagh Rathkennan A2 243.2 South-Eastern 16004 Suir Thurles A2 228.7 South-Eastern 16005 Multeen Aughnagross A2 84.0 South-Eastern
123
Station Station River Name Area (km2) River Basin District grade 16008 Suir New Bridge A2 1090.3 South-Eastern 16009 Suir Caher Park A2 1582.7 South-Eastern 18004 Awbeg Ballynamona A2 310.3 Southern 18005 Funshion Downing Br A2 378.5 Southern 19001 Owenboy Ballea A2 103.3 Southern 22071 L. Leane Tomies Pier A2 557.7 Southern 23001 Galey Inch Br A2 191.7 Mid-Western 23002 Feale Listowel A1 646.8 Mid-Western 23012 Lee (Kerry) Ballymullen A2 61.6 Mid-Western 24001 Maigue Croom A2 770.2 Mid-Western 24008 Maigue Castleroberts A2 806.0 Mid-Western 24013 Deel Rathkeale A1 438.8 Mid-Western 24082 Maigue Islandmore A2 762.8 Mid-Western 25001 Mulkear Annacotty A2 647.6 Shannon 25003 Mulkear Abington A1 399.1 Shannon 25005 Dead Sunville A2 192.6 Shannon 25006 Brosna Ferbane A1 1162.8 Shannon 25014 Silver Millbrook A1 164.4 Shannon 25016 Clodiagh Rahan A2 275.2 Shannon 25017 Shannon Banagher A1 7980.4 Shannon 25025 Ballyfinboy Ballyhooney A1 161.2 Shannon 25027 Ollatrim Gourdeen A1 118.9 Shannon 25029 Nenagh Clarianna A2 292.7 Shannon 25030 Graney Scarrif A1 280.0 Shannon 26002 Suck Rookwood A2 641.5 Shannon 26005 Suck Derrycahill A2 1085.4 Shannon 26007 Suck Bellagill A1 1207.2 Shannon 26008 Rinn Johnston’s Br A1 280.3 Shannon 26009 Black Bellantra Br A2 98.2 Shannon 26012 Boyle Tinacarra A1 519.9 Shannon 26019 Camlin Mullagh A1 253.0 Shannon 26021 Inny Ballymahon A2 1098.8 Shannon 26022 Fallan Kilmore A2 61.9 Shannon 27001 Claureen Inch Br A2 46.7 Mid-Western 27002 Fergus Ballycorey A1 564.3 Mid-Western 29001 Raford Rath-gorgin A1 115.5 Western 29004 Clarinbridge Clarinbridge A2 121.4 Western 29011 Dunkellin Kilcolgan A1 354.1 Western 124
Station Station River Name Area (km2) River Basin District grade 30004 Clare Corrofin A1 699.2 Western 30005 Robe Foxhill A1 237.8 Western 30007 Clare Ballygaddy A2 469.9 Western 30061 Corrib Estuary Wolfe Tone Br A2 3136.1 Western 34001 Moy Rahans A2 1974.8 Western 34009 Owengrave Curraghbonaun A2 117.1 Western 34018 Castlebar Turlough A1 95.4 Western 35001 Owenmore Ballynacarrow A2 299.4 North-Western 35002 Owenbeg Billa Br A2 88.8 North-Western 35005 Ballysadare Ballysadare A2 639.7 North-Western 35071 L. Melvin Lareen A2 247.2 North-Western 36010 Annalee Butlers Br A1 771.7 North-Western 36011 Erne Bellahillan B 320.5 North-Western 36015 Finn Anlore A1 153.1 North-Western 36019 Erne Belturbet A2 1491.8 North-Western 36021 Yellow Kiltybarden A2 23.4 North-Western 36027 Woodford Bellaheady A2 333.8 North-Western 39009 Fern O/L Aghawoney A2 1974.8 North-Western
Table A.2: Details of the flow data used (see also Table 7.2) Period of Flow data studied % of 15-min # of AM QMED Station arterial drainage data missing events (m3s-1) From To works 06011 01/10/1972 01/02/2001 1.6 29 15.45 06012 01/10/1972 11/01/2004 0.7 32 12.5 06013 29/10/1975 21/12/2004 1.5 30 27.75 06014 23/10/1975 22/10/2002 2.2 28 21.06 1950-57 06026 01/01/1972 01/02/2001 1.6 29 12.33 1950-57 07001 21/05/1975 30/09/2001 3.7 26 19.95 1971-73 07002 01/10/1970 01/09/2004 17.4 30 18.88 07004 26/10/1982 30/09/2005 0.9 23 22.56 07006 05/11/1956 01/10/2005 9.8 46 19.8 07007 09/04/1979 05/04/2004 2.1 25 35.41 1973-78 07009 05/11/1976 03/10/2005 0.8 30 144.91 07010 08/12/1986 21/05/2003 2.7 17 69.61 1982-86 07011 21/12/1983 30/09/1998 2.9 15 31.95 1980-82 07012 01/10/1986 01/01/2006 0.5 20 261.05 1969-86 07033 10/01/1980 01/10/2005 0.9 26 13.32 09001 15/10/1956 01/01/2006 5.8 50 33.77 125
Period of Flow data studied % of 15-min # of AM QMED Station arterial drainage data missing events (m3s-1) From To works 11001 07/03/1972 17/11/2005 4.2 34 45.7 14004 01/01/1972 01/07/2001 0.1 29 21.7 14006 01/01/1972 02/01/2006 0.4 34 78.34 14007 01/02/1980 01/10/2001 1.5 22 15.21 14009 01/01/1980 01/11/1999 2.7 20 6.78 14011 27/09/2000 12/11/2004 3.0 5 11.08 14018 01/01/1972 10/10/2005 1.5 34 147.91 15001 01/01/1972 01/08/2005 1.9 33 84.51 15002 01/10/1965 24/08/2001 1.3 36 198.19 15003 01/01/1972 01/05/2005 3.2 33 145.27 15005 01/01/1972 21/03/2005 1.3 33 27.68 15006 01/01/1972 01/01/2006 0.6 34 294.38 16001 01/10/1972 28/04/2005 7.2 33 15.41 16002 01/10/1954 21/11/2001 1.4 48 53.14 16003 01/10/1954 21/11/2001 1.4 48 29.07 16004 01/10/1954 13/06/2000 1.5 46 21.03 16005 01/10/1954 30/09/2001 16.0 47 20.4 16008 01/10/1954 01/06/2005 4.3 51 92.02 16009 14/01/1940 30/09/2004 0.5 64 163.63 18004 01/01/1972 23/04/2005 1.1 33 31.02 18005 01/01/1972 29/07/2002 0.2 30 54.83 19001 01/10/1972 01/01/2005 3.7 33 15.47 22071 01/10/1973 30/09/2004 5.2 31 105.20 23001 01/01/1960 20/12/2004 2.8 45 105.04 23002 01/10/1959 06/01/2006 7.6 46 371.84 1951-59 23012 04/04/1974 31/12/1991 4.6 18 15.88 24001 21/10/1976 01/01/2006 0.9 30 107.84 1975-76 24008 28/11/1973 01/01/2006 6.2 33 117.24 24013 01/10/1972 23/04/2003 4.9 31 108.99 24082 26/10/1977 21/02/2001 6.2 25 138.93 1975-76 25001 20/10/1977 01/01/2004 7.5 27 125.27 25003 03/04/1995 01/01/2002 4.2 7 66.71 25005 01/10/1972 01/04/1999 65.1 11 29.17 25006 05/09/1953 26/09/2005 1.7 52 81.04 1948-53 25014 01/10/1972 19/09/2005 3.2 33 16.52 1948-56 25016 01/10/1951 29/05/2005 3.1 54 24.31 1948-53 25017 02/01/1989 20/03/2003 2.1 14 481.44
126
