## 1. Introduction

Clouds and radiation interact strongly with each other. On one hand, clouds reflect solar radiation and absorb and emit thermal radiation, leading to a net cooling effect on climate (e.g., Kiehl and Trenberth 1997). On the other hand, radiation can affect cloud formation (e.g., Markowski and Harrington 2005; Frame and Markowski 2010). In large-eddy simulation (LES) and numerical weather prediction (NWP) models, the radiation is at best parameterized using computationally inexpensive one-dimensional radiative transfer solvers. The need of a good radiative transfer scheme that considers three-dimensional effects becomes more and more important when going to higher model resolutions, however. Then, the net radiation flux between neighboring model columns can no longer be neglected.

A widely employed method for calculating irradiances in NWP or LES models is the so-called independent column approximation (ICA), also called independent pixel approximation (IPA)—in the following, we will only use the term “column.” In these approaches, the radiative transfer calculations are performed in single, vertical columns (see Fig. 1, upper right) with a 1D solver (e.g., two-stream or discrete ordinate), assuming horizontally homogeneous, infinitely extended model layers. The ICA method does not account for typical three-dimensional phenomena, however, thus leading to four kinds of errors [also see discussions in Marshak and Davis (2005), Tompkins and Di Giuseppe 2007), and Hogan and Shonk (2013)]:

There is misplacement of direct radiation: the shadow of the cloud erroneously falls underneath the cloud and is neither shifted nor elongated if the sun is not in the zenith.

There is misplacement of diffuse radiation: There is no photon scattering into neighboring columns. Thus, ICA does not account for the intercolumn photon transport that would lead to radiative smoothing or roughening.

There is a bias of direct radiation: In cloud-free columns, direct radiation easily reaches the surface in the ICA approximation. For larger sun zenith angles, however, direct radiation in reality is intercepted by clouds with increasing probability; hence, ICA tends to overestimate direct radiation at the surface.

There is a bias of diffuse radiation: missing intercolumn photon transport—in particular, between cloudy and cloud-free columns—and the bias of direct radiation lead to a bias in diffuse radiation.

To consider the slant incidence of the sun, a modified ICA was suggested in the past: the tilted independent column approximation (TICA). Two different versions of the TICA have been suggested. First, there is what we call “TICA DIR,” in which the direct radiation is calculated in a single, independent column that is slanted toward the sun, that is, according to the solar zenith and azimuth angles: *θ* and *ϕ*. The diffuse radiation, however, is calculated in the vertical column having the direct radiation field as input (see Fig. 1, lower right). TICA DIR was developed for two dimensions by Gabriel and Evans (1996), who called it “independent pixel approximation modified source (IPAMS).” For IPAMS, a full two-dimensional computation of the direct beam was performed and was then used as the pseudosource in an independent pixel diffuse radiative transfer calculation. IPAMS and IPA were compared with an accurate two-dimensional radiative transfer model, but only for domain-averaged irradiances, radiances, and heating rates. Also, Zuidema and Evans (1998) used this TICA method for their studies of 2D radiative transfer in boundary layer clouds, calling it “3-D direct beam IPA (3dbIPA).”

The second TICA approach, which we call “TICA DIRDIFF,” is illustrated in the lower-left panel of Fig. 1. There, the direct as well as the diffuse radiation is calculated in the tilted, independent columns that are slanted according to *θ* and *ϕ*. This version of TICA was introduced by Várnai and Davies (1999), who studied the effects of cloud heterogeneities on shortwave radiation. It was used also for radiative transfer studies by, inter alia, Wapler and Mayer (2008) and Frame et al. (2009). Várnai and Davies note that this TICA approach “ … can also be regarded as an extension of Gabriel and Evans' (1996) IPA modified source (IPAMS), in which direct and diffuse radiation are separated … Since this artificial separation at the first scattering event can cause problems in situations involving low-order scattering, TIPA is considered more appropriate than IPAMS for the purpose of calculating the one-dimensional heterogeneity effect.”

Although both TICA methods can eliminate the first error made with ICA, they do not account for the diffuse radiation transport between the tilted columns since TICA is still a 1D approach. The scattering into neighboring columns can be considered with a second parameterization called the “nonlocal independent column approximation” (NICA), which was first developed by Marshak et al. (1995). The concept is to use the ICA results, for example, the diffuse downward radiation, and to form a convolution product for each column, thereby smoothing the original radiation field. Marshak et al. used a gamma distribution as convolution kernel, and Zuidema and Evans (1998) employed a Gaussian distribution and convolved the TICA reflectances, thus performing “NTICA.”

After these studies with TICA and NTICA, the following open questions still remain: Which of the two TICA methods gives a more realistic diffuse downward irradiance field and should be used in the future? What is a good but simple convolution kernel that can be used to convolve the radiation field—thus accounting for the intercolumn photon transport—and that is applicable for any possible cloud scene? What role does the surface albedo or the horizontal resolution of the model play when deciding which (N)TICA approach should be used?

The aim of this study is to answer these questions and, thus, to find the most accurate 3D radiative transfer parameterization for the solar spectral range that can also be utilized in NWP models. In this work we address only surface irradiance. Correct treatment of surface irradiance in models should cause an improved representation of differential heating of the ground and thus of convection. A parameterization of 3D atmospheric heating rates will be described in a separate paper.

By using the Monte Carlo code for the physically correct tracing of photons in cloudy atmospheres (MYSTIC; Mayer 2009), which is part of the radiative transfer package known as “libRadtran” (Mayer and Kylling 2005), both TICA methods are compared with each other, with ICA, and with the exact solution obtained by 3D Monte Carlo simulations. We address the question of whether the convolution can be carried out with a parameterized kernel that is suitable for different model resolutions and cloud scenes but still fast enough to be applicable in NWP models. We thereby focus on the resolutions of current NWP models, in particular Consortium for Small-Scale Modeling (COSMO-DE), the NWP model of the German weather service (DWD), with 2.8-km resolution, and higher resolutions to be expected within the next 10 years. Furthermore, an efficient technique for determining the tilted profiles will be explained that can also be applied in NWP models.

