Hydraulic River Models From ICESat‐2 Elevation and Water Surface Slope

Forecasting flood and drought events requires accurate modeling tools. Hydraulic river models are based on estimates of riverbed geometry which are traditionally collected in situ. The novel Ice, Cloud and Land Elevation Satellite 2 [ICESat‐2] lidar altimetry mission with 6 simultaneous high‐resolution laser beams provides the opportunity to define river cross‐section geometries as well as observe water surface elevation [WSE] and water surface slope spatially resolved along the river chainage. This paper describes a method to utilize terrain altimetry and water surface slope estimates to define complete river geometries from ICESat‐2 data products, using the diffusive wave approximation to calculate depth in the submerged section not penetrated by the lidar. Exemplifying the method, cross‐sections are defined for a stretch of the Mekong River. Hydrodynamic model results of the stretch are compared with ICESat‐2 WSE estimates and in situ gauging station time series. Insights in river characteristics from satellite imagery and the ICESat‐2 slope estimates allow for fine‐tuning of the cross‐sections using spatially varying Manning numbers. The final model achieves a root mean square error against the ICESat‐2 WSE of 0.676 m and average Kling‐Gupta Efficiency against gauging station time series of 0.880. The method is limited by the diffusive wave approximation resulting in inaccurate cross‐section estimates in sections with supercritical flow or significant acceleration. Errors can be identified from ICESat‐2 WSE estimates and reduced with additional cross‐sections. Combined with hydrological models, the method will allow for cross‐section definition without in situ data.