Period of Flow data studied % of 15-min # of AM QMED Station arterial drainage data missing events (m3s-1) From To works 25025 01/10/1972 17/08/2003 6.9 31 9.85 25027 01/01/1972 15/01/2002 2.9 30 23.5 1955-65 25029 01/01/1972 01/01/2005 9.7 33 52.81 25030 01/10/1972 15/10/2005 4.3 34 41.66 26002 01/10/1972 01/11/2005 7.0 34 55.14 26005 01/10/1954 01/01/2003 1.7 49 94.56 26007 01/10/1972 13/08/2003 2.6 31 85.95 26008 26/09/1979 08/08/2003 3.7 24 23.69 26009 01/10/1972 18/02/2002 0.4 30 13.02 26012 01/01/1991 22/01/2002 3.0 11 46.72 1982-92 26019 16/09/1953 21/01/2002 4.4 49 21.04 26021 01/10/1972 31/07/2003 73.0 10 64.84 26022 01/01/1972 22/01/2002 10.1 30 6.35 27001 01/10/1972 23/04/2003 8.1 31 20.03 27002 03/05/1954 31/12/2005 1.8 52 32.22 29001 07/10/1957 01/01/2001 20.0 36 13.03 29004 10/07/1973 02/01/1986 1.8 13 11.24 29011 11/02/1983 06/01/2003 2.6 20 29.26 30004 01/10/1964 01/01/2005 9.9 41 89.83 1958-64 30005 01/10/1978 31/12/2004 10.3 26 37.89 1973-78 30007 20/11/1974 30/06/2005 3.6 31 64.13 30061 01/10/1950 12/02/2004 6.1 54 228.44 34001 01/10/1972 01/01/2006 1.8 37 174.27 1960-71 34009 01/01/1972 01/01/2000 4.5 28 28.05 34018 12/08/1976 27/04/2004 3.6 28 11.51 1960-71 35001 08/11/1955 01/10/1997 3.7 43 31.34 35002 19/01/1972 12/11/2002 3.1 31 52.88 35005 01/01/1946 30/09/2004 17.2 51 74.92 35071 04/12/1974 15/01/2003 4.0 29 26.69 36010 01/10/1972 01/11/1998 0.3 27 61.61 36011 01/10/1972 01/07/1998 2.1 26 18.32 36015 29/10/1956 01/02/2001 0.9 45 22.41 36019 19/12/1597 01/09/1998 0.3 41 88.75 36021 14/03/1978 01/12/1999 0.6 22 23.59 36027 08/08/1974 23/10/1992 0.5 19 25.12 39009 05/09/1972 09/01/1982 0.0 10 43.24
127
Appendix B Précis of UCC research on flood event analysis
University College Cork (UCC) analysed rainfall-runoff behaviour for a range of Irish catchments. The research was intended to remind FSU users that – whilst river floods in the temperate climate of Ireland generally arise from heavy rainfall – there are important differences between catchments in the conditions that give rise to flooding.
Some details of the 12 catchments studied are given in Table B.1. Nine of the catchments are listed in ascending order of size. The remaining three form a nested set on the Munster Blackwater. The catchments range in size from 15 km2 for the Dripsey at Coachford to 1605 km2 for the Nore at John’s Bridge. Mainstream slopes (S1085) range from 0.32 m km-1 for the Fergus at Ballycorey to 10.3 m km-1 for the Dripsey. Average annual rainfalls range from 913 mm for the Boyne at Trim to 1470 mm for the Dripsey.
[Editorial note: Brune (2007) expands the work by considering a further ten catchments, chiefly in the north-west and in the greater Dublin area.]
Table B.1: Stations subjected to rainfall-runoff analysis S1085 Sites from which AREA SAAR Station (m hourly rainfall data Comment (km2) (mm) km-1) taken Dripsey at 15 10.3 1470 Dripsey Upland Coachford† 19001 Owenboy at 106 3.79 1248 Cork Airport 100% rural Ballea 25014 Silver at 165 5.55 992 Birr Rural Millbrook 07002 Deel at 1975, post- 285 12.72 960 Mullingar Killyon drainage 06013 Dee at Dundalk + 307 2.37 1096 Charleville Weir Bailiborough 27002 Fergus at Karst + 562 0.32 1252 Shannon Airport Ballycorey minor lakes 16008 Suir at 1120 0.96 1030 Bansha* + Dundalk* Newbridge 07005 Boyne at 1975, post- 1282 0.43 913 Mullingar + Navan Trim drainage 15002 Nore at Kilkenny + Coon + 1605 0.85 979 Up to 2002 John’s Bridge Oakpark 18050 Blackwater Nested 245 3.9 1456 Millstreet + 32** at Duarrigle (upper) 18048 Blackwater Millstreet + Nested 881 2.7 1356 at Dromcummer Freemount + 32** (middle) 18006 Blackwater Millstreet + Free + Nested 1186 2.1 1303 at Mallow Mallow + 32** (lower) † UCC research catchment, subcatchment of Station 19028 Dripsey at Dripsey * Daily data only ** Special network of 32 raingauges in Blackwater catchment, monitored by UCC for 2005-2006
128
Volume IV presents physical catchment descriptors (PCDs) developed for general use in the FSU. UCC attempted a more detailed description of the catchments. The overall goal was to illustrate rather than codifying flood event analysis in Ireland.
Twelve flood events were analysed for each of the 12 catchments. The times series data analysed are river flows and rainfall depths drawn from several years of record. The main flood event analysis undertaken was based on the unit hydrograph method but did not consider volumetric aspects of the rainfall-runoff response to any great degree. The main output was discussion of the catchment-average unit hydrographs (UHs) derived for the 12 catchments. The UHs shown in Figure B.1 have been standardised by dividing by catchment area to facilitate inter-catchment comparisons.
Figure B.1: Catchment-average unit hydrographs standardised by area
The 12 catchments were tentatively allocated to five catchment types:
Type 1 The lowest UH peak magnitude and longest UH duration response was for the Fergus at Ballycorey, with an area-normalised UH peak of ≈1.510-3 m3s-1 mm-1 km-2 and a UH timebase (i.e. the duration of flood response to a unit net rainfall) of ≈12 days; Type 2 The second lowest UH peak magnitude (≈710-3 m3s-1 mm-1 km-2) and a timebase of ≈36 to 60 hours, as in the four rivers: the Deel, the Suir, the Dee and the Boyne; Type 3 The third lowest UH peak magnitude (≈1510-3 m3s-1 mm-1 km-2) and a timebase of ≈12 to 30 hours, as in the four rivers: the Dripsey, the Owenboy, the Silver and the Blackwater at Mallow; Type 4 The fourth lowest UH peak magnitude (≈2010-3 m3s-1 mm-1 km-2) and a timebase of ≈24 hours, as in the two rivers: the Nore and the Blackwater at Dromcummer; Type 5 The highest UH peak magnitude (≈2710-3 m3s-1 mm-1 km-2) and a timebase of ~18 hours, as in the Blackwater at Duarrigle.
