1. Introduction
Clear sky satellite radiances have become the most important observations for numerical weather prediction (NWP) systems, and there is a rising interest to use also cloud-affected radiances. Satellite observations are not only utilized for data assimilation (see Bauer et al. 2011a,b, and references therein) but also for model evaluation (see e.g., Bikos et al. 2012; Heinze et al. 2017) and retrieval algorithm validation (see e.g., Bugliaro et al. 2011). All these applications require a forward operator, which computes synthetic satellite images from the NWP model state. Forward operators have to be sufficiently accurate and consistent with the assumptions made in the NWP model (see the discussion in Senf and Deneke 2017). Moreover, for operational purposes, forward operators also have to be very fast and are strongly constrained in terms of memory consumption. Suitable forward operators for thermal infrared and microwave satellite instruments have been available for some time and have been continuously improved. In most of the weather prediction centers, either the RTTOV radiative transfer package (Saunders et al. 1999) or the Community Radiative Transfer Model (Han et al. 2006) is used. In contrast, sufficiently fast forward operators for solar channels are only in an early development state.
Thermal satellite channels provide mainly temperature and humidity information with high spatial and temporal resolution and have proven to provide valuable observations for data assimilation (DA). While clouds also affect infrared radiances and this cloud effect could be exploited for data assimilation (see, e.g., Harnisch et al. 2016), more detailed and complementary information about clouds is contained in visible satellite images. However, these images have not been exploited for data assimilation so far, mainly because generating their model equivalents was computationally too expensive. This is related to two fundamental differences between visible and infrared images. First, radiances in the solar part are dominated by multiple scattering, while scattering can usually be neglected in the thermal part. Therefore, computationally cheap methods developed for thermal infrared channels cannot be applied for visible channels. Second, infrared radiances are determined by thermal emission of the atmosphere and the surface, whereas visible radiances are determined by the reflection of directional radiation from the sun by the atmosphere and the surface. The position of the sun is not required for the computation of synthetic infrared images and assuming a plane-parallel atmosphere is a sufficiently good approximation in this case. In contrast, the sun–satellite geometry is decisive and 3D radiative transfer (RT) effects are much more important for the visible channels.
Accurate RT solvers taking 3D effects and multiple scattering into account are available for the visible spectral range [e.g., the Monte Carlo code for the physically correct tracing of photons in cloudy atmospheres (MYSTIC) 3D Monte Carlo solver; see Mayer B. 2009], but they typically take several CPU days to generate one satellite image. Forward operators based on standard 1D RT methods like the one developed by Kostka et al. (2014) in the framework of the Hans-Ertel Centre for Weather Research (HErZ) project (Weissmann et al. 2014) are less accurate and faster, but they are still too slow for operational purposes. Only recently sufficiently fast 1D RT methods for visible wavelengths based on lookup tables have become available (Jonkheid et al. 2012; Scheck et al. 2016a). While these methods allow for efficient treatment of multiple scattering, they do not yet provide a full solution for the problem of generating accurate and consistent synthetic visible satellite images. The two presumably most important aspects still missing are 3D RT effects and the consistent treatment of subgrid-scale clouds.
In this work we discuss and evaluate methods to improve the accuracy and consistency of a forward operator based on the fast 1D RT method of Scheck et al. (2016a) without strongly degrading its performance. The first topic we addresses is a 3D RT effect related to the fact that the radiance observed for a cloud-top surface changes when it is tilted toward or away from the sun. The second topic is the overlap of subgrid clouds. While subgrid cloud water content assumptions consistent with the NWP model have already been used in Kostka et al. (2014), the overlap of the subgrid clouds was not taken into account. Several articles have discussed how to include cloud overlap in large-scale models with the aim to improve heating rates and surface irradiances. However, it is not clear what is the most consistent and efficient way to implement cloud overlap in the context of synthetic visible satellite images generated from convection-permitting models with higher resolution. Here we compare several methods and discuss the differences between stochastic (STO) or deterministic (DET) implementations and the impact of 3D effects related to the slant satellite viewing angle.
