We use temperature and precipitation observations from the E-OBS database . The data are available on a 0.1×0.1∘ grid from 1950 to 2018. As an estimate of temperature and precipitation in northern France we average these two fields over a rectangle spanning 45.5–51.5∘ N and 1.5∘ W–8.0∘ E (see Fig. ). This region also includes parts of the UK, Germany, Belgium, and Switzerland and therefore does not exactly match the studied area of . The seasonal meteorological conditions we study here are related to large-scale events, and averaging over a larger rectangle therefore seems appropriate.

We use the reanalysis data of the National Centers for Environmental Prediction (NCEP) for the analysis of atmospheric circulation. We consider the geopotential height at 500 hPa (Z500) and mean sea level pressure (SLP) over the North Atlantic region for computation of circulation analogues and a posteriori diagnostics. We used the daily averages between 1 January 1950 and 31 December 2018. The horizontal resolution is 2.5∘ in longitude and latitude. The rationale of using this reanalysis is that it covers 70 years and is regularly updated.

One of the caveats of this reanalysis dataset is the lack of homogeneity of assimilated data, especially before the satellite era. This can lead to breaks in pressure-related variables, although such breaks are mostly detected in the Southern Hemisphere and the Arctic region and marginally impact the eastern North Atlantic region.

Z500 patterns are well correlated with western European temperature and precipitation because those quantities and their extremes are related to the atmospheric circulation . Since Z500 values depend on temperature, we detrend the Z500 daily field by removing a seasonal average linear trend from each grid point. This preprocessing is performed to ensure that the analogue selection is not influenced by atmospheric trends.

The idea behind importance sampling is to simulate trajectories of a physical system that optimize a criterion in a computationally efficient way. used such an algorithm to simulate extreme heatwaves with an intermediate complexity climate model.

The procedure of importance sampling algorithms, say to simulate extreme heatwaves with a climate model, is to start from an ensemble of S initial conditions and compute trajectories of the climate model from those initial conditions.

An optimization “observable” is defined for the system. In this case, it can be the spatially averaged temperature or precipitation over France. The trajectories for which the observable (e.g. daily average temperature) is lowest during the first steps of simulation are deleted and replaced by small perturbations of remaining ones. In this way, each time increment of the simulations keeps the trajectories with the highest values of the observable. At the end of a simulation, one obtains S trajectories for which the observable (here average temperature over France) has been maximized. Since those trajectories are solutions of the equations of a climate model, they are necessarily physically consistent (given that the perturbations are small).

argue that the probability of the simulated trajectories is controlled by a parameter that weighs the importance to the highest observable values: if one trajectory is deleted at each time step, the simulation of an ensemble of M-long trajectories has a probability of 1-1/SM. Hence, one obtains a set of S trajectories with very low probability after M time increments at the cost of the computation of S trajectories.

For comparison purposes, if one wants to obtain S trajectories that have a low probability (p) observable, then the number of necessary “unconstrained” simulations is of the order of M/p, so that most of those simulations are left out. Systems like weather@home that generate tens of thousands of climate simulations are just sufficient to obtain S=100 centennial heatwaves, and the number of “wasted” simulations is very high. Therefore, importance sampling algorithms are very efficient ways to circumvent this difficulty. The major caveat of this approach is that one needs to know the equations that drive the system and be able to simulate them. We use an alternative method that does not require such knowledge of the system.

We use two SWG-based circulation analogues to simulate events of either warm temperature in December or high precipitation in spring. These SWGs resample daily weather observations in a plausible manner to simulate new weather events .

Circulation analogues are computed on SLP (or detrended Z500) from NCEP between 1950 and 2018. For each day in 1950–2018, K=20 best analogues are determined by minimizing a spatial Euclidean distance between SLP (or Z500) maps.

As explained by , the SWG randomly samples analogues by weighting the analogue days with a criterion that favours high temperatures or high precipitation. Hence, the importance sampling is summarized by the procedure of giving more weight to analogues that yield temperature (or precipitation) properties. There are two types of importance sampling for the analogues, which are illustrated in Fig. .

The two main types of analogue SWGs are described by and are as follows:

A “static” weather generator replaces each day with one of its K circulation analogues or itself. With this type of SWG, simulated trajectories are perturbations (by analogues) of an observed trajectory.

An importance sampling is applied while selecting an analogue at each time step by weighing probabilities with the variable to be optimized (temperature or precipitation). The K=20 best analogues and the day of interest are sorted by daily mean temperature or precipitation. The probability weights are determined by . If R(k) is the rank (in terms of temperature or precipitation) of day k in decreasing order and ωk the probability of day k to be selected, we set 2ωk=Ae-αR(k), where A is a normalizing constant so that the sum of weights over k is 1. The α parameter controls the strength of this importance sampling for temperature or precipitation.

