## 1. Introduction

A key challenge in atmospheric modeling for both weather and climate prediction is to improve the interaction of clouds with solar and thermal infrared radiation. While the representation of subgrid cloud horizontal structure and vertical overlap is often now quite sophisticated (e.g., Pincus et al. 2003; Hill et al. 2015; Di Giuseppe and Tompkins 2015), a process missing from all operational models is the horizontal transport of radiation within grid boxes. This was characterized by Hogan and Shonk (2013) as entirely associated with flow of radiation through cloud sides, and led to the development of the Speedy Algorithm for Radiative Transfer through Cloud Sides (SPARTACUS; Hogan et al. 2016; Schäfer et al. 2016). This solver is now available as an option in the radiation scheme of the forecast model used by the European Centre for Medium-Range Weather Forecasts (ECMWF; Hogan and Bozzo 2018).

In the shortwave, the main effect of transport through cloud sides is “side illumination”: the enhanced interception of direct sunlight when the sun is low in the sky, which increases the reflectance of the scene and equivalently the magnitude of the cloud radiative effect (CRE). However, Barker et al. (2015) reported Monte Carlo calculations showing that in realistic cloud scenes, the effect of introducing 3D transport was more typically to *reduce* the magnitude of the CRE, particularly when the sun is high in the sky.

Várnai and Davies (1999) characterized 3D solar radiative effects in single-layer cloud scenes in terms of four mechanisms, two of which reduce the reflectance of a cloudy scene and are therefore candidates to explain this behavior. Their “downward escape” mechanism explains how forward-scattered sunlight inside a cloud has a chance to escape through the side of a cloud and reach the surface, whereas in the corresponding independent column approximation (ICA) case it would remain within the cloud and have more chance of being scattered back to space. Welch and Wielicki (1984), Hogan and Shonk (2013), and Barker et al. (2016) argued that this escape mechanism explains why 3D effects reduce the reflectance of cumulus, stratocumulus, aircraft contrails and stochastically generated cloud fields, for high-sun conditions. This process is represented by SPARTACUS, but in this paper we present evidence to show that it is not significant enough to explain the results of Barker et al. (2015), which were for a wide range of realistic and often multilayered cloud scenes.

The second candidate mechanism from Várnai and Davies (1999) is “upward trapping,” which incorporates all light rays that (i) are reflected back to space in the ICA case but not when 3D transport is included and (ii) have a longer pathlength in 3D than ICA. Their diagram to explain how this mechanism typically acts in single-layer cloud scenes depicted an upward-traveling light ray passing horizontally through the side of a cloud above, a process that is already represented by SPARTACUS. However, it is also possible, particularly in multilayered scenes, for trapping to occur without any transport through cloud sides, but rather as a consequence of horizontal transport entirely within a clear or cloudy region and the upward-reflected ray then intercepting the base of a cloud above. This process is not explicitly handled by SPARTACUS. It was alluded to by Barker and Davies (1992) who considered idealized single-layer clouds over a reflective surface, but not studied in detail for realistic multilayered cloud scenes.

In this paper, we seek to quantify the importance of this mechanism, which we refer to as “entrapment.” In section 2, we describe it in more detail and present a simple mathematical example to illustrate how it reduces the scene reflectance. In section 3, we describe how the limits of zero and maximum entrapment may be represented in SPARTACUS. This is followed by two sections on the more complex “explicit” entrapment calculations: section 4 describes how we estimate the horizontal distance traveled by reflected radiation, with validation against monochromatic Monte Carlo simulations, while section 5 describes how the distance traveled is used to compute how much entrapment occurs, accounting for the fractal nature of clouds. Readers uninterested in the internal workings of SPARTACUS may wish to skip sections 3–5. Then in section 6, estimates of the broadband shortwave 3D radiative effect by the new SPARTACUS solver are evaluated by comparing to Monte Carlo calculations performed on 65 diverse high-resolution scenes from a cloud-resolving model.

