2.1. SRT solution

The azimuthally averaged solar radiative transfer equation [1–4, 10–12] is

$$\mu \frac{dI\_S(\tau,\mu)}{d\tau} = I\_S(\tau,\mu) - \frac{\omega(\tau)}{2} \int\_{-1}^{1} I\_S(\tau,\mu) P(\tau,\mu,\mu') d\mu' - \frac{\omega(\tau)}{4\pi} F\_0 P(\tau,\mu,-\mu\_0) e^{-\frac{\tau}{\mu\_0}} \tag{1}$$

where μ is the cosine of the zenith angle (μ > 0 and μ < 0 refer to upward and downward radiation, respectively), P τ; μ; μ<sup>0</sup> � � is the scattering phase function, τ is the optical depth (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 be written as

$$F\_S^{\pm}(\tau) = 2\pi \int\_0^{\pm 1} I\_S(\tau, \mu) \mu d\mu \tag{2}$$

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 albedo and asymmetry factor are written as

$$
\omega(\mathbf{r}) = \widehat{w} + \varepsilon\_{\omega} \left( e^{-a\_1 \mathbf{r}} - e^{-a\_1 \mathbf{r}\_0/2} \right) \tag{3a}
$$

$$\mathbf{g}(\tau) = \widehat{\mathbf{g}} + \varepsilon\_{\mathcal{S}} \left( e^{-a\_2 \tau} - e^{-a\_2 \tau\_0/2} \right) \tag{3b}$$

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

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

$$I\_S(\tau,\mu) = I\_{S0}(\tau) + I\_{S1}(\tau)\mu \tag{4}$$

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

solutions are based on this assumption of internal homogeneity [1–4], which cannot resolve

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

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

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

within-layer vertical inhomogeneity.

142 Perturbation Methods with Applications in Science and Engineering

plane-parallel counterparts can be as much as 10%.

with plane-parallel homogeneous clouds [12].

method. A summary is given in Section 4.

<sup>d</sup><sup>τ</sup> <sup>¼</sup> IS <sup>τ</sup>; <sup>μ</sup> � � � ω τð Þ

2.1. SRT solution

μ

dIS τ; μ � �

2. SRT/IRT solution for an inhomogeneous layer

The azimuthally averaged solar radiative transfer equation [1–4, 10–12] is

ð1 �1

IS <sup>τ</sup>; <sup>μ</sup> � �<sup>P</sup> <sup>τ</sup>; <sup>μ</sup>; <sup>μ</sup><sup>0</sup> � �dμ<sup>0</sup> � ω τð Þ

where μ is the cosine of the zenith angle (μ > 0 and μ < 0 refer to upward and downward radiation, respectively), P τ; μ; μ<sup>0</sup> � � is the scattering phase function, τ is the optical depth

4π

F0P τ; μ; �μ<sup>0</sup> � �e

� τ

<sup>μ</sup><sup>0</sup> (1)

2

$$\frac{dF\_S^+(\tau)}{d\tau} = \gamma\_1(\tau)F\_S^+(\tau) - \gamma\_2(\tau)F\_S^-(\tau) - \gamma\_3(\tau)\omega(\tau)F\_0 e^{-\frac{\tau}{\mu\_0}} \tag{5a}$$

$$\frac{dF\_S^-(\tau)}{d\tau} = \gamma\_2(\tau)F\_S^+(\tau) - \gamma\_1(\tau)F\_S^-(\tau) + \left[1 - \gamma\_3(\tau)\right]\omega(\tau)F\_0 e^{-\frac{\tau}{\mu\_0}}\tag{5b}$$

$$F\_S^-(0) = 0,\\ F\_S^+(\tau\_0) = R\_{dif} F\_S^-(\tau\_0) + R\_{dir} \mu\_0 F\_0 e^{-\frac{\tau\_0}{\mu\_0}} \tag{5c}$$

where <sup>γ</sup>1ð Þ¼ <sup>τ</sup> <sup>1</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> <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>4</sup> 2 � 3gð Þτ μ<sup>0</sup> � �; 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 Eq. (3) and ignoring the small second-order parameters εω2, ε<sup>g</sup> 2, and εωεg, we get

$$\gamma\_1(\tau) = \gamma\_1^0 + \gamma\_1^1 \varepsilon\_a \left( e^{-a\_1 \tau} - e^{-a\_1 \tau\_0/2} \right) + \gamma\_1^2 \varepsilon\_\mathcal{g} \left( e^{-a\_2 \tau} - e^{-a\_2 \tau\_0/2} \right) \tag{6a}$$

$$\gamma\_2(\tau) = \gamma\_2^0 + \gamma\_2^1 \varepsilon\_w \left( e^{-a\_1 \tau} - e^{-a\_1 \tau\_0/2} \right) + \gamma\_2^2 \varepsilon\_\mathcal{S} \left( e^{-a\_2 \tau} - e^{-a\_2 \tau\_0/2} \right) \tag{6b}$$

$$
\gamma\_3(\tau) = \gamma\_3^0 + \gamma\_3^2 \varepsilon\_\beta \left( e^{-a\_2 \tau} - e^{-a\_2 \tau\_0/2} \right) \tag{6c}
$$

where  $\gamma\_1^0 = \frac{1}{4} [7 - (4 + 3\hat{\mathcal{g}})\hat{\mathcal{w}}]$ ,  $\gamma\_2^0 = \frac{-1}{4} [1 - (4 - 3\hat{\mathcal{g}})\hat{\mathcal{w}}]$ ,  $\gamma\_3^0 = \frac{1}{4} (2 - 3\hat{\mathcal{g}}\mu\_0)$ ,  $\gamma\_1^1 = \frac{-1}{4} (4 + 3\hat{\mathcal{g}})\gamma\_2^1 = -\frac{3}{4} \hat{\mathcal{w}}$ , and  $\gamma\_3^2 = -\frac{3}{4} \mu\_0$ .

$$F\_{\mathbb{S}}^{+} = F\_{\mathbb{S}0}^{+} + \varepsilon\_{a} F\_{\mathbb{S}1}^{+} + \varepsilon\_{\mathbb{S}} F\_{\mathbb{S}2}^{+} \tag{7a}$$

$$F\_{\mathbb{S}}^{-} = F\_{\mathbb{S}0}^{-} + \varepsilon\_{\omega} F\_{\mathbb{S}1}^{-} + \varepsilon\_{\mathbb{S}} F\_{\mathbb{S}2}^{-} \tag{7b}$$

$$\frac{dF\_{S0}^{+}}{d\tau} = \gamma\_1^0 F\_{S0}^{+} - \gamma\_2^0 F\_{S0}^{-} - \gamma\_3^0 \widehat{\alpha} F\_0 e^{-\frac{\tau}{\tau\_0}} \tag{9a}$$

$$\frac{d\bar{F}\_{\rm S0}^{-}}{d\tau} = \gamma\_2^0 \bar{F}\_{\rm S0}^{+} - \gamma\_1^0 \bar{F}\_{\rm S0}^{-} + \gamma\_4^0 \widehat{\omega} \, F\_0 \mathfrak{e}^{-\frac{\mathfrak{r}}{\mu\_0}} \tag{9b}$$

$$F\_{S0}^{-}(0) = 0,\\ F\_{S0}^{+}(\tau\_0) = R\_{d\tilde{\eta}} F\_{S0}^{-}(\tau\_0) + R\_{d\tilde{\nu}} \mu\_0 F\_0 e^{-\frac{\tilde{\nu}}{\tilde{\mu}\_0}} \tag{9c}$$

$$F\_{S0}^{+} = K\_1 e^{k\tau} + \Gamma K\_2 e^{-k\tau} + G\_1 e^{-\frac{\tau}{\mu\_0}} \tag{10a}$$

$$F\_{\rm S0}^{-} = \Gamma K\_1 e^{k\overline{\tau}} + K\_2 e^{-k\overline{\tau}} + G\_2 e^{-\frac{\tau}{\mu\_0}} \tag{10b}$$

$$\frac{dF\_{Si}^{+}}{d\tau} = \gamma\_1^0 F\_{Si}^{+} - \gamma\_2^0 F\_{Si}^{-} + \left(e^{-a\_i \tau} - e^{-a\_i \tau\_0/2}\right) \left(\gamma\_1^i F\_{S0}^{+} - \gamma\_2^i F\_{S0}^{-}\right) - \gamma\_3^{i-1} F\_0 \left(e^{-a\_i \tau} - e^{-a\_i \tau\_0/2}\right) e^{-\frac{\tau}{\mu\_0}} \tag{11a}$$

$$\frac{d\mathcal{F}^-\_{Si}}{d\tau} = \gamma\_2^0 F^+\_{Si} - \gamma\_1^0 F^-\_{Si} + \left(e^{-a\tau} - e^{-a\tau\_0/2}\right) \left(\gamma\_2^i F^+\_{S0} - \gamma\_1^i F^-\_{S0}\right) - \gamma\_4^{i-1} F\_0 \left(e^{-a\tau} - e^{-a\tau\_0/2}\right) e^{-\frac{\tau}{\tau\_0}} \tag{11b}$$

$$F\_{\rm Si}^-(\mathbf{0}) = \mathbf{0}, F\_{\rm Si}^+(\tau\_0) = R\_{d\tilde{f}} F\_{\rm Si}^-(\tau\_0) \tag{11c}$$

$$\begin{split} \frac{dM\_{i}}{d\tau} &= \left(\chi\_{1}^{0} + \chi\_{2}^{0}\right)N\_{i} + \left(\psi\_{i}^{+} + \psi\_{i}^{-}\right)e^{-(k+a\_{i})\tau} + \left(\zeta\_{i}^{+} + \zeta\_{i}^{-}\right)e^{(k-a\_{i})\tau} + \left(\chi\_{i}^{+} + \chi\_{i}^{-}\right)e^{-\left(a + \frac{1}{\mu\_{0}}\right)\tau} \\ &- e^{-a\_{i}\eta\_{0}/2} \left[\left(\psi\_{i}^{+} + \psi\_{i}^{-}\right)e^{-k\tau} + \left(\zeta\_{i}^{+} + \zeta\_{i}^{-}\right)e^{k\tau} + \left(\chi\_{i}^{+} + \chi\_{i}^{-}\right)e^{-\frac{\tau}{\mu\_{0}}}\right] \end{split} \tag{12a}$$

