On 26 May 2018 a mid-level low at 500 hPa was located in the Bay of Biscay (Fig. ). This induced a moderate southerly flux over France: the Bordeaux sounding at 23:00 UTC on 25 May observed a 11 m s-1 southerly wind at 500 hPa. Two positive (cyclonic) upper-level potential vorticity anomalies circulated during the day in the southerly flux observed near the tropopause (Fig. a). At the surface, a shallow pressure low around 1010 hPa in the bay headed north very slowly during the day. Pressure gradients were weak all over the western part of France. Over the south-west of France, the air was mild and humid due to the convective activity that occurred the day before and in the early hours of 26 May. Indeed, a first mesoscale convective system (MCS) evolved mainly on the Atlantic Ocean, its edges affecting the French Atlantic coast from the Basque Country to Brittany between 25 May at 22:00 UTC and 26 May at 16:00 UTC. The Bordeaux 25 May 23:00 UTC sounding, the closest available before the event, exhibited only 416 J kg-1 surface-based convective available potential energy (SBCAPE) and strong 245 J kg-1 convective inhibition (CIN) because of the decrease in temperature near the surface due to the diurnal cycle. However it recorded a 1583 J kg-1 most-unstable CAPE (MUCAPE) from the lifting of a 435 m a.g.l. (above ground level) air parcel (25 J kg-1 CIN for this parcel). This shows the presence of unstable levels above the stable nocturnal boundary layer. The sounding observed a moderate 0–6 km a.g.l. 16 m s-1 wind shear, and the hodograph exhibited a clockwise rotation of the winds in the 0–1 km a.g.l. layer, resulting in a 69 m2 s-2 helicity.

At the rear of the first MCS, thunderstorms formed in the north of Spain, west of the Pyrenees, between 06:00 and 08:00 UTC. These cells, advected by the mid-troposphere southern flux, crossed the Pyrenees and headed north towards Bordeaux. A squall-line organization of the thunderstorms appeared at around 10:00 UTC. This MCS transitioned into a bow echo at around 13:00 UTC according to radar reflectivities and crossed the west of France, moving in a south–north orientation from the Bordeaux region towards Normandy and Great Britain (Fig. a). The system was still active when it left the French territory at 23:00 UTC. The path followed by the bow echo can be seen west of France in Fig. a, marked by the area of high-speed, diverging wind gusts. Also, to the north of the MCS, a severe isolated thunderstorm, identified as a supercell in the radar imagery, developed at around 10:30 UTC and merged with the MCS at around 15:00 UTC.

The supercell produced damaging hail up to 4 cm in diameter and rain up to 22 mm in 6 min in the centre of Bordeaux. The bow echo produced mainly strong wind gusts up to 31 m s-1: 13 SWSs recorded gusts higher than 25 m s-1, and 4 of them recorded gusts higher than 28 m s-1 (Fig. a). The two systems resulted in one fatality, 1555 rescue operations and 15 000 homes being without power. This produced also local flash floods in Bordeaux and hail damage in the Bordeaux vineyards.

