Research papersModeling spatial distribution of flow depth in fluvial systems using a hybrid two-dimensional hydraulic-multigene genetic programming approach
Introduction
Estimating flow depth of fluvial systems is very important for river engineering applications, such as river rehabilitation, and most importantly, flood risk mitigation (Hosseiny et al. 2020). Flooding is one of the most common and widespread weather-related natural disasters, usually causing serious consequences, such as human casualties and building collapses (Li et al., 2020, Ming et al., 2020). Due to global climate change, more extreme climatic events and resultant flood events are expected or already experienced, increasing the threats from floods. For example, what people thought were “100-year floods” inundated some communities in Ottawa twice within a three-year period (2017 and 2019, respectively). Inundation forecasting can provide necessary information on flood hazards to the decision-makers and stakeholders, and is thus the crucial component of flood disaster control planning (Burn and Whitfield, 2016). Inundation forecasting primarily focuses on two variables: inundation region and inundation depth (Jamali et al., 2018). These two variables can both be derived from the spatial distribution of flow depth. For example, a location can be regarded as inundated if the water depth exceeds a threshold, such as 0.3 m (Ming et al. 2020). Therefore, estimating flow depth is the core task of inundation analyses. Furthermore, flow depth is one of the most important variables that characterize the hydrological and hydrodynamic properties of a river, partially because water surface elevations can be easily calculated from river depth by adding up the river bed elevations, and the averaged velocity at a cross section can be roughly estimated by dividing the discharge by the flow area, which is a function of the river depth.
Flow depth in fluvial systems has been a topic of significant research. Generally, most of the existing models for flooding processes can be classified into two categories: hydrological or hydraulic. Hydrological models are currently the most popular choice for inundation forecasting, especially when a timely prediction is required and the study area is a large-scale catchment (Bermúdez et al., 2018). Traditional hydrological models mainly calculate the time series of discharges in fluvial systems, but recent distributed hydrological models based on digital elevation models (DEM) can also delineate the inundation extends and depths (Zhang et al., 2014, Li et al., 2020, Ming et al., 2020), making them widely used nowadays for flood forecasting. However, hydrological models typically ignore the strict conservation of momentum, and can only provide a simplified representation of the hydrological responses. Thus, the predictions obtained by hydrological models cannot meet the desired requirements of prediction accuracies in many cases (Ming et al., 2020). For example, a calibrated hydrological model often performs poorly for certain flow conditions (such as extreme flows) because most of hydrological models are largely dependent on calibrations instead of strict physical principles (Unduche et al., 2018). The hydrological models based on the water level methods, such as the flood-connected domain algorithm, cannot be employed in ungauged areas (Li et al., 2020). Those hydrological models based on the Bulk method typically perform poorly in catchments with steep elevation differences (Li et al., 2020). Therefore, physics-based hydraulic models, which can overcome most of the disadvantages of hydrological models, are often required to provide predictions with higher accuracies (Ming et al., 2020).
