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A Deep State Space Model for Rainfall-Runoff Simulations

Yihan Wang, Lujun Zhang, Annan Yu, N. Benjamin Erichson, Tiantian Yang

TL;DR

This paper introduces S4D-FT, a frequency-tuned diagonal state-space model, as a novel deep learning approach for rainfall–runoff simulations and benchmarks it against LSTM and Sac-SMA across 531 CAMELS watersheds in the CONUS. Using 32 input variables and a 365-day look-back, the authors demonstrate that S4D-FT achieves higher median NSE and KGE and reduced ensemble variability than LSTM, with regional gains evident in the Pacific Southwest and Mid-South but deficits along the East Coast and in snow-influenced regions for certain metrics. The work includes attribution analyses linking performance to hydrologic signatures, discusses the need for standardized evaluation, and argues for integrating physics-aware and probabilistic DL methods to improve interpretability and uncertainty estimation. Overall, S4D-FT expands hydrologic DL toolsets and challenges LSTM’s dominance, providing a new benchmark and guidance for where SSM-based approaches offer the most benefit in rainfall–runoff forecasting.

Abstract

The classical way of studying the rainfall-runoff processes in the water cycle relies on conceptual or physically-based hydrologic models. Deep learning (DL) has recently emerged as an alternative and blossomed in hydrology community for rainfall-runoff simulations. However, the decades-old Long Short-Term Memory (LSTM) network remains the benchmark for this task, outperforming newer architectures like Transformers. In this work, we propose a State Space Model (SSM), specifically the Frequency Tuned Diagonal State Space Sequence (S4D-FT) model, for rainfall-runoff simulations. The proposed S4D-FT is benchmarked against the established LSTM and a physically-based Sacramento Soil Moisture Accounting model across 531 watersheds in the contiguous United States (CONUS). Results show that S4D-FT is able to outperform the LSTM model across diverse regions. Our pioneering introduction of the S4D-FT for rainfall-runoff simulations challenges the dominance of LSTM in the hydrology community and expands the arsenal of DL tools available for hydrological modeling.

A Deep State Space Model for Rainfall-Runoff Simulations

TL;DR

This paper introduces S4D-FT, a frequency-tuned diagonal state-space model, as a novel deep learning approach for rainfall–runoff simulations and benchmarks it against LSTM and Sac-SMA across 531 CAMELS watersheds in the CONUS. Using 32 input variables and a 365-day look-back, the authors demonstrate that S4D-FT achieves higher median NSE and KGE and reduced ensemble variability than LSTM, with regional gains evident in the Pacific Southwest and Mid-South but deficits along the East Coast and in snow-influenced regions for certain metrics. The work includes attribution analyses linking performance to hydrologic signatures, discusses the need for standardized evaluation, and argues for integrating physics-aware and probabilistic DL methods to improve interpretability and uncertainty estimation. Overall, S4D-FT expands hydrologic DL toolsets and challenges LSTM’s dominance, providing a new benchmark and guidance for where SSM-based approaches offer the most benefit in rainfall–runoff forecasting.

Abstract

The classical way of studying the rainfall-runoff processes in the water cycle relies on conceptual or physically-based hydrologic models. Deep learning (DL) has recently emerged as an alternative and blossomed in hydrology community for rainfall-runoff simulations. However, the decades-old Long Short-Term Memory (LSTM) network remains the benchmark for this task, outperforming newer architectures like Transformers. In this work, we propose a State Space Model (SSM), specifically the Frequency Tuned Diagonal State Space Sequence (S4D-FT) model, for rainfall-runoff simulations. The proposed S4D-FT is benchmarked against the established LSTM and a physically-based Sacramento Soil Moisture Accounting model across 531 watersheds in the contiguous United States (CONUS). Results show that S4D-FT is able to outperform the LSTM model across diverse regions. Our pioneering introduction of the S4D-FT for rainfall-runoff simulations challenges the dominance of LSTM in the hydrology community and expands the arsenal of DL tools available for hydrological modeling.
Paper Structure (16 sections, 2 equations, 2 figures, 7 tables)

This paper contains 16 sections, 2 equations, 2 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Datasets and Methodology
  3. State Space Models for rainfall-runoff simulations
  4. Attribution analyses of S4D-FT's relative performance over LSTM
  5. Results
  6. Overall Performance of SSMs and Existing Benchmarks
  7. Regional Performance Comparison of S4D-FT and LSTM
  8. Investigating factors behind regional variability in S4D-FT performance
  9. Discussion
  10. Conclusions
  11. Text S1: Evaluation strategy
  12. Input variables for SSMs and LSTM training
  13. Model Structure and Training Hyperparameters
  14. Evaluation Metrics
  15. Description of hydrologic signatures
  16. ...and 1 more sections

Figures (2)

  • Figure 1: Simulation performance of S4D-FT relative to the LSTM model across study watersheds in CONUS. Panel a: Spatial distribution of NSE skill scores, with red dots (positive NSE skill score) indicating S4D-FT outperformance and blue dots (negative NSE skill score) indicating LSTM outperformance. Panel b: Spatial distribution of KGE skill scores, following the same color scheme as Panel a. Panel c: Boxplots of NSE and KGE skill scores by state, ordered east to west, with red boxes for positive median scores and blue for negative medians. Rhode Island and Minnesota are excluded due to no study watersheds.
  • Figure 2: Analysis of S4D-FT’s performance relative to LSTM considering multiple evaluation statistics and hydrological signatures. Panel a: Scatter plots overlaid with contour density plots of NSE and KGE skill scores against improvements in FHV, Pearson correlation, and PBias for Group 1 (red) and Group 2 (blue) watersheds. Solid red and blue lines represent regression lines. Correlation coefficients are displayed at the bottom left of each plot. The red and blue triangles highlight two specific example watersheds (i.e., 09430600 and 14400000) from Group 1 and Group 2, respectively. Panel b: Simulated 8-member ensemble hydrographs for LSTM (red) and S4D-FT (blue), along with observed streamflow (black stars), for the highlighted watersheds (i.e., 09430600 and 14400000). Panel c: heatmaps of percentage differences in the eight selected hydrologic signatures between Group 1 and Group 2 watersheds (only$> \pm 10\%$ differences shown) with values labeled in each cell. Panel d: heatmaps of correlation coefficients between the selected hydrologic signatures and NSE/KGE skill scores for Group 1 and Group 2 watersheds, with values labeled in each cell.