The ATL03 photon cloud product is used to define riverbed geometry. To ensure that as large a portion of the riverbed as possible can be extracted from ICESat-2 observations, only the low-flow season crossings are taken into account, corresponding to the period between November and April. In addition, to select good quality cross sections, a visual inspection is also performed. The strong beam and weak beam data, separated by 90 m, were observed to behave differently, depending on the acquisition date. For the cases in which the strong beam signal was very strong, the data can present noise that complicates the processing steps, so the weak beam crossing is selected. For acquisitions with weaker signals, the topography is better mapped by the strong beam signal.

The water surface in the ATL03 data is characterized by a dense cloud of photons. A Gaussian kernel distribution is used to identify the water surface by taking the peak density value of the distribution. The points between 20 cm from the peak value are also identified as water surface and averaged. The river width is computed as the along-track extent of the water surface, taking into account the crossing angle between ICESat-2 ground track and river centerline.

To remove outliers from the photon cloud, two different methods are used. First, a Hampel filter is applied, in which the median absolute deviation (MAD) is computed for a sliding window in a 3.5 m ground track interval. A value will be considered an outlier if it exceeds the MAD. The MAD is defined as in Eq. (1): 1MAD=median|Xi-X̃|, with X̃=median(X). After the Hampel filter, a median filter is applied for the same 3.5 m ground track intervals. The ground track interval is a “window” that slides, entry by entry, and replaces each entry value with the median of the neighboring entries. Finally, in order to obtain a well-defined shape of the river cross section, a simple moving average (SMA) is computed for ground track intervals of 3.5 m.

In the submerged portion of the channel, the ATL03 product does not detect the riverbed elevation. ATL03 product can successfully map bathymetry for clear waters, as shown by Parrish et al. (2019), who present results mapping the seafloor bathymetry. The portion of the area upstream of the Yellow River that we focus on (see Sect. 2) is characterized by fast and turbulent flow, which explains why bathymetry cannot be detected by ICESat-2 in this area. To cope with this limitation, the riverbed elevation in the submerged portion of the river is extrapolated using the approach proposed by Dingman (2007), which has been previously used for discharge estimates in Bjerklie et al. (2018). The shape of the submerged portion depends on a cross section form exponent r, with r=1 corresponding to a triangle and r→∞ corresponding to rectangular shape. The height above the lowest channel elevation, z, is approximated by Eq. (2), as follows: 2z=Ym*2W*rxr,0≤x≤W*/2, where Ym* is the low-flow depth, W* is the bank width, and x is the horizontal distance from the river centerline. Figure 3 shows the different processing steps applied to ATL03 data to define the riverbed geometry. In the last step, the submerged portion is interpolated; thus, for each cross section, there are two free parameters (Ym* and r) that will be estimated in the hydraulic inversion (see Sect. 4.3).

The model calibration needs an adequate selection of WSE data from ATL13 or other suitable altimetry missions. Radar altimetry missions, such as CryoSat-2 or Sentinel-3 cannot provide high-accuracy measurements in this area due to their low spatial resolution (250–300 m). The river stretch we focus on has a river width varying between 80 m in the low-flow season and a few hundred meters in the high-flow season. The ATL13 product offers a variable resolution depending on the acquisition date, but it is normally of the order of few meters (Jasinski et al., 2021a). This ICESat-2 product offers a variety of data quality flags to discard invalid measurements. In Xu et al. (2021), the following flags are considered for WSE assessments: qf_bckgrd, qf_bias_em, qf_bias_fit, stdev_water_surf, and snow_ice. These flags control the density of background photons, electromagnetic bias and bias in the fit, standard deviation of WSE, and presence of snow or ice. After applying the flags, Xu et al. (2021) recommend an extended outlier filtering method. Moreover, the ATL13 data product presents known issues, such as including land area adjacent to WSE or land measurements in the low-flow season, that need to be considered (Jasinski et al., 2021b).