On 4 July 2018, a mid-level low at 500 hPa was located over the Atlantic Ocean and was extended by a trough across the Iberian Peninsula. The trough moved west towards France during the day, inducing a moderate south-westerly flux at the mid-level over the west of France: the Bordeaux sounding observed a south-westerly 12 m s-1 wind at 23:00 UTC on 3 July and south–south-westerly 16 m s-1 wind at 11:00 UTC on 4 July. An upper-level potential vorticity anomaly circulated during the afternoon over France at the rear of the convective area and on the left side of a jet stream branch (Fig. b). At the surface, a shallow low was located south-west of England, and pressure gradients were weak all over France. Isolated thunderstorms affected the south-west of France on the night of 3 to 4 July. In the morning of 4 July, at around 09:00 UTC, numerous thunderstorms developed in the north of Spain, over the Bay of Biscay and in the south-west of France. They aggregated in several multicellular systems. Embedded in one of these systems over the south-west of France, one of the storms exhibited supercellular characteristics in the radar imagery. At around 12:00 UTC, another multicellular system formed north of Spain, strengthened over the Atlantic Ocean and transitioned into a squall line. The squall line headed north-east while isolated storms formed in its southern part. It finally merged with other storms at around 17:00 UTC, and the isolated cells south of it merged in clusters, most of them evolving along with a strong MSLP gradient area (Fig. b). The sounding of Bordeaux at 11:00 UTC, before the arrival of the squall line at 14:00 UTC, exhibited large 2155 J kg-1 SBCAPE, weak 12 J kg-1 CIN and a moderate clockwise-rotating wind hodograph, resulting in 0–3 km a.g.l. 79 m2 s-2 helicity. The system generated peak wind gusts up to 34 m s-1. A large area was affected by strong wind gusts: more than 30 SWSs recorded gust speeds higher than 25 m s-1, and 11 of them recorded gusts higher than 28 m s-1 (Fig. b). Flash floods were observed, with rain rates up to 41.6 mm in 18 min. This resulted in one fatality, six injuries, 2500 rescue operations and 185 000 homes being without power. Tennis-ball-sized hail (>6 cm in diameter) was reported: in a village named Saint-Sornin, 800 houses were seriously damaged. This hail was caused by the storm identified as a supercell, in which reflectivities up to 70 dBZ were measured by radar.

On 15 July 2018, a mid-level trough at 500 hPa circulated from Portugal towards the west of France, inducing west–south-westerly winds in mid-troposphere. An upper-level potential vorticity anomaly circulated over the south-west of France during the afternoon in a north-eastward direction (Fig. c). At the surface, pressure gradients were weak over France. The sounding of Bordeaux at 11:00 UTC exhibited a SBCAPE of 1790 J kg-1 and a CIN of 0 J kg-1, showing ideal conditions for the development of surface-based convection. A sea breeze established near the Atlantic shore ,and its effects on cloud coverage were visible on satellite images at 12:39 UTC (Fig. a). A frontier appeared between the coastal band where temperatures reached 27 to 30 ∘C with clear sky and the inland area where temperatures reached 32 to 33.5 ∘C and cumulus clouds were developing. Surface observations and satellite images showed the wind convergence due to the breeze moving eastwards between 12:30 and 13:45 UTC. At around 13:10 UTC, towering cumuli turned into cumulonimbi at the south-east of Arcachon Bay (Fig. b), where SWSs measured the strongest temperature gradient with a 5 ∘C difference in a 40 km distance. The initiation happened along the wind convergence line. The first cell triggered secondary cell development west and north of it. Two main cells split and evolved in different directions: the first one headed north–north-east and the other one headed east (Fig. c). The cell evolving north–north-east caused high wind gusts up to 34 m s-1 at the Bordeaux airport at 15:05 UTC, and the temperature dropped by 15.5 ∘C in 23 min. Hail diameters of up to 3 cm were observed in the Bordeaux region under the thunderstorm. This event caused 84 rescue operations and happened in the context of public gatherings due to the football World Cup Final.

On 28 August 2018, a mid-level trough at 500 hPa concerned the west of France and moved slightly eastwards during the day, resulting in a south–south-easterly flux. At the surface a low centred north of the Iberian Peninsula deepened during the day (Fig. d), helped by a potential vorticity anomaly at upper levels and located to the left of the jet, near a diffluent exit region visible at 18:00 UTC (not shown). The sounding of Bordeaux at 11:00 UTC exhibited a SBCAPE of 1283 J kg-1 and a strong CIN of 300 J kg-1. The hodograph showed a strong unidirectional 0–6 km a.g.l. wind shear reaching 25 m s-1. Thunderstorms formed south of the low, i.e. over sea and in the north of Spain; they crossed the Pyrenees and the Bay of Biscay between 15:00 and 17:00 UTC and reached French south-western territory between 17:00 and 18:00 UTC as multicellular systems. The northern part of the MCS evolved in squall line between 18:00 and 19:00 UTC, while the southern part formed a second squall line at the rear (Fig. d). The two lines generated gusts up to 31 m s-1; 15 SWSs recorded wind gusts higher than 25 m s-1, and 6 of them recorded gusts higher than 28 m s-1 (Fig. c). This resulted in two people being slightly injured, 28 000 homes being without power, around 200 rescue operations and nine forest fires being generated by lightning. Hail up to 8 mm in diameter was reported near the coast.

