The current version of European geostationary satellites is the Meteosat Second Generation, which has provided operational data since 2004 . The SEVIRI imager is installed aboard the Meteosat-8 to Meteosat-11 platforms, which are positioned above longitudes of 9.5∘ E and 0.0∘, respectively. One SEVIRI instrument samples the full disk of the Earth from 0.0∘ longitude with a temporal resolution of 15 min. However, a backup satellite positioned at 9.6∘ E also scans a northern subregion with a temporal resolution of 5 min (the so-called Rapid Scan Service). These samples – in our case from Meteosat-9 – provide the observational SEVIRI data set for the following analysis.

This study mainly considers observations from SEVIRI's solar reflectance channels 1–3, as well as from the spectrally broader HRV band. These channels cover the VNIR and shortwave-infrared (SWIR) spectral wavelength ranges. The two VNIR reflectances (r06 and r08) are sampled in bands 1 and 2, respectively, and are centered around wavelengths λ=0.635 and λ=0.810µm. SWIR reflectances (r16) are provided by channel 3 observations, which are centered around λ=1.640µm. The horizontal resolution of the channel 1–3 samples is 3km×3km at the subsatellite point and increases with higher sensor zenith angles. Conversely, the broadband reflectances rHV are sampled at SEVIRI's HRV channel at a horizontal scale of 1km×1km at the subsatellite point. These observations cover the spectral range of 0.4–1.1 µm.

As context for the present study, the reader is reminded that the spatial resolution of geostationary satellites is significantly reduced at higher latitudes due to the oblique viewing geometry. For Germany and Central Europe as considered in this paper, the pixel size is effectively increased by a factor of 2 in the north–south direction as a result. In addition, the distinction between sampling and optical resolution needs to be acknowledged. While the former determines the distance between recorded samples, the latter is given by the effective area of the optical system, which is larger by a factor of 1.6 than the sampling resolution for SEVIRI . The spatial response of optical systems is commonly characterized by their modulation transfer function, which describes the response of the optical system in the frequency domain.

Further information about the spectral width of each SEVIRI channel, as well as the respective spatial response and modulation transfer functions, can be found in .

The 36-band scanning spectroradiometer MODIS, which was launched aboard NASA’s Earth Observing System satellites Terra and Aqua, has a viewing swath width of 2330km, yielding global coverage every 2 d. MODIS collects data in the spectral region between 0.415 and 14.235 µm, covering the VNIR to thermal-infrared spectral wavelength range. In general, the spatial resolution at nadir of a MODIS pixel is 1000 m for most channels, although the pixel dimensions increase towards the edges of a MODIS granule. Only observations from the Terra satellite launched in 1999 are used here, due to broken detectors of the 1.64µm channel of the MODIS instrument on the Aqua satellite. Information on MODIS and its cloud product algorithms is given in , , and . The current version of the level 1b radiance and level 2 cloud products used is data collection 6.1 (C6.1).

In this study, data from a subregion of the full SEVIRI disk has been selected. This region, which is located within the European subregion described in , is illustrated by the red borders in Fig. . It is centered around Germany due to its intended domain of application (thus, from here on it is referred to as the Germany domain) and comprises the latitude and longitude ranges of ≈44.30–57.77∘ and ≈-0.33–21.65∘, respectively. This domain includes 240×400 lower-resolution pixels (i.e., the native SEVIRI resolution of channels 1–3) and is far away from the edges of the full SEVIRI disk, ensuring that the observed viewing zenith angles are <70∘.

Due to the increased sensor zenith angles the spatial resolution of each SEVIRI pixel is degraded. The average pixel size is 6.20km×3.22km and 2.06km×1.07km for channels 1–3 and the HRV channel, respectively. To avoid confusion, we will use the designations LRES (abbreviation for lower resolution) and HRES (abbreviation for higher resolution) scales to refer to the 3km×3km and 1km×1km pixel resolutions from here on.

A relatively small domain was chosen, because the number of pixels to be processed will expand by a factor of 3×3, increasing the computational costs of the subsequent cloud property retrievals by roughly 1 order of magnitude. Except for some regional dependencies introduced by changes in the prevalence of specific cloud types, we expect results of our study to also be valid for other domains.