$$\begin{split} \frac{d\mathbf{N}\_i}{d\tau} &= \left(\boldsymbol{\upchi}\_1^0 - \boldsymbol{\upchi}\_2^0\right) \mathbf{M}\_i + \left(\boldsymbol{\uppsi}\_i^+ - \boldsymbol{\uppsi}\_i^-\right) \mathbf{e}^{-(k+a)\tau} + \left(\boldsymbol{\upzeta}\_i^+ - \boldsymbol{\upzeta}\_i^-\right) \mathbf{e}^{(k-a)\tau} + \left(\boldsymbol{\upchi}\_i^+ - \boldsymbol{\upchi}\_i^-\right) \mathbf{e}^{-\left(a + \frac{1}{\mu\_0}\right)\tau} \\ &- e^{-a\tau\_0/2} \left[ \left(\boldsymbol{\uppsi}\_i^+ - \boldsymbol{\uppsi}\_i^-\right) \mathbf{e}^{-k\tau} + \left(\boldsymbol{\upzeta}\_i^+ - \boldsymbol{\upzeta}\_i^-\right) \mathbf{e}^{k\tau} + \left(\boldsymbol{\upchi}\_i^+ - \boldsymbol{\upchi}\_i^-\right) \mathbf{e}^{-\frac{\tau}{\mu\_0}} \right] \end{split} \tag{12b}$$

$$\frac{d^2M\_i}{d\tau^2} = k^2M\_i + \eta\_{1i}^+e^{-(k+a)\tau} + \eta\_{2i}^+e^{(k-a)\tau} + \eta\_{3i}^+e^{-\left(a\_i^+ + \frac{1}{\mu\_0}\right)\tau} + \eta\_{4i}^+e^{-k\tau} + \eta\_{5i}^+e^{k\tau} + \eta\_{6i}^+e^{-\frac{\tau}{\mu\_0}}\tag{13a}$$

$$\frac{d^2N\_i}{d\tau^2} = k^2N\_i + \eta\_{1i}^-e^{-(k+a)\tau} + \eta\_{2i}^-e^{(k-a)\tau} + \eta\_{3i}^-e^{-\left(a+\frac{1}{\mu\_0}\right)\tau} + \eta\_{4i}^-e^{-k\tau} + \eta\_{5i}^-e^{k\tau} + \eta\_{6i}^-e^{-\frac{\pi}{\mu\_0}} \tag{13b}$$

$$M\_i = A\_i^+ e^{-k\tau} + B\_i^+ e^{k\tau} + P\_i^+ e^{-(k+a)\tau} + Q\_i^+ e^{(k-a)\tau} + R\_i^+ e^{-\left(a + \frac{i}{\mu\_0}\right)\tau} - \frac{\eta\_{4i}^+}{2k} e^{-k\tau} + \frac{\eta\_{5i}^+}{2k} e^{k\tau} + \frac{\eta\_{6i}^+ \mu\_0^2}{1 - \mu\_0^2 k^2} e^{-\frac{\tau}{\mu\_0}} \tag{14a}$$

$$N\_i = A\_i^- e^{-k\tau} + B\_i^- e^{k\tau} + P\_i^- e^{-(k+a\_i)\tau} + Q\_i^- e^{(k-a\_i)\tau} + R\_i^- e^{-\left(a + \frac{1}{\mu\_0}\right)\tau} - \frac{\eta\_{4i}^-}{2k} e^{-k\tau} + \frac{\eta\_{5i}^-}{2k} e^{k\tau} + \frac{\eta\_{6i}^- \mu\_0^2}{1 - \mu\_0^2 k^2} e^{-\frac{\mu\_0^2}{2k} (k+a\_i)}\tag{14b}$$

$$\text{where } P\_i^{\pm} = \frac{\eta\_{li}^{\pm}}{(k+a)^2 - k^2} \mathbf{Q}\_i^{\pm} = \frac{\eta\_{2i}^{\pm}}{(k-a)^2 - k^2} \text{ and } \mathbf{R}\_i^{\pm} = \frac{\eta\_{li}^{\pm}}{\left(a + \frac{1}{\mu\_0}\right)^2 - k^2} e^{-\frac{\tau}{\mu\_0}}. \text{ Finally, we can obtain } F\_{\text{Si}}^{-} \text{ and } F\_{\text{Si}}^{+} $$

The Planck function is approximated lineally as a function of optical depth [2] as

According to the two-stream approximation, the intensities can be written as I<sup>I</sup> τ; μ<sup>1</sup>

II <sup>τ</sup>; <sup>μ</sup> � �<sup>P</sup> <sup>τ</sup>; <sup>μ</sup>; <sup>μ</sup><sup>0</sup> � �dμ<sup>0</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>3</sup>gð Þ<sup>τ</sup> μμ<sup>1</sup>

<sup>d</sup><sup>τ</sup> <sup>¼</sup> <sup>γ</sup>1ð Þ<sup>τ</sup> <sup>I</sup>

<sup>d</sup><sup>τ</sup> <sup>¼</sup> <sup>γ</sup>2ð Þ<sup>τ</sup> <sup>I</sup>

, <sup>γ</sup>2ð Þ¼ <sup>τ</sup> ω τð Þ½ � <sup>1</sup>�gð Þ <sup>τ</sup> 2μ<sup>1</sup>

þ

þ

�a1<sup>τ</sup> � <sup>e</sup>

�a1<sup>τ</sup> � <sup>e</sup>

<sup>3</sup> <sup>þ</sup> <sup>γ</sup><sup>1</sup>

factors are introduced for simplifying original expressions, in which γ<sup>0</sup>

Same as in Eq. (7), the upward and downward intensity can be written as I þ <sup>I</sup> ¼ I þ I0 þ εωI þ I1 þ εgI þ

> I � <sup>I</sup> ¼ I � I0 þ εωI � I1 þ εgI �

the medium.

and <sup>I</sup><sup>I</sup> <sup>τ</sup>; <sup>μ</sup>�<sup>1</sup>

where γ1ð Þ¼ τ

In the above formula, γ<sup>0</sup>

, γ<sup>0</sup> <sup>3</sup> <sup>¼</sup> <sup>1</sup>�b<sup>ω</sup> μ1 , γ<sup>1</sup>

γ0

<sup>2</sup> <sup>¼</sup> <sup>b</sup><sup>ω</sup> <sup>1</sup>�b<sup>g</sup> � � 2μ<sup>1</sup>

� � <sup>¼</sup> <sup>I</sup>

ð1 �1 �

where gð Þτ is the asymmetry factor. Using Eqs. (17) and (19), we can obtain

> dI<sup>þ</sup> <sup>I</sup> ð Þτ

> dI<sup>þ</sup> <sup>I</sup> ð Þτ

ignoring the second order of the small parameters of ε<sup>2</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>i</sup> , and γ<sup>2</sup>

<sup>1</sup> ¼ � <sup>1</sup><sup>þ</sup>b<sup>g</sup> 2μ<sup>1</sup> , γ<sup>1</sup> <sup>2</sup> <sup>¼</sup> <sup>1</sup>�b<sup>g</sup> 2μ<sup>1</sup> , γ<sup>1</sup> <sup>3</sup> ¼ � <sup>1</sup> μ1 , γ<sup>2</sup> <sup>1</sup> <sup>¼</sup> <sup>γ</sup><sup>2</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>

1-ω τð Þð Þ <sup>1</sup>þgð Þ<sup>τ</sup> <sup>=</sup><sup>2</sup> μ1

<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>i</sup> , γ<sup>1</sup>

radiative intensity to flux [17]. Ð <sup>1</sup>

where β ¼ ð Þ B<sup>1</sup> � B<sup>0</sup> =τ<sup>0</sup> and τ<sup>0</sup> are the total optical depth of the medium. The Planck functions B<sup>0</sup> and B<sup>1</sup> are evaluated by using the temperature of the top (τ ¼ 0) and the bottom (τ ¼ τ0) of

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

� �I

<sup>I</sup> ð Þ� τ γ2ð Þτ I

<sup>I</sup> ð Þ� τ γ1ð Þτ I

For IRT, we also use Eq. (3) to represent an inhomogeneous medium such as cloud or snow, in which ω τð Þ and gð Þτ vary with τ. By substituting Eq. (3) into γ1ð Þτ , γ2ð Þτ , and γ3ð Þτ and by

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

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

<sup>3</sup>εω e

�a1<sup>τ</sup> � <sup>e</sup>

<sup>I</sup> ð Þ<sup>τ</sup> , respectively, where <sup>μ</sup><sup>1</sup> ¼ �μ�<sup>1</sup> <sup>¼</sup> <sup>1</sup>=1:66 is a diffuse factor that converts

þ

�

�

, and γ3ð Þ¼ τ

<sup>ω</sup>, ε<sup>2</sup>

<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>

<sup>i</sup> (i = 1, 2, 3) are the known factors of <sup>ω</sup><sup>b</sup> and <sup>b</sup>g. These known

<sup>2</sup> ¼ � <sup>b</sup><sup>ω</sup> 2μ<sup>1</sup>

�<sup>1</sup> II <sup>τ</sup>; <sup>μ</sup> � �<sup>P</sup> <sup>τ</sup>; <sup>μ</sup>; <sup>μ</sup><sup>0</sup> � �dμ<sup>0</sup> can be written as

B T½ �¼ ð Þτ B<sup>0</sup> þ βτ (18)

<sup>I</sup> ð Þþ <sup>τ</sup> <sup>1</sup> <sup>þ</sup> <sup>3</sup>gð Þ<sup>τ</sup> μμ�<sup>1</sup> � �I

> 1�ω τð Þ μ1 .

� � <sup>¼</sup> <sup>I</sup>

<sup>I</sup> ð Þτ (19)

�

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

<sup>I</sup> ð Þ� τ γ3ð Þτ Bð Þτ (20a)

<sup>I</sup> ð Þþ τ γ3ð Þτ Bð Þτ (20b)

<sup>g</sup>, and εωεg, we can obtain

�a2τ0=2

�a1τ0=<sup>2</sup> � � (21c)

, and γ<sup>2</sup>

I1 (22a)

I1 (22b)

<sup>3</sup> ¼ 0.