The useful property of this formulation of weights is that the values of ωk do not depend on time t because the rank values R(k) are integers between 1 and K+1. The weight values do not depend on the unit of the variable either, and thus this procedure is the same for temperature or precipitation. If α=0, this is equivalent to a stochastic weather generator described by .

Combining the weights of the calendar day and the intensity of the climate variable, the probability of day k to be selected becomes 3ωk′=Ae-αR(k)e-αcalRcal(k).

The generators thus give more weight to the warmest or wettest days when computing trajectories of December temperature or spring precipitation. We thereby simulate extreme events, e.g. warm Decembers and wet springs (May to July).

The parameters of the SWG depend on the variables and the seasons to be simulated. We determine those parameters experimentally and detail them hereafter. Table  lists all parameters used for the simulation of December temperature and spring precipitation. These parameters were set after performing a number of sensitivity tests that are going to be discussed in Sect. . Table  lists all values tested for α and αcal. Most figures related to these tests can be found in the Appendix.

The procedure we follow is as follows.

Start and end day of simulations. For each year from 1950 to 2018, 1000 simulations are started independently for temperature in December and precipitation in spring. The temperature simulations start on 1 December and end on 31 December. Precipitation simulations start on 1 April and end on 31 July. This results in 68 000 independent simulations of December temperatures and spring precipitation.

The winter preceding the 2016 crop loss was abnormally warm, with only a few cold days. Here, cold days are defined as days with daily maximal temperatures between 0 and 10 ∘C. This December was the hottest in the observational record and also the December with the fewest cold days.

Figure a shows the observed averages of daily maximal temperatures and the results from static and dynamic SWG simulations. The observed December temperatures fluctuate around 6 ∘C, with a small warming trend of 0.2 ∘C per decade over the whole time series (p value =0.03). Simulations from the static SWG are consistently around 3.5 ∘C warmer and follow the year-to-year variability of the observations. With an average of 12 ∘C, the dynamic SWG simulations are significantly warmer than the static SWG simulations, and inter-annual variability is strongly reduced. This is to be expected as the dynamic SWG evolves freely from the starting day and is therefore less bound to each year's circulation.

In years with higher December temperatures, the number of cold days with maximal temperatures between 0 and 10 ∘C is reduced (see Fig. b). Over the period 1950–2018 no trend in the number of cold days is observed and the number of cold days fluctuates around 25 d. As the SWG simulates warmer Decembers the number of cold days is on average 8 d lower in the static SWG and 16 d lower in the dynamic SWG. Nearly half of the simulations of the dynamic SWG thus have fewer cold days than what was observed in December 2015.

December 2015 was unprecedented in terms of missing cold days, and we simulate a number of warm Decembers with even fewer cold days. To estimate the probability of such an extreme December, we fit a beta-binomial distribution to the observations and find that 2015 was a 1-in-4000-year event and that 25% of our dynamic SWG simulations are 1-in-1000-year events or even rarer (see Fig. ).

As shown in Fig. , December 2015 was characterized by a persistent anticyclonic circulation with its centre over the Alps. The circulation in the coldest December (1969) was the opposite of 2015, with negative Z500 anomalies over Europe and positive anomalies over the Atlantic. In 2008, the December with most cold days in the observations, the data resemble 1969 but have less pronounced anomalies.

For all example years, the circulation in the static SWG simulations exhibits the same features as the observed circulation. The dynamic SWG always simulates high-pressure anomalies over France irrespective of the starting conditions. These anomalies are, however, more pronounced in 2015 where the starting circulation favours the anticyclonic pattern over France.

The simulations of warm Decembers are most sensitive to the weighting of the calendar date. If this parameter is chosen too loosely, simulations would include days from other seasons, which are generally warmer. As shown in Fig. , for αcal≥6 over 70% of all days in the simulations are sampled from the November–February period. Increasing the weighting of the calendar day further does not show a significant effect.

The simulations are also sensitive to the weighting of daily maximal temperatures α (Fig. ). For α≥0.75 we simulate a large number of Decembers that are more extreme than 2015.

Finally, the choice of geopotential height or mean sea level pressure to classify circulation analogues does not influence the simulations (see Fig. ).

An extremely wet period from April to July 2016 followed the warm December in 2015, with an average precipitation of 2.7 mm per day and 332 mm for the whole period. This is more than the long-term 75th percentile, but it is topped by some years including 1983, 1987, and 2012.

Figure  shows the daily mean precipitation for April–July periods over 1950–2018. Accumulated April–July precipitation fluctuates around 256 mm with a strong year-to-year variability. Over the observed period no trend is detected.

Simulations from the static weather generator (blue boxplots in Fig. ) also show a strong inter-annual variability but have significantly larger amounts of precipitation. The average seasonal precipitation for all simulations and all years is around 487 mm–190 % of the observed average. Single simulations even reach daily mean precipitation of 6 mm for April–July, which is 3 times as high as the observed precipitation in 1983.

