Generation of High‐Resolution Water Surface Slopes From Multi‐Mission Satellite Altimetry

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.


Introduction
Water Surface Slope (WSS) is the difference in water surface elevation (WSE) between an upstream and downstream point on a river divided by the length of the reach (Ozga-Zielińska & Brzeziński, 1997).It is an important parameter in geomorphic and hydrologic modeling: the WSS determines the transport and erosion capacity of a river (Migoń, 2006), and is required to calculate the flow velocity (Manning, 1891) and the river discharge (e.g., Bjerklie et al., 2003;Durand et al., 2014;Gleason & Durand, 2020;Rantz, 1982;Tarpanelli et al., 2013).In general, a longitudinal river profile has the shape of a concave parabola, but the younger the river and the less uniform the structure of the river bed, the more this profile deviates from the parabolic shape (Dębski, 1970).
WSS can be calculated using several approaches.Continuous measurement of the WSE with a GNSS receiver mounted on a boat allows for an accurate WSS determination for the entire studied reach (e.g., Altenau et al., 2017;Habel, 2010;Pitcher et al., 2019).WSS can also be determined using airborne lidar, radar, or photogrammetry (e.g., Bandini et al., 2020;Jiang, Bandini, et al., 2020).However, these methods are mostly used on a local scale because of the high cost of a field campaign.The recently launched Surface Water and Ocean Topography (SWOT) satellite is expected to provide accurate WSS measurements even for rivers less than 100 m wide.So far, there have been several examples of the use of SWOT-like data from an airborne wide-swath altimeter (AirSWOT), which showed a promising ability to calculate WSS with a Root Mean Squared Error (RMSE) of 15 mm/km (Pitcher et al., 2019), 16 mm/km (Altenau et al., 2019) or 32 mm/km (Tuozzolo et al., 2019).
The WSS of a river can also be determined using a digital elevation model (DEM), such as the Shuttle Radar Topography Mission (SRTM) (LeFavour & Alsdorf, 2005;Paz & Collischonn, 2007) or the ALOS PALSAR RTC-DEM (Lamine et al., 2021).Cohen et al. (2018) developed a global river slope database using the HydroSHEDS DEM.Using the same DEM, Ruetenik (2022) developed a web application to generate longitudinal river profiles.However, since the vertical errors of global DEMs are considerable (e.g., the vertical error of the SRTM DEM is of several meters (Rodríguez et al., 2006)) and the spatial resolution of global DEMs is usually low, DEM-based WSS should only be calculated for long sections of large rivers (LeFavour & Alsdorf, 2005).Often DEMs such as the SRTM do not provide WSE for smaller rivers, but only the surrounding topography or averaged water levels for larger rivers.Furthermore, the inaccuracies of SRTM-based WSE significantly exceed the errors of WSE determination based on lidar data (Schumann et al., 2008).In addition, the data acquisition for a DEM is usually done in short time periods (e.g., a 10-day period in February 2000 for the SRTM DEM), but the WSS varies in time (Paris et al., 2016) so the observations may not represent the average WSS.
WSS can also be calculated from the WSE measured at neighboring gauges (Durand et al., 2014).The main advantage of this approach is its high accuracy and the possibility to observe the temporal variability of WSS.This approach also allows the calculation of an average WSS value for a given river section.However, the number of gauges has been decreasing over the last decades (Calmant & Seyler, 2006;Vorosmarty et al., 2001), and the spatial distribution of gauges is uneven (Hannah et al., 2011).In addition, some gauges are not referenced to a vertical datum, so the vertical difference between them cannot be calculated accurately.On poorly gauged rivers, the distance between neighboring gauges can be even hundreds of kilometers, making it impossible to capture the spatial variability of the river profile.Furthermore, this approach is not applicable to river sections with flow disturbances, such as waterfalls, dams, or weirs.
The gap in gauge measurements is partly filled by satellite altimetry, which has been providing WSE of oceans, wetlands, lakes, and rivers for more than 30 years (Abdalla et al., 2021).Currently operating altimetry missions can observe even small rivers (width < 100 m) with an RMSE of 20-30 cm (e.g., Deidda et al., 2021;Halicki & Niedzielski, 2022;Jiang, Nielsen, et al., 2020;Kittel et al., 2021).Using satellite altimetry, the WSE of rivers is observed at so-called virtual stations (VS), which are located at the intersection of the satellite ground track and the river channel.The quality of a VS's WSE time series can be improved by correcting it for the WSS bias that results from the orbit variation and thus a changing location of an altimeter measurement.This bias has been observed by Santos da Silva et al. (2010) and Boergens et al. (2016).Halicki et al. (2023) proposed two corrections based on gauge data and on Sentinel-3 altimetry observations and showed, that both corrections applied on 16 VS on the middle Oder River resulted in an average accuracy improvement of 25% (RMSE decrease from 22 to 16 cm).In some cases the RMSE reduction exceeded 50%.Also, Scherer et al. (2022a) corrected altimetry observations on rivers using ICESat-2 based WSS and obtained an improvement in RMSE up to 30 cm or 66%.
Since altimetry observations from a given mission are referenced to a common vertical datum, multiple VSs can be used to determine WSS (Birkett, 2002).However, WSE measurements at different VSs are observed at different times, so WSE variations can introduce errors in the derived WSS.Therefore, WSE averages at virtual stations (Halicki et al., 2023;Tarpanelli et al., 2013;Tourian et al., 2016) or monthly means (O'Loughlin et al., 2013;Paris et al., 2016) are used.Satellite observations can also be used to model the longitudinal profile of the river.Using a least squares approach based on multi-mission altimetry to derive a linear model of the Mississippi River yielded an average absolute median WSS error of 12 mm/km (Scherer et al., 2020).WSS can also be determined using laser altimetry (e.g., Hall et al., 2012;O'Loughlin et al., 2013).Using the unique measurement geometry of ICESat-2 with six parallel laser beams, Scherer et al. (2022a) derived reach-scale WSS both along and across the satellite ground track with a median absolute error of 23 mm/km.

