Discharge uncertainty

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Typical quantitative results of discharge uncertainty studies: Discharge Uncertainty.

This table originated from McMillan et al. (2012) but is now open to the community to add to and use as a resource.

PDF = probability density function; RMSE = root mean square error; SD = standard deviation

Uncertainty Type Estimation Method Magnitude Location Reference
Instantaneous Discharge Uncertainty
Single discharge measurement uncertainty when using method of verticals with current meter SD of relative discharge error calculated from individual uncertainty components 2.3 % using 30 verticals with measurements at 0.2 & 0.8 depth points; other combinations also given Columbia River, USA (5 sites) Carter & Anderson (1963)
Single discharge measurement uncertainty using velocity-area method 95 % confidence bounds on relative uncertainty, from literature review 4-17 % for 35-5 verticals at 0.25 m s-1; 5-40 % for velocities 0.5-0.05 m s-1. Various Pelletier (1988)
Single discharge measurement uncertainty under ice Difference between USGS & Water Survey of Canada instantaneous flow measurements attributed to different setup of current meter on rod or in suspension 2-17 % Red river at Emerson, Manitoba, Canada (104000 km2). Slope 0.04-0.25 m km-1, mean discharge 94.2 m3 s-1, when under ice 20 m3 s-1, drains glacial plain with moraines. Pelletier (1989)
Combination of stage error & components of discharge error for wading or cable methods Standard error computed by root-mean-square of component uncertainties: those derived from previous studies, manufacturer citations and expert knowledge. 2.4 % (Good Cable); 4.0 % (Good Wading); 19 % (Poorest measurements) Sauer & Meyer (1992)
Single discharge measurement uncertainty: effect of reducing number of verticals Halving number of verticals Approx. 5 % (given as graph relating to % reduction in verticals) 23 sites in UK North-East Whalley (2001)
Epistemic single discharge measurement uncertainty using current meter for velocity-area method Combined uncertainty values from expert opinion & previous studies 6 % Typical example Herschy (2002)
Single discharge measurement uncertainty: Salt dilution gauging SD of instantaneous discharge measured using salt dilution, deviation from rating curve developed using both salt dilution and current metering. 5 % Stephanie Creek, Vancouver Island, BC, Canada (8.6 km2). Steep rocky creek. Hudson & Fraser (2002)
7.1% Flume Creek, Sunshine Coast, BC, Canada (118 ha). Steep creek.
±42-84 % South Fork catchment (780 km2), Iowa, USA
Single discharge measurement uncertainty Typical bias determined from replicates <-4 % Hamilton & Moore (2012)
Rating Curve and Combined Uncertainty
Random errors associated with power law rating curves RMSE of component uncertainties 1.9 % in instantaneous or average daily discharge, 0.5 % in average monthly discharge Mangawhero at Ore Ore, New Zealand. Mean discharge 13m3 s-1 Dymond & Christian (1982)
Deviation between theoretical & measured rating curve (with current meter) 20 % at low flows (0.2 m above station datum), 10 % at higher flows Sprint, UK. Flat-vee crump profile weir structure. Whalley (2001)
Deviation between theoretical rating curve accounting for non-steady flow & measured discharge (also given for empirical rating curve) Coefficient of variation calculated from 55 discharge measurements 10 % (in-bank flows); 36% (including out-of-bank flows) Illinois River, USA. Low gradient river, discharge 38-3480 m3 s-1, two gauge (slop-stage-discharge) rating station. Schmidt & Yen (2008)
Total instantaneous discharge uncertainty caused by interpolation / extrapolation of rating curve, unsteady flow conditions & seasonal changes in roughness 95 % uncertainty bounds for relative error calculated through combination of three error components 6.2 % at 1000 m3 s-1 to 42.8 % at 12000 m3 s-1, average 25.6 % Po River, Italy (70000 km2). Channel width 200-500 m, depth 10-15 m, slope 0.02, floodplain width 1000-3000 m. Di Baldassarre & Montanari (2009)