The paper is organized as follows. In section 2, we give a brief description of the setup and the different cloud scenes that are used in the studies. In section 3, we describe the two TICA methods and elaborate on how they were implemented in MYSTIC. Existing methods and a new parameterization for convolution (called “paNTICA”) are presented in section 4. The results of the paNTICA studies are described and discussed in section 5. A summary and the conclusions are given in section 6. The abbreviations used for the different methods are summarized and explained in Table 1, which also notes the sections in which the results are discussed.

Abbreviations used for the different methods for calculating the direct and diffuse radiation, and the sections in which the results obtained with these methods are discussed. See also Fig. 1.

## 2. Setup of scenes for studies

To find the most accurate 3D radiative transfer parameterization that can also be used in NWP models like COSMO, studies were carried out with the MYSTIC radiative transfer solver (Mayer 2009). In the following we briefly describe the settings and cloud scenes used in the studies (see sections 4 and 5).

The experiments were carried out for different solar zenith angles and for a surface albedo of 0.05, corresponding to, for example, deep water, asphalt, or a fallow field (Bowker et al. 1985). To test the sensitivity and applicability of the new parameterizations, some of the experiments were repeated with an albedo of 0.5, representing, for example, old snow. The correlated-*k* distribution according to Kato et al. (1999) was used to account for molecular absorption. In all experiments, the sun was exactly in the south (*ϕ* = 0°) and periodic boundary conditions were applied.

Seven cloud scenes were used in the studies, and their characteristics are listed in Table 2. The scenes cover a range of horizontal grid sizes from 33 m to 2.8 km. Three scenes (Faure et al. 2009) were extracted from COSMO-DE, the NWP model of DWD: cosmo1, cosmo2, and cosmo3 (see Fig. 2). These cloud fields have a horizontal grid size of 2.8 km. The cosmo1 scene includes low water clouds, which can mainly be classified as stratocumulus, and optically thin cirrus clouds in the north. The cosmo2 scene has water and ice clouds only in the southwestern corner while additional cirrus clouds cover the region from the northwest to the southeast; the northeastern part is cloud free. The cosmo3 scene has stratocumuli, and the entire domain is covered with cirrus clouds. Since the subgrid-scale cloud parameterization of COSMO-DE cannot be taken into account in a 3D radiative transfer model such as MYSTIC, only grid-scale clouds were considered.

Cloud scenes used in the MYSTIC studies. Given are the name of the scene, cloud type(s), height of cloud base *z*_{cb} and cloud top *z*_{ct}, whether the clouds consist of water and/or ice, horizontal grid size (*dx* = *dy*), number of pixels in the horizontal plane, domain size, minimum grid spacing in the vertical direction *dz*_{min}, number of vertical layers, and whether the cloud scene is shown in a figure.

The (left) cosmo1, (center) cosmo2, and (right) cosmo3 scenes: vertically integrated optical thickness of (top) water and (bottom) ice clouds. The size of the grid cells is 2.8 × 2.8 km^{2}.

Citation: Journal of Applied Meteorology and Climatology 52, 8; 10.1175/JAMC-D-12-0227.1

The (left) cosmo1, (center) cosmo2, and (right) cosmo3 scenes: vertically integrated optical thickness of (top) water and (bottom) ice clouds. The size of the grid cells is 2.8 × 2.8 km^{2}.

Citation: Journal of Applied Meteorology and Climatology 52, 8; 10.1175/JAMC-D-12-0227.1

The (left) cosmo1, (center) cosmo2, and (right) cosmo3 scenes: vertically integrated optical thickness of (top) water and (bottom) ice clouds. The size of the grid cells is 2.8 × 2.8 km^{2}.

Citation: Journal of Applied Meteorology and Climatology 52, 8; 10.1175/JAMC-D-12-0227.1

The scene Cb, which is shown in Fig. 3, is a cumulonimbus from the Goddard Cumulus Ensemble cloud data by Zinner et al. (2008). It has a horizontal grid size of 250 m. The Cb scene was upscaled (extinction coefficient and liquid/ice water content were averaged over 10 × 10 columns in the horizontal plane and the effective radius was adapted accordingly) so that a low-resolved field (CbUp, not shown) with a horizontal grid size of 2.5 km was available for sensitivity studies.

The (left) Cb, (center) CuMed, and (right) St scenes: vertically integrated optical thickness of (top) water and (bottom) ice clouds. The size of the grid cells is 0.25 × 0.25 km^{2} (Cb), 0.05 × 0.05 km^{2} (CuMed), and 0.03 × 0.03 km^{2} (St), respectively.

Citation: Journal of Applied Meteorology and Climatology 52, 8; 10.1175/JAMC-D-12-0227.1

The (left) Cb, (center) CuMed, and (right) St scenes: vertically integrated optical thickness of (top) water and (bottom) ice clouds. The size of the grid cells is 0.25 × 0.25 km^{2} (Cb), 0.05 × 0.05 km^{2} (CuMed), and 0.03 × 0.03 km^{2} (St), respectively.

Citation: Journal of Applied Meteorology and Climatology 52, 8; 10.1175/JAMC-D-12-0227.1

The (left) Cb, (center) CuMed, and (right) St scenes: vertically integrated optical thickness of (top) water and (bottom) ice clouds. The size of the grid cells is 0.25 × 0.25 km^{2} (Cb), 0.05 × 0.05 km^{2} (CuMed), and 0.03 × 0.03 km^{2} (St), respectively.

Citation: Journal of Applied Meteorology and Climatology 52, 8; 10.1175/JAMC-D-12-0227.1

The two high-resolution water cloud scenes are CuMed and St; see Fig. 3. These scenes were created with the Eulerian–Lagrangian (EULAG) LES model. CuMed is the cumulus mediocris field used in the studies of Wapler and Mayer (2008). The field St is an overcast stratus scene (not shown).

## 3. TICA in MYSTIC

Wapler and Mayer (2008) and Frame et al. (2009) each demonstrate a method of how to calculate the geometrical information for TICA. We present a general method that is suitable for models with terrain-following coordinates. The latter means that the height of the surface and—with it—the height of the model levels change from column to column (see, e.g., Fig. 4). In sections 3a and 3b, the two TICA methods will be described for a rectangular grid. The generalization to latitude–longitude coordinates is then straightforward.

