The assessment of the adverse social, economic and environmental impacts associated with a combination of multiple climate hazards has recently become a focus of high interest . Such compound events often lead to larger impacts compared to when hazards occur separately . For instance, compound dry and hot conditions reduce carbon uptake more strongly compared to the sum of the individual hazards . Dry and hot conditions often co-occur. For instance in Europe, the extreme 2003, 2010 and 2018 heatwaves were accompanied by strong soil moisture deficits . In 2010, the compound event was particularly strong in Russia , while in 2003 the extreme drought and heatwave affected mostly central Europe, extending to west Mediterranean countries like Portugal and Spain , with critical consequences in several sectors. In 2010, widespread crop yield declines and failures occurred over the major grain-producing regions of Russia, northeastern Ukraine and northwestern Kazakhstan . Previously, the shortages in crop yields in 2003 also caused major financial losses in the agricultural sector, and when compared to the previous year, the cereal productions in the European Union (EU) decreased 23×106 t . The decline in the harvests was both in quantity and quality, as was the case for winter cereals whose maturation was accelerated due to compound extreme dry and hot conditions, forming grains with insufficient water content . The 2018 event strongly impacted pastures and arable land north of the Alps . As the occurrence of climate extremes such as heatwaves, droughts, and compound dry and hot events is expected to increase in intensity and frequency in many land regions due to climate change , associated adverse impacts such as widespread harvest failures threatening global cereal supplies may also increase.

Among the panoply of multivariate approaches applied to assess the impacts of multiple climate hazards, the use of copulas has become quite popular in studies focused on analysing the social, environmental and economic risks associated with adverse climate conditions . With copulas nonlinear dependency structures can be modelled, which offers more flexibility and possibly a more adequate fit for different dependence types at the extremes . Among all types of copulas described in the literature, the popularity of the class of elliptical copulas comes from the fact that they are derived from well-known distributions associated with the widely used Pearson's correlation, but the elliptical dependence is only able to capture radial symmetry and the respective mathematical expressions do not have a closed form. One of the copula classes that overcomes this drawback is the Archimedean, which has a simpler mathematical form and can capture different kinds of tail dependence and radial symmetry or asymmetry.

Archimedean copulas (ACs) are exchangeable, which means that the copula is the same if we permute the respective margins. For the bivariate case this may not be a limitation, but as the number of dimensions increase, it is unlikely that exchanging across the involved variables allows for the “true” dependence structure to be well-defined. To avoid exchangeability, nested Archimedean copulas (NACs) have been proposed , also referred to as hierarchical Archimedean copulas (HACs), obtained by nesting lower-dimensional Archimedean copulas into each other and/or using marginal distributions. introduced NACs where all copulas belong to the same family with a nesting condition that requires decreasing dependence strength from the highest to the lowest hierarchical level. Here we make use of this NAC approach, taking advantage of the balance between flexibility (modelling different types of dependence structures) and usability in higher dimensions (limiting the number of parameters).

The present work aims to identify how compound dry and hot conditions affect wheat and barley yields over two regions of provinces in Spain based on the trivariate dependence between precipitation, maximum temperature and yields using a NAC approach. In particular, we are interested in quantifying the additional risk associated with compound dry and hot conditions compared to only dry or only hot conditions. Wheat and barley are chosen as they are two of the major rainfed crops in the Iberian Peninsula . Moreover, we build here on prior work which has estimated wheat and barley losses in the same area but related to a single hazard, namely droughts .

Using NACs, we estimate the conditional probabilities of crop loss under different severity levels of dry and hot conditions based on the full trivariate joint distribution. We focus on annual wheat and barley yield data at the sub-national scale, thus overcoming drawbacks related to assessing climate-related crop risks at the national scale. Based on the proposed approach we (i) characterise the dependence structures between the dry and hot conditions and the crop yields, (ii) estimate the conditional probability of crop loss under different compound dry and hot severity levels, and (iii) evaluate how much the compound dry and hot conditions increase the risk of crop failure in comparison to the individual hazards.

