# Joint distribution of upstream runoff governs downstream river-discharge prediction uncertainty in distributed ML models

arXiv: https://arxiv.org/abs/2607.03217 · HTML: https://mlsanity.com/paper/2607.03217/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 25 references that are in the map.
- First posted on arXiv 3 Jul 2026.

## Builds on (most cited first, 4 of 4)

- [Enhancing streamflow forecast and extracting insights using long-short term memory networks with data integration at continental scales](https://mlsanity.com/paper/1912.08949/) (2019)
- [Uncertainty Quantification in Complex Simulation Models Using Ensemble Copula Coupling](https://mlsanity.com/paper/1302.7149/) (2013)
- [Uncertainty Estimation with Deep Learning for Rainfall-Runoff Modelling](https://mlsanity.com/paper/2012.14295/) (2020)
- [Hydra-LSTM: A semi-shared Machine Learning architecture for prediction across Watersheds](https://mlsanity.com/paper/2410.16343/) (2024)

## Built on by (most cited first, 1 of 1)

- [Distributed Hydrological Modeling in the Feature Space](https://mlsanity.com/paper/2609.32971/) (2026)

## Abstract (arXiv)

Uncertainty quantification of hydrological predictions is necessary to inform operational decisions. Recent generative machine-learning methods have advanced probabilistic streamflow prediction, but have remained confined to lumped models that predict a basin outlet directly. At the same time, deterministic LSTM runoff models are increasingly applied at grid or catchment scale and routed through river networks to produce spatially continuous, physically consistent discharge fields. This technical note argues that moving probabilistic prediction from lumped to distributed models introduces a specific new requirement: the joint distribution of upstream runoff generation must be sampled jointly. In lumped inference, the model predicts the outlet distribution directly and can modulate spread from basin attributes. In distributed inference, downstream discharge is obtained by routing many upstream runoff predictions, so independent local sampling averages uncertainty away. Using Japan as a case study, we train two probabilistic basin-scale runoff LSTMs and route their runoff through a Hayami routing scheme. Randomly matching upstream ensemble members produces severely under-dispersed downstream ensembles, whereas a simple quantile matching strategy restores much of the spread of the direct basin-scale reference. The shift from lumped to distributed probabilistic hydrology therefore requires explicit attention to the spatial joint structure of runoff uncertainty.