Modified Rational Unit Hydrograph Method and Applications

Water Management Proceedings of the Institution of Civil Engineers Modified rational unit hydrograph method and applications Dhakal, Fang, Thompson and Cleveland https://dx.doi.org/10.1680/wama.13.00032 Paper 1300032 Received 04/03/2013 Accepted 30/09/2013 Keywords: design methods & aids/hydrology & water resource/models (physical) ICE Publishing: All rights reserved Modified rational unit hydrograph method and applications 1 j Nirajan Dhakal PhD 3 j David B. Thompson PhD, PE, DWRE 2 Xing Fang PhD, PE, DWRE j 4 Theodore G. Cleveland PhD, PE j Post-Doctoral Fellow, Sustainability Solutions Initiative, University of Maine, Orono, ME, USA Professor, Department of Civil Engineering, Auburn University, Auburn, AL, USA 1 j 2 j Director of Engineering, R. O. Anderson Engineering, Inc., Minden, NV, USA Associate Professor, Department of Civil and Environmental Engineering, Texas Tech University, Lubbock, TX, USA 3 j 4 j The modified rational method (MRM) is an extension of the rational method to develop triangular and trapezoidal runoff hydrographs. A trapezoidal unit hydrograph (UH) was developed from the MRM for the case when the duration of rainfall is less than the time of concentration of the watershed and is called the modified rational unit hydrograph (MRUH). The MRUH method was applied to 1400 rainfall-runoff events at 80 watersheds in Texas. Application of the MRUH method involved three steps: (a) determination of rainfall excess using the runoff coefficient; (b) determination of the MRUH using drainage area and time of concentration; and (c) simulating event runoff hydrographs. The MRUH performed as well as the Gamma function UH, Clark-HEC-1 UH and NRCS curvilinear UH methods when the same rainfall loss model was used. The MRUH method can be applied to time-variable rainfall distributions and at watersheds with drainage areas greater than typically used for the rational method (a few hundred acres). Notation A AI C Clit Cvbc D Dw EF I i ie ¼ Ci L mo Q(t ) Qp QB  pm Q  po Q 2 2 drainage area in (ha, acres, or km , or mile ) cumulative area as a fraction of watershed area runoff coefficient composite literature-based runoff coefficient back-computed volumetric runoff coefficient storm duration (min or h) watershed equivalent diameter (km) Nash–Sutcliffe efficiency (dimensionless) average rainfall intensity (mm/h or in/h) with the duration equal to time of concentration gross rainfall intensity (mm/h or in/h) effective rainfall intensity (mm/h or in/h) main channel length (mile) dimensional correction factor (1.008 in imperial units, 1/360 ¼ 0.00278 in SI units) direct runoff hydrograph (DRH) ordinates derived by discrete convolution (m3 /s or ft3 /s) peak discharge of DRH (m3 /s or ft3 /s) relative error in observed and simulated DRH peak discharges mean of the modelled DRH peak discharges (subscript ‘m’ stands for modelled) QpD QpR QpUG QpUM QpUN QuG (t ) QuI (t ) QuM (t ) R2 RRMSE S So mean of the observed DRH peak discharges (subscript ‘o’ stands for observed) peak discharge of the modified rational method’s DRH for the case when D , Tc peak discharge of the rational method (m3 /s or ft3 /s) peak discharge of the Gamma unit hydrograph (GUH) (m3 /s or ft3 /s) peak discharge of the modified rational unit hydrograph (MRUH) (m3 /s or ft3 /s) peak discharge of the Natural Resources Conservation Service unit hydrograph NRCS UH (m3 /s or ft3 /s) GUH ordinates (m3 /s or ft3 /s) instantaneous unit hydrograph (IUH) ordinates (m3 /s or ft3 /s) MRUH ordinates (m3 /s or ft3 /s) coefficient of determination the root mean squared error of DRH ordinates normalised by observed Qp main channel slope (ft/mile) channel slope (m/m or ft/ft) for equations in Appendix 2 1 Water Management TB Tc TI Tp TpU TpUG TpUN W Æ 1. relative error in observed and simulated DRH times to peaks time of concentration (min or h) fraction of time of concentration time to peak of DRH (min or h) time to peak of UH (min or h) time to peak of the GUH (min or h) time to peak of the NRCS UH (min or h) watershed width (km) shape parameter of GUH Introduction The rational method was originally developed for estimating peak discharge QpR for sizing drainage structures, such as storm drains and culverts (Kuichling, 1889). The QpR (in m3 /s or ft3 /s) is computed using 1: QpR ¼ mo CIA where C is the runoff coefficient (dimensionless), I is the average rainfall intensity (mm/h or in/h) over a critical period of storm duration (i.e. time of concentration Tc ), A is the drainage area (hectares or acres), and mo is the dimensional correction factor (1/360 ¼ 0.00278 in SI units, 1.008 in Imperial units). Kuichling (1889) and Lloyd-Davies (1906) are credited with independent development of the rational method (Singh and Cruise, 1992). Incorporation of detention basins to mitigate effects of urbanisation on peak flows requires design methods to include the volume of runoff as well as the peak discharge (Rossmiller, 1980). Poertner (1974) developed the modified rational method (MRM) to use when designing hydraulic structures