The rest of the paper is structured as follows: In section 2 we discuss the NWP model, the RT method, the satellite data, and the test periods used in this work. Section 3 contains a description of the newly implemented operator features that take cloud-top inclination and subgrid cloud overlap into account. In section 4 we compare images generated by the operator to Monte Carlo 3D RT results, and in section 5 we compare synthetic images based on operational forecasts to 0.6-μm observations of the SEVIRI instrument on Meteosat. Last, section 6 contains a summary and conclusions.
2. Data and methods
a. COSMO forecasts
The NWP model states for which we compute synthetic satellite images are 3-hourly operational forecasts of the German Weather Service [Deutscher Wetterdienst (DWD)]. These forecasts have been generated using the nonhydrostatic limited-area COSMO community model in its German-focused convection-permitting model configuration COSMO-DE. This model configuration will also be employed for future DA experiments we intend to perform with the Kilometre-Scale Ensemble Data Assimilation (KENDA) system (Schraff et al. 2016) using an implementation of the local ensemble transform Kalman filter (LETKF; see Hunt et al. 2007). The model domain consists of 421 × 461 grid points with a horizontal grid spacing of 2.8 km and 50 layers in the vertical. The model domain covers Germany, Switzerland, and Austria, and parts of their neighboring countries. The model allows for deep convection on the grid scale and has a parameterization for shallow convection (Baldauf et al. 2011). The heating and cooling rates due to radiation are computed according to Ritter and Geleyn (1992) using a two-stream solver. In this process subgrid clouds are assumed to exist in grid cells where the relative humidity exceeds a height-dependent critical value (see Senf and Deneke 2017).
The period from 28 May to 30 June 2016 is used to evaluate the operator. For some additional tests we consider also a shorter period in 2012 (10–28 June).
b. SEVIRI observations
The Spinning Enhanced Visible and Infrared Imager (SEVIRI) aboard the geostationary Meteosat Second Generation (MSG) satellites operated by EUMETSAT provides full-disc observations in 11 spectral channels in the visible and infrared spectral range every 15 min. Here we focus on the 0.6-μm satellite images from Meteosat-10, which was located at 0° longitude during the 2016 test period. The spatial resolution of the visible and near-infrared channels is 3 km × 3 km at the subsatellite point and about 6 km × 3 km in the COSMO-DE domain.
To derive albedo values required for the RT calculation, we use also the SEVIRI Clear Sky Reflectance Map (CRM) product (EUMETSAT 2015, 266–267), which is computed as the average reflectance value for all cloud-free cases found in the past seven days. If no cloud-free cases occurred in this period, then data from the next earlier 7-day period or a climatological value is used. The clear sky product is generated for every day for 1200 UTC and on Wednesdays also for every even hour between 0600 and 2000 UTC. We derive a time-dependent albedo from the CRM by starting from a value of zero and iteratively adjusting the albedo value until the resulting reflectance agrees with the clear sky reflectance. Linear interpolation is used to compute albedo values for times where no CRM is available. The algorithm used to compute the CRM product often causes patches with significantly too-high reflectances. Using the minimum CRM value within the test period (at the same time of the day) for each pixel strongly reduces these artifacts.
c. Radiative transfer and optical properties
The operator version of Kostka et al. (2014) made use of the discrete ordinate method (DISORT; Stamnes et al. 2000) to compute reflectances. Here we use the newly developed Method for Fast Satellite Image Synthesis (MFASIS; Scheck et al. 2016a), which is based on compressed reflectance lookup tables (LUTs) computed with DISORT. The 21-MB
For each atmospheric column, MFASIS requires the vertically integrated optical depths and mean effective particle sizes of water and ice clouds as input variables. These variables are computed in the same way as in Kostka et al. (2014). The effective droplet radius is determined following Martin et al. (1994) and for the effective ice particle size we use the parameterization of Wyser (1998). Water and ice contents are converted to optical depths using the parameterizations of Hu and Stamnes (1993) and Fu (1996), respectively. We used subgrid cloud water and ice contents and cloud fractions that are consistent with the assumptions used in the internal RT code in the COSMO Model. In the current version, aerosols are not yet taken into account.
