Deep Learning-Based Probabilistic Hierarchical Reconciliation for Hydrological and Water Resources Forecasting

Abstract

Accurate, probabilistic, and consistent forecasts at different timescales (e.g., daily, weekly, monthly) are important for effective water resources management. Considering the different timescales together as a hierarchical structure, there is no guarantee that when forecast models are developed independently for each timescale in the hierarchy, they will result in consistent forecasts. For example, there is no guarantee that one-seven day(s) ahead forecasts from one model will sum to a weekly forecast from another model. Significant efforts have been made in the time-series forecasting community over the last two decades to solve this problem, resulting in the development of temporal hierarchical reconciliation (THR) methods. Until recently, THR methods had yet to be explored for hydrological and water resources forecasting. The main goal of this research is to introduce THR to the field of hydrological and water resources forecasting and to merge it with the latest advancements in deep learning (DL) to provide researchers and practitioners with a state-of-the-art model that can be used to produce accurate, probabilistic, and consistent multi-timescale forecasts. To achieve this goal, this research follows three interconnected objectives, each including a main contribution to the field of DL-based hydrological forecasting. In the first main contribution of this research, the potential of THR to produce accurate and consistent hydrological forecasts was verified for the first time in hydrology through a large-scale precipitation forecasting experiment using 84 catchments across Canada. Three THR methods were coupled with three popular time-series forecasting models (exponential time-series smoothing, artificial neural network, and seasonal auto-regressive integrated moving average) for annual precipitation forecasting at monthly (12-steps ahead), bi-monthly (6-steps ahead), quarterly (4-steps ahead), 4-monthly (3-steps ahead), semi-annual (2-steps ahead), and annual (1-step ahead) timescales. It was confirmed that not only does utilizing THR guarantee forecast consistency across all timescales, but it can also improve forecast accuracy. DL models are increasingly being used for hydrological modeling, particularly for lumped simulation, due to their ability to capture complex non-linear relationships within hydrological data as well as their efficiency in deployment. Likewise, the application of DL for hydrological forecasting has gained momentum recently. DL models can extract complex patterns from meteorological forcing data (e.g., precipitation) to forecast future streamflow, often leading to forecasts that are more accurate than current conceptual models. However, due to uncertainty in the phenomena affecting hydrological processes, it is necessary to develop accurate probabilistic forecast models to provide insights for informed water management decisions. In the second main contribution of this research, two novel state-of-art sequence-to-sequence probabilistic DL (PDL) models were developed, tested, and applied for short-term (one-seven day(s) ahead) streamflow forecasting in over 75 catchments with varied hydrometeorological properties across both the continental United States (CONUS) and Canada. The two designed models, namely quantile-based encoder-decoder and conditional variational auto-encoder (CVAE) showed superior performance compared to the benchmark long-short-term memory (LSTM) network considering forecast accuracy and reliability. Specifically, CVAE, a generative DL model that can estimate magnitudes of different sources of uncertainty (e.g., aleatoric, epistemic), proved to be effective in producing reliable forecasts for longer forecast lead times (three-seven days ahead). Given the introduction of THR to the field of hydrological forecasting through the first main contribution, there is no guidance on how to couple THR with the latest advancements in DL, especially PDL, to produce accurate, and consistent probabilistic hydrological forecasts. Furthermore, existing methods for combining THR with DL models, particularly PDL models, suffer from several limitations. Firstly, almost all approaches treat THR as a post-processing step. Secondly, existing THR methods often lack adaptability, meaning they are unable to adjust properly to changing data distributions or new information. Finally, there is limited research on implementing probabilistic THR, a crucial aspect for making probabilistic forecasts consistent. As the third main contribution, a hierarchical DL model (HDL) was introduced where THR was integrated directly into the DL model. Specifically, a custom THR layer was developed that can be combined with any DL model, much like a LSTM layer or a linear layer, to produce the proposed HDL. This integrated approach (via the new THR layer) allows any DL model to leverage temporal information across multiple timescales during training, perform probabilistic THR, and be efficient for real-time application. Furthermore, the proposed HDL is based on auto-regressive normalizing flows, a state-of-the-art generative DL model that is more flexible than CVAE in that it can non-parametrically estimate the probability distribution of the target variable (e.g., streamflow). HDL was tested on more than 400 catchments across CONUS for weekly streamflow forecasting at daily (seven-steps ahead) and weekly (one-step ahead) timescales. The performance of HDL was benchmarked against LSTM variants. HDL produced forecasts that had substantially higher accuracy than the LSTM variants and simultaneously generated consistent forecasts at both daily and weekly timescales, without the need for post-processing (as in the vast majority of THR methods). The implementation of THR as a neural network layer allows it to be seamlessly combined with other DL layers. For example, the new THR layer can be coupled with physics-based differentiable routing layers for multi-timescale distributed hydrological forecasting. It is expected that HDL will serve as a benchmark upon which future THR methods will be compared for streamflow forecasting. Furthermore, given the generality of the approach, HDL can be used for forecasting other important variables within hydrology (e.g., soil moisture) and water resources (e.g., urban water demand), as well as other disciplines, such as renewable energy (e.g., solar power).