Rating Curve Development at Ungauged River Sites using Variable Parameter Muskingum Discharge Routing Method

Water Resour Manage (4) 8:3783 38 DOI.7/s69-4-79-9 Rating Curve Development at Ungauged River Sites using Variable Parameter Muskingum Discharge Routing Method Bhabagrahi Sahoo & Muthiah Perumal & Tommaso Moramarco & Silvia Barbetta Received: 6 December 3 / Accepted: June 4 / Published online: 8 June 4 # Springer Science+Business Media Dordrecht 4 Abstract A physically based simplified discharge routing method, namely, the variable parameter Muskingum discharge-hydrograph () routing method, having the capability of estimating the stage hydrographs simultaneously in channels with floodplains is presented herein. The upstream discharge hydrograph is routed using this method in different two-stage symmetrical trapezoidal compound cross section channel reaches. The performance of the method is evaluated by numerical experiments using the benchmark MIKE hydrodynamic model and the field data of the Tiber River in central Italy. The proposed method is capable of accurately routing the discharge hydrographs, corresponding stage hydrographs and synthesizing the normal rating curves at any downstream ungauged river site which is not affected by any downstream effects. This study can be helpful for various planning and management of river water resources in both the diagnostic and prognostic modes. Keywords Hydrograph. Rating curve. Routing. Ungauged river. Variable parameter. routing method B. Sahoo (*) School of Water Resources, Indian Institute of Technology Kharagpur, Kharagpur 73 West Bengal, India e-mail: bsahoo3@yahoo.com M. Perumal Department of Hydrology, Indian Institute of Technology Roorkee, Roorkee 47 667, India e-mail: p_erumal@yahoo.com T. Moramarco: S. Barbetta Istituto di Ricerca per la Protezione Idrogeologica, National Research Council, Perugia, Italy T. Moramarco e-mail: t.moramarco@irpi.cnr.it S. Barbetta e-mail: silvia.barbetta@irpi.cnr.it

3784 B. Sahoo et al. Introduction Development of gauge discharge relationship, commonly known as the rating curve, at a river site is required for various hydrological analyses purposes such as assessment of water availability, establishing the rainfall runoff relationship, design flood estimation, environmental flow assessment, and for various water resources development activities. Establishing a rating curve at any river site requires continuous measurement of flow stage and the corresponding discharge from year to year barring a few exceptional cases. This is, however, very time consuming, costly, and frequently dangerous due to manual discharge measurements during the flood events. Hence, even in a well-gauged river basin, only a few gauge discharge measurement sites are established. However, the necessity for a water resources development project or an operational hydrology project may arise in future at any location of a river site which is not necessarily being gauged now. This necessitates for the evolution of methods for development of rating curves at such ungauged river sites. For overcoming the non-availability of discharge data at an ungauged river site, one of the ways could be by routing the upstream discharge hydrograph up to the downstream ungauged site with the known upstream rating curve. By relating the routed discharge estimate with the available concurrent stage measurement, the rating curve at the ungauged site may be established (Birkhead and James 998; Franchini et al. 999). Subsequently, Moramarco and Singh () proposed an alternate approach for establishing discharge hydrograph at an ungauged site by relating the local river stage measured at the ungauged site with the velocity distribution recorded at a remote upstream section using the entropy concept. Both the methods advocated by Birkhead and James (998) and Franchini et al. (999) are of limited use as these methods have been developed using a one-to-one relationship between stage and discharge, and involve the estimation of many parameters using the observed data. Therefore, these methods are not parsimonious in model parameters and also unsuitable for using in streams wherein the unsteady flow is not characterized by a one-to-one relationship between stage and discharge at a river site. Moreover, the currently available streamflow routing packages such as the NWS- DAMBRK (Fread 99), HEC-RAS (U.S. Army Corps of Engineers (USACE) 6a), HEC-HMS (USACE 6b), and MIKE (Danish Hydraulic Institute (DHI) 3) are not popular for establishing rating curves at ungauged sites due to the requirement of channel cross section details and roughness information at closer spatial intervals in which solution of the full dynamic wave (DW) models are subjected to downstream boundary condition. The DW models, in general, are subjected to numerical instability problems, and particularly so while routing in steep slope channels (e.g., Ferrick 985); and for this reason, besides geometric and hydraulic parameters, other parameters have been added in the programming code in order to limit the above mentioned instabilities. By way of example, the MIKE model controls the numerical fluctuations by a dissipative interface which incorporates different parameters. The motivation for developing the simplified routing models over the DW models can be found in the works of Price (973), Natural Environment Research Council (NERC) (975), Weinmann (977), Weinmann and Laurenson (979), and others. To overcome the practical limitation of using cross sectional data available at sparse intervals, one may prefer to use the simplified flood routing methods as followed by Birkhead and James (998) and Franchini et al. (999). As the rating curve sites, where the extension or development of rating curves are sought, are usually located on main streams, the simplified routing techniques employed should also be capable of routing flows in compound channels consisting of a main channel and an adjoining floodplain channel. Considering these requirements, when only a stage hydrograph is available at the upstream river section, Perumal et al. (7, ) proposed the variable parameter