In both cereals and both regions the most dependent pair of variables corresponds to crop yields and PMAM; hence the pair of variables u1,u2 defines the optimal NAC structure (Fig. ). Results for all possible variable pairs and the respective bivariate copulas are shown in Table .

Once the bivariate copula C12(u1,u2) of yields and PMAM are known, the NAC models are constructed (Table ). The Frank copula provides the best fit of C12(u1,u2) (Table ) for both cereals and both regions, and thus the parameters of the trivariate nested copulas are all from the Frank family. Nevertheless, despite Frank being the best family to characterise the nested copulas, we also constructed NAC models with Gumbel, Clayton and Joe copulas for comparison, as well as trivariate Archimedean copulas with one parameter where we selected the best structure between one-parameter and two-parameter AC copulas via the AIC (Table ). In all but one case the NAC model with Frank copulas is the best model. The only exception is barley in Region 2 whose AIC of Cθ(u1,u2,u3) is slightly lower than the AIC of Cθ1(u3,Cθ12(u1,u2)) (Table ). This feature may suggest that a structure favouring the dependence between yield and precipitation (u1,u2) may not be as relevant as in the other regions and yields due to a less dominant role of drought individually in this case. Nevertheless, in terms of the Cramér–von Mises distance (Sn) the nested copula is closer to the empirical trivariate copula. For this reason, we modelled the trivariate joint distribution based on nested Frank copulas for all cases. For all fitted models, the empirical cumulative distribution corresponds well to the theoretical cumulative distributions (Fig. ).

Bivariate dependencies as measured by Kendall's τ are captured well by the fitted models (Fig.  for wheat, Fig.  for barley). Among all possible pairs, the correlation between Tmax⁡ and yield is the weakest for both cereals (Table A1), and likely for this reason it is the pair in Figs.  and  with observational τ closest to the lower bound of the 95 % confidence intervals (Figs. f, h and f, h). Nevertheless, in both Figs.  and , the simulated level of dependence is inside the 95 % confidence level and the magnitude of correlations among the pairs is also reasonably preserved by the models i.e. such that τu1,u2>τu2,u3>τu1,u3.

The cumulative conditional probabilities of yield under moderate (+), severe (++) and extreme (+++) compound dry and hot conditions demonstrate that the probability of crop loss increases with the severity of compound dry and hot conditions for both regions and both cereals (Fig. a–d). Moreover, the likelihood of crop loss is higher in Region 2 for both cereals, particularly in the case of barley. Under extreme dry and hot conditions (+++dry+++hot, purple), the likelihood of crop loss is 68 % and 71 % for wheat and barley, respectively, in Region 2, in contrast to 62 % and 63 % in Region 1 (Fig. e, purple bars). In addition, the differences in crop loss are higher between moderate (+dry+hot) and severe (++dry++hot) conditions compared to the differences between severe and extreme (+++dry+++hot) conditions. More precisely, when the compound dry and hot conditions aggravate stress from moderate to severe levels, crop loss increases 5 % to 6 %, and when the compound dry and hot conditions aggravate stress from moderate to extreme levels, crop loss increases 6 % to 8 % (depending on the cereal and region). For comparison, conditional cumulative probability distributions for single stressors compared with the compound stressors are shown in Fig.  for all three severity levels.

While Fig.  illustrates the same severity levels for the different hazards, Fig.  illustrates crop loss for a range of different combinations of severity levels of dry and hot conditions (e.g. extreme dry conditions combined with moderate, severe and extreme hot conditions and vice versa) starting from the 50th percentile of PMAM and Tmax⁡MAM. When PMAM or Tmax⁡MAM are below or above the median, the probability of crop loss is always higher than 40 %. Similarly to Fig. , the increase in crop loss with the severity of drought and heat stress is evident (Fig. ). The higher likelihood of crop loss in Region 2, particularly for barley, is also consistent with Fig. . Moreover, the results indicate that droughts are typically associated with higher probabilities of crop loss than heatwaves at the same severity level. This finding suggests that drought stress causes more damage to crop yields than heat stress, even for lower values of stress.

