observed river discharge and EO-based river width, this method can reconstruct discharge time series that are consistent with long-term flow characteristics. The flowchart below outlines the core steps of the NPQM algorithm for estimating discharge from river width: First, the algorithm generates multiple realizations (simulations) of both the in-situ discharge and satellite-derived river width time series using Monte Carlo simulation. These simulations account for the uncertainties in the original measurements.
Then, for each realization, it builds a set of mapping functions by pairing all possible combinations of discharge and width quantile values (see Eq.4a, Eq.4b, and Eq.5, note that in Eq.4b instead of wse, we use river width). Once the mapping functions are created, the algorithm calculates the mean river width- discharge mapping function for each percentile, as well as the associated uncertainty. The non-parametric mapping function is obtained without assuming a specific functional form. To assess the accuracy of the mapping, the model compares the estimated discharge values with observed discharge values.
A "3-sigma test" is used, which checks whether the differences between estimated and observed discharges fall within three standard deviations—capturing about 99.7% of the values in a normally distributed dataset. If simultaneous discharge and width data are available, they are used directly; otherwise, the comparison is made using data that fall within the same quantile range. Based on the outcome of the 3-sigma test, the algorithm adjusts the uncertainty of the measurements to maintain consistency in an iterative process. The algorithm then checks whether the Root Mean Squared Error (RMSE) has changed significantly compared to the previous iteration. If not, the model stops.
If it has, the process adjusts the uncertainty and repeats from the first step. At the beginning of the process, a default uncertainty of 10% of the signal is assumed for both discharge and width time series, due to the limited availability of formal uncertainty estimates. As the iterations proceed, this uncertainty is refined based on the model's performance. Once the model has converged, it can be used to estimate discharge values—with associated uncertainties—based solely on the input river width time series. Further information on the uncertainty treatment and validation of the NPQM method is available in Elmi et al.
(2021, 2024), where the algorithm is applied and analysed in depth using both water level and width datasets. Figure 5: Flowchart of the stochastic quantile mapping function algorithm (adapted from Elmi et at., 2021, and Saemian et al., 2024) 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. 22/31