2.1 The Doppler W-band radar RASTA during the HyMeX-SOP1

The airborne cloud radar RASTA is a monostatic Doppler multi-beam antenna system operating in the W band at 95 GHz (Bouniol et al., 2008; Protat et al., 2009; Delanoë et al., 2013). The aircraft platform used is the French Falcon 20 research aircraft from the SAFIRE unit (Service des Avions Français Instrumentés pour la Recherche en Environnement). RASTA has six Cassegrain antennas to measure the reflectivity and radial velocity in three non-collinear directions above and below the aircraft. The maximum unambiguous distance is 15 km with an unambiguous velocity of 7.8 m s⁻¹ (the pulse repetition frequency equals 10 kHz).

After processing, the Doppler velocities of the three upward-looking and downward-looking antennas are combined to retrieve the three components of the wind field above and below the aircraft (Bousquet et al., 2016). The measurements are collected at a time resolution of 250 m s⁻¹ (i.e. 1.5 s between two measurements of the same antenna) and at a vertical resolution of 60 m. In addition, this study takes advantage of the reflectivity measurements collected by the nadir- and zenith-pointing antennas. The zenith-pointing antenna is slightly less sensitive than the nadir-pointing antenna (−26 dBZ versus −27 dBZ at 1 km).

Therefore, this unique instrument allows for the recording of the microphysical and dynamic properties of clouds in the vertical at a high resolution of 60 m and quasi-continuously in time (≈1.5 s) during the flights. In particular, during the HyMeX-SOP1, RASTA collected data during 18 flights in stratiform (72.6 %), convective (14.3 %) and clear-sky (13.1 %) columns over land, sea and complex terrain (Borderies et al., 2018). RASTA flight tracks during the HyMeX-SOP1 are represented by the black lines in Fig. 1. Further details about the RASTA configuration during the HyMeX-SOP1 are given by Bousquet et al. (2016).

Figure 1The Falcon 20 flight tracks (black lines) during the HyMeX first special observing period over the AROME-WMed domain. The Falcon 20 flight track during IOP7a (flight 15) is indicated by the red line. IOP7a case study is delimited by the red box. The altitude of ground above sea level (in metres) is represented by the colour shades. The rain gauges are represented by the blue markers. The black box shows the domain used for the impact study in Sect. 6.1.

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2.2 The AROME-WMed NWP model

This study is performed with a special version of the Météo-France operational kilometre-scale NWP model AROME (Seity et al., 2011), named AROME-WMed (Fourrié et al., 2015). AROME-WMed, which covers the entire northwestern Mediterranean Basin, was specially designed for the HyMeX-SOP1 and ran in real time to plan the airborne operations, especially in the mesoscale convective systems. The AROME-WMed domain is displayed in Fig. 1. AROME-WMed runs at a horizontal resolution of 2.5 km with 60 vertical levels ranging from approximately 10 m a.g.l. to 1 hPa. The deep convection is explicitly resolved and the microphysical processes are governed by the ICE3 one-moment bulk microphysical scheme (Pinty and Jabouille, 1998). Six water species are predicted by AROME-WMed (water vapour, rain, cloud liquid droplets, snow, pristine ice and graupel). The particle size distributions (PSDs) are expressed as generalized gamma distributions multiplied by the total number concentrations. PSDs are reduced to exponential distributions for snow, graupel and rain.

The analyses of the ARPEGE global operational NWP model are used to provide boundary conditions. AROME-WMed has a 3 h 3D variational (3DVar) data assimilation system (Brousseau et al., 2014) based on an incremental formulation (Fischer et al., 2005). The control variables of this system are temperature, specific humidity, surface pressure, vorticity and divergence. The resolution of the analysis grid is the same as that of AROME-WMed. Following the results of Brousseau (2012), the incremental analysis update (IAU; Bloom et al., 1996) is not used for the 3DVar assimilation scheme. Background error covariances were computed using a period characterized by convective systems in October 2010 over the northwestern Mediterranean region (Fourrié et al., 2015). Every 3 h, an analysis is computed by using all observations available within a ±1 h 30 min assimilation window and a 3 h forecast is produced to provide a background for the next cycle. The assimilation system ingests a wide variety of observations from satellite, ground-based GPS, aircraft, radiosondes, drifting buoys, balloons and wind profilers, automatic land and ship weather stations, and ground-based precipitation radars of the French network ARAMIS (reflectivity and radial velocity). The purpose of this study is to assess the impact of the assimilation of RASTA data in addition to this already dense observing network.

