Abstract The calculation of discharge fromstage measurement is a fundamentalprocedure in surface hydrology. Thestage-discharge relationship at agauging station is usually representedby a rating curve determined from aseries of stage-discharge measure-ments. However, the establishment ofa reliable rating curve is timeconsuming and often the larger floodevents are not captured, such that therating curve is extrapolated. InNorway about 14 % of the dischargedata has to be extrapolated. Extra-polating beyond the upper limits ofthe established rating curve canclearly result in large errors. Toovercome these difficulties involvedwith the traditional method forderiving rating curves, the use of ahydraulic model in the form of theHEC-RAS package is proposed forconstructing a stage-discharge re-

lationship. We set out to investigatethe reliability and accuracy of themethod, how to measure the profilesefficiently and to gain experience inchoosing the best Manning’scoefficient. Survey data and onedischarge measurement from fourNorwegian gauging stations that havereliable rating curves were consideredin order to assess the adequacy of anHEC-RAS modelled rating curve.Generally, the results show that HEC-RAS modelling of stage-dischargecurves calibrated according to anassumed Manning’s number can beused as a useful and reliable tool athigh water levels but can beinaccurate at lower stages.

IntroductionThe amount of water passing agauging station, the discharge, istypically calculated by transformingthe recorded stage data to discharge

Setting up rating curves usingHEC-RAS

By Kari Svelle Reistad, Asgeir Petersen-Øverleir and Leif Johnny Bogetveit

Kari Svelle Reistad is senior engineerAsgeir Petersen-Øverleir is senior engineer

Leif Johnny Bogetveit is senior engineerHydrology Department, Norwegian Water Resources and Energy Directorate

P.O. Box 5091 Majorstua, N-0301 Oslo, Norway

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by the means of an estimated stage-discharge relationship. This relation-ship, referred to as the rating curve, isassumed to be a one-to-one relation-ship between the measured stage andthe discharge passing the gaugingstation. The rating curve is con-structed by measuring the dischargeand the corresponding stage atconvenient times. The measuredstage-discharge points are then fittedto a smoothing function, typically apower-law, which makes it easy totransform the automatically recordedstage into discharge time series.There are some problems associatedwith the traditional rating curvemethod: (1) Because it often takesseveral years before enough stage-discharge measurements are availablefor estimating an accurate ratingcurve, the procedure is expensive andtime consuming; (2) Due to theconcerns about the safety andapplicability during flood conditionsof the standard discharge measure-ments methods such as current meter,dilution and ADCP, large flood peaksdischarges are often beyond the upperlimits of the rating curve; (3) In smallcatchments the stage and dischargecan change very rapidly, especiallyduring floods, such that it becomesdifficult to link the measureddischarge to a unique stage value.

An alternative to the traditionalmethod is to determine the ratingcurve using indirect procedures basedon open-channel hydraulics. Thegeneral procedure for this is: (a) tomeasure channel cross-sections in thevicinity of the gauging station; (b)note the water surface at each cross-

section to define the slope betweenthe cross-sections; and (c) to definethe roughness of the reaches.Although the indirect approach isconsidered less accurate than thetraditional method because of theerrors in the surveying and in theevaluation of the channel roughness(Bathurst 1984, Jarret 1987, Kirby1987), it has sometimes beensuccessful (e.g. Kean and DunganSmith 2005).

This study presents the results fromfour gauging stations in Norwaywhere rating curves have been derivedusing an indirect method in the formof the general-case computer modelHEC-RAS 3.1.3, developed by theHydrological Engineering Center ofthe U.S. Army Corps of Engineers(USACE 2005). The accuracy of theHEC-RAS rating curves is assessedby comparing them to the measuredrating curves.

Data and model calibrationThe data were collected during field-work July-August 2005. The heightsin each section were measured with alevelling telescope. At the Strømstøastation we used an Acoustic Dopplerinstrument to measure the depths inthe sections in the riverbed. Thelongitudinal water surface was onlyrecorded once, at the same time as theprofiling. The measured water surfaceis also connected to a water dischargemeasurement in the modelling. Thefollowing table 1 gives a shortdescription of the gauging stationsand photos i figures 1-4 shows typicalsegments of the investigated sections.

