A two-dimensional blocking index (BI) derived from daily Z500 was used to identify instantaneously blocked grid points. This blocking index is calculated according to the one-dimensional index proposed by but expanded to every latitude and longitude . Similar to , we apply a spatio-temporal filter that allows the exclusion of small-scale and short-term blocking situations accounting for large-scale and persistent systems between 35 and 80∘N. Thus, we select contiguously blocked regions with a minimum zonal and meridional extension of 15∘ and an area of at least 1.5 × 10×106 km2. A persistent blocking event is considered if the duration of the blocking system lasts a minimum of 4 d. In addition, the tracking algorithm includes possible merging and splittings of the blocking event in time by adopting a blocking overlap area criterion of 7.5 × 106 km2 between 2 consecutive days and a maximum distance between blocking centres of 1000 km . The blocking index was calculated using the “Free Evaluation System Framework” , which is a framework for scientific data processing designed for atmospheric applications that includes (among other features) software for the calculation of the BI; details about this method are given in .

Recently, copula-based approaches have become very popular for assessing interrelations between several random variables . A copula is a joint distribution function in which the marginal distributions are independent of the dependence structure and can be modelled separately . For two random variables X (temperature) and Y (MDA8O3) with marginal distributions FX(x)=Pr⁡(X≤x) and FY(y)=Pr⁡(Y≤y) respectively, a copula function allows their joint cumulative distribution to be constructed as follows: 1FXY(x,y)=C(FX(x),FY(y))=C(u,v), where FXY is the joint distribution function of X and Y, C is the copula function and u=FX and v=FY are the uniformly distributed marginals. According to Sklar's theorem, if the marginal distributions are continuous, then the copula function C is unique . The main advantage then of using copula functions is the flexibility to model the dependence between multiple random variables that follow arbitrary univariate marginal distributions. For each station, we use bivariate copulas to model the dependence between temperature and ozone and estimate their joint probability distribution under two different synoptic situations: (1) when there is a co-located block in the same location of MDA8O3 and temperature (i.e. BI =1), (2) without the presence of blocking (i.e. BI =0). We fit a total of four commonly used copulas: Student t (from the Archimedean family), Clayton, Gumbel and Joe (from the elliptical family) (Table 1). The Archimedean copulas are able to describe asymmetric tail behaviour, whereas elliptical copulas capture symmetric dependence . Among the different copulas, we selected the structures that are able to capture tail dependence; Gumbel and Joe copulas model upper tail dependence, whereas Clayton can capture lower tail dependence . The Student t copula allows dependence in both upper and lower tails. Before modelling the joint probability distribution, we fit the most appropriate marginal distribution for both temperature and MDA8O3, including Gaussian, gamma, Weibull and lognormal distributions. The parameters for the marginal distribution were obtained by the maximum likelihood method separately for each site. The marginal distributions were selected using the Kolmogorov–Smirnov goodness-of-fit test. Then, for each station and synoptic case (BI =1 and BI =0), the bivariate copulas were selected based on Akaike's Information Criteria (AIC) and the copula parameters were estimated via maximum likelihood estimation (MLE). The copula analyses were carried out with the VineCopula and the copula R packages .

The copula models were used to assess the relationship between temperature and ozone exceedances under blocking (BI =1) and non-blocking (BI =0) conditions by constructing the corresponding joint probability distribution (P(X≤x, Y≤y)). Apart from the general dependence structure, some copulas can measure the dependence of the extremes through the tail dependence parameter (λu) . As linear or rank dependence measures might not be accurate when focusing on extremes , we have further assessed the upper tail dependence of ozone and temperature extremes derived from the copulas under blocking (BI =1) and non-blocking (BI =0). We estimate the probability of a compound event at each station, in which Tmax⁡ and MDA8O3 exceed the 95th percentile of their respective distributions. It is important to note that the 95th percentile for each variable is calculated over the whole distribution of the ozone season, from April to September, for the period of study, i.e. 1999–2015. Therefore, the compound extremes at each station are defined based on a relative threshold value, defined for each station as a function of the 95th percentile over the whole distribution (i.e. including non-blocked and blocked days) of the corresponding variable (i.e. temperature and ozone). This choice was made in order to quantify the impact of blocks on the probability of exceedances, for which the definition of extreme changes depending on the station. Precisely, the use of absolute thresholds allows us to quantify the impact of blocks on the probability of exceedances. The probability of exceedances over a certain multivariate threshold was examined based on three different hazard scenarios described by the following joint and conditional joint probabilities, which can be expressed using copula notation see further details in: 2PAND=P(U>u∩V>v)=1-u-v+C(u,v),3POR=P(U>u∪V>v)=1-C(u,v),PCOND=P(U>u|V>v)4=(1-u-v+C(u,v))/(1-u).