2.3 RASTA data pre-processing

RASTA data are discarded between 250 m above and 250 m below the aircraft, which is the minimal measuring range of the zenith- and nadir-pointing antennas. Ground clutter is also removed. To reduce observation and representativeness errors, RASTA data are interpolated in the model vertical and horizontal resolutions. For the reflectivity measurements, this interpolation is done by taking the average value (in mm⁶ m⁻³) of all data available along the aircraft track within a box of 2.5 km length between the two half model levels surrounding each model level. From a given range from the radar, when the aircraft roll and/or pitch angles are greater than a threshold (|roll| >7^∘ at 10 km range), some measurements might come out of the grid box of interest (Borderies et al., 2019). Therefore, these data are removed from the interpolation. The same interpolation is done for the retrieved horizontal wind component except that a median filter is employed. Indeed, applying a median filter instead of averaging allows us to reduce the influence of outliers, due to the difficulty of having high-quality measurements for airborne Doppler radar (Bosart et al., 2002).

After this pre-processing, a thinning is applied to RASTA data to decrease observation density and satisfy assumptions about observation error covariances, which are supposed to be 0 dB². It is particularly true for measurements made by different instruments, which have independent physical errors. However, this hypothesis might be no more valid if the observations are collected very close to each other by the same instrument. Applying a thinning to the observations is therefore necessary for having satisfactory assumptions about observation error covariances (Rohn et al., 2001; Liu and Rabier, 2002). Therefore, RASTA data are assimilated every 3 time steps, which is equivalent to a distance of approximately 5 to 9 km depending on the aircraft speed. The data are not thinned vertically because the vertical forecast error covariances are less marked than the horizontal ones (Brousseau et al., 2011) and it is thus not useful to apply any thinning in that case (Jacques and Zawadzki, 2014).

3.1 The 1D+3DVar assimilation method

Here, we employ the 1D+3DVar assimilation method (Caumont et al., 2010; Wattrelot et al., 2014) used operationally to assimilate ground-based precipitation radar data in AROME. This data assimilation technique allows us to shift a pattern that was well simulated by the model but at a wrong location. It relies on the ability of the model to create consistent moisture and reflectivity profiles. Indeed, cloudy areas are generally associated with relative humidity close to the saturation and high reflectivity values. This method is particularly well suited for vertically pointing radars because the first step of the assimilation method is based on the differences between different vertical profiles of reflectivities. For instance, since March 2016, this assimilation method is operationally employed to assimilate vertical profiles of DPR reflectivity data in the Japanese kilometre-scale NWP model (JMA-NHM) (Ikuta, 2016).

The first step consists of a 1-D Bayesian retrieval of the best estimate of relative humidity (RH) profiles, named hereafter pseudo-observations (PO), given the observed vertical profile of reflectivity Z_o. For each observed column of reflectivity Z_o, the corresponding vertical profile of RH pseudo-observation $y_{\mathrm{PO}}^{\mathrm{RH}}$ is given by

with

where

• subscript i denotes the index of the model profile in the vicinity of the observed profile of reflectivity;

• $x_i^{\mathrm{RH}}$ is the vertical column of relative humidity from the model background;

• H_z(x_k) is the simulated reflectivity (in dBZ) at the model level k, given the model state x_k; H_z being the forward operator;

• n_o is the number of valid observed reflectivity data in the column;

• b_k is the bias correction value used at the altitude k (in dB), described in Sect. 3.2; and

• σ_o is the standard deviation of observation and forward operator errors (in dB).

The W-band reflectivity forward operator H_z described by Borderies et al. (2018) is used to simulate the reflectivity. It is consistent with the ICE3 one-moment microphysical scheme of AROME and takes the hydrometeor contents of the five hydrometeor species (rain, snow, graupel, cloud liquid water and pristine ice), temperature, pressure and relative humidity as input parameters. The T-matrix method (Mishchenko et al., 1996) is employed to compute the single scattering properties. Following the results of Borderies et al. (2018), the graupel axis ratio is set to 0.8, snow axis ratio to 0.7 and pristine ice axis ratio to 1. The forward operator returns the simulated reflectivity at each range gate from the radar and accounts for hydrometeors and water vapour attenuation.

According to Eq. (1), for each observed vertical profile Z_o, the vertical column of RH pseudo-observation is a linear combination of the neighbouring RH profiles taken from the model background $x_i^{\mathrm{RH}}$.