The morphological characteristics of the river make the selection of good WSE observations challenging. The river stretch presents braided structures in selected areas in which the ATL13 product can mix the land in between channels with water surface. To avoid this issue, we first apply the water occurrence map from Global Surface Water Explorer (Pekel et al., 2016), only including observations that fall over pixels with water occurrence larger than 93 %, a threshold that is exceeded in very few pixels. Taking into account the river delineation extracted from the MERIT database (Yamazaki et al., 2019; see Sect. 3.3), we only consider WSE points that are less than 15 m away from the river centerline, to ensure that ATL13 observations fall in the water surface in the low-flow period, when we have flow widths around 80 m for selected areas. The points that are within 15 m from the river centerline are averaged to have a unique observation for different dates at every chainage point. In addition, all the WSE measurements within the same acquisition date that are less than 500 m away from each other, and where the WSE variation is less than 15 times the slope times the distance between observations, are averaged. This last criterion is taking into account the computational grid resolution of the hydraulic model, which is 300 m and refined to 100 m in selected areas (see Sect. 4.2).

We compared the performance of the filtering process with the quality flags, taking the values proposed by Xu et al. (2021), qf_bckgrd < 6, |qf_bias_em| < 2, |qf_bias_fit| < 2, and constraining the water surface standard deviation, stdev_water_surf to 0.5 m. The snow_ice flag was also applied but with no presence of ice for the area of interest. The results are better when we filter with respect to the river centerline and water occurrence map than when using the ATL13 quality flags. After applying the quality flags only, some of the points that fall on the adjacent land area are not removed.

Cross-sectional data from ATL03 are selected for the available dates in the low-flow season. Depending on the acquisition dates, the discharge varies and, accordingly, the reference WSE of the cross section changes. This can lead to unreasonable variations in the bottom elevation when constant depth is assumed, thus leading to simulation errors. To avoid such errors and have a realistic reference bottom level (BL), we define a reference WSE. The reference WSE is taken from the ATL03 cross section in which the acquisition date has the lowest discharge value from the in situ data. The discharge variation is calculated between the lowest in situ discharge value and the in situ discharge values for ATL03 cross section acquisition dates. Since the discharge variation should be proportional to the WSE variation, we define a correction factor α relating the discharge variation to the WSE variation as ΔWSE=α⋅ΔQ. The correction factor α is a calibration parameter, and we add α⋅ΔQ to the corresponding cross section depth value.

To confirm that the discharge variation and the WSE between ATL03 and ATL13 are consistent, we compare the WSE measured by the ATL03 and the ATL13 for the values that are closer than 300 m, according to the resolution of our model, and the variation in discharge between the ATL03 and ATL13 acquisition dates. The analysis can be seen in Fig. 4a. The method we use to detect the WSE in the ATL03 product is different from the one used to produce the ATL13 product (see Sect. 4.1.1); hence, we can find differences between our detected WSE and the one provided by ATL13 product for the same acquisition data. An example of the difference between WSE detected in ATL03 and ATL13 WSE can be seen in Fig. 4e, where there is an offset between elevations of about 0.41 m. This would explain some of the points that lie close to ΔQ=0, for which ΔWSE is different from 0, in Fig. 4a. To avoid this problem, we substitute the WSE at each ATL03 cross section by the value given in the ATL13 product, and the submerged portion is interpolated based to this value. By substituting these values, some of the discharge variations that were not in accordance with the WSE variation were corrected; for example, the value lying close to ΔWSE=0 and ΔQ=400 in Fig. 4a lies closer to ΔWSE=1 in Fig. 4b.

In addition, the Jimai rating curve presents some anomalies (Fig. 4c) in which the discharge–WSE pair does not follow the power law (points with low discharge values have a high WSE value) that could be explained by the freezing of the water surface in the station area. We observed that the outliers in Fig. 4a were related to the rating curve anomalies (ΔWSE around 2 m and ΔQ close to -200 m3 s-1), and the ATL03 and ATL13 data taken at the corresponding dates were rejected. After this analysis, the resulting relation between ΔQ and ΔWSE can be seen in Fig. 4b.

