Surface velocity radar for rivers: Doppler physics, the velocity coefficient and noise

An AWVLR, an automatic river water level and velocity recorder, adds one measurement to a level station: how fast the surface is moving. The instrument is a microwave radar mounted over the river that never touches the water. It is simple to install, which makes it easy to forget that the number it reports is not the river's mean velocity and is not discharge. This article explains the physics, the geometry, the conversion to mean velocity, the failure modes, and the uncertainty you should attach to the result.

What a surface velocity radar measures

A Doppler radar transmits a continuous microwave signal at frequency f0 towards the water. Whatever it reflects off is moving, so the returned frequency is shifted by:

fd = 2 · v · f0 · cos θ / c

where v is the speed of the reflecting surface, θ the angle between the beam and the direction of motion, and c the speed of light. The radar measures fd and solves for v. Typical shifts for a 45° beam show why this works at all:

These are audio-range frequencies, easily resolved, which is why inexpensive radars measure velocity so well. A lower velocity limit comes not from the arithmetic but from the water, as the next section explains.

The radar does not see the water. It sees ripples.

Water is a good reflector of microwaves, but a perfectly flat surface reflects the beam away from a radar looking at an oblique angle, and returns almost nothing. What returns the signal is surface roughness: short ripples whose size matches the radar wavelength, a mechanism known as Bragg scattering. The WMO Manual on Stream Gauging states it directly: the radar receives scattered signals from random short waves that roughen the surface, the motion of those waves produces the Doppler shift, and "for surface velocities to be successfully measured by this technique, some water surface roughness must be present", generated by turbulent boils, wind or rain [1].

Three consequences follow:

1. The radar measures ripple speed, assumed equal to water speed. In practice the ripples are carried by the water but may also be pushed by wind and have their own propagation speed, which matters most when the water is slow [2].
2. No ripples, no signal. Smooth, slow, deep reaches can return too little. Fast, turbulent reaches return plenty, which is part of why the technique works well on hill rivers. One field study of fixed radars notes that mild water slopes, low stage and strong wind all degrade the signal, producing conspicuous noise and spikes [2].
3. A minimum measurable velocity exists. It is a property of the radar and the site. As one example, a widely used 24 GHz fixed radar is specified for 0.15 to 15 m/s [2]. Below the lower limit the data should be treated as unreliable.

Geometry: the cosine error

The radar measures the component of velocity along its beam, so the reported value depends on cos θ. If the angle used in the calculation differs from the true angle, the velocity is wrong by a fixed proportion. The sensitivity is d(cos θ)/cos θ = −tan θ · dθ, and it grows with the angle:

Two practical rules: measure the mounting angle with a calibrated inclinometer, not by eye, and make sure the beam is aligned with the flow in plan (a beam that crosses the current adds a second cosine term). A mount that settles by a couple of degrees over a monsoon changes every reading by several percent without any visible fault, so check the angle at every visit.

From surface velocity to mean velocity: the coefficient α

The velocity profile in a river runs from near zero at the bed to a maximum near the surface. The mean velocity in a vertical is therefore lower than the surface velocity, and the ratio is the velocity coefficient:

V_mean = α · V_surface

A value of about 0.85 is the commonly used default. The WMO Manual gives "a coefficient of about 0.85" for converting surface velocity to mean velocity, and notes that float gauging under favourable conditions may be accurate to within ±10 % [1]. A USGS study that compared radar systems with conventional measurements confirmed that mean velocity was approximately 0.85 times measured surface velocity across a range of flows, and reported agreement within about 5 % on the San Joaquin River after an initial adjustment period, and within 5 % for flood measurements on the Cowlitz River [3].

The coefficient is not a constant of nature. A field assessment of portable surface velocity radars found that 0.85 is a valid default but that site calibration is preferred, because α depends on the roughness of the bed relative to the flow depth: using the default tends to under-estimate discharge in very smooth channels (relative roughness below about 0.001) and over-estimate it in very rough channels (above about 0.05). With theoretically derived or site-calibrated values, the study reported discharge within about 10 % for low and intermediate roughness flows [4]. Older hydraulic work used values of 0.90 for large streams and 0.85 for smaller ones [5].

In steep, rough hill channels, α is likely to differ from 0.85, may change with stage, and should be derived from gaugings.

From velocity to discharge: what else you need

Discharge is the integral of velocity over the cross-section area:

Q = α · Σ ( v_s,i · A_i )

A single fixed radar measures the surface velocity at one point, usually near the middle of the channel, so converting to discharge uses an index-velocity relation: gauge the river at several stages, relate the radar velocity and stage to the true discharge, and use that relation afterwards. The WMO Manual describes the same logic: mean velocity multiplied by the channel area yields discharge, so "a means of computing channel area is needed", for example a surveyed cross-section combined with a stage-area relation [1]. The area comes from a survey, not from the radar.