## 2. The concept of entrapment

The schematic in Fig. 1 illustrates how entrapment can change the reflectance of a cloud scene. Figure 1a depicts the behavior assumed in the ICA, in which horizontal transport is ignored: incoming solar radiation scattered upward by the first cloud layer it encounters is likely to escape to space since it passes back through the same clear-sky atmosphere (similar to the “opposition effect” in vegetation; e.g., Hapke et al. 1996). Figure 1b illustrates the process of entrapment by clouds when 3D transport is permitted: radiation passing down through a clear-sky (or less optically thick) part of the atmosphere may be reflected back upward at a slantwise angle and encounter the base of a cloud due to horizontal transport within either the clear-sky or cloudy region. The depiction of upward trapping by Várnai and Davies (1999) was similar except that the two cloud layers were part of the same cloud, and the reflected ray was intercepted by the edge rather than the base of the upper layer. Since the area presented by the base of a cloud is usually much larger than its edge, we would expect the impact of trapping by the base to be greater, on average. Note that entrapment can also occur over reflective surfaces where the upward reflection is by the surface rather than a cloud.

The interception of radiation by the upper cloud layer reduces the reflectance of the scene, but the magnitude of this effect depends on how far the radiation migrates horizontally in the gap between the two cloud layers relative to the size of the clouds in the upper layer. Figure 1c depicts the extreme case in which radiation is completely horizontally homogenized in clear-sky layers. This “maximum entrapment” is actually the behavior of the original shortwave implementation of SPARTACUS described by Hogan et al. (2016), as well as other solvers such as the three-region solver in the original Edwards and Slingo (1996) radiation scheme that was adapted by Shonk and Hogan (2008) to become the “Tripleclouds” solver. Shonk and Hogan (2008) described this radiative homogenization as “anomalous horizontal transport,” which is not really accurate, as at least some of this transport occurs in reality. Nonetheless, their method to remove it and thus to move from maximum entrapment (Fig. 1c) to zero entrapment (Fig. 1a) provides the starting point for representing more realistic explicit entrapment (Fig. 1b) in SPARTACUS.

*R*, and scatter conservatively so that their transmittance is

*R*, and the final column consists of two cloud layers, which the Adding Method (Lacis and Hansen 1974) predicts to have a reflectance of

*R*= 1), the scene reflectance becomes

*R*by

## 3. Representing entrapment scenarios in SPARTACUS

Here we explain how SPARTACUS may be modified to represent zero and maximum entrapment, illustrated in Fig. 1, as well as the first step in representing explicit entrapment. The symbols used in more than one equation in sections 3–5 are defined in appendix B. SPARTACUS uses the Tripleclouds approach of splitting each cloudy layer into three regions, one clear (denoted *a*) and two cloudy (denoted *b* and *c*) with different optical depths. The radiation problem can then be written in terms of vectors and matrices; for example, *j* per unit area of the entire grid box, not per unit area of region *j*.

*direct*radiation of the entire atmosphere and surface below half level

*diffuse*radiation. (As shown in Fig. 1a, we index full atmospheric layers by

*i*, counting down from the highest layer

*i*− 1 and

*i*.) This change mirrors the application by Hogan et al. (2018) of SPARTACUS to vegetation. Both of these albedos are matrices of the form

*j*that is reflected up in region

*k*. These definitions ensure that

**u**at any given height is equal to the sum of reflection of the downward diffuse irradiance

**v**and the downward direct irradiance

**s**[see (40) of Hogan et al. 2018]:

*i*. Likewise,

*i*− 1. Equation (30) of Hogan et al. (2016) relates the two according to the maximum-entrapment assumption:

*A*is the albedo of the atmosphere below half level 3.5. Applying (5) yields

**a**is a column vector containing the reflectances of each region with the assumption that light is always reflected up from the same region it enters. Since Tripleclouds neglects lateral radiation flows between regions,

**a**simply contains its diagonal elements. To apply the zero-entrapment assumption to a SPARTACUS simulation that includes lateral flows between regions,

*j*th element contains the sum of the

*j*th column of

*i*− 1, so can be treated by maximum entrapment. By contrast, the destination of reflected radiation that does not pass through a cloud edge (the diagonal elements of

*j*th diagonal of

*j*th row of

Matrix *j* of layer *i*, but also accounting for radiation passing down through the layers below. Its elements quantify the weight of each of the arrows in Fig. 3b. Since it repartitions radiation between regions without changing the total energy, its columns sum to 1 (i.e., it is a left stochastic matrix). If we wished for the diagonal elements of

