## 3 Results  
### Historical Reanalysis  
The relationship between the DOWKI values and yield losses for grassland is illustrated in Figure 3. The damage model is the statistical relation between climatic index and yield losses at the department scale. It is a combination of two predictive models:  
* Prediction of the occurrence of a claim, i.e., yield loss exceeding 5% at the department scale;
* Prediction of the yield loss value at the department scale.  
When a DOWKI value exceeded \(-475\) mm, the departmental yield loss was equal to 42% with a probability of occurrence close to 90%. For DOWKI values close to zero, the probability of claims was significantly lower (\(\sim\)30%), as was the yield loss value (10%).  
The calibration generated false positives and false negatives not explained by our index. False positives are departments where the index indicated, for a given year, an intense drought but without consistent yield loss. A false negative, on the contrary, is a case where high yield loss could not be explained by the index value. Several hypotheses were formulated to explain these errors (Table 2).  
Figure 3: Damage function for grassland yield loss simulations: frequency of claims, percentiles 10 and 90, and average of yield losses according to the DOWKI values.  
To validate our damage model, back testing was performed by comparing, at the national scale, the observed yield losses and the simulated yield losses (Figure 4).  
The back-testing relative error at the national scale was 5.5% for grasslands (14.6% for soft winter wheat and 20.4% for winter barley). The 2011 and 2003 intense droughts were explained by the model with an underestimation of 24% (2003) and overestimation of 2% (2011), but the highest simulated yield losses remained, as expected. The lowest yield losses at the national scale (years 2000, 2001, 2002, and 2008) were overestimated by the model, but the simulated yield losses were still the lowest in the distribution. The two droughts in 2003 and 2011 were characterized by a lack of precipitation and an augmentation of evapotranspiration rates. In addition, for the 2003 drought, record extreme temperatures were experienced during the summer. The main difference between these two droughts is that they did not begin at the same period of the year. The 2011 drought was a spring drought and the extreme values of DOWKI were computed in June. For the 2003 drought, extreme values of DOWKI were computed in August. The DOWKI values were more extreme for grassland than for cereals because the drought lasted all of August and the vulnerability period of cereals is shorter.  
The most difficult issue with these model results is the case of the drought in 2018: High yield losses due to an extreme drought that occurred in the northeast region of France were not detected by our model. This was due to multiannual drought cycles. The DOWKI index value was initialized at 0 on 1 January of each simulated year, whereas in 2017, the soils were abnormally dry in December.  
As shown in Figure 4, the back testing of the model showed its capacity to simulate extreme drought events and predict the national yield loss.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**False Positives** & **False Negatives** \\ \hline Adaptation of agricultural practices: modifying & \\ the sowing period or harvest period, choice & \\ of varieties & \\ \hline Protection measures (irrigation) & Combination of several climatic events, \\  & including droughts \\ \hline \hline \end{tabular}
\end{table}
Table 2: False positives and false negatives.  
Figure 4: Average grassland yield losses at the national scale (%) computed in the AGRESTE database and simulated by the model with DOWKI values.