involving storage. The MRM approximates a direct runoff hydrograph (DRH) resulting from a design storm as being either triangular or trapezoidal in shape (Smith and Lee, 1984; Viessman and Lewis, 2003; Walesh, 1989) depending on the relation between the storm duration D and time of concentration Tc : Smith and Lee (1984) revisited the rational method that implied a rectangular response function, which is an instantaneous unit hydrograph (IUH), and developed DRHs using IUH for both constant and variable rainfall intensity events. Singh and Cruise (1992) analysed the rational formula using a systems approach and concluded that watershed’s IUH is a rectangular distribution with the base time equal to Tc of the watershed if a watershed can be represented as a linear, timeinvariant system. They used the convolution to derive the S-hydrograph and D-hour unit hydrograph (UH) from application of the rational method. Guo (2000, 2001) developed a rational hydrograph method (RHM) for continuous, time-variable rainfall events. Bennis and Crobeddu (2007) developed an improved RHM for small urban catchments using a rectangular impulse response function. However, with the exception of Smith and Lee (1984) and Bennis and Crobeddu (2007), all studies related to MRM consider MRM producing DRHs from constant rainfall distributions (Rossmiller, 1980; Viessman and Lewis, 2003). All 2 Modified rational unit hydrograph method and applications Dhakal, Fang, Thompson and Cleveland of the methods were developed and tested for small watersheds with limited data. Similarly, none of the studies has tested the sensitivity of the proposed methods to C and Tc. In this study, MRM was applied to develop a trapezoidal UH that is termed the modified rational unit hydrograph (MRUH). The purposes of the study were: (a) to evaluate the applicability of the method to watersheds of size greater than typically used with either the rational method or the MRM (that is, a few hundred acres); and (b) to study the effects of the runoff coefficient and the time of concentration on prediction of DRHs when the MRUH method is used. The MRUH method was used to compute DRHs for 1400 rainfall-runoff events at 80 watersheds in Texas, USA. The DRHs obtained from the MRUH were compared with those obtained from three other UH models: the Clark UH developed for the HEC–1 generalised basin (Clark, 1945; USACE, 1981), Gamma function UH for Texas watersheds (Pradhan, 2007), and Natural Resources Conservation Service (NRCS) curvilinear UH (NRCS, 1972). 2. Modified rational unit hydrograph First, let us revisit the MRM. If D ¼ Tc , the resulting DRH from the MRM is triangular with a peak discharge Qp ¼ QpR ¼ CIA at time t ¼ Tc ; that is case (a) in Figure 1. If D . Tc , the resulting DRH is trapezoidal with a constant maximum discharge Qp ¼ CIA from time D to Tc ; that is case (b) in Figure 1. The linear rising and falling limbs have a duration of Tc , as shown in Figure 1 (e.g. from Viessman and Lewis, 2003; Walesh, 1989). If D , Tc , then the resulting DRH is trapezoidal with a constant maximum discharge of QpD (Equation 2) from the end of the storm duration D to Tc as reported by Smith and Lee (1984) and Walesh (1989) 2: QpD ¼ CIA(D=T c ) ¼ QpR (D=T c ) Smith and Lee (1984) and Singh and Cruise (1992) noted that if the rate of change of the contributing area is constant so that the accumulated tributary area increases and decreases linearly and symmetrically with the time, then the IUH or impulse response function (Chow et al., 1988) QuI (t ) is of rectangular shape given by 3: QuI (t) ¼ dA A ¼ dt Tc (0 , t , T c ) Using the rectangular response function (Equation 3), Smith and Lee (1984) and Singh and Cruise (1992) derived the resulting DRH ordinates Q(t ) by convolution as ðt Q(t) ¼ ie (ô) QuI (t  ô) dô 4: 0 Water Management Modified rational unit hydrograph method and applications Dhakal, Fang, Thompson and Cleveland Using MRM’s DRH (case C in Figure 1) for a D-h rainfall event, the modified rational unit hydrograph or MRUH can be developed if DRH’s ordinates are divided by the effective rainfall depth (i.e. C I D) based on the UH derivation method (Viessman and Lewis, 2003). The MRUH is trapezoidal in shape with constant peak discharge QpUM ¼ QpD /(C I D) ¼ A/Tc from D to Tc : The time base for the MRUH is D + Tc and MRUH ordinates can be computed from Equation 5: Discharge: m3/s D ⫽ Tc QpR Tc Tc QuM (t) ¼ A t Tc D 0<t<D QuM (t) ¼ A Tc D < t < Tc QuM (t) ¼ A Tc þ D  t Tc D Tc < t < Tc þ D (a) Discharge: m3/s D 5: Tc The D-h MRUH results from a constant excess rainfall intensity of 1/D in/h over D h and has a peak discharge of A/Tc in ft3 /s when drainage area A is in acres and Tc is in hours for 1 in of rainfall excess (taking into account that 1 acre in/h is nearly equal to 1 ft3 /s). If SI units are used (drainage area A in ha and rainfall intensity in mm/h), the peak discharge from the MRUH should be equal to A/(360Tc ) in m3 /s for 1 mm of rainfall excess. Three examples of the MRUH developed for three watersheds used in this paper are shown in Figure 2. It is worth mentioning that