To correct the SEVIRI retrievals for parallax, we use the Satpy modifiers.parallax version 0.49.0 Python library . Here, the parallax correction is briefly recapped and graphically illustrated in Fig. . For more details, refer to and references therein. The correction starts with the computation of the satellite viewing zenith angle (θs). This requires a transformation of the Earth-centred inertial coordinate system to a topocentric coordinate system, in which the local observer horizon is used as the fundamental plane. The transformation is achieved by applying two rotations. In the topocentric coordinate system, the vectorial distance between the satellite and the uncorrected pixel of interest (Δxsat2surf, Δysat2surf, Δzsat2surf) can also be calculated, as well as the corresponding slant distance (Δssat2surf). Then, from Hc, the slant distance between the cloud top and the uncorrected pixel (Δscld2surf) and the parallax distances (Δxpllx, Δypllx) can be calculated using geometric similarity: 1Δscld2surfΔssat2surf=ΔxpllxΔxsat2surf=ΔypllxΔysat2surf=HcΔzsat2surf. Finally, the parallax distances are converted to spherical coordinates, yielding the parallax shift in latitude and longitude (Δlatpllx, Δlongpllx).

For accurate retrieval of surface GHI, the positioning of the cloud shadow location is highly relevant. The SEVIRI GHI retrieval relies on an assumption of one-dimensional (1D) radiative transfer. A consequence of this approach is that, for the GHI retrieval, the cloud shadow is assumed to be directly beneath the cloud. The cloud shadow correction aims to shift the pixel to the actual position of the shadow.

The cloud shadow displacement (Δlongshdw, Δlatshdw) can be computed from the cloud top height (Hc), the solar zenith angle (θ0), the solar azimuth angle (ϕ0), and the Earth's radius (Rearth), as shown by Eqs. () and () and as graphically illustrated in Fig. . 2Δlongshdw=Hctan⁡θ0sin⁡ϕ0Rearth3Δlatshdw=Hctan⁡θ0cos⁡ϕ0Rearth

Initially, we calculate a parallax and shadow correction for every pixel that has been flagged as cloudy. However, uncertainties in the retrieved cloud mask and Hc can lead to inaccuracies in the magnitude of the applied parallax correction, for instance, when pixels are falsely identified as either clear sky or cloudy. To get an estimate of the effect of these inaccuracies, two additional experiments are performed that evaluate the sensitivity of the parallax and shadow corrections to variations in Hc. 1.

In the first experiment, we create a dataset in which the parallax and shadow corrections are calculated using the median Hc value over 55 × 55 HR or 19 × 19 SR pixels surrounding the Jülich study domain, which corresponds to an area of about 110 × 58 km². This is termed the area-based approach as opposed to the pixel-based approach.

The experiments listed above are performed three times at HR and at SR. The first time, only a parallax correction is performed. The second time, only the shadow position is corrected. Finally, we combine both the parallax and shadow correction by first computing the magnitude of the parallax and then applying the shadow correction to the parallax-corrected cloud position.

Besides the geometric correction for parallax and shadow displacement described in the previous subsections, we also use an empirical method to improve the GHI geolocation accuracy. This method relies on optimizing the correlation between the GHI measured by the pyranometer network and the GHI from the SEVIRI-derived time series for all of the data between 06:15 and 17:15 UTC. The procedure to determine this mean optimal shift method is as follows. For each day of the field campaign, the SEVIRI grid is shifted by multiples of 500 m along the north–south and/or west–east axes, after which the correlation between the SEVIRI GHI and the pyranometer network GHI is calculated. An optimal collocation shift for the whole period of the HOPE campaign is then determined based on the highest mean correlation over all of the days. For the HR retrieval, this daily mean optimal shift is achieved by moving the SEVIRI grid 3.0 km south and 0.5 km east. For the SR retrieval, a nearly identical mean optimal shift of 3.0 km south and 1.0 km east is obtained.

The mean optimal shift method does have some drawbacks. For instance, variations in Hc are not considered in the correction. Another shortcoming of the daily mean optimal shift method is that the diurnal variation of the shadow position remains unaccounted for. Throughout the day, the optimal collocation shift is not constant. For that reason, a mean optimal shift per time step is also calculated (Fig. ). To illustrate the diurnal variation in this time step optimal shift, it is computed separately for each week of the field campaign. In Fig. , small-temporal-scale variations have been smoothed out by applying a rolling mean with a width of 1 h to show the general trends of the time step mean optimal shift. The longitude of the optimal shift moves from west to east, in line with the shadow position (Fig. a). Accordingly, the optimal shift for latitude moves slightly northward in the early morning and slightly southward in the late afternoon (Fig. b). Especially in the early morning and in the afternoon, the time step mean optimal shift deviates strongly from the daily mean optimal shift.