Introduction
The inland water cycle is susceptible to climate extremes.Earth system models indicate that water availability on land will be impacted by future increases in frequency and magnitude of both flood and drought events (Gudmundsson et al., 2021).Since freshwater is essential for both human communities and biodiversity, accurate observations and forecasts of this scarce resource have never been more important (Best, 2019).
Runoff to rivers is estimated with meteorological parameters including precipitation, evaporation, and soil moisture.Numerical weather predictions of these parameters force rainfall-runoff models.Combined with simple river network routing models, discharge can be forecast throughout the network.Global or continental hydrological models have been developed using ensemble weather predictions with global coverage.GloFAS (Alfieri et al., 2013), WW HYPE (Arheimer et al., 2020) and DHI's Global Hydrological Model [GHM] (Murray et al., 2023) provide medium range and seasonal forecasts of discharge (Chew et al., 2020;Emerton et al., 2018).
But flooding hazard and impact assessment require water level prediction for flood area and volume estimates (Winsemius et al., 2013).Hydraulic modeling, to predict water levels as well, requires additional parameters of riverbed geometry and hydraulic roughness.
In contrast to the case of modeling complexity, water level is more easily monitored than discharge is.Pressure gauges at gauging stations provide accurate and continuous observations (Durand et al., 2016).But coverage is limited in long and remote rivers by expense and accessibility (Lao et al., 2022).As an added resource, altimetry measurements from satellites can provide water level observations.Radar altimetry missions, originally designed for global sea-surface observation (Banerjee, 2021), can monitor water surface elevation [WSE] of lakes, rivers, and wetlands (Crétaux et al., 2011).At virtual stations, river sections regularly crossed by satellites, time series of WSE can be produced.Due to the original scope of ocean observations, the large ground footprints of lowresolution satellite altimeters have until recently limited the application to large rivers and lakes (Ryan et al., 2020).Synthetic Aperture Radar [SAR] has become the state of the art in satellite altimetry (Kittel, Jiang, et al., 2021).Using the Sentinel-3 SAR altimetry with a spatial resolution of 300 m, Halicki and Niedzielski (2022) and Deidda et al. (2021) found root mean square error [RMSE] of 0.22 and 0.23 m for Polish and Italian rivers, respectively.Fully focused SAR techniques can improve the along-track resolution to 0.5 m (Egido & Smith, 2017;Kleinherenbrink et al., 2020).The recently launched Surface Water and Ocean Topography mission [SWOT] will provide observations of WSE, river width, and water surface slope for rivers wider than 50-100 m with WSE and slope accuracy goals of 10 cm and 17 mm/km, respectively (Biancamaria et al., 2016).
The Ice, Cloud and Land Elevation Satellite 2 [ICESat-2] became operational in 2018 and provides novel opportunities with its laser altimetry system.Improving on the single-beam lidar of the original ICESat mission, the ATLAS instrument onboard the ICESat-2 emits six beams of green (532 nm) laser pulses at 10 kHz, equal to once every 0.7 m along ground tracks (Neumann et al., 2019).Returning photons are recorded as photon events.In addition to the high spatial resolution, the ground footprint of only 17 m in diameter allows for more precise measurements and of smaller rivers than what is possible from radar altimetry.The WSE observations over rivers have been found accurate, with and RMSE of 0.24 m for the Mekong River (Lao et al., 2022), and an RMSE of 0.17 m for rivers in the Ohio River Basin, and 0.24 m for rivers narrower than 50 m specifically (Li et al., 2023).
The 6 parallel beams of the ATLAS instrument provide simultaneous measurements in three pairs over a 6 km span.Novel for the ICESat-2 mission, this allows for the estimation of water surface slope in the downstream direction of the river.Surface slope is a crucial factor of water flow in rivers, influencing the discharge to WSE relationship.Slope varies along the stream, impacted by local characteristics like cascades, pools and tributaries (Scherer et al., 2022).Liu et al. (2022) used ICESat-2 slope estimates to define stage/slope/discharge curves, showing how backwater effects can periodically change river dynamics in affected reaches.The potential for global application is exemplified by Scherer et al. (2023a) creating a global reach-scale water surface slope data set using ICESat-2 called IRIS.They tested the accuracy of average water surface slope for selected reaches, with a median absolute error of 23 mm km 1 (Scherer et al., 2022).Additionally, Christoffersen et al. (2023) have created an R-package "ICE2WSS" for automatic slope estimation from ICESat-2.Both data sets assign slopes to reaches from the SWOT Mission River Database [SWORD] (Altenau et al., 2021).
The accuracy and resolution of ICESat-2 river crossings provides the opportunity to map exposed river crosssections for hydrodynamic models.As the laser altimetry does not penetrate river water reliably, a submerged section must be estimated.But for crossings in low-flow periods, the submerged section is narrow and most of the cross-section is directly visible to the satellite.Coppo Frias et al. (2023) used ICESat-2 to define riverbed geometry and estimate the submerged section by calibrating a steady-state hydraulic model before implementing the cross-sections in hydrodynamic simulations.However, hydraulic inverse modeling is challenging due to the high number of calibration parameters and highly non-linear sensitivity patterns.
While river depth and channel shape are required for hydrodynamic modelling and discharge estimation from WSE, the parameters cannot be observed remotely.When estimating geometry, resistance and discharge, the problem of equifinality occurs, with multiple parameter sets reproducing the observed WSE and slope (Garambois & Monnier, 2015).Neal et al. (2015) illustrates the importance of channel geometry, showing that assuming simple rectangular river geometries leads to unrealistically high resistance numbers, slowing wave propagation.Several studies have explored methods to estimate bathymetry from remote observations.2D Models can be updated with SAR and optical remote sensing (Grimaldi et al., 2018;Wood et al., 2016).

Water Resources Research
10.1029/2023WR036428 WSE, slope and width from the SWOT satellite will be used to estimate river bathymetry and discharge.Synthetic SWOT observations have been used to improve hydraulic models with data assimilation algorithms (Brêda et al., 2019;Durand et al., 2008;Yoon et al., 2012).Discharge has been derived from synthetic SWOT observations of WSE, slope and river width with only few other known parameters (Bjerklie, 2007;Durand et al., 2014;Garambois & Monnier, 2015;Gleason et al., 2014).Durand et al. (2016) compares the methods.Andreadis et al. (2020) builds on the previous methods and incorporates the model channel geometries proposed by Dingman (2007) which are also used by Coppo Frias et al. (2023) and in this paper.
Building on the work of Coppo Frias et al. (2023), the aim of this paper is to estimate cross-sectional geometry of a river channel from ICESat-2 crossings and best available discharge.Coppo Frias et al. (2023) estimates riverbed geometry by calibrating a steady-state hydraulic model for reach-covering parameters of depth and resistance, similarly to Neal et al. (2021) using airborne LiDAR and Kittel, Hatchard, et al. (2021) using CryoSat-2.Calibrating such models is computationally heavy and requires regularization to avoid issues of equifinality.
In this paper, the parameters of each cross-section are calculated individually by incorporating the slope estimates made possible by the multiple beams.The performance of a hydrodynamic model built with cross-sections from this novel approach is tested against the ICESat-2 water surface product and time series from in situ gauging stations in a stretch of the Mekong River.