A worked example. A channel 30 m wide with mean depth 2.0 m and a radar surface velocity of 2.4 m/s:

An uncertainty of ±0.05 in α alone is about ±6 % of discharge. Add a 2 % error in width and 3 % in depth (about 3.6 % in area) and 2 % in velocity, and the combined uncertainty, combining independent terms in quadrature, is about ±7 %: √(6² + 3.6² + 2²) ≈ 7.3 %. That is an honest, respectable figure for a noncontact measurement. It is not the ±1 % of the velocity specification, and publishing the smaller number as if it were the discharge uncertainty would be wrong.

Noise: what real radar data looks like

Radar velocity records are not smooth. A study of two bridge stations in Taiwan, with fixed 24 GHz radars sampling every minute over five years, showed what real data does: strong wind or low water at a mild slope produces spikes and noise in the frequency shift, and the nature of each spike is "too complex to easily identify filtering criteria", so a simple threshold is unsuitable. Even two radars a few metres apart over the same thalweg can differ, and a nearby opened gate or pipe can disturb the signal [2]. The study evaluated moving average, robust LOESS and related filters and found that robust local regression methods gave satisfactory results, in particular when the filter used the relationship between two radars or between velocity and water level to identify implausible values [2].

Practical guidance that follows from the literature and field practice:

• Look at wind. Record wind speed at the site and flag velocity data taken in strong wind. Published tests of radar surface velocimetry report that wind of a few metres per second can distort the reading at low flow by tens of percent of the surface velocity, depending on wind direction, while light wind has a far smaller effect [6]. The effect is largest where the river is slow.
• Choose the reach. Guidelines for operating radars advise avoiding wind-dominated reaches, eddies, secondary flows and macro-turbulence [2].
• Average and take the median. Report a median or mean over a burst, not a single sample.
• Use stage as a plausibility check. In a given reach, velocity and stage are related, so a velocity far from the usual value for that stage is suspect.
• Flag, do not delete. Keep the raw value and attach a quality flag.
• Know the limit on the high side too. At very high stage the water can reach the sensor. In the Taiwanese study a rising level that reached the bridge soffit made the radar data questionable for the event, a reminder that mounting height must exceed the highest flood [2].

Calibrating and checking an AWVLR

1. Gauge the discharge with a current meter or float method at several stages, including as high a flood as you can safely reach. Floats are the standard way to extend measurement into floods [1].
2. Compute α for each gauging as mean velocity divided by radar surface velocity, and look at how it varies with stage and roughness.
3. Fit the index relation between velocity, stage and discharge, and keep the gaugings that produced it.
4. Re-check each season and after floods, because the bed and cross-section change.
5. Compare two instruments where you can. Disagreement exposes a fault faster than any single instrument can.

Where ClimMaTech fits

Pravaah is our non-contact radar for AWVLR stations. It measures water level and surface velocity together, with a velocity range of 0.02 to 20 m/s and ±1 % of the measured velocity in the sensor itself, works from -40 °C to +70 °C, and reports over GSM (2G / 3G / 4G). What the radar measures is surface velocity. Turning that into discharge needs a surveyed cross-section and a calibrated velocity coefficient, which is why we plan the calibration with each client. See also can radar measure river discharge, rating curves and discharge uncertainty, and AWLR and AWVLR stations.

References

1. World Meteorological Organization, Manual on Stream Gauging, Volume I: Fieldwork, WMO-No. 1044 (2010), sections 8.9 and 8.10 and the discussion of float measurements. WMO publication page.
2. H.-W. Wang, G.-F. Lin, S. S. Tfwala, J.-H. Hong, Filtering Continuous River Surface Velocity Radar Data, Water 11(4), 764 (2019). mdpi.com.
3. J. E. Costa and others, Use of radars to monitor stream discharge by noncontact methods, Water Resources Research (2006). USGS.
4. Field assessment of noncontact stream gauging using portable surface velocity radars (SVR), Water Resources Research (2016). HAL.
5. 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, which reviews Jones's use of surface-to-mean velocity ratios of 0.90 and 0.85. hess.copernicus.org.
6. Conference paper on wind effects in radar surface velocimetry, Environment. Technology. Resources, Rezekne Academy of Technologies. journals.rta.lv.

Worked values (Doppler shifts, angle sensitivities, discharge example and error combination) are computed from the stated relations. They show the size of each effect and are not a specification of any product.