Figure  also indicates the weekly variation in the optimal shift, as shown by 25th to 75th and 5th to 95th percentiles. The time step optimal shift depends on solar position and, therefore, is not constant throughout the year. Furthermore, variations in present weather conditions and cloudiness account for some of the observed spread in the week-to-week time step mean optimal shift as well. Especially earlier in the morning and later in the afternoon, an increased variation in optimal shift can be observed compared to at noon. Finally, assuming an identical cloud field, with the sun being lower in the sky during the morning and afternoon (i.e. larger solar zenith angles), the magnitude of the shadow displacement will be larger than that around noon, allowing for an increased range of possible optimal shifts.

To ensure the most robust estimate, the time step mean optimal shift is, in the remainder of this article, determined using all of the available data from the field campaign for each specific time step. Note that the same pyranometer data are used for computation of the optimal shift and evaluation of the accuracy of the empirical collocation shift method, which makes the data not fully independent. Ideally, data from previous years would be used to establish the optimal shifts, but these are not available for the field campaign. Yet, to derive each optimal shift, large volumes of data are used, representing a wide range of weather conditions and cloud types. Therefore, we expect the optimal shifts to remain largely insensitive to time-to-time variability in GHI measured by the pyranometers.

Figure  shows the mean root mean square error (RMSE) between satellite and ground-based GHI for the parallax, shadow, and combined corrections as a function of the relative Hc at HR (Fig. a) and SR (Fig. b). For both resolutions, the daily and time step mean optimal shift are also shown; these are independent of the relative Hc.

Since geometric corrections are highly dependent on the type of clouds, the analysis will be further refined using the CRAAS cloud regimes. Figure  shows the mean HR RMSE (Fig. b) and the difference in RMSE between HR and SR (Fig. c) for each of the nine cloud regimes. In the following subsections, the main characteristics of this figure are presented and discussed.

In Fig. , the relative mean HR RMSE (Fig. a) and the HR improvement (Fig. b) divided into hourly bins are displayed. The division into hourly bins shows a diurnal cycle, with the smallest errors for the time blocks between 08:15 and 11:10 UTC and increasing errors more towards the morning or afternoon (Fig. a). Note that, here, the relative RMSE has been plotted to better compare the various time slots. In terms of absolute error, the RMSE usually peaks around solar noon when GHI is maximized. The increase in relative RMSE values for the first and last time blocks compared to those in the middle of the day is expected as larger solar zenith angles cause an increased uncertainty in the retrieval. Still, for the time step mean optimal shift and the combined geometric correction method, the diurnal variation in RMSE remains more limited when compared to the uncorrected or daily mean optimal shift retrieval. This effect can be explained when the diurnal variation in cloud shadow position is considered. With both the combined geometric correction and the time step mean optimal shift method, the diurnal variation in solar position and, thus, shadow location is accounted for. The daily mean optimal shift only considers the daily averaged cloud shadow position, which approximately resembles the situation around noon but does not represent the cloud shadow position in the early morning or late afternoon well. Hence, there is an increase in RMSE for the first and last time blocks, while, around solar noon, the difference between the various methods remains more limited.

Another observation made from Fig. a is that, towards the afternoon, errors are larger than in the morning. This is possibly due to variations in the diurnal occurrence of clouds types. Since convectively driven clouds are more likely to develop in the afternoon , this might lead to increased retrieval errors for those time blocks. Although modest diurnal variations in the relative occurrence of different CRAAS cloud types and the corresponding RMSE are present, we have not been able to pinpoint the asymmetric diurnal cycle in terms of total RMSE in relation to these variations.

Analysis of the diurnal variation in the magnitude of the HR improvement compared to SR (Fig. b) shows that the largest improvements occur in the afternoon from 13:15 to 14:10 UTC. The diurnal variation in HR improvement is largest for the daily mean optimal shift method. In the period from 09:15 to 14:10 UTC, the median HR RMSE is smaller than the SR RMSE, while, for the remaining time blocks, the SR retrieval gives more accurate results. The combined geometric correction and the time step mean optimal shift methods both yield HR improvements at nearly all times of the day. From these two methods, the time step mean optimal shift produces the largest HR improvements. Remarkably, without spatial correction, the HR retrieval is outperformed by the SR retrieval throughout the entire day. Overall this illustrates the growing importance of accurate geolocation at increasing spatial resolutions. This also makes it increasingly relevant for the current generation of geostationary satellites like the GOES Advanced Baseline Imager GOES ABI; and the Meteosat Third Generation Flexible Combined Imager MTG-FCI;, which enable retrievals of GHI down to scales of 500 m.

