In this study the LAM used is the WRF model in its Advanced Research WRF (ARW) version 3.9.1.1. A single WRF domain is applied over a part of the southwest Indian Ocean including Réunion Island and Mauritius in its centre (Fig. ). A horizontal grid spacing of 12 km and 61 vertical levels stretching from the surface up to 50 hPa are used. We chose not to perform convection-permitting simulations or nesting as our own investigations over Réunion Island, and studies of other regions , have shown that increasing the WRF grid spacing does not necessarily improve the performance of irradiance forecasts. A potential reason for this is that nonlinearity increases with increasing model grid spacing . Moreover, DA with a two-way nested domain is considerably more complex since the updated analyses of the nested domains must be physically consistent with those of the parent domains.

The Global Ensemble Forecast System (GEFS), which has a grid spacing of 0.5∘, 27 vertical levels, and a temporal resolution of 3 h, consists of 21 ensemble members and provides WRF with boundary conditions (BCs) and initial conditions (ICs). In line with previous DA studies with LAMs, which use around 40 members , an ensemble consisting of 41 WRF members is applied in this study. The BCs and ICs of members 1–20 of the WRF ensemble are generated from the original 20 GEFS members. In order to obtain 40 members using GEFS, the BCs and ICs of members 1–20 are perturbed with the WRF Data Assimilation (WRFDA; ) system using the standard NCEP background error covariance to generate WRF members 21–40. The Global Forecast System (GFS) with a grid spacing of 0.25∘, 32 vertical levels, and a temporal resolution of 1 h is used for the BCs and ICs of the 41st WRF member. Initial conditions for all members are only generated from GEFS and GFS at the start of the DA cycling after which they are fed in from the previous cycling step.

All members use the same model configuration; this includes the Thompson microphysics scheme , the Dudhia scheme for shortwave radiation , and the Kain–Fritsch scheme for cumulus parameterisation. These schemes are among the most commonly used in WRF configurations according to a WRF physics survey , and this configuration also performed well for Réunion Island in preceding experiments (not shown).

Due to the higher spatio-temporal resolution of the BCs of the GFS member, after each DA step the first 40 members of the ensemble are recentred on the GFS member (member 41). The ensemble mean is subtracted from each member to obtain the perturbations from the mean for each member. These perturbations are then individually added to the 41st member to regenerate the 40-member ensemble, which now has member 41 as its mean.

The DA cycling interval is 6 h, leading to four cycle steps per day; i.e. DA is performed at 00:00, 06:00, 12:00, and 18:00 UTC, leading to new analyses (also referred to as “posteriors”) at these times from which first-guess forecasts (also referred to as “priors”) for a lead time of 6 h are performed. As satellite observations are available more frequently than every 6 h it is possible to reduce this interval to provide updated analyses for free forecasts (FFs) at a higher update rate than 6 h. For reasons of simplicity, this study uses 6-hourly cycling.

Since member 41 uses the BCs with the highest spatio-temporal resolution this member is chosen for free forecasts that are realised every 24 h at 00:00 UTC (04:00 local time) using its updated analyses (ICs) from the DA cycling. The BCs for the free forecasts are likewise provided by GFS and are generated for lead times up to 16 h.

Two cycling and free forecast experiments were performed: a cycling experiment with CWP DA (CWPDA) and a control cycling experiment without DA wherein all observations are only evaluated by DART but not assimilated (CTRL). The respective free forecast experiments that are run using the 00:00 UTC analyses of the two cycling experiments are labelled CWPDA-FF and CTRL-FF.

The experiments of this study are performed for the austral summer of 2017 and 2018, i.e. between 9 December 2017 and 1 March 2018. Some periods are excluded in the simulations due to gaps in CWP data availability and cyclonic activity. Therefore, cycling is performed for several smaller periods, listed in Table ). The convective activity is generally pronounced during these periods and in each case at least two DA steps were performed before using a 00:00 UTC analysis for the free forecast periods listed in Table .

The DA cycling is performed using the DART and its ensemble adjustment Kalman filter , and the CWP forward operator developed by is used to assimilate CWP observations. describe a few updates to the forward operator, which is close to the version used in this study.

