## 2 Methodology
### Solving the Precipitation-Meteorological Yield Integral Regression Function of Each Subregion  
Using the meteorological yield of many years and the corresponding years' precipitation in each ten days of whole crop growth period, the integral regression curve of rainfall meteorological yield of typical counties in each subregion was calculated to determine the optimal irrigation period. The principle of precipitation-meteorological yield integral regression is to divide the whole growth period of crop into several infinitely small periods. The relationship between precipitation and meteorological yield in each small period can be expressed using the following regression integral:  
\[\hat{y}_{w}=C+\int_{0}^{T}a(t)r(t)dt \tag{3}\]  
where \(\hat{y}_{w}\) is the estimated value of the meteorological yield of crop; \(C\) is the constant; \(T\) is the number of days of the whole growth period; \(r(t)dt\) is the precipitation from \(t\) to \(t+dt\); and \(a(t)=\Delta y/\Delta r\) is the relationship between precipitation at \(t\) and the meteorological yield, that is, the effect of each \"additional\" precipitation on meteorological yield at different growth stages. \(\Delta y\) is the meteorological yield increment, and \(\Delta r\) is the precipitation increment. This function can provide a basis for determining the irrigation period of the crop. Because \(a(t)\) is usually a continuous function that is slow to change, it can be quite precisely approximated to within a few terms of a polynomial expressed in time. Specifically, it can be expressed as follows by the orthogonal function \(\varphi_{i}(t)\) (\(i\) = 0, 1, 2,...... ):  
\[a(t)=a_{0}\varphi_{0}(t)+a_{1}\varphi_{1}(t)+\cdots\cdots, \tag{4}\]  
where \(a_{i}\) (\(i\) = 0, 1, 2...... ) is a partial regression coefficient. Substituting Equation (4) into Equation (3), the following equation is obtained:  
\[ \begin{array}{ll}\hat{y}_{w}&=C+\int_{0}^{T}(a_{0}\varphi_{0}(t)+a_{1} \varphi_{1}(t)+\ldots\ldots)r(t)dt=\\ &C+a_{0}\int_{0}^{T}\varphi_{0}(t)r(t)dt+a_{1}\int_{0}^{T}\varphi_{1}(t)r(t)dt+ \ldots\ldots.\end{array}  \tag{5}\]  
Precipitation distribution coefficient \(\rho_{i}\) is defined as follows:  
\[\rho_{i}=\int_{0}^{T}\varphi_{i}(t)r(t)dt\;(i=0,\;1,\;2,\;\ldots\ldots). \tag{6}\]  
Substituting Equation (6) into Equation (5), according to the definition of the orthogonal standard function, the following equation can be obtained:  
\[\hat{y}_{w}=C+a_{0}\rho_{0}+a_{1}\rho_{1}+\ldots\ldots. \tag{7}\]  
The problem that can be seen from Equation (7) is determining \(a_{i}\) (\(i\) = 0, 1, 2,...... ). According to the precipitation distribution coefficient and meteorological yield of a region over the years, multivariate linear regression method can be used to derive \(a_{i}\). Then, using Equation (4), the influence value \(a(t)\) of the \"additional\" precipitation on meteorological yield at different periods can be obtained. The calculation process of determining the optimal irrigation period of regions crops was as follows: First, the trend yield of crops in the region was calculated by using logistic fitting method, and the trend yield is subtracted from the actual yield to obtain the meteorological yield over the years. Second, according to the discretized orthogonal function \(\varphi_{i}(t)\) and the amount of precipitation in different periods, the precipitation distribution coefficient \(\rho_{i}\) over several years was deduced from Equation (6). Then, \(a_{i}\) was calculated according to the meteorological yield and the precipitation distribution coefficient \(\rho_{i}\) over several years using Equation (7). Finally, according to the calculated \(a_{i}\), the influence coefficient \(a(t)\) of precipitation on the meteorological yield in different stages of crop growth was calculated using Equation (4). If \(a(t)\) was larger at a certain crop growth stage (indicating that \"additional rainfall\" had a significant effect on crop yield at this stage), it could also be considered that irrigation can significantly improve crop yield at this stage.