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  • View in gallery

    (first row) Reflectance vs optical depth and (second row) the corresponding relative errors with the inhomogeneous and homogeneous solutions. The asymmetry factor g is held constant, and the single-scattering albedo ω is a function of optical depth. (third row),(fourth row) As in the first and second rows, resepectively, but for absorptance.

  • View in gallery

    As in Fig. 1, but the single-scattering albedo ω is held constant, and the asymmetry factor g is a function of optical depth.

  • View in gallery

    Relative errors with the (left) homogeneous solution and (right) inhomogeneous solution for the (first row),(third row) reflectance and (second row),(fourth row) absorptance vs , , and optical depth .

  • View in gallery

    For the band of 0.25–0.69 μm, (a) cloud asymmetry factor and (b) single-scattering albedo vs cloud optical depth; (c) reflectance and (e) absorptance vs optical depth; and (d),(f) relative errors with the inhomogeneous and homogeneous solutions.

  • View in gallery

    As in Fig. 4, but for the wavelength 0.94 μm.

  • View in gallery

    For the band of 0.25–0.69 μm, (a) snow asymmetry factor (b) single-scattering albedo vs cloud optical depth; (c) reflectance and (e) absorptance vs optical depth; and (d),(f) relative errors with the inhomogeneous and homogeneous solutions.

  • View in gallery

    As in Fig. 6, but for the wavelength 0.94 μm.

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A New Radiative Transfer Method for Solar Radiation in a Vertically Internally Inhomogeneous Medium

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  • 1 Key Laboratory of Meteorological Disaster, Ministry of Education, Nanjing University of Information Science and Technology, Nanjing, China
  • | 2 Center for Atmospheric and Oceanic Studies, Graduate School of Science, Tohoku University, Sendai, Japan
  • | 3 Canadian Centre for Climate Modelling and Analysis, Environment and Climate Change Canada, University of Victoria, Victoria, British Columbia, Canada
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Abstract

The problem of solar spectral radiation is considered in a layer-based model, with scattering and absorption parallel to the plane for each medium (cloud, ocean, or aerosol layer) and optical properties assumed to be vertically inhomogeneous. A new radiative transfer (RT) method is proposed to deal with the variation of vertically inhomogeneous optical properties in the layers of a model for solar spectral radiation. This method uses the standard perturbation method to include the vertically inhomogeneous RT effects of cloud and snow. The accuracy of the new inhomogeneous RT solution is investigated systematically for both an idealized medium and realistic media of cloud and snow. For the idealized medium, the relative errors in reflection and absorption calculated by applying the homogeneous solution increase with optical depth and can exceed 20%. However, the relative errors when applying the inhomogeneous RT solution are limited to 4% in most cases. Observations show that stratocumulus clouds are vertically inhomogeneous. In the spectral band of 0.25–0.69 μm, the relative error in absorption with the inhomogeneous solution is 1.4% at most, but that with the homogeneous solution can be up to 7.4%. The effective radius of snow varies vertically. In the spectral band of 0.25–0.69 μm, the relative error in absorption with the homogeneous solution can be as much as 72% but is reduced to less than 40% by using the inhomogeneous solution. At the spectral wavelength of 0.94 μm, the results for reflection and absorption with the inhomogeneous solution are also more accurate than those with the homogeneous solution.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Feng Zhang, feng_zhang126@126.com