The statistical downscaling algorithm for the two SEVIRI VNIR reflectances was first reported in and assumes a least-squares linear model that links r06 and r08 to the reflectances in the HRV channel (see ) in the form 1r̃HV=a⋅r06+b⋅r08. Here, the HRV channel observations are first smoothed with the modulation transfer function of the lower-resolution channels, which yields reflectances r̃HV at the same HRES horizontal resolution, adjusted to the low-frequency variability at the spatial scale of the channel 1–3 observations. Subsampling the central pixel of each 3×3=9 pixel block subsequently yields r̃HV at the same LRES horizontal resolution as r06 and r08 (here, the subsampling of the field is denoted by ). The variables a and b are fit coefficients that are determined empirically by a least-squares linear fit. In order to derive a statistically significant and stable linear model, the coefficients a and b are calculated hourly between 08:00 and 16:00 UTC within 16 d intervals. Results for the time step 08:00 UTC are derived from 5 min SEVIRI rapid-scan data between 08:00 and 08:25 UTC, while the 16:00 UTC time step is comprised of SEVIRI observations between 15:30 and 16:00 UTC. For all time steps in between, data are from all samples after minute 25 of the prior hour up to minute 25 of the current hour (e.g., fit coefficients for time step 09:00 UTC are calculated from SEVIRI observations between 08:30 and 09:25 UTC).

Values of hourly derived fit coefficients for the Germany domain between 1 April and 31 July 2013 are shown in Fig. a and b for a and b, respectively. Here, circles represent the respective fit coefficient for each 16 d interval, which is indicated by the first Julian day in the time period. Colors highlight the different UTC time steps. It is obvious that both coefficients a and b are very stable and show no noticeable variation from hour to hour, as well as from one 16 d interval to another. Considering all hourly data and each 16 d interval, the median fit coefficients are 0.63 (for a) and 0.40 (for b), with low interquartile ranges (IQR) of 0.03. The only exceptions are the fit coefficients derived for the first time period of 1–17 April 2013, especially for the morning and afternoon hours of 08:00–09:00 and 13:00–16:00 UTC. Here, a and b deviate significantly from the other results, with values of ≈0.50 and ≈0.52, respectively, likely due to an abundance of observations with large solar zenith angles of θ0>60∘ in the eastern part of the domain.

The high-frequency reflectance variations for the SEVIRI HRV channel (δrHV) are calculated as the difference between the observed rHV and r̃HV, which only resolve the low-frequency variability: 2δrHV=rHV-r̃HV. Following the linear model in Eq. (), the high-frequency variations of the channel 1 and 2 reflectances (δr06 and δr08) are linked to δrHV via 3δr06=S06⋅δrHV,δr08=S08⋅δrHV. The optimal slopes S06 and S08, which minimize the least-squares deviations, can be derived from bivariate statistics: 4k1=b2⋅var(r08)a2⋅var(r06),S06=1+k1⋅cor(r06,r08)a⋅1+k12+2k1⋅cor(r06,r08),k2=a2⋅var(r06)b2⋅var(r08),S08=1+k2⋅cor(r08,r06)b⋅1+k22+2k2⋅cor(r08,r06). Here, cor(r06,r08) is the linear correlation coefficient between the channel 1 and 2 reflectances, while var(r06) and var(r08) are the spatial variances of the respective samples. Note that the sampling resolution of all reflectances is the LRES scale (i.e., ≈3km×3km).

As a result, the high-resolution reflectances r^06 and r^08, which include the high-frequency variations, can be derived from the interpolated reflectances as 5r^06=r̃06+δr06,r^08=r̃08+δr08. Note that only r^06 is used for the retrieval.

Similar steps can be applied for the calculation of r^16. Again, a simple linear model is assumed to connect r16 to the lower-resolution r̃HV at the spatial scales of the channel 1–3 observations: 6r̃HV=c⋅r16. The symbol c is used to denote the respective fit coefficient, which needs to be determined empirically. Similar to the coefficients a and b from the linear model for the VNIR reflectances, c is calculated hourly between 08:00 and 16:00 UTC within 16 d intervals. It has to be noted, however, that in contrast to the VNIR reflectances this fit does not have a clear physical motivation, as there is no spectral overlap with the HRV channel.

