Fully distributed hydrological models take into account the spatial variability of a catchment, and allow for assessing its hydrological response at virtually any location. However, these models can be time-consuming when it comes to model runtime and calibration, especially for large-scale catchments. Meanwhile, deep learning models have shown great potential in the field of hydrological modelling, but a multivariable, fully distributed hydrological deep learning model is still lacking. To address the aforementioned challenges associated with fully distributed models and deep learning models, we explore the possibility of developing a fully distributed multivariable deep learning model by using Graph Neural Networks (GNN), an extension of deep learning methods to non-Euclidean topologies. We develop a surrogate model of wflow_sbm, a fully distributed, physics-based hydrological model, by exploiting the similarities between its underlying functioning and GNNs. The GNN model uses the same input as wflow_sbm: gridded static parameters based on physical characteristics of the catchment and gridded dynamic meteorological forcings. The GNN model is trained to approximate wflow_sbm outputs, consisting of multiple gridded hydrological variables such as streamflow, actual evapotranspiration, subsurface flow, saturated and unsaturated groundwater storage, snow storage, and runoff. Our results show that the GNN model accurately predicts multiple hydrological variables in unseen catchments (median KGE=0.76), and can serve as an emulator of wflow_sbm with a shorter runtime. We furthermore demonstrate how the GNN model can function up to a prediction horizon of a full year, using physical system states to account for system memory, as well as a curriculum learning strategy combined with a multi-step ahead loss function during training. Overall, this study contributes to the field of fully distributed modelling using a deep learning approach.

Modeling the relationship between rainfall and runoff is a longstanding challenge in hydrology and is crucial for informed water management decisions. Recently, Deep Learning models, particularly Long short-term memory (LSTM), have shown promising results in simulating this relationship. The Transformer, a newly proposed deep learning architecture, has also demonstrated the ability to outperform LSTM in machine translation, text classification, etc. However, there has been limited research on applying Transformers for rainfall-runoff modeling.

The research examined the performance of using Transformer architecture, including its time series forecasting variants, to develop rainfall-runoff models using the CAMELS (US) data set. These models were compared to the LSTM regional rainfall-runoff models, with a particular focus on snow-driven basins as the attention mechanism in Transformer is believed to allow it to attend to the earlier precipitation events in the meteorological forcing. Additionally, the Transformer's potential as a global rainfall-runoff model was also tested using the global Caravan data to determine if it could learn and generalize a wide range of rainfall-runoff behaviors, allowing it to potentially be applied in ungauged basins.

The results suggest that while Transformer and its variants may not be able to fully replace LSTM for rainfall-runoff modeling, the variant called Reformer has shown promise for daily discharge forecasting in snow-driven basins, particularly in terms of peak flow and low flow prediction. However, using the global Caravan data for building a global rainfall-runoff model was not successful due to uncertainty in the forcing data, particularly precipitation. The code for Transformer-based rainfall-runoff modeling is available publicly at https://github.com/Numpy-Panda/neuralhydrology_Transformer.