## 1 Introduction  
### Consequences of Climate Change on Agriculture  
Agriculture in France represents an important economic activity (leading producer in the European Union). In 2014, of the EUR 373 billion of gross agricultural products (GAP) produced in the European Union, France produced EUR 67 billion, representing 18% of the GAP [1, 2]. France is the main producer of wheat and cattle in the European Union, and these two activities cover a large part of its utilized agricultural land [3]. In 2003, a severe drought caused a massive decrease in agricultural production and income (30% of the production was lost [4]), despite the rise in prices that some crops experienced [5]. Grassland yields were also greatly reduced and public support (via the Calamity Fund System) was necessary to allow farmers to get through the year, especially in the milk production community. These elements indicate that despite technological progress, crop production remains highly dependent on water resources and climatic conditions. In this context, increasing our scientific knowledge of the intensity and frequency of these extreme dropsthav is necessary to evaluate their impact on agricultural production. There is a global consensus in the scientific community that the climate will be very different in the middle of the 21 st century [6; 7; 8; 9]. The main cause of climate change is the increase in atmospheric concentrations of several greenhouses gases as a result of human activity [10; 11]. Several models are used in the community to study different scenarios of climate change and its consequences on agriculture. The following studies highlight several important points:  
* Annual temperatures are expected to increase [12; 13]. As a consequence, the cropping period will be shorter and the grain protein concentration will decrease [14; 15]. In addition, a high frequency of severe drought events during summer periods is to be expected [7; 11; 16]. These events may cause crop yield reduction or stagnation (depending on areas and species) [17; 18]. Some studies show that in the Mediterranean region, yields could greatly decrease compared to historical yield trends [2; 19]. * A decrease in annual precipitation, especially in the Mediterranean region, such as the South of France, will contribute to an increase in winter droughts [11]. * Many studies highlight that cereal crops (in particular wheat and corn, which are the most studied crops) will experience a decrease in yield due to an increased number of days above 30 \({}^{\circ}\)C [12; 13; 18; 20; 21; 22; 23]. * Other studies underline that high temperatures have direct and indirect negative impacts on dairy production [24; 25] by affecting animal health and grassland yields, in particular. Finally, several studies demonstrate that heat waves like the 2003 one, which particularly affected crop yields, will be more frequent in the future [2; 6; 26; 27; 28]. Extreme drought events are difficult to study because there are by definition rare events that occur very infrequently, so an archive of historical data may contain just a few extreme events [29]. The IPCC defines the concept of \"extreme\" as \"the occurrence of a value of a weather or climate variable above (or below) a threshold value near the upper (or lower) ends ('tails') of the range of observed values of the variable\" [8]. Thus, the notion of an extreme drought event is dependent on the value of the climatic index chosen to characterize the climate. Therefore, the link between the climate index and the impact on agriculture (yield losses) has to be explicit. However, annual yield losses are due to a set of phenomena (diseases, climatic events, changes in cropping practices) and it is not easy to assess the weight of one phenomenon independently. The best-known drought indicators are:  
1. Standardized Precipitation Index (SPI) developed by [30] and based on precipitation data to determine the exact period and duration of meteorological droughts.