cases (a), (b) and (c) of the MRM in Figure 1 are DRHs and none is UH, although cases (b) and (c) have the same shape as MRUH in Figure 2. Tc QpR (b) D Discharge: m3/s Tc D D QpD Time: t (c) Figure 1. The modified rational hydrographs or DRHs for three different cases: (a) D ¼ Tc , (b) D . Tc , and (c) D , Tc where ô is the time with respect to which the integration is carried out and ie (ô) ¼ Ci is the effective rainfall intensity with i as gross rainfall intensity. Two types of DRHs, triangular and trapezoidal shape (Figure 1), were obtained from Equation 4 for constant rainfall intensity, depending on the storm duration. The assumption and restriction for the application of the rational method and original MRM include constant rainfall intensity throughout the storm duration (Rossmiller, 1980) and for small catchments, that is drainage areas less than 0.8 km2 or 200 acres (TxDOT, 2002). Application of the MRUH method involves three steps as stated in the abstract. Because the MRUH method is an UH method, the approach establishes a continuity of hydrograph-development methods from very small watersheds to relatively large watersheds. The UH for a watershed can be used to predict the DRH for any given rainfall excess hyetograph (constant or time-variable rainfall distribution) using the UH discrete convolution (Chow et al., 1988; Viessman and Lewis, 2003). In summary, application of the MRUH method is straightforward and similar to the application of other UH methods using discrete convolution; the assumption and restriction for the MRM are no longer necessary, which will be demonstrated through this study. The MRUH method was first tested using rainfall-runoff data obtained for concrete surfaces from Yu and McNown (1964). The first dataset was based on a test bed with an area of 152.4 m by 0.3 m (500 ft by 1 ft), surface slope of 0.02 (dimensionless), and constant rainfall intensity. The second 3 0·006 0·04 For 152·4 m ⫻ 0·3 m (Yu and McNown 1964) 0 4·0 8·0 For 76·8 m ⫻ 0·3 m (Yu and McNown 1964) 0·03 Discharge: m3/s 0·005 Discharge: m3/s Modified rational unit hydrograph method and applications Dhakal, Fang, Thompson and Cleveland 0·004 0·003 0·002 Incremental rainfall Modelled DRH Observed DRH 0·02 0·01 Incremental rainfall: mm Water Management 0·001 0 0 0 0 1 2 3 Time:min (a) 4 5 4 8 6 0·0010 Incremental rainfall Modelled DRH Observed DRH 30 For Waller Creek, Austin, Texas 3 Discharge: m /s 25 20 Discharge: m3/s 0·0008 0 2·0 4·0 0·0006 0·0004 0·0002 15 0 10 0 5 0 0 0·2 0·4 0·6 0·8 1·0 Time:h (b) 1·2 1·4 1·6 Figure 2. The MRUHs developed for: (a) two laboratory settings from Yu and McNown (1964) and (b) for the watershed associated with USGS streamflow-gauging station 08157000 Waller Creek, Austin, Texas. Tc values used for MRUHs were computed using the Kirpich method (Equation 18) dataset was based on a test bed with an area of 76.8 m by 0.3 m (250 ft by 1 ft), surface slope of 0.005, and variable rainfall intensity. The Tc of about 5 min was computed using the Kirpich method (Kirpich, 1940) for both experiments. A trapezoidal 1 min MRUH was developed for each experiment (Figure 2(a)). The runoff coefficient was taken to be unity. For both cases, the modelled DRHs using MRUH match the observed DRHs well (Figure 3). The Nash–Sutcliffe efficiency EF (Equation 12) was 0.93 and 0.80 for the experiments using the constant (Figure 3(a)) and time-variable rainfall intensity (Figure 3(b)), respectively. According to Bennis and Crobeddu (2007), a good agreement between the simulated and the measured data is reached when EF is higher than 0.7 for hydrograph simulation; therefore, the large EF values given above indicated a good fit between modelled and observed DRHs for both experiments. 4 12 16 20 24 28 32 36 40 (a) 4 8 12 16 20 24 28 32 36 40 Time: min (b) Figure 3. Incremental rainfall hyetograph and observed and modelled DRHs using the MRUHs for the two laboratory tests on concrete surfaces: (a) 152.4 m 3 0.3 m with 2% slope and (b) 76.8 m 3 0.3 m with 0.5% slope reported by Yu and McNown (1964) 3. Applications of the MRUH method in Texas watersheds 3.1 Watersheds studied and rainfall-runoff database Watershed data taken from a larger dataset (Asquith et al., 2004) accumulated by researchers from the United States Geological Survey (USGS) Texas Water Science Center, Texas Tech University, University of Houston and Lamar University were used for this study. Location and geographic distribution of the stations are shown in Figure 4. The drainage areas of the 80 study watersheds ranged from approximately 0.8 to 65.0 km2 (0.3 to 25 mile2 ), with a median value of 15.8 km2 (6.1 mile2 ); 50 watersheds (62.5% of the 80 watersheds) have drainage areas less than 20 km2 (7.7 mile2 ). The stream slope of study watersheds ranged from 0.0026 to 0.0196 (dimensionless), with a median value of 0.0079. The main channel lengths estimated were approximately 2–80 km (1.2–49.7 miles). The percentage of impervious area (IMP) of the 80 study watersheds ranged from 0.0 to 74.0%, with a median value of 26.0%. About 40% of the watersheds were rural watersheds