Rating Curve Development using Routing Method 3785 Muskingum stage hydrograph (VPMS) routing method as an extension of the work by Perumal and Ranga Raju (998a, 998b). However, when only a discharge hydrograph is available at the upstream river section, there is a need to develop another routing method using a similar approach, for simplicity, for establishing rating curves at ungauged river sites. Hence, in this study, the variable parameter Muskingum discharge hydrograph () routing method (Perumal 994a, b) is extended for floodplain flow conditions which is used for establishing the rating curve at the required downstream river site. It is assumed that the discharge hydrograph and a reliable rating curve are available at the upstream gauged site. The routing parameters are related to measurable channel and flow characteristics with the only parameter to be calibrated in this method being the Manning s roughness coefficient, same as in the case of the hydraulic routing methods, being used for ungauged rivers. Note that there is no unsteady flow method, including the full Saint-Venant equations based methods available in the literature, which can estimate the steady stage-discharge relationship, namely, the normal rating curve at any downstream ungauged river section. The normal rating curve is the one which is commonly used by the river engineers and field hydrologists in practice. However, the method has the capability to establish the normal rating curve as has been demonstrated in this manuscript. Moreover, Heatherman (4) has pointed out that more insight into the Muskingum method could be gained by using the interpretation proposed by Perumal (994a, 994b). Note that the HEC-RAS model cannot estimate the normal rating curve at any downstream point, and the MIKE model, which uses the six-point Abbot- Ionescu scheme to solve the full Saint Venant equations, cannot estimate both the stage and discharge at the same river section. However, even if one obtains the stage or the discharge at the same location by using interpolation procedure, then the resulting pairs of stages and discharges at a location would result in a loop-rating curve and not a normal rating curve as required, except for the cases of routing in steep river reaches. Some studies have been conducted in the past for routing discharge hydrographs in compound channels using the variable parameter Muskingum Cunge method and its variants (Garbrecht and Brunner 99; Tang et al. 999). Some recent advances in the Muskingum based discharge routing methods can also be found in Todini (7), Price (9), Perumal et al. (9), Sahoo (3), and Perumal and Price (3). While demonstrating the field applicability of the method, Perumal et al. () showed in a preliminary study about the possibility of extending the method for routing floods which inundated the floodplains. However, a detailed development of the method for exclusively routing floods inundating the floodplain has not yet been presented. Moreover, the methods proposed by Garbrecht and Brunner (99) and Tang et al. (999) are not capable of estimating the stage hydrograph corresponding to the routed discharge hydrograph. In light of the above discussion, in this study, the routing method is extended for routing floods that also inundate the floodplains, to estimate the routed discharge hydrograph and the corresponding stage hydrograph at a downstream ungauged site of a river reach. The reach with floodplains is characterized by a two-stage symmetrical trapezoidal compound cross section consisting of a main channel section and a floodplain channel section. The procedure involves three steps: i) The method is used to calibrate the average channel roughness coefficient by reproducing the rating curve available at the upstream site; ii) The calibrated roughness coefficient is used to route the upstream discharge hydrograph to a downstream ungauged site; and iii) Using the routed discharge hydrograph and the corresponding stage hydrograph computed by the method, the rating curve at the downstream ungauged site is established. The proposed methodology is verified using the hypothetical data sets and field data sets of the Tiber River in central Italy. The novelty in this study lies with the development of a simplified unsteady flow hydraulic routing method amenable

3786 B. Sahoo et al. for floodplain flow for establishment of normal rating curve at any downstream ungauged river site under limited cross-sectional data-availability situations. Method: Theoretical Background The governing differential equation of the routing method derived as a simplification to the full Saint-Venant equation can be given by (Perumal 994a) Q u Q d ¼ Δx Q c 3 t d þ L ðq Δx u Q d Þ ðþ where Q u and Q d denote the discharges at the upstream and downstream of the reach, respectively (see the definition sketch of the method in Fig. ), Δx is the reach length, c 3 is the celerity at section 3 of Fig., corresponding to the dynamic normal discharge section, and L is the dynamic distance between the sections M and 3 in Fig. at any instant of time. Equating Eq. () of the routing method with the classical Muskingum difference scheme of McCarthy (938), the recursive routing equation of the method can be given by Q d; jþ ¼ C Q u; jþ þ C Q u; j þ C 3 Q d; j ðþ where the coefficients C, C,andC 3 are expressed as Kθ þ :5Δt C ¼ Kð θþþ:5δt ; C Kθ þ :5Δt ¼ Kð θþþ:5δt ; C 3 ¼ Kð θþ :5Δt ð3a; b; cþ Kð θþþ:5δt where the travel time, K, of the routing method can be expressed as K ¼ Δx ð4þ c 3 and the weighting parameter, θ, after neglecting the inertial terms (e.g., Henderson 966; Price 985) can be given by y u M 3 Q u Q M Q 3 ym y 3 y d Q d L Fig. Definition sketch of the routing method