�a2τ0=<sup>2</sup> � � (21a)

� � (21b)

<sup>1</sup> <sup>¼</sup> <sup>1</sup>�b<sup>ω</sup> <sup>1</sup><sup>þ</sup>b<sup>g</sup>

� �=<sup>2</sup> μ1

,

þ <sup>I</sup> ð Þτ 147

as

$$F\_{Si}^{+} = D\_{1i}^{+}e^{-k\tau} + D\_{2i}^{+}e^{k\tau} + \varphi\_{1i}^{+}e^{-(k+a\_{i})\tau} + \varphi\_{2i}^{+}e^{(k-a\_{i})\tau} + \varphi\_{3i}^{+}e^{-\left(a\_{i} + \frac{\Delta}{\tau\_{0}}\right)\tau} + \varphi\_{4i}^{+}\tau e^{-k\_{1}\tau} + \varphi\_{5i}^{+}\tau e^{k\tau} + \varphi\_{6i}^{+}e^{-\frac{\tau}{\tau\_{0}}} \tag{15a}$$

$$F\_{Si}^- = D\_{1i}^- e^{-k\tau} + D\_{2i}^- e^{k\tau} + \rho\_{1i}^- e^{-(k+a)\tau} + \rho\_{2i}^- e^{(k-a)\tau} + \rho\_{3i}^- e^{-\left(a + \frac{1}{\tau\_0}\right)\tau} + \rho\_{4i}^- \tau e^{-k\_{1\tau}} + \rho\_{5i}^- \tau e^{k\tau} + \rho\_{6i}^- \sigma^{-\frac{\tau}{\tau\_0}} \tag{15b}$$

$$\text{where } D\_{1i}^{\pm} = A\_i^+ a^{\mp} \pm X\_i, D\_{2i}^{\pm} = B\_i^+ a^{\pm} \pm Y\_i, a^{\pm} = \frac{1}{2} \left( 1 \pm \frac{k}{\gamma\_1^0 + \gamma\_2^0} \right), Y\_i = \frac{e^{-a\_0 \gamma\_0^0 \left( \gamma\_1^+ + \gamma\_2^- \right)}}{2 \left( \gamma\_1^0 + \gamma\_2^0 \right)} - \frac{\eta\_i^+}{4k \left( \gamma\_1^0 + \gamma\_2^0 \right)}.$$

Yi <sup>¼</sup> <sup>e</sup> �aiτ0=<sup>2</sup> <sup>ζ</sup><sup>þ</sup> <sup>i</sup> <sup>þ</sup>ζ� <sup>i</sup> ð Þ 2 γ<sup>0</sup> <sup>1</sup>þγ<sup>0</sup> ð Þ<sup>2</sup> <sup>þ</sup> <sup>η</sup><sup>þ</sup> 5i 4k γ<sup>0</sup> <sup>1</sup>þγ<sup>0</sup> ð Þ<sup>2</sup> , ϕ� <sup>1</sup><sup>i</sup> <sup>¼</sup> <sup>1</sup> <sup>2</sup> P<sup>þ</sup> <sup>i</sup> � P� i � �, ϕ� <sup>2</sup><sup>i</sup> <sup>¼</sup> <sup>1</sup> <sup>2</sup> Q<sup>þ</sup> <sup>i</sup> � Q� i � �, ϕ� <sup>3</sup><sup>i</sup> <sup>¼</sup> <sup>1</sup> <sup>2</sup> R<sup>þ</sup> <sup>i</sup> � R� i � �, ϕ� <sup>4</sup><sup>i</sup> ¼ � <sup>η</sup><sup>þ</sup> 4i �η� 4i <sup>4</sup><sup>k</sup> , ϕ� <sup>5</sup><sup>i</sup> <sup>¼</sup> <sup>η</sup><sup>þ</sup> 5i �η� 5i <sup>4</sup><sup>k</sup> , and ϕ� <sup>6</sup><sup>i</sup> <sup>¼</sup> <sup>η</sup><sup>þ</sup> 6i �η� ð Þ6<sup>i</sup> <sup>μ</sup><sup>2</sup> 0 <sup>1</sup>�μ<sup>2</sup> 0k<sup>2</sup> . Bi and Ai are determined by the boundary conditions as

B<sup>þ</sup> i

¼ � <sup>ϕ</sup>� <sup>1</sup><sup>i</sup> þ ϕ� <sup>2</sup><sup>i</sup> þ ϕ� <sup>3</sup><sup>i</sup> þ ϕ� 6i <sup>Þ</sup> <sup>α</sup>� � Rdif <sup>α</sup><sup>þ</sup> � �e�kτ<sup>0</sup> <sup>þ</sup> <sup>α</sup><sup>þ</sup> <sup>ϕ</sup><sup>þ</sup> <sup>1</sup><sup>i</sup> � Rdif ϕ� 1i � �e�ð Þ <sup>k</sup>þai <sup>τ</sup><sup>0</sup> <sup>þ</sup> <sup>ϕ</sup><sup>þ</sup> <sup>2</sup><sup>i</sup> � Rdif ϕ� 2i � �eð Þ <sup>k</sup>�ai <sup>τ</sup><sup>0</sup> � � <sup>α</sup>� <sup>α</sup>� � Rdif <sup>α</sup><sup>þ</sup> � �e�kτ<sup>0</sup> � <sup>α</sup><sup>þ</sup> <sup>α</sup><sup>þ</sup> � Rdif <sup>α</sup>� � �ek<sup>τ</sup><sup>0</sup> <sup>þ</sup> <sup>α</sup><sup>þ</sup> <sup>ϕ</sup><sup>þ</sup> <sup>3</sup><sup>i</sup> � Rdif ϕ� 3i � �e � ai<sup>þ</sup> <sup>1</sup> μ0 � �<sup>τ</sup><sup>0</sup> þ ϕ<sup>þ</sup> <sup>4</sup><sup>i</sup> � Rdif ϕ� 4i <sup>Þ</sup>τ0e�kτ<sup>0</sup> <sup>þ</sup> <sup>ϕ</sup><sup>þ</sup> <sup>5</sup><sup>i</sup> � Rdifϕ� 5i � �τ0e<sup>k</sup>τ<sup>0</sup> <sup>þ</sup> <sup>ϕ</sup><sup>þ</sup> <sup>6</sup><sup>i</sup> � Rdif ϕ� 6i � �e �τ0 μ0 � <sup>α</sup>� <sup>α</sup>� � Rdif <sup>α</sup><sup>þ</sup> � �e�kτ<sup>0</sup> � <sup>α</sup><sup>þ</sup> <sup>α</sup><sup>þ</sup> � Rdif <sup>α</sup>� � �ek<sup>τ</sup><sup>0</sup> <sup>þ</sup> ð Þ Xi <sup>þ</sup> Yi <sup>α</sup>� � Rdif <sup>α</sup><sup>þ</sup> � �e�kτ<sup>0</sup> <sup>þ</sup> Xi <sup>þ</sup> Rdif Xi � �αþe�kτ<sup>0</sup> <sup>þ</sup> Yi <sup>þ</sup> RdifYi � �αþek<sup>τ</sup><sup>0</sup> <sup>α</sup>� <sup>α</sup>� � Rdif <sup>α</sup><sup>þ</sup> � �e�kτ<sup>0</sup> � <sup>α</sup><sup>þ</sup> <sup>α</sup><sup>þ</sup> � Rdif <sup>α</sup>� � �ek<sup>τ</sup><sup>0</sup> (16a)

$$A\_i^+ = \frac{1}{\alpha^+} \left( X\_i + Y\_i - B\_i \alpha^- - \phi\_{1i}^- - \phi\_{2i}^- - \phi\_{3i}^- - \phi\_{6i}^- \right) \tag{16b}$$

All detailed calculation about solar radiation can be found at [16].

#### 2.2. IRT solution

The azimuthally averaged infrared radiative transfer equation for intensity I<sup>I</sup> τ; μ � � is [1–4, 10–12]

$$\mu \frac{dI\_{\mathrm{I}}(\tau,\mu)}{d\tau} = I\_{\mathrm{I}}(\tau,\mu) - \frac{\omega(\tau)}{2} \int\_{-1}^{1} I\_{\mathrm{I}}(\tau,\mu) P\Big(\tau,\mu,\mu'\big) d\mu' - [1 - \omega(\tau)]B(T) \tag{17}$$

where μ, τ, P τ; μ; μ � �<sup>0</sup> , and ω τð Þ are same as in Eq. (1). B Tð Þ is the Planck function at temperature T, which represents the internal infrared emission of the medium.

The Planck function is approximated lineally as a function of optical depth [2] as

$$B[T(\tau)] = B\_0 + \beta \tau \tag{18}$$

where β ¼ ð Þ B<sup>1</sup> � B<sup>0</sup> =τ<sup>0</sup> and τ<sup>0</sup> are the total optical depth of the medium. The Planck functions B<sup>0</sup> and B<sup>1</sup> are evaluated by using the temperature of the top (τ ¼ 0) and the bottom (τ ¼ τ0) of the medium.

According to the two-stream approximation, the intensities can be written as I<sup>I</sup> τ; μ<sup>1</sup> � � <sup>¼</sup> <sup>I</sup> þ <sup>I</sup> ð Þτ and <sup>I</sup><sup>I</sup> <sup>τ</sup>; <sup>μ</sup>�<sup>1</sup> � � <sup>¼</sup> <sup>I</sup> � <sup>I</sup> ð Þ<sup>τ</sup> , respectively, where <sup>μ</sup><sup>1</sup> ¼ �μ�<sup>1</sup> <sup>¼</sup> <sup>1</sup>=1:66 is a diffuse factor that converts radiative intensity to flux [17]. Ð <sup>1</sup> �<sup>1</sup> II <sup>τ</sup>; <sup>μ</sup> � �<sup>P</sup> <sup>τ</sup>; <sup>μ</sup>; <sup>μ</sup><sup>0</sup> � �dμ<sup>0</sup> can be written as

$$\int\_{-1}^{1} I\_{l}(\tau,\mu) \mathbb{P}(\tau,\mu,\mu') d\mu' = \left[1 + 3\mathfrak{g}(\tau)\mu\mu\_{1}\right] I\_{l}^{+}(\tau) + \left[1 + 3\mathfrak{g}(\tau)\mu\mu\_{-1}\right] I\_{l}^{-}(\tau)\tag{19}$$

where gð Þτ is the asymmetry factor.

where P�

as

F<sup>þ</sup> Si ¼ D<sup>þ</sup> 1i e �k<sup>τ</sup> <sup>þ</sup> <sup>D</sup><sup>þ</sup> 2i e <sup>k</sup><sup>τ</sup> <sup>þ</sup> <sup>φ</sup><sup>þ</sup> 1i e

F� Si ¼ D� 1i e �k<sup>τ</sup> <sup>þ</sup> <sup>D</sup>� 2i e <sup>k</sup><sup>τ</sup> <sup>þ</sup> <sup>φ</sup>� 1i e

where D�

Yi <sup>¼</sup> <sup>e</sup>

ϕ� <sup>4</sup><sup>i</sup> ¼ � <sup>η</sup><sup>þ</sup> 4i �η� 4i <sup>4</sup><sup>k</sup> , ϕ�

