Rating curves: from stage to discharge, and how far to trust them

Almost every discharge number reported for a river is calculated, not measured. A level sensor records stage, and a rating curve converts it to discharge. The curve is built from a small number of manual measurements and then used, unattended, for years, including in floods bigger than any it was built from. That is convenient and often accurate. It is also where hydrology hides its largest, least visible errors. This article explains how a rating curve works, where it fails, and how to use it honestly.

What a rating curve is

For a stable section with a well-defined control (a riffle, a rock sill, a channel constriction that fixes the relation between water level and flow), discharge is a smooth function of stage. The standard form is a power law:

Q = C · (h − h0)^b

where h is the stage, h0 the stage at which flow stops, C a constant and b an exponent set by the shape of the control, from about 1.5 for a rectangular weir to about 2.5 for a triangular one, with natural controls in between. The WMO Manual on Stream Gauging stresses that the quality of the relation depends on the control. Where the control is stable, the relation is stable. Where it is "subject to change", such as a sand or gravel riffle that shifts, the relation must be recalibrated, and the Manual notes that gravel and sand bar riffles are likely to shift and that vegetation growth also affects the control [1].

Building one from gaugings

You measure discharge directly at a range of stages, by current meter, float or radar, and fit the curve to those gaugings. A worked illustration, with eight gaugings that include about ±4 % random scatter (to mimic real measurements):

Taking logarithms turns the power law into a straight line, ln Q = ln C + b · ln(h − h0). Scanning for the h0 that gives the best straight line and fitting by least squares gives h0 = 0.28 m, b = 1.98 and C = 13.2. The data were generated with h0 = 0.30, b = 1.9 and C = 14.0, so the fit has recovered the control well within the scatter, over the gauged range.

The danger zone: extrapolating beyond the gaugings

A flood is, by definition, higher than most of the days the gaugings were made on. It is also when no one can safely stand in the river with a meter. So the curve is extrapolated. In the example, the highest gauging is at 2.6 m. At a stage of 6 m, the fitted curve gives 418 m³/s against a true value of 382 m³/s, an error of +9 % (and +12 % at 8 m), and the discharge is 5.6 times what was ever measured.

The reason is the exponent. The fitted b carries an uncertainty, and the effect of an error in b grows with the logarithm of how far the depth is from the middle of the gauged range. With a pivot (the geometric mean depth over the control) of about 0.9 m in the example:

A change in the way water spreads onto a floodplain, a bridge or a bend that begins to control the flow at high stage changes the curve itself, and that is not captured by any statistic from the gaugings. Published reviews of the topic list extrapolation beyond the measured range among the largest sources of error in indirect discharge estimates [2], and one numerical study of the River Po found that using a steady rating curve in the presence of significant flood waves produced discharge errors that may exceed 15 % [2].

Rule of thumb. Treat discharge above the highest gauging as an estimate, flag it as extrapolated in the record, and aim to gauge the biggest floods you can safely reach. Floats, surface velocity radar and indirect methods exist for exactly this purpose [1].

Shifts: when the control changes

A rating curve describes the channel as it was when gauged. The channel changes:

• Scour and fill of the bed during floods, which is common in steep, sediment-rich hill rivers.
• Vegetation and debris growing or lodging at the control.
• Ice in cold regions.
• Backwater from a downstream confluence, dam or bridge.
• Human changes such as sand mining or river training.

Published reviews list these as the main reasons stage-discharge relations change with time [2][3]. Practices to manage them:

1. Gauge regularly, and always after a major flood, and compare each gauging with the current curve.
2. Track the shift (the stage or discharge difference between a gauging and the curve) over time, and apply it as a dated correction.
3. Keep every version of the rating with the dates it was valid.
4. Make the stage record trustworthy first. An error in stage carries into every discharge. See radar, ultrasonic, float and pressure sensors compared.

Hysteresis: the loop in the rating

During a flood wave, the relation between stage and discharge is not single-valued. On the rising limb the water surface is steeper and discharge is higher than the steady curve gives for the same stage. On the falling limb it is lower. Plotted, the points trace a loop. The USGS notes that "all rivers and streams have some form of hysteresis" and that it matters most for low-gradient streams, streams with variable backwater, large channel or overbank storage, and highly unsteady flow [3].