The downstream boundary is defined as a free, uniform outflow boundary condition (Li et al., 2017). This boundary is defined 10 km away from the downstream station by duplicating the ATL03 cross section defined closest to the Mentang station. To initialize the steady-state solver, a value of WSE at the downstream point is calculated from in situ discharge values, using Manning's equation.

The lateral inflow q is distributed along the chainage proportionally to the contributing area (see Fig. 1), taking into account the discharge variations from upstream and downstream in situ data. The lateral inflow at chainage x is computed as in Eq. (6). 6q=Qx-Qx-1Δx To calculate the discharge at chainage x, Qx, we assume that the runoff in the catchment is uniformly distributed, i.e., boundary inflow is proportional to the increment in contributing area, as defined by the term upstream drainage area (UPA), which is available in the MERIT dataset (see Sect. 3.3). Instead of calculating a rainfall–runoff model, we use in situ discharge measurements at two available stations Jimai (upstream discharge Qup) and Mentang (downstream discharge Qdown; see Sect. 2). For the downstream station, Mentang, the available data are between May and November, while for the upstream station, Jimai, data are available for the whole period. Hence, two possible scenarios are taken into account.

If in situ observations are available at both upstream and downstream stations, then the distributed discharge at chainage point x is as in Eq. (7).7Qx=Qup+UPAx-UPAupUPAds-UPAupQds-Qup

The calibrated model reproduces WSE–discharge relationships in the form of rating curves. We produce a series of rating curves along the river stretch for a given downstream discharge value to create a lookup table between river chainage, WSE, and discharge. This relation can be used for different applications.

Densification of WSE. This is the aim of several studies that use different methods to interpolate altimetry WSEs, i.e., by using Kriging methods (Boergens et al., 2017; Yoon et al., 2013) or by interpolation using WSE (Villadsen et al., 2015). In our case, we use the calibrated 1D hydraulic model to calculate WSE along the river stretch. Using ATL13 observations at different acquisition times, we can predict WSE at any river point using the calibrated model.

The optimal parameters from the 1D hydraulic calibration can subsequently be transferred to a full hydrodynamic model simulation if predictions of dynamics of discharge and WSE are of interest. To demonstrate the ability of the method in the parameterization of full hydrodynamic simulations in the transient state, we use MIKE HYDRO River (Havnø et al., 1995). This is a 1D hydrodynamic model that uses a dynamic wave solver of the 1D Saint-Venant equations. To run the hydrodynamic model, we use daily in situ discharge data as an upstream boundary for a period of 3 years. MIKE HYDRO River uses the calibrated cross sections to obtain information on calibrated channel geometry and optimal Manning's roughness coefficient. The runoff increments are distributed according to the contributing area, which is obtained from the MERIT flow accumulation map.

The cross sections are visually inspected to identify if the land and water topography are well defined, and there are no signs of cloud formation or weather effects. In total, 80 ATL03 cross sections are selected which define the riverbed geometry along the river reach between Jimai and Mentang (see Fig. 5, bottom), including measurements of land elevation of the dry portion of the riverbed. Measurements are provided every 3.5 to 20 m. Closely spaced cross sections are removed to reduce the computational cost of the model, since only cross sections whose chainage differ by more than 500 m provide meaningful information due to the resolution of the 1D hydraulic model. In areas with scarce ICESat-2 data, there can be a separation between cross sections of up to 10 km. The gaps in the data at the scale of our investigation are mostly due to the poor quality of acquisitions in the period in which weather conditions are having an impact on the ICESat-2 observations. For the river stretches with little cross section information, we interpolate cross section in between using the MIKE HYDRO River cross section module; however, model performance is expected to be worse here.