10.1029/2023WR034907
Although the accuracy of satellite altimetry has improved significantly over the past decades, observations are still limited by low spatial coverage (e.g., equatorial track spacing of 311 km for the Jason satellites) and low temporal resolution (e.g., a revisit time of 27 days for the Sentinel-3 satellites).Since WSS can have strong temporal and spatial variability, altimeter observations from a single satellite may be too sparse to accurately determine the WSS variability along an entire river.However, by using observations from many different satellites (multi-mission approach), the temporal and spatial resolution of altimeter observations can be increased (e.g., Bogning et al., 2018;Normandin et al., 2018;Tourian et al., 2016).
In this paper, we present a new cross-calibrated multi-mission approach to determine the WSS of a river.Using altimeter observations from CryoSat-2, Envisat, ERS-1, ICESat-1/-2, Jason-2/-3, Sentinel-3A/-3B/-6A, and SARAL ranging from 1994 to 2023, we aim to obtain non time-varying high-resolution WSS (every kilometer) of the largest Polish rivers within the accuracy requirement recommended for the SWOT mission (17 mm/km).We will assess the accuracy of this method using WSS derived from in situ water levels, airborne lidar, ICESat-2, and DEMs.
This article is structured as follows: Section 2 describes the study area, which includes the 11 Polish river.In Section 3, the used altimeter data, SWORD data, and validation data are presented.In Section 4, the methodology for estimating WSS from satellite altimetry using a weighted least squares adjustment is explained.The WSS results are then presented and a quality assessment is performed in Section 5.In Section 6, the WSS results of this study are discussed in the context of WSS from other sources.The paper concludes with a summary and an outlook.

Study Area
The study area includes 11 rivers in the Vistula and Oder basins, which are located in Central Europe and cover most of Poland (Figure 1).We selected only those rivers, whose centerlines are included in the "SWOT Mission River Database" (SWORD, see Section 3.2).The southern part of the study area is characterized by mountain ranges (Sudetes and Carpathians), whose heights do not exceed 2,500 m.North of them is an area of highlands, while in the central and northern part of Poland lowlands predominate.The river network in this area is characterized by a right-sided asymmetry: both the Vistula and the Oder rivers have many more tributaries from the east than from the west (Pociask-Karteczka, 2018).This asymmetry is closely related to the history of the development of the river network, which was shaped by numerous regressions and transgressions of the Scandinavian ice sheets and changes in the level of the Baltic Sea (Andrzejewski & Starkel, 2018).
The characteristics of the rivers studied are presented in Table 1.These rivers range in length from 173 km (Wisłoka) to 1,022 km (Vistula).The Vistula has the highest discharge (over 1,000 m 3 /s).The discharge of the Oder is almost twice as low and amounts to 567 m 3 /s.The area of the studied basins is more than 313,000 km 2 , of which is about 62% and 38% for the Vistula and Oder basins respectively.Due to limited data availability (i.e., the SWORD data set does not include upper river sections) and the presence of hydraulic structures, not all river sections are considered in this work.For large, lowland rivers, almost all sections are included (91%, 80%, and 72% for the Bug, Warta and Vistula rivers, respectively).Due to the large number of hydraulic structures, many sections of the Oder and Noteć rivers were excluded from this study.The average river width of the investigated sections, calculated on a basis of the SWORD database, ranges from 46 m (Noteć) to 299 m (Vistula).The narrowest sections are 42 m wide, while the widest sections were recorded on Bug (716 m) and Vistula (640 m).It should be noted, however, that fluvial lakes have been excluded from the river width calculations, as they may bias the river width values.
According to the world map of the Köppen-Geiger climate classification (Peel et al., 2007), the climate of the study area can be classified as humid continental, with an average annual precipitation of 610 mm (Miętus et al., 2022).The flow regime of Polish rivers has been proposed by Wrzesiński (2018), who followed the criteria of Dynowska (1997), using the relation of the average flow in spring or summer to the annual flow.In most of the studied reaches, the river regime is nival, with a high flow in the spring months.The mountain rivers in the south are characterized by the nival-pluvial regime, with high flows in the spring and summer months.The high spring flows are due to snowmelt, while the high summer flows are due to the intense precipitation.
Figure 2 shows the 11 altimeter missions divided into 17 orbit phases used in this study.The mission colors are chosen according to their orbit phase.The data used in this study were measured between the years 1994 and 2023.
The variety of satellite altimetry missions on different orbits contributes to a dense coverage of WSE observations along the rivers.In particular, missions with long repeat cycles or drifting orbits.These missions are ERS-1E (168 days), ERS-1F (168 days), CryoSat-2 (369 days), SARAL (DP, 35 days, drifting) and Jason-2 (GM, 16 days, drifting).Other missions with a short repeat cycle such as Jason-2/-3 (10 days), Sentinel-3A/-3B (27 days) or Envisat (35 days) without a drifting orbit, monitor the same river crossings with high temporal resolution but poor spatial resolution.ICESat-1 and ICESat-2 are a compromise between the two orbits mentioned above, with a lower repeat cycle of only 90 days, but a higher spatial resolution between the satellite tracks.
Overall, the combination of the different types of altimeter missions is essential in this study to derive a high resolution WSS along the river.

SWORD Data
The "SWOT River Database" (SWORD) (Altenau et al., 2021), developed for the "Surface Water and Ocean Topography" (SWOT) satellite mission, provides the spatial framework for this study.SWORD contains highresolution river centerlines (30 m) and widths from the "Global River Widths from Landsat" (GRWL, Allen & Pavelsky, 2018) data set.The centerlines are segmented into approximately 10 km long reaches and nodes with 200 m spacing.The reaches and nodes contain additional metadata, such as information on the location of artificial or natural river obstructions (i.e., dams and waterfalls).In addition, SWORD contains WSE and WSS data from MERIT Hydro (Yamazaki et al., 2019), a multi-error-removed improved-terrain DEM based on SRTM, which we use for comparison with the results of this study.In this study, the latest version 15 of SWORD is used.

Lidar-Based WSS
We use airborne laser scanning (ALS) lidar data to extract an in situ river profile for validation.The lidar data are provided by the Polish Head Office of Geodesy and Cartography (Główny Urząd Geodezji i Kartografii) via geoportal.gov.pl(Kurczyński, 2015).The ALS campaigns started in 2010 with reference to the height system "PL-KRON86-NH."From 2018 to 2021 (the latest available data), the lidar point clouds are referenced to the European vertical reference frame "PL-EVRF2007-NH" height system.The study areas are not completely covered by a single ALS campaign, and the lidar data were acquired on different dates within 1 year.Since the water level of the studied rivers varies significantly, the WSS can only be calculated in reaches with lidar data from the same date and not all reaches are covered.For each point along the SWORD river centerline, class 9 (water) records are extracted from the lidar point cloud within a 15 m radius of the centerline.This was selected in order to avoid using lidar measurements contaminated by the river shore.If more than 500 records can be extracted, the median elevation is assigned to the centerline point.Additionally, the standard deviation of the elevations of the extracted points is used for outlier detection.However, the results can still be affected by land contamination.The quality of such lidar data over water bodies shows that airborne LiDAR systems, which are known for their high-accuracy measurements of land surfaces, tend to provide poorer returns over open water surfaces due to the absorption of the laser beam within the water column, low signal-to-noise ratios, and high occurrences of specular reflection (Schumann et al., 2008).Furthermore, temporal WSS variations can affect the lidar WSS so that it does not represent the mean WSS.