## 4. Explicit entrapment: Horizontal distance traveled by reflected radiation

The white-headed arrow in Fig. 1b illustrates the horizontal distance traveled by a single light ray reflected below half-level 3.5, and includes the horizontal distance associated with both the downward and upward parts of the journey. This section deals with the task of estimating the *mean* horizontal distance traveled by reflected radiation below a particular half-level

### a. Method

*i*is

*i*: given the layer reflectance

*i*and the atmosphere below. This is a geometric series that reduces to

*i*. The first term on the right-hand side contains

*i*, rather than penetrating the layer and being reflected by the layers below. We assume that, on average, such radiation penetrates to the center of the layer before being reflected back out (hence traveling a distance

The first line in the curly brackets in (15) represents radiation that passes down through the entire layer *i* and back up again, so the horizontal distance associated with transiting the layer is twice that of radiation reflected by the layer (the *i* and the layers below half level

*direct*radiation. The equivalent expression to (14) for the albedo to direct radiation is

**x**and

**y**whose

*j*th elements contain the horizontal distances associated with region

*j*. As illustrated in Fig. 3b, each region is considered independently, so we may still use these four equations to step the elements of

**x**and

**y**from the base of the layer to the top. The other inputs to these equations,

*i*; radiation that passes laterally between regions in layer

*i*was dealt with by (10). The final aspect to deal with partially cloudy profiles is to translate

**x**and

**y**from the regions below half-level

**y**.

### b. Evaluation

Here we evaluate the estimates of mean horizontal distance traveled by reflected radiation as a function of height (the values of *x* and *y* above), using Monte Carlo calculations by the model of Villefranque et al. (2019, manuscript submitted to *J. Adv. Model. Earth Syst**.*), which implements ray-tracing techniques from computer graphics and permits the paths of individual photons to be tracked. The results are shown in Fig. 4 for four cloud scenes and three solar zenith angles in simulations using periodic boundary conditions in the horizontal. All are at a single wavelength in vacuum with idealized cloud optical properties over a Lambertian surface with an albedo of 0.2. The first profile (Figs. 4a–d) consists of a plane-parallel cloud layer containing isotropic scatterers with an optical depth of 1 and a single-scattering albedo of 0.999 999. Beneath the cloud, all reflection is from the surface so the mean horizontal distance traveled increases linearly with height above the surface *z*. The direct mean horizontal distance *y* increases with *z* as diffuse radiation. Within the cloud a fraction of downwelling radiation is reflected by the cloud, rather than the surface, and so the mean horizontal distance is reduced. We see that the SPARTACUS estimates using the method described above are accurate to around 10% for *x* and 3% for *y*. The second profile (Figs. 4e–h) is the same but with an optical depth of 5. The SPARTACUS errors are somewhat larger at around 18%.

The last two profiles contain more realistic clouds. Both assume an asymmetry factor of 0.86, appropriate for liquid clouds in the midvisible. SPARTACUS then performs the usual delta-Eddington scaling, treating some of the forward scattered light as if it had not been scattered at all. To achieve a fair comparison in terms of the definition of direct and “diffuse” radiation, the Monte Carlo model takes the delta-Eddington-scaled extinction coefficient, and assumes a Henyey–Greenstein scattering phase function using the delta-Eddington-scaled asymmetry factor value of 0.462. Figures 4i–l show the results for a 6.4 km × 6.4 km large-eddy simulation of cumulus clouds from Brown et al. (2002), which was also used by Hogan et al. (2016) and is based on an observed case from the Atmospheric Radiation Measurement (ARM) program. Figures 4m–p show the results for a 100 km × 100 km scene from a 250-m simulation by the Canadian Global Environmental Multiscale (GEM) model of a multilayer liquid cloud (Pacific scene 16). The GEM scenes are described in detail in section 6. In both the ARM and the GEM cases, the typical SPARTACUS errors are 25% for *x* and 6% for *y*. Given the simplifications involved, SPARTACUS performs very well in estimating horizontal distance traveled, and should be adequate to feed into the final step for computing entrapment.