In comparison to other cloud property assimilation methods that do not take into account multilevel cloud information, one strength of this forward operator is that it accounts for cases when the model and CWP observations contain clouds that are localised at different altitudes. The forward operator does this through adjustments to the modelled CWP. For example, if the model contains a low-level cloud and the observations indicate high-level cirrus, the integrated CWP value might be similar and the impact to the model analysis would be small if the cloud altitude was not considered. In this case the forward operator constrains the model CWP to the level of the observed cloud, leading to a larger impact in the analysis, and a cloud is introduced at the correct vertical location.

The SatCORPS cloud products are retrieved using algorithms originally developed to analyse the MODerate-resolution Imaging Spectroradiometer (MODIS) aboard Terra and Aqua for the NASA Clouds and the Earth's Radiant Energy System (CERES) project . These algorithms have been adapted to other imagers aboard geostationary and other low-Earth-orbit satellites . Among other parameters, the SatCORPS data set includes both a pixel and a gridded CWP product.

Depending on the cloud-top phase, as determined from cloud-top temperature (CTT), a given pixel is defined as ice, supercooled liquid, or liquid under the assumption that the cloud phase and particle size are vertically homogenous. The actual CWP is derived as a function of total optical depth and particle size retrievals. At night, the retrievals of cloud optical depth are not as accurate as during daytime due to the lack of visible data.

The global gridded data have a grid spacing of 0.25∘ (approximately 28 km) latitudinally and 0.3125∘ (approximately 34 km) longitudinally and are independent of the respective satellite sensor resolution (i.e. the resolution of the pixel product). The data are available at hourly resolution. For this study, the retrievals of Meteosat-8 are used. The satellite has been operational over the Indian Ocean since February 2017 and has a sub-satellite position of 41.5∘ E.

SatCORPS provides water path (WP) retrievals in three phases: liquid water path (LWP), supercooled water path (SWP), and ice water path (IWP). The supercooled phase is determined using a post-retrieval classification. If the pixel phase is liquid and has a CTT below 273.15 K the pixel phase is defined as supercooled liquid. The retrievals of the different phases are assimilated as independent observations using the same forward operator, which can be considered multilayer cloud property assimilation.

Each of the three-phased WP retrievals are bound between a cloud-base pressure (CBP) and a cloud-top pressure (CTP) that are also included in the SatCORPS product. The forward operator requires information about the cloud effective pressure (CEP) for vertical localisation. Following we set this to be the mean of CTP and CBP.

As this is the first time that SatCORPS gridded Meteosat-8 retrievals are assimilated, a horizontal localisation radius has to be defined for these observations. The Gaspari–Cohn covariance cutoff method is used for horizontal localisation . In this study, the half-width of the localisation radius (also called cutoff) is set to 90 km, a value that performed well in sensitivity runs (not shown). Consequently, there is a factor of 3 difference between observation grid spacing (approximately 30 km) and cutoff. This factor corresponds approximately to the factor of 3.3 applied by , who use a cutoff of 20 km for satellite observations at 4 km nominal resolution.

While in radiance assimilation the historical approach is to assimilate only clear-sky observations, avoid cloud-affected observations, and gradually move towards all-sky assimilation , an opposite strategy is followed in this work on cloud property assimilation. No clear-sky retrievals are assimilated in the experiments in this study. Test experiments showed that the assimilation of clear-sky retrievals led to a strong reduction in the amount of cloud in WRF simulations with the current experimental setup. This means that defining a minimum threshold for “cloudy” WP retrievals allows the EnKF to predominantly assimilate observations over cloudy locations. To address cases when WRF tends to underestimate cloud presence we use a minimum threshold of 0.04 kgm-2 to define cloudy observations of all phases.