B<sup>þ</sup> i

<sup>1</sup><sup>i</sup> ¼ A<sup>þ</sup>

�aiτ0=<sup>2</sup> <sup>ζ</sup><sup>þ</sup> <sup>i</sup> <sup>þ</sup>ζ� <sup>i</sup> ð Þ 2 γ<sup>0</sup> <sup>1</sup>þγ<sup>0</sup> ð Þ<sup>2</sup>

conditions as

¼ � <sup>ϕ</sup>�

<sup>þ</sup> <sup>α</sup><sup>þ</sup> <sup>ϕ</sup><sup>þ</sup>

2.2. IRT solution

μ

where μ, τ, P τ; μ; μ � �<sup>0</sup>

dI<sup>I</sup> τ; μ � �

<sup>1</sup><sup>i</sup> þ ϕ�

<sup>2</sup><sup>i</sup> þ ϕ�

<sup>3</sup><sup>i</sup> � Rdif ϕ� 3i � �e

<sup>3</sup><sup>i</sup> þ ϕ� 6i

> � ai<sup>þ</sup> <sup>1</sup> μ0 � � τ0 þ ϕ<sup>þ</sup>

<sup>þ</sup> ð Þ Xi <sup>þ</sup> Yi <sup>α</sup>� � Rdif <sup>α</sup><sup>þ</sup> � �e�kτ<sup>0</sup> <sup>þ</sup> Xi <sup>þ</sup> Rdif Xi

A<sup>þ</sup> <sup>i</sup> <sup>¼</sup> <sup>1</sup>

<sup>i</sup> <sup>¼</sup> <sup>η</sup>� 1i ð Þ <sup>k</sup>þai <sup>2</sup>

�<sup>k</sup>2, <sup>Q</sup>�

<sup>i</sup> <sup>α</sup> <sup>∓</sup> � Xi, <sup>D</sup>�

<sup>þ</sup> <sup>η</sup><sup>þ</sup> 5i 4k γ<sup>0</sup> <sup>1</sup>þγ<sup>0</sup> ð Þ<sup>2</sup>

<sup>5</sup><sup>i</sup> <sup>¼</sup> <sup>η</sup><sup>þ</sup> 5i �η� 5i <sup>4</sup><sup>k</sup> , and ϕ�

<sup>i</sup> <sup>¼</sup> <sup>η</sup>� 2i ð Þ <sup>k</sup>�ai <sup>2</sup> �k

146 Perturbation Methods with Applications in Science and Engineering

2, and R�

2i e

2i e

<sup>i</sup> <sup>α</sup>� � <sup>Y</sup>, <sup>α</sup>� <sup>¼</sup> <sup>1</sup>

<sup>1</sup>�μ<sup>2</sup>

<sup>Þ</sup> <sup>α</sup>� � Rdif <sup>α</sup><sup>þ</sup> � �e�kτ<sup>0</sup> <sup>þ</sup> <sup>α</sup><sup>þ</sup> <sup>ϕ</sup><sup>þ</sup>

<sup>4</sup><sup>i</sup> � Rdif ϕ� 4i

<sup>α</sup>� <sup>α</sup>� � Rdif <sup>α</sup><sup>þ</sup> � �e�kτ<sup>0</sup> � <sup>α</sup><sup>þ</sup> <sup>α</sup><sup>þ</sup> � Rdif <sup>α</sup>� � �ek<sup>τ</sup><sup>0</sup>

<sup>α</sup><sup>þ</sup> Xi <sup>þ</sup> Yi � Biα� � <sup>ϕ</sup>�

The azimuthally averaged infrared radiative transfer equation for intensity I<sup>I</sup> τ; μ � � is [1–4, 10–12]

I<sup>I</sup> τ; μ � �P τ; μ; μ

<sup>0</sup> � �

, and ω τð Þ are same as in Eq. (1). B Tð Þ is the Planck function at temper-

All detailed calculation about solar radiation can be found at [16].

2

ature T, which represents the internal infrared emission of the medium.

ð1 �1

<sup>d</sup><sup>τ</sup> <sup>¼</sup> II <sup>τ</sup>; <sup>μ</sup> � � � ω τð Þ

�ð Þ <sup>k</sup>þai <sup>τ</sup> <sup>þ</sup> <sup>φ</sup><sup>þ</sup>

�ð Þ <sup>k</sup>þai <sup>τ</sup> <sup>þ</sup> <sup>φ</sup>�

<sup>1</sup><sup>i</sup> <sup>¼</sup> <sup>1</sup> <sup>2</sup> P<sup>þ</sup> <sup>i</sup> � P� i � �, ϕ�

<sup>6</sup><sup>i</sup> <sup>¼</sup> <sup>η</sup><sup>þ</sup> 6i �η� ð Þ6<sup>i</sup> <sup>μ</sup><sup>2</sup> 0

<sup>2</sup><sup>i</sup> ¼ B<sup>þ</sup>

, ϕ�

<sup>i</sup> <sup>¼</sup> <sup>η</sup>� 3i ai<sup>þ</sup> <sup>1</sup> μ0 � �<sup>2</sup>

ð Þ <sup>k</sup>�ai <sup>τ</sup> <sup>þ</sup> <sup>φ</sup><sup>þ</sup>

ð Þ <sup>k</sup>�ai <sup>τ</sup> <sup>þ</sup> <sup>φ</sup>�

�k 2 e � τ

3i e � ai<sup>þ</sup> <sup>1</sup> μ0 � � τ þ φ<sup>þ</sup> 4i τe

3i e � ai<sup>þ</sup> <sup>1</sup> μ0 � � τ þ φ� 4i τe

<sup>2</sup> <sup>1</sup> � <sup>k</sup> γ0 1þγ<sup>0</sup> 2

� �

<sup>2</sup><sup>i</sup> <sup>¼</sup> <sup>1</sup> <sup>2</sup> Q<sup>þ</sup>

<sup>1</sup><sup>i</sup> � Rdif ϕ� 1i � �e�ð Þ <sup>k</sup>þai <sup>τ</sup><sup>0</sup> <sup>þ</sup> <sup>ϕ</sup><sup>þ</sup>

> <sup>5</sup><sup>i</sup> � Rdifϕ� 5i � �τ0e<sup>k</sup>τ<sup>0</sup> <sup>þ</sup> <sup>ϕ</sup><sup>þ</sup>

� �αþek<sup>τ</sup><sup>0</sup>

<sup>3</sup><sup>i</sup> � ϕ� 6i

<sup>α</sup>� <sup>α</sup>� � Rdif <sup>α</sup><sup>þ</sup> � �e�kτ<sup>0</sup> � <sup>α</sup><sup>þ</sup> <sup>α</sup><sup>þ</sup> � Rdif <sup>α</sup>� � �ek<sup>τ</sup><sup>0</sup>

<sup>α</sup>� <sup>α</sup>� � Rdif <sup>α</sup><sup>þ</sup> � �e�kτ<sup>0</sup> � <sup>α</sup><sup>þ</sup> <sup>α</sup><sup>þ</sup> � Rdif <sup>α</sup>� � �ek<sup>τ</sup><sup>0</sup>

� �αþe�kτ<sup>0</sup> <sup>þ</sup> Yi <sup>þ</sup> RdifYi

<sup>Þ</sup>τ0e�kτ<sup>0</sup> <sup>þ</sup> <sup>ϕ</sup><sup>þ</sup>

<sup>1</sup><sup>i</sup> � ϕ�

<sup>2</sup><sup>i</sup> � ϕ�

� � (16b)

dμ 0

, Xi <sup>¼</sup> <sup>e</sup>

<sup>i</sup> � Q� i � �, ϕ�

0k<sup>2</sup> . Bi and Ai are determined by the boundary

� �

<sup>μ</sup><sup>0</sup> . Finally, we can obtain F�

�k1<sup>τ</sup> <sup>þ</sup> <sup>φ</sup><sup>þ</sup> 5i τe <sup>k</sup><sup>τ</sup> <sup>þ</sup> <sup>φ</sup><sup>þ</sup> 6i e � τ μ0 (15a)

�k1<sup>τ</sup> <sup>þ</sup> <sup>φ</sup>� 5i τe <sup>k</sup><sup>τ</sup> <sup>þ</sup> <sup>φ</sup>� 6i e � τ μ0 (15b)

> � <sup>η</sup><sup>þ</sup> 4i 4k γ<sup>0</sup> <sup>1</sup>þγ<sup>0</sup> ð Þ<sup>2</sup> ,

<sup>3</sup><sup>i</sup> <sup>¼</sup> <sup>1</sup> <sup>2</sup> R<sup>þ</sup> <sup>i</sup> � R� i � �,

<sup>2</sup><sup>i</sup> � Rdif ϕ� 2i � �eð Þ <sup>k</sup>�ai <sup>τ</sup><sup>0</sup>

� ½ � 1 � ω τð Þ B Tð Þ (17)

<sup>6</sup><sup>i</sup> � Rdif ϕ� 6i � �e

�τ0 μ0 �

(16a)

�aiτ0=<sup>2</sup> <sup>ψ</sup><sup>þ</sup> <sup>i</sup> <sup>þ</sup>ψ� <sup>i</sup> ð Þ 2 γ<sup>0</sup> <sup>1</sup>þγ<sup>0</sup> ð Þ<sup>2</sup>

Si and F<sup>þ</sup> Si

Using Eqs. (17) and (19), we can obtain

$$\frac{dI\_{\rm I}^{+}(\tau)}{d\tau} = \gamma\_{1}(\tau)I\_{\rm I}^{+}(\tau) - \gamma\_{2}(\tau)I\_{\rm I}^{-}(\tau) - \gamma\_{3}(\tau)B(\tau) \tag{20a}$$

$$\frac{dI\_{\rm I}^{+}(\tau)}{d\tau} = \gamma\_{2}(\tau)I\_{\rm I}^{+}(\tau) - \gamma\_{1}(\tau)I\_{\rm I}^{-}(\tau) + \gamma\_{3}(\tau)B(\tau) \tag{20b}$$

where γ1ð Þ¼ τ 1-ω τð Þð Þ <sup>1</sup>þgð Þ<sup>τ</sup> <sup>=</sup><sup>2</sup> μ1 , <sup>γ</sup>2ð Þ¼ <sup>τ</sup> ω τð Þ½ � <sup>1</sup>�gð Þ <sup>τ</sup> 2μ<sup>1</sup> , and γ3ð Þ¼ τ 1�ω τð Þ μ1 .