Abstract

The problem of solar spectral radiation is considered in a layer-based model, with scattering and absorption parallel to the plane for each medium (cloud, ocean, or aerosol layer) and optical properties assumed to be vertically inhomogeneous. A new radiative transfer (RT) method is proposed to deal with the variation of vertically inhomogeneous optical properties in the layers of a model for solar spectral radiation. This method uses the standard perturbation method to include the vertically inhomogeneous RT effects of cloud and snow. The accuracy of the new inhomogeneous RT solution is investigated systematically for both an idealized medium and realistic media of cloud and snow. For the idealized medium, the relative errors in reflection and absorption calculated by applying the homogeneous solution increase with optical depth and can exceed 20%. However, the relative errors when applying the inhomogeneous RT solution are limited to 4% in most cases. Observations show that stratocumulus clouds are vertically inhomogeneous. In the spectral band of 0.25–0.69 μm, the relative error in absorption with the inhomogeneous solution is 1.4% at most, but that with the homogeneous solution can be up to 7.4%. The effective radius of snow varies vertically. In the spectral band of 0.25–0.69 μm, the relative error in absorption with the homogeneous solution can be as much as 72% but is reduced to less than 40% by using the inhomogeneous solution. At the spectral wavelength of 0.94 μm, the results for reflection and absorption with the inhomogeneous solution are also more accurate than those with the homogeneous solution.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Feng Zhang, feng_zhang126@126.com

1. Introduction

Radiative transfer (RT) is a key issue in climate modeling and remote sensing. In most numerical radiative transfer algorithms, the atmosphere is divided into many homogeneous layers. The inherent optical properties (IOPs) are then fixed within each layer, and variations of IOPs inside each layer are ignored, effectively regarding each layer as internally homogeneous. The standard RT solutions are based on this assumption of internal homogeneity (Lenoble 1985; Toon et al. 1989; Fu et al. 1997; Li et al. 2005; Zhang and Li 2013), which cannot resolve within-layer vertical inhomogeneity.

It has been well established by observation that cumulus and stratocumulus clouds (hereinafter collectively referred to as cumulus clouds) are inhomogeneous, both horizontally and vertically (Vane et al. 2006; Boutle et al. 2014; Young et al. 2013; Luo et al. 2014; Cheng et al. 2015). Inside a cumulus cloud, the liquid water content (LWC) and the cloud droplet size distribution vary with height, so the IOPs of cloud droplets depend on vertical height. For snow, the mean grain size of snow packs generally increases with time (Nakaya 1954; LaChapelle 1969; Flanner and Zender 2006). A wide variety of research (Arai 1965; Colbeck 1975; Gow 1969; Hobbs 1974; Mellor 1977; Sommerfeld and LaChapelle 1970) suggests that the average grain radius is in the range of 20–100 μm for new snow, 100–300 μm for fine-grain older snow, and 1.0–1.5 mm for old snow near the melting point. Therefore, snow grain size varies with snow depth, and the IOPs of snow are vertically inhomogeneous. In addition, the aerosol concentrations in the atmosphere are vertically inhomogeneous (Li et al. 2008; von Salzen et al. 2013; Guo et al. 2016; Qin et al. 2016, 2017; Wang et al. 2015).

How to deal with vertical internal inhomogeneity in RT models is an interesting topic for researchers. Li et al. (1994) developed a Monte Carlo cloud model that can be used to investigate photon transport in inhomogeneous clouds by considering internal variation of the optical properties. Their model showed that when overcast clouds become broken clouds, the difference in reflectance at large solar zenith angles between vertically inhomogeneous clouds and their plane-parallel counterparts can be as much as 10%. However, the Monte Carlo method is time consuming and not applicable to climate models or remote sensing (Liou 2002). The albedo of inhomogeneous mixed-phase clouds at visible wavelengths could be obtained by using a Monte Carlo method to compare such clouds with plane-parallel homogeneous clouds (Macke et al. 1998).

