Text S1 – S6
Text S1 Derivation of the rating curves
Flow in rivers is governed by the Saint-Venant equations. The conservation form of momentumequation is:
\begin{equation} \begin{matrix}\frac{\text{dQ}}{\text{dt}}+\frac{\partial}{\partial x}\left(\frac{\beta Q^{2}}{A}\right)+gA\frac{\text{dH}}{\text{dx}}-gA(S_{0}-S_{f})=0\ \#\left(1\right)\\ \end{matrix}\nonumber \\ \end{equation}
In equation (1), \(Q\) is river discharge (m3/s), t is time (s), \(x\) is river chainage (m), β is momentum coefficient (-),\(A\) is the cross-sectional area (m2), \(H\) is the flow depth (m), \(g\) is the gravitational constant (9.81 m/ s-2), \(S_{0}\) is river bed slope (-), \(S_{f}\) is the friction slope (-).
If the inertia terms (\(\frac{\text{dQ}}{\text{dt}}+\frac{\partial}{\partial x}\left(\frac{\beta Q^{2}}{A}\right)\)) of the momentum equation can be neglected, we obtain the so-called diffusive wave approximation:
\begin{equation} \begin{matrix}\text{gA}\frac{\text{dH}}{\text{dx}}-gA\left(S_{0}-S_{f}\right)=0\#\left(2\right)\\ \end{matrix}\nonumber \\ \end{equation}
The water surface elevation (WSE) of a river section can be measured by gauging stations or satellite altimetry. The changes of WSE along river chainage (\(\frac{\partial wse}{\partial x}\)) is the water surface slope/fall. The relationship between the water surface fall, changes in depth, and riverbed slope is:
\begin{equation} \begin{matrix}S_{0}-\frac{\text{dH}}{\text{dx}}=-\frac{\partial wse}{\partial x}\#\left(3\right)\\ \end{matrix}\nonumber \\ \end{equation}
Manning’s equation is an empirical formula for the friction slope:
\begin{equation} \begin{matrix}Q=\ \frac{1}{n}\bullet A\bullet{(\frac{A}{P})}^{\frac{2}{3}}{{\bullet S}_{f}}^{\frac{1}{2}}\#\left(4\right)\\ \end{matrix}\nonumber \\ \end{equation}
\(n\) is Manning’s roughness coefficient (\(\frac{s}{m^{\frac{1}{3}}}\)), is the wetted perimeter of the flow (m).
Assuming that the river is wide and the cross section is rectangular with constant river width (\(w\)), Manning’s equation can be written as:
\begin{equation} \begin{matrix}Q=\ \frac{1}{n}\bullet H^{\frac{5}{3}}\bullet w{{\bullet S}_{f}}^{\frac{1}{2}}\#\left(5\right)\\ \end{matrix}\nonumber \\ \end{equation}
For the river sections without backwater effects, i.e. in uniform flow conditions, river depth variations along the channel are insignificant, and \(\frac{\text{dH}}{\text{dx}}=0\). From equation (2), we obtain the so-called kinematic wave approximation:
\begin{equation} \begin{matrix}S_{0}-S_{f}=0\ \#\left(6\right)\\ \end{matrix}\nonumber \\ \end{equation}
River flow depth can be expressed as the difference between water surface elevation and bed elevation. From equations (5) and (6), we have:
\begin{equation} \begin{matrix}Q=\ \frac{1}{n}\bullet{(WSE-z_{0})}^{\frac{5}{3}}\bullet w{{\bullet S}_{0}}^{\frac{1}{2}}\#\left(7\right)\\ \end{matrix}\nonumber \\ \end{equation}
Assuming \(n\), \(w\), and \(S_{0}\) are constant values at a cross section. This equation is exact for wide rectangular rivers. For the general case, we assume a similar relationship with fitting parameters\(a\), \(b\), and \(z_{0}\) between discharge and WSE, which is the stage-discharge rating curve:
\begin{equation} \begin{matrix}Q=a\bullet\left(WSE-z_{0}\right)^{b}\ \#\left(8\right)\\ \end{matrix}\nonumber \\ \end{equation}
For river reaches affected by backwater, depth changes with chainage are significant, and \(\frac{\text{dH}}{\text{dx}}\neq 0\). Combing equations (2) and (3), we have:
