Figure 4. Influence of alpha coefficient on Mean Absolute Error
(observed vs LSPIV-estimated discharge)
The choice of a depth averaging coefficient (α ) had a significant
influence on the accuracy of our discharge estimates (Figure 4). At both
reaches A and B, we experimented with values between 0.5 – 1.0 to
depth-average our satellite-based PIV velocity estimates, previous
studies have found that α values of between 0.8 – 1 are
appropriate for computing depth-averaged velocities in natural rivers
with a depth of greater than 2 m (Hauet et al., 2018; Vigoureux et al.,
2022). At reach A, α values in the range 0.8-0.9 minimize the
difference between PIV-derived discharge and gauged discharge to within
15% (Figure 4a). At reach B a narrow band of α values in the
range 0.94 - 0.97 minimize the error, and values in the range 0.9 - 1.0
result in MAE < 10%.
Discussion
LSPIV velocity estimation
Our sensitivity analysis (section 3.2.1) highlighted the fundamental
significance of frame sampling frequency when computing LSPIV
velocities, similar to other LSPIV field studies (e.g. Legleiter and
Kinzel, 2021; Muste et al ., 2008; Pearce et al ., 2020). In
lieu of reference field measurements to assess the accuracy of our
velocity estimates, we conducted a direct comparison to those of a 2D
model (HEC-RAS) simulation of the flood event at Tilpa.
Statistical analysis of LSPIV velocity deviations (using our best-case
scenario of 0.25 Hz processed using the FFT algorithm) showed that LSPIV
tended to underestimate velocities as compared to the 2D model
predictions. However, we propose that this approach of assessing PIV
velocities enabled us to sample the spatial patterns of velocity equally
and capture a diversity of velocities when making quantitative
comparisons. We acknowledge the inherent limitations of 2D models due to
assumptions and simplifications of shallow water equations as well as
documented uncertainties of subgrid scale turbulence (Dewals et al.,
2023; Pasternack, 2011). In addition to uncertainties associated with
boundary conditions as well as choice of model parameters (primarily the
Manning’s roughness coefficient), 2D models have also been reported to
underrepresent velocity distributions due to errors in terrain and
bathymetry data (Bates, 2022), although these were minimized here as
LiDAR was acquired when the river bed was dry. Nevertheless, calibrated
2D models are still a viable means to assess PIV velocities for cases
where flows exceed the safe operating ranges of conventional sensors.
Thus, our approach presents the opportunity to avoid extrapolation of
rating curves for high river flows which cannot be measured directly
using conventional instruments. Although optical space-based video
sensors are still constrained by cloud cover and limited in spatial
resolution, advances in computer vision techniques, including image
super-resolution and deep-learning based cloud removal present avenues
to further refine satellite based LSPIV workflows.
Discharge accuracy assessment
LSPIV-based surface velocities, combined with preexisting, independent
information on channel bathymetry, have been successfully used to obtain
river discharge estimates in previous studies (e.g. Le Coz et
al ., 2010; Lewis et al ., 2018). Using the velocity-area
technique, we find our discharge estimates, on average, to be as close
as within 0.3% of gauged discharge (Table 2), assuming that our
topographic data accurately captures channel geometry. Absolute river
discharges obtained solely from satellite-based LSPIV velocities yielded
acceptable results, with a maximum mean absolute error of 35% which
could be reduced to 0.3% by tuning α . The accuracy and precision
of our reported discharge estimates compare favorably with those
documented by Sun et al . (2010) and Lewis et al . (2018)
who computed river discharges using LSPIV-based measurements to within
-5 to 7% and < 20% respectively.