In principle, the vertical-inhomogeneity problem of the RT process can be solved by increasing the number of layers of the climate model. However, it is time consuming to increase the vertical resolution of a climate model. Typically, there are only 30–100 layers in a climate model (von Salzen et al. 2013), which is not high enough to resolve the cloud vertical inhomogeneity. For snow-covered ground, most climate models consider only the change in surface albedo due to the snow. Recently, the RT process has been extended to snow by treating it as either a single layer or a double layer (Namazi et al. 2015), even though the parameterization of snow-layer IOPs was proposed a long time ago (Wiscombe and Warren 1980; Warren and Wiscombe 1980). To completely address the problem of vertical inhomogeneity with only a limited number of layers in a climate model, the standard RT method must be extended to deal with the vertical inhomogeneity internal to each model layer. The main purpose of this study is to develop a new inhomogeneous RT solution. This solution follows a perturbation method: the zeroth-order solution is the Eddington approximation for homogeneous layers, with a first-order perturbation to account for the inhomogeneity effect. In section 2, the basic theory of radiative transfer is introduced, and the new inhomogeneous RT solution is presented. In section 3, the accuracy of the new solution is examined for a wide range of idealized cases. Furthermore, this inhomogeneous RT solution is applied to cloud and snow as realistic examples to demonstrate the practicality of this new method. A summary is given in section 4.

2. RT solution for an inhomogeneous layer

The azimuthally averaged solar radiative transfer equation is
e1
where μ is the cosine of the zenith angle ( and refer to upward and downward radiation, respectively), is the scattering phase function, τ is the optical depth ( and refer to the top and bottom of the medium, respectively), is the single-scattering albedo, and is the incoming solar flux. For the Eddington approximation, for . For the scattering atmosphere, the irradiance fluxes in the upward and downward directions can be written as
e2
To simulate a realistic medium such as cloud or snow, we consider and to vary with τ, and we use exponential expressions here to simplify the process. The single-scattering albedo and asymmetry factor are written as
e3a
e3b
where is the optical depth of the layer, is the single-scattering albedo at , and is the asymmetry factor at the same place. Both and are small parameters that are far less than and , respectively, in a realistic medium.
By perturbation theory (Kato 1966), the corresponding flux can also be expanded by using the perturbation coefficients and :
e4a
e4b
By using the Eddington approximation, we can obtain differential equations for and ; a detailed derivation is presented in appendix A. These equations [see Eq. (A4) for the precise formulation] can be rewritten as separate equations for , , and . We obtain the following equations for the scattered flux :
e5a
e5b
e5c
e5d
where is defined in the appendix A. Equation (5) is the standard RT equation for a homogeneous layer (Meador and Weaver 1980) and has the following solution:
e6a
e6b
where K1, K2, G1, G2, Γ, and k are defined in appendix B. According to Eq. (A4), the equations for the perturbation terms are
e7a
e7b
e7c
where , , , and are defined in appendix A. The solutions (the details are in appendix C) are
e8a
e8b
Finally, we can obtain and as
e9a
e9b
where , , , , , , , and are defined in appendix D.

3. Results and discussion

a. Idealized medium

We analyze the accuracy of the new inhomogeneous RT solution by applying it to a single-layer inhomogeneous scattering medium. In the benchmark calculations, the inhomogeneous medium is divided into 100 sublayers, and the IOPs vary from layer to layer to resolve their vertical variation. The Eddington approximation is used in the benchmark calculation. The optical depth of the medium varies from 0.1 to 50.

First, we consider the case of vertical variation in single-scattering albedo with . The asymmetry factor is constant, with . Different values of a control the degree of vertical variation in . The results of reflection are shown in the top two rows of Fig. 1 for a = 0.01, 0.05, and 0.25: the larger the value of a, the stronger the vertical variation of . Figure 1 shows that the reflection calculated with the inhomogeneous solution is always much more accurate than that with the homogeneous solution. For , the relative error in the reflection with the homogeneous solution is about 5.8% at , whereas the error with the inhomogeneous solution is less than 0.4%. For , the relative error in the reflection with the homogeneous solution increases to 20%, but it is limited to 4.5% with the inhomogeneous solution. All the results show that the new algorithm is significantly more accurate than the homogeneous RT solution, especially for large optical depths. There is an interesting phenomenon whereby the reflectance decreases after reaching its maximum. This is because the single-scattering albedo in layers having nearly the same optical depth decreases with the optical depth . The absorptance results are shown in the bottom two rows of Fig. 1. For , the relative error with the homogeneous RT solution is 2.3% at , whereas the error is less than 0.2% with the inhomogeneous solution. For , the relative error with the homogeneous solution increases to 7%, whereas the relative error with the inhomogeneous solution does not exceed 1.5%. All results with the homogeneous solution underestimate the solar absorption.