The probabilities in Eqs. (2) and (3) have been widely applied in the literature to assess compound extremes . Equation (2) represents the scenario in which both variables, temperature (Tmax⁡) and ozone (MDA8O3), exceed the 95th percentile, whereas Eq. (3) considers a situation wherein the events occurred when temperature or ozone or both exceed their respective threshold (95th). As blocks normally lead to persistent positive surface temperature anomalies during summer over Europe , it is of interest to evaluate the influence of blocking on the probability of ozone exceedances given high temperatures, which is assessed in the COND scenario.

To quantify the significant impact of blocks on the compound ozone and temperature events, we estimated the differences between the probabilities derived from the copulas (i.e. ΔP=P1-P0). Then, we assessed whether the difference between the probabilities when BI =1 and BI =0 are significantly different from zero. To do so, we apply a bootstrap procedure for each probability scenario (i.e. AND, OR, COND) in which we drew 100 bootstrapped samples and derived the respective probabilities P1 and P0 when BI =1 and BI =0, respectively. For the null hypothesis (H0), there is no difference between the probabilities obtained from the cases BI =1 and BI =0, while the alternative hypothesis indicates that the probability of an extreme event conditioned to a blocking situation is significantly different from the probability under non-blocking conditions.

For the period of study (1999–2015) a total of 3111 d were analysed during the ozone season (April–September). The blocking frequency (% of days) ranges between 5 % in the southern latitudes (30–45∘ N) and 14 % in the northern latitudes (60–70∘ N) (Fig. S1). In central Europe, where the density of stations is higher, the frequency of blocked days is ∼ 8 %. Typically, blocking presents a well-established climatology in terms of frequency in the North Hemisphere, being more frequent in winter and spring, and less frequent in autumn . During summer, blocking events have shown a tendency to occur at high latitudes over continental areas . In contrast to previous related studies analysing the seasonal responses of air pollution to blocks , our study focuses on the whole ozone season during which the compounding effect of ozone and temperature is particularly relevant for human health . Moreover, atmospheric blocking events are likely to have a major impact owing to their connection with heatwaves in spring and summer .

We start by examining the individual impacts of blocks on the anomalies of ozone and temperature, in order to establish a comparison of anomalies across different stations. MDA8O3 anomalies were calculated as the difference between MDA8O3 values and the average of MDA8O3 over all the days in April–September during the period of study, i.e. 1999–2015. This average is obtained individually for every station. Tmax⁡ anomalies were similarly calculated with respect to the average value over all Tmax⁡ values from April to September during the same period. It is important to highlight that all calculations were applied separately for each station, and therefore, the number of blocking days might differ across the different stations. Figure 1 illustrates the composites of the anomalies of both MDA8O3 and Tmax⁡ during blocking days (BI =1). In general, most of the stations show positive anomalies of MDA8O3 under blocking days (Fig. 1a). The strongest positive anomalies > 30 µgm-3 are observed over the south of Germany, north-east of France and north-west of Italy, whereas weaker anomalies are found over Scandinavia, the west of the UK and the north of Spain. Similarly, reported strong positive anomalies over large areas of central and northern Europe in spring and summer respectively. In the case of temperature anomalies, blocks led to positive anomalies of Tmax⁡ over all of the stations included in this study (Fig. 1b). The largest values of temperature anomalies (> 7 ∘C) are observed over the central and western stations (north-east of France, Austria and the south of Germany). This is consistent with the radiative heating due to enhanced clear sky conditions over continental areas under atmospheric blocking conditions, especially in summertime .