\begin{equation} \begin{matrix}S_{f}=\frac{\partial H}{\partial x}-S_{0}=\frac{\partial wse}{\partial x}\#\left(9\right)\\ \end{matrix}\nonumber \\ \end{equation}
From equation (5), we have the following relationship:
\begin{equation} \begin{matrix}Q=\ \frac{1}{n}\bullet{(WSE-z_{0})}^{\frac{5}{3}}\bullet w\bullet{(\frac{\partial wse}{\partial x})}^{\frac{1}{2}}\#\left(10\right)\\ \end{matrix}\nonumber \\ \end{equation}
This equation is exact for wide rectangular rivers. For the general case, we assume a similar relationship with fitting parameters \(c\),\(d\), \(z_{s}\), and \(d\) between stage, fall, and discharge, which is the stage-fall-discharge rating curve:
\begin{equation} \begin{matrix}Q=c\bullet\left(\frac{\partial wse}{\partial x}\right)^{d}\bullet\left(WSE-z_{s}\right)^{e}\ \#\left(11\right)\\ \end{matrix}\nonumber \\ \end{equation}
Text S2. Hydrodynamic model description
MIKE Hydro River (MIKE hereinafter), a one-dimensional (1D) hydrodynamic model based on shallow-water equations (Vreugdenhil, 1994), is used to simulate stage and fall along the chainage. The required inputs for MIKE are river reaches, cross sections, Manning-Strickler coefficient (Ks), and boundary conditions of discharge. The following texts describe cross section delineation (Text S3), model parameterization (Text S4), and estimation of boundary discharge (Text S5). A summary of model performance is presented in Text S6.
Text S3. Cross section delineation
ICESat-2 ALT03 provides detailed measurements of land surface height with specific passing dates. The river bank and water surface can be monitored from space, which forms the exposed cross section shape (Ma et al., 2020; Neumann et al., 2019; Xu et al., 2021). ICESat-2 measures a larger portion of the river cross section during low-flow seasons, and the submerged portion of the cross section is smaller. We processed ICESat-2 ALT03 products in low-flow seasons, and the processed data are used to delineate exposed parts of the cross sections. ALT03 products are the laser points with noise. Hampel Filter is used for data smoothing (Pearson et al., 2016).
The submerged part of the cross section is assumed to follow the power-law hydraulic geometry relationship (Lawrence, 2007). The power-law relationship needs two parameters, i.e., a depth (distance between the ICESat-2 measured water surface elevation and river bed) and a shape parameter (beta), which can be calibrated (Vatankhah, 2020) by hydraulic inversion. The determination of submerged depth and beta are introduced in the following section. Fig. S3 shows one of the cross sections in Amur River with the exposed part monitored by ICESat-2 ALT03 and the submerged part delineated by the power-law relationship with two candidate parameters.
Text S4. Parameterization
As described above, each cross section should determine a submerged depth and a shape parameter beta. Considering the limited WSE measurements, we assume that the submerged depth is uniform both upstream and downstream of the river confluences and that beta is constant along the entire river channel. Thus, we calibrate two effective submerged depths in each river reach, one for the reach upstream river confluence and one for the reach downstream river confluence, and a shape parameter beta. Additionally, the roughness parameter, i.e., Ks, needs to be determined for the hydrodynamic model. These parameters are determined by hydraulic inversion, which uses optimization methods to search a set of optimum parameters matching, in a least-squares sense, observed and simulated WSE (Frias et al., 2022).