## 5. Explicit entrapment: How much radiation is trapped?

### a. Method

*j*of layer

*i*that lies beneath region

*k*in layer

*i*− 1. It is denoted as a function of the mean horizontal distance traveled,

*x*, since the radiation entered the layer. Matrix

*x*.

**Γ**contains the rates of radiation exchange between the “subregions” of region

*j*(with subregions defined by the regions of layer

*i*− 1 that they are beneath, illustrated by the dashed line in Fig. 3b), and

*k*to

*l*, per unit increase in horizontal distance traveled

*x*. The solution to (25) is (24) but with

### b. Representing fractal behavior

To test the validity of this approach, we use the contrasting binary cloud scenes shown in Figs. 5a–d, which have been generated by applying an optical-depth threshold to four of the GEM simulations described in section 6. A scalar field is defined containing a value of 1 in the clear (black) areas and 0 in the cloudy (white) areas, which can be thought of as solar radiation that has passed through the gaps between the clouds. Gaussian smoothing is then applied to the field with varying smoothing scales *x*, representing horizontal radiation transport beneath the cloud. Previous studies of the interaction of radiation and clouds have found a Gaussian to be reasonably good at describing the horizontal distribution of diffuse radiation originating from a point source (e.g., Hogan and Battaglia 2008; Wissmeier et al. 2013). The fraction of the total scalar field that is then in the cloudy parts of the domain is the “trapped fraction,” and is shown by the black lines in Figs. 5e–h. The dotted lines show the cloud cover, which corresponds to the trapped fraction one would expect if the radiative energy were completely homogenized horizontally (maximum entrapment).

*k*and

*l*(i.e., the perimeter length of the clouds) per unit area of the domain, and

*k*, both of which can be obtained by analyzing the binary cloud fields. We then apply (26) and (27) to obtain

*x*, but for larger

*x*it overestimates entrapment significantly.

*S*[see (20) of Hogan et al. 2018] such that normalized perimeter length is

*S*can be thought of as the size that equally sized squares would need to have if they were placed randomly on a grid and their fractional cover and total perimeter length were equal to the values for the actual cloud field. The

*S*values for the scenes in Fig. 5 are shown above panels e–h. Substitution of (29) into (28) gives

*S*. Thus, if all clouds indeed had a diameter of around

*S*then we would expect the trapped fraction to quickly approach the asymptotic value of the cloud cover for

In reality the clouds span a wide range of scales, and the presence of very large clouds reduces the trapped fraction for larger *x*. Another way of looking at this is to recognize that since clouds are fractal, the *effective* perimeter length *D* is the fractal dimension. Many studies have estimated the fractal dimension of clouds, with *D* = 1.5 being a reasonable representative value (e.g., Cahalan and Joseph 1989; Gotoh and Fujii 1998; Wood and Field 2011), implying

*x*in this way, with the formula leading to the best fit given by

*x*.

### c. Treatment of overhanging clouds

Applying these findings in SPARTACUS presents one further issue to resolve, since as shown in Fig. 3b we are not dealing with radiative exchange between regions, but exchange between the subregions of region *j* in layer *i*, defined according to the regions above them in layer *i* − 1. Unfortunately the perimeter length of the interface between these subregions is not completely defined by the variables available to SPARTACUS. Consider the case of two layers, each with a cloud fraction of 0.5, an overlap parameter of *a* in layer *i*, which in this 2D diagram is illustrated by the number of overhanging clouds shown by the dashed lines.

*j*could be written as

*j*in layer

*i*that is maximally overlapped, so divide through by

*j*that is randomly overlapped:

*k*and

*l*, while the fewest overhangs (Fig. 6b) is obtained by using

*ζ*that varies the effective perimeter length linearly between most overhangs

## 6. Results

In this section we evaluate the shortwave 3D radiative effect predicted by the new SPARTACUS implementation in the ecRad radiation scheme (Hogan and Bozzo 2018), and investigate the impact of various different treatments of entrapment. We have used 65 scenes generated from simulations by Environment and Climate Change Canada’s GEM model (Girard et al. 2014), using the configuration described by Leroyer et al. (2014) with the Milbrandt and Yau (2005) double-moment bulk microphysics cloud scheme.