We further examined the impact of blocking on individual extremes of ozone and temperature. To this end, we do not work with anomalies but, instead, fix a threshold for MDA8O3 as well as Tmax⁡. The absolute values for these thresholds vary among stations. A transparent way to set these thresholds are quantiles, i.e. values with a specified non-exceedance probability. For each station, we use the 0.95 quantile (or 95th percentile) from the sample restricted to April to September and thus get individual thresholds for all stations reflecting their local climatology. Thus, we define days with MDA8O3 exceeding this threshold as extreme days. Then, we obtain relative frequencies by dividing the number of extreme days with a simultaneous blocking by the number of total days in the data set restricted to April to September (i.e. p^=extremes with blocking/3111). A similar approach was applied in and calculated the percentage of blocking days with MDA8O3 values above the 90th percentile. It must be noted that the spatial variability of high levels of ozone is very heterogeneous (Fig. S2). The 95th percentile of MDA8O3 exceeds the European target value of MDA8O3 120 µgm-3, in a large number of stations in central and southern Europe. Only in the case of northern stations (UK and Scandinavia), would MDA8O3 not often exceed the mentioned target value (Fig. S2).

The same procedure using the 95th percentile was applied to identify days of Tmax⁡ exceedances and days above the 95th percentile of the Tmax⁡ (Fig. S2) were classified as exceedances. We acknowledge that, in the case of Tmax⁡, the number of exceedances above the 95th percentile might not be equally distributed across the ozone season (i.e. this threshold is more likely to be exceeded in July and August than it is in April and September). Although this could be corrected by either using a threshold that varies seasonally or by removing the seasonal trend in the data, we would like to stress that the main goal of this study is to quantify the impacts of blocking on the upper-tail dependence between MDA8O3 and Tmax⁡ over the entire ozone season. Our main interest is in the physiological effects of such compound events, for which only absolutely high temperatures (as they tend to occur in July or August) are relevant. Similar to other studies e.g., we use the 95th percentile over the period between April and September. A lower threshold, e.g. the 90th percentile, would lead to many temperature values not being physiologically relevant; a higher threshold, e.g. the 99th percentile, on the other hand, would lead to a strong reduction in the data available for the subsequent copula modelling. The 95th percentile-based definition for examining the individual impacts of blocks on MDA8O3 and Tmax⁡ is also justified to be consistent with the joint probability analysis, for which the 95th percentile is applied for the risk assessment (see below). Moreover, earlier studies used a similar threshold percentile-based definition to assess the links between temperature extremes and atmospheric blocking .

Figure 2 illustrates the frequency of blocked single extremes of ozone and temperature (i.e. the percentage of exceedances of MDA8O3 and Tmax⁡ with respect to the total number of blocked days). More than 40 % of MDA8O3 extremes coincide with blocked days over most of the central stations. The frequency of blocked days of MDA8O3 extremes is generally lower in the southern stations. The percentage of Tmax⁡ extremes coincident with blocks increases northwards and eastwards, which is consistent with subsidence processes and the clear-sky radiative forcing associated with summer blocking events . Moreover, as stated above, the strong seasonal variability of the blocking activity in the Northern Hemisphere must be noted with a reduced number of occurrences late in summer and autumn but being considerably more frequent in eastern Europe than in the Euro-Atlantic region . This pattern is also reflected in our results that show the largest number of blocked temperature extremes north and eastwards.

To investigate the impact of blocks on the relationship between MDA8O3 and Tmax⁡, Kendall's tau coefficient (τ) was calculated during blocking and non-blocking days as well as the difference between the correlations obtained when BI =1 and BI =0 (Fig. 3a and b). The correlations are weaker under non-blocking days (BI =0) and a few northern stations show negative values (Fig. 3a). In general, the dependence between MDA8O3 and Tmax⁡ is higher under the influence of blocks. The positive differences between the correlation values (Fig. 3b) clearly reflect the strong impact of blocks at most of the stations. The largest differences are found over the north-west of Europe. Consistent with previous work, the central stations show the strongest relationship between ozone and temperature , which significantly increases when BI =1 with the largest correlation values (> 0.6). Blocks seem to have a great influence over the north-west of Europe in particular, the UK, the north of France and the Benelux countries (i.e. Belgium, Netherlands and Luxembourg), where the correlations are higher and mostly positive under blocking conditions, whereas negative correlations are found under non-blocking conditions. A similar pattern was found when calculating the correlations for the respective extremes based on the 95th percentile that showed the strongest relationship under the influence of blocks over a large number of stations of France, Germany and the UK (Fig. S3). The impact of blocks in the relationship between MDA8O3 and Tmax⁡ is smaller in the south and north-east, which is reflected by non-significant and weaker correlations that show similar magnitude values when BI =1 and BI =0.