3. Operator improvements
a. Cloud-top structure
Synthetic satellite images in the visible and near-infrared obtained by 1D RT methods show much less structure within the clouds than images generated using 3D RT methods or real satellite images (see Fig. 6 in Kostka et al. 2014), in particular for large solar zenith angles (SZAs). The missing structure can thus be considered as a systematic error that is mainly related to 3D RT effects. Three-dimensional RT methods like Monte Carlo ray tracing are orders of magnitude too slow for operational purposes. However, the 3D RT effect dominating the cloud structure is rather simple and can be approximately taken into account with a computationally cheap method. The clouds are darker where the cloud-top surface is tilted away from the sun, and they are brighter where the surface is tilted toward the sun. This effect is simply related to the fact that the same number of photons from the sun is distributed over a large or a smaller cloud surface area. The orientation of the cloud surface with respect to the satellite is also important, as it determines whether the surface is visible from the satellite and which solid angle it occupies.
This variation in brightness is a local effect in the sense that to first order it depends on only the direct radiation from the sun and the local morphology of the cloud. Only when the cloud surface is tilted so far away from the sun that the diffuse illumination of the cloud is of similar importance as the direct radiation, nonlocal effects must not be neglected anymore and the RT problem becomes much more complicated. However, for not-too-extreme SZAs, this is a rather rare case in geostationary satellite images.







Illustration of the tilted cloud problem in a (a) standard frame and (b) rotated frame of reference. For the sake of simplicity, we show here the case where the sun, the satellite, and the cloud surface normal vector all lie on the same plane.
Citation: Journal of Atmospheric and Oceanic Technology 35, 3; 10.1175/JTECH-D-17-0057.1























In principle another correction term accounting for deviation of the actual cloud-top height from the one used in the calculation of the MFASIS LUT could be added to Eq. (5). This would require an additional LUT dimension and for the 0.6-μm channel we focus on in this work, the reflectance correction would be very small. Therefore, we omit this term here. For channels more strongly affected by water vapor absorption or Rayleigh scattering, it could be necessary to include such a term.

























b. Point spread function

Applying the point spread function is particularly important when the cloud-top inclination is taken into account. Otherwise, sharp features with a too-high contrast related to locally tilted cloud tops appear. Applying the PSF blurs these sharp features and leads to minimum and maximum reflectance values that are in better agreement with the observations.
c. Cloud overlap schemes
NWP and climate models cannot resolve clouds on scales smaller than their grid length. As the influence of these clouds is not negligible, the models rely on parameterizations to account for these unresolved clouds on subgrid scales. Often these parameterizations generate cloud water and cloud ice content in cells where the grid-scale cloud water variable is zero but the relative humidity is higher than a height-dependent critical value (for an overview, see Quaas 2012). For global and convection-permitting models these partially cloudy grid cells, in which the subgrid cloud fraction variable γ is between zero and one, constitute an important contribution to the total cloud cover.
When several partially cloudy grid cells are present in the same column of the model grid, the radiative transfer calculation requires assumptions on how subgrid clouds overlap. The operator version discussed in Kostka et al. (2014) made use of the subgrid cloud water and ice content parameterizations implemented in COSMO. However, in contrast to the RT implementation in COSMO, it was assumed that the subgrid clouds fill the whole grid cell, that is,





While more sophisticated cloud overlap assumptions exist (see, e.g., Di Giuseppe and Tompkins 2015, and references therein), we restrict ourselves here to the maximum-random assumptions that are consistent with the internal two-stream RT code in COSMO. The response of a global model to changes between these assumptions that were used in the computation of heating rates has been investigated by Morcrette and Jakob (2000). Here we are interested in convective scales and the impact of taking the slant satellite viewing angle into account, which can be ignored in global models. We implemented four different methods to take maximum-random cloud overlap into account. These methods differ in the assumed subgrid cloud size distribution, the required computational effort, and whether the stochastic nature of the subgrid clouds and the slant satellite viewing angle are considered. In addition, an implementation of the random overlap assumption will serve as a reference. In each of these methods, the column containing partially cloudy layers is subdivided into N subcolumns in which the cloud fraction is either 0 or 1 in each layer. The reflectance for the column is computed as the weighted average of the reflectances for the subcolumns. Examples of subcolumn cloud configurations generated by the different methods are given in Fig. 2 for a column containing four partially cloudy layers.