We apply the same retrieval errors as (Table ). These were defined for pixel data of the SatCORPS product over the US for both IWP and LWP derived from the Geostationary Operational Environmental Satellite (GOES). There has yet to be a study assessing the errors of the gridded product for Meteosat-8-based retrievals. Therefore, these errors serve as a first estimate for our study. The true uncertainty in CWP varies with cloud conditions, solar and viewing geometry, and other factors which need to be assessed more thoroughly. Defining more region-specific errors as well as independent errors for each phase may be an objective of future work.

Spatially varying state-space adaptive covariance inflation is applied to the first guess at each cycling step . This is a commonly used method to increase prior ensemble spread and prevent the ensemble from collapsing to a single solution.

Independent observations are used to evaluate the DA method in addition to the assimilated observations. The WRF domain in this study contains one radiosonde station at Gillot-Aéroport (Fig. ) where radiosondes are launched daily at 12:00 UTC. These soundings provide valuable independent in situ observations that are used for observation space diagnostics. The soundings are available via the National Center for Environmental Prediction (NCEP) Global Data Assimilation System (GDAS) data set and are evaluated within DART.

An evaluation of the GHI forecasts produced is performed using pyranometer observations provided by Météo-France from 12 locations spread across Réunion Island (Fig. ) at various altitudes between 5 m (Pointe des Trois-Bassins) and 2149 m (Piton–Maido). The raw observations have a temporal resolution of 6 min, and a linear interpolation to 15 min is performed to match the WRF output.

The quality control measures for GHI observations consist of a visual verification and the application of the sub-hourly data quality control procedures proposed by that detect extrema, rare observations, and maximum steps for two following measures.

It is common to perform spatial averaging of WRF solar irradiance output around the site of interest rather than using only the GHI forecasted for the closest grid box . This reduces the variability of the forecasted GHI and thus typically reduces forecast errors in terms of the standard metrics, root mean square error (RMSE), and mean absolute error (MAE). For each GHI forecast at a given location and lead time we apply inverse distance weighting (IDW) according to the formula 1GHI(bc)^=∑i=1nGHI(bi)di-1∑i=1ndi-1, where di is the distance between the grid box containing the observation site, bc, and another grid box, bi. The best results were seen when bi was considered within a 75 km radius of the observation site. Besides IDW, no further post-processing is applied to the WRF solar irradiance forecasts.

Aerosol optical thickness in the study region rarely exceeds 0.2 , leading to a stable clear-sky irradiance on Réunion Island compared to other regions of the world where aerosol optical thickness is highly variable. Moreover, the influence of volcanic aerosols on solar irradiance can be neglected as there were no volcanic eruptions on Réunion Island during the overall study period. We therefore assume that fluctuations between forecasted and observed GHI caused by aerosols are negligible compared to the influence of clouds.

The clear-sky model of the European Solar Radiation Atlas (ESRA) is used for clear-sky normalisation of solar irradiance forecasts.

Various metrics are considered to evaluate the performance of WRF outputs compared to CWP retrievals, radiosoundings, and GHI observations. For n predictions, y, of the observation (y can be either a prior or posterior) and the observations, o, these metrics are defined as follows. The RMSE is defined as 2RMSE=1n∑i=1n(yi-oi)2. The MAE is defined as 3MAE=1n∑i=1n|yi-oi|. MAE becomes the mean bias error (MBE or simply bias) if the absolute of yi-oi is not taken in Eq. ().

The correlation (or Pearson correlation coefficient) is defined as 4corry,o=cov(y,o)σyσo, with “cov” being the covariance and σ being the standard deviation.

The total spread (TSPRD) is defined as the pooled spread of the ensemble and observation errors: 5TSPRD=σy+σo2, with σo being the standard deviation of the observation error and σy being the spread of the 41-member ensemble, which is defined as 6σY=141∑m=141(Ym-Y‾)2, where Y‾ is the ensemble mean.

This subsection evaluates the implementation of the CWP DA methodology.

As a first step, the RMSE for WP at different altitudes during all cycling periods listed in Table  is calculated and shown in Fig.  in which the impact of the three-phased CWP assimilation can be seen. For each phase and altitude, the majority of the available observations (green circles) are assimilated (green asterisks), which indicates that the DART quality check only excludes a reasonable amount of observations and that the defined WP errors (Table ) are realistic. Depending on the mean CEP of the different phases, the maximum number of WP observations, approximately 20 000 per phase, are localised around certain pressure-level bins. These bins are 400 hPa for IWP, 500 hPa for SWP, and 850 hPa for LWP.