Materials and Methods
Data products from the ICESat-2 altimetry mission are processed to create cross-sections for a 1-D hydrodynamic model of the river stretch.The best available discharge data is used in conjunction with terrain and water surface slope estimated from the ICESat-2 crossings.The workflow is shown in Figure 1.Processing consists of geoid correction, noise filtering, and chainage assignment (Section 2.2).The submerged section of the cross-section is estimated using Manning's equation of flow with the diffusive wave approximation (Section 2.3).The limitations to the application of Manning's equation in this context were explored by Neal et al. (2021).

Case Study-Stretch of the Mekong River
The methods presented in this paper are exemplified on a hydrodynamic model of an approximately 346 km long stretch of the lower Mekong River between Chiang Khan and Paksane.At Chiang Khan, daily average discharge varies between 892 m 3 s and 16,100 m 3 s in the modeling period between 1 January 2017 and 4 June 2022.The stretch can be seen in Figure 2 along with the three gauging stations at Chiang Khan (CK), Nong Khai (NK) and Paksane (PS), as well as three radar altimetry virtual stations.A Unified River Basin Simulator [URBS] rainfall-runoff model (Carroll, 2016) is in use in the area by the Regional Flood and Drought Forecasting Center of the Mekong River Commission (MRC, 2018).The relevant URBS catchments are shown in Figure 3.The upstream section of the stretch, covered by catchment 16 in Figure 3, is characterized by mountainous terrain, a narrower span, and rock protrusions in the river.The rest of the stretch is wider with sand banks.

ICESat-2 Data Processing
Version 5 of the ICESat-2 data products ATL03, ATL08, and ATL13 are retrieved over the area of interest.The data is provided in granules, containing all six tracks.Each product is retrieved and processed separately.

ATL03-Global Geolocated Photons
The ATL03 data product consists of geolocated photon event ellipsoidal heights.The data points are provided with time, latitude, and longitude indexes and additional signal confidence and quality parameters.The points are reprojected to UTM zone 48N for distancing.To ensure compatibility with other data products, the product was externally referenced to the EGM2008 geoid described by Pavlis et al. (2008).The spherical harmonic expansion series is calculated for cells on a 1 km resolution grid over the area of interest.ATL03 data points are then sampled from the resulting grid, as it is not computationally possible to calculate the geoid anomaly for each point individually.Points with signal confidence below three (medium confidence) and quality attribute value different from 0 (nominal) are removed.Tracks are assigned a reference chainage as the position of the data point closest to the river centerline.The track is split into separate cross-sections where the river course results in multiple crossings.

ATL08-Land and Vegetation Height
ATL08 contains terrain and canopy heights derived from ATL03 photons in 100 m segments, using the Ground Finding Filter described in (A.Neuenschwander, Pitts, Jelley, Robbins, Markel, et al., 2021).We extract best fit ground height and mid-point location data for all granules in the area of interest.Poor coverage over the river crossing indicates cloud cover.As good coverage in the ATL08 data indicates a suitable ATL03 track for crosssection definition, we retrieve all ATL08 granules first for screening.Only granules in the low flow period of November to March are considered.The ATL08 tracks are referenced and assigned a chainage similarly to the ATL03 tracks.