Some consideration was also given to artificial neural network (ANN) analysis, and to the potential application of ANN models in flood warning.
129
Appendix C Performance of HWA methods on verification events
Figure C.1: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 06011 130
Figure C.2: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 06012
131
Figure C.3: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 06013
132
Figure C.4: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 06014
133
Figure C.5: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 06026
134
Figure C.6: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 07007
135
Figure C.7: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 07009
136
Figure C.8: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 07010
137
Figure C.9: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 07012
138
Figure C.10: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 09001
139
Figure C.11: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 14004
140
Figure C.12: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 14006
141
Figure C.13: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 14007
142
Figure C.14: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 14011
143
Figure C.15: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 14018
144
Figure C.16: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 15005
145
Figure C.17: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 23002
146
Figure C.18: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 24013
147
Figure C.19: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 25003
148
Figure C.20: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 25006
149
Figure C.21: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 25014
150
Figure C.22: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 25017
151
Figure C.23: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 25025
152
Figure C.24: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 25027
153
Figure C.25: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 25030
154
Figure C.26: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 26007
155
Figure C.27: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 26008
156
Figure C.28: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 26012
157
Figure C.29: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 26019
158
Figure C.30: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 27002
159
Figure C.31: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 29001
160
Figure C.32: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 29011
161
Figure C.33: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 30004
162
Figure C.34: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 30005
163
Figure C.35: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 34018
164
Figure C.36: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 36010
165
Figure C.37: Verification of median hydrograph + UPO-ERR-Gamma methods at Station 36015
166
Appendix D HWA results and their estimates from PCDs
Table D.1: Hydrograph width analysis results for all 89 stations
Estimates in the right-hand columns derive from PCD regression models that do not use BFI, i.e. from the models reported in Table 6.7
From HWA From PCD models of Table 6.7 (no BFI) Width descriptors UPO-ERR-Gamma params Width descriptors UPO-ERR-Gamma params Station # River Station Name W75 (h) W50 (h) n Tr (h) C(h) W75 (h) W50 (h) n Tr (h) C(h) 06011 Fane Moyles Mill 114.35 1.30 29.69 360.83 131.72 2.67 99.81 220.11 06012 Fane Clarebane 136.25 243.02 2.46 93.31 280.95 143.58 275.96 2.35 92.91 270.81 06013 Dee Charleville 41.99 70.87 7.78 59.99 102.86 20.32 37.60 5.63 24.52 53.83 06014 Glyde Tallanstown 95.87 154.30 3.01 89.91 113.85 36.71 69.03 4.21 35.31 88.45 06026 Lagan (Glyde) Aclint 82.47 146.16 5.00 102.18 140.97 28.95 53.96 4.43 29.03 67.84 07001 Tremblestown Tremblestown 18.36 34.20 5.01 23.31 38.36 24.88 45.71 5.44 34.50 48.64 07002 Deel Killyon 46.74 93.14 2.77 32.74 151.96 41.90 82.28 4.14 42.49 92.89 07004 Kells Blackwater Stramatt 110.14 171.87 3.01 101.98 116.78 134.85 274.14 2.38 88.18 274.11 07006 Moynalty Fyanstown 15.42 30.38 7.78 20.87 58.15 18.91 33.47 6.51 27.49 36.21 07007 Boyne Aqueduct 26.22 52.53 3.75 26.84 69.88 20.42 40.55 5.86 27.13 56.14 07009 Boyne Navan Weir 27.75 54.14 5.28 33.04 95.53 26.51 52.30 5.41 34.31 91.07 07010 Blackwater Liscartan 30.42 119.06 3.46 25.78 222.32 47.19 89.45 4.08 48.25 119.06 07011 Kells Blackwater O’Daly’s Br 108.21 170.42 2.99 92.96 116.78 123.25 246.43 2.57 85.30 238.80 07012 Boyne Slane Castle 27.75 49.10 6.11 36.41 88.20 33.89 65.87 4.98 40.20 116.29 07033 Kells Blackwater Virginia Hatchy 46.41 84.10 6.11 60.82 105.05 65.21 118.23 3.93 66.92 81.00 09001 Ryewater Leixlip 11.21 22.64 8.40 21.75 22.97 28.54 52.78 5.56 44.47 50.93 11001 Owenavorragh Boleany 7.54 10.84 26.36 25.24 5.38 18.13 34.51 6.52 36.26 35.12 14004 Figile Clonbulloge 48.17 74.35 5.41 66.48 53.75 53.65 100.57 5.04 96.62 65.75 14006 Barrow Pass Bridge 57.46 83.02 4.71 75.83 41.29 27.45 50.07 5.91 49.03 69.84 14007 Stradbally Derrybrock 9.34 17.14 21.41 31.79 11.24 32.10 59.83 5.06 32.68 52.78 14009 Cushina Cushina 26.27 46.20 3.88 27.75 46.42 32.62 59.38 5.84 53.95 35.04 14011 Slate Rathangan 45.96 89.57 5.28 54.73 116.78 61.11 121.30 4.21 86.32 82.24 14018 Barrow Royal Oak 65.08 121.78 2.94 57.73 116.78 46.55 88.89 5.66 68.27 93.39
167