SWSs are all automatic Météo-France operational weather stations sampling atmospheric parameters at a time step of 1 min. These weather stations have been installed, maintained and quality-controlled by Météo-France. The requirements in terms of accuracy for Météo-France weather stations are ±0.5 ∘C in temperature and ±6 % in relative humidity . They are taken as a reference in this study. The maximum number of weather stations measuring each physical parameter during the cases of 2018 is shown in Table . The SWS least-measured parameter over France is surface pressure, with only 19 % of SWSs equipped. The number of humidity and wind sensors equipping SWSs is respectively 3.7 to 3.8 times larger than the number of pressure sensors. Also, there are 5.4 and 5.2 times as many temperature and precipitation sensors as pressure sensors. Additional automatic weather stations, owned by Météo-France or its partners with only 5 min, 6 min or hourly measurements, are not part of the SWS dataset but are used for verification. These represent approximately 800 stations measuring temperature and 250 measuring relative humidity.

A PWS dataset made of all Netatmo automatic weather stations available over metropolitan France is used. During the case studies of 2018, a maximum of 44 115 different PWSs recorded at least one observation which is approximately 15 times the total number of professional automatic weather stations currently available at Météo-France. Among these PWSs, 95 % recorded pressure measurements, 83 % temperature and relative-humidity measurements, 27 % rain measurements, and 13 % wind measurements. On 15 July, for example, PWSs provided a total of 5 625 137 surface pressure observations, 4 837 133 temperature observations and 4 836 843 relative-humidity observations during the case study.

The metadata associated with each station are quite basic: a unique identification number, the latitude, the longitude and the altitude. The altitude of 17 % of PWSs is missing. During the year 2017, the number of PWSs recording at least once in a month increased from around 37 800 in January to around 44 000 in December, showing the rapid development of this network.

The transmission of data by these PWSs is based on radio waves between outdoor and indoor modules, on Wi-Fi between the indoor module and the personal Internet box, and then on different methods but essentially wires between the personal Internet box and the Internet service provider. At each step, technical failures or user-related shutdowns can occur. In each file transmitted by the PWS's manufacturer, 10 % to 15 % of the total number of PWSs do not provide measurements. This can be due to disconnection between station modules, disconnection of the personal Internet box, or power or Internet outages.

PWS measurements are irregular in time, whereas meteorological networks are usually designed to perform them at regular time steps. The mean time step between two measurements indicated by the manufacturer is 5 min; it may sometimes vary because PWS owners can also perform additional on-demand measurements. However, Netatmo provided, in near-real time, only 10 min time step measurements, which is the minimum time step used in this study. On average, most of the measurements are done at the minutes 5, 15, 25, 35, 45 and 55 of each hour. Also, the mean spacing between PWSs is not regular, whereas the average separation of SWSs is about 30 km. The spatial density of PWSs is highly correlated to the population density (Fig. ).

The operational weather radar network between May and August 2018 in metropolitan France is composed of 30 radars; 5 radars in the south of France are S-band radars, 20 are C-band radars and 5 are X-band radars. In this study, the French operational base reflectivity, i.e. measured at the lowest elevation angle of the radar, mosaicked from these 30 radars, is used. It has a 1 km×1 km spatial resolution and a 5 min temporal resolution, with reflectivities ranging from -9 to 70 dBZ with a 0.5 dBZ step. For every pixel in the mosaic, the maximum base reflectivity from radars distant by 180 km or less is taken. If the pixel is distant by more than 180 km to every radar, the maximum base reflectivity of radars at a distance between 180 and 250 km is taken. More details on the French radar network are given by .

