Hydraulic river models from ICESat-2 elevation and water surface slope

The water surface slope (WSS) of rivers is essential for estimating flow velocity and discharge. It is also helpful as a correction applied to range measurements of satellite altimetry missions to derive water level time series at a virtual station. Using radar altimetry, WSS can only be roughly estimated and is limited to wide rivers because of its coarse spatiotemporal resolution. In contrast, the lidar sensor onboard Ice, Cloud, and Land Elevation Satellite 2 (ICESat‐2) can also observe small rivers. Using ICESat‐2's unique measurement geometry with six parallel laser beams, we derive instantaneous WSS along and across the satellite's ground track, time‐variable WSS (with an average of 5 days of records in the studied epoch between October 2018 and October 2021), and average WSS on reach‐scale. Although the method can be applied globally, this study is limited to 815 reaches in Europe and North America where sufficient validation data is available. We compare the ICESat‐2 WSS with time‐variable WSS derived from multiple gauges and constant data from the “SWOT River Database.” For 89% of the studied reaches, ICESat‐2 can be used to estimate the average WSS with a median absolute error of 23 mm/km. We also show the possible performance gain at multiple virtual stations (VS) in the “Database for Hydrological Time Series of Inland Waters” (https://dahiti.dgfi.tum.de), applying the WSS as a correction for altimetry satellites' ground track variability. We correct 137 VS for the derived ICESat‐2 WSS and yield improvements in the root mean square error by up to 30 cm or 66%.

Water surface slope (WSS) of rivers is a key parameter in hydrological modelling, which allows for estimation of the transport and erosion capacity of a river, its flow velocity and discharge. On a local scale, WSS can be measured with a GNSS receiver installed on a boat, using remote sensing techniques (e.g. airborne lidar) or from a Digital Elevation Model (DEM). The most accurate method to measure WSS avoiding high-cost field campaigns is based on Water Surface Elevations (WSE) measured at in-situ stations. However, in poorly gauged rivers the neighboring gauges can be up to hundreds of kilometers apart, which inhibits a proper river profile observation. The gap in decreasing number of gauge readings is partially filled with satellite altimetry over rivers. Altimetry based WSE can be used to estimate WSS between neighboring measurements. Here, we present an innovative approach for estimating high-resolution WSS derived from multi-mission satellite altimetry for the largest Polish rivers.In this study, we used measurements from 9 altimetry missions: CryoSat-2, Envisat, ICESat-2, Jason-2/-3, SARAL, Sentinel-3A/-B, and Sentinel-6A. These observations cover the years from 2002 to 2022. We extracted the river centerlines from the global “SWOT Mission River Database” (SWORD). In order to validate the obtained results, we used WSE from 81 gauges, which are maintained by the Institute of Meteorology and Water Management – National Research Institute (Instytut Meteorologii i Gospodarki Wodnej – Państwowy Instytut Badawczy, IMGW-PIB). These measurements are referenced to the Kronsztadt’86 vertical datum and they range from 01.2016 to 05.2022. Additionally, we used the reach-scale “ICESat-2 River Surface Slope” (IRIS) and the DEM-derived WSS values from SWORD.To obtain WSS, we first determined WSE at each satellite pass crossing the studied river. Next, we split rivers into sections without dams and reservoirs. The Support Vector Regression (SVR) has been applied to reject outliers. Then, water levels were assigned to a given river kilometer (bin). For each of them a median WSE has been calculated. Finally, WSS were calculated at river sections between bins, excluding those disrupted by hydraulic structures. Finally, we weighted the section-wise WSS inversely proportional to the length of each section and applied a Least Square Adjustment with an additional Laplace condition to obtain bin-wise WSS for each river kilometer.To assess the accuracy of the proposed approach, we compared the obtained WSS with the slopes between IMGW-PIB gauges. For large rivers (Vistula, Odra, Warta), the multi-mission approach revealed high accuracy with preliminary Root Mean Squared Error (RMSE) below 30 mm/km. For smaller, mountain rivers (San, Dunajec) the preliminary errors were slightly larger (RMSE ~100 mm/km). We also compared our accuracies with those of the slopes based on DEM models, lidar data, ICESat-2 altimetry, and SWORD database. In general, the multi-mission approach revealed the highest accuracy. The research is supported by the National Science Centre, Poland, through the project no. 2020/38/E/ST10/00295.

