Understanding the Problem Structure of Optimisation Problems in Water Resources

Abstract

Optimisation algorithms are widely used in water resources to identify the optimal solutions for problems with multiple possible solutions. Many studies in this field focus on the development and application of advanced optimisation algorithms, making significant contributions in improving optimisation performance. On the other hand, the performance of optimisation algorithms is also related to the features of the problems being solved, therefore, selecting appropriate algorithms for corresponding problems is also a key to the success of optimisation. Although a number of metrics have been developed to assess these features, they have not been applied to problems in the water resources field. The primary reason for this is that the computational cost associated with the calculation of many of these metrics increases significantly with problem size, making them unsuitable for problems in water resources. Consequently, there is a lack of knowledge about the features of problems in the water resources field. This PhD thesis aims to understand the features of problems in water resources, and the process can be split into two stages. The first stage is to identify metrics that can be applied within an affordable computational cost. This is addressed in the first content chapter (Paper 1). The second stage is to apply metrics identified in the first stage to understand the features of problems in the water resources field, including the calibration of artificial neural network models (Paper 2) and conceptual rainfall runoff models (Paper 3). This includes the understanding of optimisation difficulty of these problems according to their features, and how their features change through the change of their problem structure and the types of problems to which they are applied. In the first paper, the computational cost of fitness landscape metrics (explanatory landscape analysis (ELA) metrics) used in computer science is tested and metrics that are suitable for application to water resources problems are identified. Each metric used to understand the features of problems requires a given number of samples, which usually increases with an increase in problem size (dimensionality). Consequently, metrics which require a big increase in sample size through the increase of problem size are not suitable for real-world water resources problems. To identify ELA metrics that have low dependence on problem size, 110 metrics in total are tested on a range of benchmark functions and a number of environmental modelling problems, and 28 are identified to be able to be applied to complex problems without significant increase in computational cost. This finding provides us a new approach to better understand the problem structure of optimisation problems in water resources and has the potential to provide guidance in optimisation algorithm selection for problems in the water resources field. In the second paper, metrics identified to have low dependence on problem size in the first paper are applied to Artificial Neural Network (ANN) model calibration problems. ANN models for different environmental problems with different number of inputs and hidden nodes are used in the test. The environmental problems considered include Kentucky River Catchment Rainfall‐Runoff Data (USA), Murray River Salinity Data (Australia), Myponga Water Distribution System Chlorine Data (Australia), and South Australian Surface Water Turbidity Data (Australia). It is demonstrated that ELA metrics can be used successfully to characterize the features of the error surfaces of ANN models, thereby helping to explain the reasons for an increase or decrease in calibration difficulty, and in doing so, shedding new light on findings in existing literature. Results show that the error surfaces of ANNs with relatively simple structures have a more well-defined overall shape and have fewer local optima, while the error surfaces of ANNs with more complex structures are flatter and have many distributed, deep local optima. Consequently, ANNs with simpler structures can be calibrated successfully using gradient-based methods, such as the back-propagation algorithm, whereas ANNs with more complex structures are best calibrated using a hybrid approach combining metaheuristics, such as genetic algorithms, with gradient-based methods. In the third paper, the ELA metrics identified to have low dependence on problem size in the first paper are applied to Conceptual Rainfall Runoff (CRR) model calibration problems. Different CRRs with different model types, error functions, catchment conditions and data lengths are tested to identify how they affect the features of problem structure, which are related to their model calibration and parameter identification difficulty. It is suggested that ELA metrics can be used to quantify key features of the error surfaces of CRR models, including their roughness and flatness, as well as their degree of optima dispersion. This enables key error surface features to be compared for CRR models with different combinations of attributes (e.g. model structure, catchment climate conditions, error metrics and calibration data lengths and composition) in a consistent, efficient and easily communicable fashion. Results from the application of these metrics to the error surfaces of 420 CRR models with different combinations of the above attributes indicate that model structure differences result in the differences in surface roughness and relative optima dispersion. Additionally, increasing catchment wetness increases the relative roughness of error surfaces, it also decreases optima dispersion. This suggests that model structure and catchment climate conditions can be key issues in affecting the calibration difficulty, efficiency and parameter uniqueness. The experiments conducted in this study also encourage further tests on further CRR models and catchments to identify general patterns between calibration performance, model structure and catchment characteristics.