ICESat-2 River Surface Slope
The reach-scale "ICESat-2 River Surface Slope" (IRIS, Scherer et al., 2022bScherer et al., , 2023) ) data set is used to evaluate the results of this study.IRIS is derived for each SWORD (Version 15) reach (Altenau et al., 2021) from observations of the spaceborne lidar sensor ATLAS onboard ICESat-2.Since ICESat-2 measures synchronously along six beams, the WSS can be calculated across all beams intersecting the respective reach (Scherer et al., 2022a).In addition, due to the high accuracy and precision of the ICESat-2 observations, the WSS is also calculated along a single beam if it intersects the river nearly parallel.In this study, we use the combination of the across-and alongtrack methods for comparison.Compared to the results of this study, the spatial resolution of IRIS is lower as it corresponds to the SWORD reach length of about 10 km.However, IRIS data are homogeneously distributed along the river and are therefore available where in situ data may be missing.IRIS has been validated against 815 reaches in Europe and North America with a median absolute error of 23 mm/km (Scherer et al., 2022a).

DEM-Based WSS
To assess the accuracy of our results, we also use WSS data sets based on DEM models.The WSS from the SWORD database have already been described in Section 3.2.Furthermore, we use the "Global River Slopes" (GloRS) database, developed by Cohen et al. (2018).Here, the authors calculated the WSS based on the 15 arc-sec resolution (∼460 × 460 m) "SHuttle Elevation Derivatives at multiple Scales" (HydroSHEDS) DEM and streamnetwork (Lehner et al., 2008).The proposed approach consisted of calculating the maximum and minimum elevations of each river segment and dividing the elevation difference by the length of the segment.For a global analysis, the authors upscaled the 15 arc-sec DEM to a 6 arc-sec model (1 arc-sec ∼ 30 m at the equator).
Another DEM-based analysis of river profiles was recently presented by Ruetenik (2022), who developed the "RiverProfileApp" (https://riverprofileapp.github.io,accessed on 2023-01-25).This tool allows an almost global analysis of river profiles with a resolution of 90 m.The "RiverProfileApp" offers two DEM models.To extract river profiles, we use the default HydroSHEDS flow direction grid for flow routing.In addition, a smoothing Water Resources Research window size of 10 km is applied to the calculated profiles.To obtain WSS based on the river location and elevation, we perform the following calculations: (a) for each river coordinate, the nearest SWORD centerline and chainage is assigned, (b) due to the amount of data noise, we average the elevations for each river kilometer using a 30 km window (15 km upstream and 15 km downstream), (c) elevations with a dam or river lake within the window are discarded, (d) for each river kilometer, the WSS is calculated by comparing its elevation to the neighboring river kilometer elevation.These values (30 km window and 1 km distance) were obtained by minimizing the noise of slope variations and comparing the obtained slopes with in situ data.

Methodology
In this section, the new innovative approach for the generation of high-resolution water surface slopes from crosscalibrated multi-mission satellite altimetry is described in detail.
The approach consists of six processing steps which are shown in the flowchart in Figure 3 and described in the following sections.The method is explained using an example section of the Vistula River between chainage 0 and 211 km.

SWORD River Centerline
For each river, a high-resolution centerline is derived from the latest SWORD (Version 15) (Altenau et al., 2021) data set described in Section 3.2.It provides reaches (∼10 km), nodes (∼200 m) and centerlines (∼30 m) for rivers worldwide.In this approach, we estimate the mean non time-varying slope of the water surface with a spatial resolution of 1 km along the river centerline.The spatial resolution of 1 km was chosen as a compromise between available altimeter measurements and computational limitations.This approach can be easily modified to any user-defined bin size.For example, a higher spatial resolution, for example, 100 m, could be used to better account for satellite orbit shifts of 1-2 km, but the impact on the final water surface slopes is expected to be minimal.For this purpose, the high-resolution centerlines are grouped into 1 km bins, which serve as reference points in this approach.In addition, each centerline point is mapped to its reference point, so that each altimeter crossing can be mapped exactly to the corresponding reference point, but also the centerline point on the river.Figure 4 shows an example section of the Vistula River between chainage 52 and 88 km with the extracted SWORD centerline highlighted in black and the reference points as black dots along the centerline.

Area of Interest (AOI)
To extract the relevant altimeter data across the river, we use the SWORD centerline from the last step as input.
Since there are valid altimeter measurements not only over the river, but also several hundred meters close to the river due to the size of the altimeter footprint (Boergens et al., 2016;Schwatke, Dettmering, Börgens, & Bosch, 2015), we create an AOI with a boundary of 1,000 m from the SWORD centerline.This allows us to extract altimeter data that most probably measures the river and not land or adjacent waters.Remaining outliers are rejected later.The AOI derived from the SWORD centerline is shown in Figure 4.It is highlighted in white in the background.