Rating Curve Development using Routing Method 3787 θ ¼ Q 3 S o ðda=dyþ 3 c 3 Δx ¼ Q 3 S o ð Q= yþ 3 Δx ð5þ Using the assumption that Q/ x over the Muskingum computational subreach of length Δx, the relationship between the downstream stage, y d, corresponding to the routed discharge, Q d, can be given by y d ¼ y 3 þ Q d Q 3 ð6þ ð Q= yþ 3 Similarly, the stage at the upstream gauging site, y u, corresponding to a given discharge, Q u, can be expressed as y u ¼ y 3 þ Q u Q 3 ð7þ ð Q= yþ 3 The relationships between y d and Q d,andy u and Q u form the downstream and upstream looped rating curves, respectively. 3 Extension of the Method for Routing in a Two-Stage Compound Cross Section Channel Reach A two-stage uniform compound cross section channel reach with a symmetrical trapezoidal main channel flow section and an extended trapezoidal floodplain channel section (shown in Fig. ) as adopted by Ackers (993), Perumal et al. (7, ) and O Sullivan et al. () is considered for the present study. The routing reach considered is assumed to be characterized by a representative Manning s roughness coefficient irrespective of the main or floodplain channel sections. This assumption may not be strictly valid in practice. However, as the main aim herein is to develop a simplified hydraulic routing method using discharge as the main routing variable, such an assumption helps to reduce complications in the development of the method and, also, circumvents the subjectivity involved in selecting roughness values for the in-bank and over-bank channel sections. The discharge hydrograph routing using the method, either in a single section (main channel section) or in a compound section channel reach, involves the use of Eqs. (), (3a) (3c), (4), and (5). Estimation of the parameters K and θ, given by Eqs. (4) and (5) at b m +y m z z z b f MAIN b m y m Fig. Partitioning of the compound channel section into a main channel section (shaded) and two floodplain channel sections and for computing the flood wave celerity, when y > y m (after Sahoo 7)

3788 B. Sahoo et al. every routing time interval, involves the variables Q 3 and c 3. The wave celerity, c 3,forthe main channel and floodplain channel sections can be computed as a function of the flow depth. Note that the celerity flow depth relationship is not unique during unsteady flow due to differing relationships established when the flood is in the rising and falling stages. Further, the celerity versus stage relationship have a discontinuity at the intersection of the bankfull flood level (corresponding to the top width of the main channel cross section) and the bottom width of the floodplain cross section, due to sudden increase of the wetted perimeter. 3. Estimation of celerity for the main channel and for the floodplain channel Using the Manning s friction equation for unsteady flows, the celerity of flow in the trapezoidal main channel can be expressed as (Perumal et al. 7) c main ¼ dq main=dy da main =dy ¼ 5 3 3 R main ðdp main =dyþ Qmain ðda main =dyþ A main ; when < y y m ð8þ where Q main =discharge in the main channel; y m = stage at the bank-full level; and A main, P main, and R main are the area, wetted perimeter and hydraulic mean radius of the flow in the main channel. Conversely, when the flow is above the bankfull level (y>y m ),thewavecelerityfor the compound channel section, c comp, considering the compartmentalized flow in the main channel and in the two symmetrical compartments above bankfull level (Fig. ) canbe expressed as (Perumal et al. 7; Sahoo7) " c comp ¼ dq sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# ð main þ Q þ Q Þ=dy 5 da main ¼ v main; y = da comp da comp =dy 3 dy S o x dy " þ 5 da 3 dy sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# A dp v ; y = da comp ð9þ 3 P dy S o x dy " þ 5 da 3 dy sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# A dp v ; y = da comp 3 P dy S o x dy where A, A = flow area of the two symmetrical floodplains; P, P = wetted perimeter for the two symmetrical floodplains; A comp = total flow area; v main = velocity in the main channel section; v,, v, = normal velocities in the floodplain channels; and S o = channel bed slope. 3. Rating curve development procedure The step-by-step procedure for discharge routing using the method is given below. ) The unrefined discharge Q d,j+ for the current routing time interval is computed using the values of K and θ, estimated at the previous time step and, subsequently, using this estimate and Q u,j+, the flow at the middle of the reach is estimated as Q M ¼ :5 Q u; jþ þ Q d; jþ ðþ The initial values of the routing parameters K and θ are estimated using the initial steady flow in the reach.

Rating Curve Development using Routing Method 3789 ) The normal discharge at section 3 of Fig. is computed as Q 3 ¼ θq u; jþ þ ð θþq d; jþ ðþ 3) Using Q 3, estimated at step, the stage at the midsection is estimated using the Newton Raphson method, by solving the following equations depending on whether y M is within the main channel section or in the compound channel section: Q 3 ¼ A M n A =3 M ffiffiffiffiffi S o P M p ; when ym y m ðaþ pffiffiffiffiffi S h i o Q 3 ¼ A main R =3 main n þ A R =3 þ A R =3 ; when y M > y m M where A main,m,anda,m and A,M are evaluated at the midsection of the reach. 4) The stage at section 3 of Fig. is computed as ðbþ y 3 ¼ y M þ Q 3 Q M ð3þ ðdq=dyþ 3 in which (dq/dy) 3 =(da/dy) 3 c 3, depending on y M y m or y M y m. 6) Using the values of Q 3, c 3, and (dq/dy) 3, the refined values of K and θ are estimated using Eqs. (4) and (5), respectively, at the current routing time interval, which are, subsequently, used to estimate the refined discharge hydrograph estimate using Eqs. () and (3a) (3c). 7) The relationships between y M - Q 3 and y d - Q d yield the normal and looped rating curves, respectively, where y d is the computed stage corresponding to Q d. 4 Performance Evaluation Measures The performance of the proposed extended model was evaluated through the following performance measures: i) Nash Sutcliffe criteria of variance explained (Nash and Sutcliffe 97; ASCE993), η y and η q, employed both for stage and discharge hydrograph reproductions, respectively; ii) Percentage errors of stage and discharge peaks defined by y per = (computed peak stage/observed peak stage ) and q per = (computed peak discharge/ observed peak discharge ), respectively; iii) Errors of time to routed peak stage and computed peak discharge defined by t per = (time to computed peak stage time to observed peak stage) and t pqer = (time to computed peak discharge timetoobservedpeakdischarge), respectively; and iv) Percentage error in flood volume, EVOL=(volume of the routed hydrograph/inflow flood volume ). 5 Numerical Application A number of different compound channels were selected for this study with b m =5. m, y m =.5 m, and z =z =. (see Fig. ). Each of the channel was characterized by a varied size of floodplain channel sections with six different b f /b m ratios, two Manning s roughness coefficients, n, and six varying channel bed slopes, S o, comprising of 7 channel types in all