The results presented in this study are valid for a limited domain in the mid-latitudes. Both the geometric correction method and the collocation shift method, which does require ground observations, can be applied globally. However, the relevance of the corrections will vary depending on the location, time of day, and day of year. This section discusses the generalizability of our results to other geographical areas.

First, we focus on the parallax correction. Due to the fixed position of geostationary satellites above the Equator, the satellite zenith angle increases with increasing latitude. This means that the magnitude of the north–south parallax increases with latitude, and the effect of parallax on the retrieval accuracy will, therefore, remain more limited at lower latitudes compared to at higher latitudes. On the other hand, the location of the current study has almost the same longitude as the Meteosat satellite. For regions at longitudes that are further away from the satellite longitude, the east–west parallax will be larger and thus have more impact on the retrieval accuracy. The relative importance of the north–south and east–west parallax could be studied in more detail by comparing retrievals from the MSG Prime or RSS service, for which the satellite is positioned at 0/9.5° E, to retrievals from the MSG Indian Ocean Data Coverage (IODC) service, for which the satellite is positioned further east at 41.5/45.5° E. However, this comparison is not possible for the dates of the HOPE field campaign as the MSG-IODC service became operational in 2016.

The second aspect that needs to be considered is the solar position and its effect on cloud shadow location. The solar position is described by the solar zenith and azimuth angles. The diurnal and seasonal variations of these angles are such that cloud shadow displacements are smallest around noon and in summer, when the sun is high in the sky. In the early morning and late afternoon, as well as during winter, cloud shadow displacements are larger, and spatial corrections are more relevant. In terms of geographic location, the overall magnitude of cloud shadow displacements is smallest near the Equator and increases toward higher latitudes. However, near the Equator, the variation in solar azimuth angle still causes a strong diurnal cycle in east–west cloud shadow displacement, and corresponding corrections are required. assessed parallax and cloud shadow corrections for locations with varying latitudes and longitudes and showed that, overall, larger reductions in the RMSE of satellite-observed GHI against ground-based measurements are obtained for higher satellite viewing zenith angles.

Thirdly, regional variations in cloud occurrence are highly relevant in assessing the generalizability of the results. For instance, in subtropical land regions, the climatological mean total cloud fraction is very small , making parallax or shadow corrections, to a large extent, obsolete. As shown in this article, the importance of the parallax and shadow correction depends on the cloud regime through cloud heterogeneity and cloud top height. In , the annual and diurnal variabilities of cloud regimes were investigated for Europe. The authors show, for instance, that the fair-weather cloud regime is mainly present in summer, while the alto- and nimbo-type clouds are mainly observed during wintertime. Another example is the shallow-cumulus regime, which mainly has an oceanic character and therefore will weigh more heavily on the overall retrieval accuracy over ocean domains than it does on the Jülich domain in the present study.

Finally, our results show the limitations of the applicability of the geometric correction for multilayered clouds, which, based on the satellite retrievals during the field campaign, mainly occurred in the cirrostratus cloud regime. quantified how the probability of occurrence of multilayer clouds depends on cloud type and latitude. These findings are relevant to assess the utility of the applied corrections at different locations. The statistical evaluation by shows that high clouds, altostratus, altocumulus, and cumulus often coexist with other cloud types. Stratus, nimbostratus, and convective clouds are more likely to occur without other cloud types being present. The observation that high clouds tend to coexist with other clouds agrees well with the high degree of multilayer clouds observed in the cirrostratus cloud regime in this study. Furthermore, the limited occurrence of multilayer clouds for the stratocumulus regime can also be observed in Fig. . On the other hand, no large degree of multilayer clouds is observed for the cumulus regime in this study. This might be explained by the limitation of passive sensors like SEVIRI in distinguishing different cloud levels, for instance, when optically thin cirrus clouds appear above thick shallow cumulus. Multilayer clouds can be much better captured by the active sensor measurements from CALIPSO and CloudSat, as used by . In terms of latitudinal variation, the fraction of multilayer cloud peaks in the tropics, while it shows a dip in the subtropics and further minor peaks at mid-latitudes . This suggests larger uncertainties due to parallax and shadow corrections in the tropics. However, effects due to the presence of multilayer clouds are counteracted by the overall smaller parallax and shadow displacements at lower latitudes, as discussed previously.

In summary, the impact of spatial displacements on the accuracy of satellite-retrieved GHI is influenced by various factors and varies in space and time. Overall, this impact increases towards higher latitudes, but the predominant cloud types play an important role as well. We expect that the results of this study in terms of cloud type dependencies are relatively general. However, the magnitude of parallax and shadow displacement effects on GHI remains location-specific.