The $x_i^{\mathrm{RH}}$ neighbouring profiles are located in a 160 km wide square centred on the aircraft location. For the AROME-WMed model, this size is sufficient to reduce the effects of spatial mismatches between model and observations (Borderies et al., 2018) and to gather a database of $x_i^{\mathrm{RH}}$ which are consistent with the meteorological situation. In addition, the $x_i^{\mathrm{RH}}$ profiles would become less representative with a larger size since meteorological environments can change over ≈100 km.

In Eq. (1), the $x_i^{\mathrm{RH}}$ profiles are weighted by a function (J_PO) of the difference between the observed Z_o and simulated H_z(x_i) column of reflectivities (cf. Eq. 2). Thus, larger weights are given to the neighbouring columns that most closely resemble the observations. To ensure equivalent weights regardless of the number of altitude levels used for each neighbourhood profile, the square difference in Eq. (2) is divided by the number of valid data over the observed column of reflectivity.

The square difference is also divided by the observation error variance ${\mathit{\sigma }}_{\mathrm{o}}^{\mathrm{2}}$. A small σ_o will favour the neighbouring columns that most closely resemble the observation. However, if there is no simulated profile of reflectivity which is close enough to the observed one, there will be no retrieval since the weight tends towards a value close to 0. Hence, a small σ_o either leads to an accurate retrieval or to no retrieval at all. On the other hand, a large σ_o would give similar weights and smooth the neighbourhood $x_i^{\mathrm{RH}}$ profiles, regardless of the extent to which they resemble the observed profile of reflectivity (Caumont et al., 2010). Therefore, a sensitivity study is performed in Sect. 3.3 to σ_o values.

The Bayesian retrieval is not applied in the case of clear sky, i.e. when all the reflectivities over the whole vertical column are below the radar sensitivity in both the simulations and the observations. However, if the simulations indicate cloud or precipitation, the closest “clear-sky” profile in the vicinity of the radar is selected for the retrieval.

In the second step of the 1D+3DVar assimilation approach, the retrieved vertical profiles of relative humidity pseudo-observations $y_{\mathrm{PO}}^{\mathrm{RH}}$ are assimilated in the 3DVar assimilation system of AROME-WMed, like any other conventional observations.

3.2 Bias correction

The Bayesian retrieval can also be applied to other variables using the same weights as those from the retrieval of RH profiles (in Eq. 2), for example to retrieve reflectivity pseudo-observations $y_{\mathrm{PO}}^Z$:

The reflectivity pseudo-observation $y_{\mathrm{PO}}^Z$ can be used as an indicator to evaluate the quality of the 1-D Bayesian retrieval and to estimate the biases that can arise between observations and simulations. Indeed, in data assimilation it is necessary to have unbiased quantities and to remove these systematic errors (Janisková, 2015; Okamoto et al., 2016). These biases can arise from the observations, the ICE3 microphysical scheme and/or the forward operator formulations. Janisková (2015) showed that the biases between observations and simulations depend on the temperature and on the altitude. Since this study is focused on one specific area of interest during the same season, the bias is mainly a function of the altitude. The bias is also a function of the error standard deviation σ_o. Indeed, while a small value of σ_o favours the simulated columns that most closely resemble the observations, a larger value of σ_o smooths the simulated reflectivity profiles.

Therefore, a bias b was determined from the statistics between Z_o and $y_{\mathrm{PO}}^Z$ using the altitude and the error standard deviation σ_o as predictors. Calculations were performed for all flights during the HyMeX-SOP1 every 4 time steps. The background states of a control (CTRL) experiment were used as a reference to simulate the reflectivity pseudo-observations and to estimate the biases. The CTRL experiment was run during a 45-day cycled period from 00:00 UTC 24 September 2012 – which is the day when the Falcon 20 first flew during HyMeX-SOP1 – to 5 November 2012, after the last flight. It includes all the observations that are operationally assimilated (see Sect. 2.2).

The bias between RASTA observations and the reflectivity pseudo-observations is depicted in Fig. 2 as a function of the altitude for different values of σ_o (see legend). Calculations were only performed if both the observation and the reflectivity pseudo-observation are above the radar sensitivity. The number of observations used for the calculations is also shown in Fig. 2b. This number is smaller for small values of σ_o (red curve) because it constrains the amount of retrieved profile of reflectivity pseudo-observations to only those which most closely resemble the observations.