379 B. Sahoo et al. (Table ). For each of the 7 channel types studied, two test runs were carried out corresponding to routing by the method in a 4 km channel reach, considering it as ten (Δx = km) and twenty (Δx =4 km) equal subreaches. A uniform routing time interval, Δt = 3 min was used for all the test runs, except in the case of routing in steep channel reaches with bed slopes of. and.3, wherein a routing time interval, Δt = 5 min was used. Use of a larger Δt in such cases resulted in wiggles while simulating the recession of the discharge hydrograph at 4 km. 6 Results and Discussion 6. Discharge Hydrograph Reproduction It is evident from the discharge hydrograph routing results, illustrated in Fig. 3, and the typical results as illustrated in Fig. 4a and b that the proposed routing method is able to reproduce the MIKE solutions closely for channel types with lower b f /b m values. With the increase in size of the floodplain channel section from b f /b m =.5 to 4., the discharge routing capability of the method reduces slightly, but not to the extent of rendering the results unusable for practical purposes. Similarly, it is also revealed that the η q estimate decreases slightly when the bed slope decreases or when the roughness increases or for both. It is inferred from Fig. 3 that the method fails to route hydrographs in channels characterized by small bed slopes such as S o =. due to the violation of the assumption of discharge varying approximately linearly over the routing reach. Since the longitudinal gradient of Q is directly linked to the longitudinal gradient of y as: Q/ x=bc y/ x (Perumal 994a; Perumal and Ranga Raju 999), where B = A/ y, the applicability limit of Q/ x for the successful application of the method may be defined by the magnitude of (/S o )( y/ x). In a study on the applicability criterion of the method, Perumal and Sahoo (7) have estimated this criterion as (/S o )( y/ x) max.43 for simple section channel reaches. This study reveals that when the assumption of Q varying approximately linearly is satisfied fully while routing a discharge hydrograph in a channel, the method works successfully howsoever small the subreach length may be. However, when the assumption is not fully satisfied, use of smaller length subreach may lead to failure of the method in completely routing the given flood hydrograph. Moreover, one may still succeed in achieving the routing of the given discharge hydrograph by using a slightly longer subreach length, but at a cost of estimating slightly inferior solution which may be still acceptable from practical considerations. Such successful runs achieved by using the next non-failing subreach length for the failed cases shown in Fig. 3 are presented in Table. When the number of subreaches are reduced from, as adopted in Fig. 3, all the failed routing cases could be successfully run as seen from Table. If the result of a routing case with η q 95 % is considered as practically acceptable, then it is inferred from Table that only four runs may be considered as failures. From the test run results corresponding to routing in steep slope channels with Δx =4km, it is evidenced that there is a tendency of formation of wiggles in the simulated hydrograph, whenever there is a sudden rise or fall of the hydrograph. Weinmann and Laurenson (979) have well documented this feature of the Muskingum routing solution, which is termed as the initial dip or negative initial flow. The size of the wiggle depends directly on the steepness of the bed slope, steepness of the rise or fall of the input discharge hydrograph and the magnitude of the roughness coefficient (Perumal 99). In essence, when the input discharge hydrograph experiences a large variation in the magnitude of (/S o )( y/ x), while routing in steep slope channels, the tendency for the formation of wiggle is strong. The effect of the

Rating Curve Development using Routing Method 379 Table Combination of different numerical experiments S o n b f /b m Channel type S o n b f /b m Channel type.3.4 3. 37.3.4.5. 38.. 39. 3.5 4.5 4.3 4.3 5. 4. 6.3.5 3. 43.3.5.5 7. 44. 8. 45. 9.5 46.5.3 47.3. 48..3.4 3.5 49.3.4. 3. 5. 4. 5. 5.5 5.5 6.3 53.3 7. 54. 8.3.5 3.5 55.3.5. 9. 56.. 57..5 58.5.3 59.3 3. 6. 4.3.4 4. 6.3.4.5 5. 6. 6. 63. 7.5 64.5 8.3 65.3 9. 66. 3.3.5 4. 67.3.5.5 3. 68. 3. 69. 33.5 7.5 34.3 7.3 35. 7. 36 wiggle on the hydrograph can be minimized by increasing the number of subreaches used for routing as in the case of routing with Δx = km. However, the effect of wiggle impacts the solution only locally and does not affect the entire solution significantly, such as the peak and time to peak characteristics of the hydrograph. Further, the size of the wiggle is less pronounced in the discharge hydrograph routing solution, as compared to that in the estimated stage hydrograph. Hence, the overall performance of the method may be considered

