Eq.6 N is the number of measurements, and p i is the empirical probability associated with each sorted value. This produces two ordered series of quantiles (WSE and Q), which are then paired to create a scatter plot of discharge quantiles versus WSE quantiles. The quantile functions can be constructed: using mean monthly discharge and WSE (based on historical daily discharge and satellite observations at altimeter passes), as in Tourian et al. (2013), or by matching altimetric WSE values at satellite overpasses with daily discharge values in the historical dataset, to better reflect short-term flow variability, including extreme events.
This second variant has been explored in this study to overcome the limitations of monthly mean values, which may obscure significant hydrological fluctuations. By doing so, the derived rating curve incorporates more of the inherent non-stationarity of river flow, offering a better match between satellite-derived WSE and hydrological dynamics. 4.2.4.3 Application and Comparison Both the monthly-based and daily-based quantile pairing approaches are tested in this work to build rating curves during periods without temporal overlap between WSE and Q datasets. These curves are evaluated in Work Package 4.2. In both cases, the quantile space is discretized using an equidistant step of 5%.
The resulting quantile– quantile scatter plots are used as input for a Bayesian inversion, following the same procedure described in Section 4.2.1, to derive the final rating curve parameters. This method is applied both to merged multi-mission altimetry WSE time series and to each individual altimetry mission, to evaluate the contribution and uncertainty associated with each data source. The mission-specific analysis may enhance discharge estimation accuracy for missions with better precision or denser temporal coverage. 4.3 Uncertainties Uncertainty propagation through mathematical models plays a crucial role in estimating the reliability of derived results in various scientific fields.
In the context of hydrology and discharge estimations, the propagation of uncertainties in parameter estimation, such as those in the parameters of the discharge equation, becomes essential for assessing the reliability of the calculated discharge values. Utilizing a Gaussian error propagation method provides a systematic approach to quantify the uncertainties associated with parameters a, WSE, b, and z0 (and s if available) from the power law function defined in section 4.2.1 to express the relation between Q and WSE. This method involves employing statistical principles to propagate uncertainties through the mathematical relationships between the parameters and the discharge equation.
By considering the Gaussian distribution of errors in these parameters, this approach enables a more comprehensive evaluation of the overall uncertainty in discharge estimations (e.g., McMahon and Peel, 2019, Tourian et al., 2017). D.3 River Discharge ATBD. Reference: RD_CCI-0006-ATBD - Issue 2.0 – 31/07/2025 Open/Public © 2019 CLS. All rights reserved. Proprietary and Confidential. 19/31