The temporal behavior of the fit coefficient c for the Germany domain for the time period between 1 April and 31 July 2013 is shown in Fig. c. In contrast to the coefficients a and b, there is a noticeable trend in the data, both diurnally and during the transition from 1 April to 31 July. For each 16 d interval the variability in the hourly derived c values ranges between IQR=0.05 and 0.15, while the median 16 d value varies between 1.04 and 1.25. Overall, the median c is 1.16, with an IQR of 0.08 (i.e., almost 3 times larger than the one for the coefficients a and b). The observed trends and larger IQR in the c data set shown in Fig. c illustrate that the linear model in Eq. () is not ideal and is expected to introduce significant uncertainties in the calculation of r̃16. This behavior is expected, as the relationship between VNIR and SWIR reflectance can usually not be described by a linear function (see discussions in , as well as the LUT examples in Fig.  later in this study). For a constant reff there is a linear increase in r16 with increasing r06, as the cloud optical thickness increases. However, the slope of this linear relationship increases with decreasing reff. For τ>10 the relationship between r16 and r06 is characterized by a prominent curvature, while for τ≫10 the r16 values become independent of r06. Therefore, the fit coefficients c depend on the distribution of cloud optical and microphysical parameters, which varies widely with cloud type, meteorological conditions, and different dynamic processes.

Values of r̃16 can be derived similarly to Eqs. (–) for the channel 1 and 2 observations: 7δr16=S16⋅δrHV,S16=cov(r16,r̃HV)var(r16),r^16=r̃16+δr16. Note that the use of linear models and bivariate statistics means that the downscaling algorithm described in this section is an example of statistical downscaling techniques, which are common in climate science applications (e.g., ). While for the VNIR channels the spectral overlap with the HRV channel and the spectrally flat properties of clouds provide a sound physical justification for this technique, this is not the case for the SWIR channel.

The reliability of the linear model in Eq. () depends upon the correlation between channel 1 and 2 reflectances (i.e., cor(r06,r08)), as well as the stability of the fit coefficients a and b. The analysis in concludes that the explained variance in the estimates of r^06 and r^08 is close to 1, corresponding to low residual variances, which indicates that the linear model is robust. Moreover, the two fit coefficients are found to exhibit very low variability, as shown in Fig. a–b.

To verify the reliability of the linear model with a large SEVIRI data set, a joint PDF of the actually observed r̃HV and the results from Eq. () is shown in Fig. a; data are from all SEVIRI observations within the Germany domain during June 2013. In case of an ideal linear model, as well as a perfect correlation between the two reflectances, Eq. () would replicate the r̃HV observations. Conversely, deviations from these assumptions will yield different results from the sampled SEVIRI reflectances. It is clear that the linear model can reliably reproduce r̃HV, as most of the observations lie on the 1:1 line, and Pearson's product-moment correlation coefficient (R) is R=0.999. While some larger deviations exist, such occurrences are significantly less likely (i.e., the joint probability density is several orders of magnitude lower than the most-frequent occurrences along the 1:1 line). Regarding r16, the assumption of a linear model is evidently flawed, because the relationship between VNIR and SWIR reflectances depends on the optical and microphysical cloud properties. As a result, a single linear slope, which describes the whole relationship between the two reflectances for all distributions of cloud properties, will introduce significant uncertainties. This is illustrated in Fig. b, where the joint PDF of r̃HV and the results from the linear model in Eq. () are shown. The comparison between the two data sets reveals a much larger spread around the 1:1 line and a lower correlation coefficient. Overall, the relationship resembles the shape of a LUT, displayed in the form of the well-known diagram introduced by , where changes in reff result in a spread in the observed SWIR reflectances (see, e.g., ).