Figure 2Bias (a) between RASTA reflectivity and the reflectivity pseudo-observation as a function of the altitude for different values of error standard deviation σ_o (see legend box). Panel (b) shows the number of observations used for the calculations as a function of the altitude for the different values of error standard deviation σ_o.

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Figure 2 shows that the bias increases with the altitude, which is consistent with the existence of model biases in cloudy areas in the ICE3 microphysical scheme of AROME-WMed (Borderies et al., 2018; Taufour et al., 2018). Figure 2 highlights the fact that, because of the smoothing effect, the bias increases with the error standard deviation σ_o. Indeed, at approximately 6 km altitude, the bias can reach up to 6 dB if σ_o equals 9 dB, and only ≈1.5 dB if σ_o equals 2 dB.

The effect of the bias correction is shown in Fig. 3, in which a contoured frequency by altitude diagram (CFAD) of the differences between the observed reflectivity and the bias-corrected reflectivity pseudo-observations are shown for a σ_o of 2 dB. The residual bias is indicated by the black line. Figure 3 demonstrates that, after applying the bias correction in Eq. (2), the residual bias is close to 0 dB except above an altitude of approximately 10 km, which is probably due to the smaller number of points used to calculate the bias correction. As explained by Janisková (2015), the use of additional predictors, such as temperature or hydrometeor contents, could lead to an improvement in the bias correction at higher altitude.

Figure 3Contoured frequency by altitude diagram (CFAD) of the differences between the observed reflectivity and the bias-corrected reflectivity pseudo-observations. The residual bias after applying the bias correction is indicated by the black line.

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3.3 Observation error within the Bayesian inversion

As explained in Eq. (2), the quality of the 1-D Bayesian retrieval relies on the specification of standard deviation of observation and forward operator errors σ_o (in dB). In-flight water vapour mixing ratio measurements, r_o, are available at the flight level and can be used to estimate σ_o and to evaluate the quality of the retrieval. These data present the advantage of being completely independent from the retrieval and they allow the evaluation of the humidity pseudo-observations which will then be assimilated in the 3DVar assimilation system of AROME-WMed.

The 1-D Bayesian retrieval is applied to the CTRL background states for error standard deviations σ_o ranging from 0.6 to 9 dB. The bias correction, which has been calculated for each σ_o in Sect. 3.2, is applied in Eq. (2). The retrieved pseudo-observations, r_m, of water vapour mixing ratio at the flight level are then compared with the in-flight measurements, r_o, over 32 flights of the HyMeX SOP1. The comparison is done as follows. First, a manual data quality control is applied to in situ humidity observations in order to remove the poor-quality measurements that can arise from instabilities or periods of malfunctioning during the flights. After this quality control, 24 flights out of 32 remain. Second, water vapour mixing ratio measurements are averaged over 12 time steps to reduce observation noise and representation errors.

Figure 4 shows the standard deviations (Fig. 4b) and biases (Fig. 4a) between the observed in-flight water vapour mixing ratio and the retrieved ones (red curve) as a function of the error standard deviations σ_o. The standard deviations between in-flight measurements and the background water vapour mixing ratio are also represented by the black data points.

Figure 4Standard deviations (b, in g kg⁻¹) and biases (a, in g kg⁻¹) of the water vapour mixing ratio differences between in-flight measurements, r_q, and the retrieved ones, r_m (red), as a function of the error standard deviations σ_o (in dB). The standard deviations and biases before applying the 1-D Bayesian retrieval are represented by the black data points.

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The standard deviation values in Fig. 4 demonstrate that the retrieved water vapour mixing ratios are always in better agreement with the in-flight measurements compared to the background state. This improvement highlights the ability of the 1-D Bayesian method to retrieve humidity fields that are closer to independent observations. The variation in the standard deviation indicates the existence of an optimal value of σ_o of approximately 2 dB. Indeed, below 2 dB, the standard deviation increases with decreasing σ_o. This is due to the tendency of the retrieval to be more selective for small values of σ_o, which results in using the background state instead of applying the retrieval. On the contrary, above 2 dB, the standard deviation increases with σ_o. Indeed, a large σ_o increases the number of successful inversions, but smooths them to produce the resulting humidity pseudo-observations. Finally, it should be noted that the bias is also improved with a σ_o of 2 dB (Fig. 4a). Hence, we decided to use an error standard deviation σ_o of 2 dB for the rest of this study.