The results from the individual impacts of atmospheric blocking on MDA8O3 and Tmax⁡ are consistent with previous studies that showed the impacts of blocking on ozone and the association between blocking and temperature (e.g. ). Consistent with these works, we found notable spatial differences, with the largest blocking effects on the north-western and central stations and weaker impacts on the southern stations. showed that sub-tropical ridges, an extension of the sub-tropical high pressure belt extending northwards , had a major impact on surface ozone in the central-southern European sectors, especially in summer, whereas blocks showed a stronger impact in central and northern Europe in spring and summer, respectively. Nevertheless, they pointed out that the influence of ridges for the build-up of ozone pollution is not as clear as in the case of blocks, and its impact is more sensitive to the location. Using a similar catalogue to detect blocks and sub-tropical ridges, showed that blocks play an important role in warm temperature anomalies in spring and summer over central Europe, whereas the impact is generally lower over the south, mostly because of the position of the block . It must be noted that our detection method only focuses on blocks and, unlike the cited works, sub-tropical ridges were not included in this analysis. Despite this, our results are in good agreement with Sousa et al.'s findings and they also point out the spatial variability of the blocking effects on both MDA8O3 and Tmax⁡. Previous work has shown a strong effect of NOx levels on the temperature sensitivity of ozone . Owing to the relatively short atmospheric lifetime of NOx (order of hours), the combined effect of blocking and high temperature on ozone would be larger in areas close to strong NOx sources, such as large urban areas. Thus, we might anticipate spatial differences in the impact of blocks on the compound extremes of ozone and temperature and their joint distribution.

In the previous section, we have shown, separately, the effects of atmospheric blocking on both MDA8O3 and Tmax⁡, as well as the influence on their relationship through the correlation coefficients. Here, we present the results from the copula modelling analysis, which allow us not just to confirm the impacts of blocks on the structure dependence between MDA8O3 and Tmax⁡ previously shown but also to quantify the impacts of blocks on the compounding effect of both variables by estimating the joint probability of exceedances.

Among the different types of copulas presented in the literature, a total of four copulas (Table 1) were tested to find the most appropriate fit that characterises the relationship between MDA8O3 and Tmax⁡ at each station. Our copula choice was mainly motivated by their ability to represent joint tail dependence (upper and/or lower). After modelling the dependence between the two variables, when BI =1 and BI =0 separately, we quantify the effect of atmospheric blocking on compound extremes of ozone and temperature through the differences between probabilities derived from the cases mentioned above (BI =1 and BI =0) for each probability scenario. The impact of blocks on the joint behaviour between MDA8O3 and Tmax⁡ is reflected in the selected copula (Fig. S4). When BI =0, a large number of stations are characterised by an asymmetric dependence structure, as is the case of Joe and Gumbel copulas. The Gumbel copula is also selected in a number of stations when BI =1, but, in this case, the t copula is representative of a major number of stations. Contrary to the Gumbel and Joe copulas, the t copula belongs to the elliptical and radially symmetric copulas, but captures dependence in the extremes in both the lower and the upper tail . We further investigated the influence of blocks on the upper tail dependence parameter,λu, obtained from the chosen copulas. λu measures the tendency of concurrent extremes of MDA8O3 and Tmax⁡ exceeding the 95th percentile. According to Fig. 4, the strongest upper tail dependence occurs under the influence of blocks over north-west and central Europe (e.g. the UK, France, the Benelux countries and the north of Germany). The impact of blocks is particularly noticeable over the UK and the Benelux countries with an increase in the dependence of extremes when BI =1, which is well observed when plotting the differences between the values of λu obtained for both cases (Fig. 4b). This pattern is in agreement with the relationship obtained by Kendall's τ (Fig. S3), which shows a stronger relationship of extremes under the influence of blocking conditions.

We use three hazard scenarios (AND, OR, COND) to quantify the impacts of blocks on compound extremes of MDA8O3 and Tmax⁡. The probabilities associated with each type of hazard scenario are defined based on the domain where they are estimated and the critical region related to the probability type (see Fig. 5 for an illustrative example as shown in ).