To test the impact of changes in a and b on the derived r^06 and r^08, two experiments are conducted: (i) the fit coefficients are derived only from cloudy pixels and are compared to the higher-resolution results from a and b, which are derived for all pixels; and (ii) the Germany domain is divided into 100km×100km subscenes, and the fit coefficients are derived more locally within each subscene instead of globally from the full domain. Subsequently, statistics from the difference between the two data sets are calculated. Data are from 14 June 2013 at 14:05 UTC. For experiment (i), the 1st, 50th, and 99th percentiles of the relative difference in r^06 (defined as the difference between the reflectances from only cloudy data and the full data set, normalized by the full data set) are -0.08, -0.02, and 0.03%, while for r^08 the analysis yields -0.04, 0.02, and 0.19 %. Similarly, experiment (ii) yields relative differences of -0.08, 0.03, and 0.36% and -0.17, 0.00, and 0.19 % for r^06 and r^08, respectively. These deviations are negligible compared to the measurement uncertainty, and naturally the correlation coefficients between the different data sets are R≈1.00. This confirms the robustness of the linear model described in Eq. (). For the derivation of r^16 from Eq. (), a slightly increased sensitivity to the fit coefficient c is observed. Here, experiment (i) yields percentiles of the relative difference of -0.16, 0.08, and 0.86 %, whereas experiment (ii) results in -0.39, -0.01, and 0.40 %. While slightly higher deviations are observed compared to the linear model for the VNIR reflectances, the uncertainty in r^16 induced by the variability in c is still significantly lower than the measurement uncertainty.

Compared to the downscaling approach in Sect. , where fit coefficients for a linear model are derived over a large temporal and spatial domain, this second method uses local relationships (i.e., on the pixel level) between the SEVIRI reflectances. The constant reflectance ratio approach was introduced by and is based on the assumption that the inhomogeneity index of the HRV reflectance (Hσ,HV, defined as the ratio of standard deviation σHV to the average, pixel-level reflectance r̃HV) equals that for the channel 1 reflectance (Hσ,06). This implies a spectrally consistent subpixel reflectance variability. The relationship can be written as 8Hσ,06=Hσ,HV,σ06r06=σHVr̃HV,19-1⋅∑i=1i=9(r^06,i-r06)2r06=19-1⋅∑i=1i=9(rHV,i-r̃HV)2r̃HV, where the index i=1,2,…,9 indicates any one of the nine available HRES subpixels within a lower-resolution SEVIRI pixel (i.e., at the LRES scale of channels 1–3). This relationship can be further simplified, assuming that this relationship is also true for individual pixels: 9r^06,i-r06r06=rHV,i-r̃HVr̃HV,r^06,irHV,i=r06r̃HV.

The relationship in Eq. () suggests that the ratio of channel 1 and HRV reflectances (i.e., narrowband and broadband VNIR reflectances) remains constant for different scales. Thus, this approach is called the constant reflectance ratio approach.

Finally, we can mitigate some of the scale effects by substituting the lower-resolution variables with the higher-resolution reflectances that resolve the low-frequency variability (i.e., r̃06 and r̃HV) and solve for r^06: 10r^06=rHV⋅r̃06r̃HV.

Similarly, higher-resolution SWIR reflectances r^16 can be derived from 11r^16=rHV⋅r̃16r̃HV. As before, the relationship implies that the ratio of VNIR and SWIR reflectances remains constant for different scales. This assumption has been shown to be reasonable, at least for optically thin (i.e., τ≤10) liquid water clouds over the ocean .

A third method to derive high-resolution cloud property retrievals for SEVIRI utilizes an iterative approach to determine δr06 and δr16 independently, based on the shape of the LUT, while constraining the observed reff to that of the baseline approach (i.e., simple trigonometric interpolation, which yields reflectances r̃06 and r̃16 that only resolve the large-scale variability). While the previous approaches can be implemented as a preprocessor outside the actual retrieval, this method requires access to the LUT and has thus been implemented through modifications of the CPP retrieval algorithm.