Figure 5Time–height cross section of the reflectivity observed by RASTA (a), simulated from the background (b) and pseudo-observations (c). The differences between the RH pseudo-observations and the relative humidity from the background state are shown in (d). Aircraft altitude is indicated by the black line.

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5.1 1-D Bayesian retrieval

As explained in Sect. 3.1, the first step to assimilate the reflectivity consists of a 1-D Bayesian retrieval of RH pseudo-observation profiles, given the vertical profile of reflectivity observed by RASTA. Since no direct RH observations are available with such a high vertical resolution as the one of RASTA data, the method is validated by comparing the reflectivity pseudo-observations $y_{\mathrm{PO}}^Z$ with RASTA Z_o observations. Figure 5 shows RASTA observations (interpolated on the vertical grid model; (a), the simulated profile of reflectivities from the background (b) and the retrieved reflectivity pseudo-observations (c). The differences between the RH pseudo-observations and the background RH profiles are also shown in panel (d). Differences are displayed in red (blue) if RH pseudo-observations are larger (smaller) than the background.

Figure 5 highlights the capability of the 1-D Bayesian method to retrieve profiles which are in better agreement with the observations than the background. For example, at approximately 06:30 UTC, the observation profiles indicate clouds below an altitude of 6 km, as opposed to the simulated profiles from the background which only indicate clear-sky profiles. This has been rectified in the reflectivity pseudo-observation profiles, and in the corresponding RH pseudo-observation profiles. Indeed, the RH pseudo-observation values are larger than the background RH values (red values in d), and are thus more representative of the presence of cloud. Inversely, at approximately 06:25 UTC, the Bayesian retrieval has been able to remove the low-level clouds present in the background, and to add clouds above an altitude of about 4 km. Between 06:50 and 07:00 UTC, the reflectivity pseudo-observations are also in much better agreement with the observations than the background. The corresponding RH pseudo-observations values are also larger than the background, which is consistent with the fact that larger RH values are usually associated with larger reflectivity values. Hence, Fig. 5 demonstrates the ability of the Bayesian retrieval to pick up vertical profiles in the neighbourhood which are more consistent with the observations. Indeed, this retrieval successfully dried areas associated with low reflectivity values, and moistened areas associated with high reflectivity values. These retrieved RH pseudo-observation profiles are then assimilated in the 3DVar assimilation system of AROME-WMed in the Z^IOP7 and ZV^IOP7 experiments.

5.2 Impact on analyses

Figure 6 shows (from the top to the bottom panels), the relative humidity for the pseudo-observations, the CTRL^IOP7, the Z^IOP7, the V^IOP7 and the ZV^IOP7 analyses. Similarly, Fig. 7 represents the wind speed for the observations and the different experiments. The four different analyses were computed using the same background state.

Figure 9Temperature N_T (a), surface pressure N_p (b), humidity N_v (c) and kinetic N_s (d) contribution terms of the exergy distance as a function of the altitude for the Z^SOP1 (red curve), V^SOP1 (blue curve) and ZV^SOP1 (black curve) analyses.

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As shown in Figs. 6a and 7a, the number of RH pseudo-observations which have been assimilated is larger than the number of RASTA wind data. Indeed, contrary to RASTA wind data, the reflectivity is also assimilated in the case of clear sky. Besides, airborne Doppler velocity measurements are contaminated by the aircraft motion (roll/pitch/drift angles, ground speed, etc.). Therefore, because of the difficulty to have high-quality measurements (Bosart et al., 2002), RASTA wind data have been more frequently rejected (cf. between 06:42 and 06:50 UTC in Fig. 7). In addition, contrary to the W-band reflectivity measurements, RASTA horizontal wind components are obtained through a retrieval, which might also explain the smaller number of assimilated horizontal wind data. Finally, since RH pseudo-observations are assimilated in the same way as radiosonde observations are (cf. Sect. 4), they are rejected above an altitude of approximately 9 km because the values are very small.