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The simulation was performed usingHEC-RAS 3.1.3 (USACE 2005). Thisprogram performs one-dimensionalhydraulic calculations for i.a. steadyflow water surface profiles. It canmodel both sub- and supercritical aslong as mixed flow regime watersurface profiles. In our case study wewanted to model the water surface atthe gauging station, which means onehas to model the entire river reach thatis believed to affect the water level atthe gauging station. We thereforeincluded one cross-section upstreamof the gauging station as well asseveral cross-sections downstream.The river reach in most casescontained both sub- and supercritical

sections and was therefore modelledas a mixed flow regime. Themodelling was performed three timesfor each station: 1) The Manning’sroughness constant was selected onthe basis of observed type of channelcharacteristics (substrate, vegetation,etc) which were compared withdescriptions in the HEC-RAShydraulic manual (USACE 2002) andcase-studies by USGS (USGS 1987),2) the model was calibrated to best fitthe water surface measured in thecross-sections and 3) the model wascalibrated to best fit the stage-discharge curve at the gauging station.The results were then compared to seeif there are any common features.

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Strømstøa Magnor Nautsundvatn Bjørnegårdssvingen

Mean annual flow 300 39 170 42[m3/s]

Description of High degree ofriver bed embeddedness,

stones 10 cm + Outlet from small Mountain stream Clean channel, flat.rocks up to >1m lake (supercritical clean and steep artificially dammedmid-river. Flat flow), mouth bedrock channel. with large rocks.

gradient, no artificially dammed Determining profile Determining profilesupercritical flow with large rocks supercritical. supercritical.

sections.

Description of leftriver bank Dense forest Forest and brush Light brush Light brush

Description of Light forest and Brush and some Concrete wall andright river bank brush trees Brush and trees brush

Downstream reachboundary Normal depth Normal depth Critical depth Normal depthcondition

Table 1. Keywords for describing the hydrology, river bed and river banks and selectedboundary conditions at the 4 gauging stations.

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Figure 2. Determining profile at Bjørnegårdssvingen gauging station. Supercritical flowin profile at low water level.

Figure 1. Determining profile at Magnor gauging station. Supercritical flow in profile atlow water level.

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Figure 4. Determining profile at Strømstøa gauging station. Subcritical flow in profile.

Figure 3. Determining profile at Nautsundvatn gauging station. Supercritical flow inprofile at low water level.

Results and discussionMagnorAs showed in figure 5, the result ofthe simulation with pre-selectedManning’s n overestimates dischargeat low water levels and slightly under-estimates discharge at stages abovemean annual flow. The Manning’s nafter calibration simulating the

measured water surface only givesabout 1/3 of the discharge comparedto the stage-discharge curve at annualmean flow level. We were able toobtain a good fit to the stage-discharge curve at this station bychoosing a higher friction number inthe determining section and the nextsection upstream.

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Pre-selected Manning’s n Manning’s n after calibration Manning’s n after calibrationaccording to measured water according to stage-discharge

surface curve

Section LOB* Channel ROB* LOB Channel ROB LOB Channel ROBno.

5 0.073 0.045 0.06 0.07 0.05 0.06 0.07 0.035 0.064 0.073 0.045 0.06 0.07 0.05 0.06 0.07 0.04 0.063 0.073 0.045 0.06 0.07 0.1 0.06 0.07 0.08 0.062 0.073 0.055 0.06 0.07 0.11 0.06 0.07 0.11 0.061 0.073 0.045 0.06 0.07 0.06 0.06 0.07 0.04 0.060 0.073 0.045 0.06 0.07 0.14 0.06 0.07 0.035 0.06

*LOB: Left overbank channel, ROB: Right overbank channel

Table 2. Summary of Manning’s n selected at the Magnor gauging station before and aftercalibration. The station is situated in section 4. Section 2 was, at the time we did fieldwork(low water level) the determining section for the water level in section 4.

Figure 5. Comparison of the 3 calibrated models and the stage-discharge curve for thegauging station situated in section 4 at Magnor.

StrømstøaBoth the two first calibrations over-estimates discharge at low waterlevels and underestimates discharge atstages above mean annual flow. It wasalso impossible to match the curve atthe lowest level in the final calibrationbut it makes a good fit to the upper

part of the rating curve. The finalManning’s n values are generallylower than the pre-selected ones,which could be caused by a very highdegree of embeddedness and alsosmoothness caused by algae growth.This situation was not described inany of our reference literature.

