1. Introduction

Solving the radiative transfer equation (RTE) is a key issue in radiation scheme for climate model 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 the variations of IOPs inside each layer are ignored, effectively regarding each layer as internally homogeneous. The standard solar/infrared radiative transfer (SRT/IRT)

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

solutions are based on this assumption of internal homogeneity [1–4], which cannot resolve within-layer vertical inhomogeneity.

(τ ¼ 0 and τ ¼ τ<sup>0</sup> refer to the top and bottom of the medium, respectively), ω τð Þ is the single-scattering albedo, and F<sup>0</sup> is the incoming solar flux. For the Eddington approximation, <sup>P</sup> <sup>τ</sup>; <sup>μ</sup>; <sup>μ</sup><sup>0</sup> � � <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>3</sup>gð Þ<sup>τ</sup> μμ<sup>0</sup> (�1 < <sup>μ</sup> <1) and <sup>g</sup>ð Þ<sup>τ</sup> are the asymmetry factors. For the scattering atmosphere, the irradiance fluxes in the upward and downward directions can

Perturbation Method for Solar/Infrared Radiative Transfer in a Scattering Medium with Vertical Inhomogeneity…

ð�<sup>1</sup> 0

To simulate a realistic medium such as cloud or snow, we consider ω τð Þ and gð Þτ to vary with τ, and we use exponential expressions here to simplify the process. The single-scattering

where <sup>τ</sup><sup>0</sup> is the optical depth of the layer, <sup>ω</sup><sup>b</sup> is the single-scattering albedo at <sup>τ</sup>0=2, and <sup>b</sup><sup>g</sup> is the asymmetry factor at the same place. Both <sup>ε</sup><sup>g</sup> and εω are small parameters that are far less than <sup>b</sup><sup>g</sup>

According to the Eddington approximation, the radiative intensity IS τ; μ � � can be written as

<sup>S</sup> ð Þ� τ γ2ð Þτ F�

<sup>S</sup> ð Þ¼ τ<sup>0</sup> Rdif F�

τ<sup>0</sup> is the optical depth of the single layer; and Rdif (Rdir) is the diffuse (resp., direct) reflection from the layer below or the diffuse (direct) surface albedo. Substituting γ1ð Þτ , γ2ð Þτ , and γ3ð Þτ into

> �a1τ0=2 � � <sup>þ</sup> <sup>γ</sup><sup>2</sup>

> �a1τ0=<sup>2</sup> � � <sup>þ</sup> <sup>γ</sup><sup>2</sup>

<sup>S</sup> ð Þ� τ γ1ð Þτ F�

�a1<sup>τ</sup> � <sup>e</sup>

�a1<sup>τ</sup> � <sup>e</sup>

<sup>3</sup> <sup>þ</sup> <sup>γ</sup><sup>2</sup> <sup>3</sup>ε<sup>g</sup> e

�a1<sup>τ</sup> � <sup>e</sup>

�a2<sup>τ</sup> � <sup>e</sup>

IS τ; μ � �μdμ (2)

http://dx.doi.org/10.5772/intechopen.77147

143

�a1τ0=<sup>2</sup> � � (3a)

�a2τ0=<sup>2</sup> � � (3b)

� τ

�τ0

2, and εωεg, we get

�a2τ0=2

�a2τ0=<sup>2</sup> � � (6c)

� � (6a)

�a2τ0=<sup>2</sup> � � (6b)

� τ

<sup>μ</sup><sup>0</sup> (5a)

<sup>μ</sup><sup>0</sup> (5b)

<sup>4</sup> 2 � 3gð Þτ μ<sup>0</sup> � �;

<sup>μ</sup><sup>0</sup> (5c)

IS <sup>τ</sup>; <sup>μ</sup> � � <sup>¼</sup> IS0ð Þþ <sup>τ</sup> IS1ð Þ<sup>τ</sup> <sup>μ</sup> (4)

<sup>S</sup> ð Þ� τ γ3ð Þτ ω τð ÞF0e

<sup>S</sup> ð Þþ <sup>τ</sup> <sup>1</sup> � <sup>γ</sup>3ð Þ<sup>τ</sup> � �ω τð ÞF0<sup>e</sup>