Illustration of the TICA DIRDIFF method. The vertical model columns nx = from (*i* − 1) to (*i* + 2) are shown with their different layers nz (the surface layer corresponds to nz = 1). The column boundaries are numbered from (*i* − 3/2) to (*i* + 5/2). The beam toward the sun starts at the center point of the surface pixel of column *i*. At the right, the new column *i*_{NEW} after the TICA DIRDIFF approach is shown with its new layers (1–6) and the corresponding optical properties (op).

Citation: Journal of Applied Meteorology and Climatology 52, 8; 10.1175/JAMC-D-12-0227.1

Illustration of the TICA DIRDIFF method. The vertical model columns nx = from (*i* − 1) to (*i* + 2) are shown with their different layers nz (the surface layer corresponds to nz = 1). The column boundaries are numbered from (*i* − 3/2) to (*i* + 5/2). The beam toward the sun starts at the center point of the surface pixel of column *i*. At the right, the new column *i*_{NEW} after the TICA DIRDIFF approach is shown with its new layers (1–6) and the corresponding optical properties (op).

Citation: Journal of Applied Meteorology and Climatology 52, 8; 10.1175/JAMC-D-12-0227.1

Illustration of the TICA DIRDIFF method. The vertical model columns nx = from (*i* − 1) to (*i* + 2) are shown with their different layers nz (the surface layer corresponds to nz = 1). The column boundaries are numbered from (*i* − 3/2) to (*i* + 5/2). The beam toward the sun starts at the center point of the surface pixel of column *i*. At the right, the new column *i*_{NEW} after the TICA DIRDIFF approach is shown with its new layers (1–6) and the corresponding optical properties (op).

Citation: Journal of Applied Meteorology and Climatology 52, 8; 10.1175/JAMC-D-12-0227.1

### a. MYSTIC

MYSTIC (Mayer 2009) is the Monte Carlo solver of libRadtran. Equipped with advanced variance reduction methods (Buras and Mayer 2011), it allows 3D radiative transfer simulations in cloudy atmospheres without simplifying assumptions. Since a Monte Carlo code is orders of magnitude too slow to be directly operated in LES or NWP models, its output can only serve as the “truth” with which a new parameterization can be compared. For our studies, both TICA methods were implemented into MYSTIC so that the results can be compared with that of an exact 3D solution. It might seem a bit exaggerated at first to use a Monte Carlo code to perform calculations in single columns. By using Monte Carlo as a solver for TICA DIRDIFF, TICA DIR, and real 3D simulations, however, all observed differences are caused by the new parameterization and are not due to differences between solvers.

### b. TICA DIRDIFF

The technical method of TICA DIRDIFF is illustrated in Fig. 4. For simplicity, we consider a vertical cross section through the domain at a constant latitude *y* and with the sun in the east. The vertical model columns from (*i* − 1) to (*i* + 2) are shown with their respective layers. For calculations with terrain, the levels in the model columns are located at different heights *z*. For the calculation of the irradiance at the surface of column *i*, the optical properties along the beam toward the sun are needed. For computational efficiency we decided to trace only one beam, which starts at the center point of column *i*.

First, the new levels (dashed light-gray lines in Fig. 4; *z _{x}*

_{1}and

*z*

_{x}_{2}) are determined, which arise because of the intersection of the beam (solid diagonal line) with the column boundaries in the

*x*direction (here,

*i*+ 1/2 and

*i*+ 3/2). Together with these new height levels, the coordinates of the grid box where the beam is leaving are saved [here (nx, nz) = (

*i*, 2) and (nx, nz) = (

*i*+ 1, 5)]. In case the sun is not exactly in the west or east, the same is done for the intersection of the beam with column boundaries in the

*y*direction. Then, all of these new height levels (

*z*

_{x}_{1},

*z*

_{x}_{2}, … and

*z*

_{y}_{1},

*z*

_{y}_{2}, …) that arose because of the change into the next model column are sorted by height.

As a next step, we look for the original height levels that lie between the new levels (*z _{x}*

_{1},

*z*

_{x}_{2}, …,

*z*

_{y}_{1},

*z*

_{y}_{2}, …) that were determined before. For example, between the intersection level of the beam with column boundaries

*i*+ 1/2 (

*z*

_{x}_{1}) and

*i*+ 3/2 (

*z*

_{x}_{2}), the beam is located in column (

*i*+ 1). In this column, the beam crosses two original model levels. These levels (gray dotted lines) along with the coordinates of the grid box below are saved also and sorted into the levels (

*z*

_{x}_{1},

*z*

_{x}_{2}, …,

*z*

_{y}_{1},

*z*

_{y}_{2}, …) determined at the beginning.

In our example, the new vertical column *i*_{NEW} (shown at the right of Fig. 4) now has six layers into which the original optical properties are copied (“op” stands for extinction coefficient, single scattering albedo, and phase function or asymmetry parameter). With this procedure we obtain the necessary information for the column *i*_{NEW} in which the direct and diffuse radiation are then calculated with a standard 1D radiative transfer code (e.g., two stream).

We have to keep in mind that our implementation of the TICA is not an exact method, not even for the calculation of the direct radiation if the solar zenith angle is different from 0. This is because the beam along which the geometrical information is used for the TICA calculation starts at the center of each surface pixel *i* and the optical properties along this beam are assumed to be representative for the entire tilted column (gray shading in Fig. 4). In a full 3D calculation, the surface irradiance in a column is the average over various parallel beams. The deviations in direct radiation between the approximate TICA method and the exact 3D calculation, however, are dwarfed by the deviations in diffuse radiation (see section 5b) such that an improvement—for example, by antialiasing methods—will not be considered here.

### c. TICA DIR

For TICA DIR only the direct radiation is calculated in tilted columns, and the diffuse radiation is computed in the vertical (ICA) column. The computation of the direct component is carried out as described for TICA DIRDIFF (see section 3b). To calculate the diffuse irradiance for TICA DIR at the surface of column *i*, the direct radiation at every original model level *k* in column *i* is needed as input. Figure 5 shows a schematic with beams (denoted by “beam lev*k*”) starting at every level *k*. At these levels, the direct radiation is calculated as described in section 3b. Thereby, each beam encounters different column boundaries and levels of other columns. This is indicated by the four columns on the right of Figure 5, where the first column *i*_{NEW,lev1} shows the geometrical information needed for the calculation of the direct radiation at level *k* = 1, the second column *i*_{NEW,lev2} shows the geometrical information needed for the calculation of the direct radiation at level *k* = 2, and so on. Once the direct radiation at every level *k* of column *i* is available, it is used as source for the computation of the diffuse radiation.