Total instantaneous discharge uncertainty caused by rating curve uncertainty Relative error compared to manual measurements 1-20 % (average 8.76 %), negatively related to stage Hillslope (172 m2), WS10 catchment, HJA Experimental Forest, Oregon, USA. Stilling well with 30° V-Notch Weir. Graham et al. (2010); values calculated from original figures
Average 3.6 %, not related to stage WS10 catchment (10.2 ha), HJA Experimental Forest, Oregon, USA. 90° V-Notch Weir
Total instantaneous discharge uncertainty caused by gauging errors & rating curve form / extrapolation Estimate of upper & lower discharge bounds for any given stage through combination of component errors Relative error from 100 % (low flows) to 10 % (low-mid flows) to 20 % (high flows) Rowden Experimental Research Platform (1 ha fields), Devon, UK. 250 x 37 cm weir box, stainless steel 45° V-Notch, bucket method & electromagnetic flowmeter (Magflo Mag 5100, Siemens), ave. annual precipitation 1055 mm. Krueger et al. (2010)
Total instantaneous discharge uncertainty caused by gauging error, rating curve form / extrapolation & instability of rating curve Estimate of complete instantaneous discharge PDF for any given stage Relative error from 46 % (low flows) to 10 % (mid-high flows) to 15 % (flood flows), average 22 % Wairau River, New Zealand (3825 km2). Elevation 0-2309 m a.s.l., braided reach, 100 m width. McMillan et al. (2010)
Total instantaneous discharge uncertainty caused by gauging error & instability of rating curve Estimates of upper & lower instantaneous discharge bounds for any given stage using uncertain time-varying rating curve Difference from constant rating curve ranged from -60 to 90 % (low flows) to ±20 % (mid-high flows); mean daily discharge error -43 % to +73 %. Effect of using only 3 stage measurements / day to calculate mean daily discharge: ±17 % Choluteca River, Honduras (1766 km2). Mountainous, 660 – 2320 m a.s.l., precipitation mainly convective. Westerberg et al. (2011)
Time-averaged Discharge Uncertainty
Total uncertainty of daily discharge PDF, mean, SD Normal, 0, 10 % Odense basin (1190 km2), Denmark. Low rolling hills, elevation 0-100 m a.s.l. Refsgaard et al. (2006)
Relative uncertainty of daily & annual discharge estimates in rivers subject to icing Statistical analysis of uncertainty in the parameters of the fitted quadratic rating curves & ice correction coefficients Where cross sections assumed stable: 8-25 % for low flows, 2-5 % for high flows (variation for different rivers); where cross section not stable (e.g. with ice): 10-21 % with high frequency gaugings, 15-45 % under the worst conditions in the record 6 largest Eurasian Arctic Rivers (248000-2950000 km2). Mean discharge 2200-18400 m3 s-1. Shiklomanov et al. (2006)
Monthly discharge uncertainty Probable error range ±42 % Small watershed near Riesel, Texas, USA Harmel & Smith (2007) based on Harmel et al. (2006)
Daily discharge uncertainty ±42 %; ±100-200 % for low flows; ±100 % for high flows Reynolds Creek catchment (239 km2), Idaho, USA
Storm discharge uncertainty Total probable error based on RMSE propagation method 2-19 % Various in USA (2.2-5506 ha) Harmel et al. (2009) based on Harmel et al. (2006)
Deep seepage uncertainty in steady state (as residual water balance component) Relative uncertainty based on propagation of component uncertainties 57 % (under steady state); 32 % (during irrigation); 34 % (during irrigation + 5 days); 35 % (during irrigation + 10 days) Hillslope (172 m2), WS10 catchment, HJA Experimental Forest, Oregon, USA. Stilling well with 30° V-Notch Weir. Graham et al. (2010); values calculated from original figures
84 % (under steady state); 62 % (during irrigation); 93 % (during irrigation + 5 days); 155 % (during irrigation + 10 days) WS10 catchment (10.2 ha), HJA Experimental Forest, Oregon, USA. 90° V-Notch Weir
Daily discharge; effect of manual stage reading Manually minus automatically derived discharge Up to ±10-50 % Lillooet River near Pemberton, British Columbia, Canada. Nivo-glacial. Hamilton & Moore (2012)
Monthly discharge; effect of manual stage reading Up to 5-10 %


Carter, R.W., Anderson, I.E., 1963. Accuracy of current meter measurements. Journal of the Hydraulics Division, 89(4): 105-115.

Di Baldassarre, G., Montanari, A., 2009. Uncertainty in river discharge observations: a quantitative analysis. Hydrology and Earth System Sciences, 13(6): 913-921.