<sup>4</sup> f g <sup>1</sup> � ½ � <sup>4</sup> � <sup>3</sup>gð Þ <sup>τ</sup> ω τð Þ , and <sup>γ</sup>3ð Þ¼ <sup>τ</sup> <sup>1</sup>

<sup>S</sup> ð Þþ τ<sup>0</sup> Rdirμ0F0e

<sup>1</sup>ε<sup>g</sup> e

<sup>2</sup>ε<sup>g</sup> e

�a2<sup>τ</sup> � <sup>e</sup>

�a2<sup>τ</sup> � <sup>e</sup>

�a2<sup>τ</sup> � <sup>e</sup>

F�

albedo and asymmetry factor are written as

and <sup>ω</sup>b, respectively, in a realistic medium.

Using Eqs. (1), (2), and (4), we obtain

dF� <sup>S</sup> ð Þτ

where <sup>γ</sup>1ð Þ¼ <sup>τ</sup> <sup>1</sup>

dF<sup>þ</sup> <sup>S</sup> ð Þτ

<sup>d</sup><sup>τ</sup> <sup>¼</sup> <sup>γ</sup>1ð Þ<sup>τ</sup> <sup>F</sup><sup>þ</sup>

<sup>S</sup> ð Þ¼ 0 0, F<sup>þ</sup>

<sup>4</sup> f g <sup>7</sup> � ½ � <sup>4</sup> <sup>þ</sup> <sup>3</sup>gð Þ<sup>τ</sup> ω τð Þ , <sup>γ</sup>2ð Þ¼ <sup>τ</sup> �<sup>1</sup>

Eq. (3) and ignoring the small second-order parameters εω2, ε<sup>g</sup>

<sup>1</sup> <sup>þ</sup> <sup>γ</sup><sup>1</sup>

<sup>2</sup> <sup>þ</sup> <sup>γ</sup><sup>1</sup>

<sup>1</sup>εω e

<sup>2</sup>εω e

<sup>γ</sup>3ð Þ¼ <sup>τ</sup> <sup>γ</sup><sup>0</sup>

<sup>d</sup><sup>τ</sup> <sup>¼</sup> <sup>γ</sup>2ð Þ<sup>τ</sup> <sup>F</sup><sup>þ</sup>

F�

<sup>γ</sup>1ð Þ¼ <sup>τ</sup> <sup>γ</sup><sup>0</sup>

<sup>γ</sup>2ð Þ¼ <sup>τ</sup> <sup>γ</sup><sup>0</sup>

<sup>S</sup> ð Þ¼ τ 2π

ω τð Þ¼ <sup>ω</sup><sup>b</sup> <sup>þ</sup> εω <sup>e</sup>

<sup>g</sup>ð Þ¼ <sup>τ</sup> <sup>b</sup><sup>g</sup> <sup>þ</sup> <sup>ε</sup><sup>g</sup> <sup>e</sup>

be written as

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 [5–9]. Inside a cumulus cloud, the liquid water content (LWC) and the cloud droplet size distribution vary with height, and so the IOPs of cloud droplets depend on vertical height.

How to deal with vertical internal inhomogeneity in SRT/IRT models is an interesting topic for researchers. Li developed a Monte Carlo cloud model that can be used to investigate photon transport in inhomogeneous clouds by considering an internal variation of the optical properties [10]. 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 very expensive in computing and not applicable to climate models or remote sensing [11]. 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 [12].

In principle, the vertical inhomogeneity problem of the SRT/IRT 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 [13], which is not high enough to resolve the cloud vertical inhomogeneity. To completely address the problem of vertical inhomogeneity by using a limited number of layers in a climate model, the standard SRT method must be extended to deal with the vertical inhomogeneity inside each model layer. The primary purpose of this study is to introduce a new inhomogeneous SRT/IRT solution presented by Zhang and Shi. This solution follows a perturbation method: the zeroth-order solution is the standard Eddington approximation for SRT and two-stream approximation for IRT, with a first-order perturbation to account for the inhomogeneity effect. In Section 2, the basic theory of SRT/IRT is introduced, and the new inhomogeneous SRT/IRT solution is presented. In Section 3, the inhomogeneous SRT/IRT solution is applied to cloud as realistic examples to demonstrate the practicality of this new method. A summary is given in Section 4.