379 B. Sahoo et al. EVOL (%) 5 5 b f /b m.5.5...5.5 3. 3. 3.5 3.5 4. 4. η q (%) 95 9 85 η y (%) 95 9 85 q per (%) 5-5 7.5 y per (%) 5.5 t pqer (h) t pyer.5 -.5 -.5 -.5 - dx=4 km dx= km 6 8 4 3 36 4 48 54 6 66 7 Channel Type Fig. 3 Reproduction of the pertinent characteristics of the MIKE solutions by the method satisfactory given the fact that even when the input discharge hydrograph has completely entered into the channel reach, typically as seen in Fig. 4a, with no significant response at the outlet of the reach, the method could closely reproduce the MIKE solutions with η q >98 %. Only in a very few cases of channel routing with Δx =4 km, η q <95 %. The MIKE solutions considered herein were subjected to attenuation to the extent of 5 % and 5 %, respectively, for the stage and discharge hydrograph reproductions, and they could be reproduced very well by the method.

Rating Curve Development using Routing Method 3793 Discharge (m 3 /s) 8 6 4 (a) Channel Type-7 Inflow ( subreach) ( subreach) MIKE 3 4 5 Stage (m ) 4. 3.. (b) Channel Type-7 Input stage ( subreach) ( subreach) MIKE. 3 4 5 Time (h) Fig. 4 Typical routed discharge and computed stage hydrographs for channel type 7 Similarly, the solutions have the errors in peak discharge estimates and its timings with less than 5 % for most of the test run cases, except for the case of routing in channel reach with S o =. and.3 with higher b f /b m ratios. However, the volume error, EVOL<5 % and 5 %< EVOL < % for the cases of routing in channel reaches with S o. and S o <., respectively. It may be surmised that the overestimation of the flood volume for very Table Summary of performance criteria showing reproduction of pertinent characteristics of the MIKE solutions by the method for the failure cases with Δx =4km Channel type b f /b m n S o Subreaches Discharge Routing Stage Computation t pqer (h) q per (%) EVOL (%) η q (%) t pyer (h) y per (%) η y (%) 6.5.4. 7.5 4.4 4.73 96.48.5 6.69 9.9 8..4. 7. 6.94 6.46 93.9. 7.79 89.4 4..5. 9.5.8 6.54 98.64. 3.8 97.58 3.5.4. 6.5.3 7.9 95.93.5 5.4 9.5 4 3..4. 5. 3.47 5.46 93.96. 5.95 88.39 54 3.5.4. 4.5 6.98.97 85.39. 7.6 74.9 66 4..4. 5. 6.83 9.9 87.5.5 6.77 8.63

3794 B. Sahoo et al. gentle slope channels is expected because of the violation of the assumption of approximately linearly varying discharge along the routing reach. However, considering many uncertainties involved in the practical measurement of overbank floods, one may find even a volume error up to % may be acceptable. On the basis of the past studies of the routing method (Perumal 994b), it is surmised that the method may have to be applied with caution for routing in channels characterized by very gentle slopes as seen herein with S o =.. Only in such situations, one may invoke the use of more descriptive models like the MIKE, HEC-RAS, etc. Overall, the results demonstrate the capability of the method for routing floods in moderate slope channels wherein attenuation of a passing flood wave may be significant. 6. Stage Hydrograph Reproduction Generally, the simplified hydraulic routing methods available in the literature exclusively route either the stage hydrograph or the discharge hydrograph. If one of these variables is used for routing, the other is generally estimated using steady flow rating curve pertaining to that routing section. However, the routing method is different from these existing simplified methods in that the stage variable can be estimated at any section of the channel reach by using Eq. (6). Routing results shown in Fig. 3, corresponding to Δx =4 km, reveal that for most of the steeper slope cases with S o >., the reproduction of the MIKE stage hydrographs by the method may be considered close, with η y >99 %. However, in channel reaches with S o <., the variance explained estimates in reproducing the MIKE stage hydrograph decreases with decreasing channel slope, in a similar manner as in the case of discharge hydrograph routing which is discussed in section 6.. The variance explained estimate, η y varies in the range of 99.9 %, at its best, to 88.7 %, at the lowest (Fig. 3); and a minimum of η y =74.9 % for the failure cases given in Table. One of the factors for the poor reproduction of stage hydrograph in Table may be attributed to the failure of the assumption of approximately linearly varying discharge in smaller slope channels which can be addressed by considering smaller Δx values of km. It is inferred from the solutions of the stage hydrographs, computed using Eq. (6), that the errors in peak stage estimates and its timings to reproduce the benchmark stage hydrographs were found to be<5 %, except in few test run cases with S o =.. Conclusively, the results reveal the capability of the method to estimate the stage hydrograph corresponding to the routed discharge hydrograph within the applicability limit of the method. 6.3 Field Application The discharge hydrograph routing and the corresponding stage hydrograph computation capability of the method in a river reach was investigated using the field data of the Tiber River in central Italy, for the 5 km long river reach between the gauged sections of Pierantonio (upstream) and Ponte Felcino (downstream) with an average bed slope of.6, determined from topographical surveys. For approximating the Pierantonio Ponte Felcino natural river reach by a two-stage trapezoidal compound section geometry, the following steps were followed (Perumal et al. ): i) Using the surveyed cross-sections at the Pierantonio and Ponte Felcino sections, the stage flow area relationship was developed for each section; ii) The stage mean flow area relationship for this river reach was developed considering the mean flow area as the average of the flow areas of the Pierantonio and Ponte Felcino sections at the same stage; iii) The value of the main channel depth y m was marked at which the flood likely to traverse the floodplains; iv) The values of b m and z of the compound trapezoidal channel