Abstract. Calculating contributing area (often used as a proxy for surface water discharge) within a digital elevation model (DEM) or landscape evolution model (LEM) is a fundamental operation in geomorphology. Here we document the fact that a commonly used multiple-flow-direction algorithm for calculating contributing area, i.e., D∞ of Tarboton (1997), is sufficiently biased along the cardinal and ordinal directions that it is unsuitable for some standard applications of flow-routing algorithms. We revisit the purported excess dispersion of the multiple-flow-direction (MFD) algorithm of Freeman (1991) that motivated the development of D∞ and demonstrate that MFD is superior to D∞ when tested against analytic solutions for the contributing areas of idealized landforms and the predictions of the shallow-water equation solver FLO-2D for more complex landforms in which the water surface slope is closely approximated by the bed slope. We also introduce a new flow-routing algorithm entitled IDS (in reference to the iterative depth- and slope-dependent nature of the algorithm) that is more suitable than MFD for applications in which the bed and water surface slopes differ substantially. IDS solves for water flow depths under steady hydrologic conditions by distributing the discharge delivered to each grid point from upslope to its downslope neighbors in rank order of elevation (highest to lowest) and in proportion to a power-law function of the square root of the water surface slope and the five-thirds power of the water depth, mimicking the relationships among water discharge, depth, and surface slope in Manning's equation. IDS is iterative in two ways: (1) water depths are added in small increments so that the water surface slope can gradually differ from the bed slope, facilitating the spreading of water in areas of laterally unconfined flow, and (2) the partitioning of discharge from high to low elevations can be repeated, improving the accuracy of the solution as the water depths of downslope grid points become more well approximated with each successive iteration. We assess the performance of IDS by comparing its results to those of FLO-2D for a variety of real and idealized landforms and to an analytic solution of the shallow-water equations. We also demonstrate how IDS can be modified to solve other fluid-dynamical nonlinear partial differential equations arising in Earth surface processes, such as the Boussinesq equation for the height of the water table in an unconfined aquifer.

The applicability of artificial neural networks (ANNs) and the adaptive neuro-fuzzy inference system (ANFIS) for determination of mean velocity and discharge of natural streams is investigated. The 2,184 field data obtained from four different sites on the Sarimsakli and Sosun streams in central Turkey were used in the study. ANNs and ANFIS models use the inputs, water surface velocity and water surface slope, to estimate the mean velocity and discharges of natural streams. The accuracies of both models were compared with the multiple-linear regression (MLR) model. The comparison results showed that the ANFIS model performed better than the ANNs and regression models for estimating mean velocity and discharge. The ANN model also showed better accuracy than the MLR model. The root mean square errors (RMSE) and mean absolute relative errors (MARE) of the MLR model were reduced by 88 and 91 % using the ANFIS model in estimating discharges, respectively. It is found that the optimal ANFIS model with RMSE of 0,063, MARE of 3,47 and determination coefficient (R2) of 0,996 in the test period is superior in estimation of discharge than the MLR model with RMSE of 0,532, MARE of 38,9 and R2 of 0,776, respectively. The study reveals that the ANFIS technique can be successfully used for estimating the mean velocity and discharge of natural streams by using only the inputs of water surface velocity and water surface slope.

Satellite altimetry has emerged as a key alternative for inland water level measurement in addition to ground observations. Water surface slope (WSS) is one of the basic parameters of river morphology for discharge calculation. Estimation of WSS can also avoid systematic bias in satellite water levels relative to gauged data. A range of satellite data products are available to provide accurate river water level measurements and estimates of river WSS on a worldwide scale. Nonetheless, satellite-based observation of river water surface remains challenging in small rivers, such as the mountainous river reaches with narrow water surfaces. In this study, we examined the accuracy of the ICESat-2 ATL03 photon height data in estimating WSS over the mountainous river reach of Yongding River flowing across Hebei Province and Beijing City in northern China. With minimum along-track sampling interval of 0.7m, the ICESat-2 ATL03 data provided reliable estimation of WSS over narrow river reaches which are 50 to 100m wide. Satellite virtual stations were located mainly with a histogram-based statistical method, seeking for photon height that corresponds to the peak frequency. The twelve groups of satellite virtual stations chosen for river WSS estimation finally show an overall correlation coefficient of 0.96 in validation. Relative error of WSS estimation ranges from 0.13% to 14.51%. Findings of this study provide further implications for satellite-based river water surface measurement in small mountain river basins that lack of ground observation conditions, bringing in reliable estimation of key hydrological parameters based on satellite observation.

