Reservoirs play a substantial role in meeting water demands in arid and semi-arid regions especially with increasing changes in global rainfall patterns. Kurau catchment located at Northwest Perak, Malaysia, is the largest source of water to Bukit Merah Reservoir. Based on climate change effects on rainfall pattern, it is important to develop a model to simulate rainfall occurrence and amount that flow into the reservoir. To address this problem, a stochastic rainfall generator model based on first-order two-state Markov chain approach was developed to simulate long-term daily rainfall series. Rainfall time series for a 30-year period (1976-2006) was assessed. The observed time series data were used as input to the stochastic model to generate a new set of daily time series data. The statistical properties of the new set of data including monthly mean, standard deviation, dry and wet spell lengths were compared with the observed data to gauge the model accuracy. The results obtained were satisfactory, giving motivation in applying the model for generating future rainfall series under different climate change scenarios.
Operation of a water storage tank in a specific environment motivates mathematical studies on a discrete-time deterministic dynamic programming problem. The operator decides whether or not to open the valve releasing the water in the tank to a drip irrigation system, based on the information on the storage volume of the tank. Two cases of functional regularity, which are Lipschitz continuous and of bounded variations, are considered for the reward defining the performance index to be maximized. Firstly, it is shown that the value function inherits the Lipschitz continuity of the reward in the infinite time horizon problem with discounting. Then, time periodic value functions are discussed in terms of the fixedpoint theorem. Discrete approximation of value functions is discussed as well, to conduct numerical experiments with a-posteriori error estimation applied to the real-world problem where the discount rate approaches to unity. It is found that a Skiba point appears as a threshold of valve opening for each day in an optimal policy for operation. Practically, setting a constant threshold throughout the period is quite reasonable and acceptable for the operator of the water storage tank to irrigate the farmland.
The occurrence of prolonged dry spells and the shortage of precipitation are two different hazardous factors affecting rainfed agriculture. This study investigates a multi-state Markov chain model with the states of dry spell length coupled with a probability distribution of positive rainfall depths. The Nineveh Plains of Northern Iraq is chosen as the study site, where the rainfed farmers are inevitably exposed to drought risks, for demonstration of applicability to real-time drought risk assessment. The model is operated on historical data of daily rainfall depths observed at the city Mosul bordering the Nineveh Plains during the period 1975-2018.The methodology is developed in the context of contemporary probability theory. Firstly, the Kolmogorov-Smirnov tests are applied to extracting two sub-periods where the positive rainfall depths obey to respective distinct gamma distributions. Then, empirical estimation of transition probabilities determining a multi-state Markov chain results in spurious oscillations, which are regularized in the minimizing total variation flow solving a singular diffusion equation with a degenerating coefficient that controls extreme values of 0 and 1. Finally, the model yields the statistical moments of the dry spell length in the future and the total rainfall depth until a specified terminal day. Those statistical moments, termed hazard futures, can quantify drought risks based on the information of the dry spell length up to the current day. The newly defined
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