Fig. 1.
Fig. 1.

(first row) Reflectance vs optical depth and (second row) the corresponding relative errors with the inhomogeneous and homogeneous solutions. The asymmetry factor g is held constant, and the single-scattering albedo ω is a function of optical depth. (third row),(fourth row) As in the first and second rows, resepectively, but for absorptance.

Citation: Journal of the Atmospheric Sciences 75, 1; 10.1175/JAS-D-17-0104.1

Second, we consider the case of vertical variation in the asymmetry factor, with , while the single-scattering albedo is constant at . Different values of b represent different degrees of vertical variation in ; we set b = 0.01, 0.05, and 0.25. The reflection results are shown in the top two rows of Fig. 2. The maximum relative error in refection with the homogeneous solution can be over 4%, whereas the error is not more than 0.6% with the inhomogeneous solution. The corresponding absorptance results are shown in the bottom two rows of Fig. 2. The maximum relative error with the homogeneous solution can reach 8%, whereas the relative error with the inhomogeneous solution is roughly 1% at large optical depths. Overall, the inhomogeneous solution can dramatically improve the accuracy in both reflection and absorption from that given by the homogeneous solution.

Fig. 2.
Fig. 2.

As in Fig. 1, but the single-scattering albedo ω is held constant, and the asymmetry factor g is a function of optical depth.

Citation: Journal of the Atmospheric Sciences 75, 1; 10.1175/JAS-D-17-0104.1

Third, we examine the sensitivity to the two other parameters, and . In the first and second rows of Fig. 3, we take the case of , , and , where varies from 0.05 to 50 and varies from −0.2 to −0.002. For the reflectance (first row), the relative error with the homogeneous solution can be as high as 20% for large optical depths with large but only around 4% with the inhomogeneous solution at the same optical depth and value of . The same applies to the absorptance (second row): the relative error can be up to 7% with the homogeneous solution but is limited to 1.5% with the inhomogeneous solution. The accuracy is much higher with the inhomogeneous solution than with the homogeneous solution.

Fig. 3.
Fig. 3.

Relative errors with the (left) homogeneous solution and (right) inhomogeneous solution for the (first row),(third row) reflectance and (second row),(fourth row) absorptance vs , , and optical depth .

Citation: Journal of the Atmospheric Sciences 75, 1; 10.1175/JAS-D-17-0104.1

In the third and fourth rows of Fig. 3, we take the case of , , and , where varies from 0.05 to 50 and varies from −0.2 to −0.002. The relative errors with the inhomogeneous solution are more than one order of magnitude smaller than those with the homogeneous solution for both reflectance and absorptance. The relative errors in reflectance (Fig. 3, third row) with the homogeneous solution are greater than 0.5%, whereas with the inhomogeneous solution, they are less than 0.3% in most regions. The relative errors in absorptance (Fig. 3, fourth row) with the inhomogeneous solution are less than 0.1% in most regions, whereas with the homogeneous solution, they are greater than 0.1%.