Schematic that illustrates the TICA DIR method (for explanation see also Fig. 4). The beams starting at the center of every level *k* of column *i* are denoted by “beam lev*k*,” whereby *k* = 1, …, 5. At the right, the new columns *i*_{NEW,levk} after the TICA DIR approach are shown with their new layers and the corresponding optical properties (op).

Citation: Journal of Applied Meteorology and Climatology 52, 8; 10.1175/JAMC-D-12-0227.1

Schematic that illustrates the TICA DIR method (for explanation see also Fig. 4). The beams starting at the center of every level *k* of column *i* are denoted by “beam lev*k*,” whereby *k* = 1, …, 5. At the right, the new columns *i*_{NEW,levk} after the TICA DIR approach are shown with their new layers and the corresponding optical properties (op).

Citation: Journal of Applied Meteorology and Climatology 52, 8; 10.1175/JAMC-D-12-0227.1

Schematic that illustrates the TICA DIR method (for explanation see also Fig. 4). The beams starting at the center of every level *k* of column *i* are denoted by “beam lev*k*,” whereby *k* = 1, …, 5. At the right, the new columns *i*_{NEW,levk} after the TICA DIR approach are shown with their new layers and the corresponding optical properties (op).

Citation: Journal of Applied Meteorology and Climatology 52, 8; 10.1175/JAMC-D-12-0227.1

### d. Implementation in MYSTIC

In an NWP model like COSMO, the tilting procedure described in sections 3b (for TICA DIRDIFF) and 3c (for TICA DIR) has to be performed for every single column because angles *θ* and *ϕ* and the surface heights vary from column to column. If we assume plane-parallel geometry with constant (*θ*, *ϕ*) across the scene and constant surface altitude, the computation becomes considerably faster. Constant viewing angles (*θ*, *ϕ*) are obviously a good assumption for a sufficiently small model domain. As a consequence, the geometrical calculations explained above have to be performed just once for some column so as to obtain the new *z* levels with the corresponding gridbox coordinates. The determined *z* levels are the same for all model columns, and only the corresponding optical properties then need to be copied into each column.

The implementation of TICA DIR in MYSTIC is straightforward: As long as the photons from the sun are defined as direct radiation, they move freely in the entire model domain. Once the photon is scattered, however, it is trapped inside the vertical column.

## 4. NTICA: Methods for smoothing

In this section we concentrate on the diffuse downward irradiance and on the question of how to improve the ICA and TICA results. Neither method considers the horizontal diffusion of photons into neighboring columns (see Fig. 1). In the following, examples are given for calculations with overhead sun (*θ* = 0), for which TICA and ICA give identical results, and low sun (*θ* = 60°), for which we used the TICA DIRDIFF approach for calculating the unsmoothed irradiances (for a discussion of the quality of the TICA methods see the next section).

The top panels of Fig. 6 show the integrated diffuse downward shortwave irradiance at the surface in the Cb scene, calculated with 3D Monte Carlo (Fig. 6, left) and ICA (Fig. 6, center). It is apparent that the full 3D field is much smoother than the ICA field. This can be seen also from the irradiance histograms (bottom panels in Fig. 6): In the simple ICA calculation, the clear-sky columns have values of about 55 W m^{−2}, which causes a large peak (height of 22 048, cut off for presentation). The latter is not present in the full 3D simulation (left histogram) because of the intercolumn photon transport that leads to a smoother field and, thus, to a broader distribution in the histogram.

(top) Integrated diffuse downward shortwave irradiance (W m^{−2}) at the surface, underneath the Cb: (left) 3D Monte Carlo, (center) ICA, and (right) paNTICA (see section 4b), with overhead sun and surface albedo = 0.05. (bottom) Corresponding irradiance histograms.

Citation: Journal of Applied Meteorology and Climatology 52, 8; 10.1175/JAMC-D-12-0227.1

(top) Integrated diffuse downward shortwave irradiance (W m^{−2}) at the surface, underneath the Cb: (left) 3D Monte Carlo, (center) ICA, and (right) paNTICA (see section 4b), with overhead sun and surface albedo = 0.05. (bottom) Corresponding irradiance histograms.

Citation: Journal of Applied Meteorology and Climatology 52, 8; 10.1175/JAMC-D-12-0227.1

(top) Integrated diffuse downward shortwave irradiance (W m^{−2}) at the surface, underneath the Cb: (left) 3D Monte Carlo, (center) ICA, and (right) paNTICA (see section 4b), with overhead sun and surface albedo = 0.05. (bottom) Corresponding irradiance histograms.

Citation: Journal of Applied Meteorology and Climatology 52, 8; 10.1175/JAMC-D-12-0227.1

### a. Existing methods

*E*

_{TICA}, it is convolved with a smoothing kernel

*G*(

*x*,

*y*) and a nonlocal TICA irradiance is obtained:

*C*being a constant, obtained from normalization:

*x*−

_{b}*x*) and (

_{a}*y*−

_{b}*y*) are the edge lengths of the kernel in the

_{a}*x*and

*y*directions, respectively.

To find the optimum *σ* for Eq. (2) for a certain solar zenith angle *θ*, Zuidema and Evans (1998) varied *σ* and compared the resulting convolved field with the “truth,” that is, the output of a full 3D simulation. The *σ* giving the smallest error or the best correlation *σ*_{best} was then used for NTICA. Applying this best-*σ* approach to our different cloud scenes shows that *σ*_{best} is not only a function of the solar zenith angle *θ*.

