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.