For temperature, relative humidity, MSLP and surface pressure, all gridded analyses derived from observations are built at a 10 min time step and 0.01∘ resolution in latitude and longitude (≈1.1 km N–S and ≈0.8 km E–W at 45∘ N) by interpolating, for each grid point, weather stations available in the vicinity. The gridding method used is the inverse-distance weighting (IDW) with a power factor of 2. Weather stations too far away from a grid point are not used in the computation.

Every maximum range is a trade-off between the smallest possible range that limits the extrapolation of small-scale features and a larger range keeping enough stations to limit the influence of a single station over the surrounding grid points. For our cases, the maximum distance between every pair of closest SWS sensors is 46 km for relative humidity and 42 km for temperature; for the combined network of SWS and PWS after processing (see Sect. 4) it is 28 km for relative humidity and 21 km for temperature. These values are lower bounds of the maximum ranges in order to prevent inland grid points from having the value of the closest SWS even if they are not at the same location. For the sake of simplicity, an identical radius is chosen for temperature and relative humidity. Thus, for temperature and relative humidity, SWSs distant by more than 60 km are not taken into account; this radius is set to 30 km for PWSs. The choice of 60 km instead of a 50 km radius for example is done to take into account more SWSs at each grid point (for a given inland grid point, interpolation uses 9.8 SWSs on average with a 60 km radius compared to 8.6 SWSs on average with a 50 km radius for the 26 May case). For MSLP and surface pressure, SWSs distant by more than 100 km are not taken into account; this radius is also set to 100 km for PWSs. The radius is larger for pressure because it is the minimum radius for covering the entire metropolitan area of France. This is due to the small number of SWS pressure sensors and because stations with an altitude higher than 750 m are discarded (see Sect. ). A maximum of 10 SWSs and 30 PWSs are used at each grid point in the IDW, an arbitrary limit set to diminish the program execution time.

MSLP and relative humidity are directly gridded (Fig. a; method [1]). For temperature and surface pressure, because they vary strongly with altitude, a linear regression of the SWS observations used in the gridding with respect to the altitude is performed first. After that, the residuals (i.e. the difference between the values obtained by linear regression and the observations) are gridded as shown in Fig. a (method [2]) and then added to the grid derived from the linear regression. For temperature, the linear regression is adapted: to diminish the predominant weight of the low altitude SWSs over the highest SWSs, SWSs are binned in vertical layers of 100 m height. The mean temperature and the mean altitude of SWSs comprised in each layer are computed. A linear regression is then performed over these vertical layers. This choice was made to be closest to the observed temperature lapse rate rather than using a constant lapse rate.

The reference analyses, called SWS analyses, used in the following sections are built only with SWS data.

Even when the altitude of the Netatmo PWS (zPWS) is unknown, the PWS still provides a pressure value. In fact, under the name of pressure, Netatmo provides two different quantities.

It provides a MSLP (MSLPPWS) computed from the hydrostatic equation assuming a constant 15 ∘C temperature and a 0 % relative humidity at sea level if zPWS is known (83 % of cases): 1MSLPPWS=P1-ΓzPWST0-gMΓR0,where P is the surface pressure measured at the PWS (in hPa), T0=288.15 K is the sea level temperature of the International Civil Aviation Organization (ICAO) standard atmosphere, Γ=0.0065 K m-1 is the ICAO environmental lapse rate in the troposphere below 11 km, g=9.80665 m s-2 is the standard acceleration of gravity, M=0.0289644 kg mol-1 is the molar mass of dry air and R0=8.31447 J mol-1 K-1 is the ideal gas constant.

The motivation to compare Netatmo measurements to SWS analyses is to eliminate systematic errors. Some of them are due to the PWS itself, such as sensor quality or the impossibility of maintenance by design; some are due to the environmental conditions where the PWS is set up, but some are due to PWS owners who can calibrate sensors as they wish. The mobile-phone application allows users to calibrate the temperature sensor and modify the altitude, which has an influence on pressure. All sensors can be calibrated by personal requests to Netatmo.