The difference between the first guesses (solid lines) and the analyses (dashed lines) is a measure of the impact of the respective phase on the analysis, and it can be seen that IWP has the greatest impact and LWP has the lowest impact. This is to be expected since IWP usually exhibits the largest absolute values of WP. This difference in absolute values also affects the RMSE, which at 400 hPa is highest for IWP with averages of 0.43 and 0.13 kgm-2 in the first guesses and analyses, respectively. In comparison, the RMSE for LWP at 850 hPa is reduced from 0.09 kgm-2 in the first guesses to 0.08 kgm-2 in the analyses.

The evolution of RMSE, MBE, and TSPRD for the ensemble mean first guess and analysis is shown in Fig. . Cycling period C (Table ) is chosen as an example and the pressure levels identified above are chosen for the respective phase.

The classical sawtooth pattern caused by the differences between the first guess and analysis is most clearly visible for IWP. This indicates that the retrievals of this phase have the most impact in the filter. While a reduction of RMSE between the first guess and analysis is visible at most assimilation times for SWP, this is not the case for LWP. In fact, the smallest impact is achieved for LWP, which is in line with the findings shown in Fig. .

TSPRD and RMSE are roughly in phase for IWP; the values generally increase to around 0.3 kgm-2 for IWP in the 6 h first-guess forecasts. Relatively high RMSE values of more than 0.6 kgm-2 are reached twice for IWP during this period, but these are linked to exceptionally large biases in both cases. In the analyses RMSE and TSPRD are mostly around 0.1 kgm-2.

RMSE and TSPRD are less in phase for SWP, which is in line with the expected lower impact of this phase. For LWP, TSPRD never assumes values lower than 0.05 kgm-2, which is the defined observation error for the lowest WP observations (Table ). This underlines once more the importance of determining phase-dependent errors that should ideally be less than 0.05 kgm-2 for low LWP retrievals. As can be seen in Fig. , a clear difference between the first guess and analysis becomes visible only at times when TSPRD is larger than 0.05 kgm-2. Thus, the relatively large observation error for the smallest observations is likely the reason for the comparably small impact of LWP.

The number of assimilated observations may be very different than the number of available observations due to quality control within DART, as can be seen in Fig. . The example of SWP in Fig.  shows that the number of assimilated observations is not necessarily correlated with an improvement in terms of RMSE. For example, on 12 January the number of assimilated observations fluctuates between 80 and 300, but the RMSE remains relatively stable. For all three phases, the number of assimilated observations per assimilation time varies heavily from zero to a few hundred observations depending on the time of day.

The MBE for IWP fluctuates around zero with a positive correction in each analysis. For SWP and LWP there is a continuously negative MBE, mostly between -0.1 and -0.05 kgm-2. This indicates that WRF underestimates clouds in the middle and lower troposphere. DA corrects for MBE for all three phases but is far from achieving MBEs close to zero for SWP and LWP.

Figures  and show evaluations of DA experiments compared to the assimilated observations and therefore demonstrate the ability of the DA cycling to constrain the WRF simulations. Although this can be used to assess the performance of the assimilation itself, this cannot act as an exhaustive validation as no independent observations are used.

Consequently, an evaluation of the MBE for temperature and specific humidity using independent radiosonde measurements is shown in Fig. . All 43 radiosoundings at Gillot-Aéroport during the complete study period (Table ) are considered. The total number of evaluated observations per pressure bin are shown in green and reach up to 80 at around 400 hPa. The lack of radiosonde stations in the model domain and the fact that only one station in the centre of the domain is considered are compensated for, to some extent, by the duration of the study period that includes various cloud and weather situations.