Maximum-random cloud overlap examples for a cloud layer between
Citation: Journal of Atmospheric and Oceanic Technology 35, 3; 10.1175/JTECH-D-17-0057.1
The first method is deterministic and aims to estimate the mean reflectance resulting from all subgrid cloud realizations that are allowed according to the cloud overlap rules. Method DET is the “streams” approach (Matricardi 2005) used in the RTTOV package. Here the total cloud cover is computed for each layer according to Eq. (15). Adopting a dimensionless coordinate
In addition to this deterministic cloud overlap scheme, several stochastic schemes were implemented. These schemes generate only one random subgrid cloud realization compatible with the overlap rules. It should be noted that by design these stochastic schemes are not differentiable, which makes them unsuitable for variational DA methods. However, there is a general trend toward DA methods that do not require an adjoint model and exact differentiability and use information derived from ensembles instead (see, e.g., Gustafsson et al. 2018). These methods are based on the assumption that changes in the model state and the operator output can be described approximately by a linear relation. This may not be the case for the reflectance of a single pixel that depends on random numbers used in a stochastic scheme. However, one can use the standard approach of averaging over several pixels to obtain a “superobservation” that is nearly free of stochastic noise and depends on a sufficiently linear way on the model state.
Methods STO-N and STO-C (refer to the appendix) use a uniform decomposition of the column into N subcolumns, where, in contrast to the DET scheme, N can be chosen a priori. Clouds are placed into the subgrid cells according to nondeterministic rules. Details on the algorithms used in these methods are given in the appendix. Here we discuss only the most important differences. Method STO-N computes the probability
Finally, the last cloud overlap scheme, STO-C3D, is also stochastic and takes into account that the maximum-random overlap assumption, as used in the internal RT code in COSMO, applies for vertical columns, whereas the operator makes use of TICA (see Wapler and Mayer 2008) to account for the satellite viewing angle θ. A group of maximum overlapping cloud layers should be stacked on top of each other in the vertical direction, not along the tilted column. Descending in the tilted column by one layer of height z causes a horizontal displacement
Example of a subgrid cloud distribution generated by the STO-C3D method for a satellite viewing angle of 45°. (a) Projection onto the x–z plane. Subgrid columns tilted toward the satellite (solid). Subgrid cells for which
Citation: Journal of Atmospheric and Oceanic Technology 35, 3; 10.1175/JTECH-D-17-0057.1
It should be noted that according to the reciprocity principle not only is the slant viewing path important but also the slant path of the photons arriving from the sun, which means in particular that cloud shadows must be taken into account. With the current version of MFASIS this is not yet possible. Thus, STO-C3D is more consistent than the other schemes, but it does not yet provide a full solution for the problem. However, STO-C3D would provide a full solution in this sense for thermal channels, for which the direct radiation from the sun is irrelevant.
4. Comparison with 3D Monte Carlo results
a. Cloud-top inclination
To verify that the cloud-top inclination correction discussed in section 3a improves the results, synthetic MFASIS images generated with and without the correction are compared to reference results obtained with the 3D Monte Carlo code MYSTIC (Mayer B. 2009).
For this comparison output from the Icosahedral Nonhydrostatic (ICON) model runs with a 150-m grid resolution performed in the framework of the High Definition Clouds and Precipitation for Advancing Climate Prediction [HD(CP)2] project was mapped onto a regular latitude–longitude grid with a grid spacing of about 1 km and used to compute input data for the RT. Details on the model run settings and a model evaluation can be found in Heinze et al. (2017). Several simplifications were adopted for the RT computations. Cartesian geometry and constant viewing and sun angles are assumed. As MYSTIC does not support subgrid clouds, the latter were removed. For this purpose, in grid cells with a cloud fraction higher than 0.5 the cloud fraction was set to one and in all other cells to zero. Because of the computational effort required for 3D Monte Carlo simulations, we considered only 7 days (listed in Fig. 7a). The cloud situation varies strongly from day to day and also within the domain and includes both water and ice clouds. For each day only the 1200 UTC model output was used, but we considered three different sun angle combinations that roughly correspond to the situation at 0600, 0900, and 1200 UTC in June. In the following 0600 UTC means





Synthetic images for the model state from (left) 15 Aug 2014 and the 0600 UTC solar angles computed with MYSTIC, (middle) MFASIS, including the inclination correction; and (right) pure 1D MFASIS.