Each scene measures 100 km × 100 km, has a horizontal resolution of 250 m and employs 56 vertical layers. The simulations were originally performed to generate synthetic satellite data from two swaths: an Atlantic swath on 7 December 2014 from Greenland to the Dominican Republic, from which 39 scenes were extracted, and a Pacific swath on 24 June 2015 from Hawaii to Tonga, from which a further 26 scenes have been extracted. Thus, the scenes span a wide range of cloud conditions.

Both ICA and 3D Monte Carlo shortwave radiative transfer calculations have been performed on these scenes using the model of Barker et al. (2003), which tracks photons through sequences of scattering events until they are either absorbed by a particle, molecule, or the surface, or exit the domain’s top. Calculations were performed for solar zenith angles at 5° intervals between 0° and 85°, but with random solar azimuth angle, and assuming a periodic domain. The Rapid Radiative Transfer Model for GCMs (RRTM-G) of Iacono et al. (2008) was used to represent gas absorption, the Yi et al. (2013) scheme for ice optical properties and Mie theory for liquid droplets. Scattering by air molecules and cloud particles were handled by the Rayleigh and Henyey–Greenstein phase functions, respectively. To simplify the comparison with 1D radiation schemes, all calculations assumed a Lambertian surface with an albedo of 0.05.

We first compare the TOA cloud radiative effect between Monte Carlo calculations run in an ICA mode (Monte Carlo ICA) and Tripleclouds (the SPARTACUS control) for the same scenes, that is, in the absence of 3D radiative transfer. In addition to cloud fraction and gridbox-mean liquid and ice mixing ratio, Tripleclouds takes as input the overlap parameter at each half-level and the fractional standard deviation of in-cloud water content, FSD, in each layer. It was found that the original implementation of Tripleclouds was not capable of accurately representing the effect of horizontal heterogeneities for FSD > 1.5, which occurs in many of these scenes. Appendix A describes an improvement to Tripleclouds and SPARTACUS that has overcome this problem. Figures 7a–c reveal that the resulting root-mean-square error (RMSE) in CRE predicted by Tripleclouds is around 10% and its bias is only 1%–2%.

The differences between 3D and ICA Monte Carlo calculations of CRE for the 65 scenes are summarized by the black box-and-whisker plots in Figs. 8a and 8b, and the mean by the thick black line. We see that for ^{−2} at

^{−2}at

^{−2}at

^{−2}at TOA for overhead sun, but is over twice as strong as the reference Monte Carlo calculations.

The other two SPARTACUS simulations in Figs. 8a and 8b are much closer to the reference calculations: the red and pink lines show results using explicit entrapment described in sections 4 and 5, with the two treatments of cloud overhangs illustrated in Fig. 6. It is clear that the least-overhang scenario

Figure 8c shows the change to total atmospheric absorption when 3D effects are included. The Monte Carlo calculations show an increase in absorption by around 1 W m^{−2} at most solar zenith angles, which is comparable to the findings of Barker et al. (2016). Both the main 3D mechanisms contribute to this effect: side illumination at large solar zenith angle enhances the interception and hence absorption by clouds, while Fig. 3b shows that entrapment increases the pathlength of radiation in clear skies beneath cloud, enhancing water vapor absorption. SPARTACUS with explicit entrapment leads to around 2 W m^{−2} greater atmospheric absorption than Tripleclouds, on average, which is twice the 3D effect in the Monte Carlo simulations. This is related to the presence of a handful of outliers among the SPARTACUS simulations (shown by red dots in Fig. 8c); indeed, if we were to look at the median rather than the mean of the 65 cases then it would suggest instead that SPARTACUS tends to underestimate the 3D effect on atmospheric absorption.