The ensemble mean first guess (solid lines) and analysis (dashed line) are shown for both experiments CTRL (orange line) and CWPDA (black lines), with the first guess and analysis being the same for CTRL as no DA is performed. For both experiments an overall negative MBE for specific humidity and a positive MBE for temperature are visible throughout the troposphere. The fact that the MBE for specific humidity is largest in the lower troposphere, with more than 1 gkg-1, confirms that WRF tends to underestimate low clouds. At the same time, the difference between the two experiments for specific humidity is the largest in the lower troposphere around 850 hPa. This shows that the assimilation corrects for the lack of humidity in the analyses to some extent and thus has the effect of a bias correction.

Although Figs.  and indicate that the largest impact of the DA is in the higher troposphere from IWP observations, the effect on bias regarding radiosonde specific humidity is smaller at these altitudes than in the low troposphere. This may be explained by the fact that absolute values of specific humidity are generally largest in the low troposphere. Moreover, the evaluated radio soundings are valid only for the centre of the domain, while Figs.  and include information about the whole model domain. Furthermore, local thermal circulations likely cause more low clouds at this coastal location than in the rest of the domain, which lacks other landmasses.

The difference between CWPDA prior and posterior is more distinct for specific humidity than for temperature, leading to an improvement of humidity bias in the analyses compared to the first guesses. As the objective here is to improve cloud prediction, the improvement in humidity, a field strongly related to cloud, is more significant than an improvement in temperature. It is, however, favourable that the CWP DA does not have a negative impact on the temperature profile.

Regarding the DA configuration, a number of parameters such as the covariance localisation radius are known to largely impact the DA outcome . Concerning the gridded multiphase SatCORPS retrievals used in this study, the sensitivity of the assimilation to the localisation radius might be assessed in detail in future work. This is especially true for experiments at convective-scale resolutions that have yet to be performed.

Moreover, the ensemble spread could be modulated in various ways by adjusting the WRF ensemble generation method. For example, multiple sets of physics options could be applied in the ensemble members, an optimal compromise between ensemble size and spread could be determined, the method for WRF ensemble member generation from the 21-member GEFS ensemble ICs and BCs might be optimised, and different settings for adaptive inflation could be tested. The WRF ensemble may also be applied to the free forecasts for probabilistic solar irradiance forecasting.

In summary, the operational correctness of the DA methodology is confirmed. The largest impact is found for the ice-phase retrievals and the lowest impact for the liquid-phase retrievals. Independent radiosoundings indicate a humidity bias reduction between first guesses and analyses. Any improvement of the utilised cloud products is expected to positively influence the DA outcome. Future cloud products are aiming to better account for vertical heterogeneity and thus produce multiphase CWP estimates that are closer to reality and more similar to what NWP models produce regarding deep overlapping cloud systems. Moreover, a precise definition of phase-dependent errors for the SatCORPS Meteosat-8 products does not yet exist. Once this information is obtained, the performance of the applied system using these observation errors can be assessed.

Having demonstrated the correct implementation of the DA methodology, which shows a generally positive impact on the cloud analyses, this subsection focuses on a case study of one particular day. The influence of the applied DA strategy on the cloud analysis and the subsequent free forecast with respect to solar irradiance is analysed.

On 12 January 2018 the large-scale flow in the study region was governed by an anticyclone south of Madagascar and two depressions located at the northern boundary of the WRF domain as indicated by synoptic surface analyses published by Météo-France and the South African Weather Service (not shown). The first depression was located approximately 700 km north of Réunion Island and the second approximately 1500 km northeast of Réunion Island. This situation led to a northwesterly flow throughout the model domain and the creation of a convergence zone extending diagonally across the model domain from the northwest to the southeast.

Such large convective cloud systems associated with low-pressure systems north of Réunion Island typically produce the lowest GHI values during austral summer . Hence, the day considered here has one of the lowest observed GHI values throughout the combined study period listed in Table  and therefore most distinctly shows the impact of the DA on the GHI forecast.

