## 1 Introduction  
In the context of global warming, the average global surface temperature is approximately 1 \({}^{\circ}\)C higher than that during industrialization, resulting in frequent extreme climate events. Among them, heat waves and drought events have had serious impacts on society and human life [1, 2, 3, 4]. Studies have shown that the frequency and severity of global terrestrial droughts and high temperatures are increasing, and the resulting climate fluctuations are bringing a variety of compound impacts to different regions [5, 6, 7]. As the impact of compound dry and hot events is far greater than that of single extreme events, a comprehensive analysis of the characteristics of compound dry and hot events can provide more effective information for mitigating global warming and maintaining ecological stability. Vegetation plays an important role in the energy exchange of different layers of the earth and is an important part of the ecological environment of the earth system and an important member of the biological carbon cycle [8; 9; 10]. Vegetation change is the result of the combined action of the interior and exterior of the Earth. The vegetation index covers a wealth of surface vegetation information, which is of great significance for the study of hydrology, ecology, and regional changes [11; 12]. NDVI is the most often used and is an operational, global-based vegetation index, partly due to its \"ratio\" properties, which enable the NDVI to cancel out a large proportion of the noise caused by changing sun angles, topography, clouds or shadow, and atmospheric conditions [13; 14]. The NDVI has been widely used to study vegetation growth in recent years [15; 16; 17]. The extraction of vegetation information using NDVI is commonly used to indicate the quantitative characteristics of vegetation and for monitoring seasonal changes in vegetation and land cover studies [18; 19]. Drought and heat waves often lead to reduced carbon uptake and productivity in ecosystems, affecting the growth and development of vegetation [20; 21]. However, the factors that have the greatest impact on vegetation often do not originate from a single extreme weather event but are more likely to be compounded by feedback associations between single extreme weather events; therefore, focusing only on traditional single extreme weather events may underestimate such impacts. In particular, the frequency of simultaneous droughts and heat waves has increased significantly in recent years, and the effects of the resulting compound dry and hot events on vegetation growth can be exacerbated by soil moisture-atmosphere coupling. These events often directly impact the carbon cycle by reducing water availability, reducing vegetation productivity, and regulating soil respiration and indirectly impact it through time-lagged responses that simultaneously trigger forest fires and tree mortality [22; 23]. Complex and large-scale changes in the carbon, water, and energy cycles result from these non-linear and combined impacts and small changes in terrestrial ecosystems due to climate extremes. Therefore, it is essential to understand the mechanisms by which vegetation responds to extreme climatic events, especially compound extreme events [24; 25]. In recent years, global climate and ecological issues have attracted widespread attention from the community and research scholars, and a series of studies have been conducted using bivariate distributions of drought (precipitation) and high temperature (temperature) to model the occurrence of compound dry and hot events. [PERSON] and [PERSON] et al. quantified the occurrence of compound dry and hot events on a global scale by establishing a joint distribution function based on copula's principle and found that when summer rainfall and temperature are negatively correlated, a compound event occurs [26]. [PERSON] et al.