ATL13-Along Track Inland Surface Water Data
The ATL13 granules contain along-track surface water elevation for lakes larger than 0.1 km 2 and rivers wider than 50-100 m (M.Jasinski et al., 2021).Data is derived from ATL03 photons in short segments, containing 75-100 consecutive photon events.Short segment water surface elevation ("ht_water_surf") and mid-point location are extracted from all granules available in the area of interest in the period 15 October 2018 until 1 April 2022.Points are individually geoid referenced with the spherical harmonic series.Chainage is assigned as for the other ICESat-2 products.Before WSE estimation, ATL13 points not over the river must be removed.The points are sampled on the Global Surface Water Occurrence map by Pekel et al. (2016), and points with a surface water occurrence <85% of the time are removed to filter out points on banks and over land.This filter does not catch points over nearby lakes, so points more than 1,500 m from the river centerline are removed as well.Rock protrusions and islands can skew the WSE estimate, so statistical outlier WSE points are identified using the interquartile range method, or Tukey's fences.Points more than 1.5 times the interquartile range above the third quartile or below the first quartile are removed.An WSE estimate for each crossing is found following the method described by Scherer et al. (2022), as a weighted average of remaining points, weighted by inverse distance to the river centerline.Slope is calculated between all possible crossing pairs from the same date, except between crossings with a mutual distance less than 1,000 m.The slope is found as the WSE difference divided by chainage difference.Negative slopes are discarded.A combined WSS estimate for each crossing date is found as the weighted average between remaining slope estimates, weighted by the inverse sum of the standard deviations of the WSE estimates used in the pair.

Riverbed Geometry
Processed ATL03 crossings are inspected and manually selected based on the following criteria, in order of priority: 1. Spatial distribution along the river stretch for full coverage.2. Low WSE to retrieve as much exposed geometry as possible.

Perpendicularity to river, reducing angle correction.
A cross-section spacing of 10 river-km was achieved.Multiple crossings in proximity can be combined into a single cross-section.This highlights terrain elevation and reduces impact of vegetation and buildings.Thirty-eight cross-sections are created from 76 manually selected crossings.Cross-sections are processed to remove outlier photons, buildings, and vegetation.An example cross-section with processing steps is shown in Figure 4.The processing is modified from the approach by Coppo Frias et al. (2023).ATL03 photon events are graphed by distance to the center-point of the crossing, in the downstream perspective.Initially, the corresponding ATL08 terrain heights are used to filter the photon events, removing ATL03 data points outside of a manually adjusted window, usually ±10 m of the ATL08 terrain height.Next, a Hampel filter is applied to remove remaining outliers.The median and median absolute deviation [MAD] is found in a moving window x.The MAD is defined as Points are removed if they differ more than 1 standard deviation from the median, using the estimator σ = 1.4826 ⋅ MAD.The windows size is manually selected for each cross-section to ensure the desired outlier filtering, with a smaller window removing more points.A rolling average is found on the remaining points to delineate a cross-section surface.The rolling average window is similarly adjusted for each cross-section to follow the terrain as closely as possible.A larger window results in more averaging, removing smaller undulations in terrain height.
The submerged section is estimated with the approach proposed by Dingman (2007).A symmetrical geometry is approximated with Equation 1. Water Resources Research Where z d is the height above the deepest point in the channel, d and W are maximum depth and width of the submerged section, and y is the horizontal distance from the center of the submerged section.r is the shape parameter, kept at r = 2 in this study, resulting in a parabolic shape of the submerged geometry, seen in Figure 4d.
The width is manually estimated from the ATL03 crossings, while the depth is estimated from discharge at the time of crossing in the section below.
The area A and wetted perimeter P of the submerged section can be formulated as functions of the maximum depth and width of the channel: As is the case in the example cross-section, the area and wetted perimeter is summed when multiple submerged sections occur in a braided river.