*N*and

_{x}*N*are the numbers of columns in the

_{y}*x*and

*y*directions, respectively, and

*σ*is shown in Fig. 7a for overhead sun for the different cloud scenes. It is apparent that the optimum

*σ*varies considerably from cloud field to cloud field, even if the zenith angle is held constant (here

*θ*= 0 for all cases). It is smallest for CuMed (

*σ*

_{best}= 625 m) and largest for Cb (

*σ*

_{best}= 3700 m). Moreover,

*σ*

_{best}depends also on the horizontal resolution. It is 3700 m for Cb and only 1475 m for the lower-resolved cumulonimbus CbUp, which are based on the same cloud scene.

Comparison between the 3D Monte Carlo calculations and ICA convolved with the best-*σ* approach for (a) *θ* = 0° and (b) *θ* = 60° and showing RMSE vs *σ* for all seven cloud fields; the surface albedo = 0.05.

Citation: Journal of Applied Meteorology and Climatology 52, 8; 10.1175/JAMC-D-12-0227.1

Comparison between the 3D Monte Carlo calculations and ICA convolved with the best-*σ* approach for (a) *θ* = 0° and (b) *θ* = 60° and showing RMSE vs *σ* for all seven cloud fields; the surface albedo = 0.05.

Citation: Journal of Applied Meteorology and Climatology 52, 8; 10.1175/JAMC-D-12-0227.1

Comparison between the 3D Monte Carlo calculations and ICA convolved with the best-*σ* approach for (a) *θ* = 0° and (b) *θ* = 60° and showing RMSE vs *σ* for all seven cloud fields; the surface albedo = 0.05.

Citation: Journal of Applied Meteorology and Climatology 52, 8; 10.1175/JAMC-D-12-0227.1

In Table 3 the normalized bias^{1} and the RMSE made with pure ICA are compared with those made with the different NICA methods: NICA with *σ*_{best}, NICA with simple averaging over the whole domain (Wapler and Mayer 2008), and the NICA method with the parameterized *σ* (paNICA), which is a new parameterization that will be explained in section 4b. From Table 3 it is apparent that NICA with the best-*σ* approach would reduce the error considerably relative to the pure standard ICA calculation. For example, the error reduces from 70.9% to 32.7% for the Cb, and for the high-resolved CuMed the RMSE decreases from 107.9% to as low as 16.2%. The simple averaging (“NICA avg” in Table 3), however, is useful only for the cloud scenes CuMed and St for which the domain size is small and the cloud types and shapes are constant over the domain. Note that for overhead sun the direct radiation is calculated correctly by the ICA, thus reducing the RMSE even more when the total irradiance (sum of direct and diffuse downward irradiance) is considered (see values in parentheses in Table 3).

RMSE and bias of the diffuse downward irradiance at surface for pure ICA, for when ICA is combined with convolution using the best-*σ* approach (NICA *σ*_{best}), for NICA averaging (NICA avg), and for the new NICA method in which *σ* is parameterized (paNICA; see section 4b). The RMSE and bias of the total (=direct + diffuse) irradiance are given in parentheses. All cases assume an overhead sun and that the surface albedo = 0.05.

The best-*σ* approach yields similar results for a nonzenithal sun, for example, for *θ* = 60°; see Fig. 7b. The RMSE is, naturally, generally larger than in the *θ* = 0° case even after applying *σ*_{best} since 3D effects get stronger when the sun illuminates cloud sides. For the COSMO scenes, the RMSE is approximately 2 times as large; for CuMed, however, it is even smaller. In this situation, *σ*_{best} has different values than for overhead sun; it increases by factors of ~1.6–3. For St there is no *σ*_{best} and the simple averaging yields the smallest RMSE.

The results show that NICA with best *σ* would be a good method to correct the diffuse downward irradiance. The problem with this approach, however, is that *σ*_{best} cannot be determined unless an expensive 3D simulation is performed for each scene. Furthermore, *σ*_{best} depends on the cloud field and the model resolution.

### b. New method: paNTICA

*σ*. This is the reason why we will call the method parameterized NTICA (paNTICA) in the following. Now,

*σ*is different for every surface pixel, since

*θ*is the solar zenith angle,

*d*

_{cb}is the distance from the center of the surface pixel to the center of the base of the closest cloud (see Fig. 8), and

*f*is a factor to be determined. In the case in which the column of the pixel to be considered is clouded,

*d*

_{cb}corresponds to the height of the cloud base

*z*

_{cb}(see Fig. 8). The kernel is chosen to be circular with a radius

*r*of 20 km; that is, only the surrounding columns whose centers lie inside the 20-km radius are considered for convolution.

_{k}Schematics illustrating the definition of the (left) height of the cloud base *z*_{cb} and (right) distance to the base of the closest cloud *d*_{cb}. The shaded areas represent clouded grid boxes.

Citation: Journal of Applied Meteorology and Climatology 52, 8; 10.1175/JAMC-D-12-0227.1

Schematics illustrating the definition of the (left) height of the cloud base *z*_{cb} and (right) distance to the base of the closest cloud *d*_{cb}. The shaded areas represent clouded grid boxes.

Citation: Journal of Applied Meteorology and Climatology 52, 8; 10.1175/JAMC-D-12-0227.1

Schematics illustrating the definition of the (left) height of the cloud base *z*_{cb} and (right) distance to the base of the closest cloud *d*_{cb}. The shaded areas represent clouded grid boxes.

Citation: Journal of Applied Meteorology and Climatology 52, 8; 10.1175/JAMC-D-12-0227.1

The dependence of *σ* on the solar zenith angle *θ* is motivated by studies using TICA with the best-*σ* approach, in which we found that *σ*_{best} increases with increasing *θ*. A possible physical motivation for this dependence may be the fact that radiation scattering on thin clouds will in part maintain the approximate direction of the direct radiation and thus be spread over larger distances.

The dependence on the distance to the closest cloud base is included because we assume that the higher the cloud base *z*_{cb} is, the more the scattered photons are spread. Furthermore, the closer a cloud (side) is, the more photons will be scattered into the column considered.

## 5. N(T)ICA studies: Results and discussion

In this section, the results obtained with the two TICA methods and with the new parameterization for the diffuse downward radiation (paNTICA) are discussed. In section 5a we concentrate first again on overhead sun, thus quantifying the improvement made by the new convolution method. In section 5b, experiments with oblique sun are presented and the two (N)TICA methods are compared. In section 5c we discuss the bias of the different methods.