Figure a, c, and e show SatCORPS WP retrievals at the time of the analysis (00:00 UTC) of the free forecasts performed for this day. The clouds induced by the convergence zone are visible, especially in the ice and liquid phase, with WP values exceeding 0.3 kgm-2 in some areas. The highest values in the vicinity of Réunion Island can be observed in the ice phase, indicating high clouds. These ice clouds persist during the day (not shown) and contribute largely to the observed low solar irradiance on Réunion Island throughout the day.

Figure b, d, and f show observation space diagnostics of the WP difference in each phase between the two experiments (CWPDA minus CTRL) in terms of the posterior ensemble mean WP. More cloud water is present in all phases over the convergence zone in CWPDA when compared with the CTRL experiment. In the ice phase, distinct gradients between areas of increased and decreased WP are visible that indicate corrections to the cloud location resulting from the DA. Maximum and minimum values are -0.9 and 1.9 kgm-2 for IWP (b), -0.2 and 0.19 kgm-2 for SWP (d), and -0.06 and 0.19 kgm-2 for LWP (f).

The effect of these corrections on the free forecast experiment in terms of cloud fraction is shown in Fig. . The ice clouds induced by the convergence zone are visible in both CTRL-FF and CWPDA-FF, and in both cases clouds are located above Réunion Island 15 min after the analysis time (a and b). A northeastward relocation of the clouds around Réunion Island is visible for CWPDA-FF. As the clouds move eastward over the course of the day, this has the effect of causing high clouds to persist over Réunion Island in CWPDA-FF (d and f), while clouds are further west in CTRL-FF (c and e), leaving the model levels around 300 hPa cloud-free above Réunion at 08:00 UTC (corresponding to noon local time). In CWPDA-FF, clouds are still present at 300 hPa over Réunion Island at 08:00 UTC, leading to more realistic forecasts of solar irradiance when compared with ground observations as shown in the following.

A large variability of GHI is observed during the day for all pyranometer sites as well as between the different sites (Fig. ). GHI values are overall low and mostly below 400 Wm-2 at all sites; this is mainly caused by the high clouds during that day as determined from satellite images. A distinct difference is visible between CTRL-FF and CWPDA-FF as a consequence of the DA. Although the forecasted GHI is largely reduced in CWPDA-FF, the values around noon are still too high compared to the observations for most sites. Further improvements might be found by analysing the interplay between DA, post-processing, and the configuration of WRF in terms of grid spacing, nesting, and the choice of parameterisation schemes. In this study we focus on the influence of DA only, which is clearly visible in this example.

This comparison between forecasts and observations of GHI also illustrates the difficulty of forecasting ramp events. The chosen grid spacing of 12 km and IDW post-processing results in a smoothing of ramps in the WRF forecasts, which is why the widely used metrics RMSE and MAE are suitable for a quantification of the DA impact. If one wants to study the impact of DA on ramp forecasts specifically, experiments at convection-resolving resolutions, a focus on parameterisation schemes, and specific ramp metrics and post-processing methods are required.

The previous subsection illustrates that cloud property DA can have a considerable positive impact on short-term forecasts of cloud-related parameters such as CWP, specific humidity, and solar irradiance in the SWIO. Nevertheless, as in many other proofs of concept for DA methods, only one weather situation has been considered so far in terms of solar irradiance. An evaluation over a period of more than a few days is rare in peer-reviewed literature . Therefore, an evaluation of the free forecasts for a total of 44 d (Table ) is performed in this subsection in order to quantify the impact of DA on GHI forecasts more meaningfully.

Figure  shows an evaluation of GHI forecasts from CTRL-FF and CWPDA-FF for the 12 considered sites at Réunion Island. The spread between the sites for all metrics can be explained by the location of the sites and the local meteorological conditions. The six sites with the best performance in terms of RMSE (between 230 and 280 Wm-2) and MAE (between 180 and 220 Wm-2) are Gillot-Aéroport, Pierrefonds-Aéroport, Pointe des Trois-Bassins, Piton Sainte-Rose, Le Port, and Le Baril. All of these sites are located on the coastline of Réunion (Fig. ) where the influence of clouds that are induced by orographic uplift and thermal circulations caused by the mountains is lowest.