In all cases, the additional effect of compound dry and hot conditions is larger when starting from only hot conditions, compared to when starting from only dry conditions (Fig.  for moderate stress, Fig. a and b for severe and extreme stress). The estimates are based on Eqs. () and (). Depending on the cereal and region, the difference between drought stress and compound conditions may vary from 8 % (barley in Region 1) to 11 % (barley in Region 2). In contrast, the difference between heat stress and compound conditions may vary between 19 % (barley in Region 2) to 29 % (wheat in Region 2). Uncertainties are large for these estimates and increase with the severity of the events (Fig. ). Consistent with Fig.  these findings suggest that drought stress is the major driver of crop loss associated with compound drought and heat.

We have modelled the trivariate relationship between Tmax⁡MAM and PMAM and wheat and barley yields in two regions in Spain using nested copulas. We found that the likelihood of crop loss increases with the severity of the compound dry and hot conditions and that compound drought and heat always increases the likelihood of crop loss. Moreover, our findings suggest that drought stress is not required to be as extreme as heat stress to cause the same adverse impact on crop yields. Hence drought is the more dominant driver of crop loss, when considering compound drought and heat.

Although the use of different methodologies and spatio-temporal scales and the focus on different cereals and regions make a comparison between studies difficult, our findings are consistent with previous work. Using bivariate return periods of combined climate conditions, have shown how linear models based directly on precipitation and temperature (and not the respective bivariate return period) may underestimate the explained variability in crop yields and that in several countries maize yields decrease with dry and hot conditions. Based on a meta-Gaussian model at the national level, have also shown that compound dry and hot extremes lead to larger impacts on maize yields than the individual hazards across five major maize-producing countries.

In terms of the relative contributions of drought and heat conditions, a variety of studies at the national scale have found that the response varies from country to country. have found that in China, France and Romania, higher chances of maize loss under dry conditions with normal temperatures (rather than under hot conditions with normal precipitation) can be expected, while in the USA and Argentina, higher chances of maize loss under hot conditions with normal precipitation (rather than under dry conditions with normal temperatures) can be expected. In contrast, have found that in countries such as Lithuania, Luxembourg and the UK, maize yields increase under hot and wet conditions, likely because of the importance of summer precipitation for the crop vegetative cycle and the relatively cooler climate in those countries.

Although previous studies have discussed that maximum temperature might be the best predictor variable for yield variability in most countries , our study highlights that in Spain crop loss of wheat and barley is more sensitive to dryness than to hot conditions. This finding agrees with the rainfed practices adopted in the wheat and barley cultivation in Spain. In fact, the nesting structure of the trivariate models adopted in the present study privileges the stronger dependency between yields and precipitation, rather than between yields and temperature or between precipitation and temperature (Fig. ). Though irrigated crops typically produce higher yields, the pressure in water resources is already increasing the deficit between water supplies and water demand in Spain . Hence, understanding climate risks for rainfed crops is crucial to address the current water management challenges for agricultural practices in Mediterranean regions.

Higher probabilities of crop loss under drought and/or heat stress are generally expected in the southern region of Spain, in comparison to the northern region (Figs.  and ), in agreement with the higher temperatures and lower rainfall amounts observed in the south . In the case of wheat losses, this finding is in agreement with previous work which focused on drought risks for the same crops and the same region (assessed based on remote sensing and hydro-meteorological drought indicators; ). However, identified a higher likelihood of barley loss with drought in the northern region. This discrepancy underlines the importance of addressing the interaction between compound dry and hot conditions and the associated impacts on vegetation. For instance, compound dry and hot conditions have a larger impact on the carbon uptake potential than the sum of the individual impacts , highlighting the relevance of interactions between multiple stressors.