The top panels of Fig. 5 give examples of the cross section geometry measured with ATL03 together with submerged portion that is not observable and has been calibrated. These cross sections give quite a reasonable representation of the river geometry. An advantage of the method is that most of the cross section points are from observed data, since the data are selected for the low-flow season. This gives a more accurate representation of the hydraulic geometry. In Fig. 5a and b, which correspond to the most upstream part, we can observe the presence of floodplains, with a braided structure of few kilometers, while the rest of the stretch has a confined structure.

Initially, we use observations from 236 ATL13 data products, providing measurements in low-flow and high-flow seasons. The 236 ATL13 products contain 3199 WSE points, as illustrated by red crosses in Fig. 6. The ATL13 quality flags described in Sect. 4.1.3 are applied to the data after removing the largest outliers. The data, after applying the quality flags, are represented by light blue dots in Fig. 6. After applying the extra filtering methods, as explained in Sect. 4.1.3, we obtain 81 WSE observations. These observations cover the calibration period, between November 2018 and September 2020, and are illustrated in Fig. 6 by the blue circles. The processing steps provide a good sample of WSE observations, with good spatial coverage of the entire river reach. Moreover, the temporal coverage of the dataset includes low-flow and high-flow season observations, with a slight predominance in the low-flow season, which is related to the quality of ICESat-2 data for the different seasons. The filtering criteria are not biased against high-flow season observations.

The validation data consist of ATL13 observations for the period between 13 December 2020 and 19 October 2021 in which in situ discharge is used to force the hydraulic model in this period, assuming uniform runoff and boundary inflow distributed according to the MERIT flow accumulation map. In the validation period, the RMSE of the model is 44.3 cm, which is in good agreement with the calibration RMSE. The error in the validation data can be seen in Fig. 9. The deviation from the ATL13 WSE minus inferred BL goes up to 1.5 m (Fig. 9a). We observe again that a larger error is present between chainage 650 000 and 680 000 m (Fig. 9b), as previously seen in the calibration data.

The rating curves define the relation between WSE and discharge along the river reach. Rating curves depend on hydraulic characteristics of the channel, so an accurate parameterization is necessary. The WSE–discharge relation generally follows a power law. We calculate rating curves every 50 m from the calibrated hydraulic model. This defines a lookup table relating discharge, WSE, and river chainage. Examples of rating curves in the lookup table are illustrated in Fig. 10, where the flow depth, computed as simulated WSE minus calibrated BL, is compared to the discharge. The computed rating curves follow a power law. We can also see that, for the more upstream point at 491 020 m, the discharge values are lower than the ones further downstream.

Note that, for the relation at chainage 565 990 m, the increase in flow depth is larger than for other examples. This effect is produced due to the narrower river width at this river location that produces an increase in flow depth for the same discharge values. This is a clear example of the river geometry effect in building WSE–discharge relationships.

The results of the validation are presented as the percentage of flow exceedance for a given discharge value and are compared with in situ data (Fig. 11a and b). The estimated discharge values are for selected ATL13 WSE observations. The validation dataset, corresponding to the period from 13 December 2020 to 19 October 2021, is ingested together with the calibration data in the hydraulic model to interpolate the corresponding discharge at the time of acquisition at a given river location (Jimai and Mentang gauging stations). The estimation using the MERIT flow accumulation, uniform boundary inflow, and the prior guess is presented. The percentage of flow exceedance for the estimated discharge is in good agreement with the in situ data for the MERIT flow accumulation case. For Mentang station (Fig. 11b), the observed Q60 is about 400 m3 s-1, while the simulated Q60 is around 250 m3 s-1, which is a considerable difference. For the case using uniform inflow and the prior guess, the difference with in situ data is larger, mostly for the larger values of discharge, which is from 400 m3 s-1 at Jimai and from 500 m3 s-1 at Mentang. In the simulated discharge, we are also calculating low-flow values that are missing in the in situ measurements, which explains this mismatch.