Water Levels at River Crossings Using Satellite Altimetry
Using the AOI of the river of interest, we extract the high-frequency altimeter measurements of the 11 altimeter missions introduced in Section 3.1 from OpenADB (Schwatke, Dettmering, et al., 2023).The combination of measurements from altimeter missions on different orbits increases the number of river crossings and thus the spatial resolution along the river.
Since the altimeter missions have different orbits, the crossing of the river of interest is random, which also depends on the river topology.Rivers that flow in an east-west direction have a higher probability of being crossed than rivers that flow in a north-south direction, because the altimeter tracks also run in a north-south direction.Figure 4 shows the distribution of crossing altimeter tracks within the AOI for the example section of the Vistula River.It clearly shows the missions with a short repeat cycle between 10 and 35 days such as Envisat, Jason-2/-3, Jason-2/-3 (EM), Sentinel-3A/-3B, SARAL, and Sentinel-6A, where many altimeter tracks cross the river side by side.More important for our approach are altimeter missions that fill in the data gaps along the river.Therefore, altimeter missions with long repeat cycle (CryoSat-2, ICESat-1/-2) or a drifting orbit (ERS-1E/-1F, SARAL (DP), Jason-2 (GM)) are more suitable.By combining both types of altimeter missions, a good data coverage along the river can be achieved, as shown in Figure 4. Table 2 gives an overview of the used river crossings per mission and river.The number of valid river crossings depends on the length of the river, but also on the width of the river.A comparison between the Dunajec (161 km studied river length) and the Oder (481 km studied river length), which is 3 times longer, shows that about 15 times more valid river crossings are available for the Oder (6,808) than for the Dunajec (451).This is mainly due to the data quality for small river crossings, but the river course can also have an influence.
To estimate the water levels at the river crossings, the necessary altimeter measurements, geophysical corrections and models are extracted from OpenADB.When processing the water levels, an individual analysis of the radar echoes, called retracking, is applied.Therefore, the Improved Threshold Retracker (Hwang et al., 2006) is used, which is optimized for inland waters.The combination of water levels from different altimeter missions requires the consideration of range biases caused by systematic effects, which are computed by a multi-mission crossover analysis (Bosch et al., 2014).In this study, we use mission-dependent mean range biases calculated by the MMXO16 which vary between 0.06 m (ICESat-1) and 0.66 m (ERS-1) using the Jason series as a reference.
Equation 1 shows the formula and parameters used to estimate the water levels of each altimeter measurement along the crossing altimeter tracks.The WSE is computed by subtracting the retracked altimeter range (R ralt ), geoid height (N), geophysical corrections and range bias (Δh rbias ) from the satellite height H sat to obtain the physical heights used in the next processing steps.The altimeter range is corrected by the geophysical corrections However, an outlier rejection is necessary before using the water levels in our new approach.There are several reasons for outliers, such as off-nadir measurements (Boergens et al., 2016), adjacent waters, or waveforms distorted by land contamination.Therefore, we apply an iterative outlier rejection on each crossing altimeter track in order to use only the most accurate altimeter measurements.To do this, we estimate the median water level for the altimeter track and the standard deviation of the differences.Then, water levels are rejected as long as the standard deviation is greater than 10 cm or the number of along-track altimeter measurements is greater than 5. Using a minimum of 5 altimeter measurements ensures that the later water level of the river crossing is based on multiple altimeter measurements and is therefore more accurate.After the outlier rejection, the median water level and the corresponding standard deviation of the water levels are assigned to the river crossing and used as input data in the next processing steps.

Water Levels for Each River Section With Least-Squares Adjustment
In this section, the approach for estimating the water levels along the river with a spatial resolution of 1 km is described.We demonstrate this approach, which is based on a weighted least squares adjustment, in detail on a river section of the Vistula River between chainage 0 and 211 km.Rivers are split into sections where discontinuities such as dams or reservoirs occur.This information is provided in SWORD reaches and verified manually.There may still be erroneous water levels in the data at this point because the consistency of neighboring water levels has not yet been considered in the along-track outlier rejection step above.For this purpose, we apply a Support Vector Regression (SVR, Smola & Schölkopf, 2004) to the water levels of each river section to rejected clear outliers of several meters.Figure 5 shows the valid water levels at the Vistula River section colorcoded by altimeter mission.One can clearly see the influence of the different altimeter missions on the data distribution along the river.For example, Jason-2/-3 and Sentinel-6A cross the river only near the 12 km river chainage.However, ICESat-1/-2 and CryoSat-2 are more evenly distributed along the river than the other missions.As mentioned before, a combination of water levels from different altimeter missions is essential for an accurate estimation of WSE and WSS, respectively.
In the next step, we describe the applied weighted least squares adjustment to estimate the water level for each 1 km bin.In the example of the Vistula River reach between 0 and 211 km, water levels are calculated for 211 nodes n every kilometer.In addition, 1,690 water levels from altimeter measurements m at the river crossing are used as input data.
In the general least squares adjustment formula, only observations l in the design matrix A without weighting are used to estimate the unknown water levels at each node x (Niemeier, 2008).Equation 2shows the modified weighted least squares adjustment formula compared to the general least squares adjustment described in Niemeier (2008) which is used to estimate the water levels at each reach river node.
In this study, however, we extended the design matrix A by two additional conditions, so that the design matrix A finally consists of three sections, which are introduced as follows.• Altimeter measurements: In the first section of the design matrix A, the water levels of the altimeter measurements are assigned to the corresponding node.In the design matrix A, the corresponding node is set to 1 and the value of the water level is added to the observation vector l.• Laplace condition: Since water levels are not available for all nodes, an additional Laplace condition was added to the design matrix A to ensure that it is not singular and still solvable.This Laplace condition can be thought of as an interpolation and smoothing filter that minimizes the differences between the water level of the current node and the previous and next nodes.In the design matrix A a filter of [1 2 1] is applied to each node, except for the first and last node.The value in the observation vector l is set to 0. However, this may result in constant water levels at the boundaries of the river sections if no data is available.• A priori gradient condition: To avoid the problem at the boundaries caused by the Laplace condition, an additional a priori gradient condition has been added to the design matrix A. In the design matrix A, a filter of [ 1 1] is applied to each node and the a priori water surface gradient is added to the observation vector l.The a priori water surface gradient is calculated by estimating a linear trend within a 20 km moving window along the river.This condition ensures that the resulting water levels at the boundaries do not converge to constant water levels, but take into account the a priori water surface gradient.
The dimension of the design matrix A consists of k rows and n columns where k = 2n + m 2, m is the number of altimeter measurements, n 2 is the number of rows of the Laplace condition and n is the number of rows of the a priori gradient condition.
Additionally, a weighting of the three sections is applied in the matrix P.This is necessary to control the impact of the altimeter measurements, the a priori gradient condition, but also the smoothing of the Laplace condition along the river.For the weighting of the altimeter measurement σ 2 m , a mission-depending weighting is applied.Table 3 shows the precision used for each mission, derived from the median of all standard deviations of the measurements used.
Equation 3 shows the applied formula for the weighting of altimeter measurements.
The weighting of the Laplace condition w 2 l and a priori gradient condition w 2 g is performed relative to the weighting of the altimeter measurements w 2 m .
The weighting factors for the Laplace condition and a-priori gradient condition have been derived by a closedloop simulation shown in Figure 6 and taking into account in situ data and lidar data for a quality assessment.
Here, we have created a river section of 100 km where, in the ideal case, altimeter "crossings" are available every 10 km and cover the full seasonal variation (red + gray dots).The averaged water level every 10 km is known and is used in a closed loop test for quality assessment.To be more realistic, the seasonal variations and the slope of the water surface change along the river.In this simulation, all altimeter measurements are assumed to have an uncertainty of 0.15 m.In the closed-loop test, we estimate the least squares fit using only the red data as input.The ignored measurements are highlighted in gray.For six scenarios, we calculated the average water level of the river reach by varying the weighting of the Laplace condition (1-10,000) and the a-priori gradient condition (0.5-5,000) shown in Figure 6.
This closed-loop test clearly illustrates the challenge of choosing the optimal weights.Using a low weighting (blue line) for the Laplace condition (e.g. 1) and the a-priori condition (e.g., 0.5) with respect to the altimeter measurements leads to an overweighting of the altimeter measurements and the seasonal signal is included in the final averaged water levels (see Figure 6, 35 km-70 km).On the other hand, using a high weighting (brown line) for the Laplace condition (e.g., 100,000) and the a-priori condition (e.g., 50,000) with respect to the altimeter measurements leads to an overweighting of the a-priori gradient condition and the seasonal signal is smoothed, but also the water level variability along the river is removed in the final averaged water level.The simulation shows that the factor of 100 for the Laplace condition and 50 for the a-priori gradient performed best, resulting in the smallest RMSE of 17 cm compared to the known simulated average water levels and is finally used in this study.The used weights in this study have been additionally supported by using in situ data and lidar data.The same derived factors of 100 for the Laplace condition and 50 for the a-priori gradient are used for all rivers in this study and can be used for any other river of interest.
The Laplace condition is more important and therefore weighted higher than the a priori gradient condition because it is used to interpolate data gaps, but also to reduce the influence of seasonal signals (smoothing).The gradient condition is therefore weighted less as it is only used to provide information to the WSS in the case of data gaps or near the boundaries of the river segments.This condition also prevents the presence of negative WSS.It can be considered as a secondary condition.For the weighting, we estimate a mean standard deviations σ section of all used altimeter measurement within a river section first.Then, we use the formulas of Equations 4 and 5 for the weighting of the Laplace condition w 2 l and a priori gradient condition w 2 g .
The weightings w 2 m , w 2 l and w 2 g are set to the diagonal values of the identity matrix P.
The advantage of the weighted least squares adjustment is that the associated WSE uncertainties for each node can be estimated by computing the covariance matrix Σ xx using the formula described in Niemeier (2008).
Figure 5, shows the resulting water levels (black line) of the introduced least squares approach for the river section along the river.It can be clearly seen that the estimated water levels describe the average water level of the river very well.The seasonal water level variations and the uneven distribution of water levels are also well captured.
The resulting WSE uncertainties for this river section vary between 0.05 and 0.24 m with an average of 0.11 m.
The WSE uncertainties are used in the next section to estimate the WSS uncertainties.