For IRT, we also use Eq. (3) to represent an inhomogeneous medium such as cloud or snow, in which ω τð Þ and gð Þτ vary with τ. By substituting Eq. (3) into γ1ð Þτ , γ2ð Þτ , and γ3ð Þτ and by ignoring the second order of the small parameters of ε<sup>2</sup> <sup>ω</sup>, ε<sup>2</sup> <sup>g</sup>, and εωεg, we can obtain

$$
\gamma\_1(\tau) = \gamma\_1^0 + \gamma\_1^1 \varepsilon\_o \left( e^{-a\_1 \tau} - e^{-a\_1 \tau\_0/2} \right) + \gamma\_1^2 \varepsilon\_g \left( e^{-a\_2 \tau} - e^{-a\_2 \tau\_0/2} \right) \tag{21a}
$$

$$
\gamma \gamma\_2(\tau) = \gamma\_2^0 + \gamma\_2^1 \varepsilon\_\omega \left( e^{-a\_1 \tau} - e^{-a\_1 \tau\_0/2} \right) + \gamma\_2^2 \varepsilon\_\mathcal{g} \left( e^{-a\_2 \tau} - e^{-a\_2 \tau\_0/2} \right) \tag{21b}
$$

$$\gamma\_3(\tau) = \gamma\_3^0 + \gamma\_3^1 \varepsilon\_w \left( e^{-a\_1 \tau} - e^{-a\_1 \tau\_0/2} \right) \tag{21c}$$

In the above formula, γ<sup>0</sup> <sup>i</sup> , γ<sup>1</sup> <sup>i</sup> , and γ<sup>2</sup> <sup>i</sup> (i = 1, 2, 3) are the known factors of <sup>ω</sup><sup>b</sup> and <sup>b</sup>g. These known factors are introduced for simplifying original expressions, in which γ<sup>0</sup> <sup>1</sup> <sup>¼</sup> <sup>1</sup>�b<sup>ω</sup> <sup>1</sup><sup>þ</sup>b<sup>g</sup> � �=<sup>2</sup> μ1 , γ0 <sup>2</sup> <sup>¼</sup> <sup>b</sup><sup>ω</sup> <sup>1</sup>�b<sup>g</sup> � � 2μ<sup>1</sup> , γ<sup>0</sup> <sup>3</sup> <sup>¼</sup> <sup>1</sup>�b<sup>ω</sup> μ1 , γ<sup>1</sup> <sup>1</sup> ¼ � <sup>1</sup><sup>þ</sup>b<sup>g</sup> 2μ<sup>1</sup> , γ<sup>1</sup> <sup>2</sup> <sup>¼</sup> <sup>1</sup>�b<sup>g</sup> 2μ<sup>1</sup> , γ<sup>1</sup> <sup>3</sup> ¼ � <sup>1</sup> μ1 , γ<sup>2</sup> <sup>1</sup> <sup>¼</sup> <sup>γ</sup><sup>2</sup> <sup>2</sup> ¼ � <sup>b</sup><sup>ω</sup> 2μ<sup>1</sup> , and γ<sup>2</sup> <sup>3</sup> ¼ 0.

Same as in Eq. (7), the upward and downward intensity can be written as

$$I\_{\rm I}^{+} = I\_{\rm I0}^{+} + \varepsilon\_{\rm w} I\_{\rm II}^{+} + \varepsilon\_{\rm 8} I\_{\rm II}^{+} \tag{22a}$$

$$I\_{\rm I}^{-} = I\_{\rm I0}^{-} + \varepsilon\_{\omega} I\_{\rm II}^{-} + \varepsilon\_{\rm \mathcal{S}} I\_{\rm II}^{-} \tag{22b}$$

$$\frac{dI\_{10}^+}{d\tau} = \gamma\_1^0 dI\_{10}^+ - \gamma\_2^0 dI\_{10}^- - \gamma\_3^0 B(\tau) \tag{24a}$$

$$\frac{dI\_{\rm I0}^{-}}{d\tau} = \gamma\_2^0 I\_{\rm I0}^{+} - \gamma\_1^0 I\_{\rm I0}^{-} + \gamma\_3^0 B(\tau) \tag{24b}$$

$$I\_{10}^{-}(0) = 0, I\_{10}^{+}(\tau\_0) = (1 - \varepsilon\_s)I\_{10}^{-}(\tau\_0) + \varepsilon\_s B(T\_s) \tag{24c}$$

$$\mathbf{J}\_{\rm l0}^{+} = \boldsymbol{\alpha}^{+} K\_{0} \boldsymbol{\varepsilon}^{-k(\boldsymbol{\tau}\_{0} - \boldsymbol{\tau})} + \boldsymbol{\alpha}^{-} H\_{0} \boldsymbol{\varepsilon}^{-k\boldsymbol{\tau}} + G\_{1} \boldsymbol{\tau} + G\_{2}^{+} \tag{25a}$$

$$I\_{\rm I0}^{-} = \alpha^{-} K\_0 e^{-k(\tau\_0 - \tau)} + \alpha^{+} H\_0 e^{-k\tau} + G\_1 \tau + G\_2^{-} \tag{25b}$$

$$\frac{dI\_{\rm li}^{+}}{d\tau} = \gamma\_1^0 I\_{\rm li}^{+} - \gamma\_2^0 I\_{\rm li}^{-} + \left(e^{-a\_i \tau} - e^{-a\_i \tau\_0/2}\right) \left[\gamma\_1^i I\_{\rm I0}^{+} - \gamma\_2^i I\_{\rm I0}^{-} - \gamma\_3^i B(\tau)\right] \tag{26a}$$

$$\frac{d\overline{I\_{\rm li}}}{d\tau} = \gamma\_2^0 I\_{\rm li}^+ - \gamma\_1^0 I\_{\rm li}^- + \left(e^{-a\_7 \tau} - e^{-a\_7 \tau\_0/2}\right) \left[\gamma\_2^i I\_{\rm I0}^+ - \gamma\_1^i I\_{\rm I0}^- + \gamma\_3^i B(\tau)\right] \tag{26b}$$

$$I\_{\rm li}^{-}(0) = 0, I\_{\rm li}^{+}(\tau\_0) = (1 - \varepsilon\_s)I\_{\rm li}^{-}(\tau\_0) \tag{26c}$$

$$\frac{dM\_i}{d\tau} = \left(\gamma\_1^0 + \gamma\_2^0\right) N\_i + \chi\_{1i}^+ e^{-k\tau\_0 + (k - a)\tau} + \chi\_{2i}^+ e^{-(k + a)\tau} + \chi\_{3i}^+ e^{-k(\tau\_0 - \tau)} + \chi\_{4i}^+ e^{-k\tau} + \chi\_{5i}^+ + \chi\_{6i}^+ e^{-a\tau} \tag{27a}$$

$$\frac{dN\_i}{d\tau} = \left(\gamma\_1^0 - \gamma\_2^0\right) M\_i + \chi\_{1i}^- e^{-k\tau\_0 + (k - a)\tau} + \chi\_{2i}^- e^{-(k + a)\tau} + \chi\_{3i}^- e^{-k(\tau\_0 - \tau)} + \chi\_{4i}^- e^{-k\tau} + \chi\_{5i}^- + \chi\_{6i}^- e^{-a\tau} \tag{27b}$$

$$+ \chi\_{7\tau}^- + \chi\_{-7\tau}^- e^{-4\tau}$$

$$\begin{aligned} \frac{d^2M\_i}{d\tau^2} &= k^2M\_i + \phi\_{1i}^+e^{-k\tau\_0 + (k-a)\tau} + \phi\_{2i}^+e^{-(k+a)\tau} + \phi\_{3i}^+e^{-k(\tau\_0 - \tau)} + \phi\_{4i}^+e^{-k\tau} + \phi\_{5i}^+ + \phi\_{6i}^+e^{-a\_i\tau} \\ &+ \phi\_{7i}^+\tau + \phi\_{8i}^+\tau e^{-a\_i\tau} \end{aligned} \tag{28a}$$
 
$$\begin{aligned} \frac{d^2N\_i}{d\tau^2} &= k^2N\_i + \phi\_{1i}^-e^{-k\tau\_0 + (k-a)\tau} + \phi\_{2i}^-e^{-(k+a)\tau} + \phi\_{3i}^-e^{-k(\tau\_0 - \tau)} + \phi\_{4i}^-e^{-k\tau} + \phi\_{5i}^- + \phi\_{6i}^-e^{-a\_i\tau} \\ &+ \phi\_{7i}^-\tau e^{-a\_i\tau} \end{aligned} \tag{28b}$$

$$\begin{split} M\_{i} &= K\_{1i}e^{-k(\tau\_{0}-\tau)} + H\_{1i}e^{-k\tau} + P\_{1i}^{+}e^{-k\tau\_{0} + (k-a)\tau} + P\_{2i}^{+}e^{-(k+a)\tau} + P\_{3i}^{+}\tau e^{-k(\tau\_{0}-\tau)} + P\_{4i}^{+}\tau e^{-k\tau} \\ &\quad + P\_{5i}^{+} + P\_{6i}^{+}e^{-a\_{i}\tau} + P\_{7i}^{+}\tau + P\_{8i}^{+}\tau e^{-a\_{i}\tau} \end{split} \tag{29a}$$

$$\begin{split} N\_{l} = K\_{2} \varepsilon^{-k(\mathbf{r}\_{0} - \boldsymbol{\tau})} + H\_{2} \varepsilon^{-k\boldsymbol{\tau}} &+ P\_{1i}^{-} \varepsilon^{-k\mathbf{r}\_{0} + (k - a\_{i})\boldsymbol{\tau}} + P\_{2i}^{-} \varepsilon^{-(k + a)\boldsymbol{\tau}} + P\_{3i}^{-} \varepsilon e^{-k(\mathbf{r}\_{0} - \boldsymbol{\tau})} + P\_{4i}^{-} \varepsilon e^{-k\boldsymbol{\tau}} \\ &+ P\_{5i}^{-} + P\_{6i}^{-} \varepsilon^{-a\boldsymbol{\tau}} + P\_{8i}^{-} \varepsilon e^{-a\boldsymbol{\tau}} \end{split} \tag{29b}$$