Abstract. The advent of low-cost pressure transducers capable of directly measuring water surface elevation enables continuous measurements of dynamic water surface slopes. This opens up a new possibility of dynamically monitoring unsteady flows (i.e., hysteresis) during the course of flood wave propagation. Hysteresis in this context refers to a looped stage–discharge rating caused by unsteadiness of flows. Hysteresis is monitored in this study using a continuous slope area (CSA) method, which uses Manning's equation to calculate unsteady discharges based on continuously measured water surface slopes. In the rising stage, water surface slopes become steeper than a steady water surface slope, resulting in higher discharges than steady-based discharges, while the trends are reversed in the falling stage. The CSA method applied to Clear Creek near Oxford (Iowa, USA) estimates the maximum differences of peak discharges by 30–40 %, while it shows sound agreements for a low to medium range of discharges against USGS steady-based records. The primary cause of these differences is the use of a single channel bed slope in deriving Manning's roughness coefficients. The use of a single channel bed slope (conceptually equal to the water surface slopes at every stage in uniform flow conditions) causes substantial errors in estimating the channel roughness, specifically at high stages, because non-uniformities of natural channels result in varying (non-uniform) steady water surface slopes at each stage. While the CSA method is promising for dynamically tracking unsteady water surface slopes and flows in natural streams, more studies are still needed to increase the accuracy of the CSA method in future research.

For nearly three decades, satellite radar altimetry has provided measurements of the water surface elevation (WSE) of rivers. These observations can be used to calculate the water surface slope (WSS), which is an essential parameter for estimating flow velocity and river discharge. In this study, we calculate a non time‐varying high‐resolution WSS of 11 Polish rivers based on multi‐mission altimetry observations from 11 satellites in the period from 1994 to 2023. The proposed approach is based on a weighted least squares adjustment with an additional Laplace condition and an a priori gradient condition. The processing is divided into river sections not interrupted by dams and reservoirs. After proper determination of the WSE for each river kilometer (bin), the WSS between adjacent bins is calculated. To assess the accuracy of the estimated WSS, it is compared with slopes between gauge stations, which are referenced to a common vertical datum. Such gauge stations are available for 8 studied rivers. The root mean squared error (RMSE) ranges from 4 mm/km to 77 mm/km, with an average of 27 mm/km. However, the mean RMSE decreases to 11 mm/km when the 2 mountain rivers are excluded. The WSS accuracies are also compared with slope data sets based on digital elevation models, ICESat‐2 altimetry, and lidar. For 6 rivers the estimated WSS shows the highest accuracy. The improvement was particularly significant for mountain rivers. The proposed approach allows an accurate, non time‐varying high‐resolution WSS even for small and medium‐sized rivers and can be applied to almost any river worldwide.

Monitoring global river discharge is fundamentally limited by the sparse distribution and ongoing decline of ground-based gauging stations. This observational gap is most critical in "ungauged basins," where lack of data hinders water resource management and flood forecasting. The Surface Water and Ocean Topography (SWOT) satellite mission serves as a transformative solution, providing the first global inventory of Earth’s surface water by measuring river height, width, and slope from space.Estimating discharge from satellite observations is a complex inverse problem because key hydraulic parameters, such as river bathymetry (bottom elevation) and bed roughness, are unknown in ungauged regions. This research explores various advanced methodologies to bridge this gap by integrating satellite-derived measurements of water surface elevation, width, and slope into hydraulic models. These approaches allow for the simultaneous estimation of unknown parameters and river flow, providing a globally consistent, observation-based record of discharge even in basins where ground-based statistics are entirely unavailableThe studies demonstrate that the unique spatial coverage of SWOT allows for a transition from traditional single-point calibration to a "multi-point" parameter selection approach. This strategy uses observations from numerous points across a river network, which significantly improves the model's ability to identify the correct hydrologic parameters compared to relying on a single virtual gauge.While the satellite data contains inherent measurement noise and systematic biases, the spatially distributed nature of the observations helps compensate for these errors. The research indicates that SWOT is particularly effective at resolving temporal variations in discharge, providing a reliable record of hydrologic events even when absolute discharge values carry uncertainty. However, the effectiveness of the mission can be influenced by river "flashiness"—basins with very rapid changes in flow may be harder to characterize due to the satellite's specific overpass timing.SWOT observations provide a vital resource for constraining hydrological processes globally. By enabling the calibration of hydrologic models in previously unmonitored regions, the mission allows for a better understanding of how streamflow responds to rainfall. This capability is expected to lead to transformative science in global hydrology, offering a consistent and observation-based measure of discharge that far exceeds the accuracy of existing uncalibrated global models.