Compared to the RH pseudo-observations in Fig. 6, RH is overestimated in the CTRL^IOP7 and in the V^IOP7 analyses. Except at approximately 8 km of altitude, the RH profiles are much more similar to the RH pseudo-observations in the Z^IOP7 and ZV^IOP7 analyses. Conversely, in Fig. 7, the V^IOP7 and ZV^IOP7 analyses are in much better agreement with RASTA wind observations compared to the CTRL^IOP7 and the Z^IOP7 analyses. Figure 6 shows that the V^IOP7 analysis is very similar to the CTRL^IOP7 one in terms of humidity. Similarly, in Fig. 7, the Z^IOP7 analysis is very similar to the CTRL^IOP7 one in terms of wind speed. Therefore, the assimilation of RASTA wind data (respectively RH pseudo-observations) does not impact the humidity (respectively wind) field in the analysis, probably because wind and humidity are not highly correlated in the assimilation process through the background error covariances. However, the joint assimilation of the RH pseudo-observations with RASTA wind data (ZV^IOP7 experiment) results in a positive impact in terms of both the wind and the humidity fields.

5.3 Impact on rainfall forecasts

Figure 8 shows the 12 h accumulated rainfall between 06:00 and 18:00 UTC 26 September 2012 (IOP7a) for radar observations, the CTRL^IOP7, the Z^IOP7, the V^IOP7 and the ZV^IOP7 experiments.

First, the predicted rainfall pattern is well reproduced in the four different experiments. As shown by Borderies et al. (2019), the maximum rainfall accumulation is overestimated in the CTRL^IOP7 experiment (≈142 mm versus 93 mm in the radar observations), but is better reproduced in the Z^IOP7 experiment (130 mm). In addition, the assimilation of RH pseudo-observations jointly with RASTA wind data in the ZV^IOP7 experiment also results in a decrease (133.5 mm) of the predicted maximum rainfall accumulation. Finally, the experiment in which RASTA wind data are assimilated alone in the V^IOP7 leads to the better agreement with the radar observations. Indeed, the maximum rainfall forecast accumulation has been reduced to only 118 mm.

6.1 Impact study using an exergy-distance-based approach

The moist-air available-enthalpy (exergy) distance (Marquet et al., 2019) is first briefly described, and then calculated to measure the relative impact of the assimilation of RASTA data on analyses and short-term forecasts.

6.1.1 The moist-air available-enthalpy (exergy) distance

Traditionally, the impact of the assimilation of a new observation type and its synergistic effect with other observations are assessed through verification scores of a long data-denial assimilation experiment (Storto and Randriamampianina, 2010). These approaches are very expensive from a numerical point of view. The new type of observation needs to be assimilated in a large number of analysis cases, which is not affordable for airborne radar measurements since the availability of the new observations depends on the aircraft flights. By contrast, energy-based approaches (Ehrendorfer et al., 1999; Marquet et al., 2019) are cost-effective methods for evaluating the impact of the assimilation of a new observing system in a NWP model. The idea is to combine thermodynamic variables of the atmosphere into a model space-based measure (Storto and Randriamampianina, 2010), which avoids the use of long data-denial experiments and adjoint-based methods that rely on strong linearity assumptions which are not valid at the convective scale. These approaches provide a measure of the relative impact of the observations on the analysis and forecast fields. For example, Storto and Randriamampianina (2010) employed the moist total energy norm (MTEN; Ehrendorfer et al., 1999) to evaluate the loss of quality in the forecasts when an observation type is not assimilated. A similar methodology was employed by Fabry and Sun (2010) to characterize model errors in winds, temperature, humidity and precipitation.

Based on results of Marquet (1993), Marquet et al. (2019) defined a moist-air available-enthalpy (exergy) distance, which provides a more general and comprehensive metric between a perturbed thermodynamic state (here the RASTA experiments) and a reference one (here the CTRL experiment). It is defined by the integration over the 2-D domain of the sum of four quadratic terms in horizontal wind components U,V (N_s), temperature T (N_T), surface pressure p_s (N_p) and water vapour mixing ratio r_v (N_v). The four contribution terms of the exergy distance are then given by