Due to the recent advances in computing techniques and resources, modeling river channels and floodplains using fully physics-based hydraulic models has become feasible (Teng et al., 2017, Ming et al., 2020). For example, the parallel computations and graphics processing units (GPUs) can significantly improve the computing speeds (Smith and Liang, 2013, Ming et al., 2020). 1D models may not be capable of sufficiently predicting detailed spatial distribution of hydraulic variables because they solve highly simplified forms of the equations. In recent years, 2D and 3D models have been widely reported and validated (e.g., Yan et al., 2019b, Yan et al., 2020a, Yan et al., 2019a, Yan et al., 2020b, Yan et al., 2020c). Examples of some popular 2D or 3D hydraulic modeling system include MIKE DHI, Delft3D, OpenFOAM, and TELEMAC-MASCARET. Solving the 3D partial differential equations for mass continuity and momentum conservation for a large area is very computationally expensive, and is thus still not very practical for some river engineering applications, especially those needing real-time forecasts. The horizontal inundation length scale in fluvial systems typically are much greater than the water depth, and thus the flow in fluvial systems can be reasonably resolved using 2D shallow water equations. Recently, Ming et al. (2020) has executed the High-Performance Integrated Hydrodynamic Modelling System (HiPIMS) on a server fitted with 8 × NVIDIA Tesla K8- GPUSs to simulate a catchment in the United Kingdom with an area of 2500 km2 and grid resolution of 10 m. The run time of the simulation was completed within 2 h, indicating that the efficiency has been significantly enhanced compared to traditional hydraulic simulations (Ming et al., 2020). Compared to most hydrological models, which can rapidly complete calculations on personal computers, the computing machines and calculations for 2D hydraulic simulations are still more expensive and time-consuming, hindering its wider usage in practical applications. Previous studies (e.g., Bruwier et al., 2018; Mustafa et al., 2020) have investigated the influence of urban characteristics on inundation extents and water depths by integrating hydraulic models and urban procedural models, and the inundation properties can be efficiently predicted by using the developed relationships. However, the integrated model is specifically designed for modeling urban flooding events and is thus not very suitable for modeling flow depths in rivers. Therefore, proposing a new approach for modeling spatial distribution of flow depth in fluvial systems that is highly efficient and accurate is still a significant challenge and requires further investigation.
Supplementing physically-based hydraulic models, machine learning or artificial intelligence techniques has been successfully applied in water-related problems (e.g., Mehr and Nourani, 2017, Safari and Danandeh Mehr, 2018, Jamei and Ahmadianfar, 2020, Jamei et al., 2020a, Jamei et al., 2020b, Kabir et al., 2020, Mehr and Safari, 2020, Safari, 2020, Yan and Mohammadian, 2019, Yang et al., 2020d, Yan and Mohammadian, 2020). These techniques can relate input and output variables without requiring a modeler to pre-define the model structures, and thus can eliminate the imperfections of model assumptions and can figure out some deeply hidden relationships. The mathematical models obtained by these techniques are typically very efficient and are thus quite suitable for real-time forecasts and comparative studies. The models developed by machine learning techniques are typically easy to use, and thus can reduce the requirements of expertise of the users. Although the models obtained by these techniques may only be valid within a certain data range depending on the training dataset, similar to most data-driven techniques, these models can be continuously improved with data availability. A common limitation of machine learning techniques is the extensive data requirements, which often gives rise to the insufficient training problem (Yang et al., 2020d). Typically, flowrates or water depths are only sparsely recorded at a few gauges in a catchment, and monitoring detailed spatial distribution of flow depth in fluvial systems is not a common practice. Therefore, it is nearly impossible to employ a traditional machine learning technique to study the distributed flow depths due to the insufficiency of training data.
To overcome the insufficient training problems, recent studies have proposed physical process and machine learning combined models for different problems (Ding et al., 2019, Guo et al., 2021, Hosseiny et al., 2020; Yang et al., 2020d, Zahura et al., 2020). In these studies, a physically-based model was first calibrated and validated, and then the model was assumed to reflect the real physical process and used to provide sufficient data, which was in turn adopted by a machine learning technique to train models. For example, Ding et al. (2019) combined a computational fluid dynamics model and a feed-forward artificial neural network model to develop a multiscale data-driven model for atomic layer deposition (ALD) of SiO2. The study showed that the model can efficiently describe the dynamic relationships for input and output variables for the SiO2 thermal ALD process, and can significantly reduce the computational costs. Yang et al. (2020d) have combined a physically-based distributed hydrological model (the Geomorphology-based Hydrological Model) with machine learning techniques to provide a model for daily streamflow simulations. The combined model was then applied so as to predict daily streamflow in the upper Chao Phraya Basin in Thailand, and the results demonstrated that the combined approach can significantly improve the predictions in data-limited watersheds, compared to the traditional machine learning techniques trained merely by the observational data. Zahura et al. (2020) performed urban flooding simulations in the coastal city of Norfolk, USA, and trained a surrogate model using the Random Forest (RF) machine learning technique. The results demonstrated that using the RF approach to develop surrogate models is promising. Hosseiny et al. (2020) integrated a physically-based model, the International River Interface cooperative (iRIC), with the RF classification and the multilayer perceptron (MLP) machine learning methods for modeling the flow depth in a segment of the Green River in the United States. The results showed that the RF and MLP techniques can satisfactorily replicate the predictions obtained by the iRIC model. The study of Hosseiny et al. (2020) has provided a starting point for estimating flow depth using an integrated numerical and machine learning model. However, the approach requires further improvements in two aspects: first, the machine learning techniques adopted by Hosseiny et al. (2020) were “black-box”; namely, the techniques cannot provide a compact explicit arithmetic equation describing the relationships between the input and output variables, and thus the model cannot be readily transferrable (i.e., it is difficult to incorporate the model into another program); and second, the study trained a single machine learning-based model for an entire large domain, and thus the training process would be time consuming and the trained model would be extremely complicated.