The classical correction is the Jones formula, derived from the momentum equation under a kinematic-wave assumption [2]:

Q = Q_s · √( 1 + (1 / (S0 · c)) · ∂h/∂t )

where Q_s is the steady-state discharge from the standard curve, S0 the bed slope, c the flood-wave celerity and ∂h/∂t the rate of change of stage. The term in brackets shows the whole story: the correction is large when the slope is small and the rate of change is large. Worked values:

Steep hill rivers, even with rapid rises of several metres per hour, show a small loop, because the large slope dominates. In those rivers the larger source of error is a shifting bed. In gentle rivers the loop is the larger error and a steady rating can be badly wrong during a flood. The USGS has developed dynamic rating methods that capture hysteresis using stage, channel geometry, roughness and slope, and tested them at ten streamgages, with mean discharge percent errors between about 0.2 and 14 % across sites [3].

Putting uncertainty on discharge

A defensible discharge value carries an uncertainty that combines several terms:

• Stage: ΔQ/Q ≈ b · Δh / (h − h0), large at low flow. A 5 mm stage error at a section with a stage sensitivity of 0.5 mm per percent of discharge gives 10 % [1].
• Gaugings: the gauged discharges themselves have uncertainty. Published studies commonly assign a few percent to a good current-meter gauging, and more to floats, radar and gaugings in poor conditions [4].
• The curve: scatter of gaugings about the fitted curve, and the uncertainty of b, h0 and C.
• Extrapolation, as above.
• Shifts and hysteresis not captured by the curve.

Combine independent terms in quadrature, as in √(a² + b² + c²), and state the result with the value (for example, "±8 % within the gauged range, ±20 % above 2.6 m"). Good practice also shows a band around the curve and marks values beyond the highest gauging.

For flood warning, use stage, not discharge

A warning does not need a rating curve. People are threatened by the water level reaching their house, their bridge or their field. A stage threshold, with a rate-of-rise trigger, is directly observable and independent of every uncertainty above, and the record includes what the river actually did. Discharge matters for water accounting, reservoir operation, flood frequency analysis and hydrological models, and there the curve, its version history and its uncertainty matter. A well-run station reports both, and says clearly which is measured and which is derived.

Alternatives that avoid the rating curve

• Index-velocity. Measure velocity at a fixed point as well as stage, and relate the velocity to mean velocity. This removes the dependence on a stable control and handles backwater and hysteresis better, at the cost of more calibration. A radar can supply the velocity, as described in surface velocity radar for rivers.
• Dynamic ratings that include slope or unsteady terms [3].
• Stage-fall-discharge with two gauges where backwater is common.

Where ClimMaTech fits

Tarang provides the stage record, with ±1 mm accuracy and 0.5 mm resolution so that the first term in the uncertainty is small. Pravaah adds surface velocity for index-velocity work, and DeviceCloud keeps readings and metadata together. Turning stage into discharge needs gaugings and a surveyed cross-section for each site, which we help plan. See can radar measure river discharge for what is and is not possible.

References

1. World Meteorological Organization, Manual on Stream Gauging, Volume I: Fieldwork, WMO-No. 1044 (2010), Chapter 3 (gauging station controls, sensitivity) and discussion of floats and radar. WMO publication page.
2. F. Dottori, M. L. V. Martina, E. Todini, A dynamic rating curve approach to indirect discharge measurement, Hydrology and Earth System Sciences 13 (2009), 847-863. hess.copernicus.org.
3. U.S. Geological Survey, Dynamic Rating Method for Computing Discharge and Stage from Time-Series Data, Scientific Investigations Report 2024-5129. pubs.usgs.gov.
4. A comparison of methods for streamflow uncertainty estimation, U.S. Geological Survey. usgs.gov.

The rating-curve example uses synthetic gaugings generated with a known curve plus random scatter, so that the effect of extrapolation can be shown against a known truth. Other worked values are computed from the stated relations.