Citation: Journal of Atmospheric and Oceanic Technology 35, 3; 10.1175/JTECH-D-17-0057.1
Reflectance histograms computed for the three methods for all days and three solar angle combinations. (a) 0600, (b) 0900, and (c) 1200 UTC cases. A reflectance bin size of 0.05 was used.
Citation: Journal of Atmospheric and Oceanic Technology 35, 3; 10.1175/JTECH-D-17-0057.1
Histogram error [Eq. (16)] of the reflectance fields computed using MFASIS with (blue) and without (red) the CTI correction with respect to the MYSTIC reference solution for all test cases. Solar angles are used in the (a) 0600, (b) 0900, and (c) 1200 UTC cases. Numbers on the x axis indicate the date and region of the case; see Fig. 7a.
Citation: Journal of Atmospheric and Oceanic Technology 35, 3; 10.1175/JTECH-D-17-0057.1
As in Fig. 6, but for the reflectance RMSE. Dates of the test cases are provided in (a). Last character indicates around which point the region for the RT calculations was centered, where A is located at 51°N, 7°E and B is located at 51°N, 12°E.
Citation: Journal of Atmospheric and Oceanic Technology 35, 3; 10.1175/JTECH-D-17-0057.1
As in Fig. 7, but for the reflectance bias. Mean biases is denoted (dashed lines).
Citation: Journal of Atmospheric and Oceanic Technology 35, 3; 10.1175/JTECH-D-17-0057.1
Larger differences between MFASIS and MYSTIC often result from the violation of the assumptions made for independent column 1D RT or the inclination correction. In 1D RT it is assumed that the clouds extend to infinity in horizontal direction. No photons can enter or leave the cloud through the cloud’s sides. In reality and in 3D Monte Carlo simulations, there is a photon flux through the cloud’s sides, which can lead to a positive or negative reflectance bias, compared to 1D RT. In particular, clouds with a high aspect ratio (height divided by horizontal size) and intermediate optical thickness are problematic in this respect, as the cloud’s sides contribute a larger fraction to the total cloud surface area and photons entering through the sides have a higher probability of reaching the cloud top than in very thick or very thin clouds. Moreover, in the independent column approach, clouds in one column cannot cast shadows on other columns. In particular for large SZAs and clouds with a high aspect ratio, not accounting for the extinction of direct radiation by clouds can lead to a positive bias, which can also be seen here in Fig. 8. Finally, the inclination correction assumes there is a sufficiently well-defined cloud surface and that the inclination angle is smaller than 90°. These assumptions can be invalid, for example, in pixels where mainly cloud sides are visible.
While these problems exist and should be addressed by future developments, our results indicate that for most optically thick clouds and clouds with a not-too-extreme aspect ratio, 1D RT together with the inclination correction is a reasonable and computationally much cheaper approximation to Monte Carlo 3D results and thus presents a significant improvement compared to pure 1D RT. It should also be noted that the seven test days used here were chosen to be simulated in the framework of the HD(CP)2 project because they are characterized by strong convection or other extreme weather events. Therefore, cases that should be less sensitive to the missing 3D RT effects discussed above—for example, mainly stratiform clouds—are underrepresented in this sample. The mid- to low-reflectance bias discussed above should thus on average be less severe for samples randomly drawn from the climatology (see section 5).
b. Cloud overlap
It would be interesting to test also the overlap schemes discussed in section 3c by means of a comparison with Monte Carlo results. Such a comparison could indicate how important it is to take not only the slant viewing path into account but also the slant path to the sun for partially cloudy columns. This would require constructing an ensemble of high-resolution model states with known cloud overlap statistics below a certain spatial scale, to compute MYSTIC images for this ensemble and to compare them to MFASIS images computed for a version of the model state that has been coarsened to that scale. However, because of the bias related to missing 3D effects discussed above, in particular the missing cloud shadows, such a comparison would favor the overlap implementation that best compensates for the existing bias and not the one providing the best representation of the cloud overlap statistics for the high-resolution data. Therefore, we refrain from a comparison with MYSTIC here and will just estimate the potential impact of taking the slant viewing path into account in the next section.