Diffusive Wave Approximation
Manning's equation is derived from the St Venant equations, two partial differential equations expressing the conservation of mass and momentum, respectively (St Venant, 1871).
The mass or continuity equation: The momentum equation: Where Q is the discharge, x is chainage along the river, A is the flow area, t is time, β is the Boussinesq coefficient, g is gravity, d is mean water depth, S 0 is the bed-slope and S f is flow resistance as friction slope.
Bed slope S 0 describes the change in bed elevation z with chainage x: The friction slope S f is here described with Manning's Equation Where R is the hydraulic radius R = A P and n is Manning's n, a resistance coefficient with the unit s m 1 3 .
The first two terms of the momentum equation, the inertia terms, represent acceleration over time and space respectively.In the diffusive wave approximation, the inertia terms are neglected under the assumption that friction and gravity forces dominate with no significant flow acceleration or deacceleration occurring.
Equation 5 simplifies to Water Resources Research Change in depth and bed elevation can be collected as change in WSE: With the diffusive wave approximation, the friction slope can now be assumed equal in magnitude to the water surface slope: The relationship between discharge, riverbed geometry, and resistance can now be described with Manning's Equation 7 and water surface slope: Using the area and wetted perimeter functions Equations 2 and 3, discharge in the cross-section can be estimated as a function of depth for a given Manning's n.
The estimated discharge is compared to the best available observed discharge at the time of crossing.This allows us to iteratively solve for depth in the crossing, resulting in a stage/slope/discharge relationship in the created cross-section as observed.Observed discharge at the cross-section point is interpolated between the nearest gauging stations.In this stretch, gauged discharge observation at daily resolution is available at Chiang Khan and Nong Khai.As no discharge time series is available at the downstream gauging station Paksane, measurements from a station further downstream, Mukdahan, are utilized.Figure 3 shows the URBS model catchments and the gauging stations.Instead of interpolating discharge proportionally to chainage, discharge between gauging stations is interpolated proportionally with average runoff in the URBS catchments and drainage area from the MERIT Hydro Flow Accumulation map by Yamazaki et al. (2019) based on the MERIT DEM.This approach assumes that discharge increase between gauging stations is purely due to runoff, disregarding possible water storage changes in the reach.The approach is explained in further depth in Text S3.1 in Supporting Information S1.

Hydrodynamic Modeling
A MIKE Hydro River 1D hydrodynamic model setup is prepared with the defined cross-sections.The MIKE model setup and parameters are further described in Supporting Information S1.The model is forced with the Chiang Khan discharge time series as upstream boundary, and the URBS catchments 16, 19, 23, 24, and 25 are implemented as distributed source boundaries over their respective sections of the branch, see Figure 3.

Manning's n Calibration
Manning's n must be chosen for cross-section definition and hydrodynamic modeling to optimally simulate observed WSE.We use manual trial-and-error adjustment instead of an automatic calibration of Manning's n.
Where N is the number of ATL13 WSE estimates not used for cross-sections, WSE ref and WSE sim are reference and simulated WSE, respectively.
Performance at gauging stations is measured as King-Gupta Efficiency [KGE]: Where ρ is the linear correlation, and σ and μ are standard deviation and mean of the respective time series.KGE = 1 indicates perfect agreement between simulations and observations.
Comparing with ATL13 gives an indication of spatial differences in performance, while the time series performance shows the model's ability to recreate the full amplitude of the WSE dynamics.A pareto optimum is sought between the two objective functions, creating hydrodynamic models where adjustment in Manning's n cannot improve one objective function without worsening the other.A final model will be chosen that minimizes the distance to the optimal scenario of RMSE = 0 m and KGE = 1.
After running models defined with a global Manning's n parameter, the ATL13 residuals should be investigated for spatial bias.If such exist, Manning's n can be varied in sections of the modeled stretch, with as few sections as necessary in the pursuit of parsimony.Finally, specific cross-sections with poor fit can be investigated for improvement.Uncharacteristically deep cross-sections indicate that the diffusive wave approximation does not hold for the time and place of the cross-section definition.Defining additional cross-sections from different satellite passes in proximity to the erroneous ones can mitigate modeling error if the flow is more accurately described by the diffusive wave approximation at other times.Looking at slopes estimated from ATL13, areas with slope changing in time can indicate complicated flow dynamics.

ATL13 WSE and Slope Estimates
In the ≈346 km stretch, 906 WSE estimates are produced from ATL13 crossings, resulting in 215 slope estimates.The WSE estimates are found in Figure 5.The ATL13 WSE estimates adequately cover the stretch with very few gaps.The additional filtering steps described in Section 2.2.3 are required to remove outliers and reduce variance.Points over banks are especially unreliable if not removed with the surface water occurrence filter.
Slope estimates are found in Figure 6 along with the IRIS slope averages and ICE2WSS slope estimates on relevant SWORD reaches.All slope products show similar tendencies.The upstream section shows larger variation and steeper slopes than the downstream section.The drastic change in slope within the section implies local heterogeneities causing rapid change along the chainage or temporal variability.