The two sites with the worst results in terms of RMSE, MAE, and correlation are Colimaçons and Petite-France. These sites are both located in the west of the island at 800 and 1200 m, respectively, meaning these sites are typically in the lee of the trade winds at altitudes at which thermally driven convective clouds often form. This leads to lower GHI values and produces the most complex solar irradiance conditions compared to the other sites. This explains the high positive MBE for these two sites.

A positive bias can be seen for most sites, which confirms that WRF tends to overestimate GHI on Réunion Island during summertime with pronounced convective activity. There is a shift to lower values of MBE between CTRL-FF and CWPDA-FF with approximately the same amplitude for all sites. This illustrates that CWPDA-FF generally includes more clouds than CTRL-FF, but it also means that for sites with a negative MBE, typically the ones that are not located in the mountains, there is a degradation towards more negative values of MBE.

An improvement of GHI forecasts between CTRL-FF and CWPDA-FF is visible for almost all sites in terms of RMSE, MAE, and correlation. On average across all sites, RMSE improves by 11 Wm-2 (4 %), MAE by 6 Wm-2 (3 %), and correlation by 0.03. The only exception to this is the site at Le Port where the RMSE is degraded by 1 Wm-2 and MAE by 2 Wm-2. In terms of RMSE and MAE, there is less improvement at sites for which GHI is predicted most accurately (Gillot-Aéroport and Pierrefonds-Aéroport). This may be explained in the same way as the large improvement at the sites with the least accuracy: as found above, DA leads to both a better representation of cloud location and generally increased lower tropospheric moisture. Both of these effects have an impact on the sites with the least accuracy. Being located far from mountain slopes, thermally induced convective clouds are rarer at Gillot-Aéroport and Pierrefonds-Aéroport. The effect of low-level clouds at the mountain slopes is thus of less consequence here compared to improved large-scale cloud system locations.

Figure  shows the averaged intraday values for each evaluation metric over all forecasted lead times. The RMSE per lead time as a mean over all sites is shown in Fig.  in order to evaluate the impact of the applied DA method on the free forecast of GHI with respect to lead time.

GHI observations are not always available for all 12 sites for a given lead time and day, which complicates comparisons of different lead times. Consequently, the number of considered days per lead time shown in the figure is the number of days for which observations are available for all 12 sites. These are the days that have been considered in the calculation of RMSE for a given lead time. The first lead time is no earlier than 5 h (05:00 UTC or 09:00 am local time) since the observations in the morning did not pass the quality control for several stations, which is often due to shadowing, mainly caused by the mountains.

The free forecasts that have been initialised with CWPDA have a lower RMSE than CTRL-FF throughout the day. Both the absolute (a) and the clear-sky normalised RMSE (b) are shown in Fig. . The representation of absolute RMSE does not correct for the diurnal cycle of GHI, which leads to the characteristic curve. It does, however, allow for the identification of the absolute difference in RMSE between CTRL-FF and CWPDA-FF that reaches up to 60 Wm-2 at 08:00 UTC (noon local time).

The normalisation with the clear-sky irradiance removes the diurnal cycle, showing an expected increasing forecast error with increasing lead time for both experiments. The maximum absolute difference of 60 Wm-2 between CTRL-FF and CWPDA-FF translates to approximately 2 % of normalised RMSE (Fig. b).

With increasing lead time, the predictability of cloud evolution decreases and the influence of the boundary conditions is expected to become larger compared to that of the initial conditions. This means that the differences between CTRL-FF and CWPDA-FF are expected to be larger for short lead times than for longer ones, indicating that most of the DA impact occurs close to the analysis time. According to Fig.  this is not the case in the present experiments; rather, in the first 14 h of free forecast the difference between CTRL-FF and CWPDA-FF does not change remarkably except for the first two lead times (04:45 and 05:00 UTC), for which RMSE is almost identical.

Together with the above findings this indicates that the DA of CWP generally adds more clouds to WRF, reduces the overestimation of GHI, and therefore acts like a sophisticated bias correction. On average, the additional clouds do not vanish within the first 14 h of forecast, leading to improved GHI forecasts throughout the day.