b. Cloud

The optical properties of ice clouds depend not only on the effective size but also on the complex particle habits (Letu et al. 2012, 2016; Yang et al. 2015). Thus, the ice cloud is not to be considered in this study. We just discuss the water cloud. Observations show that the cloud LWC and droplet radius tend to increase with height above the cloud base (Noonkester 1984). This phenomenon is attributed to the process of adiabatic water vapor condensation. To take into account the internal inhomogeneity of cloud optical properties, we assume that LWC (g m−3) and droplet cross-sectional area (DCA; cm−2 m−3) increase linearly from the cloud base (0 m) to a position near the cloud top (1000 m):
e10a
e10b
where , z is the height in meters from the cloud base, and is the height of the cloud top. We can obtain the cloud effective radius (μm) from the given LWC and DCA:
e11
where ρ (g m−3) is the liquid water density. Therefore, the LWC varies from 0.22 to 0.30 g m−3, and re varies from 2.06 to 16.50 μm. This is consistent with observations (Chen et al. 2008). The liquid water path (LWP) is defined as . In this case, the LWP is 260 g m−2, which represents low cloud (Fu et al. 1997). In the benchmark calculations, is divided into 100 internal homogeneous sublayers, although other numbers can be chosen (e.g., 200). In principle, more internal sublayers should result in more accurate results. We use 100 internal sublayers throughout this study because having any more makes little difference to the calculated results. Using 100 sublayers is sufficiently accurate to resolve the vertical internal inhomogeneity of the medium. We use the optical properties of a water cloud in the spectral band of 0.25–0.69 μm and at 0.94 μm. We choose the 0.25–0.69-μm band because it contains roughly 46% of the solar radiation energy; the 0.94-μm wavelength is at the center of channel 19 of the Moderate Resolution Imaging Spectroradiometer (MODIS), which is used in this paper to illustrate the effect of vertical internal inhomogeneity related to remote sensing. The surface albedo is set to zero. We calculate the asymmetry factor and single-scattering albedo of each sublayer as the benchmark values, which we then use in Eq. (3) to obtain the relevant parameters.

In Figs. 4a and 4b, the benchmark values of the inhomogeneous IOPs and the parameterized results for the spectral band of 0.25–0.69 μm are shown. The parameterized inhomogeneous IOPs are and , where . The corresponding results for reflection and absorption are shown in Figs. 4c–f. For reflection, the relative error with the homogeneous solution increases from 0.25% to 0.71% as increases from 0.01 to 1, whereas the relative error with the inhomogeneous solution increases from 0.05% to 0.14%. For absorption, the relative error is not sensitive to ; it is around 7.4% with the homogeneous solution but around only 1.4% with the inhomogeneous solution.

Fig. 4.
Fig. 4.

For the band of 0.25–0.69 μm, (a) cloud asymmetry factor and (b) single-scattering albedo vs cloud optical depth; (c) reflectance and (e) absorptance vs optical depth; and (d),(f) relative errors with the inhomogeneous and homogeneous solutions.

Citation: Journal of the Atmospheric Sciences 75, 1; 10.1175/JAS-D-17-0104.1

In Figs. 5a and 5b, the benchmark values of the inhomogeneous IOPs and the parameterized results for the wavelength 0.94 μm are shown. The parameterized inhomogeneous IOPs are and , where . For both and , the parameterized results match well with the corresponding benchmark values. Figures 5c–f show the corresponding results for reflection and absorption. For reflection, the relative error with the homogeneous solution increases from 1.1% to 3.0% as increases from 0.01 to 1, whereas the relative error with the inhomogeneous solution increases from 0.7% to 2.0%. For absorption, the relative error is not sensitive to ; it is around 10% with the homogeneous solution but around only 5.7% with the inhomogeneous solution.

Fig. 5.
Fig. 5.

As in Fig. 4, but for the wavelength 0.94 μm.

Citation: Journal of the Atmospheric Sciences 75, 1; 10.1175/JAS-D-17-0104.1

c. Snow

Generally, the effective snow grain size re (μm) and the equivalent depth of liquid water in a snowpack L (g cm−2) can be assumed to increase linearly with depth:
e12a
e12b
where h is the snow thickness in centimeters from the snow top (, with the thickness of the whole snow layer) and ρsnow = 0.15 g cm−3 is the density of snow. We assume is equal to 100 cm; thus, the effective size of snow varies from 50 to 550 μm. The snow layer is divided into 100 internal homogeneous sublayers in the benchmark calculations.