Again, a simple linear relationship between δrHV, δr06 and δr08 based on Eq. () is assumed: 12δrHV=a⋅δr06+b⋅δr08, where the fit coefficients a and b are determined from the same techniques as described in Sect. . The variation δrHV of the HRV channel is obtained from the observations following Eq. (), while δr08 is calculated as the difference between r08 from high- and low-resolution optical thickness τ based on the functional relation F of the reflectances and cloud properties stored in the LUT (which motivates the name of this method). Therefore, δr06 can be derived from 13δr06=1a⋅δrHV-b⋅δr08,r^06=r̃06+δr06,δr08=F08τ^,r^eff-F08τ̃,r̃eff. Note that the addition of δr08 in the calculation of δr06 helps to account for the noticeable increase in surface albedo of vegetation-like surfaces at λ>700 nm (i.e., the vegetational step). This should improve the estimation of δr06 for thin clouds (i.e., τ<10) and cloud-edge pixels. For the SWIR reflectance, instead of relying on the imperfect linear model in Eq. () or assumptions about the inhomogeneity index Hσ,16, the adjustment δr16 is determined iteratively to conserve the coarse-resolution, pixel-level (i.e., LRES scale of channels 1–3) value of the effective droplet radius. To reduce some of the associated uncertainties, the effective droplet radius based on the reflectances from triangular interpolation can be used instead of the LRES result. If τ̃ and r̃eff are the cloud properties based on trigonometric interpolation, and τ^ and r^eff are the higher-resolution retrievals, which are derived from an inversion of the functional relationship (F) between the high-resolution reflectances r^06 and r^16 following 14τ^,r^eff=F-1r̃06+δr06,r̃16+δr16, then δr16 can be determined as 15δr16=F16τ^,r^eff=r̃eff-F16τ̃,r̃eff. This implies that a positive or negative δr06 is connected to a positive or negative δr16 using the LUT to adjust the SWIR subpixel reflectance variations in such a way as to be representative of the respective standard-resolution r̃eff. As a result, we do not expect any improvement for the reff retrieval during the transition to smaller scales. Instead, we try to find a physically reasonable constraint for δr16 to achieve a reliable retrieval of the higher-resolution τ^, while retaining the accuracy of the standard-resolution retrieval of r̃eff.

The LUT approach is illustrated in Fig. a, where an example SEVIRI liquid-phase LUT for a specific solar zenith angle (θ0=40∘), sensor zenith angle (θ=20∘), and relative azimuth angle (φ=60∘) is shown. Vertical dashed lines and values below the grid denote fixed τ̃, while the horizontal dashed lines and values to the right of the grid denote fixed r̃eff in units of micrometers. The green dot highlighted by the capital letter A represents an example SEVIRI reflectance pair of approximately r̃06=0.33 and r̃16=0.34, which maps to τ̃=8 and r̃eff=12µm (i.e., the retrieval result for the high-resolution reflectances from trigonometric interpolation). The red line highlights the r̃eff=12µm isoline. The two horizontal blue arrows indicate a positive (δr06,1) and negative (δr06,2) adjustment to r̃06 based on Eq. (). Without an adjustment to r̃16, these newly derived higher-resolution r^06 values map to significantly larger and lower effective droplet radii of about r^eff=29 and r^eff=5µm, respectively. The adjustments δr16,1 and δr16,2 simply assure that the prior effective radius retrieval is preserved (i.e., r^eff=r̃eff). Due to the curvature of the isolines of fixed r̃eff given by the LUT, small deviations of the coarse-resolution average from r̃16 can still occur.

Note that the LUT approach requires a prior cloud phase retrieval (either from the lower-resolution or interpolated reflectances) to determine the correct LUT for either liquid water or ice.

In order to improve the estimation of δr16 in the LUT approach, two modifications to the previous assumption are introduced in this section. The first one aims to provide a more realistic estimate of r̃eff compared to the coarser LRES result, which subsequently is used to determine δr16. The value of r̃eff is derived from adiabatic theory, which provides a physically sound relationship between the derived high-resolution cloud variables: 16r^eff=r̃effτ^τ̃a. Based on observations, the study by confirmed the value of a≈0.2 predicted by theory for marine stratocumulus, so this is the value also adopted here. This approach is illustrated in Fig. b, where the r̃eff retrieval based on the interpolated reflectances at point A is indicated by the red reff isoline. During the first iteration step δr06 is derived from Eq. () and δr16=0, which maps to τ^1 in the LUT (the exponent 1 indicates the first iteration step). This value is highlighted by the vertical blue line. Based on Eq. () the corresponding adiabatic r^eff1 is calculated (highlighted by the horizontal blue line). This value determines the adjustment δr16. Note that the resulting reflectances at point B do not exactly map to τ̃1 after the first iteration. As a result, multiple iterations are necessary to derive the final cloud properties. It has however been relatively simple to merge this iteration into the iterative retrieval loop of the CPP retrieval.