Rating Curve Development using Routing Method 3795 were computed by fitting a quadratic equation in the form of A main =b m y+z y (where y y m )to the stage mean flow area relationship; v) Similarly, the values of b f and z were estimated by fitting another quadratic equation in the form of A f =(b m+ y m z )y m +b f (y y m )+z (y y m ) (where y>y m, b f b m +y m z ) to the stage mean flow area relationship. Hence, the Pierantonio Ponte Felcino river reach was approximated by a two-stage trapezoidal compound section geometry with b m =7.3 m, b f =57.6 m, y m =5 m, z =.98, and z =3.8 (see Fig. 5). Six flood events recorded in December 996, April 997, November 997, February 999, December, and November 5 with minimum lateral flow contribution (< % of the inflow volume) were used for the application of the method. It is presumed that for discharge routing by the method in field situations, only the normal rating curve at the upstream gauging site is known. This facilitates the calibration of the reach averaged Manning s roughness coefficient by matching the observed normal upstream rating curve with the routed normal discharge, Q 3 and corresponding stage, y M computed by Eqs (a) and (b). The approximated normal rating curve is synthesized for the upstream site by establishing the relationship between Q 3, estimated during the first subreach routing, and the corresponding stage y M at the upstream site. To reduce this approximation considerably to closely reproduce the actual rating curve existing for the year 996 997 pertaining to that of the upstream Pierantonio site, a routing reach of m with Δx =5 m is used herein. By a trial and error approach, the observed discharge and stage values of the normal rating curve at the Pierantonio section were matched closely with the routed stage and discharge values by the method. The calibrated n for this best matching was found as.43. Note that use of a single reach-averaged roughness may be considered a better option for field applications than considering a depth varying roughness that involves more parameter uncertainty. The calibrated rating curve for the Pierantonio section is illustrated in Fig. 6, which shows that the reproduction of the rating curve is very good except at higher flows above 7 m 3 /s wherein there is an error up to % in discharge estimation, which may be considered acceptable under field condition. The pertinent characteristics of the simulated events for the 5 km river reach of the Tiber River by using the calibrated Manning s n =.43, with a spatial resolution of Δx = km and a temporal resolution of Δt = 3 min, are illustrated in Table 3. Figure 7a l demonstrate the comparisons between the routed discharge and the computed stage hydrographs with the respective observed hydrographs at the downstream Ponte Felcino station. It is evidenced from Fig. 7a l and Table 3 that the discharge hydrographs of those events with no lateral flow Elevation (m) 8 7 6 5 4 3 - Pierantonio section Ponte Felcino section Trapezoidal section 3 4 5 6 7 8 9 Distance (m) Fig. 5 Cross sections of the upper Tiber River at Pierantonio (upstream) and Ponte Felcino (downstream) gauging stations (after Sahoo 7)

3796 B. Sahoo et al. Stage (m) 7 6 5 4 3 Existing (996-997) 4 6 8 Discharge (m 3 /s) Fig. 6 Calibrated upstream rating curve existing for the year 996 997 at the Pierantonio section contribution could be simulated well in comparison with the discharge hydrographs of those events where lateral flow occurs. However, corresponding to the initial routed baseflow at the downstream section, the computed stage is underpredicted. The plausible reason for the underprediction may be that the stage is a localized variable and is a function of the shape of the river cross section at any location and, hence, sensitive to minor local variations of the cross section shapes. Conversely, discharge is not much sensitive to the local variation of the river cross section. It can be seen from the trapezoidal section assumed for the reach as shown in Fig. 5 that between the flow depths.. m, the cross sectional area of the fitted trapezoidal section area is greater than the actual area at the downstream section. Hence, when the routed discharge is within this range of depth, the computed stage by the method, which is sensitive to this excess cross sectional area, estimates a low computed stage. Another reason for lower estimation of stage at the downstream section corresponding to the baseflow may be attributed to the use of unique Manning s roughness coefficient for all ranges of flow in a section. The unique n value has been determined from the consideration of overall reproduction of the downstream discharge hydrograph, and not the stage hydrograph. Generally, in natural rivers, the lower flow depth is subjected to more resistance due to rock outcrops and big boulders and the effect of which gets reduced when these resistant causing elements gets well submerged in the water. If the n value is slightly increased in the range of flow Table 3 Summary of performance criteria showing reproduction of pertinent characteristics of the flood events of the Tiber river by the method Events Discharge Routing Stage Computation Lateral inflow (%) t pqer (h) q per (%) EVOL (%) η q (%) t pyer (h) y per (%) η y (%) December 996.5 3.38.85 99.58. 4. 98.45. April 997..58.9 99.5..48 95.8. November 997. 8.8.37 99.8. 6.36 93.. February 999.5.34.6 95.8. 8.5 9.88. December. 9.45.3 9.85. 6.63 9.86 Flooding a November 5 5.5..98 9.44 5. 6.56 93. Flooding a a No discharge measurement was made