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Figure 6. Comparison of the 3 steps of calibrated models and the stage-discharge curvefor the gauging station situated in section 10.

Pre-selected Manning’s n Manning’s n after calibration Manning’s n after calibrationaccording to measured water according to stage-discharge

surface curve

Section LOB* Channel ROB* LOB Channel ROB LOB Channel ROBno.

11 0.1 0.04 0.06 0.1 0.04 0.06 0.07 0.04 0.0610 0.1 0.04 0.07 0.1 0.04 0.07 0.07 0.04 0.069 0.1 0.04 0.07 0.1 0.04 0.07 0.07 0.04 0.068 0.1 0.055 0.07 0.1 0.03 0.07 0.07 0.03 0.067 0.1 0.055 0.07 0.1 0.037 0.07 0.07 0.04 0.066 0.1 0.055 0.07 0.1 0.04 0.07 0.07 0.04 0.065 0.1 0.055 0.07 0.1 0.04 0.07 0.07 0.04 0.064 0.1 0.04 0.07 0.1 0.06 0.07 0.07 0.04 0.063 0.1 0.055 0.07 0.1 0.10 0.07 0.07 0.04 0.062 0.1 0.04 0.07 0.1 0.04 0.07 0.07 0.04 0.060 0.1 0.04 0.07 0.1 0.04 0.07 0.07 0.035 0.06

*LOB: Left overbank channel, ROB: Right overbank channel

Table 3. Summary of Manning’s n chosen before and after calibration at the Strømstøagauging station, situated in section 10. Section 8 was, at the time we did fieldwork (lowwater level) the determining section for the water level at the gauging station. There wereno sections with supercritical flow in this river reach.

Pre-selected Manning’s n Manning’s n after calibration Manning’s n after calibrationaccording to measured water according to stage-discharge

surface curve

Section LOB* Channel ROB* LOB Channel ROB LOB Channel ROBno.

300 0.04 0.03 0.045 0.04 0.045 0.045 0.055 0.05 0.0689.5 0.04 0.03 0.045 0.04 0.045 0.045 0.055 0.06 0.0667.5 0.04 0.03 0.045 0.04 0.045 0.045 0.055 0.08 0.0663.5 0.04 0.03 0.045 0.04 0.25 0.045 0.055 0.11 0.0644.5 0.04 0.03 0.045 0.04 0.25 0.045 0.055 0.06 0.06

0 0.04 0.03 0.045 0.04 0.2 0.045 0.055 0.11 0.06*LOB: Left overbank channel, ROB: Right overbank channel

Table 4. Summary of Manning’s n selected before and after calibration at the Nautsund-vatn gauging station, situated in section 89.5. Section 67.5 was, at the time we didfieldwork (low water level) the determining section for the water level at the gaugingstation. There was supercritical flow in both this section and section 0.

NautsundvatnThis station is situated in a typicalNorwegian mountain river. Theriverbed is comprised of smoothlyscoured bedrock. This situation wasalso not described in any of thereferences; the chosen Manning’snumbers are therefore based on thesuggested friction number for a built-up channel with concrete bottom fromthe HEC-RAS table (USACE, 2002).The calibration with this pre-decided

Manning’s n matches the curve bothat the bottom and the top, which isabove mean flow level but overesti-mates the discharge in the middle ofthe curve. The result after the secondcalibration to simulate the measuredwater surface generally underesti-mates the discharge. The final fittingmatched the curve at all levels. FinalManning’s n was higher than expec-ted, especially in the sections wheresupercritical flow was observed.

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Figure 7. Comparison of the 3 steps of calibrated models and the stage-discharge curvefor the gauging station situated in section 89.5.

Pre-selected Manning’s n Manning’s n after calibration Manning’s n after calibrationaccording to measured water according to stage-discharge

surface curve

Section LOB* Channel ROB* LOB Channel ROB LOB Channel ROBno.