For relative humidity, PWS time series are compared to the SWS analyses at the closest grid point. For surface pressure and temperature, because they vary rapidly with altitude, which itself varies rapidly with spatial distance in mountainous regions, the value at the PWS closest grid point may be really different of the actual PWS value. That is why PWS time series are not compared directly to the SWS analyses at the closest grid point. A more precise calculation is performed: the altitude z previously defined, considered to be the closest to the actual PWS altitude, is used in the computation. Residuals time series are taken from the closest grid point residuals. This precise calculation corresponds to SWS analyses having an accurate ground altitude at PWS locations.

For each PWS, the median of the errors between the time series derived from SWS analyses at its location (xa) and its raw PWS time series (xr) is obtained. The corrected PWS time series (xc) is computed by removing the median of the errors from the raw PWS time series: 4xc=xr-medxr-xal, with xc, xr, xa and l column vectors gathering a single PWS time series of dimension n, being equal to the number of time steps of a case. Here, l={1, …, 1}.

The choice of the median is explained by the observation of large variations in temperature, humidity or pressure due to deep convection. Because of the lower density of the SWS network compared to the PWS network, some of these variations that are actual signals affect the calculation of mean error. Using the median allows for ignoring a major part of these physical deviations while identifying systematic errors affecting PWSs. This procedure is close to the one followed by , which is performed over periods of several months.

In the following parts, all PWS time series refer to corrected PWS time series. The steps leading to these PWS corrected time series, i.e. the computation of PWS MSLP and the PWS systematic error correction, are referred to as PWS preprocessing.

Two common filters are applied to pressure, temperature and humidity. A PWS is removed if it has the same latitude and longitude as another and less than half of the measurements are available. For the computation of MSLP, PWSs with altitude higher than 750 m are discarded, as recommended by the . Then a last filter is applied in order to discard PWSs that do not provide accurate measurements.

For temperature and relative humidity, the last filter is based on the assumption that the larger the differences between PWS time series and SWS analyses during the case study, and the longer they last, the less confidence is put in PWS measurements. For each PWS, the root-mean-square error (RMSE) of PWSs temperature and relative-humidity time series (xc) against time series derived from SWS analyses (xa) is computed. It is hereafter called RMSEPWS, with n being the number of time steps: 5RMSEPWS=1n∑j=1nxc[j]-xa[j]2. The filter eliminates PWSs with RMSEPWS higher than an adaptive threshold called RMSEthresh: 6RMSEPWS>RMSEthresh. To determine the RMSEthresh, an automatic algorithm based on leave-one-out cross validation (LOOCV; see Fig. b) was built. Consider p surface stations (PWSs and SWSs) producing observations including m validation stations (SWSs only; p≥m). For a given time step j∈[1;n], the LOOCV removes one validation observation k∈[1;m]. Using p-1 observations (all except the observation k), an estimate at the removed observation location Ek(p) is computed through the gridding method described in Sect. . Then, the estimate is compared to the actual observation Sk, giving an error ϵj,k(p): 7ϵj,k(p)=Ek(p)-Sk. The process is reproduced over the m validation stations and the n time steps of the case study, giving an array of m×n errors, from which the LOOCV RMSE is computed: 8RMSELOOCV(p)=1n1m∑j=1n∑k=1mϵj,k(p)2. The lower the errors, the closer to the observations the estimates are. Thereby, RMSELOOCV(p) can be chosen as a metric evaluating the accuracy of the p surface stations from which the estimates are built.

Let x be the unknown RMSEthresh. Then, p(x) is the number of PWSs and SWSs which verify RMSEPWS≤x (m is the number of SWSs). The RMSEthresh chosen is the x that minimizes RMSELOOCV(p(x)): 9RMSEthresh=argminxRMSELOOCV(p(x)). For large values of x, p(x) tends to the total number of PWSs and SWSs remaining after the two common filters, and so RMSELOOCV(p(x)) tends toward large values because almost all PWSs are kept, including those exhibiting abnormal behaviours. For small values of x, p(x) approaches m, the number of SWSs, and RMSELOOCV(p(x)) approaches quite large values because of the small number of SWSs and their large spacing.