Water Surface Slopes for Each River Section
In the final step, the water levels along the river are converted to WSS. Between two neighboring river nodes, the difference in WSE is calculated and divided by the length of the river from the SWORD centerline between them.
The WSE uncertainties computed with the weighted least squares adjustment are used to compute the WSS uncertainties for which error propagation is applied.
Figure 7 shows the resulting WSS and uncertainties for the example section of the Vistula River.

Results and Quality Assessment
The new, innovative approach for generating high-resolution water surface slopes from multi-mission satellite altimetry is based on global, freely available data: river centerlines from SWORD and altimetry measurements from OpenADB.Therefore, this approach can be applied globally to almost any river.In this study, we present the WSS analysis of 11 Polish rivers, including sections located in lowland, upland, and mountainous areas (Section 5.1).Due to the dense network of gauges, referenced to a common vertical datum, we are able to assess the WSS accuracy by comparing it with the river slopes between adjacent gauges (Section 5.2).Furthermore, we perform a quality assessment based on cross validation (Section 5.3).Finally, to prove the usefulness of the WSS, we apply the river altimetry slope bias correction (Halicki et al., 2023) to the Sentinel-3B water level time series over two virtual stations (VS-intersections of satellite ground tracks and river channels) located in mountainous areas (Section 5.4).

WSS of Polish Rivers
Figure 8 shows the WSS of 11 Polish rivers.These results are also provided as NetCDF and shapefile, freely available at https://doi.org/10.5281/zenodo.7709474(Schwatke, Halicki, & Scherer, 2023).For most of the rivers, the WSS ranges from 0 to 500 mm/km.The steepest rivers occur in the southern, mountainous area-the WSS of Dunajec, Poprad and San (in their upper part) ranges from 1,000 mm/km to 4,000 mm/km.In general, the WSS of each river decreases in the downstream direction.On the contrary, the slope of the Noteć River slightly increases toward its mouth, but it is a highly regulated, lowland river with low WSS values on the whole studied section.It is also worth mentioning, that the WSS of most of the rivers is strongly variable in the spatial domain.
For example, the WSS of the Vistula River changes by up to 200 mm/km every few kilometers.The most stable WSS can be found on the Pilica River, for which the slope values vary in the range of 350 mm/km to 500 mm/km almost along the whole studied section.
WSS variations can also be clearly seen in Figure 9, which shows the Vistula river (a), and Oder river (b).
Figure 10 shows the WSS variations of the Warta river (c), and Dunajec (d) river.Vistula, Oder, and Warta are the longest rivers in Poland.On the other hand, Dunajec is mainly located in a mountainous area with the highest WSS.The graphs showing the WSS variation of the other investigated rivers are presented in the appendix (Figures A1-A4).The WSS of the Oder and Warta rivers (Figures 9b and 10a) varies by about 50-100 mm/km.The WSS variations on the Vistula (Figure 9a) are even stronger with up to 250 mm/km.These variations are less significant on the Dunajec (Figure 10b), compared to its total WSS of up to 4,000 mm/km.database, and (d) WSS calculated from lidar data (see Section 3.3.2).A comparison between the different WSS will be made in the following sections.Table 4 shows additionally the mean WSS uncertainties of the river section and river.