50 0.05 0.035 0.045 0.05 0.04 0.045 0.05 0.035 0.04549 0.05 0.035 0.045 0.05 0.04 0.045 0.05 0.035 0.04548 0.05 0.035 0.045 0.05 0.04 0.045 0.05 0.035 0.04544 0.05 0.035 0.045 0.05 0.04 0.045 0.05 0.035 0.04534 0.05 0.035 0.045 0.05 0.04 0.045 0.05 0.035 0.04525 0.05 0.035 0.045 0.05 0.04 0.045 0.05 0.035 0.04510 0.05 0.035 0.045 0.05 0.17 0.045 0.05 0.035 0.0452.5 0.05 0.055 0.045 0.05 0.1 0.045 0.05 0.055 0.0450 0.05 0.035 0.045 0.05 0.04 0.045 0.05 0.035 0.045

-20 0.05 0.035 0.045 0.05 0.04 0.045 0.05 0.035 0.045*LOB: Left overbank channel, ROB: Right overbank channel

Table 5. Summary of Manning’s n selected before and after calibration at the Bjørne-gårdssvingen gauging station, which is situated in section 48. Section 2.5 was, at the timewe did fieldwork (low water level) the determining section for the water level at thegauging station.

BjørnegårdssvingenThis riverbed consists partly ofsmoothly scoured bedrock but in thedetermining profile (section 2.5) thereare also sharp rocks, while down-stream from the gauging station(section 48) the riverbed is covered bysmaller rocks and gravel. TheManning’s n was chosen from theHEC-RAS table (USACE 2002) witha higher friction number in the

determine profile. This station doesnot have a stage-discharge curve; theresults are therefore compared withthe discharge-measurements at thestation. The calibration with this pre-selected Manning’s n matched thecurve; we therefore chose these valuesas the final friction numbers. Theresult after the second calibration tosimulate the measured water surfaceunderestimates the discharge.

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Figure 8. Comparison of the 2 steps of calibrated models and the discharge measurementsfor the gauging station Bjørnegårdssvingen.

The simulation results after the cali-brations with pre-selected Manning’sn generally overestimated thedischarge at the lowest stagescompared to the original stage-discharge curve at all four stations.The results after calibration tosimulate the measured water levelwere unreliable at all of the stations.The fieldwork was done during thesummer that is the dry season;therefore all water surfaces are at lowwater levels. If calibrated to simulatethe water surface at higher waterlevels, the results would probably bedifferent. The pre-selected Manning’snumbers did not vary much from thefinal Manning’s n, but we underesti-mated the Manning’s numbers in thesections with supercritical flow.

ConclusionsCompared to the original stage-discharge curve, calibrations with pre-selected Manning’s roughnessnumbers have a tendency to giveslightly more water at low water-levels and slightly less water at stagesabove mean annual flow. Calibrationsaccording to the one measured watersurface gave, in every case, signifi-cantly underestimated discharge. TheManning’s n selected from the HEC-RAS Manning’s table (USACE 2002)or from the book “RoughnessCharacteristics of Natural Channels”(USGS 1987) provided goodestimates, except in sections withsupercritical flow were the valueswere underestimated. Generally, theresults show that HEC-RASmodelling of stage-discharge curves

calibrated according to assumedManning’s numbers can be used as auseful and reliable tool at high waterlevels but can be inaccurate at lowerstages.

Acknowledgements Considerable and generous help withsurveying was given by Kai Fjeldstadand Bjarne Kjølmoen. Beathe Sætherprovided excellent advice about theHEC-RAS program.

ReferencesBathurst, J.C. 1984. Slope-areadischarge gauging in mountain rivers.J. Hydraulic Eng. 112, 376-391.

Jarret, R.D. 1987. Errors in slope-areacomputations of peak discharges inmountain streams. J. Hydrol. 96, 3-67.

Kean, J.W., & Dungan Smith, J. 2005.Generation and verification of theo-retical rating curves in the WhitewaterRiver basin, Kansas. Wat. Resour. Res.,

Kirby, W.H. 1987. Linear erroranalysis of slope-area dischargedeterminations. J. Hydrol., 96, 125-138.

USACE, Hydrologic EngineeringCenter, Hydrologic Modeling SystemHEC-HMS, Technical ReferenceManual, September 2002.http://www.hec.usace.army.mil

USACE, Hydrologic EngineeringCenter, River Analysis System HEC-RAS, Hydraulic Reference Manualversion 3.1, September 2002.http://www.hec.usace.army.mil

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USACE, Hydrologic EngineeringCenter, River Analysis System, HEC-RAS program, version 3.1.3 May2005. http://www.hec.usace.army.mil/

USGS, US. Geological Survey,Roughness Characteristics of NaturalChannels, United States GovernmentPrinting Office,1987. http://wwwrcamnl.wr.usgs.gov/sws/fieldmethods/Indirects/nvalues/index.htm

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