with IMP less than 5%, and about 29% of the watersheds were urbanised with IMP greater than 60%. Water Management Modified rational unit hydrograph method and applications Dhakal, Fang, Thompson and Cleveland N Texas Fort Worth Dallas Austin San Antonio 0 320 640 1280 km Explanation US Geological Survey streamflow-gauging stations (watershed) location Figure 4. Map showing the USGS streamflow-gauging stations (triangles) associated with the watershed locations in Texas, USA The rainfall-runoff dataset comprised about 1400 rainfall-runoff events recorded during 1959–1986. Event rainfall depths ranged from 3.56 mm (0.14 in) to 489.20 mm (19.26 in), with a median value of 57.66 mm (2.27 in). About 41 and 86% of the events had a storm depth less than 50.8 mm (2 in) and 101.6 mm (4 in), respectively. The base flow separations for observed runoff hydrographs were not done. This is because the majority of the gauging stations are on small ephemeral streams; base flow represents a small component of the total flow at the station. The streamflow for the watershed frequently was zero at the beginning of the storms (Asquith et al., 2004). 3.2 Time of concentration and runoff coefficients Time of concentration, Tc , and the runoff coefficient, C, are the required parameters for the MRUH method. The Tc values were estimated by Fang et al. (2008) using four empirical equations (see Appendix 2): (1) Williams equation (Williams, 1922); (2) Kirpich equation (Kirpich, 1940); (3) Johnstone and Cross equation (Johnstone and Cross, 1949); and (4) Haktanir and Sezen equation (Haktanir and Sezen, 1990). The excess rainfall or the net rainfall is obtained from the product of the incremental rainfall and C (the volumetric interpretation, Dhakal et al., 2012), similar to Smith and Lee (1984). Two estimates of C were examined for the application of the MRUH method. The first C is a watershed composite, literature-based coefficient (Clit ) derived from land-use information for the watershed and published values of Clit for appropriate land uses (Dhakal et al., 2012). The second C is a back-computed, volumetric runoff coefficient (Cvbc ) determined by preserving the runoff volume using observed rainfall and runoff data. Cvbc was estimated by the ratio of total runoff depth to total rainfall depth for an individual observed storm event. The determination and comparison of Clit and Cvbc for the study watersheds was documented by Dhakal et al. (2012). 3.3 DRHs derived using the MRUH method For the 80 Texas watersheds, observed rainfall hyetograph and runoff hydrograph data were tabulated using a time interval of 5 min. Therefore, a 5 min MRUH was developed for each of the 80 study watersheds. The 5 min MRUH duration was less than Tc for all study watersheds. The observed and simulated DRHs for the event on 8 July 1973 at the USGS streamflow-gauging station 08157000 Waller Creek, Austin, Texas are presented in Figure 5 as an illustrative example. The watershed drainage area was 5.72 km2 (2.21 mile2 ). The Cvbc was 0.29. The Tc values estimated using the Kirpich, Haktanir and Sezen, Johnstone and Cross, and Williams equations were 1.7, 2.2, 1.4 and 3.4 h, respectively. Peak discharges QpUM of the 5 min MRUH using 1 in (or 25.4 mm) rainfall excess for the watershed are 23.7, 18.3, 28.8 and 11.9 m3 /s using Tc values estimated from the Kirpich, Haktanir and Sezen, Johnstone and Cross, and Williams equations, respectively. Figure 2(b) shows an example MRUH for the watershed developed using Tc estimated from the Kirpich method (Equation 18); and the other three MRUHs developed from other Tc methods are trapezoids with different peaks and time bases (D + Tc ), but the area under each trapezoid is the same because MRUH is a UH. The duration of the rainfall event was 19 h. Three distinct rainfall episodes resulted in three distinct peaks. These were reasonably represented by the DRHs derived from the MRUH using Tc estimated by the Kirpich, Haktanir and Sezen, and Johnson and Cross equations. The DRH developed from the MRUH using the Williams equation appears to over-estimate Tc for the watershed, and discharge peaks of the DRH were then underestimated (Figure 5). When the MRUHs were developed using Tc values estimated from the Kirpich, Haktanir and Sezen, Johnstone and Cross, and Williams equations, the EF (Equation 14) values derived between observed DRH and modelled DRHs using the above corresponding MRUHs are 0.83, 0.86, 0.70 and 0.63, respectively. Simulated times to peak (Tp ) agree reasonably well with observed values (Figure 5) when using Tc estimated by Kirpich, Haktanir and Sezen, and Johnson and Cross equations for the MRUHs. However, using Tc estimated by the Williams equation for the MRUH resulted in the computed Tp exceeding the observed Tp : Different combinations of Tc and C were used for applications of the MRUH method to predict the DRHs and to determine the 5 Water Management Modified rational unit hydrograph method and applications Dhakal, Fang, Thompson and Cleveland 0 8 2 7 Modelled DRH using Haktanir–Sezen Tc equation Modelled DRH using Johnstone–Cross