4.2.4 Method 2 – Bayesian estimation using quantile-based matching – alternative approach The quantile-based approach is a statistical modelling technique used to derive a rating curve by matching quantile functions of water surface elevation (WSE) and discharge time series. Unlike the Bayesian approach, it does not require a temporal overlap between altimetry observations and in situ discharge measurements. This makes it especially valuable when satellite data and discharge records originate from distinct periods.
The method has been successfully applied across a range of river basins in various climatic zones—including the Amazon, Brahmaputra, Danube, Niger, and Ob rivers (Tourian et al., 2013). 4.2.4.1 Theoretical Framework This method assumes that river flow behavior is stationary over time and that the bathymetry remains stable both at the virtual altimetry station and the in-situ gauging station. Under these conditions, historical in situ discharge data can be validly used to estimate present-day discharges derived from satellite altimetry. The foundation of the approach lies in the construction of quantile functions: Eq.4a Eq.4b where:
Q R (p), is the quantile function of in situ discharge values X R Q W (p), is the quantile function of altimetric WSE values X W F(⋅) denotes the cumulative distribution function (CDF), and p∈(0,1) is the exceedance probability. The quantile function Q(p) returns the maximum value of the variable that is not exceeded with probability p . It provides a distribution-free description of the data, enabling the definition of a functional transformation T(⋅) between the two datasets: Eq.5 This transformation is assumed to be monotonic and non-decreasing (Gilchrist, 2000), a key requirement that allows for a meaningful functional relationship between WSE and discharge.
Since this formulation does not rely on time-matching between datasets, it is ideal for scenarios without temporal overlap. 4.2.4.2 Practical Implementation To empirically estimate the quantile functions, each dataset (WSE and Q) is sorted in ascending order. The rank k i of each value is normalized as: 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. 18/31