From HWA From PCD models of Table 6.7 (no BFI) Width descriptors UPO-ERR-Gamma params Width descriptors UPO-ERR-Gamma params Station # River Station Name W75 (h) W50 (h) n Tr (h) C(h) W75 (h) W50 (h) n Tr (h) C(h) 15001 Kings Annamult 15.74 28.49 9.42 30.46 23.70 20.33 35.66 6.87 33.68 42.10 15002 Nore John’s Br 8.12 19.23 17.64 24.97 21.50 31.62 58.18 6.91 60.23 58.77 15003 Dinin Dinin Br 5.77 8.79 10.54 12.42 6.11 14.41 25.68 8.36 41.24 27.00 15005 Erkina Durrow Foot Br 60.06 114.00 5.27 83.23 106.52 36.07 65.31 5.58 54.18 59.22 15006 Nore Brownbarn 19.95 35.64 13.83 53.25 22.97 31.07 56.89 6.73 56.72 68.11 16001 Drish Athlummon 30.61 46.41 5.55 40.16 39.83 28.69 52.40 7.11 69.12 29.22 16002 Suir Beakstown 58.04 102.32 8.34 91.27 114.58 31.72 57.04 6.47 58.76 48.16 16003 Clodiagh Rathkennan 39.53 77.32 2.53 16.15 188.60 18.62 32.01 7.03 27.76 34.51 16004 Suir Thurles 63.79 102.80 6.39 90.24 99.19 28.94 51.57 6.44 49.91 39.89 16005 Multeen Aughnagross 13.76 21.02 28.19 51.75 5.38 10.77 18.20 9.93 26.89 14.46 16008 Suir New Bridge 88.08 146.96 3.15 44.16 410.67 31.21 55.98 6.74 52.23 55.12 16009 Suir Caher Park 39.50 92.10 6.11 49.99 163.69 30.29 54.38 6.92 52.13 57.84 18004 Awbeg Ballynamona 38.18 73.27 5.00 43.89 105.05 23.77 41.61 8.21 41.88 28.22 18005 Funshion Downing Br 15.63 26.35 24.60 52.75 15.64 27.52 47.41 6.73 48.80 41.96 19001 Owenboy Ballea 26.34 44.94 2.79 12.68 112.38 32.67 52.61 6.50 32.46 31.84 22071 L. Leane Tomies Pier 96.29 190.22 1.88 50.56 243.57 103.53 202.15 2.51 67.99 286.23 23001 Galey Inch Br 6.98 11.84 12.54 16.65 5.38 12.30 21.01 9.13 21.59 20.64 23002 Feale Listowel 5.81 9.27 30.27 23.00 3.18 15.08 25.04 10.05 37.07 24.21 23012 Lee (Kerry) Ballymullen 11.67 18.29 18.23 32.13 11.24 14.00 21.82 8.57 24.43 17.01 24001 Maigue Croom 18.24 27.88 4.44 19.16 35.43 20.86 37.55 6.39 22.81 55.25 24008 Maigue Castleroberts 19.08 30.54 3.88 19.40 28.10 21.15 38.13 6.28 23.27 57.71 24013 Deel Rathkeale 19.66 28.77 12.33 48.75 6.85 21.06 37.01 6.75 27.70 43.26 24082 Maigue Islandmore 18.66 27.51 5.14 21.42 38.36 20.76 37.37 6.39 22.70 55.06 25001 Mulkear Annacotty 15.51 29.51 9.45 31.94 35.43 9.48 16.65 10.11 19.85 23.86 25003 Mulkear Abington 15.38 29.57 4.30 16.51 50.09 10.65 18.62 8.96 18.87 25.76 25005 Dead Sunville 14.00 33.17 5.00 13.14 69.88 16.17 28.73 7.11 20.71 31.80 25006 Brosna Ferbane 35.86 63.91 5.28 42.42 94.06 33.85 66.59 4.64 28.77 109.20 25014 Silver Millbrook 17.63 30.05 8.97 30.83 36.16 10.69 19.40 7.27 16.99 29.60 25016 Clodiagh Rahan 20.26 36.19 3.88 18.40 62.55 16.72 31.01 5.68 22.13 52.82 25017 Shannon Banagher 231.91 1.50 109.75 374.76 285.21 2.35 142.58 719.78
168
From HWA From PCD models of Table 6.7 (no BFI) Width descriptors UPO-ERR-Gamma params Width descriptors UPO-ERR-Gamma params Station # River Station Name W75 (h) W50 (h) n Tr (h) C(h) W75 (h) W50 (h) n Tr (h) C(h) 25025 Ballyfinboy Ballyhooney 85.33 187.22 4.14 111.29 222.32 41.39 74.47 4.98 64.07 58.50 25027 Ollatrim Gourdeen 7.66 14.25 19.41 23.42 19.31 18.66 33.56 6.35 24.45 34.51 25029 Nenagh Clarianna 15.87 29.50 8.89 28.70 50.09 18.27 32.82 6.36 24.67 43.40 25030 Graney Scarrif 70.22 122.12 4.22 81.86 94.06 27.49 52.55 5.74 54.81 46.96 26002 Suck Rookwood 118.67 163.98 6.58 186.50 39.83 84.62 142.63 5.47 87.48 69.54 26005 Suck Derrycahill 136.04 209.83 3.60 148.51 84.53 86.15 147.63 5.46 93.33 80.02 26007 Suck Bellagill 156.97 257.02 3.04 159.81 107.99 88.32 151.33 5.44 94.60 83.12 26008 Rinn Johnston’s Br 106.69 206.81 3.88 124.50 177.61 110.70 200.38 3.61 80.01 115.85 26009 Black Bellantra Br 26.52 38.47 8.47 49.00 14.18 58.62 98.28 4.76 62.44 53.74 26012 Boyle Tinacarra 177.27 285.94 3.85 195.50 257.50 187.00 366.65 2.98 122.83 202.74 26019 Camlin Mullagh 72.81 114.01 3.36 71.39 70.61 64.94 109.55 5.85 72.53 48.30 26021 Inny Ballymahon 52.68 179.79 3.48 24.80 527.93 185.77 397.77 2.47 108.40 384.16 26022 Fallan Kilmore 41.75 74.82 5.00 53.64 66.94 75.77 129.73 4.18 79.86 64.81 27001 Claureen Inch Br 14.87 20.26 5.00 17.72 11.24 10.02 16.57 11.91 33.86 9.06 27002 Fergus Ballycorey 270.72 493.96 5.28 429.25 187.87 153.55 316.40 2.59 139.23 292.53 29001 Raford Rath-gorgin 49.94 86.19 5.33 74.75 46.42 29.52 48.72 7.48 51.37 25.59 29004 Clarinbridge Clarinbridge 76.87 130.05 2.77 79.50 76.47 58.11 99.97 4.70 77.98 60.86 29011 Dunkellin Kilcolgan 138.65 242.46 4.01 182.75 114.58 50.90 88.36 5.18 66.23 65.87 30004 Clare Corrofin 44.33 68.49 5.30 59.44 46.42 49.13 84.48 5.33 46.89 75.02 30005 Robe Foxhill 39.87 56.46 7.40 77.00 22.97 44.37 74.76 5.67 43.71 50.26 30007 Clare Ballygaddy 38.52 59.31 6.11 67.75 20.77 53.24 89.71 5.35 50.11 67.04 30061 Corrib Estuary Wolfe Tone Br 2.91 41.90 1.56 168.16 34001 Moy Rahans 95.52 2.79 58.35 829.75 103.33 3.52 64.53 197.49 34009 Owengrave Curraghbonaun 19.67 26.03 9.10 39.70 8.31 32.28 49.18 7.44 38.86 25.88 34018 Castlebar Turlough 1.27 35.58 187.87 2.43 66.80 196.27 35001 Owenmore Ballynacarrow 103.62 154.99 3.88 116.94 105.79 111.98 188.17 5.12 111.33 61.88 35002 Owenbeg Billa Br 7.62 11.39 17.77 24.01 10.56 15.38 13.72 23.90 35005 Ballysadare Ballysadare 38.18 85.88 5.00 36.80 187.87 69.53 116.80 5.34 68.86 68.27 35071 L. Melvin Lareen 123.67 210.68 4.22 99.46 456.84 28.70 44.00 7.54 42.51 30.62 36010 Annalee Butlers Br 75.15 155.52 3.04 73.10 152.69 107.09 196.64 3.30 76.90 183.22
169
From HWA From PCD models of Table 6.7 (no BFI) Width descriptors UPO-ERR-Gamma params Width descriptors UPO-ERR-Gamma params Station # River Station Name W75 (h) W50 (h) n Tr (h) C(h) W75 (h) W50 (h) n Tr (h) C(h) 36011 Erne Bellahillan 316.02 613.41 2.21 295.25 229.65 195.91 380.64 2.46 107.26 277.04 36015 Finn Anlore 40.08 67.19 6.19 68.00 35.43 54.95 90.55 4.92 51.43 57.30 36019 Erne Belturbet 295.51 491.91 2.48 254.50 302.20 234.64 468.71 2.20 109.37 540.31 36021 Yellow Kiltybarden 3.11 4.90 14.70 7.66 2.45 5.89 8.84 18.78 22.48 3.67 36027 Woodford Bellaheady 332.71 539.54 2.61 300.50 213.52 233.97 476.48 2.30 112.89 314.24 39009 Fern O/L Aghawoney 45.10 86.66 9.10 74.10 115.31 59.63 102.78 4.04 49.19 81.79
170
Appendix E Application of the HWA software
E1 Troubleshooting the installation
The installation process may fail because of a number of conflicts or inadequacies in the computer in which the HWA program is to be installed. If that happens, before exiting the installation process, the installation wizard will display a message informing the user on the cause of failure to install. The user may then take appropriate action to satisfy the requirements to successfully install the program. Some of the most likely causes of failure of the installation process and possible remedies are given below.