5. Comparison with SEVIRI observations
a. Setup and metrics
The impact of the operator improvements described in section 3 is investigated using 0.6-μm SEVIRI observations and model equivalents computed from COSMO-DE forecasts for a 34-day test period in May–June 2016. To reduce potential lateral boundary effects, we consider only the part of the COSMO-DE domain that is at least 12 grid cells away from the boundaries for the evaluation of the results. Moreover, to avoid problems with snow and steep orography, we exclude the region with longitudes between
Several metrics are considered to evaluate how well the synthetic satellite images agree with the observed ones. We define the cloudiness C as the fraction of pixels in which the reflectance is higher than a critical value Rc of 0.2, which is an upper limit for the clear-sky reflectance in the considered domain for the 0.6-μm SEVIRI channel. This simple definition is not perfect, as it may miss some thin clouds over darker ground. However, the error should be rather small and this definition has the advantage that cloudiness can be computed in a consistent way for both observed and synthetic images. The cloudiness bias
b. Cloud-top inclination
An example of images computed with and without taking the cloud-top inclination (CTI) into account for a 0600 UTC case can be found in Fig. 9. To make it easier to distinguish clouds from the ground, not only the 0.6-μm but also the 0.8-μm channels have been used to generate these images (see figure caption). Without CTI, dense clouds are visible as uniformly white areas without features (Fig. 9c). With the inclination, much more structure is visible in the synthetic image (Fig. 9b). Although some variability on the finest scales is still missing, the synthetic image is qualitatively much more similar to the observation than the pure 1D result. For instance, a band of high clouds starting at the middle of the left domain boundary is barely visible without the inclination correction and much easier to see with the correction. A similar feature is present in the SEVIRI image. Moreover, it is now possible to distinguish convective from stratiform clouds and to see boundaries between low and high clouds. For smaller SZAs, the impact of the CTI correction is weaker but still present.
(a) SEVIRI observation for 0600 UTC 3 Jun 2016 (mean SZA: 66.1°) (b) Synthetic image with CTI. (c) Synthetic image without CTI. To generate these red–green–blue (RGB) images, the 0.6-μm reflectance was used for the red channel, the 0.8-μm reflectance for the green channel, and the mean value of the 0.6- and 0.8-μm reflectances for the blue channel.
Citation: Journal of Atmospheric and Oceanic Technology 35, 3; 10.1175/JTECH-D-17-0057.1
Taking the CTI into account also leads to improved reflectance distributions, in particular for larger SZA. As is visible from the reflectance histograms for 0600 and 1800 UTC in Fig. 10, the slope of the histogram curve at high reflectances (
Reflectance histogram for all days of the May–June 2016 period at (a) 0600, (b) 0900, (c) 1500, and (d) 1800 UTC computed from SEVIRI observations (gray) and operator results with (blue) and without (red) taking the CTI into account. Experiments in which the inclination correction term was multiplied by 3.
Citation: Journal of Atmospheric and Oceanic Technology 35, 3; 10.1175/JTECH-D-17-0057.1
While the slope of the histograms in Fig. 10 is improved by the inclination correction at high reflectances, in particular for 0600 and 1800 UTC, other problems are still present. The dense cloud peak is still located at too-high reflectances for the model results. Moreover, for 0600 and 1800 UTC there are too many clouds in the model, which is indicated by ground peaks at
Figure 11a shows more quantitatively that in particular for large SZA
(a) Histogram error, (b) cloudiness bias, and (c) reflectance bias as a function of the time of the day averaged over all days of the May–June 2016 test period with (blue) and without (red) taking the CTI into account and for case in which the inclination correction term was multiplied by 3 (green). Standard deviations are denoted (black lines).