We focus on reflection and absorption by snow in the spectral band of 0.25–0.69 μm and at the wavelength 0.94 μm. The surface albedo is set at zero. We calculate the asymmetry factor and single-scattering albedo of each sublayer as the benchmark values (Wiscombe and Warren 1980) and then use these in Eq. (3) to obtain the relevant parameters.

Figures 6a and 6b show the benchmark values of the inhomogeneous IOPs and the parameterized results for the spectral band 0.25–0.69 μm. The parameterized inhomogeneous IOPs are and , with . Figures 6c–f show the results for reflection and absorption with the homogeneous and inhomogeneous solutions and their relative errors against the benchmark calculations. For reflection, the relative error with the homogeneous solution is 0.8% for , rising to 1.9% for . The error is reduced by using the inhomogeneous solution, with the relative error increasing from only 0.4% to only 1.0%. For absorption, the relative error with the homogeneous solution can be in excess of 72%, whereas it is no more than 40% with the inhomogeneous solution. However, because the absorption is very small in this case, the absolute error with the inhomogeneous solution is less than 0.008.

Fig. 6.
Fig. 6.

For the band of 0.25–0.69 μm, (a) snow asymmetry factor (b) single-scattering albedo vs cloud optical depth; (c) reflectance and (e) absorptance vs optical depth; and (d),(f) relative errors with the inhomogeneous and homogeneous solutions.

Citation: Journal of the Atmospheric Sciences 75, 1; 10.1175/JAS-D-17-0104.1

Figures 7a and 7b show the benchmark values of the inhomogeneous IOPs and the parameterized results for the wavelength 0.94 μm. The parameterized inhomogeneous IOPs are and , with . Figures 7c–f show the results for reflection and absorption with the homogeneous and inhomogeneous solutions. For reflection, the relative error with the homogeneous solution is 10% for , rising to 27.5% for . The inhomogeneous solution produces smaller relative errors, ranging from 1.2% to 3.7%. For absorption, the relative error with the homogeneous solution can be in excess of 120%, whereas the relative error with the inhomogeneous solution is not more than 15%. Although the relative error with the inhomogeneous solution can be as high as 15%, the absorption is very small in this case, with the absolute error with the inhomogeneous solution being less than 0.04.

Fig. 7.
Fig. 7.

As in Fig. 6, but for the wavelength 0.94 μm.

Citation: Journal of the Atmospheric Sciences 75, 1; 10.1175/JAS-D-17-0104.1

4. Summary and conclusions

In the above, we have considered the vertically inhomogeneous structures of only cloud and snow, whereas all physical quantities in the atmosphere are vertically inhomogeneous (e.g., the concentrations of all types of gases and aerosols). In current climate models, the vertical-layer resolution is far from that required to resolve such vertical inhomogeneity. In this study, we have proposed a new inhomogeneous RT solution to address the vertical inhomogeneity by introducing an internal variation of IOPs inside each model layer. This scheme is based on standard perturbation theory and allows us to use the standard RT solution for homogeneous layers to identify a zeroth-order equation and a first-order equation that includes the inhomogeneous effect. The new RT solution can accurately express the inhomogeneous effect in each model layer, and it reduces to the standard Eddington solution when the medium is homogeneous.

The accuracy of the new inhomogeneous RT solution has been investigated systematically for both an idealized medium and realistic samples of cloud and snow. For the idealized medium, the relative errors in reflection and absorption calculated with the inhomogeneous RT solution are significantly smaller than those obtained with the standard homogeneous RT solution. Generally, the relative error with the homogeneous RT solution increases with optical depth, and the relative error can be over 20%. However, the relative error with the inhomogeneous RT solution is not particularly sensitive to optical depth, and the relative error is limited to 4% in most cases.