The resulting RMSEthresh picked up by the algorithm depends on the case, varying from 1.10 to 1.45 ∘C in temperature and from 5.5 % to 7.5 % in relative humidity.

For MSLP and surface pressure, instead of a threshold, PWSs providing suspicious measurements are eliminated one by one by an algorithm. This consists of a LOOCV using SWSs and PWSs as validation stations (m=p) that eliminates one suspicious PWS at each step s. PWSs are used in the validation dataset this time because SWS coverage is quite sparse. A one-by-one elimination is possible because only few PWS errors remain after the first three filters in pressure. The suspicious PWS is identified by computing the RMSE associated with all validation stations k, k∈[1;m], which is 10RMSELOOCV,k(p)=1n∑j=1nϵj,k(p)2. The PWS with the highest RMSELOOCV,k(p) is physically the one which disagrees the most in RMSE with all neighbour PWSs and SWSs during the case study, which is suspicious. This station is eliminated. The algorithm stops when RMSELOOCV(p) increases at step s+1 compared to step s. Physically, an increase means that a PWS which was in strong agreement with at least one neighbour station (PWS or SWS, called k′) was eliminated at step s. At step s+1, k′ captures some physical process (local low or high) but is alone in doing it, and so RMSELOOCV,k′(p) increases as well as the RMSE of some PWS around it. As a consequence, the resulting RMSELOOCV(p) taking into account all PWS contributions increases. This algorithm is well fitted for pressure because most of the errors affecting PWSs are uncorrelated, and few PWSs provide erroneous values. This will probably not work for other parameters like temperature, whose errors may be spatially correlated (errors because of direct radiation for example). Each step of quality control in MSLP is detailed in Table .

The result of PWS processing is illustrated for temperature in Fig. a, for relative humidity in Fig. b and for MSLP in Fig. . PWS measurements are compared at different time steps to the SWS analyses before and after processing. In terms of temperature (Fig. a), the distribution of departures before processing exhibits systematic positive departures with a diurnal cycle. The daily minimum of the median departures is reached in the morning between 08:00 and 10:00 UTC, after sunrise, and the daily maximum is reached in the evening or the night between 17:00 and 06:00 UTC in the 4 July case but also in the other cases not shown. In terms of relative humidity (Fig. b), the distribution of departures before processing also exhibits a diurnal cycle, with positive departures during the day and negative departures during the night in all cases. In terms of MSLP (Fig. ), the distribution of departures before processing seems to exhibit a small diurnal cycle, with departures increasing in the morning and decreasing in the evening. For all parameters, the processing shifts the distribution of departures near zero and strongly decreases the width of the interquartile range of departures. This shows the efficiency of the algorithm in diminishing departures to SWS analyses while keeping features associated with deep convection, as it was designed for (see Sect. ).

In real time, we do not have access to the complete time series. A variation in the method that could be applied in real time is using time series over a rolling period of the 24 last hours ending at the time of the analysis instead of the time series over a complete event. Then, every 10 min the automatic processing would be launched for each analysis produced. This method implies that the algorithm runs in less than 10 min, which is not the case for the current algorithm. It takes around 1 h to perform the quality control over 24 h of measurements on a computer with a central processing unit (CPU) with four cores and 16 GB of random-access memory. To increase the processing speed, one or several available computing nodes with 24 cores each could be used because the algorithm is partially parallel. Parts of the algorithm that are still sequential could be parallelized. In addition, the LOOCVs in the quality control could be modified because there are the most time-consuming parts of the algorithm. For temperature and humidity, LOOCVs provide thresholds, and for pressure the LOOCV eliminates a small number of PWSs one by one. Thus, the algorithm at a given time can use the temperature and humidity thresholds as well as the list of PWSs eliminated that were computed by the algorithm launched 1 h before.