Validation With In Situ Slopes
In order to assess the accuracy of the derived WSS of Polish rivers, we compare it with the in situ WSS between gauging stations.This comparison is not possible for Wisłoka, Noteć, and Poprad, due to the lack of connected gauges undisturbed by hydraulic structures.The median in situ slopes of the Vistula, Oder, Warta, and Dunajec are shown in Figures 9 and 10.To properly compare the high-resolution, altimetry based WSS with in situ slopes, we calculate the median WSS for each river section between selected gauges.These values for sections between neighboring gauges are presented in Figures 9 and 10 with black, horizontal lines.At the Vistula River, the lower and middle sections agree better than the upper section, but the differences do not exceed 50 mm/km.The WSS of the Oder and Warta are almost identical to the in situ slopes, with very small differences.Also for the Dunajec River the agreement is very high for most of the sections, except for the most upstream section, where the difference exceeds 200 mm/km.
The accuracy of the estimated WSS from satellite altimetry of Polish rivers is presented in Table 4 (In situ RMSE).The RMSE value for each river (except for Wisłoka, Noteć, and Poprad) is given for each river section between flow disturbances, as well as for the entire river.The values in brackets refer to the number of gauged sections included in the RMSE calculation.The RMSE is calculated by using the differences between the median slope between two in situ stations and the averaged high-resolution WSS.The RMSE for the whole rivers ranges from 4 mm/km to 77 mm/km, with an average of 27 mm/km.The RMSE of more than half of the rivers studied (5 out of 8) is less than 15 mm/km.The lowest RMSE values of 4 mm/km and 5 mm/km could be archived for the Bug and Oder rivers respectively, both based on 45 gauging sections.The derived WSS of the largest Polish river (Vistula) also shows a very good agreement with the in situ WSS (RMSE: 13 mm/km).However, the accuracy is significantly higher in the lower and middle sections (13 mm/km and 10 mm/km RMSE for the 0-211 km and 255-647 km sections, respectively) than in the upper section (30 mm/km RMSE).The only two rivers with RMSE above 30 mm/km are Dunajec (74 mm/km) and San (77 mm/km), which are located in a mountainous and upland areas and their slopes can locally reach between 2,000 mm/km and 4,000 mm/km.

Internal Cross-Validation of WSS
Using the method described in Section 5.2, we can only compare the average WSS between two gauges.In this section, we perform an internal cross-validation of the derived WSE and WSS to evaluate the quality of the river sections not covered by gauges.It is also used to estimate the accuracy of the variability of the WSS along the river.
For the cross-validation, we calculate a WSS between each possible combination of two altimeter heights from Section 4.3 and compare them with our mean WSS between the two river crossings.Due to the large number of combinations (e.g., Warta: 300,000) and the different track lengths, this allows a robust internal validation of the WSS.Based on the WSS differences of all pairwise comparisons, the root mean square deviation (RMSD) is calculated for each river section and for the entire river.
Table 4 shows the results of the cross-validation (Cross-val.RMSD) for each studied river section and for the whole river.For the Vistula, Oder, Warta, Bug, Narew, San, and Pilica rivers, the RMSE of the cross-validation varies between 16 mm/km and 68 mm/km.However, for the rivers Wisłoka, Dunajec, Noteć, and Poprad, the  RMSD of the cross-validation is significantly larger and varies between 87 mm/km and 371 mm/km.This is mainly influenced by the smaller river width and the mountainous regions where three of the rivers are located.Table 4 clearly shows that the RMSD increases in the upstream direction.

Correcting Water Level Time Series From Satellite Altimetry for the Ground Track Shift Bias
Orbit perturbations cause a shift of the satellite ground tracks, which, for example, for Sentinel-3 can vary up to ±1 km.Therefore, the locations of radar altimetry measurements for a single VS are not stationary.Since rivers are inclined water bodies, the altimeter measurements are subject to a bias that depends on (a) the local WSS and (b) the distance between the actual measurement and the VS reference position.The WSS described in this study is estimated for each river kilometer, therefore it is possible to correct the WSE time series for the bias using the WSS for the river section exactly at the VS location.Determining the exact location of an altimetry measurement can be challenging when a river section is parallel to the satellite ground track.Since the footprint size of radar altimetry measurements is generally greater than one km, some WSE may be biased by off-nadir measurements.In these cases, the exact location of the satellite measurement cannot be accurately determined, and thus the WSE time series cannot be properly corrected for the WSS.Since the aim of this analysis is to prove the usefulness of the estimated WSS, we select two VS of the Sentinel-3B satellite from DAHITI, located on mountainous stretches of the San (DAHITI-ID: 41491) and Dunajec (DAHITI-ID: 41492) rivers, where the problem described above does not occur.We correct these VS for the WSS bias using the results of this study, which are 550 mm/km and 1,032 mm/km for the San and Dunajec VS, respectively.
To assess the improvement of the correction, we compare the uncorrected and corrected WSE time series of each VS with measurements from adjacent IMGW-PIB gauges, which are located 3.1 and 3.  41492).Errors are reduced for most of the measurements.However, VS in mountainous areas are affected by larger errors than VS in lowland river sections, mostly due to the surrounding topography (Jiang, Nielsen, et al., 2020).Therefore, the WSE time series may still contain outliers, even though an outlier rejection has been performed in the DAHITI approach.In these cases, the bias correction does not reduce the measurement error.