A second approach to improve upon the LUT approach again utilizes the shape of the LUT to derive a local slope S=∂r16/∂r06 from the simulated LUT reflectances. The value of S is calculated at the position denoted by τ̃ and r̃eff. In the iterative CPP retrieval, this requires that both low- and high-resolution cloud properties are estimated during each iteration until convergence of both properties is achieved. This approach is illustrated in Fig. c. Again, the initial r̃eff retrieval based on the interpolated reflectances at point A1 is indicated by the red r̃eff isoline. The slope SA1 at this position in the LUT is highlighted by the solid blue line. Based on the derived slope and δr06 from Eq. () the corresponding δr16 can be calculated for each iteration step. Two additional examples for initial starting points (A2 and A3) and the respective slopes (SA2 and SA3) are also shown. These examples indicate the change in slope for different parts of the LUT. For small τ̃, the slope SA3 become steeper, which leads to a larger adjustment δr16. Meanwhile, for large τ̃>30 (for this specific viewing geometry and LUT) the τ̃ and r̃eff isolines are nearly orthogonal and both the respective slope SA2 and δr16 are close to 0.

Both approaches introduced in this section have advantages and disadvantages but promise to improve on the standard LUT approach. While physically sound, adiabatic assumptions might not always be appropriate, especially for highly convective clouds or in the presence of drizzle. Meanwhile, large δr06 adjustments might map to a point in the LUT where the derived local slopes at the position of τ̃i and r̃effi might not be representative anymore.

To obtain a reliable higher-resolution reference data set, MODIS level 1b swath observations (MOD021km) have been projected to the grid of SEVIRI's rHV samples, which corresponds to the geostationary satellite projection at the HRES scale. Initially, the native HRV grid is oversampled by a factor of 3 in each dimension (i.e., the target grid has a ≈333 m resolution), and nearest-neighbor interpolation is used for the projection. This oversampled field is subsequently smoothed with the modulation transfer function of the HRV channel as given by , to remove high-frequency variability not resolved by the sensor and, in particular, the artifacts introduced by the nearest-neighbor interpolation technique. Finally, this field is downsampled, such that only each central pixel of a 3×3 block (each pixel with a horizontal resolution of 333 m) is retained to represent the HRES value.

To perform the subsequent downscaling experiments, a second set of level 1b radiances are generated, where the spatial variability is reduced to match that of the LRES channels of Meteosat SEVIRI. This step again involves the smoothing of the respective reflectance field with the channel-specific modulation transfer function of the lower-resolution SEVIRI channels . This data set represents hypothetical SEVIRI-like observations at the native LRES.

In addition, a band-pass filter has been constructed from the difference between the modulation transfer functions of the HRV and the 0.6 and 0.8µm channels (weighted by the coefficients of a linear model; see ). This filter is used to extract the high-frequency signal of the HRV channel.

It should be noted that retrievals based upon these radiances will be different than those based upon the original MODIS C6 radiances or from an absolutely accurate representation of the (hypothetical) truly observed, high-resolution SEVIRI samples. For one, it uses the linear model of and as a proxy for the HRV channel, thereby excluding a potentially significant source of uncertainty. Moreover, MODIS acquires these reflectances under different viewing geometries (note that the true viewing angles are used in the CPP retrieval, so within the limits of plane-parallel radiative transfer, this effect is accounted for), and the spectral characteristics of the MODIS and SEVIRI channels are not entirely comparable. However, the goal of this study is to provide a consistent reference data set for a comparison of different retrieval data sets, which are derived from a single retrieval algorithm core. The actual absolute values of the retrieved cloud products are not important here.

Figure a shows a comparison of τ at the native LRES (replicated onto each subpixel) and the reference τ at the HRES scale for the example cloud field in scene 2, which is shown as an RGB composite image in Fig. b. A total of over 13000 cloudy pixels (liquid phase) are located in this scene. While for small reference τ<20 there is a reasonable agreement between the two data sets, there is increased scatter around the 1:1 line (indicated by the dashed gray line) for larger values of cloud optical thickness. For reference τ>40, a substantial underestimation of the LRES τ is observed, which yields a sizable contribution to the nRD of 15.8%. Figure b–c show similar scatter plots of τ and τ^ from both experiment 2 (constant reflectance ratio approach) and 3 (adjusted LUT approach with LUT slope adjustment), respectively. It is obvious that the results from these two downscaling techniques improve the agreement with the reference retrievals significantly. The explained variance (R2, which equals the square of Pearson's product-moment correlation coefficient R) between the data sets is increased, and the nRD is strongly reduced to values of 1.182% (experiment 2) and 1.589% (experiment 3).