Table 2 presents the RMSE using the three different approaches. The best results with the lower RMSE are from using the MERIT flow accumulation with RMSE 107 m3 s-1 at Jimai and 122.38 m3 s-1 at Mentang. When using the uniform inflow, the difference in RMSE is not significant in Mentang (127.43 m3 s-1), but it increases for Jimai discharge estimations (149.30 m3 s-1). In the case of the prior guess, it is quite clear that the results are worse than for the calibrated models.

There are two different virtual stations (VSs) available from Hydroweb in this area of the Yellow River (Normandin et al., 2018) at kilometer 4914, corresponding to chainage 545 030 m, and at kilometer 4901, corresponding to chainage 560 360 m. We compare the Sentinel-3-derived Hydroweb VS time series at these two points, with the WSE estimated by the model using the MERIT flow accumulation map (Fig. 12). The only available ATL13 observation in this area is represented by the blue star in Fig. 12a, which falls very close to the ATL13-derived WSE time series. It is clear, in Fig. 12a, that the Sentinel-3-derived WSE is not well represented, probably due to the low resolution of this mission and the mountainous terrain, which reduce the accuracy of the retrieved WSE in this area. The seasonality is not present in the Hydroweb VS time series, and some low-flow season values are above the high-flow season values, which should not be the case. The offset with the observed and estimated WSE from ATL13, which is expected to perform better due to its resolution, is more than 2 m and mostly in the low-flow season. A general bias in the data of this VS can be also observed. In Fig. 12b, ATL13-derived values match slightly better with the WSE from the VSs; however, the VS time series present large uncertainties, and the seasonality is poorly represented.

Figure 13 shows examples of WSE (Fig. 13a, c and e) and discharge (Fig. 13b, d)and e) time series for three different river locations at chainage 491 020, 565 990, and 717 320 m. The results are interpolated from ATL13 observations using the WSE–discharge relation at the selected river locations. In situ measurements for WSE to compare with the simulated values are not available. The figure shows the estimation using the MERIT flow accumulation and compares it with the estimation using the uniform inflow and the prior guess. The discharge estimated from the prior and uniform inflow are clearly overestimated compared with the MERIT flow accumulation case. This was also observed in Fig. 11 when we compared it with in situ discharge. The main difference between the prior and calibrated parameters can be seen in Fig. 13e at the more upstream example. The larger depth in the prior guess (2 m) is clearly making an impact in this area, since we expect to have a shallower depth upstream.

The calibrated Manning's roughness together with the calibrated cross sections are used in MIKE HYDRO River to run a full hydrodynamic model simulation. The model runs in a grid with 1 006 558 grid points which consist of pairs with a timestamp for chainage. We define 2 min time steps and 100 m distance. We define a boundary condition at the upstream cross section with input discharge from Jimai in situ data and a downstream boundary at the synthetic cross section relating discharge and stage. We use the dynamic wave solver that corresponds to the full 1D Saint-Venant equations. Figure 14a compares the simulated WSE by MIKE HYDRO and ATL13 WSE minus inferred BL. MIKE simulations seem to be underestimated in most of the cases, and the maximum deviation from the ATL13 WSE minus inferred BL is around 1.7 m. The RMSE is 0.59, which is in good agreement with the results from the steady-state solver.

Figure 14b presents the WSE time series at chainage 491 020 m, as simulated from MIKE and from the steady-state solver. The time series represent the seasonal variation fairly well. The simulated values from the steady-state solver appear to be above the ones from MIKE simulation. This was also observed when we compare the simulated MIKE WSE with the observed ATL13 WSE minus the inferred BL (Fig. 14a), which makes sense if we take into account that the simulated water surface from the steady-state solver is directly derived from ATL13 observations, while the simulated MIKE WSE is derived from discharge observations.