Tc equation 6 Modelled DRH using Williams Tc equation 4 Incremental rainfall 6 Observed DRH 8 Incremental rainfall: mm Discharge: m3/s Modelled DRH using Kirpich Tc equation 5 4 3 2 1 0 5 10 15 20 25 30 Time: h Figure 5. Incremental rainfall hyetograph for the event on 8 July 1973 and observed and modelled DRHs using the MRUHs with Tc estimated by four empirical equations for the watershed associated with the USGS streamflow-gauging station 08157000 Waller Creek, Austin, Texas sensitivity of the DRH peak discharges (Qp ) to different Tc and C values. Five combinations of Tc and C were used. (a) (b) (c) (d ) (e) Tc estimated Tc estimated Tc estimated Tc estimated Tc estimated using Haktanir and Sezen equation and Cvbc : using Johnstone and Cross equation and Cvbc : using Williams equation and Cvbc : using Kirpich equation and Cvbc : using Kirpich equation and Clit : Figure 6 is a plot of the observed and computed DRH peaks using Cvbc and Tc values calculated using the four different empirical equations. In comparison to observed Qp modelled Qp using Tc estimated from the Haktanir and Sezen, Johnstone and Cross and Kirpich equations not only graphically look alike (Figure 6), but also are similar with respect to three statistical parameters (Table 1): coefficient of determination R2 ; Nash– Sutcliffe efficiency EF; and relative error in peak QB (defined in Appendix 1). The results for EF using the Williams equation are inferior to the others. The fraction of modelled Qp results that are within 1/3 of a log-cycle from the 1: 1 line are summarised in Table 1 and ranged from 67.5% (Williams equation) to 88.7% (Johnstone and Cross equation) of total events. Fractions of storms with QB less than  50% (Cleveland et al., 2006) are listed in Table 1 for applications of the MRUH method with four combinations of Tc and C. Using Tc estimated from the Kirpich equation and Cvbc resulted in 75% of storms with QB less than  50%. Parameter Cvbc (back-computed from rainfall and runoff data) results in the preservation of event runoff volume, 6 and the Kirpich equation provides reliable estimations on watershed Tc values (Fang et al., 2008). Ideally, computed and observed peaks should plot precisely along the equal value line (black line in Figure 6). However, the UH is a mathematical model that is an incomplete description of the complexity of the combination of the rainfall-runoff process and runoff dynamics. Therefore, the relatively simple approach cannot fully capture the nuances of watershed dynamics and deviations from this ideal (the equal-value line) are expected. For example, Asquith and Roussel (2009) computed mean residual standard error about 1/3 of a log-cycle for annual peak discharges at 638 streamflow gauging stations in Texas. The observed Tp and computed Tp values of DRHs predicted using Cvbc and Tc values calculated using the four different empirical equations were compared using three error parameters R2 , EF and relative error in time to peak TB (Equation 16). Parameter Tc , estimated from the Haktanir and Sezen, Johnstone and Cross, and Kirpich equations, produces the similar values of the quantitative measures: R2 , EF, median value of TB and fraction of storms with TB less than  50% (Table 1). The Tp results using the Williams equation seem to be slightly inferior to the others with respect to median value of TB and percentage of storms within  1/3 of a log cycle (Table 1). In summary, for predicting Qp and Tp , use of Tc estimated from the Williams equation for the MRUH produces less accurate results than those computed using the Kirpich, Haktanir and Sezen, and Johnstone and Cross equations. Modelled DRH, Qp: m3/s Water Management Modified rational unit hydrograph method and applications Dhakal, Fang, Thompson and Cleveland 100 10 1 1:1 line ⫾1/3 of a log cycle (b) Modelled DRH, Qp: m3/s (a) 100 10 1 1 10 100 Observed DRH, Qp: m3/s (c) 1 10 100 Observed DRH, Qp: m3/s (d) Figure 6. Modelled plotted against observed DRH peak discharges Qp for 1400 rainfall-runoff events in 80 Texas watersheds. Modelled DRH peaks were developed using event Cvbc and MRUHs with Tc estimated using four different methods: (a) Haktanir and Sezen equation, (b) Johnstone and Cross equation, (c) Williams equation, and (d) Kirpich equation Simulated Qp results obtained from the MRUH method using the forward-computed (literature-based) runoff coefficient Clit are compared against the Qp results obtained using the backcomputed runoff coefficient Cvbc (Figure 7). For both the cases, Tc values were estimated using the Kirpich equation. For the peak discharges predicted using Clit , most of the values are above the equal value line (1: 1 line). Qp results computed using Cvbc are superior to those using Clit with respect to all statistical measures used to assess goodness of fit (Table 2). Use of Clit tends to generate estimates of Qp that exceed expected values (observations) when the Clit values are interpreted as volumetric coefficients. In contrast, there is no difference in five quantitative measures between the observed and predicted Tp values (Table 2), regardless of which runoff coefficient is used. Hence, the simulation results of Qp are more sensitive to the choice of C or rainfall loss model than to the choice of Tc : Furthermore, the Tp results are not related to C when the MRUH method was used and controlled by the time-variable rainfall distribution. A sensitivity analysis was performed to evaluate the sensitivity of the DRH derived from the MRUH method to Tc and C. A rainfall event on 7 May 1972 for the USGS streamflow-gauging station 08178600 Salado Creek, San Antonio (24.88 km2 or 9.61 mile2 ) was selected for the analysis. The Tc used for the MRUH was varied from 50 to +50% of Tc estimated from the Kirpich equation. Similarly, the C used for rainfall loss was varied from 50 to +50% of Cvbc : The EF computed between the observed DRH and modelled DRH derived from the MRUH method using Cvbc and Tc estimated from the Kirpich equation was 0.89. The changes in EF values as a result of the changes in Tc and C for the sensitivity analysis are presented in Table 3. The change in EF ranged from 0.01 to 0.22 for  50% change in Tc. Similarly, the change in EF ranged from 0.02 to 0.66 for  50% change in C. This analysis further supports the above conclusion that DRH value derived using the MRUH method are more sensitive to the choice of C than to the choice of Tc. 7 Water Management Modified rational unit hydrograph method and applications Dhakal, Fang, Thompson and Cleveland Using the Using the Haktanir Johnstone and and Sezen Cross equationy equation* Statistical parameters R2 for Qp EF for Qp Median value of QB Fraction of storms with 0.5 < QB < 0.5 Percentage of storms within  1/3 of a log cycle (Qp ) R2 for Tp EF for Tp Median value of TB Fraction of storms with 0.5 < TB < 0.5 Percentage of storms within  1/3 of a log cycle (Tp ) 0.75 0.66 0.19 0.70 82.4 0.75 0.74 0.00 0.72 82.1 0.80 0.79 0.00 0.72 88.7 0.72 0.71 0.05 0.73 80.5 Using the Williams equation{ Using the Kirpich equation} 0.75 0.48 0.41 0.60 67.5 0.74 0.74 0.10 0.65 78.2 0.80 0.73 0.10 0.75 88.6 0.73 0.72 0.01 0.72 82.3 * Tc computed using the Haktanir and Sezen equation ranged from 0.8 to 6.5 h in the study watersheds, with median and mean values of 2.6 and 2.9 h, respectively. y Tc computed using the Johnstone and Cross equation ranged from 0.7 to 5.0 h in the study watersheds, with median and mean values of 1.7 and 1.9 h, respectively. { Tc computed using the Williams equation ranged from 1.2 to 11.7 h in the study watersheds, with median and mean values of 4.0 and 4.5 h, respectively. } Tc computed using the Kirpich equation ranged from 0.6 to 7.1 h in the study watersheds, with median and mean values of 2.2 and 2.4 h, respectively. Modelled DRH Qp: m3/s Table 1. Quantitative measures of the success of the DRH Qp and Tp modelled using Cvbc and MRUHs with Tc estimated using four equations Regression equations were developed for 5 min GUH parameters: QpUG (in ft3 /s) and TpUG (in h) for Texas watersheds (Pradhan, 2007) 100 10 Modelled using Cvbc Modelled using Clit 1 : 1 line ⫾1/3 of a log cycle 1 1 10 Observed DRH Qp: m3/s Comparison of DRHs from different UH methods In addition to the MRUH, three other UH models – UH developed using the Clark IUH method (Clark, 1945) with the generalised basin shape of HEC–1 (USACE, 1981), the NRCS UH (NRCS, 1972), and the Gamma function UH (GUH) for Texas watersheds (Pradhan, 2007) – were used to develop the DRH for each rainfall-runoff event in the database for the comparison. 8 : : T pUG ¼ 0:55075 A0 26998 L0 42612 S 0 06032 7: QpUG ¼ 93:22352 A0 83576 L0 326 S 0 5 : : : 100 Figure 7. Observed and modelled DRH peak discharges developed using Cvbc (circles) and Clit (triangles) and MRUHs with Tc estimated using the Kirpich equation for 80 Texas watersheds 4. : 6: where A is drainage area in square miles, L is main channel length in miles, and S is main channel slope (ft/mile, elevation difference in feet divided by main channel length in miles). The ordinates of the GUH can be obtained from (Viessman and Lewis, 2003) 8: QuG (t) ¼ QpUG (t=T pUG )Æ e½1(t=T pUG )] Æ where Æ is the shape parameter of GUH, which is determined from QpUG and TpUG (Aron and White, 1982). The IUH method of Clark (1945) is based on the time–area curve method (Bedient and Huber, 2002). A synthetic time–area curve derived from a generalised basin shape was used to implement Water Management Modified rational unit hydrograph method and applications Dhakal, Fang, Thompson and Cleveland Statistical parameters Using Cvbc Using Clit 0.80 0.73 0.10 0.75 88.6 0.73 0.72 0.01 0.72 82.3 0.44 0.42 0.45 0.45 63.0 0.73 0.72 0.01 0.72 82.3 R2 for Qp EF for Qp Median value of QB Fraction of storms with 0.5 < QB < 0.5 Percentage of storms within  1/3 of a log cycle (Qp ) R2 for Tp EF for Tp Median value of TB Fraction of storms with 0.5 < TB < 0.5 Percentage of storms within  1/3 of a log cycle (Tp ) Table 2. Quantitative measures of the success of the DRH Qp and Tp modelled using MRUH with Tc estimated using the Kirpich equation and C estimated using two different methods (Cvbc and Clit ) Change in Tc : % 