The new inhomogeneous RT solution is a good way to resolve cloud vertical inhomogeneity. In the spectral band of 0.25–0.69 μm, the relative error in the inhomogeneous RT solution is no more than 1.4%, whereas the error with the homogeneous RT solution can be up to 7.4%. At the specific wavelength of 0.94 μm, the relative error with the inhomogeneous solution is not more than 5.7% but can be up to 10% with the homogeneous solution.

The new inhomogeneous RT solution can also resolve snow vertical inhomogeneity. In the spectral band of 0.25–0.69 μm, the relative error in reflection with the homogeneous solution can be up to 72%, but the error is reduced to less than 40% by using the inhomogeneous solution. At the specific wavelength of 0.94 μm, the relative error in the inhomogeneous solution is not more than 15% but can be up to 120% in the homogeneous solution.

In specific spectral bands or at specific wavelengths, the vertical variations in IOPs can typically be fitted easily into Eq. (3) to obtain the required parameters. A simple fitting program can be easily incorporated into a climate model to produce the inhomogeneous IOPs of stratocumulus clouds. If no such cloud-inhomogeneity information is available in the current climate models, the vertical variation rates of cloud LWC and DCA can be derived empirically from observations, which show that the vertical variation rates of LWC and DCA in stratocumulus clouds are not very different (Vane et al. 2006; Boutle et al. 2014; Young et al. 2013; Luo et al. 2014; Zhang et al. 2017). For snow, the current climate models typically can calculate the snow age (Namazi et al. 2015). In principle, the vertical variation information for snow could be obtained to fit the IOPs in Eq. (3).

In this study, we presented only a single-layer inhomogeneous RT solution, which corresponds to the standard Eddington-approximation solution. To implement the new solution in a climate model, the process of adding layer-to-layer connections has to be resolved (Hansen 1971; de Haan et al. 1987; Nakajima et al. 2000; Zhang et al. 2013; Zhang and Li 2013). Under the homogeneous condition, the single-layer result in reflection and transmission is the same for an upward path and a downward path, but this is not true for an inhomogeneous layer. Therefore, the adding process has to be modified. We will present an algorithm for this multilayer adding process in our next study, in which the climatic impact of inhomogeneous clouds and inhomogeneous snows will be explored. The code base for the inhomogeneous RT solution is available from the authors upon request.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (41675003, 41675056, and 91537213), a Grant-in-Aid for Scientific Research (16F16031) from the Japan Society for the Promotion of Science, the Natural Science Foundation of Jiangsu Province (BK20150931), and the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD). The ESMC contribution number is ESMC 180.

APPENDIX A

Perturbation Equations

According to the Eddington approximation, the radiative intensity can be written as
ea1
Using Eqs. (1), (2), and (A1), we obtain
ea2a
ea2b
ea2c
where , , and ; is the optical depth of the single layer; and is the diffuse (direct) reflection from the layer below or the diffuse (direct) surface albedo.
Substituting , , and into Eq. (3) and ignoring the small second-order parameters , , and , we get
ea3a
ea3b
ea3c
where , , , , , , , and .
Substituting Eqs. (4) and (A3) into Eq. (A2) yields
ea4a
ea4b
where .

APPENDIX B

Coefficients of Eq. (6)

The terms K1, K2, G1, G2, Γ, and k in Eq. (6) are
eq1

APPENDIX C

Solution of Eq. (7)

Letting and , Eqs. (7a) and (7b) yield
ec1a
ec1b
where , , , , , and .
From Eq. (C1), we obtain
ec2a
ec2b
where , , , , , and .

APPENDIX D

Coefficients of Eq. (9)

The terms D1i, D2i, φ1i, φ2i, φ3i, φ4i, φ5i, and φ6i in Eq. (9) are , , , , , , , , , , , , , and . We determine and by the boundary conditions as
ed1a
ed1b

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