April–July periods simulated by the dynamic SWG are even wetter than the simulations of the static SWG, with an average seasonal precipitation of 590 mm. As expected, the inter-annual variations are smaller in the dynamic SWG simulations than in the static SWG simulations because the dynamic SWG evolves freely, with the starting conditions their only link to the observed circulation.

We estimate the return periods of our simulated events by fitting a normal distribution to the observed April–July precipitation events. As we average over a quite large region and over 4 months, a normal distribution represents the observations well (even though the analysed variable is precipitation). We find that the 2016 April–July period was a 1-in-17-year event, while the majority of our SWGs simulations are 1-in-10 000-year events.

In April–July 2016, the atmospheric circulation was characterized by a moderate low-pressure anomaly north of France and north of the Azores (Fig. a). The North Atlantic Oscillation (NAO) index switched from slightly positive to negative in May and remained negative until the end of June ().

We next analyse the large-scale atmospheric circulation patterns that characterize our SWG simulations by comparing them to a few examples of observed events. Figure a–d shows the mean sea level composites of 2016, the driest (1976), the median (1986), and the wettest (1983) April–July periods. The main feature in the median event (Fig. c) is a low pressure anomaly north-westward of the British Isles. The wettest event (Fig. d) is characterized by a strong dipole over the North Atlantic with low pressure in the east and high pressure in the west. In the driest event (Fig. b) this dipole is reversed and slightly shifted to the east.

For all four events, the static SWG tends to create events with stronger low pressure anomalies over northern France (Fig. e–h). Similarly, the simulations from the dynamic SWG all show a strong low-pressure anomaly over northern France (Fig. i–l). For the dynamic SWG simulations, even in 1976, which was the driest April–July period, a low-pressure anomaly is simulated for northern France where a high-pressure system had been observed. In the static SWG, the high-pressure anomaly is relocated to the west, also leading to a low-pressure anomaly over northern France.

Besides a general tendency towards low-pressure anomalies over northern France, the 2016 April–July period was characterized by an increased daily pressure variability west of France (compare Figs. a and c). This indicates an enhanced storm track activity downstream of our region of interest and could explain the increased precipitation observed in 2016. In contrast to the persistent anticyclonic anomaly that led to a continuously warm December in 2015, the wet April–July period was favoured by a number of storms passing over northern France.

Our simulations of April–July periods combine 5 d chunks of observed weather into one coherent time series. By using 5 d chunks instead of combining single-day observations, we constrain our simulations to observed day-to-day variations that appear to be crucially important for precipitation events. This ensures that in our simulations storms predominantly travel eastwards and that the moisture transport in the simulations is reasonable – at least during the 5 d in question (see the animated .gif files in the Supplement).

Sensitivity tests indeed show that simulations where a new analogue is chosen every day result in significantly higher precipitation, with 7 mm per day for the dynamic SWG simulations (see Fig. ). The amount of precipitation steadily decreases with the length of the observed chunks that are assembled by the SWGs (ndays). This is to be expected, as with longer assembled chunks and fewer analogue choices the simulated weather events resemble the observations more and more. There is an especially strong decrease in simulated precipitation from 1 to 3 d, which suggests that when analogues are chosen more frequently than every third day potentially unreasonable weather events are created. Note that taking 5 d windows is a heuristic choice and that window sizes between 4 and 7 d give similar results.

The simulations are by definition sensitive to the weighting of the amount of precipitation α. As shown in Fig. , with a relatively small weight of 0.1 most dynamic simulations already bring more precipitation than what was observed in 2016. This could be due to the length of the simulations: it is rather unlikely that extreme weather endures over 4 months. However, with a weak weighting of wet weather simulations can already result in a long-lasting consistent wet periods. This increase in precipitation saturates after α≈0.5, and increasing α further has no effect on the final results.

As for the other free parameters of the SWG, this sensitivity test does not directly justify the choice of the parameter α. It instead gives guidance on the values that would be appropriate choices for our application. In the end the parameter is heuristically chosen considering the trade-off between creating high-precipitation events and keeping as much randomness as possible in our simulations.

As shown in Fig. , the weighting of the calendar day has limited influence on the amount of precipitation in northern France simulated by our SWGs.

For precipitation in northern France the weighting of the calendar day is less relevant as there is no pronounced seasonal cycle in precipitation (see Fig. ).

Finally, one feature in the simulations of April–July deserves some more attention: for both static and dynamic SWG simulations precipitation is exceptionally high in 1994 and 1998. Although observed precipitation in these years was relatively high, this cannot explain the amount of precipitation in the simulations. One explanation for these outlier years could be a loop in the simulations leading to an excessive repetition of the same (wet) sequence of days. As shown in Fig. , in 1998 one date is indeed repeated 10 times in both the static and dynamic weather generator. In most other years, repetitions of single dates are rare. As our results do not rely on simulations of single years, this feature does not affect the overall findings of the study.

These simulations show that there are many possible April–July periods that would be significantly wetter than what was observed in 2016 and also wetter than the observed record precipitation (1983).