A similar comparison between the reference reff at the HRES scale and reff at native LRES, as well as r^eff from the same downscaling experiments, is presented in Fig. d–f. Here, the native-resolution results show a much better agreement with the reference retrievals, and, compared to the cloud optical thickness, the nRD=5.505% is much lower. While experiment 2 exhibits a good agreement between reference τ and τ^, the comparison of retrieved r^eff to the reference results is less favorable. Both the reduced explained variance (R2=0.889 versus R2=0.929) and the increased scatter around the 1:1 line (nRD =6.630%) indicate that the results from experiment 2 are less reliable than the ones performed at the native LRES. Thus, the elaborate downscaling procedure actually reduces the accuracy of the r^eff retrieval. In contrast, the retrieved r^eff values from experiment 3 improve upon the native-resolution results, with slightly better values of R2=0.953 and nRD =4.402%.

Statistics of the comparison between the reference and native LRES, baseline, and experimental retrievals are presented in Fig. a–d for example scenes 1–4, respectively. The parameters which are used to quantify the individual comparisons are the median of the relative difference (abbreviated with p50) to indicate the average deviation from the reference results, the interquartile range (IQR; defined as the relative difference between the 75th and 25th percentile of the deviation to the reference retrievals) to indicate the spread between the different data sets, the nRD as a second measure of the spread of data points, and the explained variance R2 between the different retrievals and the reference. Values with a green and red background highlight the respective experiment with the best and worst comparison for the specific parameter. Yellow backgrounds, meanwhile, indicate all other experiments in between the two extreme results. The first noteworthy observation concerns the native and baseline retrievals of τ, which universally exhibit the largest median deviations and spread to the reference results, as well as the lowest R2. Still, the difference between native and baseline results indicates that the trigonometric interpolation to the HRES grid has significantly improved the comparison.

In contrast, each retrieval of τ^ that accounts for small-scale reflectance variability yields significant improvements, regardless of the approach. This is especially obvious in the parameters that characterize the spread in the deviations, i.e., IQR and nRD, which are between 2–9 and 2–10 smaller for the various experiments and example scenes, respectively. Experiments 2 and 3 seem to achieve the best agreement with the reference retrievals.

Regarding the effective droplet radius, the agreement between the native LRES and (i) baseline retrievals and (ii) the reference results is significantly better. It is worth pointing out that, similar to the optical thickness comparison, the r̃eff retrieval based on interpolating reflectances to the HRES grid performs better than the native-resolution reff retrieval for all scenes. The most reliable downscaling approach seems to be experiment 3, which performs noticeably better than experiments 1 (note the increased nRD and reduced R2 for scene 3) and 2 (increased spread and overall issues for the heterogeneous cloud field in scene 4). This indicates that the linear model in Eq. () or assumptions about a constant ratio of VNIR and SWIR reflectances are not adequate to estimate higher-resolution r^16, at least not for certain cloud conditions. In the case of experiment 2 this is understandable, because the technique was developed for partially cloudy pixels . These observations are characterized by a low cloud optical thickness, where the relationship between VNIR and SWIR reflectance can reliably be considered to be linear (see example LUTs in Fig. ).

There is a notably better performance of experiment 3, the adjusted LUT approach with LUT slope adjustment, compared to the standard LUT approach highlighted in Sect. . Of particular note is the r^eff retrieval based on the standard LUT scheme, which compares significantly worse to the reference results (R2 of 0.890, 0.648, 0.751, and 0.581 for cloud scenes 1–4, respectively). This is somewhat surprising, because the specified goal of the standard LUT approach is to maintain the accuracy of the baseline r̃eff retrieval, which has not been fully reached. We believe that this might be caused by the sensitivity of the cloud property retrieval to small reflectance perturbations, in particular for broken clouds. It is also an indication that assuming constant subpixel reff values within each LRES pixel is not sufficient. We plan to investigate this effect further in future studies. However, the second adjusted LUT approach, which determines SWIR reflectance adjustments based on adiabatic theory, performs even worse (R2 of 0.846, 0.579, 0.741, and 0.519 for cloud scenes 1–4, respectively). This suggests that the observed cloud fields do not follow adiabatic theory and the method is not adequate to estimate higher-resolution r^16.