Cross-Section Definition
The 76 crossings used for cross-section definition are shown in Figure 7. Cross-section definition is possible with reasonable spacing for the entire modeling stretch but is in some cases complicated by the limited availability of full-coverage ATL03 tracks.Cloud cover creates gaps in data coverage over the river.As the mission continues to provide new measurements, coverage will become denser, and more tracks will be available for cross-section definition over time.The river course also has an impact on the suitability of the tracks.The satellite flies north to south, with a slight angle dependent on north or south-bound direction.For generally north-south oriented rivers, crossings approximately perpendicular to the river will be rare.Natural bends and the slight angle of the satellite tracks will allow for application in dominantly north-south oriented rivers as coverage increases.
With the filtering steps and rolling average performed in Section 2.3.1, it is possible to define clear cross-sections from the ATL03 crossings.Manual adjustment of the parameters is still required, as generalization is not easily achieved.Simplification of the resulting cross-sections could improve the method, as the current resulting smoothed cross-sections contain a coordinate for every ATL03 point in the track.

Model Performance
Table 1 shows performance of initial models defined with Manning's n between 0.02 s m 1 3 and 0.055 s m 1 3 . RMSE against the ATL13 reference ranges between 0.970 and 1.074 m, with the best performing model using Manning's  .Looking at model performance at gauging stations in Table 1, higher Manning numbers perform the best at the upstream station (CK), while lower numbers perform better at the downstream stations (NK and PS).Additionally, Figure 8 indicates that the ATL13 residuals of the best performing model are not normally distributed but are chainage dependent.Both trends indicate that the river characteristics in the two sections are too different to be described by a single Manning number.In the upstream section, the model generally underestimates WSE.Model performance is improved by splitting the stretch into two sections at chainage 426 km, and using cross-sections made with different Manning's n in each section The residual bias does not indicate that more than two sections would improve the model enough to offset the added complexity.The split is chosen from satellite imagery and slope estimates.At 426 km, the mountainous area, with rock protrusions and steeper slopes, ends, and the more sandy and wide section with gentler slopes and less slope variance begins.Table 2 shows RMSE for each section.The best performing model in the section is Manning's n = 0.025 s m 1 3 , while the upstream section only improves with increasing Manning's n.As expected, the upstream section with steeper slopes and rocky terrain is better described by higher Manning numbers.The rocky terrain can be seen in the zoom of Figure 7.