Citation: Journal of Atmospheric and Oceanic Technology 35, 3; 10.1175/JTECH-D-17-0057.1
c. Cloud overlap
The deterministic and stochastic cloud overlap methods discussed in section 3c differ in important aspects. Only the methods DET and STO-N adhere to the total cloud cover formula [Eq. (15)]; only STO-C, STO-R, and STO-C3D reproduce the desired cloud fraction profile in each realization; and only STO-C3D takes the slant satellite viewing angle into account. For certain profiles of the cloud fraction and optical depths, the methods can in some cases yield strongly differing results, as the idealized examples in section 3c demonstrate. Here, we investigate how much influence the choice of the overlap method has on synthetic satellite images of the test period, that is, for profiles that can be regarded to be more realistic. As discussed in section 3c, the 3D scheme is more consistent than the 2D schemes, but it is still restricted in the sense that it does not account for cloud shadows (like all the other schemes). To minimize this effect, we will mainly focus on only the 1200 UTC cases in the following paragraphs.
We will first consider only results obtained for
(a) Cloudiness at 1200 UTC for all days of the May–June 2016 test period computed from observations (gray line) and model equivalents. In addition to the results generated using the STO-C and STO-R cloud overlap methods, we show the results obtained by neglecting the subgrid clouds and setting the cloud fraction of the subgrid clouds to 1. (b) Cloudiness bias (averaged over all days) as a function of the time of the day for the overlap methods DET, STO-N, STO-C, STO-R, and STO-C3D. Standard deviations are denoted (vertical lines).
Citation: Journal of Atmospheric and Oceanic Technology 35, 3; 10.1175/JTECH-D-17-0057.1
Reflectance histogram for all days of the May–June period at 1200 UTC computed from SEVIRI observations (gray) and operator results with different cloud overlap assumptions.
Citation: Journal of Atmospheric and Oceanic Technology 35, 3; 10.1175/JTECH-D-17-0057.1
Despite all of the differences between the different overlap methods, the results for the cloudiness are rather similar. The average cloudiness bias in the test period depends more strongly on the time of the day than on the overlap method (Fig. 12b). The 2D stochastic maximum-random overlap methods and the deterministic method lead to nearly identical cloudiness biases (Fig. 12b) and reflectance histograms (not shown). The 2D random and the 3D maximum-random images are somewhat brighter than the 2D maximum-random images. However, these relatively small differences in global metrics are somewhat misleading. To get a clearer view on the differences between the overlap methods and separate systematic from random deviations, we performed additional experiments. For each of the stochastic methods, an ensemble consisting of 100 different realizations was computed for a shorter test period (10–28 June 2012) and only at 1200 UTC. An example of the ensemble mean reflectance computed with STO-C is displayed in Fig. 14a. For this case the difference between the STO-R and the STO-C ensemble means is shown in Fig. 14b. It is obvious that for some parts of the scene, the difference of the ensemble mean reflectances can exceed 0.1. However, only in about
(a) Synthetic SEVIRI 0.6-μm image for 14 Jun 2012 computed with the STO-C maximum-random overlap method. (b) Difference between STO-R and STO-C for the same day (positive values indicate higher reflectance for STO-R). (c) Standard deviation of the reflectance for an ensemble of 100 images computed using STO-C. In all cases 256 subcolumns were used.
Citation: Journal of Atmospheric and Oceanic Technology 35, 3; 10.1175/JTECH-D-17-0057.1
Differences between the ensemble mean reflectance distributions obtained with different stochastic overlap implementations and the distribution generated by the deterministic DET method. Bias, RMS difference, 95th percentile of the deviation, and maximum deviation, averaged over all 1200 UTC images for the June 2012 period.
For data assimilation the ensemble mean value and the reflectance spread are of interest. The spread generated by a stochastic method quantifies the uncertainty related to the unknown subgrid cloud configuration. However, like the mean reflectance, the reflectance spread also can be affected by discretization errors related to the finite number N of subcolumns used in the stochastic schemes. As discussed in section 3c, the STO-N method implicitly assumes that the subgrid cloud sizes go to zero for
(a) Maximum deviation, 99th percentile of the deviation, and root-mean-square deviation in ensemble mean reflectance for the stochastic maximum-random overlap method STO-N with different numbers of streams relative to the 512 stream case computed for the June 2012 test period. Ensembles with 100 members were used. (b) As in (a), but for the STO-C implementation.
Citation: Journal of Atmospheric and Oceanic Technology 35, 3; 10.1175/JTECH-D-17-0057.1
6. Summary and conclusions
We have proposed efficient methods to increase the accuracy and consistency of a forward operator for visible satellite images based on a fast 1D RT method. Two problems were addressed: how to model the variation of reflectance related to the inclination of cloud-top surfaces and how to account for the overlap of subgrid clouds.