Retrievals of τ and reff (regardless of the resolution they are derived at) provide the means to infer other commonly used cloud variables. The WL, which describes the amount of liquid water in a remotely sensed cloud column, can be derived as the product of retrieved cloud products : 17WL≈23⋅ρL⋅τ⋅reff. Here, ρL is the bulk density of liquid water. Assuming adiabatic clouds, where the vertical structure of effective droplet radius follows the adiabatic growth model, introduces an extra factor of 5/6 and the coefficient 2/3 changes to 5/6⋅2/3=5/9. Meanwhile, ND describes the number of liquid cloud droplets in a cubic centimeter of cloudy air. Calculating ND from remote sensing products requires a number of assumptions, e.g., about the vertical cloud structure and shape of the droplet number size distribution, which are summarized and discussed in , , , and . A simplified form of the resulting equation for ND is 18ND≈α⋅τ0.5⋅reff-2.5, with α=1.37⋅10-5 (see ). Note that Eqs. ()–() can yield both baseline and downscaled results (i.e., W̃L and ÑD, as well as W^L and N^D) when they are derived from the respective cloud optical thicknesses and effective droplet radii.

Similar to the comparison in Sect. , scatterplots of the reference WL, the native LRES WL, and the results from the downscaling experiments 2 and 3 (W^L) are shown in Fig. a–c, respectively. As before, data are provided by example scene 2 sampled on 9 June 2013 at 10:55 UTC. Compared to the native LRES results, a noticeable improvement in the correlation and nRD is achieved by utilizing the two downscaling experiments. Not only are retrieved W^L values closer to the 1:1 line, but the significant underestimation of the LRES WL values for larger reference results is mitigated. Especially for experiment 3, the spread is less than one-third of the value of the LRES results (4.857% versus 15.234%). Regarding the comparison between reference and native ND, as well as N^D, downscaling experiment 2 yields less favorable results. There is a slight decrease (increase) in R2 (nRD). This is caused by the large IQR and nRD of the deviations in the retrieved r^eff, shown in Fig. e, which are amplified due to the associated power of 2.5 in Eq. (). However, the derived values from experiment 3 show a significantly better agreement with the reference ND.

Values of p50, IQR, nRD, and R2 for the WL and ND comparison from the four example scenes are illustrated in Fig. a–d. Due to the large deviations between the native τ and the reference retrievals, WL values for the LRES results almost universally show the largest deviations to the reference values and thus the largest IQR and nRD, as well as the lowest explained variance. The exception is the heterogeneous cloud field in the fourth example scene, where the large deviations between r^eff from experiment 2 and the reference retrievals yield the worst comparison for the respective W^L. The estimates based on the adjusted lookup table approach using the LUT slope adjustment (i.e., experiment 3) almost universally exhibit the best agreement with the reference results of WL.

Overall, 27 of the 32 comparisons (four cloud scenes, two cloud variables, and four statistical measures) exhibit the best performance for experiment 3. For the example scenes considered in this analysis, it is obvious that the adjusted lookup table approach with LUT slope adjustment is preferable to the other downscaling techniques and yields more reliable high-resolution cloud variables than the standard LRES results.

As before, we also tested the standard LUT approach highlighted in Sect. , as well as the second adjusted LUT approach, which determines SWIR reflectance adjustments based on adiabatic theory. Due to the poor performance of the r^eff retrieval, the N^D results based on adiabatic assumptions show a similarly poor agreement with the reference results. Meanwhile, the cloud variables based on the standard LUT approach never show the best or worst performance but are almost universally worse than the adjusted lookup table approach with LUT slope adjustment. This again illustrates that assumptions of adiabatic clouds and constant subpixel reff values within each LRES pixel are not suitable for the cloud scenes analyzed in this study.