To investigate the factors that influence the nature of 3D radiative transfer in individual cases, and the fidelity with which they are captured by SPARTACUS, we analyze the radiation fields for the four contrasting GEM scenes depicted in Fig. 9. Vertical profiles of the four main inputs to SPARTACUS are shown in Figs. 10a–d. Atlantic case 6 consists of cumulus clouds with some vertical development; the small effective cloud scale of *S* ≃ 1 km, and hence large cloud-side area, leads to significant shortwave side illumination, with Fig. 10e showing a 7 W m^{−2} increase in the reflectance of the scene at ^{−2} warming at

The three remaining scenes, by contrast, appear to be dominated by entrapment. Atlantic case 14 contains deep frontal cloud with considerable small-scale structure. The zero-entrapment simulation in Fig. 10f shows the significant cooling effect of side illumination, but the explicit-entrapment simulation shows that this is overwhelmed by entrapment and indeed the net warming by 3D effects is up to 39 W m^{−2} (a −8% change to CRE). A key factor is the large vertical extent of the cloud, which means that radiation passing down through the gaps in the clouds can travel a large distance horizontally before being reflected back up to its original level, increasing the trapping. Atlantic case 32 contains much more homogeneous and overcast boundary layer cloud. The zero-entrapment simulation has a 3D radiative effect of less than 1 W m^{−2}, confirming that cloud-side effects are weak. With entrapment included, the 3D effect is a warming of up to 6–7 W m^{−2}, with good agreement between SPARTACUS and Monte Carlo. In absolute terms this effect is significant, but this scene is the most reflective of the four and in relative terms it is only a −1% change to CRE. Additional Tripleclouds and SPARTACUS calculations in which the in-cloud heterogeneity is removed (i.e., setting FSD = 0) lead to the 3D effect almost entirely disappearing, which suggest that it is due to trapping associated with cloud heterogeneity, similar to one of the mechanisms proposed by Várnai and Davies (1999). Finally, Pacific case 25 consists of remnants of deep convection including anvils with *S* ≃ 10 km. Again the entrapment mechanism appears to dominate.

## 7. Conclusions

Cloud scenes have varied and complex structures, and consequently it can be very challenging to understand the magnitude and even sign of the differences between radiation calculations with and without horizontal transport. The simplest mechanism to understand shortwave 3D radiative transfer is side illumination, which enhances cloud reflectance. This has led many previous studies to focus on cloud types with a relatively large cloud-side area such as cumulus (Benner and Evans 2001; Pincus et al. 2005) and aircraft contrails (Gounou and Hogan 2007). However, Barker et al. (2015) analyzed a much more varied and representative set of cloud fields and found that shortwave 3D transport tends to reduce the reflectance of clouds overall and hence has a warming effect on the Earth system. In this paper we propose the mechanism of entrapment to explain this behavior. Entrapment is similar to one of the mechanisms suggested by Várnai and Davies (1999) for single-layer cloud scenes, but an important insight is that it need not involve transport through cloud sides. It tends to be strongest in deep, multilayer scenes, which are common in reality but have tended to be ignored in previous case studies, presumably due to them being regarded as too complex to interpret.

We have described modifications to the shortwave SPARTACUS solver of Hogan et al. (2016) to incorporate an explicit calculation of entrapment, making use of the effective cloud scale variable already provided as input to SPARTACUS. This involves a novel method to estimate the mean horizontal distance traveled by reflected radiation, something that could be useful in other contexts, for example in determining when the radiation scheme of a cloud-resolving model ought to represent lateral exchange of radiation between grid boxes. We have also found it necessary to explicitly represent the fractal dimension of cloud perimeters.

Evaluation against Monte Carlo calculations on 65 contrasting scenes from a cloud-resolving model reveals the new SPARTACUS scheme to be capable of predicting the “3D effect,” that is, the difference between cloud radiative effect computed with and without horizontal radiative transport, with a TOA bias of no more than 0.3 W m^{−2} for all solar zenith angles, and skill in predicting the dependence of the 3D effect on solar zenith angle in individual scenes. On average, 3D radiative effects tend to make these scenes less reflective (similar to the findings of Barker et al. 2015), implying that entrapment is a more important mechanism than side illumination. However, this result is highly dependent on the realism of the clouds simulated by the cloud-resolving model; if real clouds were smaller, on average, than those used here then the side-illumination mechanism would be relatively more important.