Discussion
Table 5 shows the accuracy of WSS results from this study with WSS derived between gauging stations.In addition, the accuracy of other WSS data sets, based on DEM models (GLoRS (Cohen et al., 2018), River-ProfileApp (Ruetenik, 2022) and SWORD (Altenau et al., 2021)), lidar (Section 3.3.2),and ICESat-2 from the IRIS data set (Scherer et al., 2022b) with WSS derived between gauging stations is shown.The GLoRS data set is the least accurate with a mean RMSE of 742 mm/km.The accuracy of the SWORD WSS is also poor, with a mean RMSE of 71 mm/km and a minimum RMSE of 19 mm/km.The RiverProfileApp is the best DEM-based approach with an average RMSE of 63 mm/km.Although the RiverProfileApp is also based on a global DEM model, the processing uses a different approach than the GLoRS and SWORD databases (see Ruetenik, 2022).The RiverProfileApp allows the parameters to be set manually via the web application.However, this application does not provide WSS directly but generates river profiles downstream of a selected point.
Based on this data, we calculate the WSE for each kilometer by averaging heights within a 30 km moving window (15 km upstream and 15 km downstream).Next, we calculate the WSS by comparing adjacent WSE.However, even though the RiverProfileApp revealed the highest accuracy among the DEM-based slopes, it was still significantly less accurate than WSS from multi-mission satellite altimetry approach.The low accuracy is probably caused by the coarse resolution of global DEM models, which in the area of small and medium-sized river channels causes large vertical errors.Furthermore, the mean RMSE values are strongly deteriorated by the high RMSE on the Dunajec River.
The RMSE of the WSS from airborne lidar is low for most of the lowland rivers.On the contrary, the RMSE for the mountain rivers is significantly higher (169 mm/km and 184 mm/km for the Dunajec and San rivers, respectively).The RMSE of the lidar-based WSS for the Pilica River is also high with 182 mm/km.The WSS from lidar is not well suited for validation because it does not represent a mean WSS but only a short temporal sample and lidar can be distorted over water.However, it has a high spatial resolution.Therefore, it can be used to interpret the quality of the spatial variations of our results, which are not visible in the WSS from gauges.The overall frequency of the spatial variations is in good agreement between our results and the lidar WSS, although the local extremes are not always in perfect agreement, possibly due to temporal variations.Specific features, such as the significantly increasing WSS between chainage 100 and 125 km at the Dunajec River or the most upstream section of the Oder river, align very well (Figures 9b and 10b).Also, a very good agreement of the WSS variations with the lidar WSS can be seen at the Vistula River between chainage 350 km and 450 m.
The results of the reach-scale IRIS WSS are comparable to this study.This is probably also due to the fact that ICESat-2 altimeter measurements are also used as input data in this study.Only at the Bug River (28 mm/km vs. 4 mm/km) and at the Dunajec River (271 mm/km vs. 74 mm/km) the IRIS data show a significantly lower accuracy.Similar to the DEM-based approaches, the high mean RMSE of 54 mm/km is strongly influenced by the Dunajec River.
The WSS derived in this study are in agreement with WSS of Polish rivers reported in literature.There is no highresolution information about WSS for short sections of Polish rivers available.However, there are several studies with general information about mean WSS for selected river sections.The WSS of the entire Vistula River (divided into 12 sections) are provided by Starkel (2001).Considering only the sections overlapping with this study, the WSS by Starkel (2001) ranges from 360 mm/km in the upstream reach to 170 mm/km in the downstream reach.These values agree well with the WSS estimated in our study (cf. Figure 9a).Although in some cases the WSS from this study exceeds the WSS by Starkel (2001), we derived the WSS for almost every kilometer of the river, whereas Starkel (2001) reported average WSS over long river sections.Habel (2010) conducted a WSS measurement campaign for the 60 km section of the Vistula between the Włocławek dam and the city of Toruń using a GNSS receiver mounted on a boat.The average slope for this section from two separate measurement campaigns is of 157 mm/km, which is identical to the mean WSS for the same section from this study (157 mm/km).The WSS derived in this study shows high accuracy not only for the lowland rivers, but also for those located in mountainous areas.The WSS of the studied sections of the Dunajec River in the literature ranges from 580 mm/km to 3,350 mm/km (Pasternak, 1968), which agrees with the WSS from this study (cf. Figure 10b).Although the WSE determination from satellite altimetry is challenging in steep-sided valleys (Jiang, Nielsen, et al., 2020), the difference between our results (2,736 mm/km) and a study by Nyka (2006) (3,200 mm/ km) is relatively low for the Dunajec River Gorge.
In addition to the comparison with in situ and other WSS data set, an internal cross-validation is performed comparing the WSS between two altimeter measurements with the WSS from this study.The resulting RMSD for the 11 Polish rivers varies between 16 mm/km and 371 mm/km, showing lower RMSD for the larger rivers and higher RMSD for the smaller mountain rivers.The cross-validation is a valuable tool to assess the WSS variation along the rivers because of the large amount of used altimeter measurements located at different river chainages.This method also allows us to assess the quality for river sections where no in situ data is available.
The WSS derived from satellite altimetry can also be useful for geomorphic and hydrologic applications.The accurate, high resolution WSS can significantly correct the altimetry-based WSE time series at virtual stations (Halicki et al., 2023;Scherer et al., 2022a).In this study, the RMSE of WSE time series is reduced by up to 42% for two virtual stations located at the San River and the Dunajec River.However, when WSE time series are affected by other errors such as the off-nadir effect, the WSS correction may be ineffective.
This new approach for estimating high-resolution WSS was developed to be applicable to any other river in the world.Since altimeter satellites fly on a fixed orbits with a certain repetition cycle, the data coverage is similar around the world and the use of drifting and geodetic orbits makes the coverage even denser.The river topology can have an impact on the data coverage, as rivers flowing in an east-west direction (e.g., Notec, Bug, Warta) are better covered than rivers flowing in a north-south direction (e.g., Dunajec, Wisloka, San).But even for rivers with "bad" data coverage the approach worked.The width of the river has no influence on the transferability of the approach, as wider rivers like Oder or Vistula were selected, but also smaller rivers (e.g., Dunajec, Poprad) or rivers far upstream, as long as the SWORD data set is available for the definition of the river topology.In this study, we selected rivers with low water surface slope (e.g., Oder, Vistula) and rivers in mountainous regions (e.g., Dunajec, Poprad, Wisloka, San) in order to cover different variations of water surface slope as good as possible.In summary one can conclude that this new approach has the potential to be applied to any other river in the world, regardless of the data coverage, river topology, river width, or water surface slope.In addition, the Water Resources Research 10.1029/2023WR034907 applied weighted least squares adjustment with its given weights of the Laplace condition and the a-priori gradient can be directly applied to other rivers as no additional adjustment is required.