Model Adjustments
Model performance for models with Manning's n = 0.025 s in the upstream section are found in Table 3. Models are named from their upstream resistance and the suffix "s." Performance measures improve in all cases, with ATL13 RSME decreasing with increasing Manning's n.KGE at CK is best in the 0.06s model, while performance at the other gauging stations is constant across models.This is expected, as the Manning's n is kept constant in that section.) used for cross-section definition and hydrodynamic modeling.Performance measured as Root Mean Square Error between model and ATL13 WSE estimates, and Kling-Gupta Efficiency of modeled time series at gauging station locations Chiang Khan (CK), Nong Khai (NK) and Paksane (PS).Significant underestimation still occurs in a part of the upstream section, shown in Figure 9. Investigating the calculated depth of each cross-section in the 0.075 s model as an example, we find that one cross-section at chainage 376 km has a calculated depth of 22.6 m, while all other cross-sections have depths in the range of 2.55-12.78m.Low slope and high discharge at the time of crossing results in a large calculated cross-sectional area and therefore depth.The cross-section is found in a section of the river with rapids and white water, where the assumption of negligible acceleration does not hold.This results in severe underestimation of WSE upstream of the deep cross-section.To improve the model, an additional cross-section is defined at chainage 374 km.At the time of crossing, the river is flowing at full width, and the cross-section is defined significantly shallower, resulting in a calculated depth of 6.3 m.The model performance after implementation is found in Table 4. Models are named from their upstream resistance and the suffix "'r."Implementation results in an increase in performance on the ATL13 RMSE objective, but a slight decrease in performance of CK KGE.The cross-section causes increased WSE in the entire upstream section, leading to overestimation at the beginning of the stretch, where the gauging station is found.Without adding the additional cross-section, increasing Manning's n improves ATL13 RMSE by inducing higher upstream WSE amplitude, compensating for the impact of the deep cross-section.After implementation, the WSE amplitude is balanced by the overestimation in the beginning of the upstream section.
The performance measures are graphed in Figure 10, showing a Pareto front between models 0.055r and 0.075r.The best performing model is 0.075r, with the shortest distance to the optimum point at (0, 1).0.075r also has the lowest ATL13 RMSE at 0.676 m.
Figure 11 shows the WSE time series for models 0.075s and 0.075r.The upwards bias from the added cross-section is clear at Chiang Khan, the upstream end of the stretch.Similar for all models is the gradually increasing forcing error, seen at the downstream gauging stations, as the RR-model impact becomes more significant.
Modeled WSE ranges and observed ranges at in situ and virtual stations ranges are shown in Figure 5. Gauging station ranges represent the full range of WSE possible in the stretch.It should be noted that the ATL13 WSE estimates do not capture the full range, as the temporal resolution is low.The ATL13 is valuable for validation of ) used for cross-section definition and hydrodynamic modeling.The uncharacteristically low depth and its impact on model performance exemplifies the limitations of the diffusive wave approximation.The full shallow water equations are required to accurately describe the flow in sections with significant acceleration.As investigated by Neal et al. (2021), such areas are characterized by low Froude and kinematic wave numbers.While the methods in this paper do not describe flow accurately in some sections, the method avoids global optimization when estimating depths, making it applicable for large scale models.
The methodology is limited by the precision of the WSE estimate.For low river slopes, the WSE difference will not be significant within the 6 km span of the satellite track at the current precision.The method is therefore not applicable where low slopes occur, for example, where large rivers approach the ocean.

Conclusions
This paper describes a method to define river cross-sections without in situ measurements of geometry or hydraulic inverse modeling.The ICESat-2 mission's novel high-resolution altimetry over river crossings and slope estimates from simultaneous WSE observations are utilized to inform a hydrodynamic model of a stretch of the Mekong.Submerged sections of crossings, where the lidar does not penetrate, are estimated with best available discharge and the diffusive wave approximation. .Models are named from their upstream Manning's n and the suffix "s."Higher resistance in the upstream section improved performance.The spatial coverage of the ATL13 WSE estimates can be used for model validation over the entire stretch of the model, but the low temporal resolution means that the full WSE range is not captured.
Model errors are largest in magnitude where the flow cannot be adequately described by diffusive wave theory.This leads to inaccurate depth estimates in cross-sections, with upstream impact on WSE estimates in the hydrodynamic model.Improvements are easily implemented in poor performing areas if additional tracks are available for cross-section definition.Automating more steps of the cross-section definition process will allow for more cross-sections without additional manual work.
The methods developed here can be applied globally for operational forecasting.Combined with global hydrological models and forecasts, hydrodynamic models without in situ data are possible. .Models are named from their upstream Manning's n and the suffix "r." Figure 10.Model performance comparison, using the ATL13 RMSE objective and the average KGE of the three gauging stations.Implementation of an additional cross-section in the "r" models reduces ATL13 RMSE.A Pareto front of Manning's n is found between 0.055r and 0.075r (square symbols).The final chosen is 0.075r (triangle symbol).