The ephemeral nature of the River Darling at Tilpa is advantageous for
acquiring high-accuracy bare-earth topography, here using airborne
LiDAR. In other ephemeral locations, lower resolution datasets with
near-global coverage could be used, such as the SRTM, MERIT and ASTER
DEMs, depending on river flows when data were acquired. In temperate and
tropical locations, direct bathymetric surveys (e.g. echo sounding) or
bathymetry derived from multispectral satellite imagery (limited to
shallow clear waters) and altimetry (which only gives information on
water surface elevation) (e.g., Liu et al ., 2020; Moramarcoet al ., 2019) would be required as a precursor to discharge
estimation using satellite video. Despite these additional data demands,
our results demonstrate that satellite-based optical video sensors could
be deployed for near real-time estimation of riverine velocity and
discharge after extreme events within tolerable uncertainties common to
traditional discharge estimation techniques.
Variability of surface coefficient values, α
Our satellite-video based LSPIV discharge estimation procedure yielded
promising results, in terms of absolute flow magnitude, but the
selection of the coefficient (α ), used to convert surface to
depth-averaged velocities, remains a key source of uncertainty in
discharge estimation (Figure 4). Fulton et al . (2020), Moramarcoet al . (2017) and Welber et al . (2016) all observed local
variability of α (0.52 – 0.78; 0.85 – 1.05 and 0.71 – 0.92
respectively) when estimating discharge using non-contact techniques,
attributable to variations in stage (especially during higher flows due
to changes in wetted channel perimeter), channel geometry, slope, and
channel alignment. Significant shifts in the absolute error of
LSPIV-based discharges due to variations in α indicated that
sufficient cross-section specificity in defining α is critical to
our technique. We observe, on average, higher values of αminimize the uncertainty of our discharge estimates in Reach B as
compared to Reach A, attributable to the fact that LSPIV velocities were
higher as compared to our benchmark velocities (from the 2D model). When
computing flood discharge using non-intrusive methods, Hauet et
al . (2018) established a proportional link between α and a
river’s hydraulic radius with a mean value of α = 0.8 (with an
uncertainty of ± 15% at 90% confidence level) being recommended for
natural rivers with depths of less than 2 meters, and α = 0.9 for
rivers of greater depth. We observe relatively modest intra-measurement
variability when varying α at the respective reaches, which can
be explained by relatively uniform flow thanks to the generally straight
and simple channel morphology of each reach. In the absence of an
empirically formulated α specific to a river channel and based on
in-situ velocity measurements, the extent to which α varies
remains poorly understood (Legleiter et al ., 2023). When
estimating flood flows in remote locations where remote sensing
instruments are the sole source of depths (i.e., derived from a DEM),
experimenting with values provided by Rantz, (1982) (α = 0.85 or
0.86), Turnipseed and Sauer (2010) (α = 0.84 – 0.90), and, in
extreme cases, α > 1 due to non-standard velocity
distributions (see, for example, Moramarco et al ., 2017) is a
sensible approach to improve the precision of flow measurements from
surface velocimetry techniques. On average, in our study the αvalues that led to the closest approximations of observed discharge were
all less than unity, indicating our velocity distributions could be well
approximated using logarithmic or power laws. The variability of our
best fitting, cross-section averaged α at our reaches implies
that the commonly used default value of 0.85 is not always appropriate
in field conditions where spatial heterogeneities in channel beds have a
significant impact on velocity profiles. Although we provide a method
for assessing the variability of α , calibration of site-specificα values based on traditional contact measurements remains the
preferred solution for accurate discharge estimation.
Conclusion
We demonstrate that river discharge can be estimated, within acceptable
error, using velocities estimated from satellite-collected video.
Discharge estimates obtained using satellite video based LSPIV
velocities and channel bathymetry data ranged between 0.3% to 35.4% of
observed discharge. Sensitivity tests affirmed the fundamental role for
the depth averaging coefficient, α , when translating surface
LSPIV velocities into depth-averaged velocity estimates. Advances in
satellite sensor technologies hold the promise of even higher
temporal/spatial resolution video which will likely enable better
approximations of river discharge. The scientific and socio-economic
implications of our study are important as the absence of in-situ river
velocity measurements and other a priori information have been a
long-standing barrier in the remote quantification of river flows in
ungauged basins. The synergy of new generation video sensors and
non-intrusive techniques for estimation of riverine velocities during
extreme flows will enrichen the availability of data for flood
forecasting and water resources management.