Let ℒ denote the set of pixels in the buffered river region and assign a binary label 𝑓 𝑝 ∈{0,1} to each pixel 𝑝 ∈ℒ , where 𝑓 𝑝 = 1 represents water and 𝑓 𝑝 = 0 represents non-water. The goal is to find a labeling that minimizes the energy: Eq.9 where: - 𝐷 𝑝 (𝑓 𝑝 ) is the data term (unry potential), encoding the likelihood that pixel p belongs to class 𝑓 𝑝 . - 𝑉 {𝑝,𝑞} (𝑓 𝑝 , 𝑓 𝑞 ) is the smoothness term (pairwise potential), enforcing spatial coherence. - λ > 0 is the regularization term.
- ℕ is the number of 4-connected pixel neighbors around the border The term 𝐷 𝑝 (𝑓 𝑝 ) is the cost function defined in the temporal domain, which can be obtained from the monthly or long-term water occurrence map from GSW that delivers the probability of water in a certain pixel 𝑃 𝑝 . So: Eq. 10 On the other hand, the smoothness term 𝑉 𝑝,𝑞 (𝑓 𝑝 , 𝑓 𝑞 ) penalizes discontinuities between neighboring pixels based on water index similarity: 𝑉 𝑝,𝑞 (𝑓 𝑝 , 𝑓 𝑞 ) = 0𝑖𝑓𝑓 𝑝 = 𝑓 𝑞 , exp(−|𝐼 𝑝 𝑞 |/𝜎) 𝑖𝑓𝑓 𝑝 𝑞 Eq. 11 To solve the energy minimization problem defined by the binary labeling function 𝑓 𝑝 , we employ the Max- Flow/Min-Cut algorithm.
The formulation is cast as a graph cut problem, where each pixel corresponds to a node in a directed graph, and two additional nodes represent the source (water) and sink (non-water). The edge weights are derived from the data and smoothness terms: terminal links (T-links) encode the data cost 𝐷 𝑝 (𝑓 𝑝 ) , while neighborhood links (N-links) encode the pairwise smoothness cost 𝑉 𝑝,𝑞 (𝑓 𝑝 , 𝑓 𝑞 ) . The Max-Flow algorithm (implemented using the PyMaxflow library) computes the minimum s–t cut that separates the graph into water and non-water regions, yielding the optimal binary segmentation f* that minimizes the total energy.
This approach guarantees a globally optimal solution for binary segmentation under submodular energy terms. 5.2 River discharge estimation using river width A complementary approach to traditional discharge estimation using hydraulic or hydrological models is the use of satellite-derived river width time series as a proxy for in-situ discharge observations. We apply a non-parametric, data-driven technique called Non-Parametric Quantile Mapping (NPQM) (Elmi et al. 2021) to estimate river discharge from width time series extracted from optical satellite imagery (section 5.1).
The approach allows for discharge estimation at river reaches near selected gauges, even in the absence of simultaneous observations. By building a statistical relationship between the distribution of 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. 21/31