Probabilistic Analysis of Drought Effects on Rainfed Wheat Yield Using Copula Functions (Case Study: Tabriz Plain)

A combined assessment of drought risk and associated impacts on crop production based on a probabilistic approach seems appropriate to understand the multivariate nature of drought risk in agriculture. To overcome the problems caused by drought impact detection, several approaches have been developed in recent decades. Among the multivariate analysis approaches, copula functions are very popular. Copulas use univariate marginal distributions to form a joint distribution. The joint distribution can be described by the corresponding marginal distributions and copula functions that describe the dependency structure. In this research, using the statistical precipitation data in the Tabriz plain in eastern part of Lake Urmia basin, and yield of rainfed wheat in this area, a model based on copula functions was developed to determine the diffrent probabilities of yield risk and different drought conditions. Also, the application of copula functions related to rainfed wheat yield in this region was performed for the first time, and the presented method will be applicable to other areas and other crop cultivation.

In the case of meteorological drought, the basis for calculating the degree of drought is determined by comparing the amount of precipitation with the long-term average or its normal values. The SPI index is considered to be an appropriate and powerful index to use as a time scale droughts monitoring. Basically, SPI was created to detect the lack of precipitation on multiple time scales. Among the reasons that make the use of this index so popular, we can mention the standard nature of this index as it can be used in regional studies and establish a temporal relationship between drought events in different parts of the same area. The SPI index is a dimensionless index and its more negative values show the more severe the drought.
Analysis of variables individually is easy and can be analyzed by statistical distribution functions; but statistical analysis joint variables is very complicated and impossible in most cases. If the correlation criterion of these variables is known, their joint probability distribution can be obtained using copula functions. Using copula functions for modeling has a high degree of flexibility as it is possible to choose different marginal distributions to create a multivariate model. Copulas are functions that form a bivariate or multivariate distribution based on two or more univariate marginal functions. Several copula functions can be used to construct a two-dimensional joint distribution of hydrological and agricultural variables, among which Archimedean and elliptical copula families are the most commonly used. In the present study, six copula functions are used and the parameters of the paired functions are determined using the two-stage maximum likelihood method, which estimates the parameters of the marginal distribution and the copula function by forming two likelihood functions. In order to investigate joint probability of rainfed wheat yield and drought index, the time series of rainfed wheat yield in the Tabriz Plain region and SPI index during the last 30 years from 1990 to 2020 was applicated in this study.

In general, the improvement in agricultural methods, investments and technological advances during this period have led to a continuous increase in crop yield, however, a sharp drop in crop yield is clearly evident at times during the reporting period. In this research, to ensure that the observed trend does not affect the results, the Copula model was built using the detrended time series data by removing the trend in the values and the variance in the yield data. Based on the results obtained, crop production decreases dramatically during severe droughts, so such sensitivity to moisture deficits caused by low rainfall after several wet years can be attributed to farmers' expectations and management policies driven by high productivity during the previous ones wet years were determined.
To use series of standardized yield and growing season SPI in copula, the most appropriate distributions were selected and used, logistic and normal, respectively. Also, according to the calculated Kendall correlation (0.35), the best fit joint was Claytons function with AIC = -11.10, RMSE = 0.036 and used to construct the joint probability distribution of the standardized yield series of rainfed wheat and SPI of growth Period. Results showed that the cumulative probability of yield risk event in mild, moderate, severe, and very severe drought was 18.42, 7.82, 4.26, and 2.32 percent, respectively,and for an overall rainfed wheat yield risk and drought condition is about 33%. Meanwhile, the probability of rainfed wheat yield risk and non-drought events is only about 7% and joint probability of yield risk and SPI>1 is so close to zero.

In the current research, the probability of occurrence of the rainfed wheat yield risk and drought conditions was extracted by copula functions. Based on the results obtained, most changes in yield occur when the drought index is in the range greater than -1.5. In other words, in the conditions of severe and very severe drought, the yield of rainfed wheat does not show noticeable changes in probability. The opposite situation can also be observed for the situation of very wet years, that when there is high amounts of precipitation and the drought index show values higher than 1.5, the yield of the crop does not show much change. Thus, the SPI must be greater than one in order to achieve the desired yield and not be at risk of rainfed wheat yield, which in this study is considered to be a standardized yield values greater than zero. Because in this range of SPI, the joint probability of yield risk is estimated to be very close to zero. Therefore, this threshold can be introduced as the rainfed wheat yield safety threshold, but depending on the situation of the region and the drought index, the drought threshold (SPI<0) contains cumulative probability about 33% of the rainfed wheat yield risk.