$$\begin{array}{ll}\textbf{where} & P\_{1i}^{\pm} = \frac{\phi\_{ii}^{\pm}}{\left(k - a\right)^{2} - k^{2}} \quad P\_{2i}^{\pm} = \frac{\phi\_{2i}^{\pm}}{\left(k + a\right)^{2} - k^{2}} \quad P\_{3i}^{\pm} = \frac{\phi\_{ii}^{\pm}}{2k} \quad P\_{4i}^{\pm} = -\frac{\phi\_{ii}^{\pm}}{2k} \quad P\_{5i}^{\pm} = -\frac{\phi\_{ii}^{\pm}}{k^{2}} \quad P\_{6i}^{\pm} = \frac{\phi\_{ii}^{\pm} \left(k^{2} - k^{2}\right) + 2\phi\_{ii}^{\pm}}{\left(k^{2} - k^{2}\right)^{2}} \\\ P\_{7i}^{+} = -\frac{\phi\_{7i}^{+}}{k^{2}} \text{ and } P\_{8i}^{+} = \frac{\phi\_{ii}^{+}}{\frac{\phi\_{ii}^{-}}{k^{2}} - k^{2}}. \end{array}$$

$$\begin{split} I\_{\rm li}^{+} &= D\_{\rm I}^{+} \varepsilon^{-k(\tau\_{0}-\tau)} + D\_{\rm 2}^{+} \varepsilon^{-k\tau} + \sigma\_{\rm 1i}^{+} \varepsilon^{-k\tau\_{0} + (k-a)\tau} + \sigma\_{\rm 2i}^{+} \varepsilon^{-(k+a)\tau} + \sigma\_{\rm 3i}^{+} \pi e^{-k(\tau\_{0}-\tau)} + \sigma\_{\rm 4i}^{+} \pi e^{-k\tau} \\ &\quad + \sigma\_{\rm 5i}^{+} + \sigma\_{\rm 6i}^{+} \varepsilon^{-a\tau} + \sigma\_{\rm 7i}^{+} \pi e^{-a\tau} \end{split} \tag{30a}$$

$$\begin{split} I\_{\text{li}}^{-} = D\_{\text{li}}^{-} \varepsilon^{-k(\tau\_{0}-\tau)} + D\_{\text{2}}^{-} \varepsilon^{-k\tau} + \sigma\_{\text{1}l}^{-} \varepsilon^{-k\tau\_{0}+(k-n)\tau} + \sigma\_{\text{2}l}^{-} \varepsilon^{-(k+a)\tau} + \sigma\_{\text{3}l}^{-} \varepsilon e^{-k(\tau\_{0}-\tau)} + \sigma\_{\text{4}l}^{-} \varepsilon e^{-k\tau} \\ + \sigma\_{5i}^{-} + \sigma\_{6i}^{-} \varepsilon^{-a\tau} + \sigma\_{7l}^{-} \varepsilon^{-a\tau} \end{split} \tag{30b}$$

$$\text{where } D\_{\text{li}}^{\pm} = K\_{\text{li}} a^{\pm} \pm X\_{i\nu} D\_{\text{2i}}^{\pm} = H\_{\text{li}} a^{\mp} \pm Y\_{i\nu} X\_{i} = \frac{p\_{\text{ji}}^{+} - \chi\_{\text{ij}}^{+}}{2\langle \gamma\_{\text{i}}^{0} + \gamma\_{\text{2}}^{0} \rangle}, Y\_{i} = \frac{p\_{\text{li}}^{+} - \chi\_{\text{qi}}^{+}}{2\langle \gamma\_{\text{i}}^{0} + \gamma\_{\text{2}}^{0} \rangle}, \sigma\_{\text{ji}}^{\pm} = \frac{1}{2} \left( P\_{\text{ji}}^{+} \pm P\_{\text{ji}}^{-} \right). \text{ (j = 1, 2, \dots, N)}$$

2, 3, 4, 5, 6, 8), <sup>σ</sup>7<sup>i</sup> <sup>¼</sup> <sup>1</sup> <sup>2</sup> P<sup>þ</sup> 7i , and K1<sup>i</sup> and H1<sup>i</sup> are determined by boundary conditions. By substituting Eq. (30) into the boundary conditions of Eq. (26c), we can obtain

$$H\_{li} = \frac{(a^{+} - R\alpha^{-})(X\_{i}e^{-k\tau\_{0}} + Y\_{i} - \sigma\_{1i}^{-}e^{-k\tau\_{0}} - \sigma\_{2i}^{-} - \sigma\_{6i}^{-}) + a^{-}e^{-k\tau\_{0}}[(R+1)X\_{i} + (R+1)Y\_{i}e^{-k\tau\_{0}}]}{a^{+}(a^{+} - R\_{dif}a^{-}) - a^{-}(a^{-} - R\_{dif}a^{-})e^{-2k\tau\_{0}}}$$

$$-\frac{a^{-}e^{-k\tau\_{0}}\left[(R\sigma\_{1i}^{-} - \sigma\_{1i}^{+})e^{-a\tau\_{0}} + (R\sigma\_{2i}^{-} - \sigma\_{2i}^{+})e^{-(k+a)\tau\_{0}} + (R\sigma\_{3i}^{-} - \sigma\_{3i}^{+})\tau\_{0} + (R\sigma\_{4i}^{-} - \sigma\_{4i}^{+})\tau\_{0}e^{-k\tau\_{0}}\right]}{a^{+}(a^{+} - R\_{dif}a^{-}) - a^{-}(a^{-} - R\_{dif}a^{-})e^{-2k\tau\_{0}}}$$

$$-\frac{a^{-}e^{-k\tau\_{0}}\left[(R\sigma\_{5i}^{-} - \sigma\_{5i}^{+}) + (R\sigma\_{6i}^{-} - \sigma\_{6i}^{+})e^{-a\tau\_{0}} + (R-1)\sigma\_{7i}\tau\_{0} + (R\sigma\_{8i}^{-} - \sigma\_{8i}^{+})\tau\_{0}e^{-a\tau\_{0}}\right]}{a^{+}(a^{+} - R\_{dif}a^{-}) - a^{-}(a^{-} - R\_{dif}a^{-})e^{-2k\tau\_{0}}}\tag{31a}$$

$$K\_{\rm li} = \frac{1}{\alpha^- e^{-k\tau\_0}} \left( X\_{\rm i} e^{-k\tau\_0} + Y\_i - \alpha^+ H\_{\rm li} - \sigma\_{\rm 1i}^- e^{-k\tau\_0} - \sigma\_{\rm 2i}^- - \sigma\_{\rm 5i}^- - \sigma\_{\rm 6i}^- \right) \tag{31b}$$

Finally, the upward and downward fluxes are obtained by

$$F\_{\mathbf{l}}^{+}(\mathbf{0}) = \pi I\_{\mathbf{l}}^{+}(\mathbf{0})\tag{32a}$$

where r (g m<sup>3</sup>

According to [24], we choose LWP = 260 (g m<sup>2</sup>

) is the liquid water density. In this case, LWC varies from 0.22 to 0.30 g m<sup>3</sup>

) to represent low cloud. In the benchmark

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

and re varies from 2.06 to 16.50 μm, in which both ranges are consistent with observation [23].

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

calculations, z0 is divided into 100 internal homogeneous sub-layers, although other numbers can be chosen (e.g., 200). In principle, more internal sub-layers should result in more accurate results. We use 100 internal sub-layers throughout this study because having any more makes little difference to the calculated results. Using 100 sub-layers are sufficiently accurate to resolve the vertical internal inhomogeneity of the medium. We use the optical properties of a

Figure 1. For the band of 0.25–0.69 μm, (a-b) show cloud asymmetry factor/single-scattering albedo versus cloud optical depth (a for asymmetry factor; b for single-scattering albedo), (c-d) show the reflectance/absorptance versus solar zenith angle (c for reflectance; d for absorptance) and (e-f) show the relative errors of the homogeneous and inhomogeneous

solutions (e for reflectance error, f for absorptance error).

,

151

$$F\_{\mathcal{I}}^{-}(\pi\_{0}) = \pi I\_{\mathcal{I}}^{-}(\pi\_{0}) \tag{32b}$$

All detailed calculation about solar radiation can be found at [18].

## 3. Results and discussion

We apply the two schemes to idealized medium to investigate its accuracy, and the result has been shown on [16] and [18].

For true cloud medium, because ice clouds' optical properties strongly depend on the complex particle habits [19–21]. Therefore, we limit our discussion here to water cloud only. According to the observation, the internal LWC (g m�<sup>3</sup> ) and droplet radius of the cloud tend to increase with height [22]. To take this feature into account, LWC and droplet cross-sectional area (DCA; cm�<sup>2</sup> , m�<sup>3</sup> ) should increase linearly from the cloud base to the position near the top of the cloud:

$$\text{LWC} = 0.22 + 0.00008z \tag{33a}$$

$$\text{DCA} = 100 + z \tag{33b}$$

where 0 < z < z0. The terms z and z0 denote the height from the cloud base and the height of the cloud top, respectively. From Eq. (33a) to (33b), the cloud effective radius (re; μm) and liquid water path (LWP; g m�<sup>2</sup> ) can be obtained:

$$r\_{\varepsilon}(z) = \frac{3}{4\rho} \frac{\text{LWC}}{\text{DCA}} 10^{10} \tag{34a}$$

$$\text{LWP} = \int\_0^{z\_0} \text{LWCdz} \tag{34b}$$

where r (g m<sup>3</sup> ) is the liquid water density. In this case, LWC varies from 0.22 to 0.30 g m<sup>3</sup> , and re varies from 2.06 to 16.50 μm, in which both ranges are consistent with observation [23]. According to [24], we choose LWP = 260 (g m<sup>2</sup> ) to represent low cloud. In the benchmark calculations, z0 is divided into 100 internal homogeneous sub-layers, although other numbers can be chosen (e.g., 200). In principle, more internal sub-layers should result in more accurate results. We use 100 internal sub-layers throughout this study because having any more makes little difference to the calculated results. Using 100 sub-layers are sufficiently accurate to resolve the vertical internal inhomogeneity of the medium. We use the optical properties of a

where D�

� <sup>α</sup>�e�kτ<sup>0</sup> <sup>R</sup>σ�

� <sup>α</sup>�e�kτ<sup>0</sup> <sup>R</sup>σ�

<sup>1</sup><sup>i</sup> ¼ K1iα� � Xi, D�

<sup>H</sup>1<sup>i</sup> <sup>¼</sup> <sup>α</sup><sup>þ</sup> � <sup>R</sup>α� ð Þ Xie�kτ<sup>0</sup> <sup>þ</sup> Yi � <sup>σ</sup>�

<sup>1</sup><sup>i</sup> � σ<sup>þ</sup> 1i � �e�aiτ<sup>0</sup> <sup>þ</sup> <sup>R</sup>σ�

<sup>5</sup><sup>i</sup> � σ<sup>þ</sup> 5i � � <sup>þ</sup> <sup>R</sup>σ�

3. Results and discussion

been shown on [16] and [18].

water path (LWP; g m�<sup>2</sup>

m�<sup>3</sup>

the observation, the internal LWC (g m�<sup>3</sup>

<sup>K</sup>1<sup>i</sup> <sup>¼</sup> <sup>1</sup>

<sup>2</sup> P<sup>þ</sup> 7i

150 Perturbation Methods with Applications in Science and Engineering

2, 3, 4, 5, 6, 8), <sup>σ</sup>7<sup>i</sup> <sup>¼</sup> <sup>1</sup>

<sup>2</sup><sup>i</sup> <sup>¼</sup> <sup>H</sup>1iα<sup>∓</sup> � Yi, Xi <sup>¼</sup> <sup>P</sup><sup>þ</sup>

substituting Eq. (30) into the boundary conditions of Eq. (26c), we can obtain

<sup>e</sup>�kτ<sup>0</sup> � <sup>σ</sup>�

<sup>α</sup><sup>þ</sup> <sup>α</sup><sup>þ</sup> � Rdif <sup>α</sup>� � � � <sup>α</sup>� <sup>α</sup>� � Rdif <sup>α</sup>� � �e�2kτ<sup>0</sup>

F<sup>þ</sup> <sup>I</sup> ð Þ¼ 0 πI

F�

1i

<sup>6</sup><sup>i</sup> � σ<sup>þ</sup> 6i

<sup>α</sup>�e�kτ<sup>0</sup> Xie

Finally, the upward and downward fluxes are obtained by

All detailed calculation about solar radiation can be found at [18].

) can be obtained:

reð Þ¼ z

LWP ¼

<sup>2</sup><sup>i</sup> � σ<sup>þ</sup> 2i � �e�ð Þ <sup>k</sup>þai <sup>τ</sup><sup>0</sup> <sup>þ</sup> <sup>R</sup>σ�

3i �χ<sup>þ</sup> 3i 2 γ<sup>0</sup> <sup>1</sup>þγ<sup>0</sup> ð Þ<sup>2</sup>

<sup>5</sup><sup>i</sup> � σ� 6i � � <sup>þ</sup> <sup>α</sup>�e�kτ<sup>0</sup> ð Þ <sup>R</sup> <sup>þ</sup> <sup>1</sup> Xi <sup>þ</sup> ð Þ <sup>R</sup> <sup>þ</sup> <sup>1</sup> Yie�kτ<sup>0</sup>

<sup>α</sup><sup>þ</sup> <sup>α</sup><sup>þ</sup> � Rdif <sup>α</sup>� � � � <sup>α</sup>� <sup>α</sup>� � Rdif <sup>α</sup>� � �e�2kτ<sup>0</sup>

<sup>2</sup><sup>i</sup> � σ�

� �

<sup>α</sup><sup>þ</sup> <sup>α</sup><sup>þ</sup> � Rdif <sup>α</sup>� � � � <sup>α</sup>� <sup>α</sup>� � Rdif <sup>α</sup>� � �e�2kτ<sup>0</sup>

� �e�aiτ<sup>0</sup> <sup>þ</sup> ð Þ <sup>R</sup> � <sup>1</sup> <sup>σ</sup>7iτ<sup>0</sup> <sup>þ</sup> <sup>R</sup>σ�

�kτ<sup>0</sup> <sup>þ</sup> Yi � <sup>α</sup>þH1<sup>i</sup> � <sup>σ</sup>�

<sup>I</sup> ð Þ¼ τ<sup>0</sup> πI

We apply the two schemes to idealized medium to investigate its accuracy, and the result has

For true cloud medium, because ice clouds' optical properties strongly depend on the complex particle habits [19–21]. Therefore, we limit our discussion here to water cloud only. According to

height [22]. To take this feature into account, LWC and droplet cross-sectional area (DCA; cm�<sup>2</sup>

) should increase linearly from the cloud base to the position near the top of the cloud:

where 0 < z < z0. The terms z and z0 denote the height from the cloud base and the height of the cloud top, respectively. From Eq. (33a) to (33b), the cloud effective radius (re; μm) and liquid

> 3 4r

> > ð<sup>z</sup><sup>0</sup> 0

LWC

� �

, Yi <sup>¼</sup> <sup>P</sup><sup>þ</sup> 4i �χ<sup>þ</sup> 4i 2 γ<sup>0</sup> <sup>1</sup>þγ<sup>0</sup> ð Þ<sup>2</sup>

, and K1<sup>i</sup> and H1<sup>i</sup> are determined by boundary conditions. By

<sup>3</sup><sup>i</sup> � σ<sup>þ</sup> 3i � �τ<sup>0</sup> <sup>þ</sup> <sup>R</sup>σ�

> 1i e

þ

�

<sup>8</sup><sup>i</sup> � σ<sup>þ</sup> 8i � �τ0e�aiτ<sup>0</sup>

<sup>2</sup><sup>i</sup> � σ�

) and droplet radius of the cloud tend to increase with

LWC ¼ 0:22 þ 0:00008z (33a)

DCA ¼ 100 þ z (33b)

DCA <sup>1010</sup> (34a)

LWCdz (34b)

�kτ<sup>0</sup> � <sup>σ</sup>�

� � (31b)

, σ� ji <sup>¼</sup> <sup>1</sup> <sup>2</sup> P<sup>þ</sup> ji � P� ji � �

� �

<sup>4</sup><sup>i</sup> � σ<sup>þ</sup> 4i � �τ0e�kτ<sup>0</sup>

> <sup>5</sup><sup>i</sup> � σ� 6i

<sup>I</sup> ð Þ0 (32a)

<sup>I</sup> ð Þ τ<sup>0</sup> (32b)

. (j = 1,

(31a)

,

Figure 1. For the band of 0.25–0.69 μm, (a-b) show cloud asymmetry factor/single-scattering albedo versus cloud optical depth (a for asymmetry factor; b for single-scattering albedo), (c-d) show the reflectance/absorptance versus solar zenith angle (c for reflectance; d for absorptance) and (e-f) show the relative errors of the homogeneous and inhomogeneous solutions (e for reflectance error, f for absorptance error).

water cloud in the solar spectral band of 0.25–0.69 μm and at 0.94 μm and in the infrared spectral band of 5–8 μm and 11 μm.

In Figure 1a and b, 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

$$1 - \omega(\tau) = 3.979 \times 10^{-7} - 1.897 \times 10^{-6} \left( e^{-0.1539\tau} - e^{-0.1539\tau\_0/2} \right) \tag{35a}$$

$$g(\tau) = 0.8359 + 0.0289 \left( e^{-0.1539\tau} - e^{-0.1539\tau\_0/2} \right) \tag{35b}$$

where τ<sup>0</sup> ¼ 110:84. The corresponding results for reflection and absorption are shown in Figure 1c–f. For reflection, the relative error with the homogeneous solution increases from 0.25 to 0.71% as μ<sup>0</sup> 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 μ0; it is around 7.4% with the homogeneous solution but around only 1.4% with the inhomogeneous

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

In Figure 2a and b, the benchmark values of the inhomogeneous IOPs and the parameterized results for the wavelength 0.94 μm are shown. The parameterized inhomogeneous

where τ<sup>0</sup> ¼ 54:46. Figure 2c–f shows the corresponding results for reflection and absorption. For reflection, the relative error with the homogeneous solution increases from 1.1 to 3.0% as μ<sup>0</sup> 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 μ0; it is around 10% with the homogeneous solution but around only 5.7% with the inhomo-

The benchmark values of IOPs and parameterized results for the band of 5–8 μm are shown in

where τ<sup>0</sup> ¼ 55:85. For upward emissivity (Figure 3c and d), the relative errors of both solutions

1% for inhomogeneous solution. For downward emissivity (Figure 3e and f), the relative error

decreases to 0 firstly but then negatively increases to around �10%. The error of inhomogeneous solution shows a similar decreasing-increasing pattern, but the negative increase only

The benchmark values of IOPs and parameterized results for the band of 11 μm are shown in

�0:0357<sup>τ</sup> � <sup>e</sup>

<sup>0</sup>:0218<sup>τ</sup> � <sup>e</sup>

�0:0142<sup>τ</sup> � <sup>e</sup>

�0:0155<sup>τ</sup> � <sup>e</sup>

<sup>I</sup> ð Þ τ<sup>0</sup> ; the errors are around �3% for homogeneous solution and around

<sup>I</sup> ð Þ τ<sup>0</sup> increasing from 0 to 5πB Tð Þ, the error of homogeneous solution

�0:1018<sup>τ</sup> � <sup>e</sup>

<sup>0</sup>:1087<sup>τ</sup> � <sup>e</sup>

�0:0357τ0=<sup>2</sup> (36a)

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

153

<sup>0</sup>:0218τ0=<sup>2</sup> (36b)

�0:0142τ0=<sup>2</sup> (37a)

<sup>0</sup>:0155τ0=<sup>2</sup> (37b)

�0:1018τ0=<sup>2</sup> (38a)

<sup>0</sup>:1087τ0=<sup>2</sup> (38b)

<sup>I</sup> ð Þ¼ τ<sup>0</sup> 0, while the error of inhomogeneous solution is

<sup>1</sup> � ω τð Þ¼ <sup>1</sup>:<sup>936</sup> � <sup>10</sup>�<sup>4</sup> � <sup>5</sup>:<sup>263</sup> � <sup>10</sup>�<sup>4</sup> <sup>e</sup>

gð Þ¼ τ 0:8321 � 0:0403 e

ω τð Þ¼ 0:6757 � 0:3697 e

gð Þ¼ τ 0:8644 þ 0:1023 e

ω τð Þ¼ 0:4623 � 0:2155 e

gð Þ¼ τ 0:9118 � 0:0083 e

solution.

IOPs are

geneous solution.

are not sensitive to F<sup>þ</sup>

only 1%. With F<sup>þ</sup>

reaches about �2%.