In MSLP (Table ), a decrease ranging from 0.01 to 0.05 hPa in absolute value of the median error is observed in SPWS experiments compared to SWS experiments, depending on the case. The absolute value of the mean error decreases in three cases and remains stable in one case: it is less than 0.02 hPa for all cases in SPWS experiments. A decrease ranging from 0.32 to 0.48 hPa in the interquartile range of errors is observed for all cases in SPWS experiments compared to SWS experiments. Also, a very substantial decrease in RMSE ranging from 73 % to 77 % is observed. These results quantitatively show that adding PWS measurements in observed MSLP analyses strongly improves their accuracy. For MSLP, the number of available observations is multiplied by 134 on average over the four cases with the SPWS network compared to the SWS network.

In temperature (Table ), a positive shift of the median error ranging from 0.73 to 1.39 ∘C is observed in SPWS_raw experiments compared to SWS experiments. Bias reaches 0.74 to 1.45 ∘C, and the increase in RMSE ranges from 41 % to 72 % compared to SWS experiments. These results show the key role of processing: without this step, adding PWSs strongly decreases the quality of analyses.

For SPWS experiments, a decrease ranging from 0.00 to 0.04 ∘C in absolute value of the median error is observed compared to SWS experiments, depending on the case. The absolute value of the mean error shows no particular trend and remains less than 0.07 ∘C for all cases in SPWS experiments. This indicates that PWSs do not introduce substantial bias or shifts in the temperature distribution. A decrease ranging from 0.06 to 0.22 ∘C in the interquartile range of errors is observed for all cases in SPWS experiments compared to SWS experiments. Also, a substantial decrease in RMSE ranging from 12 % to 23 % is observed. These results quantitatively show that adding PWS measurements in temperature analyses improves their accuracies. For temperature, the number of available observations is multiplied by 11 on average over the four cases with the SPWS network compared to the SWS network.

In relative humidity (Table ), shifts of the median error ranging from -3.3 % to 2.7 % are observed in SPWS_raw experiments compared to SWS experiments. Biases reach -2.3 % to 1.9 %, and RMSEs increase from 6 % to 31 % compared to SWS experiments. These results show the key role of processing: without this step, adding PWSs strongly decreases the quality of analyses.

For SPWS experiments, the absolute value of the median error is less than or equal to 0.2 % and the absolute value of the mean error remains less than 0.6 %. This indicates that PWSs do not introduce any substantial bias or shifts in the relative-humidity distribution. A decrease ranging from 0.0 % to 1.9 % in the interquartile range of errors is observed for all cases in SPWS experiments compared to SWS experiments. Also, a substantial decrease in RMSE ranging from 17 % to 21 % is observed. These results quantitatively show that adding PWS measurements in relative-humidity analyses improves their accuracies. For relative humidity, the number of available observations is multiplied by 14 on average over the four cases with the SPWS network compared to the SWS network.

To study the sensitivity to the gridding method, we slightly modified the gridding method for the 26 May case. The power factor of the IDW was set to 1 instead of 2. We observe little sensitivity to the change of the power factor. With a power factor of 1 (respectively 2), for SPWS the MSLP RMSE equals 0.118 hPa (0.099 hPa), the temperature RMSE equals 0.877 ∘C (0.889 ∘C) and the relative-humidity RMSE equals 5.480 % (5.375 %). Decreases in RMSE reach 70 % (75 %) in MSLP, 16 % (16 %) in temperature and 17 % (21 %) in relative humidity with SPWS compared to SWS.

The contribution of PWSs in two situations is shown. Measurements of surface pressure, temperature and relative humidity of SPWS network allow computing derived quantities such as virtual temperature, Tv, the temperature of a dry air parcel which has the same density as the humid air considered, and the virtual potential temperature associated, θv, which is related to buoyancy and is identified as pertinent to track cold pools (e.g. ). θv is computed as follows: 11θv=TvP0PR0Mcp, with P, M and R0 being defined in Eq. (); Tv being defined in Eq. (); P0=1000 hPa being the standard reference pressure; and cp=72R0M being the specific heat capacity at a constant pressure.