$$\begin{array}{}\text{(4)}& {\displaystyle }{N}_T& {\displaystyle }=\underset{D}{\int }{\displaystyle \frac{C_{p_{\mathrm{d}}}{T}_{\mathrm{r}}}{\mathrm{2}}}{\displaystyle \frac{{\left(T^{\mathrm{CTRL}}-T^i\right)}^{\mathrm{2}}}{{\stackrel{\mathrm{‾}}{T^{\mathrm{CTRL}}}}^{\mathrm{2}}}}\mathrm{d}D,\text{(5)}& {\displaystyle }{N}_p& {\displaystyle }=\underset{D}{\int }{\displaystyle \frac{R_{\mathrm{d}}{T}_{\mathrm{r}}}{\mathrm{2}}}{\displaystyle \frac{{\left(p_{\mathrm{s}}^{\mathrm{CTRL}}-p_{\mathrm{s}}^i\right)}^{\mathrm{2}}}{{\stackrel{\mathrm{‾}}{p_{\mathrm{s}}^{\mathrm{CTRL}}}}^{\mathrm{2}}}}\mathrm{d}D,\text{(6)}& {\displaystyle }{N}_{\mathrm{v}}& {\displaystyle }=\underset{D}{\int }{\displaystyle \frac{R_{\mathrm{v}}{T}_{\mathrm{r}}}{\mathrm{2}}}{\displaystyle \frac{{\left(r_{\mathrm{v}}^{\mathrm{CTRL}}-r_{\mathrm{v}}^i\right)}^{\mathrm{2}}}{\stackrel{\mathrm{‾}}{r_{\mathrm{v}}^{\mathrm{CTRL}}}}}\mathrm{d}D,\text{(7)}& {\displaystyle }{N}_{\mathrm{s}}& {\displaystyle }=\underset{D}{\int }{\displaystyle \frac{{\left(U^{\mathrm{CTRL}}-U^i\right)}^{\mathrm{2}}+{\left(V^{\mathrm{CTRL}}-V^i\right)}^{\mathrm{2}}}{\mathrm{2}}}\mathrm{d}D,\end{array}$$

where the superscript i denotes the RASTA experiments (Z^SOP1, V^SOP1 or ZV^SOP1), $C_{p_{\mathrm{d}}}$ is the specific heat of dry air, R_d is the dry air constant, R_v is the water vapour gas constant and T_r is the reference temperature (taken to be 300 K). The total exergy distance is then given by the sum of the four quadratic terms N_T, N_p, N_v and N_s.

Figure 11Standard deviation (g kg⁻¹) of the water vapour mixing ratio differences between in-flight humidity measurements r_o and the analyses r_m for the different experiments (CTRL^SOP1, Z^SOP1, V^SOP1 and ZV^SOP1) over the 24 analyses. The standard deviation differences between the measurements and the background state are also shown by the black data points (N=6307).

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In Eqs. (4), (5) and (6), the contribution terms of the exergy distance are divided by the weighting factors $\stackrel{\mathrm{‾}}{T^{\mathrm{CTRL}}}$, $\stackrel{\mathrm{‾}}{p_{\mathrm{s}}^{\mathrm{CTRL}}}$ and $\stackrel{\mathrm{‾}}{r_{\mathrm{v}}^{\mathrm{CTRL}}}$, which correspond to the average values over the 2-D domain of T^CTRL, $p_{\mathrm{s}}^{\mathrm{CTRL}}$ and $r_{\mathrm{v}}^{\mathrm{CTRL}}$, respectively. Hence, as defined by Ehrendorfer et al. (1999), the weighting factors $\stackrel{\mathrm{‾}}{T^{\mathrm{CTRL}}}$ and $\stackrel{\mathrm{‾}}{r_{\mathrm{v}}^{\mathrm{CTRL}}}$ are a function of the altitude, where an arbitrary factor “ϵ” was introduced but with unknown values between 0.1 and 10. This arbitrariness is removed by Marquet et al. (2019) where $\stackrel{\mathrm{‾}}{r_{\mathrm{v}}^{\mathrm{CTRL}}}$ varies significantly with height, since the water vapour mixing ratio decreases by 3 orders of magnitude between the surface and the stratosphere. Therefore, moisture analysis and forecast impacts between the different atmospheric levels are fully taken into account through the use of these altitude-dependent weighting factors. Hence, the use of the exergy distance is expected to more fairly rank the different observing systems through the use of more balanced contributions between wind, temperature and water vapour.

Figure 12Differences in the average Heidke skill score (HSS) of the 6 h cumulated precipitation forecasts versus rain gauge measurements, between the three RASTA experiments and the CTRL^SOP1 experiment (a) Z^SOP1, (b) V^SOP1 and (c) ZV^SOP1. Calculations were performed over the 32 runs in which RASTA data were assimilated. The error bars represent the 95 % bias-corrected and accelerated (BCa) bootstrap confidence intervals (see Efron and Tibshirani, 1994).

