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In this chapter, the application of a DLRNN is demonstrated to simulate rainfall-runoff processes and recognize the transition of UHs in hydrology. Although most neural networks are black-box models that lack physical meanings of weights, the DLRNN developed in this chapter connects its weights with UHs that reveal the physical concepts from the network based on the special structure of RNNs. Without trial and error method, the structure and the weights of DLRNN can be quickly determined through a modified form of system identification that combines indirect system identification with the subspace algorithm. Then, the DLRNN learning algorithm based on the interchange of the roles of the network state variables and the weight matrix is derived for on-line training. In this chapter, the DLRNN introduced can not only simulate rainfall-runoff processes, but also recognize the transition of UHs. Owing to the feedback connections, DLRNN performs rainfall-runoff simulations as dynamic systems, and the advantage of DLRNN's dynamic feature has been proven after the comparison between DLRNN and FNN. The investigation of the connections between weights and physical meanings is an extension of neural networks applied in hydrological field due to the linearization of the RNN. Based on the linearization, weights of DLRNN are treated as Markov parameters to realize the transition of UHs. Through on-line learning, DLRNN modifies the weights to capture the relation between rainfall and runoff every time step, and the transition of rainfall-runoff processes can be emerged based on the changes of UHs. Furthermore, a modified system identification that combines indirect system identification with subspace algorithm is described to calibrate the DLRNN. This method determines the quantity of neurons in hidden layer and the weights of the network. It overcomes the drawback of costing time by traditional trial and error search for optimum structure of DLRNN. Additionally, the different forms of DLRNN have also been discussed herein. The results show that the performances of DLRNNs in different forms are close. Hence, the transformation of canonical form can be ignored in the flowchart of simulation via DLRNN. Finally, four criteria have been applied to evaluate the performance of rainfall-runoff simulation via DLRNN. The results show that the performance is satisfactory and DLRNN is competent to simulate dynamic systems, like rainfall-runoff processes.

This study applies the discrete wavelet transform (DWT) to decompose the unit hydrograph, thereby generating parsimonious reparameterizations of the unit hydrograph. A model compression method is then employed to significantly compress the unit hydrograph requiring that fewer coefficients be estimated. Moreover, a wavelet-based linearly constrained least mean squares (WLCLMS) algorithm is also used to estimate on-line the wavelet coefficients of the unit hydrograph. The updated wavelet coefficients of the unit hydrograph, convoluted with effective rainfall input in the wavelet domain, allow for accurate prediction of one-step-ahead runoff in the time domain. The proposed approach allows the unit hydrographs to vary in time and accurately predicts runoff from a basin in Taiwan, thus making it highly promising for flood forecasting.

A preliminary method for coding random self-similar river networks and the corresponding distance calculations are proposed in a companion paper. The coding method is applied to generate random self-similar river networks, and the corresponding algorithm for calculating the geometric distances of the links is employed to determine the width function of the river networks, and thus evaluates the adaptability of the process. The width function-based geomorphological instantaneous unit hydrograph (WF-GIUH) model is then applied to estimate the runoff of the Po-bridge watershed in northern Taiwan. The results imply that the separately random self-similar generating algorithm can be used to simulate river networks during the rainfall–runoff process. It can also help analyse the variations of the river network when rainfall locations change and study the influence on hydrological responses (IUH) when the shape of river network changes.

This study proposes a preliminary method for coding random self-similar river networks as a series of numbers, and investigates the corresponding algorithm that calculates the geometric distances from the code series. The coding method, generating random self-similar patterns with various probabilities of particular pattern links, and transferring the pattern into numerical code series, is proposed, based on the wide range of stochastic characteristics of natural patterns. The heredity of self-similarity and the connecting relationships inside the generated pattern are discussed. Finally, the algorithm to calculate the geometric distances of the generated pattern are developed.