This paper focuses on spatial corrections for parallax and cloud shadow displacement in order to ensure accurately geolocated GHI retrievals. In theory, the error due to parallax and shadow displacement can be fully accounted for if the clouds are spatially homogeneous objects whose position is precisely known. However, this is, in reality, not the case, and, furthermore, additional mismatch or representativeness errors are introduced when the satellite retrievals are validated against ground observations .

A first mismatch error is the result of variations in the spectral range between SEVIRI and the pyranometers of the HOPE network. GHI values retrieved with CPP-SICCS and measured by the HOPE pyranometers are representative of the total solar irradiance. However, the sensitivity of the HOPE pyranometers is limited to wavelengths between 0.3 and 1.1 µm. A considerable amount of energy is contained in the part of the solar spectrum that the pyranometers remain insensitive to. In particular, GHI variations due to differential absorption by liquid and ice cloud particles and particles of different sizes, which occurs at wavelengths in the shortwave infrared, are not accounted for. In , the spectral errors of the pyranometers are reported to be 2 % to 5 %, which means that, for example, cloudy pixels with a GHI of 400 W m⁻² could have a 20 W m⁻² error.

Secondly, there are temporal mismatch errors. In this study, the pyranometer data are averaged over a 5 min time interval centred around the SEVIRI acquisition time. This should reduce the mismatch error, but the error increases with cloud variability and, therefore, will be relevant for heterogeneous cloud conditions.

A third error source is the spatial mismatch. The SEVIRI observations are representative of an area larger than the pixel size, with decreasing sensitivity towards the edges, as governed by the modulation transfer functions of the channels e.g.. The pyranometer measurements are point measurements, but, since the diffuse radiation originates from the surroundings, they reflect atmospheric conditions in an area. The purpose of the temporal averaging of the pyranometer data is to partly compensate for the spatial point-versus-area mismatch with the satellite observations . In addition, a Gaussian smoothing with a filter width of 1 km is applied to the satellite observations. However, the optimal filter width depends on cloud heterogeneity and, therefore, is not constant between cloud conditions, as was demonstrated by . Thus, even with this careful collocation and averaging strategy, some uncertainty due to spatial mismatch does remain.

In , the authors investigated mismatch errors by averaging BSRN measurements to wider temporal intervals (temporal mismatch) and by aggregating GHI from the SARAH‐2.1 dataset to coarser pixel grids (spatial mismatch). They found that the mean absolute deviation (MAD) for the temporal mismatch was minimized with a ±14 min temporal averaging window, while the MAD for the spatial mismatch was smallest if the retrievals were smoothed to 0.25 × 0.25° (see their Fig. 6). The width of the optimal temporal averaging window agrees well with our previous results in , where the RMSE was found to be smallest at a 20 min averaged temporal resolution. However, the results for the spatial mismatch in appear to be counterintuitive and might be related to the neglect of parallax and cloud shadow correction, which becomes more important at higher resolutions. In the current study, as well as in , a higher resolution does lead to better correspondence with ground-based observations.

Higher-resolution retrievals should indeed be able to better capture the smaller-scale variability in GHI around the pyranometer stations, leading to a reduced spatial mismatch error. However, the smallest-scale cloud variability will be too fine to be captured by current geostationary satellites. This sub-pixel cloud variability therefore remains a source of errors, leading to biased cloud property retrievals e.g.. Still, biases are generally small if pixels are overcast but grow rapidly if cloud heterogeneity increases .

Finally, three-dimensional (3D) radiative effects, not included in the SEVIRI retrieval, can introduce a spatial mismatch as well. For instance, 3D effects like cloud-side illumination might limit the accuracy of geometric parallax and shadow corrections. From a 1D perspective, radiation will always enter a cloud from the cloud top. With 3D radiative transfer, this assumption does not hold anymore. A consequence of side illumination is cloud shadow enlargement at the surface. Moreover, since the geometric corrections are based on cloud top height, the magnitude of the shadow corrections is likely to be overestimated due to side illumination. This effect might partially explain why, for the combined geometric corrections, the lowest RMSE is found below 100 % relative Hc.

For the mean optimal shift method, it can be argued that the sub-pixel variability and 3D radiative effects have a smaller influence on the accuracy of the achieved geolocation. The GHI patterns observed by the pyranometer network result from 3D cloud radiation interactions. Since the mean optimal shift method is based on a collocation shift, which optimizes the correlation with respect to ground observations, implicitly, the 3D effects, like cloud shadow enlargement, are accounted for.