50 25 10 10 25 50 Change in EF 0.18 0.02 0.01 0.03 0.09 0.22 Change in C: % 50 25 10 10 25 50 Change in EF 0.27 0.02 0.02 0.06 0.21 0.66 Table 3. Sensitivity (change in EF) of DRH derived from MRUH on Tc and C for the rainfall event on 7 May 1972 for the USGS streamflow-gauging station 08178600 Salado Creek, San Antonio, Texas Clark’s IUH in HEC-1 (USACE, 1981). The equations for the time–area curve are : 9: AI ¼ 1:414 TI 1 5 , 0 < TI < 0:5 10: : 1  AI ¼ 0:414 (1  TI)1 5 , 0:5 , TI , 1 where AI is the cumulative area as a fraction of watershed area and TI is fraction of Tc : The NRCS curvilinear UH was developed in the late 1940s (NRCS, 1972). The QpUN for the NRCS UH is computed by approximating the UH with a triangular shape having base time of 8/3TpUN and unit area (Viessman and Lewis, 2003) 11: QpUN ¼ 484A T pUN where QpUN is ft3 /s and A is the drainage area in mile2 : UHs developed using all four models, including the MRUH, for the watershed associated with the USGS streamflow-gauging station 08048520 Sycamore Creek in Fort Worth are shown in Figure 8(a). The shape of the MRUH is trapezoidal, whereas the UHs from the Clark-HEC-1, the Gamma, and the NRCS methods are curvilinear. The UH peak discharge from each model is different (Figure 8(a)). However, the area under the UH curves is the same. This is because each UH corresponds to 1 in of a uniform excess rainfall over a 5 min duration (one impulse). Gamma, Clark-HEC-1 and NRCS UHs developed for each watershed were applied to the 1400 rainfall-runoff events in the database to generate DRHs using discrete UH convolution (Chow et al., 1988). The values of Cvbc determined for each event were used. Parameter Tc , which was determined using the Kirpich method (Kirpich, 1940), was used for those methods that require Tc : As an illustrative example, observed and simulated DRHs for the rainfall event on 28 July 1973 at the USGS streamflowgauging station 08048520 (Sycamore Creek in Fort Worth, Texas) by the four models (base flow was assumed to be zero) is presented in Figure 8(b). The watershed area is 45.66 km2 (17.63 mile2 ), Tc is 3.96 h from the Kirpich method, and Cvbc is 0.20. Simulated peak discharges from the four UH methods are different, but comparable. For the particular example shown in Figure 8(b), the MRUH and the Clark-HEC-1 model appear to perform better than the other UH models with regard to prediction of Qp : The Tp , simulated values using the four methods agree reasonably well with the observed value (Figure 8(b)). Furthermore, the area under the four simulated DRHs matches that of the observed curve because the event Cvbc was used. 9 Water Management Modified rational unit hydrograph method and applications Dhakal, Fang, Thompson and Cleveland 140 Discharge: m3/s 120 shows one example to illustrate the similarity of DRHs derived from these UH models. MRUH Clark-HEC-1 UH Gamma UH NRCS UH 100 80 60 40 20 0 0 1 2 3 4 5 6 7 8 9 10 11 (a) 60 Discharge: m3/s 50 40 30 6 12 18 24 20 10 Incremental rainfall: mm 0 Incremental rainfall Observed DRH Modelled using MRUH Modelled using Clark-HEC-1 UH Modelled using Gamma UH Modelled using NRCS UH 0 18 20 22 24 26 28 30 32 34 36 38 40 42 Time: h (b) Figure 8. (a) Modified rational, Gamma, Clark-HEC-1, and NRCS UHs developed for the watershed associated with USGS streamflow-gauging station 08048520 Sycamore, Fort Worth, Texas; and (b) rainfall hyetograph, observed and modelled DRHs using the four different UHs for the rainfall event on 28 July 1973 for the same watershed Simulated DRH ordinates derived from all the four UH models were compared with observed DRH ordinates for each rainfall event, and the root mean squared error of the DRH ordinates normalised by observed Qp (RRMSE, Equation 12) was calculated for each event and then averaged for all the events in the same watershed. A statistical summary of averaged normalised root mean squared errors for 80 study watersheds is presented in Table 4. All the four UH models behave similarly to predict DRHs based on statistical parameters in Table 4, and Figure 8(b) The observed and modelled Qp results from all four UH models developed using Cvbc and Tc from the Kirpich method are presented in Figure 9 for all 1400 events. Modelled Qp results from all the four UH models are similar (Figure 9). Based on the three statistical measures (RRMSE, R2 , and EF) the authors concluded that all the four UH models perform similarly in predicting DRH Qp and Tp (Table 5) after considering possible errors in DRH prediction. Fractions and percentages of storms for each model meeting the tolerances of QB and TB are also listed in Table 5 and show that all the models perform similarly. However, the GUH developed for Texas watersheds perform slightly worse than the other three UH models (Table 5) in predicting DRH Qp : 5. Summary and conclusions The MRM is an extension of the rational method to produce simple triangular and trapezoidal DRHs that have been used in some engineering applications. MRM’s DRH for D , Tc was used to derive a trapezoidal UH termed the modified rational UH or MRUH. The MRUH