### a. NTICA (θ = 0)

The top panels in Fig. 6 show the diffuse downward irradiance underneath the Cb, calculated with 3D Monte Carlo, TICA, and paNTICA. As discussed in the previous section and as intended, the new parameterization leads to a smoothing effect that is apparent when comparing the irradiance fields and the histograms (bottom panels in Fig. 6). TICA (Fig. 6, center) shows a strong peak at values around 55 W m^{−2}, but this peak is not present in the full 3D simulation (Fig. 6, left) because of the intercolumn photon transport. The new parameterization paNTICA (Fig. 6, right) successfully accounts for the latter and leads to a histogram that is similar to that for the full 3D calculation.

Table 3 shows the RMSE for paNTICA. It is apparent that the new approach gives much better results than the pure TICA method. For the Cb, the error is reduced from 70.9% (TICA) to 35.7%. The best-*σ* approach would have brought a slightly larger improvement in this case (RMSE 32.7%). As discussed in the previous section, however, *σ*_{best} is unknown unless the full time-consuming 3D calculation is performed for the specific case, and thus the method is not applicable. In the case of cosmo1, paNTICA is even slightly better than the best-*σ* approach (14.0% vs 14.4%).

Further, sensitivity studies (not shown here) in which *r _{k}* was varied from 5 to 50 km have shown that a radius of 15–20 km gives the best results. The RMSE increases—often by more than 10%—for smaller

*r*. Choosing a kernel radius of more than 20 km does not lead to better results but increases the computational time needed for convolution. Furthermore, studies were done in which the factor

_{k}*f*in Eq. (4) was varied from 0.1 to 10 for an

*r*of 20 km and showed that the RMSE is smallest for values of

_{k}*f*between 0.5 and 1.5 (depending on the cloud scene). Hence, we chose

*f*= 1.

### b. NTICA (θ > 0)

So far, only experiments with overhead sun, and thus use only of NICA, were discussed. Now we will concentrate on the cases with a solar zenith angle *θ* > 0, for which the question arises as to which of the two TICA methods (DIRDIFF or DIR) along with the new parameterization (paNTICA) gives better results. Therefore, the calculations with the seven cloud fields were repeated for solar zenith angles of 20°, 30°, 40°, 50°, 60°, 70°, and 80° (with a surface albedo of 0.05) and for 30°, 50°, and 80° (with a surface albedo of 0.5). In the following, we show the outcome of the experiments cosmo2 (see Fig. 2) and Cb (see Fig. 3) in which the albedo was set to 0.05. The statements apply also to the other experiments, however. We will first consider the diffuse downward irradiance at the surface. The direct and total irradiance will be discussed below. The RMSE and bias for the scenes cosmo2 and Cb are listed in Table 4.

The Cb and cosmo2 cloud scenes: RMSE (%) for direct, diffuse downward, and total irradiance (*E*_{tot} = *E*_{dir} + *E*_{dd}) at the surface for different solar zenith angles (*θ* = 0°, 20°, 40°, 60°, and 80°). The bias (%) relative to the 3D Monte Carlo calculation is given in parentheses. The listed methods are ICA, TICA DIRDIFF without and with convolution, and TICA DIR without and with convolution. The surface albedo is 0.05 for all cases. For case *θ* = 0°, ICA and the two TICA methods are identical. Note that the nonzero RMSE of *E*_{dir} for *θ* = 0° is due to statistical Monte Carlo noise.

#### 1) Diffuse downward irradiance *E*_{dd}

Figure 9 shows the integrated diffuse downward shortwave irradiance underneath the Cb, calculated with 3D Monte Carlo (Fig. 9a), ICA (Fig. 9b), TICA DIRDIFF (Fig. 9c), and TICA DIR (Fig. 9d). The same is shown for the cosmo2 cloud field in Fig. 10. In both cases, the solar zenith angle is 50° and the sun is located in the south (bottom of the plot). Like for overhead sun, the Monte Carlo field (Fig. 10a) is smooth relative to the fields obtained with the pure ICA (Fig. 10b) or the TICA methods without smoothing (Figs. 10c,d).

Diffuse downward solar irradiance (W m^{−2}) at the surface underneath the Cb: (a) 3D Monte Carlo, (b) ICA, (c) TICA DIRDIFF, (d) TICA DIR, (e) paNTICA DIRDIFF, and (f) paNTICA DIR. Here, *θ* = 50° and surface albedo = 0.05.

Citation: Journal of Applied Meteorology and Climatology 52, 8; 10.1175/JAMC-D-12-0227.1

Diffuse downward solar irradiance (W m^{−2}) at the surface underneath the Cb: (a) 3D Monte Carlo, (b) ICA, (c) TICA DIRDIFF, (d) TICA DIR, (e) paNTICA DIRDIFF, and (f) paNTICA DIR. Here, *θ* = 50° and surface albedo = 0.05.

Citation: Journal of Applied Meteorology and Climatology 52, 8; 10.1175/JAMC-D-12-0227.1

Diffuse downward solar irradiance (W m^{−2}) at the surface underneath the Cb: (a) 3D Monte Carlo, (b) ICA, (c) TICA DIRDIFF, (d) TICA DIR, (e) paNTICA DIRDIFF, and (f) paNTICA DIR. Here, *θ* = 50° and surface albedo = 0.05.

Citation: Journal of Applied Meteorology and Climatology 52, 8; 10.1175/JAMC-D-12-0227.1

As in Fig 9, but for the cosmo2 cloud field.

Citation: Journal of Applied Meteorology and Climatology 52, 8; 10.1175/JAMC-D-12-0227.1

As in Fig 9, but for the cosmo2 cloud field.

Citation: Journal of Applied Meteorology and Climatology 52, 8; 10.1175/JAMC-D-12-0227.1

As in Fig 9, but for the cosmo2 cloud field.

Citation: Journal of Applied Meteorology and Climatology 52, 8; 10.1175/JAMC-D-12-0227.1

With ICA, the direct and diffuse radiation are calculated in the vertical column, thus leading to shadows directly underneath the cloud(s) (see also the schematic diagram in Fig. 1). Since we are looking at diffuse irradiance only, the cloud-free columns also appear dark because of the missing direct radiation. With TICA DIRDIFF, the direct and diffuse radiation are both calculated in the tilted column, which results in shadows that are shifted toward the north since the sun is in the south. Furthermore, the shadows are elongated, which is in particular evident for the Cb (Fig. 9c).