Dymond, J.R., Christian, R., 1982. Accuracy of discharge determined from a rating curve. Hydrological Sciences Journal-Journal Des Sciences Hydrologiques, 4(12): 493-504.

Graham, C.B., van Verseveld, W., Barnard, H.R., McDonnell, J.J., 2010. Estimating the deep seepage component of the hillslope and catchment water balance within a measurement uncertainty framework. Hydrological Processes, 24(25): 3878–3893.

Hamilton, A.S., Moore, R.D., 2012. Quantifying Uncertainty in Streamflow Records. Canadian Water Resources Journal, 37(1): 3-21.

Harmel, R.D., Cooper, R.J., Slade, R.M., Haney, R.L., Arnold, J.G., 2006. Cumulative uncertainty in measured streamflow and water quality data for small watersheds. Transactions of the ASABE, 49(3): 689-701.

Harmel, R.D., Smith, D.R., King, K.W., Slade, R.M., 2009. Estimating storm discharge and water quality data uncertainty: A software tool for monitoring and modeling applications. Environmental Modelling & Software, 24(7): 832-842.

Harmel, R.D., Smith, P.K., 2007. Consideration of measurement uncertainty in the evaluation of goodness-of-fit in hydrologic and water quality modeling. Journal of Hydrology, 337(3-4): 326-336.

Herschy, R.W., 2002. The uncertainty in a current meter measurement. Flow Measurement and Instrumentation, 13(5-6): 281-284.

Hudson R, Fraser J., 2002. Alternative methods of flow rating in small coastal streams. Forest Research Extension Note EN-014 (Hydrology). Vancouver Forest Region.

Krueger, T., Freer, J., Quinton, J.N., Macleod, C.J.A., Bilotta, G.S., Brazier, R.E., Butler, P., Haygarth, P.M., 2010a. Ensemble evaluation of hydrological model hypotheses. Water Resources Research, 46: W07516.

McMillan, H., Freer, J., Pappenberger, F., Krueger, T., Clark, M., 2010. Impacts of uncertain river flow data on rainfall-runoff model calibration and discharge predictions. Hydrological Processes, 24(10): 1270-1284.

McMillan, H., Krueger, T., Freer, J., 2012. Benchmarking observational uncertainties for hydrology: Rainfall, river discharge and water quality. Hydrological Processes 26(26): 4078–4111

Pelletier, P.M.., 1988. Uncertainties in the determination of river discharge: A literature review. Canadian Journal of Civil Engineering, 15: 834-850.

Pelletier, P. M., 1989. Uncertainties in streamflow measurement under winter ice conditions a case study: The Red River at Emerson, Manitoba, Canada, Water Resour. Res., 25(8), 1857–1867, doi:10.1029/WR025i008p01857.

Refsgaard, J.C., van der Keur, P., Nilsson, B., Mueller-Wohlfeil, D.I., Brown, J., 2006. Uncertainties in river basin data at various support scales - Example from Odense Pilot River Basin. Hydrology Earth System Sciences Discussions, 3(4): 1943-1985.

Sauer, V.B., Meyer, R.W., 1992. Determination of error in individual discharge measurements, U.S. Geological Survey Open-File Report 92–144.

Shiklomanov, A.I., Yakovleva, T.I., Lammers, R.B., Karasev, I.P., Vörösmarty, C.J., Linder, E., 2006. Cold region river discharge uncertainty - Estimates from large Russian rivers. Journal of Hydrology, 326(1-4): 231-256.

Schmidt, A.R., Yen, B.C., 2008. Theoretical development of stage-discharge ratings for subcritical open-channel flows. Journal of Hydraulic Engineering-ASCE, 134(9): 1245-1256.

Westerberg, I., Guerrero, J.L., Seibert, J., Beven, K.J., Halldin, S., 2011. Stage-discharge uncertainty derived with a non-stationary rating curve in the Choluteca River, Honduras. Hydrological Processes, 25(4): 603-613.

Whalley, N., Iredale, R.S., Clare, A.F., 2001. Reliability and uncertainty in flow measurement techniques - Some current thinking. Physics and Chemistry of the Earth Part C-Solar-Terrestial and Planetary Science, 26(10-12): 743-749.