Figure 3a and b. Here, we assume

of homogeneous solution is 4% when F<sup>þ</sup>

Figure 4a and b. In this case, we assume

Figure 2. Same as Figure 1 but for the wavelength 0.94 μm.

where τ<sup>0</sup> ¼ 110:84. The corresponding results for reflection and absorption are shown in Figure 1c–f. For reflection, the relative error with the homogeneous solution increases from 0.25 to 0.71% as μ<sup>0</sup> 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 μ0; it is around 7.4% with the homogeneous solution but around only 1.4% with the inhomogeneous solution.

water cloud in the solar spectral band of 0.25–0.69 μm and at 0.94 μm and in the infrared

In Figure 1a and b, 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

> �0:1539<sup>τ</sup> � <sup>e</sup> �0:1539τ0=<sup>2</sup>

�0:1539<sup>τ</sup> � <sup>e</sup> �0:1539τ0=<sup>2</sup> (35a)

(35b)

<sup>1</sup> � ω τð Þ¼ <sup>3</sup>:<sup>979</sup> � <sup>10</sup>�<sup>7</sup> � <sup>1</sup>:<sup>897</sup> � <sup>10</sup>�<sup>6</sup> <sup>e</sup>

gð Þ¼ τ 0:8359 þ 0:0289 e

spectral band of 5–8 μm and 11 μm.

152 Perturbation Methods with Applications in Science and Engineering

Figure 2. Same as Figure 1 but for the wavelength 0.94 μm.

IOPs are

In Figure 2a and b, the benchmark values of the inhomogeneous IOPs and the parameterized results for the wavelength 0.94 μm are shown. The parameterized inhomogeneous IOPs are

$$1 - \omega(\tau) = 1.936 \times 10^{-4} - 5.263 \times 10^{-4} \left( e^{-0.0357\tau} - e^{-0.0357\tau\_0/2} \right) \tag{36a}$$

$$g(\tau) = 0.8321 - 0.0403 \left( e^{0.0218\tau} - e^{0.0218\tau\_0/2} \right) \tag{36b}$$

where τ<sup>0</sup> ¼ 54:46. Figure 2c–f shows the corresponding results for reflection and absorption. For reflection, the relative error with the homogeneous solution increases from 1.1 to 3.0% as μ<sup>0</sup> 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 μ0; it is around 10% with the homogeneous solution but around only 5.7% with the inhomogeneous solution.

The benchmark values of IOPs and parameterized results for the band of 5–8 μm are shown in Figure 3a and b. Here, we assume

$$
\omega(\tau) = 0.6757 - 0.3697 \left( e^{-0.0142\tau} - e^{-0.0142\tau\_0/2} \right) \tag{37a}
$$

$$g(\tau) = 0.8644 + 0.1023 \left( e^{-0.0155 \tau} - e^{0.0155 \tau\_0/2} \right) \tag{37b}$$

where τ<sup>0</sup> ¼ 55:85. For upward emissivity (Figure 3c and d), the relative errors of both solutions are not sensitive to F<sup>þ</sup> <sup>I</sup> ð Þ τ<sup>0</sup> ; the errors are around �3% for homogeneous solution and around 1% for inhomogeneous solution. For downward emissivity (Figure 3e and f), the relative error of homogeneous solution is 4% when F<sup>þ</sup> <sup>I</sup> ð Þ¼ τ<sup>0</sup> 0, while the error of inhomogeneous solution is only 1%. With F<sup>þ</sup> <sup>I</sup> ð Þ τ<sup>0</sup> increasing from 0 to 5πB Tð Þ, the error of homogeneous solution decreases to 0 firstly but then negatively increases to around �10%. The error of inhomogeneous solution shows a similar decreasing-increasing pattern, but the negative increase only reaches about �2%.

The benchmark values of IOPs and parameterized results for the band of 11 μm are shown in Figure 4a and b. In this case, we assume

$$
\omega(\tau) = 0.4623 - 0.2155 \left( e^{-0.1018\tau} - e^{-0.1018\tau\_0/2} \right) \tag{38a}
$$

$$g(\tau) = 0.9118 - 0.0083 \left( e^{0.1087\tau} - e^{0.1087\tau\_0/2} \right) \tag{38b}$$

where τ<sup>0</sup> ¼ 28:23. For upward emissivity (Figure 4c and d), the relative error of homogeneous solution is �1.2%, while the error of inhomogeneous solution is less than 0.5%. For downward emissivity (Figure 4e and f), with F<sup>þ</sup> <sup>I</sup> ð Þ τ<sup>0</sup> increasing from 0 to 5πB Tð Þ, the error of homogeneous (inhomogeneous) solution varies from 3 to �11% (from 0 to �1%).

Figure 3. For the band of 5-8 μm, (a-b) show the cloud single-scattering albedo and asymmetry factor versus cloud optical depth, black dots represent the exact values and the blue lines is the fitting results (a for single-scattering albedo; b for asymmetry factor); (c-d) show the upward/downward emissivity versus the ratio of the radiation incident from the bottom to the internal infrared emission of the medium (c for upward emissivity; d for downward emissivity) and (e-f) show the relative errors of the homogeneous and inhomogeneous solutions (e for upward emissivity; f for downward emissivity).

4. Summary and conclusions

Figure 4. Same as Figure 3 but for the wavelength of 11 μm.

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 SRT/IRT solution to address the vertical inhomogeneity

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

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

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Perturbation Method for Solar/Infrared Radiative Transfer in a Scattering Medium with Vertical Inhomogeneity… http://dx.doi.org/10.5772/intechopen.77147 155

Figure 4. Same as Figure 3 but for the wavelength of 11 μm.

#### 4. Summary and conclusions

where τ<sup>0</sup> ¼ 28:23. For upward emissivity (Figure 4c and d), the relative error of homogeneous solution is �1.2%, while the error of inhomogeneous solution is less than 0.5%. For downward

Figure 3. For the band of 5-8 μm, (a-b) show the cloud single-scattering albedo and asymmetry factor versus cloud optical depth, black dots represent the exact values and the blue lines is the fitting results (a for single-scattering albedo; b for asymmetry factor); (c-d) show the upward/downward emissivity versus the ratio of the radiation incident from the bottom to the internal infrared emission of the medium (c for upward emissivity; d for downward emissivity) and (e-f) show the relative errors of the homogeneous and inhomogeneous solutions (e for upward emissivity; f for downward

neous (inhomogeneous) solution varies from 3 to �11% (from 0 to �1%).

<sup>I</sup> ð Þ τ<sup>0</sup> increasing from 0 to 5πB Tð Þ, the error of homoge-

emissivity (Figure 4e and f), with F<sup>þ</sup>

154 Perturbation Methods with Applications in Science and Engineering

emissivity).

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 SRT/IRT 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 solar Eddington solution and standard infrared two-stream solution for homogeneous layers to identify a zeroth-order equation and a first-order equation that includes the inhomogeneous effect. The new SRT/IRT solution can accurately express the inhomogeneous effect in each model layer, and it reduces to the standard solution when the medium is homogeneous.

Author details

, Feng Zhang<sup>1</sup>

\*, Jia-Ren Yan<sup>1</sup>

Canada, University of Victoria, Victoria, British Columbia, Canada

1 Key Laboratory of Meteorological Disaster, Ministry of Education, Nanjing University of

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

2 Canadian Centre For Climate Modelling and Analysis, Environment and Climate Change

[1] Lenoble J. Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Com-

[2] Toon OB, McKay CP, Ackerman TP. Rapid calculation of radiative heating rates and photodissociation rates in inhomogeneous multiple scattering atmospheres. Journal of

[3] Fu Q, Liou KN, Cribb MC, Charlock TP, Grossman A. Multiple scattering parameterization in thermal infrared radiative transfer. Journal of the Atmospheric Sciences. 1997;54:

[4] Li J, Dobbie S, Raisanen P, Min Q. Accounting for unresolved cloud in solar radiation.

[5] Vane D, Tourville N, Stephens G, Kanekiewicz A. New observations of hurricanes from

[6] Boutle IA, Abel SJ, Hill PG, Morcrette CJ. Spatial variability of liquid cloud and rain: Observations and microphysical effects. Quarterly Journal of the Royal Meteorological

[7] Young AH, Bates J, Curry J. Application of cloud vertical structure from cloudsat to investigate modis-derived cloud properties of cirriform, anvil and deep convective clouds.

[8] Luo ZJ, Jeyaratnam J, Iwasaki S, et al. Convective vertical velocity and cloud internal vertical structure: An a-train perspective. Geophysical Research Letters. 2014;41:723-729

[9] Chen MN, Lu CS, Liu YG. Variation in entrainment rate and relationship with cloud microphysical properties on the scale of 5m. Scientific Bulletin. 2015;60:707-717

[10] Li J, Geldart DJW, Chylek P. Solar radiative transfer in clouds with vertical internal

homogeneity. Journal of the Atmospheric Sciences. 1994;51:2542-2552

putational Procedures. Hampton, VA: A. Deepak Publishing; 1985. 314 pp

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the cloudsat radar. In: AGU Fall Meeting Abstracts. 2006:A13A-0885

Journal of Geophysical Research. 2013;118:4689-4699

\*Address all correspondence to: fengzhang@nuist.edu.cn

Information Science and Technology, Nanjing, China

Geophysical Research. 1989;94:16287-16301

, Qiu-Run Yu1 and Jiangnan Li<sup>2</sup>

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

157

Yi-Ning Shi1

References

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Society. 2014;140:583-594

The new inhomogeneous SRT/IRT 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 SRT solution is no more than 1.4%, whereas the error with the homogeneous SRT 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 SRT solution. In the band of 5–8 μm, the homogeneous IRT solution is not sensitive to F<sup>þ</sup> <sup>I</sup> ð Þ τ<sup>0</sup> , and its relative error may reach �3.2% for upward emissivity, whereas the error of inhomogeneous IRT solution is only 1%. With F<sup>þ</sup> <sup>I</sup> ð Þ τ<sup>0</sup> increasing from 0 to 5πB Tð Þ, the error of downward emissivity for homogeneous solution varies from 4 to �10%, while the error ranges from 1 to �2% for inhomogeneous IRT solution. In the band of 11 μm, the relative error of homogeneous IRT solution is around �1.2% for upward emissivity, and the error of inhomogeneous IRT solution is only less than 0.5%. For downward emissivity, the maximum error of homogeneous IRT solution can be up to �11%, and the maximum error of inhomogeneous IRT solution is only around �1% when F<sup>þ</sup> <sup>I</sup> ð Þ¼ τ<sup>0</sup> 5πB Tð Þ.

In specific spectral bands or at particular 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 [5, 7, 8].

In this study, we presented only a single-layer inhomogeneous SRT/IRT solution. To implement the new solution in a climate model, the adding process for layer-to-layer connections has to be solved. 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 SRT/IRT solution is available from the authors upon request.