Data Availability Statement
Processing scripts and intermediate products from this paper will be available in the DTU Data repository upon publication at https://doi.org/10.11583/DTU.24118023.ICESat-2 data products are available from the NSIDC (M.F. Jasinski et al., 2021;A. L. Neuenschwander, Pitts, Jelley, Robbins, Markel, et al., 2021;Neumann et al., 2021).The EGM2008 geoid model described in Pavlis et al. (2008) was retrieved from the NGA Office of Geomatics at https://earth-info.nga.mil/.The Global Surface Water Occurrence map used for ATL13 filtering is described in Pekel et al. (2016) and was retrieved from https://global-surface-water.appspot.com/downloadas the 10-20°N 100-110°E tile.The ICESat-2 river surface slope (IRIS) data set by Scherer et al. (2023a) is available from Scherer et al. (2023b).Relevant reaches from SWORD (Altenau et al., 2021) was extracted for chainage assignment from Altenau et al. (2022).Version 15, used for the IRIS data set above, is available from http://www.swordexplorer.com/.The R-package ICE2WSS for slope estimation (Christoffersen et al., 2023) was installed through https:// github.com/lindchr/ICE2WSS/,licensed under MIT.An archived version will be published at https://doi.org/10.11583/DTU.23268218.Figure 2 was made with public domain Natural Earth shapefiles of administrative boundaries, oceans and rivers, found at https://www.naturalearthdata.com/features/,specifically as 10m admin boundary lines, 10m Ocean, and 10m rivers and lake centerlines.Access to in situ discharge and URBS runoff time series is restricted.The Mekong River Commission can be contacted about access at https://www.mrcmekong.org/contact-us/.

Figure 1 .Figure 2 .
Figure 1.Method workflow to create full cross-sections for a hydrodynamic model and investigate performance.Orange boxes indicate an input.Processes are described in the method section indicated in italics.

Figure 3 .
Figure 3. URBS Catchments for the relevant section of the Mekong River (MRC, 2018).Yellow catchments force the hydrodynamic model.Red catchments are only used for interpolating discharge between Nong Khai and Mukdahan, as no discharge time series is available for Paksane.

Figure 4 .
Figure 4. Cross-section definition steps.(a) ATL03 photon events from four crossings, two passes of the left and right beam in the third beam pair with Pass 1 on 08 November 2019 and Pass 2 on 04 February 2021.(b) photons filtered by corresponding ATL08 terrain estimate.(c) Hampel filter and rolling average on remaining points, and (d) geometry estimation under the submerged sections.

Figure 6 .
Figure6.surface slope (WSS) estimates from ATL13 over the studied stretch of the Mekong River.Red: Slope estimates from this study, assigned to closest chainage.Black: Slope estimates from the ICE2WSS R-package(Christoffersen et al., 2023), shown at the chainage of the SWORD database reach centroid that the slope estimate is found in.Orange: SWORD reach average slope from the IRIS database byScherer et al. (2023b).

Figure 5 .
Figure 5. ATL13 Water surface elevation [WSE] estimates along with ranges of WSE from virtual and in situ gauging stations.The 0.075r model range is overlaid.

Figure 7 .
Figure 7. (Right) ATL03 crossings selected for cross-section definition.After initial modeling, performance is improved by implementing an additional cross-section in the upstream section with rapids (left).

Figure 8 .
Figure 8. Residuals of modeled water surface elevation [WSE] using the n = 0.035 s m 1 3 model against ATL13 WSE estimates.

Figure 9 .
Figure 9. Observed and simulated water surface elevation [WSE] in an upstream part of the stretch.Cross-section locations are indicated by vertical lines.The additional cross-section [XS] added for the 0.075r model improves the simulated WSE chainage 350 and 375 km.Notice the resulting overestimation of WSE upstream of chainage 350 km.

Figure 11 .
Figure 11.Modeled and observed WSE time series at three gauging stations in the stretch.The two models have the same Manning's n in the downstream section where Nong Khai and Paksane are located.
. Simulated WSE is interpolated linearly between the nearest h-points to each ATL13 estimate and gauging station.ATL13 WSE estimates from crossings used to define cross-sections are not considered for comparison.RMSE is calculated between reference ATL13 WSE estimates and simulated WSE at the time of crossing: Model performance is measured by comparison with ATL13 WSE estimates and WSE time series at gauging stations

Table 2
Model Performance of Initial Models, Splitting Observations of the Stretch Into Two Sections results spatially distributed over the reach of a hydrodynamic model but cannot be used to validate the simulated temporal WSE amplitude.Hydroweb virtual stations are also unlikely to capture the full range.The modeled WSE amplitude is however not constricted by the limited observed range. model

Table 3
Model Performance of Models With a Manning's n Split at 426 km

Table 4
Model Performance of Models With a Manning's n Split at 426 km and an Additional Cross-Section in the Rapid Section