Satellite altimetry measured WSE is only available at satellite passing dates. The outputs of MIKE models are time-continuous, but only the simulations with the same dates of satellite altimetry are used for calibration. To run the model with time-continuous simulations over simulation periods of several years is computationally demanding in an inverse parameter estimation workflow. Therefore, we simplify the calibration problem by assuming that the flow is in a steady state on satellite passing dates, and the model is run only on the dates with satellite altimetry. The simplification significantly increases computational efficiency, and the calibrated parameters can be transferred back to MIKE for unsteady simulations (Kittel et al., 2021; Liu et al., 2021). The objective function for the inversion problem is:
\begin{equation} \begin{matrix}\varphi=\sqrt{\frac{1}{N}\sum_{i=1}^{N}\left(\text{WSE}_{sim,\ \ i}-\text{WSE}_{obs,\ i}\right)^{2}}\#\left(13\right)\\ \end{matrix}\nonumber \\ \end{equation}
In the equation, \(\text{WSE}_{\text{sim}}\) is the simulated WSE where ICESat-2 observations exist (at all chainage and time),\(\text{WSE}_{\text{obs}}\) is the ICESat-2 ALT13, N is the total number of ICESat-2 ALT13 WSE observations. A global optimization package (i.e., the Shuffled Complex Evolution Algorithm) in Statistical Parameter Optimization Tool for Python (SPOTPY) is used for parameter calibration in the present study to avoid interference from improper initial parameter settings and local optima (Houska et al., 2015).
Text S5. Estimation of boundary discharge
River discharge is a necessary boundary condition for hydrodynamic models. However, obtaining high spatio-temporal resolution discharge estimates covering the operating period of ICESat-2 for the studied river reaches (for both mainstream and major tributaries) is challenging. We only found in-situ discharge for one of our cases (the Missouri River and its tributary, the Yellowstone River) from the United States Geological Survey. Therefore, the Global Flood Awareness System (GloFAS) reanalysis product is used as reference discharge for the other river sections, providing daily discharge estimates for global rivers from 1979 to the present (Harrigan et al., 2020). The time series of the daily discharge for mainstream and tributary are shown in Fig.S2.
Text. S6 Performance of hydrodynamic models
Hydrodynamic models are calibrated against the WSE from ICESat-2 ALT13 and ALT08 depending on the width of the river and validated against the WSE measurements from Sentinel-3A/B or Jason-3. The range of model misfit (in terms of RMSE) at the six case study sites is [0.62 m, 1.36 m]. The results are displayed with scatter plots in Fig.2. The calibrated parameters, i.e., upstream low-flow depth, downstream low-flow depth, beta, and Strickler coefficient, inverted by the optimization algorithm are effective values, which compensate for the irregularly varying effects of width, bed elevation, and vegetation in space and time.
The validation results indicated by the root mean square error (RMSE) of the six VS ranges from 0.83 m to 3.14 m, as shown with the curve plots in Fig 2. The simulations mismatch the satellite altimetry at VS-3 (Amazon-Negro River) and VS-6 (Niger-Benue River), with RMSE of 3.14 m and 3.02 m, respectively. For the Niger River section, we found two large dams upstream of the studied river segments (The Kainji Dam and the Jeba Dam, Lehner et al., 2011), and it appears from the GLOFAS hydrographs that those are not modeled in the GloFAS system. Because the measurements of WSE by Sentinel-3B show that the water surface is relatively stable in the low-flow seasons, we hypothesize that the natural river flow has been altered and that the alteration is not reproduced in the GloFAS discharge product. For the Amazon-Negro River, the GloFAS discharge is significantly higher than the in-situ observations in other years (Fig. S3). Thus we have a calibrated upstream depth of 37.88 m to compensate for the high discharge. Similar situations can also be found in the Amur- Zeya River (Zeya Reservoir locates upstream of Zeya), and the Missouri-Yellowstone River (Fort Peck Lake locates upstream of the Missouri River). Inaccurate discharge of the Zeya River may influence the fall of the mainstream. For the Missouri-Yellowstone River, we replaced the GloFAS discharge with in-situ observations for both the mainstream and the tributary.