E1.1 Lack of storage space
The user’s computer may not have adequate disk space for installing all files required for running the program. This problem may be solved by creating enough space in the disk either by removing unnecessary or temporary files and folders, or by adding extra physical disk space by upgrading to higher spcifications. When enough disk space is available, the installation may be tried again.
E1.2 One or more files exist in the destination folder
The installation process may fail because of the existence of one or more files in the destination folder. This problem may be solved by emptying the destination folder, either by deleting the files, if not useful, or by moving the useful files to some other folder or folders. The installation process may then be tried again.
E1.3 An out-of-date system file exists in the computer
When the Version of a dll (Dynamic Link Library) file (Oleaut32.dll) is older than that required by the program for successful installation, the following message appears and the installation process fails:
Setup cannot continue because some system files are out of date on your system. Click OK if you would like Setup to update these files now. You will need to restart Windows before you can run Setup again. Click Cancel to exit Setup without updating system files.
When the user selects the OK button on the message box thus displayed, the Setup.Exe program installs a newer version of the required dll file, which is compatible with the installation program. In order to update the file to the correct version, the operating system must be restarted by rebooting the computer. After rebooting, the application, the Setup.Exe program is to be re-run to install.
E1.4 Out-of-date system files exist in the computer (multiple errors)
During installation, the installation program delays the replacement of the in-use system files, until rebooting takes place, by saving the new files as temporary files in the Temp folder. In order to replace the existing files with the .tmp files to complete the installation process, the
171
system uses a replacing and renaming operation. If something interferes with this operation, then the in-use system files are not updated. Hence, once the computer is rebooted and the installation program is restarted, the same error message appears and the installation process fails again. The two most common causes for this to happen are that the .tmp files are deleted or that the Temp folder is on a different drive or partition from the operating system. By default, the operating system is installed to either the Windows or the Winnt folder.
The following stops may resolve this problem: a) Copy the TEMP and TMP environment variables to a folder that is in the same drive partition as the Windows system files. To do this, open a command prompt window and type the following at the prompt:
Set TMP = C:\TEMP
Set TEMP = C:\TEMP
This will save the TEMP and TMP environment variables to a folder named Temp that resides on the C: drive. The folder must exist prior to carrying out these steps. Once these environment variables are set, the application should then install. b) If the Autoexec.bat file contains the following line (or similar):
If exists c:\temp\*.tmp del c:\temp\*.tmp
this is to be commented out by placing REM in front of it. c) Disable any antivirus software (or other memory resident programs) and try running Setup again. Often the best way to accomplish this is to run Setup in Safe Mode. It may also be necessary to copy all of the Setup files to a temporary folder on the hard drive disk and run Setup.exe from there. d) Left-over files from a failed Setup attempt can also cause this problem. If found, delete the msftqws.pdw subfolder and its contents from the Temp folder. Also look for Setup1.exe in the Windows or Winnt folder and any *.CAB files from previous installs, and delete them. This should be done after each failed install. e) Some logon scripts can cause this problem, so try to run Setup before logging on to the network if the computer is connected to a network.
E1.5 Existence of out-of-date system files in the computer (on non-upgraded operating systems)
If the operating system in the user’s machine is out of date, older versions of the system files may cause problems during installation of the program and the installation process may fail. The file that causes this problem is Oleaut32.dll. In this case, the same message, as given in Section E1.3, appears repeatedly during each successive installation attempt even after carrying out the modifications suggested in Sections E1.3 and E1.4.
172
In order to solve this problem, the user has to install the latest Versions of system files in the computer. Microsoft provides freely downloadable programs called ‘Service Packs’ for this purpose. These are available in the Download Centre (or the Support Section) of the Microsoft website (www.Microsoft.com). The user has to download the relevant Service Pack and install it in his/her computer before installing the HWA program. The Service Packs replace the older Versions of the system files by the latest (updated) ones, and add additional components which were not previously available for proper functioning of the system. It is expected that after successful installation of Service Packs and hence upgrading of the operating system, the HWA program can be installed properly.
E1.6 The program icon is not created and the program is not listed under the Program menu
During the last phase of installation of the HWA program, the Installation Wizard may issue a message saying, An error occurred trying to create a program icon for “HWA”. This message box displays Abort, Retry, and Ignore buttons. The Abort button terminates the installation process, the Retry button does not help in resolving the problem, and repeats the message after each click, while the Ignore button finishes the installation process, and the message saying the HWA Setup was completed successfully appears. But, while searching for the HWA Program in the Program list under the Start task bar item, the program may not be found. This problem may be caused by some error in the required System files. In order to find a solution for this problem, the user may locate the HWA.EXE application file in the folder where the program was installed in the user’s hard-disk (usually, by default, C:\Program Files\HWA). Double clicking the .Exe file in the My Computer pane opens the HWA program and the program is then expected to work properly. The user may create a Shortcut to the application file and place the shortcut on the Desktop for quick access to the program.
E2 Graphical User Interface (GUI) concepts of HWA software
E2.1 Basic GUI components
In the Graphical User Interface (GUI) environment, the user interacts with the HWA Package using the keyboard, the mouse, the windows, the menus, the tool bar buttons and the command buttons. A window is usually a facility for entering input information, which is surrounded by a border and moveable around the screen. The main application window in the HWA Package is shown in Figure E.1.
As HWA is a Windows-based program, other Windows-based applications may also be operated at the same time without necessarily closing the HWA program. The main window of the HWA program can be minimised or maximised or used in its normal size, during working with other Windows applications. The mouse enables the user to point at and to select objects to work on, or to perform actions, by clicking the left-hand mouse button. The keyboard is used for entering input information for running the HWA program, to move between various objects, or to perform particular actions. Menus in the menu bar are used to perform certain actions which may be accessed by clicking with the mouse or pressing the relevant key on the keyboard. The buttons in the tool bar at the top of a window provide a visual representation of and quick access to the intended tasks, which the buttons are meant to
173
link. Whenever the mouse pointer is placed above a button in the tool bar, a short description (ToolTip text), briefing the intended task to be performed by the button, is displayed. To invoke an effect on a button or a menu with the keyboard keys, identical to that produced by clicking a button or menu with a mouse pointer, the user has to hold down the ALT key and press the key for the letter (i.e. character) shown underlined (i.e. underscored) in the caption of the button or menu. The buttons captioned Continue or OK or Close in any window may be invoked by pressing the Return key on the keyboard, while the buttons captioned Back or Cancel may be activated by pressing the Esc (Escape) key. A message box – shown in the window in Figure E.1 – is used to instruct, inform or warn the user as appropriate.