We start analysing the impacts of blocks in the probability of concurrent events of high ozone pollution and hot days using the scenario AND (Fig. 6a and b). Although there is a very low probability of the co-occurrence of extremes when BI =0 (< 2.5 %), the presence of blocking generally increases the probability of compound events of MDA8O3 and Tmax⁡ (Fig. 6a). Under blocking conditions (BI =1), the probability of co-occurrent extremes is ∼ 20 % in most of the stations over central Europe, and ∼ 10 % over the UK (Fig. 6a). The probability of occurrence of compound events of MDA8O3 and Tmax⁡ significantly increases by more than 18 % in a large number of stations over Europe, as shown by the ΔP (Fig. 6b). Despite ozone concentrations being generally lower over the UK (Fig. S2) than over other regions, it is interesting to note that blocks seem to play a significant role in the compounding effect ozone and temperature over the UK. analysed the influence of heatwaves on air pollution in the UK, specifically Birmingham, and found that ozone levels increased by more than 50 % with high temperature. Here, we consistently show the combined effect ozone and temperature. Our results also indicate that such a combination mainly occurs under blocks, which might be due to the clear-sky radiative forcing, as pointed out by earlier work, and subsidence processes associated with the anticyclonic circulation . The stations over the north-eastern and south-western stations (i.e. Scandinavia and Spain respectively) exhibit the lowest probability of compound of extremes of MDA8O3 and Tmax⁡. As shown in Fig. 4, those stations are characterised by low or null upper tail dependence, which already indicates a weak relationship between the extremes. In addition, the distinct response of heatwaves to blocking found in northern and southern Europe is noteworthy, especially in summer . An increasing frequency of heatwaves linked to blocks has been observed over northern Europe in summer, whereas an opposite response has been seen in southern Europe . Therefore, one could expect a smaller impact of blocks on the compounding effect of ozone pollution and high temperatures in the case of the southern regions. Our results are in agreement with the study carried out by , which found a lower number of compound ozone-heat wave events in Portugal compared with the compound identified in Germany (Bavaria).

We examine the OR scenario under the assumption that blocks might enhance the probability of either high ozone pollution levels or hot temperatures, both being relevant for health impacts . As shown in Fig. 6c, the probability obtained for the OR scenario is considerably higher when BI =1, reflecting the strong impact of blocks on single extreme events. Atmospheric blocking conditions enhance the probability that either MDA8O3 or Tmax⁡ exceeded the 95th percentile by more than 40 % in a large number of stations mostly concentrated in Germany, Austria and the east of France. For the rest of the stations, the probability of extremes in the OR scenario increases by 20 %–30 % under blocking conditions (Fig. 6d). Consistent with previous works that showed the strong association of warm temperature extremes and blocking , as well as the impact of blocks on ozone pollution over some European sectors , our results show the increasing probability of temperature OR ozone pollution extremes under atmospheric blocking.

From a risk assessment perspective, the scenario COND is also of interest as it quantifies the impact of blocks of ozone pollution extremes events conditioned on high temperature. For the COND probability, both the computation domain (i.e. the joint space where the probability of exceedances is calculated) and the critical region (i.e. the region of exceedances of ozone conditioned by temperature) evolve when moving along higher temperatures; then, the probability is computed over a reduced subset (e.g. conditioned on temperature extremes) see Fig. 5 andfor further details. As illustrated in Fig. 6e and f, blocks generally enhance the probability of extremes of ozone pollution conditioned on temperature exceedances. Blocks significantly influence the compound events in the stations over north-western and central-eastern Europe, which show positive and large values of ΔP (Fig. 6f), suggesting a higher probability of ozone pollution extremes when temperature exceeds the 95th percentile. In particular, blocks lead to an increasing probability of ozone extremes given high temperatures in the UK (> 40 %). In a few number of stations over southern and north-eastern Europe, blocks did not show a significant influence in the conditional probability of extremes, with low and non-significant values of ΔP. For some of these stations, the copula selected when BI =1 is the Clayton copula (Fig. S4), which indicates a greater probability of joint extreme low values (lower tail dependence), but not in the upper tail, as shown in Fig. 6e and f. Thus, the presence of blocks is not relevant for ozone pollution exceedances that seem to occur independently of temperature extremes. In such situations, high ozone levels are less likely to be due to the enhanced local ozone production from locally emitted precursors that comes with higher temperatures , and are more likely to be due to residual ozone left over from previous episodes of enhanced local ozone production , or long-range transport of ozone produced elsewhere .

The results from the joint probabilities derived from the copulas pointed out notable spatial differences that were consistent with the analysis presented above. The impacts of blocks on the joint probabilities corresponding to the AND and OR scenarios is significant at all stations, with a major effect (in terms of the magnitude of ΔP) in those located in central Europe. The smallest impact was found at the southern and north-eastern stations for the conditional case, COND, which did not show a significant impact of blocking. Despite not considering the sub-tropical ridge in our methodology, the results from the copula analysis are in line with those of previous studies, which showed the spatial variability in the impacts of blocking.