Rating Curve Development using Routing Method 3797 Discharge (m 3 /s) 4 3 (a) Dec. 996 Inflow Discharge (m 3 /s) 4 3 (g) Feb. 999 Inflow Stage (m) Stage (m) Discharge (m 3 /s) Stage (m) Discharge (m 3 /s) 4.5 3.5.5.5.5 5 4 3 4 3 5 4 3 3 4 5 (b) Dec. 996 Input stage 3 4 5 3 (c) April 997 Inflow 4 6 8 (d) April 997 Input stage 4 6 8 (e) Nov. 997 Inflow 3 4 (f) Nov. 997 Input stage 3 4 Time (h) Stage (m) Stage (m) Discharge (m 3 /s) Stage (m) Discharge (m 3 /s) 6 4 4.5 3.5.5.5.5 6 5 4 3 6 5 4 3 8 6 4 3 4 5 6 (h) Feb. 999 Input stage 3 4 5 6 (i) Dec. Inflow 4 6 8 (j) Dec. Input stage 4 6 8 (k) Nov. 5 Inflow 3 4 5 6 (l) Nov. 5 Input stage 3 4 5 6 Time (h) Fig. 7 Comparison plots of routed discharge and corresponding computed stage hydrographs by the method for the validated events from (a and b)december996,(c and d) April 997, (e and f) November 997, (g and h) February 999, (i and j) December, and (k and l) November 5 at the Ponte Felcino section

3798 B. Sahoo et al. 8 December 996 April 997 November 997 February 999 Decemebr November 5 Existing (996-997) Existing (998-) Normal (Manning's Eq.) 6 Stage (m) 4 5 3 45 6 75 9 Discharge (m 3 /s) Fig. 8 Reproduction of downstream rating curves by the method at the Ponte Felcino section for different flood events. The existing rating curves at this section for the years 996 997 and 998, and the normal rating curve computed by the Manning s friction law are also shown corresponding to the baseflow, the observed downstream flow depth would have been matched by the model estimated depth. However, that may affect the reproduction of the peak stage region. Therefore, use of unique value of n is acceptable for the overall reproduction of the observed discharge hydrograph, and the corresponding stage hydrograph may not be entirely reproduced in the same manner. Further, it is seen from Fig. 7a l and inferred from Table 3 that the lateral flow significantly distorts the matching between the observed stage hydrograph and the corresponding computed stage hydrograph. For example, it is seen from Fig. 7i that the volume of observed outflow is less than the inflow volume due to the detention of over-bank flood within the considered river reach and the detention storage is not accounted by the proposed method as the entire flow section is assumed to convey the floods. A similar inference may be made for the flood events shown in Fig. 7k, wherein over-bank storage takes place with a delayed release of the same. Therefore, the present extended method could be improved considering the over-bank storage in the reach. 6.4 Rating Curve Synthesis To enable the development of rating curve at the downstream ungauged Ponte Felcino section corresponding to 5 km, the given 5 km uniform routing reach is extended to 6 km, which is, then, divided into 4 subreaches of one kilometer each for the upstream portion of the reach, and the 5th subreach having a length of km with the middle of this subreach coinciding with the ungauged site located at 5 km from the upstream Pierantonio site. Routing of the given upstream discharge hydrograph in the given 5 km channel reach with such reach subdivision arrangement enables us to arrive at the conventional form of rating curve at the ungauged Ponte Felcino site. Fig. 8 shows the comparison between the simulated normal rating curve by the method and the existing normal rating curve at the Ponte Felcino hydrometric gauging station of the Tiber River. It is evidenced from Fig. 8 that the normal rating curve is well reproduced by the method for all ranges of flow.

Rating Curve Development using Routing Method 3799 7Conclusions With the involvement of high operational and maintenance costs of a hydrometric gauging station at all the locations in a river at all times, the river engineers and hydrologists confront a challenge in establishing rating curves at the gauged and ungauged river sites. This study was carried out with the objective of discharge routing, corresponding stage computation, and establishing the rating curve at an ungauged location by using the extended variable parameter Muskingum discharge hydrograph () routing method. It is assumed that only the normal rating curve at the upstream channel reach is known for calibration of the reachaveraged Manning s roughness which is, subsequently, used in the method to establish the normal rating curve at a downstream ungauged channel end. The routing results carried out in this study reveal that the method has the capability to synthesize the conventional normal rating curve being generated for the natural rivers at any location for field use. However, the numerical experiments reveal that the method should not be applied in small slope channel reaches with S o. which are subjected to backwater effects. Moreover, this method can very well estimate the stage and discharge hydrographs at any ungauged river site subject to the presence of no significant lateral flow in the river reach. Hence, the assumption of no lateral flow while routing the field observed hydrographs is a real restriction on the practical utility of this method. The extension of this method to accommodate lateral flow contribution along the routing reach forms the possible future work. This study can be helpful for planning and management of river water resources in both the diagnostic and prognostic modes which can be incorporated in the meso- and macro scale basin models and land-surface schemes of the climate change models. Acknowledgments The authors thank the Umbria Region, CNR IRPI Office of Perugia, Italy for providing the Tiber River data. References Ackers P (993) Stage-discharge functions for two-stage channels: the impact of new research. J Inst Water Environ Manag 7():5 6 ASCE (993) Task committee on definition of criteria for evaluation of watershed models of the watershed management committee, irrigation and drainage division criteria for evaluation of watershed models. J Irrig Drain Eng ASCE 9(3):49 44 Birkhead AL, James CS (998) Synthesis of rating curves from local stages and remote discharge monitoring using nonlinear Muskingum routing. J Hydrol 5( ):5 65 Danish Hydraulic Institute (DHI) (3) User s manual and technical references for MIKE (version 3b). Hørsholm, Denmark Ferrick MG (985) Analysis of river wave types. Water Resour Res ():9 Franchini M, Lamberti P, Giammarco PD (999) Rating curve estimation using local stages, upstream discharge data and a simplified hydraulic model. Hydrol Earth Syst Sci 3(4):54 548 Fread, DL (99) DAMBRK: The NWS Dam-Break Flood Forecasting Model. report, National Weather Service, Office of Hydrology, Silver Spring, Marryland, USA Garbrecht J, Brunner G (99) Hydrologic channel-flow routing for compound sections. J Hydraul Eng ASCE 7(5):69 64 Heatherman WJ (4) Muskingum Cunge revisited. World Water and Environmental Resources Congress 4, ASCE, Edited by G Sehlke, DF Hayes, DK Stevens (June July, 4), Salt Lake City, Utah, USA Henderson FM (966) Open channel flow. MacMillan and Co., New York