method was applied at 80 watersheds in Texas to determine the DRHs for 1400 rainfall-runoff events. The purposes were: (1) to evaluate the applicability of the MRUH method when applied to watersheds of larger size (0.8–65.0 km2 or 0.3–25 mile2 ), and (2) to study the effects of C and Tc on prediction accuracy of the MRUH method on DRH ordinates, DRH Qp and DRH Tp : Three other UH models; the Clark (using HEC–1’s generalised basin equations), the Gamma, and the NRCS UHs were used to compute the DRH for each rainfallrunoff event in the same database. Simulated peak discharges of DRHs from MRUH and the other three UHs agree reasonably well with observed values. The drainage area of the study watersheds (0.8–65.0 km2 or 0.3–25 mile2 ) is greater than that usually accepted for rational method application (0.8 km2 or 0.3 mile2 ). Three general conclusions for the study are: (1) being a UH, the MRUH method can be applied to time-variable rainfall events and for watersheds with drainage areas greater than typically used with either the rational method or the MRM (a few hundred Statistical parameters Using MRUH Using Gamma UH Using Clark-HEC-1 UH Using NRCS UH Maximum Minimum Mean Median 1.78 0.25 0.61 0.52 1.61 0.19 0.61 0.53 1.95 0.23 0.62 0.53 1.74 0.22 0.57 0.51 Table 4. Statistical summary of watershed-averaged root mean squared errors between modelled and observed DRHs normalised by observed peak discharges 10 Water Management Modified rational unit hydrograph method and applications Dhakal, Fang, Thompson and Cleveland Modelled DRH, Qp: m3/s 100 10 1 1 : 1 line ⫾1/3 of a log cycle (a) (b) Modelled DRH, Qp: m3/s 100 10 1 1 10 1 100 10 100 Observed DRH, Qp: m3/s (d) Observed DRH, Qp: m3/s (c) Figure 9. Observed and modelled DRH peak discharges using: (a) MRUH, (b) Gamma UH, (c) Clark-HEC-1 UH and (d) NRCS UH for 1400 rainfall-runoff events in 80 Texas watersheds Statistical parameters Using MRUH Using Gamma UH Using Clark-HEC-1 UH Using NRCS UH R2 for Qp EF for Qp Median value of QB Fraction of storms with 0.5 < QB < 0.5 Percentage of storms within  1/3 of a log cycle (Qp ) R2 for Tp EF for Tp Median value of TB Fraction of storms with 0.5 < TB < 0.5 Percentage of storms within  1/3 of a log cycle (Tp ) 0.80 0.73 0.10 0.75 88.6 0.73 0.72 0.01 0.72 82.3 0.82 0.63 0.32 0.71 80.6 0.73 0.72 0.03 0.73 84.1 0.81 0.79 0.02 0.71 88.5 0.71 0.70 0.02 0.75 81.8 0.83 0.76 0.12 0.77 90.9 0.71 0.70 0.00 0.75 82.4 Table 5. Quantitative measures of the success of DRH Qp and Tp modelled using four UH models for 1400 rainfall-runoff events in 80 Texas watersheds 11 Water Management Modified rational unit hydrograph method and applications Dhakal, Fang, Thompson and Cleveland acres); (2) the MRUH performs about as well as other UH methods used in this study for predicting Qp and Tp of the DRH, so long as the same rainfall loss model is used; (3) modelled peak discharges from application of the MRUH method are more sensitive to the selection of C and less sensitive to Tc : In predicting peak discharges and DRHs for engineering design, rainfall loss estimation results in greater uncertainty and contributes more model errors than variations of UH methods and model parameters for UH. for modelled), Q(t )o j is the observed DRH ordinate (subscript o stands for observed), N is the number of DRH ordinates for an event, Qpmi is the modelled Qp for the event i, Qpoi is the  pm and Q  po are observed Qp , n is the number of observations, Q the mean values of the modelled and observed peak discharges, Tpmi is the modelled Tp , and Tpoi is the observed Tp : Acknowledgements The authors thank TxDOT project director Mr. Chuck Stead, P.E., and project monitoring advisor members for their guidance and assistance. They also express their thanks to anonymous reviewers. This study was partially supported by TxDOT Research Projects 0–6070, 0–4696, 0–4193, and 0–4194. Appendix 1: Statistical measures to evaluate model performance Five statistical measures were used to analyse modelled DRH results against observed ones. They are the root mean squared error (RMSE) of the DRH ordinates normalised by observed DRH Qp , – that is, relative RMSE or RRMSE, the coefficient of determination R2 , the Nash–Sutcliffe efficiency EF, the relative error in peak QB, and the relative error in time to peak TB (Cleveland et al., 2006; Loague and Green, 1991; Zhao and Tung, 1994) 12: RRMSE ¼ hP 14: 15: (Q(t)mj  Q(t)oj )2 =N Qpo i0:5 32   ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5 ffiqP R2 ¼ 4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn n  2  2 i¼1 (Qpoi  Qpo ) i¼1 (Qpmi  Qpm ) Pn 2 13: N j¼1 i¼1 (Qpoi  Qpo )(Qpmi  Qpm ) P  po )2  n (Qpmi  Qpoi )2 (Qpoi  Q i¼1 Pn  po )2 (Q  Q poi i¼1 EF ¼ Pn QB ¼ Qpmi  Qpoi Qpoi TB ¼ T pmi  T poi T poi i¼1 and 16: where Q(t )m j is the modelled DRH ordinate (subscript m stands 12 Appendix 2: Empirical equations used to estimate Tc Four empirical equations Williams (1922), Kirpich (1940), Johnstone and Cross (1949) and Haktanir and Sezen (1990) used to estimate Tc (in min) by Fang et al. 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