In the case of TICA DIR, the direct radiation, which is calculated in the tilted column, is used as input for the computation of the diffuse radiation in the vertical column. If the sun is in the south, this setup results in an enhancement of diffuse radiation at the southern side of the cloud, since many photons are scattered there and then descend in the vertical column. This effect is more obvious in the case of the Cb (Fig. 9d) than in the cosmo2 case (Fig. 10d) because of the large anvil of the deep convective cloud. The enhancement at the southern side leads to a large RMSE in the case of TICA DIR (see *E*_{dd} in Table 4). The RMSE of TICA DIR is sometimes even larger than that made with ICA. This can be seen, for example, by comparing the *E*_{dd}-RMSE of the Cb in Table 4: for a solar zenith angle of 40°, the error for ICA is 94.7%, whereas that for TICA DIR is 130.5%. Thus, when no additional correction (e.g., smoothing via convolution) is applied to the diffuse downward field, the TICA DIRDIFF approach gives definitely better results than the TICA DIR method.

Figures 9e,f and 10e,f show the convolved TICA fields, that is, paNTICA DIRDIFF and paNTICA DIR, respectively. The convolution leads to a considerable reduction of the errors made with the TICA methods (see Table 4). For example, for a solar zenith angle of 80°, the RMSE of *E*_{dd} in the cosmo2 case is 66.2% with the TICA DIR method. After convolution the error is only 24.2% (paNTICA DIR) and is smaller than that of the alternative approach paNTICA DIRDIFF (38.1%). In general, the errors of TICA DIR are reduced considerably by convolution so that the RMSE is now comparable to or slightly smaller than that of paNTICA DIRDIFF. This result means that TICA DIR should only be applied if the resulting diffuse downward irradiance field is convolved with a smoothing kernel. Then, for most cases, paNTICA DIR gives slightly better results.

#### 2) Direct irradiance *E*_{dir}

Table 4 shows that the RMSE of the direct irradiance calculated with the ICA approximation becomes considerably larger as the solar zenith angle increases. This again demonstrates that ICA is a good method only for overhead sun and should not be used for an oblique sun.

From Table 4 it is further apparent that both TICA methods give identical results, since in both cases *E*_{dir} is calculated in the tilted column with the same method (see Figs. 4 and 5). Using TICA instead of ICA reduces the RMSE by a factor that ranges from 14 (for *θ* = 20°) to 90 (for *θ* = 80°) for the Cb and by a factor that ranges from 2 (for *θ* = 20°) to 14 (for *θ* = 80°) for the cosmo2 scene (Table 4). This drastic reduction of the error is because with ICA the shadow erroneously falls underneath the cloud, whereas with TICA it is elongated and placed at the right position. For the sake of computational efficiency, the direct radiation calculated with TICA still is not exact since the optical properties along the central beam are assumed to be representative for the entire tilted column. We are not concerned about this effect, however, since the remaining error in *E*_{dir} using TICA is much smaller than the error in diffuse radiation; see Table 4.

#### 3) Total irradiance *E*_{tot}

In looking at the sum of the direct and diffuse downward radiation (*E*_{tot} for the Cb in Table 4 and Fig. 13a, described below), it is apparent that paNTICA reduces the errors considerably relative to the pure ICA method. The ICA approach leads to an RMSE of 44.8% for *θ* = 20°, while the RMSEs made with paNTICA DIRDIFF and paNTICA DIR are only 12.2% and 12.7%, respectively. For lower sun, the improvement caused by paNTICA is even larger: for *θ* = 80°, the RMSE reduces from almost 100% for ICA to 41% for paNTICA DIRDIFF, and even to 27% for paNTICA DIR. Looking at the distribution of the total surface irradiance (Fig. 11), we see that the features seen in the 3D simulation (Fig. 11a) are nicely reproduced by paNTICA DIRDIFF (Fig. 11e) and paNTICA DIR (Fig. 11f). Merely the enhancement of the irradiance on the sunny side of the Cb is too smooth for paNTICA DIRDIFF, while it is too structured for paNTICA DIR.

As in Fig. 9, but for total downward solar irradiance, i.e., including direct irradiance.

Citation: Journal of Applied Meteorology and Climatology 52, 8; 10.1175/JAMC-D-12-0227.1

As in Fig. 9, but for total downward solar irradiance, i.e., including direct irradiance.

Citation: Journal of Applied Meteorology and Climatology 52, 8; 10.1175/JAMC-D-12-0227.1

As in Fig. 9, but for total downward solar irradiance, i.e., including direct irradiance.

Citation: Journal of Applied Meteorology and Climatology 52, 8; 10.1175/JAMC-D-12-0227.1

For the cosmo2 scene (see Table 4 and Figs. 12 and 13b) the RMSE for ICA is smaller because the resolution is lower and thus 3D effects are of minor importance. Still, using the paNTICA approach significantly reduces the RMSE. Also here, paNTICA DIR yields better results than does paNTICA DIRDIFF for large *θ*.

As in Fig. 10, but for total downward solar irradiance, i.e., including direct irradiance.

Citation: Journal of Applied Meteorology and Climatology 52, 8; 10.1175/JAMC-D-12-0227.1

As in Fig. 10, but for total downward solar irradiance, i.e., including direct irradiance.

Citation: Journal of Applied Meteorology and Climatology 52, 8; 10.1175/JAMC-D-12-0227.1

As in Fig. 10, but for total downward solar irradiance, i.e., including direct irradiance.

Citation: Journal of Applied Meteorology and Climatology 52, 8; 10.1175/JAMC-D-12-0227.1

Comparison of the RMSE (%) for total irradiance at the surface as a function of *θ* (SZA) for different methods for (a) scene Cb and (b) scene cosmo2. The surface albedo is 0.05 for all cases.

Citation: Journal of Applied Meteorology and Climatology 52, 8; 10.1175/JAMC-D-12-0227.1

Comparison of the RMSE (%) for total irradiance at the surface as a function of *θ* (SZA) for different methods for (a) scene Cb and (b) scene cosmo2. The surface albedo is 0.05 for all cases.