Using a transformation into a frame of reference aligned with the tilted cloud top, we reduce the 3D RT problem to a nearly one-dimensional problem that can be approximately solved with a fast 1D RT method. A comparison with 3D Monte Carlo RT calculations based on high-resolution model runs confirms that the inclination correction reduces errors, in particular for dense clouds. A comparison of synthetic images based on operational COSMO Model forecasts with 0.6-μm SEVIRI observations show that the information content of synthetic images is increased significantly and that the reflectance histograms are improved when the cloud-top inclination is taken into account. In this case a tuning factor had to be introduced to compensate for the fact that the lower model resolution leads to gradients that are too weak. The additional information visible in synthetic images computed with the cloud-top inclination 3D correction could be a significant advantage for DA. Further studies are required to investigate how this information can be exploited.
The comparison with 3D Monte Carlo RT calculations also demonstrates that several other 3D effects still not accounted for can cause significant errors. Photon transport through the cloud sides can lead to a positive of negative reflectance bias and is especially problematic for clouds with intermediate optical depths. Moreover, cloud shadows were not taken into account, which leads to a positive reflectance bias. In particular for high clouds or larger SZA, a way to simulate cloud shadows could reduce errors and provide additional features that could be useful for DA. A preliminary version of a cloud shadow approximation was already used to generate synthetic satellite images for Heinze et al. (2017). For more accurate approaches, methods to estimate the diffuse radiation similar to the one of Wissmeier et al. (2013), who considered surface irradiances, could also be useful for top-of-atmosphere reflectances. The development of methods accounting for further 3D RT effects in an efficient way could be facilitated once fast 3D RT methods (see, e.g., Jakub and Mayer 2015) are applied within the NWP model and provide information about the diffuse radiation field.
Ignoring the problem of subgrid cloud overlap causes large errors in the cloudiness. Several different implementations of maximum-random cloud overlap yielded similar results and a much better agreement with the observed cloudiness. A 3D maximum-random approach taking the slant satellite viewing path into account is the most consistent method and leads to results that are similar to those of 2D schemes for the random overlap assumptions. The 3D scheme is still limited in the sense that it does not take cloud shadows into account. However, the results still support the conclusion that taking the slant viewing path into account can be of similar importance as changing the fundamental overlap assumption. The reflectance spread generated by stochastic methods is probably too small to be relevant for DA. As our goal was to maximize consistency with the RT code in COSMO, we did not consider newer suggestions for cloud overlap assumptions that are often based on a decorrelation length (see Hogan and Illingworth 2000) and vary with the seasons or parameters, like vertical wind shear (Di Giuseppe and Tompkins 2015) or the SZA (Tompkins and Di Giuseppe 2007). However, it should be relatively easy to adapt the 3D stochastic overlap method to more complicated assumptions. A second subgrid effect that could also be included in a future version of the stochastic scheme is the subgrid variation of water content, which is already taken into account in broadband radiation schemes (see, e.g., Pincus et al. 2003) and was found to be important by Shonk et al. (2010).
Acknowledgments
The study was carried out in the Hans-Ertel Centre for Weather Research (Simmer et al. 2016). This German research network of universities, research institutes, and DWD is funded by the Federal Ministry of Transport and Digital Infrastructure (BMVI Grant DWD2014P8). Data generated in the research program High Definition Clouds and Precipitation for Advancing Climate Prediction [HD(CP)2], founded by the Federal Ministry of Education and Research in Germany (BMBF), was used in this study.
APPENDIX
Stochastic Cloud Overlap Algorithms
The stochastic methods STO-C, STO-N, and STO-R use for each model grid column (tilted toward the satellite) the profiles of subgrid cloud fraction
The stochastic methods must determine


Here the fill probability
.
The first rule applies for empty layers, the second for the topmost layer of a group of cloudy layers, and the last two rules for cloudy layers directly below other cloudy layers. As can be shown easily, the expectation value of











Finally, method STO-C3D is a generalization of STO-C, which takes the slant satellite viewing angle into account. For this purpose a bundle of
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