Menu bar Toolbar Toolbar Top title Close button Menu item button Maximize button Minimize button
Message-box Main application window
Figure E.1: Main application window of the HWA program showing a message-box
The HWA GUI uses three types of windows. These are:
Main application window Data window Dialog window
Some of the important features of these windows are now described.
174
Main application window
The main application window is the one shown in Figure E.1. It appears as soon as the user starts running the program. This window is comprised of a Multiple Document Interface (MDI) form which includes a menu, a toolbar, a “child” window area, and a status bar. The title bar contains the title of the program as well as the maximise, minimise and close buttons. All other windows required for entering data or displaying outputs reside within this MDI window.
Data windows
Data windows are the windows containing either data-entry fields or displays of tabular and graphical outputs. The data window in Figure E.2 contains information labels, six data-entry boxes, three pairs of “radio” (i.e. option) buttons, one check box and three command buttons which are to be used to enter input information for running the HWA program.
The data window in Figure E.3 contains information labels, a list-box showing event numbers, a picture-box exhibiting the hydrograph of the flood event shown highlighted in the list box, one grid with a vertical scroll bar providing the ordinate number, date, time and discharge related to the displayed hydrograph in a tabular form, one pair of radio (option) buttons, two check boxes, 18 command buttons and one frame shown disabled. In the case of such a data-entry window, the cursor is automatically placed at the topmost data-entry field when the data window first appears. The user can directly enter data by using the keyboard and/or mouse. In order to specify data in other fields, the user navigates by using the (Up), (Down), (Left), and (Right) buttons, by pressing the Tab button on the keyboard, or by clicking the mouse after placing the cursor into that field where the data is to be entered.
Figure E.2: Data window for entering data
175
Figure E.3: Data windows for displaying graphical and tabular outputs
Dialog window
Dialog windows are used in HWA to either request information from the user or provide information to the user. Usually an ellipsis (…) after the caption of a menu or a button indicates that invoking the intended action results in a dialog window. A dialog window is also displayed when some command buttons, e.g. that captioned ‘Browse’, are clicked, or when a data-entry field that requires a filename is double-clicked. Figure E.4 shows a dialog window for specifying a filename for storing the results summary.
Figure E.4: A dialog window showing a standard windows file-opening dialog-box
176
E2.2 Typical GUI features of the windows displaying hydrographs
In the case of a data-display window containing a hydrograph, different GUI components are provided for effective display of relevant information. For illustration, some features of the window displaying flood hydrographs are described below. These features are common to other data-display windows as well.
i Displaying the hydrograph of an event from a list of selected events: In the window displaying the observed flood hydrographs, event numbers of all selected observed floods are provided in a list-box on the left hand side of the window. By default, one item in the list, usually the first one, is shown highlighted by a blue shade, and the corresponding hydrograph is displayed in a picture-box provided on the right hand side of the list-box. The user may scroll through the list using the vertical scroll bar provided alongside the list-box to choose any item. The list of events can also be navigated using the (Up) and (Down) keys. When the user clicks on an event number in the list box, that event number is highlighted and the existing hydrograph is replaced by that corresponding to the highlighted event number. ii Displaying hydrograph in different units: By default, time is displayed in hours on the time-axis, i.e. the abscissa, of the hydrograph. However, given that the OPW data files hold data in 15-minute time intervals, an option is provided to display time in 15- minute intervals as well. A pair of radio (option) buttons, one captioned Hour and the other captioned No. of 15 minute intervals (data interval in OPW data file) is provided for this purpose. The user can view the hydrograph in either of the two time units by clicking the respective radio button. iii Showing data values in a grid: By default, the tabular values are not displayed. However, a command button captioned Show tabular values of hydrograph is provided at the bottom on the right hand side of the window. When this button is pressed, a grid appears on the right hand side of the picture-box and the caption of the button changes to Hide tabular values of hydrograph. The values related to the displayed flood hydrograph are shown in that grid in tabular form, using rows and columns. By default, one of the rows, usually the top one, is shown highlighted by a yellow shade. When the user clicks on any other row, the highlighter moves to the row thus clicked. The (Down) and (Up) keys on the keyboard can also be used to move the highlighter from one row to the next, either downward or upward. The user may scroll through rows using the vertical scroll bar provided alongside the grid. iv Showing values on the hydrograph at any data point: When the user clicks on any point inside the picture-box, an indicator appears at that point as a grey-coloured vertical line. Note that for some graphical outputs in other windows, e.g. those displaying derived flood hydrographs, an indicator having a light green colour instead of grey, is displayed. The picture-box shown in Figure E.3 shows this line at the 62nd hour, i.e. the 248th time-step in the 15-minute interval, on the receding side of the hydrograph. A label having a light yellow background simultaneously appears at the top right of the window displaying relevant information for the time-step corresponding to the clicked data point, and a check-box provided on the left hand side of the label is shown checked. The user can hide the label containing the information by unchecking the check-box. The user can place the indicator in the picture-box by clicking at any point for which s/he wishes to see the value. Alternatively, the ‘’ or the ‘’ keys may be pressed to move the indicator by one
177
time-step at a time, either to the left or to the right, in order to show the values at the desired data point. If, as described in iii) above, the grid containing the values of the hydrograph is exhibited, then the highlighter on this grid automatically moves to the row containing the information related to the data-point, either clicked on the hydrograph, or reached by pressing the ‘’ or the ‘’ arrow buttons on the keyboard. When the user double-clicks on the picture-box, the indicator and the label disappears. v Changing the background colour: If the user wishes to change the background colour of the hydrograph, for example for the purpose of presentation and/or printing, the command button BackGround Colour… needs to be clicked. The standard Windows colour-palette displaying all colours, hues and shades, appears. The user may choose any colour and press the OK button provided on the palette. The graph is redrawn with the chosen colour on the background. vi Copying a graph onto the clipboard: By pressing the command button Copy Graph, a graph can be copied onto the clipboard. The Graph, thus copied, can then be pasted in other Windows application programs such as MS-Word or MS-Excel. vii Copying tabular values onto the clipboard: By pressing the command button Copy table, the table can be copied onto the clipboard. The values, thus copied, can then be pasted into other Windows application programs. In the case of MS-Excel, the copied values are pasted in separate rows and columns; the user can thus carry out any further analysis or displays using facilities of MS-Excel to augment those provided within the HWA program. viii Saving a graph in a picture file: The hydrograph displayed in the picture-box can be saved as a picture file in bitmap format for later use, e.g. for preparing a report, or for presentation. This is achieved by clicking the command button Save graph As… when a standard Windows file-saving dialog box appears. The user can use the controls on the dialog box, thus displayed, and specify a filename to save the hydrograph as a bitmap in the chosen folder. ix Printing a graph using a printer: The hydrograph displayed in the picture-box can be printed using a printer connected to the user’s computer by clicking on the command button Print graph…, when a standard Windows printing dialog box appears. The user can select the desired settings before printing the hydrograph.