We found that for barley in Region 2, drought is the least dominant driver in comparison to the other cereals and regions. Barley in Region 2 shows the highest difference between drought and compound dry and hot conditions and the lowest difference between heat stress and compound conditions (Fig. ). This suggests that for both cereals and in both regions, barley in Region 2 is the case where the compound and possibly interacting effects of drought and heat are most relevant. Also note that in this case the CDFs between the dry and hot and dry or hot conditions are more differentiated from each other for the severe and extreme stress (Fig. ). This is consistent with a recent study at the province level, which recommended that crop production in Spain should focus more on wheat production given that most provinces displayed lower levels of wheat loss with drought in comparison to barley loss . This finding is also consistent with Figs.  and .

The uncertainties associated with the parametric statistical model were assessed with a large number of sampled distributions with the same sample size as the observations. In some of these distributions, drought or heat alone may cause more damage than concurrent drought and heat (lower uncertainty bound is below 0 in Figs.  and ). This highlights the challenges of estimating the likelihood of rare events in two- or three-dimensional probability distributions with limited sample sizes . For the same reason, the wheat loss in Region 2 when PMAM is below the 5th percentile in Fig.  slightly decreases when the threshold of Tmax⁡MAM changes from the 10th percentile to the 5th percentile (where an increase would be expected as in the other cases). These features are associated with the uncertainties in the estimation procedure, which may be particularly large for extreme values, and it would be difficult to find a physical explanation for such a feature. Note that the uncertainties increase with the increasing severity of the compound dry and hot conditions (Fig. ) due to the rapid decrease in available samples in the corners of the three-dimensional probability distribution. Nevertheless, the best estimates (bars in Figs.  and ) show indeed that compound dry and hot extremes contribute to an increase in yield loss. In the general sense, the biophysiological explanation for the combination of environmental drivers leading to stronger yield reductions relates to the crop's requirements of water and thermal conditions during the key phenological stage in the analysis. The selection of the climate variables during spring corresponds to the reproductive phase of the plants and when vegetation is photosynthetically more active, and the combined effect of water and heat stress during this period is critical for the crop's health leading to yield decrease. During this stage of formation of the grains the dry and hot extremes may accelerate the maturation, affecting the size, number and weight of the grains and consequently affecting the crop's harvests in quantity and quality .

Following the work by , here we considered nesting copulas of the same family only, as more complex structures would be difficult to implement in general. Vine copulas might offer an alternative that is also appropriate for higher dimensions , when considering for instance more driver variables. Nevertheless, in comparison with previous studies based on bivariate models only, we argue that the statistical modelling based on NACs is a good compromise between complexity and the trivariate dimension.

The present study assessed how compound drought and heat enhances losses of wheat and barley in two major dryland areas in Spain. We showed that nested Archimedean copulas can successfully model the trivariate joint distribution between spring maximum temperature, spring precipitation and yields to estimate conditional probabilities of crop loss under different severity levels of hot and dry conditions. The strongest dependence exists between spring precipitation and yields and is best captured by a Frank copula. Our results demonstrate that the probability of crop loss increases with the severity of compound dry and hot conditions. Furthermore, the likelihood of wheat and barley loss increases when drought or heat, respectively, are aggravated to form compound dry and hot conditions in both regions. Overall, the likelihood of crop loss in the southern region is larger, in particular for barley. For both cereals and regions, the likelihood of crop loss increases more with increasing drought stress than with heat stress, suggesting that drought plays a dominant role in the compound event. Our results illustrate the additional value of using trivariate copula modelling to estimate the compounding effects of dry and hot extremes on the risk of crop failure. In operational practice, this research can contribute to the design of supporting tools and provide guidance in the decision-making process in agricultural practices to minimise crop losses related to climate hazards.