Citation: Journal of Applied Meteorology and Climatology 52, 8; 10.1175/JAMC-D-12-0227.1

Comparison of the RMSE (%) for total irradiance at the surface as a function of *θ* (SZA) for different methods for (a) scene Cb and (b) scene cosmo2. The surface albedo is 0.05 for all cases.

Citation: Journal of Applied Meteorology and Climatology 52, 8; 10.1175/JAMC-D-12-0227.1

These findings apply to all seven cloud scenes used for the studies (Table 2), for the different solar zenith angles, and also when the surface albedo is increased from 0.05 to 0.5 (not shown).

### c. Bias

While the RMSE is a quality flag for the spatial distribution of the energy deposited by radiation at the surface, the (normalized) bias specifies whether the total amount of energy deposition is correct. A positive/negative bias signals that the surface is heated too much/too little.

We find that ICA generally has a positive bias on direct radiation, which increases with *θ*. On the other hand, diffuse radiation is always underestimated in the ICA approach because the diffuse radiation that is generated in clouds cannot reach the surface through the cloud-free adjacent columns. Averaging over all 56 simulations (7 scenes times 8 *θ*s), we find that the average bias in direct and diffuse irradiance is +25% and −12% respectively, but for total irradiance it is only −0.5%. This, however, is a coincidence: for small *θ* we find the bias in total irradiance to be negative and up to −5%, and for high *θ* it is positive and up to +33%. The variation of the bias, that is, the amplitude with which the bias varies between the 56 simulations, is ±4%.

For TICA, the direct irradiance has a negligible bias, which is to be expected since TICA takes account of the increased interception of direct radiation with clouds for nonzero *θ*. The diffuse radiation, however, still yields a significant negative bias: on average −9% for TICA DIRDIFF and −5% for TICA DIR. For TICA DIRDIFF this is understandable; TICA DIRDIFF has the same problems as ICA with the transport of cloud-scattering radiation. For TICA DIR, the bias is smaller since the scattered radiation no longer follows the same column as the direct radiation. The total irradiance still is biased on average over all simulations by −5% for TICA DIRDIFF and −2.2% for TICA DIR. The variation of the bias is of the same order.

Last, the convolution is not energy conserving because *σ*, which is a function of the distance to the nearest cloud base [see Eq. (4)], changes from column to column. The bias on diffuse irradiance introduced by paNTICA, however, is, on average over all simulations, negligible (<0.1%). The variation of the bias is 0.7% and 1.4% for paNTICA DIRDIFF and paNTICA DIR, respectively.

In summary, although TICA DIR and TICA DIRDIFF systematically bias the total irradiance at the surface, this bias is generally much smaller than for ICA. TICA DIR yields the smaller average bias of 2.2%. The convolution scheme paNTICA violates energy conservation, but only by 1%–2%, and not systematically. Considering that the two-stream approximation—which is currently used in all NWP models—has an uncertainty of 5%–10%, we are not concerned about the remaining bias.

## 6. Summary and conclusions

Even though NTICA was used in earlier studies, it still was unclear which TICA approach (DIR or DIRDIFF) gives more realistic results and which parameterization for convolution is applicable also in NWP models where terrain, subgrid-scale clouds, and different cloud types are present. Therefore, we carried out studies with a radiative transfer model in which the methods of NTICA and new parameterizations were compared with the truth, that is, a full 3D Monte Carlo calculation. We concentrated on the surface irradiance in the solar spectral range. Our studies answered the following open questions that were raised in the introduction:

Which of the two TICA methods gives a more realistic diffuse downward irradiance field and should be used in the future?—TICA DIR (also called IPAMS and 3dbIPA in the past) alone gives in some cases even worse results than ICA, and thus it should only be applied if the resulting diffuse downward irradiance field is smoothed afterward. With this additional smoothing (via convolution), both TICA approaches yield equally good results for small sun zenith angle. For large sun zenith angles, however, paNTICA DIR is the better solution.

What is a good but simple convolution kernel that can be used to convolve the radiation field (thus accounting for the intercolumn photon transport) and that is applicable for any possible cloud scene?—The studies have shown that the convolution of the diffuse irradiance field with a large enough Gaussian kernel reduces the error considerably relative to the pure TICA or ICA method. The width

*σ*of the kernel has been parameterized and depends on the distance to the closest cloud base and the solar zenith angle. In contrast to earlier parameterizations (best-*σ*approach; averaging), this method—which we call paNTICA—can be applied also in NWP models for which different cloud scenes are present.What role do the surface albedo or the horizontal resolution of the model play when deciding which (N)TICA approach should be used?—The new parameterization paNTICA was tested for applicability for seven different cloud scenes, horizontal grid sizes ranging from 33 m to 2.8 km, and two different surface albedos (0.05 and 0.5). In general paNTICA can reduce the RMSE made with ICA by up to 90% if the total irradiance is considered. These error reductions apply for both surface albedos that were tested.

Beside the studies, a method was presented of how to obtain the “tilted” optical properties needed for TICA. This method is suitable for spherical geometry and models with terrain-following coordinates like COSMO.

Hitherto we have not discussed the computational efficiency of the paNTICA method, which, however, is crucial if the method is to be used in operational NWP models. The paNTICA method is currently being implemented into the COSMO-DE model. First tests show that the computational time for the radiative transfer increases by a factor of roughly 2 with paNTICA, which implies an increase of total computational time of about 20%. These numbers may decrease when the code will be optimized.

In a forthcoming paper, topics are addressed that become relevant when (paN)TICA is implemented into the NWP model COSMO: This includes the parallelization and the treatment of subgrid-scale clouds. Moreover, future studies with paNTICA in the COSMO-DE model will show the effect of the improved 3D parameterization paNTICA on the cloud development.

## Acknowledgments

The first author acknowledges financial support from the German Weather Service (DWD). We also thank the reviewers for many helpful and clarifying suggestions. We also thank F. Jakub for providing numbers of the computational timing.

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^{1}

The normalized bias is defined as the difference of the mean irradiance obtained with ICA or any (N)TICA method and the mean 3D Monte Carlo irradiance