In addition to the above-mentioned GUI components, the data-display window showing the hydrographs of the observed flood events supports other specific features. Some of these are evident from diagrams presented in the main body of the volume (e.g. Figure 5.18).
E2.3 Options on start-up of the HWA program
When the user clicks the Continue button on the start-up window, a message box appears as shown in Figure E.1. This message provides information about two options of entering input information into the program.
Under the first option, the user may enter all relevant input information into data-entry fields in a window by using keyboard and mouse. Each data-entry field is provided with an appropriate label to indicate the specific type of data that the field is supposed to receive from the user. This option is essentially required when the user applies the program to a gauged station for the first time. In the data-entry window, a data-entry field is provided in which the user can specify the name of a results summary file. The default filename extension of such a
178
file is *.sum. If no filename is specified, a default filename WP3_1Sys.sum is used to hold the results summary of the Hydrograph Width Analysis each time the program is run.
The second option can be used only if the HWA program was run at least once before with the data of the gauged station and the results summary had been saved in a user-specified file. Under this option, the user has to specify the name of the previously-saved results-summary file in a data-entry field. In order to do this, s/he can browse for the required file, either by clicking on the Browse button, or by double-clicking the data-entry field. The resulting window shows the list of directories and folders which can be searched for the required file, as in any other Windows-based program. Once the desired file is located, the user can either select the file by clicking on its name and press the OK button, or double-click on the filename. The window containing the list of directories and folders automatically closes and the filename appears in the data-entry field. When the user presses the Continue button, relevant information is extracted by the program from the specified results summary file and displayed in the data-entry window. Thus, the user is saved the effort of entering input information each time the program is re-run with data of a particular station. Of, course, the user can make changes and deletions in the data-entry fields after these are displayed in the window, and then run the program with the changed or modified set of information. The default system filename of WP3_1Sys.sum can be specified if the user wishes to resume the previous analysis but had omitted to save this in a named file.
E2.4 Menu bar items
As highlighted in Figure E.1, the main application window of the HWA program contains nine menu items on the menu bar. The menu and sub-menu captions provide a brief indication of the intended tasks. Although a mouse can be used for pointing to and clicking on a menu item, the keyboard keys can also be used for invoking a similar effect. In order to do this, the user has to press the key for the letter (i.e. character) underlined in the caption of the menu or sub-menu item when the user holds down the ALT key. Some initially disabled menu bar items are later enabled. For example, the menu items Results summary and Hydrographs of flood events are enabled once hydrographs of observed flood events have been extracted by the program from the record of discharge data. Similarly, the Derived hydrographs menu item is enabled once the derived semi-dimensionless flood hydrographs are produced by the program. A brief description of the menu bar items is provided below.
‘File’ menu
This menu is activated by clicking File on the menu bar or by keying ALT+F. A list of sub- menus appears having two enabled items (Set working directory and Open) and three disabled items.
The sub-menu Set working directory facilitates setting of the working directory in the case when the user wants to use data files from and store output files into a folder other than the one in which the HWA Package is installed. When clicked, this menu item brings up a standard dialog window, showing the list of drives, folders and files, which may be used to nominate/select the working directory before running a program.
The Open sub-menu has two further sub-menus (Existing file… and New document) which can be used for either opening an existing file or a new file in ASCII format and for carrying out editing, printing and saving operations as required. The Save As…, Close and Print…
179
sub-menus under the File menu are enabled once the user opens an existing file or a new document.
‘Analysis options’ menu
This menu is activated by clicking Analysis options on the menu bar or by keying ALT+A. Sub-menus Start from scratch and Use results file stored during an earlier run then appear.
By clicking the sub-menu Start from scratch, the user can enter relevant input information into data-entry fields in a window by using the keyboard and mouse (see Section E2.2). The sub-menu Use results file stored during an earlier run can only be used if the HWA program was run at least once before, and the data and results summary saved in a user-specified file.
‘Results summary’ menu
This menu is activated by clicking Results summary on the menu bar or by keying ALT+R. It is enabled after the HWA program extracts the hydrographs of observed flood events from the record and displays these in a data window. When the user clicks on this menu, a data window appears showing the summary of results in an ASCII formatted file. It may be noted that the current version of the HWA program retains all options and procedures developed and tested in the course of the hydrograph-width research. All outputs of the Hydrograph Width Analysis are stored in the results summary file, which can be quite large.
‘Hydrographs of flood events’ menu
This menu is activated by clicking Hydrographs of flood events on the menu bar or by keying ALT+G. It is enabled after the HWA program extracts the hydrographs of observed flood events from the record and displays these in a data window. If a data window other than that containing the observed flood hydrographs is shown active within the main application window, a click on this menu activates and brings to the front the window containing the flood hydrographs.
‘Derived hydrographs’ menu
This menu is activated by clicking Derived hydrographs on the menu bar or by keying ALT+D. It is enabled after the HWA program successfully produces the derived median and mean flood hydrographs as one of the outputs of the Hydrograph Width Analysis. If a data window other than that containing the derived flood hydrographs is shown active within the main application window, a click on this menu activates and brings to the front the window containing the derived flood hydrographs.
‘Modified Gamma hydrographs’ menu
This menu is activated by either clicking Modified Gamma hydrographs on the menu bar or by keying ALT+y. It is enabled after the HWA program successfully produces the modified Gamma hydrographs as one of the outputs of the Hydrograph Width Analysis. If a data window other than that containing the modified Gamma hydrographs is shown active within the main application window, a click on this menu activates and brings to the front the window containing the modified Gamma hydrographs.
180
‘Window’ menu
The ‘Window’ menu contains four sub-menus and a list of “child” windows which are open within the main application window. It can be activated by clicking the menu title or by keying ALT+W. The first sub-menu is New Window, which allows the user to open a new document window for creating a new ASCII formatted file, if required. Any number of such windows can be opened. The other sub-menu items are Cascade, Tile Horizontal and Tile Vertical, which may be used to arrange a number of opened windows on the screen by any of the three different styles as available in any standard Windows application. When one or more child windows are kept opened during a run of HWA, a list (by title) of the windows appears at the bottom of the Window menu. The user may bring up any of these child windows by clicking the relevant title.
‘Help’ menu
This menu option is disabled in the current version of the HWA software program.
‘Exit’ menu
In order to close the session, the user can click Exit in the menu bar or key ALT+x. The user is guided through the closure procedure.
E2.5 Toolbar buttons
As shown in Figure E.1, the main application window of the HWA program contains eleven buttons on the toolbar. ToolTips provide a brief indication of the intended tasks and appear when the mouse hovers on the relevant button. Some buttons are disabled during data entry operations. The toolbar functions are summarised in Table E.1.
E2.6 Closing the HWA program
The expected options are provided to close the HWA program in a number of different ways.