Geographic Information Systems (GIS) have been recognised as a powerful means to integrate and analyse data from various sources in the context of comprehensive floodplain management. As part of this comprehensive approach to floodplain management, it is very important to be able to predict the consequences of different scenarios in terms of flooded areas and associated risk. Hydrologic and hydraulic modelling plays a crucial role and there is much to gain in incorporating these modelling capabilities in GIS. This is still a rather complex task and research is being done on the full integration of these models. Interfacing between these models and GIS may be a very efficient way of overcoming the difficulties and getting very good results in terms of engineering practice. This paper presents results based on the use of Intergraph GIS coupled with Idrisi GIS. Using these two systems substantially increased the flexibility of using GIS as a tool for flood studies. A lumped (XSRAIN) and a distributed (OMEGA) hydrologic models were used to simulate flood hydrographs. The well known HEC-2 Hydraulic model was used to compute flooded areas. These models were applied in the Livramento catchment with very good results. The computation of flooded areas for different flood scenarios, and its representation in GIS, can be used in the assessment of affected property and associated damages. This is a very useful GIS-based approach to floodplain management.

The SWOT satellite mission is the first to conduct a global survey of the Earth’s surface waters, measuring water surface height, river width, and water surface slope, based on which river discharge is estimated. At mid-latitudes, the repeat orbit design of SWOT only allows a sampling of twice per repeat cycle, which is considered too low for most hydrological applications. We develop a linear dynamic system that ingests SWOT observations for daily discharge estimation over continuous reaches in a single-branch river network to overcome this limitation. The linear dynamic system includes a process model based on a physically-based spatiotemporal discharge correlation model and observation equations utilizing SWOT products. We solve this dynamic system through a Kalman filter, which is executed in the time domain to obtain daily discharge. Building on the strong performance of the method with synthetic data, we apply this algorithm using SWOT measurements in the Rhine River, where we validate its performance by comparing the estimates against gauge discharge data. These efforts aim to unlock the potential of SWOT data for daily discharge estimation in diverse river networks globally.

Hydrodynamic models of overland flow and channel flow are based on the shallow water wave theory described by the St Venant (SV) equations. These models are derived from either the kinematic wave (KW) approximation, the diffusion wave approximation (DW), or the dynamic wave (DYW) representation of the SV equations. In studies reported to date, different criteria have been established to evaluate the adequacy of the KW and DW approximations, but no explicit relations either in time or in space between these criteria and the errors resulting from these approximations have yet been derived. Furthermore, when carrying out hydrological modelling, it is not evident if the KW and the DW approximations are valid for the entire hydrograph or a portion thereof. In other words, these criteria take on fixed point values for a given rainfall‐runoff event. This paper attempts to derive, under simplified conditions, error equations for the KW or DW approximations for space independent flows, which provide a continuous description of error in the flow discharge hydrograph. The KW, DW and DYW solutions are parameterized through a dimensionless parameter γ which reflects the effect of the initial depth of flow, channel bed slope, lateral inflow and channel roughness. By comparing the kinematic wave and diffusion wave solutions with the dynamic wave solution, equations are derived in terms of γ for the error in the kinematic wave and diffusion wave approximations. The error equations turn out to be the Riccati equation.

The RISKADAPT project addresses the growing challenges caused by extreme weather phenomena on critical infrastructure. This study contributes to the project by assessing the hydrodynamic loads on the piers and abutments of the Polyfytos Bridge, located at the Polyfytos Lake, Greece. Specifically, it evaluates the discharge rates, water flow velocities, and water levels under present and future climate projections to understand the potential risks to this critical asset from climate-induced flooding. Climate change, through its effects on temperature and precipitation patterns, disrupts the hydrological cycle, resulting in altered river runoff regimes. The study employs hydrological and hydraulic modeling techniques to assess these impacts on critical infrastructure. A hydrological model is used to convert different precipitation scenarios into river discharges, considering present and future climate projections. Next, the hydraulic model simulation provides water flow parameters, which are the basis for estimating the risk of scour formation around the Polyfytos Bridge piers. The modeling was conducted using the HEC-RAS software. For the first phase, the study utilized extreme precipitation data with three return periods (50, 100, and 1,000 years) for present and future climates. Historical data were drawn from global extreme precipitation (GPEX) datasets, and future projections were sourced from the EURO-CORDEX dataset, encompassing 48 combinations of global circulation and regional climate models. These data were used to predict the impact of future climate scenarios on extreme discharges, with some projections indicating a decrease in extreme discharges, while others predict an increase of 48%, 46%, and 30% for the events with return periods of 50, 100, and 1,000 years, respectively. In the second phase, hydrological results were used to generate hydrographs, which served as an input for the hydraulic simulations at the Polyfytos Lake inflow. The hydraulic modeling provided key parameters, such as water depth, surface elevation, flow velocity, and discharge, essential for further scour analysis. Results indicated that the hydrodynamic loads on the bridge piers were relatively low, even under extreme flood events. Water flow velocities remained below 0.5 m/s during the 100-year flood event, suggesting a low risk of scour formation that could compromise the bridge’s stability. The analysis of future climate scenarios showed varying impacts on discharge rates, with some indicating an increase in extreme discharges. However, the conclusion was that the Polyfytos Bridge is not significantly susceptible to scour, even under the most extreme projected climate conditions.