Conclusion and Outlook
In this study, we present an innovative approach to estimate high-resolution WSS of rivers based on multi-mission altimetry.We study 11 Polish rivers located in both lowland and mountainous areas.To maximize the spatial coverage of the altimetry measurements, we combine WSE from 11 satellites.The used missions are CryoSat-2, Envisat, ERS-1, ICESat-1/-2, Jason-2/-3, Saral, Sentinel-3A/-3B, and Sentinel-6A.The altimetry measurements cover the period from 1994 to 2023.In our approach, we first divide the rivers into river sections that are not interrupted by dams, waterfalls, or reservoirs.Then, we use a weighted least squares adjustment with an additional Laplace condition and an a priori gradient condition to estimate the WSE at each river kilometer from which we derive the WSS.et al., 2021), the RMSE values vary between 19 mm/km and 199 mm/km (average: 71 mm/km).The comparison of using WSS data from the IRIS database (Scherer et al., 2022b) results in RMSE values between 7 mm/km and 271 mm/km (Average: 54 mm/km).Finally, the WSS between in situ gauges are compared with lidar data, resulting in RMSE values between 10 mm/km and 184 mm/km (Average: 100 mm/km).This study shows that the accuracy of WSS from satellite altimetry is high compared to WSS from the other sources shown.The advantage of accurate WSS of rivers is that the WSE time series at VS from satellite altimetry can be improved by correcting the ground track shift bias of the altimeter missions.
For two examples at the San River and the Dunajec River, the RMSE of the WSE time series decreases by 42% and 41% respectively.
The SWOT mission, launched in December 2022, will also provide global WSS using state-of-the-art "radar interferometry," to monitor surface waters with unprecedented resolution.The scientific requirements of SWOT aim for a WSS accuracy of 17 mm/km (Biancamaria et al., 2016).The multi-mission satellite altimetry approach presented in this study shows an accuracy within the SWOT requirements for most of the rivers studied.Only the mountain rivers, that is, San and Dunajec, have significantly lower accuracies.Since the WSS estimation approach can be applied globally, it can serve as validation data for the upcoming SWOT observations.

Figure 1 .
Figure 1.Study area of the 11 investigated Polish rivers with the highlighted studied reaches and available in situ stations.

Figure 3 .
Figure 3. Flowchart of the processing steps of the new approach to derive water surface slopes for rivers from multi-mission satellite altimetry.

Figure 4 .
Figure 4. Vistula River between chainage 52 and 88 km with SWORD centerline in black and reference points every kilometer as black dots along the centerline.The locations of all high-frequency altimeter measurements prior to outlier detection within the area of interest are colored by each group of missions.

Figure 5 .
Figure 5. Input water levels and final heights at the Vistula River between chainage 0 and 211 km.

Figure 6 .
Figure 6.Simulated river section and applied least squares adjustment using different weights for the Laplace condition and a-priori condition.
Figures 9 and 10 also include WSS uncertainties (gray, vertical bars), which are related to the vertical errors of WSE in each of the 1 km bins (see Section 4.4).In general, large errors appear at the edges of the sections due to the lower number of WSE measurements.In addition, Figures 9 and 10 include (a) the median WSS between neighboring gauges, (b) WSS from the SWORD database, (c) ICESat-2 based WSS from the IRIS

Figure 7 .
Figure 7. Water surface slope (dots) and uncertainty (bars) of the Vistula River between chainage 0 and 211 km derived from multi-mission satellite altimeter.

Figure 8 .
Figure 8. High-resolution non time-varying Water Surface Slope (WSS) of the 11 investigated Polish rivers.

Figure 11 .
Figure 11.Water level time series of the virtual station (DAHITI ID: 41491) at San River without water surface slope (WSS) correction (red) and with WSS correction applied (green).The blue bars show the distance between the river crossing of each satellite track and the defined reference location along the river.The green and red bars show the water level differences with respect to the in situ station.

Figure 12 .
Figure 12.Water level time series of the virtual station (DAHITI ID: 41492) at Dunajec River without water surface slope (WSS) correction (red) and with WSS correction applied (green).The blue bars show the distance between the river crossing of each satellite track and the defined reference location along the river.The green and red bars show the water level differences with respect to the in situ station.
3 km downstream of the San and Dunajec VS, respectively.All three time series (in situ, uncorrected and corrected) are shown in the upper graph in Figures 11 and 12 for the San and Dunajec VS, respectively.The distance between the altimetry measurement and the VS reference position is presented in the middle plot (blue bars).The lower plot shows the error bars of the uncorrected (red bars) and corrected (green bars) measurements.The bias correction results in a significant reduction of the RMSE: from 0.36 to 0.21 m (42%) for the San VS (DAHITI ID: 41491) and from 0.49 to 0.29 m (41%) for the Dunajec VS (

Table 1 Characteristics of the Rivers Included in This Study and of the River Sections Studied
(Altenau et al., 2021) the SWORD data(Altenau et al., 2021).c Fluvial lakes are excluded from this statistics.SCHWATKE ET AL.
To validate the WSS obtained in this study, we use WSE data from 81 gauges of the Institute of Meteorology and Water Management-National Research Institute (Instytut Meteorologii i Gospodarki Wodnej-Państwowy Timeline of the altimeter missions used in this study.The color of the mission is chosen according to its orbit.

Table 3
Mission-Dependent Precision Used for the Weighting of Altimeter Measurements

Table 4
Quality Assessment and Validation of Estimated Water Surface Slopes From Satellite Altimetry a Number of In Situ Sections.b Number of Sections between two Water Levels from Satellite Altimetry.
Table5includes only 8 of the 11 studied rivers, because on Noteć, Wisłoka, and Poprad there are no gauge sections undisturbed by hydraulic structures.In general, the mean RMSE of the WSS derived in this study is significantly lower compared to the other approaches.The only two exceptions are the Narew River, where the accuracy of the ICESat-2 WSS (9 mm/km RMSE) slightly exceeds the accuracy of this study (14 mm/km RMSE), and the San River, where the accuracy of the WSS based on the RiverProfileApp (51 mm/km RMSE) exceeds the accuracy of this study (77 mm/km RMSE).
Cohen et al. (2018) study are the most accurate WSS for Polish rivers from remote sensing data.The RMSE values for 11 investigated Polish rivers vary between 4 mm/km and 77 mm/km.It outperforms other WSS data especially in mountain rivers.The results of this study are compared with other global WSS data sets which are, however, limited in both quality and quantity.Existing global databases based on DEM models do not provide sufficient accuracy.Using WSS data fromRuetenik (2022)results in RMSE values varying between 20 mm/km and 243 mm/km with an average of 63 mm/km.Using WSS data fromCohen et al. (2018)results in RMSE values varying between 294 mm/km and 2,741 mm/km (average: 742 mm/km).Using WSS from SWORD(Altenau

Table 5
Validation of WSS From Satellite Altimetry With In Situ WSS and Additional Quality Assessment Between WSS From DEM, SWORD, and Lidar With In  Situ WSS SCHWATKE ET AL.