181
Table E.1: Details of toolbar buttons (Letters in red show the relevant keyboard shortcuts when depressed with the ALT key.) Menu or sub- menu item Icon ToolTip text Status performing the task Set working File: Set working Enabled throughout. directory directory Open new File: Open: Enabled throughout. document Existing file… File: Open: New Open existing file Enabled throughout. document Enabled at the beginning. Disabled when the option of analysing by entering data on the Start analysing Analysis options: screen is chosen by clicking this button. from scratch Start from scratch Enabled again when this option of analysing is closed. Enabled at the beginning. Disabled when the Use results file Use results file option of analysing by reading data from a stored during an stored during an pre-saved results summary file is chosen by
earlier run earlier run clicking this button. Enabled again when this option of analysing is closed. Becomes enabled once hydrographs of Results summary Results summary observed flood events are extracted and
displayed. Becomes enabled once hydrographs of Hydrographs of Hydrographs of observed flood events are extracted and flood events flood events displayed. Becomes enabled once semi-dimensionless Derived Derived derived hydrographs are produced and hydrographs hydrographs displayed. Disabled at the beginning. Becomes enabled Modified Gamma Modified Gamma after semi-dimensionless modified Gamma hydrographs hydrographs hydrographs are produced and displayed. Disabled throughout in the current version of Help Help the HWA software. Exit Exit Enabled throughout.
182
Appendix F Further details of IBIDEM
F1 Method of optimising Tp
Optimising Tp involves minimising the objective function:
2 Σ[WFSU(i) - WFSR(i)] for i = 1 to m.
With only one variable to adjust, the minimisation is straightforward. IBIDEM adopts a simple approach based on the method of bisection.
The starting point for the search is an initial guess for Tp. This is set as the time elapsing from the first ordinate of the input FSU hydrograph to its peak value. It is not necessary to start with a particularly good guess. The search range is then set as follows:
Minimum = 0.001 hours Maximum = 10 times initial guess
This range of extreme values is likely to far exceed the realistic range in which the best parameter value lies. However, with automated calculations, there is little penalty in adopting such a wide range as a precautionary measure.
At each iteration, IBIDEM considers five trial values of Tp denoted by Tp[1], Tp[2], Tp[3], Tp[4] and Tp[5]. For the first iteration, the five trial values are:
Tp[1] = 0.001 i.e. the minimum value which will be considered Tp[2] = (Tp[1] + Tp[3]) /2 i.e. half way between Tp[1] and Tp[3] Tp[3] = initial guess Tp[4] = (Tp[3] + Tp[5]) /2 i.e. half way between Tp[3] and Tp[5] Tp[5] = 10 x initial guess i.e. the maximum which will be considered
Each of the five trial values of Tp is used to create an FSR hydrograph shape. The value, Tp[n], that gives the best fit (i.e. the smallest value of the objective function) is used as the central value for Tp in the next iteration. Trial values for Tp are reassigned as follows:
Tp[1] = Tp[n-1] i.e. updates lower bound of range in which optimum lies Tp[2] = (Tp[1] + Tp[3]) /2 i.e. half way between Tp[1] and Tp[3] Tp[3] = Tp[n] i.e. best value from previous iteration Tp[4] = (Tp[3] + Tp[5]) /2 i.e. half way between Tp[3] and Tp[5] Tp[5] = Tp[n+1] i.e. updates upper bound of range in which optimum lies
The iterative procedure continues until the range within which the optimum value is known to lie becomes acceptably small. The criterion used is Tp[5] - Tp[1] < 0.01 hours. At that point, the value giving the best fit is adopted as the optimum value of Tp.
Consider, as an example, the case where the ultimate best-fitting value is Tp = 3.0 hours and the initial guess is Tp = 5.0 hours. Table F.1 shows the trial values used in the first five
183
iterations. The best-fitting trial value of Tp (shown in red) is adopted as the central value Tp[3] in the next iteration.
Table F.1: Example of iteration to find best-fitting value of Tp Iteration First Second Third Fourth Fifth Tp[1] 0.001 0.001 1.25 2.50 2.81 Tp[2] 2.50 1.25 1.87 2.81 2.96 Tp[3] 5.00 2.50 2.50 3.12 3.12 Tp[4] 27.5 3.75 3.12 3.43 3.27 Tp[5] 50.0 5.00 3.75 3.75 3.43
It is seen that, after five iterations, the Tp value (2.96 hours) is already close to the ultimate best-fitting value of 3.0 hours.
F2 Method of fitting SPR
Having determined Tp so as to fit the characteristic hydrograph as closely as possible, IBIDEM calculates SPR by matching the peak of the FSR response hydrograph, qpeak, to the FSU required response peak from the FSU hydrograph, q peak, which is found by subtracting BF from the peak FSU flow.
The calculation of SPR follows from the fact that the response hydrograph is directly proportional to the percentage runoff, PR. In the general case of a part-urbanised catchment,
PR = PRrural (1.0 – 0.47 URBEXT) + 70 (0.47 URBEXT) F.1 This applies the FSSR16 urban adjustment, but using the FSU index URBEXT rather than the FSR index URBAN. See Step 9 of Section 9.2.
Simplifying Equation F.1 gives:
PR = PRrural (1 – 0.47 URBEXT) + 32.9 URBEXT F.2
The term PRrural is composed of three parts:
PRrural = SPR + DPRCWI + DPRRAIN F.3
Here, SPR is standard percentage runoff, DPRCWI is dynamic percentage runoff attributable to catchment wetness and DPRRAIN is dynamic percentage runoff attributable to event rainfall. The latter two quantities depend solely on catchment descriptors (SAAR, which controls CWI) and the design rainfall depth, which is a function of storm duration, which can be determined from Tp.
It is possible to calculate the response hydrograph peak qpeak without carrying out a full convolution (see Houghton-Carr, 1999), by using the “short-cut” method described in FSSR9 (IH, 1979):
qpeak = RC.(PR/100).(P/D).AREA F.4
184
where:
RC is a routing coefficient which depends on D/Tp (a function of SAAR); P is the depth of the design storm (after application of ARF); D is the duration of the design storm.
FSSR9 presents a graph showing how RC varies with D/Tp based on the 75% winter rainfall profile. A similar graph is presented by Houghton-Carr (1999), which includes a line corresponding to the 50% summer rainfall profile.
Based on Equations F.2 to F.4, IBIDEM calculates the value of SPR that yields the required qpeak. It is found from:
100.D.q peak - 32.9 URBEXT F.5 SPR RC.AREA.P DPR - DPR 1 - 0.47 URBEXT CWI RAIN
IBIDEM evaluates RC from a digitised version of the graph in Figure 3.10 of Houghton-Carr (1999), with the y axis quantities divided by 10 to correct a mistake in the FEH. The fact that RC is found graphically introduces a slight uncertainty to the calculation of SPR, but this has been found to be small.
F3 Checks and validation of outputs
IBIDEM checks the outputs and provides error, warning or information messages when necessary. A list of the checks made is in Table F.2.
185
Table F.2: Checks and outputs Criterion Status Comment Notes This could happen, for example, when the peak of the input hydrograph is unrealistically low and the catchment is urbanised. After the urban component of the PR is calculated by IBIDEM, it is sometimes found necessary to use a negative SPR in order to match the input peak flow. A Inferred SPR is SPR ≤ 0 Error typical cause will be that the imported FSU peak not positive flow is too low, e.g. due to a failure to incorporate an adequate urban adjustment in QMED. Alternatively, the catchment may not be well represented by the structure of the FSR rainfall- runoff model or by the composition of the design event used as the input to the model. This may be valid if the catchment is highly Inferred SPR is permeable. But it could otherwise be caused by the 0
186