The resolution of digital elevation model (DEM) used for watershed delineation and data extracted from the delineated watershed relevant to hydrography plays an important role in the calibration of hydrological models. The effect of DEM resolution on the drainage length, watershed area and average slope are studied in many earlier works. DEM resolution can have impact on hydrologic modeling parameters, water quality simulation results. The impacts of DEM resolution on results of the watershed response and water quality modeling are studied using HSPF (Hydrologic Simulation Program Fortran) model applied to a watershed in Kentucky in this study. Three major streams in this watershed are impaired due to pathogen pollution according to the current state of Kentucky water quality standard set for recreation. The effects of DEM resolution on water quality modeling and calibration for fecal coliform bacteria are studied using a HSPF model. Results from this study indicate that the DEM resolution affects hydrography parameters of the delineated watershed, influence the calibration process, and the non-point and point source loadings to the streams in the watershed.

This work investigates the integration of hydro-geomorphic models, traditional data (static stage gages) and novel data sources, such as remotely sensed images and Crowdsourced data (Volunteering Geographic Information or VGI), for observation-driven improvements of hydro-modelling tools. The Tiber river basin, was selected as case study with a focus domain on the approximately 120 km channel upstream of Rome for its strategic importance in the protection of the historical city centre and the coastal urbanized zone. 
\nA parsimonious hydrological modelling algorithm was implemented, calibrated and validated for calculating the flow hydrographs of the ungauged small basins contributing to the study area. Furthermore, to delineate the boundaries computational domain of the hydraulic model for the Data Assimilation application, a DEM-based hydro-geomorphic floodplain delineation algorithm adapted from literature was tested with different DEMs and considering also its parametrization varying the stream orders. 
\nThe adopted DA methodology is the Ensemble Kalman Filter (EnKF) that requires multiple simulations for representing the uncertainties related to the model and the observations errors. 
\nNew approaches were proposed for integrating, as observations in the DA method, traditional static sensors, and simultaneously remotely sensed images and VGI data. Despite the static sensor have already been adopted in literature as observations in a DA framework, some new technical measures were necessary for integrating them in Quasi-2D hydraulic model. The assimilation of satellite images resulted to be effective, since the whole computational domain is interested by the water levels correction, although the improvement of the model performance persisted for only some hours of simulation. The usefulness of VGI have been investigated considering the uncertainties related to their reliability mostly in terms of accuracy and time allocation. Results show the potential of new data for improving the performance of the flood model, partially overcoming the limitations and the potential scarce availability of the traditional sensors. Finally, the simultaneous integration of all the three types of observations gave promising results, improving the performance of the model compared to the ones obtained assimilating only Satellite images or VGI observations.

ABSTRACTHydro‐geomorphological assessments are an essential component for riverine management plans. They usually require costly and time‐consuming field surveys to characterize the spatial variability of key variables such as flow depth, width, discharge, water surface slope, grain size and unit stream power throughout the river corridor. The objective of this research is to develop automated tools for hydro‐geomorphological assessments using high‐resolution LiDAR digital elevation models (DEMs). More specifically, this paper aims at developing geographic information system (GIS) tools to extract channel slope, width and discharge from 1 m‐resolution LiDAR DEMs to estimate the spatial distribution of unit stream power in two contrasted watersheds in Quebec: a small agricultural stream (Des Fèves River) and a large gravel‐bed river (Matane River). For slope, the centreline extracted from the raw LiDAR DEM was resampled at a coarser resolution using the minimum elevation value. The channel width extraction algorithm progressively increased the centerline from the raw DEM until thresholds of elevation differences and slopes were reached. Based on the comparison with over 4000 differential global positioning system (GPS) measurements of the water surface collected in a 50 km reach of the Matane River, the longitudinal profile and slope estimates extracted from the raw and resampled LiDAR DEMs were in very good agreement with the field measurements (correlation coefficients ranging from 0 · 83 to 0 · 87) and can thus be used to compute stream power. The extracted width also corresponded very well to the channel as seen from ortho‐photos, although the presence of bars in the Matane River increased the level of error in width estimates. The estimated maximum unit stream power spatial patterns corresponded well with field evidence of bank erosion, indicating that LiDAR DEMs can be used with confidence for initial hydro‐geomorphological assessments. Copyright © 2013 John Wiley & Sons, Ltd.