38 B. Sahoo et al. McCarthy GT (938) The unit hydrograph and flood routing. Conf. North Atlantic Div, U.S. Army Corps of Engineers, New London, Conn Moramarco T, Singh VP () Simple method for relating local stage and remote discharge. J Hydrol Eng ASCE 6():78 8 Nash JE, Sutcliffe JV (97) River flow forecasting through conceptual models, Part-: a discussion of principles. J Hydrol (3):8 9 Natural Environment Research Council (NERC) (975) Flood routing studies report, Vol. III. Institute of Hydrology, Wallingford O Sullivan JJ, Ahilan S, Bruen M () A modified Muskingum routing approach for floodplain flows: theory and practice. J Hydrol 47 47:39 54 Perumal M (99) The cause of negative initial outflow with the Muskingum method. Hydrol Sci J 37(4):39 4 Perumal M (994a) Hydrodynamic derivation of a variable parameter Muskingum method: Theory and solution procedure. Hydrol Sci J 39(5):43 44 Perumal M (994b) Hydrodynamic derivation of a variable parameter Muskingum method:. verification. Hydrol Sci J 39(5):443 458 Perumal M, Price RK (3) A fully mass conservative variable parameter McCarthy Muskingum method: theory and verification. J Hydrol 5:89. doi:.6/j.jhydrol.3.8.3 Perumal M, Ranga Raju KG (998a) Variable parameter stage-hydrograph routing method: II. Evaluation. J Hydrol Eng ASCE 3():5 Perumal M, Ranga Raju KG (998b) Variable parameter stage-hydrograph routing method: I. Theory. J Hydrol Eng ASCE 3():9 4 Perumal M, Ranga Raju KG (999) Approximate convection diffusion equations. J Hydrol Eng ASCE 4():6 64 Perumal M, Sahoo B (7) Applicability criteria of the variable parameter Muskingum stage and discharge routing methods. Water Resour Res W549 43(5):. doi:.9/6wr499 Perumal M, O Connell PE, Ranga Raju KG () Field applications of a variable parameter Muskingum method. J Hydrol Eng ASCE 6(3):96 7 Perumal M, Moramarco T, Sahoo B, Barbetta S (7) A methodology for discharge estimation and rating curve development at ungauged river sites. Water Resour Res W4 43():. doi:.9/5wr469 Perumal M, Sahoo B, Moramarco T, Barbetta S (9) Multilinear Muskingum method for stage-hydrograph routing in compound channels. J Hydrol Eng ASCE 4(7):663 67. doi:.6/(asce)he.943-5584. 9 Perumal M, Moramarco T, Sahoo B, Barbetta S () On the practical applicability of the VPMS routing method for rating curve development at ungauged river sites. Water Resour Res 46(3): 9. doi:.9/ 9WR83, W35 Price RK (973) Flood routing methods for British rivers. Proc Inst Civ Engrs 55():93 93 Price RK (985) Flood routing. In: Novak P (ed) Developments in Hydraulic Engineering. chap. 4. Elsevier, New York, pp 9 73 Price RK (9) Volume conservative non-linear flood routing. J Hydraul Eng ASCE 35():838 845 Sahoo B (7) Variable parameter flood routing methods for hydrological analyses of ungauged basins. Ph.D. Thesis. Department of Hydrology, Indian Institute of Technology Roorkee, Roorkee, India Sahoo B (3) Field application of the multilinear Muskingum discharge routing method. Water Resour Manag 7:93 5. doi:.7/s69--8-5 Tang X, Knight DW, Samuels P (999) Variable parameter Muskingum-Cunge method for flood routing in a compound channel. J Hydraul Res IAHR 37(5):59 64 Todini E (7) A mass conservative and water storage consistent variable parameter Muskingum-Cunge approach. Hydrol Earth Syst Sci :645 659 US Army Corps of Engineers (6a) HEC-RAS river analysis system user s manual (version 4. beta). Rep. CPD-68, Hydrol Eng Cent, Davis, Calif US Army Corps of Engineers (6b) Hydrologic modeling system HEC-HMS user s manual (version 3..). Rep. CPD-74A, Hydrol Eng Cent., Davis, Calif Weinmann PE (977) Comparison of flood routing methods for natural rivers. Report No. /977, Department of Civil Engineering, Monash University, Victoria, Australia Weinmann PE, Laurenson EM (979) Approximate flood routing methods: a review. J Hydraul Div ASCE 5():5 56