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In this study, the four different contribution terms of the exergy distance will be studied independently in order to evaluate the respective impact of the assimilation of the RH pseudo-observations and/or RASTA wind components on temperature (Eq. 4), surface pressure (Eq. 5), water vapour mixing ratio (Eq. 6) and wind (Eq. 7) fields.

6.2 Analyses evaluation: comparisons against in situ measurements

The aim of this section is to assess the added value of the assimilation of RASTA data on the analyses. The evaluation is not shown against other conventional assimilated observations, because, as expected, the fit to observations is always better in CTRL^SOP1 than in the RASTA experimental analyses. However, in-flight humidity measurements at flight level are not assimilated in any of the experiments, and are used as independent observations to assess the impact of the assimilation of RASTA data on the humidity analyses. As explained in Sect. 3.3, poor quality measurements are removed for the comparisons. Hence, after the manual quality control, only 24 analysis cases remain. Figure 11 shows the standard deviation between humidity mixing ratio measurements and the analysed ones for the different experiments (Z^SOP1, V^SOP1 and ZV^SOP1) during the 24 analysis cases. The standard deviation between the measurements and the water vapour mixing ratios from the background state is also represented by the black data points, which is a constant value because the same background states are used in all the different experiments.

First, it should be noted that the analysed water vapour mixing ratios are always in better agreement with the observations compared to the background field, which is quite reassuring. Next, the standard deviation is slightly larger for the V^SOP1 than for the CTRL^SOP1 experiment. Hence, the assimilation of RASTA wind data alone (V^SOP1) does not improve the analysis in terms of humidity, which was expected because RASTA wind data are only slightly related to humidity, so it is likely that the humidity analysis field moves away from humidity observations. The experiment that reduces the standard deviation the most is the Z^SOP1 experiment, which indicates that the assimilation of RH pseudo-observations alone positively impacts the analysis in terms of humidity.

Even though slightly less pronounced, the joint assimilation of RH pseudo-observations with RASTA wind data (ZV^SOP1 experiment) also leads to an improvement of the analysed humidity field. The respective impacts of the Z^SOP1 and V^SOP1 experiments are both present in the ZV^SOP1 experiment. Therefore, since the standard deviation is slightly larger for the V^SOP1 experiment, it seems logical that the standard deviation in Fig. 11 is larger for the ZV^SOP1 experiment than for Z^SOP1. In addition, it has been demonstrated in Fig. 9 that the humidity field in the analysis is more impacted by the assimilation of RH pseudo-observations (Z^SOP1) than RASTA wind data (V^SOP1). Consequently, the ZV^SOP1 experiment inherits more from the benefits of the Z^SOP1 experiment than from the disadvantages of the V^SOP1 experiment in the humidity analysis.

6.3 Rainfall forecast evaluation

Forecast scores are now validated using the rain-gauges, whose locations are indicated by the blue markers in Fig. 1. For the comparisons, model outputs are interpolated to the rain-gauge station locations using a linear interpolation. Heidke skill score (HSS) is calculated for the 6 h accumulated rainfall forecasts for the CTRL^SOP1 and the three RASTA experiments (Z^SOP1, V^SOP1 and ZV^SOP1). HSS is calculated for the 32 assimilation cases in which RASTA data have been assimilated. Figure 12 represents the mean HSS differences in the 6 h accumulated rainfall forecasts between the RASTA and the CTRL^SOP1 experiment as a function of the rainfall accumulation threshold (millimetres). The bootstrap confidence intervals are also shown for each threshold. They are quite large because HSS has only been calculated over 32 cases. The impact of the assimilation of RASTA wind data is positive if the differences are above 0.

Figure 12 indicates that the benefit of the RH pseudo-observations (Z^SOP1) is neutral to slightly positive above the threshold of approximately 25 mm. Besides, the impact of the assimilation of wind vertical profiles (V^SOP1) is larger than that of RH pseudo-observations (Z^SOP1), especially for the larger rainfall accumulation thresholds. This is consistent with the fact that the impact of the RH pseudo-observations is less pronounced than the impact of RASTA wind data as the forecast term increases (see Sect. 6). Similar results were also obtained in prior studies (Pu et al., 2009; Zhao and Jin, 2008; Zhang et al., 2012). Finally, the best results are obtained for the ZV^SOP1 experiment, which suggests that the accumulated rainfall forecasts benefit more from the assimilation of the W-band reflectivity jointly with RASTA wind data. Similar results were also obtained with other categorical scores (FAR and POD), and for the 9 and 12 h rainfall accumulation forecasts.