The primary objective of this study is to propose and evaluate a new approach of estimating flow depth in fluvial systems using a hybrid 2D hydraulic-multigene genetic programming (MGGP) approach. The study was motivated by the fact that there is a lack of flow depth modeling techniques that are satisfactorily accurate and fast to run. The present study can provide a new promising tool for flow depth modeling and address the scientific question about whether a hybrid hydraulic-MGGP approach is capable of efficiently estimating flow depth in fluvial systems. In this study, the TELEMAC-2D system was used for hydraulic modeling, and the MGGP technique was used for machine learning. A key merit of the MGGP approach is that it can provide compact explicit mathematical models, and thus the evolved models can be easily transferable and used in different programs (e.g., MATLAB, Excel, or Python, etc.). Therefore, the MGGP technique is adopted in this study instead of other machine learning techniques, such as RF, MLP, and artificial neuron networks. Moreover, instead of training a single MGGP model using the numerical data for the entire study domain, the present study divided the numerical dataset for the study domain into different sub-datasets, and each sub-dataset was used to train an MGGP model for the corresponding sub-domain. This can reduce the processing time for training the models because the work can be done in parallel and the complexity of the trained models is reduced. To the best of the authors’ knowledge, this is the first time that a hybrid 2D hydraulic-MGGP approach has been proposed to develop models for estimating spatial distribution of flow depth in fluvial systems.
Section snippets
Study area
The area of interest (Fig. 1) is a segment of the Ottawa River in the vicinity of the ROPEC (Robert O. Pickard Environmental Centre) wastewater treatment plant, immediately downstream of the island named Île Kettle. The length of the segment is approximately 3.2 km, and the river width in the segment varies from approximately 1.06 km to 1.40 km. The Outaouais community near this area has been dealing with the Ottawa River flooding problem for decades, and an accurate and easy-to-use 2D model
Model validation and additional computations
A grid independence analysis was conducted to determine the optimal size of computational mesh: a total of four different meshes (Mesh 1 ~ Mesh 4) were tested and their corresponding numerical results were compared. The number of elements for Mesh 1 to Mesh 4 were 8 927, 14 025, 25 087, and 56 580, respectively. It has been found that the difference between Mesh 3 and Mesh 4 was insignificant, implying that the refinement effect on simulated results was negligible beyond the current level of
Discussion
A primary factor that limits the widespread usage of sophisticated hydraulic models in practical engineering projects is the heavy computational costs, especially for large-domain and real-time purposes. The outcomes obtained by the proposed approach were MGGP-based models, which were very fast to run. For example, estimating the river depth at a location in the study area using the hydraulic model took about 53 min, but estimating the same data using the MGGP-based model only took about
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work was partially funded by the Natural Sciences and Engineering Research Council of Canada (NSERC Discovery Grants). The first author, Xiaohui Yan, is supported by the Fundamental Research Funds for the Central Universities (China; DUT20RC(3)096). The field data were collected by Dr. Ivana Vouk. We would like thank the Editor in Chief, the anonymous Associate Editor, and the four reviewers for their careful reading of our manuscript, and their insightful comments and suggestions.
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