## 1 Introduction  
Global dryland occupies a large proportion of the total area of the earth [1], and current climate warming and rapid population growth trends are only increasing the risk of dryland area growth, especially in developing countries [2]. In many developing countries, water resources are scarce and cultivated land is fragmented, due to serious soil erosion, river channels, streams, dry gullies, poor agricultural infrastructure, and incomplete irrigation measures [3]. These problems have created severe food security challenges in these regions [4]. In regions where it is difficult to establish large-scale irrigation channels or pipelines to provide enough water for irrigation, only small-scale irrigation projects, such as water wells and rainwater harvesting projects, can be used to carry out limited irrigation with extremely limited water amount (the available total irrigation water amount in the whole growing season is commonly less than 40 mm) in the sensitive period of crop growth and water shortage to improve food production and alleviate poverty [5]. Several studies have investigated limit irrigation using small-scale water conservancy projects, such as rainwater harvesting projects [6]. Currently, research on rainwater harvesting has thoroughly explored rainwater harvesting technology and its impact on crop yield [7; 8], whereas studies focused on determining the limit irrigation schedule of small-scale water conservancy projects are rarely published. Establishing an irrigation schedule is crucial to the development of small-scale water source supplementary irrigation agriculture [9; 10; 11]. Initiating a crop irrigation schedule requires study of the rational allocation of water during crops' growing periods to achieve maximum crop yield or maximum benefit. Many methods can be used to determine an irrigation schedule, such as the empirical methods or field irrigation test methods. And among the field test methods, optimization methods have also been explored. The optimization method has experienced the development process from dynamic programming to intelligent optimization, and from single-objective optimization to multiobjective optimization. In recent years, some scholars have coupled crop models and optimization algorithms to determine irrigation schedules. [PERSON] et al. [12] used HYDRUS-2D software to simulate water and nutrient flow, combined with the multiobjective optimization program of DSSAT crop model software, to determine the optimal irrigation schedule for wheat. [PERSON] et al. [13] combined the unified nondominated sorting genetic algorithms with the DSSAT crop model to determine the best irrigation schedule for maize. [PERSON] et al. [14] evaluated the ability of the AquaCrop model to simulate cotton in the North China Plain and optimized its irrigation strategy. In addition, determining irrigation schedules based on random rainfall has been studied [15; 16]. However, these studies often depend on field experiments with long test cycles and a heavy workload. Moreover, determining the irrigation schedule over a long period is difficult. Furthermore, the topographic characteristics of farmland in a region are complex, and the representativeness of field test samples is insufficient, which makes it difficult to coordinate a final irrigation schedule with the topographic characteristics of the region. Therefore, determining a method that can overcome these shortcomings and adequately reflect crop yield with fewer variables to feasibly determine an irrigation schedule for crops over a long period is necessary. Integral regression method has been widely used in many fields because of its substantial dimension reduction and limited loss of original information. According to the utilization of seasonal climate prediction information, [PERSON] et al. [17] used the principle of integral regression as a downscaling method to refine the spatial scale and improve climate predictability. [PERSON] et al.