The modified SPARTACUS is now an option in the ecRad radiation scheme (Hogan and Bozzo 2018) used in the ECMWF model. Hogan and Bozzo (2018) reported the original SPARTACUS with maximum entrapment to be 3.5 times slower than Tripleclouds, and we find that explicit entrapment increases this to around 4.5. While too costly to use operationally, it is fast enough to use for research purposes. The next step will be to use this validated tool to estimate the global impact of 3D radiative transfer, not just in the shortwave but also in the longwave (Schäfer et al. 2016).

## Acknowledgments

We thank Zhipeng Qu for performing the GEM model simulations. NV acknowledges support from the Agence Nationale de la Recherche (Grant ANR-16-CE01-0010), and from the French Ministry of Higher Education, Research and Innovation through the doctoral school SDU2E of Université de Toulouse.

## APPENDIX A

### Improving Tripleclouds for Very Heterogeneous Scenes

*b*and

*c*) of different extinction coefficient. Shonk and Hogan (2008) reported that for FSD up to 2, predicted irradiances agreed best with ICA if the two cloudy regions had equal area, region

*b*used the 16th percentile of the full extinction distribution, and the extinction of region

*c*was chosen so as to conserve the layer-mean extinction. The implementation of Tripleclouds in ecRad (Hogan and Bozzo 2018) includes the option to represent either a lognormal distribution of optical depth, in which case the ratio of the 16th percentile to the in-cloud mean is given approximately by (44) of Hogan et al. (2016), or a gamma distribution (e.g., Barker et al. 1996) for which this ratio is approximated by

*c*to the in-cloud mean to be

Comparison of Tripleclouds to ICA calculations on the scenes described in section 6 revealed the gamma distribution to perform best, but even then Tripleclouds tended to overestimate scene reflectance for the more heterogeneous scenes, some of which have FSD values up to 4. In Fig. A1a we have repeated the analysis of Shonk and Hogan (2008) but for a gamma rather than a lognormal distribution, and considered larger values of FSD. A substantial albedo overestimate is apparent for FSD > 2. The problem arises because for large FSD,

*b*and correspondingly reduce that of region

*c*: for FSD in the range 1.5–3.75, the fraction of the cloudy area occupied by region

*b*increases linearly from 0.5 to 0.9, while outside this range it is capped at 0.5 or 0.9. The extinction of region

*c*is still chosen to conserve the layer-mean value. Figure A1b shows that these changes virtually eliminate the albedo bias up to an FSD of 4. This solution has been implemented in both the Tripleclouds and SPARTACUS solvers of ecRad, and is used in section 6.

## APPENDIX B

### List of Symbols

The following list includes symbols used in more than one equation in sections 3–5, and “PP” indicates a variable from section 4a where a plane-parallel atmosphere has been assumed.

Diffuse albedo of entire atmosphere and surface below interface | |

Diffuse albedo of entire atmosphere and surface below interface | |

Fraction of layer occupied by region | |

Same as | |

Direct albedo of entire atmosphere and surface below interface | |

Fraction of direct radiation penetrating layer | |

Rate at which radiation passes from subregion | |

Length of interface between regions | |

| |

Matrix expressing how much radiation entering region | |

Diffuse reflectance of layer | |

s | Vector of downwelling direct irradiances in each region at a particular height |

Reflectance of layer | |

Fraction of direct radiation that penetrates layer | |

S, S^{het} | Effective cloud scale, cloud heterogeneity scale |

Transmittance of layer | |

u | Vector of upwelling irradiances in each region at a particular height |

Upward overlap matrix expressing how upwelling irradiances in each region just below interface | |

v | Vector of downwelling diffuse irradiances in each region at a particular height |

Downward overlap matrix expressing how downwelling irradiances in each region just above interface | |

Vector expressing the fraction of radiation in region | |

Mean horizontal distance traveled by reflected diffuse radiation below interface | |

Mean horizontal distance traveled by diffuse radiation reflected by layer | |

Mean horizontal distance traveled by reflected direct radiation below interface | |

Mean horizontal distance traveled by direct radiation reflected by layer | |

Γ | Matrix expressing the rates of radiation exchange between the subregions of a region |

Horizontal distance traveled by diffuse radiation passing through layer | |

Horizontal distance traveled by direct radiation passing through layer | |

Thickness of layer | |

ζ | Overhang factor |

Solar zenith angle |

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