Light Scattering, Absorption, Extinction, and Propagation in the Terrestrial Atmosphere

Abstract

Radiative transfer is the physical phenomenon of energy transfer in the form of electromagnetic radiation, while the radiative transfer equation describes this phenomenon mathematically. The radiative transfer theory found its applications in astrophysics, atmospheric physics and engineering sciences. In atmospheric remote sensing, it is the basic for the so-called forward models which are used to simulate the measurements for a given set of atmospheric parameters.

Appendices

Appendix 1: On the Use of Complex Electromagnetic Fields for Computing Time-Averaged Poynting Vector

In electrodynamics, the complex representation of electric and magnetic fields is used quite extensively for mathematical convenience. Consider time-harmonic complex fields

$$\begin{aligned} \hat{\mathbf {E}}\left( \mathbf {r},t\right) =\hat{\mathbf {E}}\left( \mathbf {r}\right) e^{-j\omega t}, \end{aligned}$$

(3.265)

$$\begin{aligned} \hat{\mathbf {H}}\left( \mathbf {r},t\right) =\hat{\mathbf {H}}\left( \mathbf {r}\right) e^{-j\omega t}. \end{aligned}$$

(3.266)

Here the “hat” sign is used for complex quantities, and $j^{2}=-1$. Note that time-independent amplitudes $\hat{\mathbf {E}}\left( \mathbf {r}\right)$and $\hat{\mathbf {H}}\left( \mathbf {r}\right)$ correspond to perfectly monochromatic radiation, while slowly varying $\hat{\mathbf {E}}\left( \mathbf {r}\right)$ and $\hat{\mathbf {H}}\left( \mathbf {r}\right)$ represent the so-called quasi-monochromatic radiation.

In practice, only the real parts, i.e., $\mathbf {E}=\mathrm {Re}\{\hat{\mathbf {E}}\}$ and $\mathbf {H}=\mathrm {Re}\{\hat{\mathbf {H}}\}$ , have the physical meaning. Furthermore, letting $^{*}$ denote complex conjugate, we can express $\mathbf {E}$ as follows:

$$\begin{aligned} \mathbf {E}\left( \mathbf {r},t\right) =\mathrm {Re}\left\{ \hat{\mathbf {E}}\left( \mathbf {r},t\right) \right\} =\frac{1}{2}\left[ \hat{\mathbf {E}}\left( \mathbf {r}\right) e^{-\mathrm{j}\omega t}+\hat{\mathbf {E}}^{*}\left( \mathbf {r}\right) e^{\mathrm{j}\omega t}\right] \end{aligned}$$

(3.267)

and similarly for H:

$$\begin{aligned} \mathbf {H}\left( \mathbf {r},t\right) =\mathrm {Re}\left\{ \hat{\mathbf {H}}\left( \mathbf {r},t\right) \right\} =\frac{1}{2}\left[ \hat{\mathbf {H}}\left( \mathbf {r}\right) e^{-\mathrm{j}\omega t}+\hat{\mathbf {H}}^{*}\left( \mathbf {r}\right) e^{\mathrm{j}\omega t}\right] . \end{aligned}$$

(3.268)

Substituting Eqs. (3.267) and (3.268) into Eq. (3.39) yields the expression for the complex Poynting vector:

$$\begin{aligned} \begin{array}{l} {\hat{\mathbf {S}}=\dfrac{1}{4}\left[ \hat{\mathbf {E}}\left( \mathbf {r}\right) \times \hat{\mathbf {H}}\left( \mathbf {r}\right) e^{-2\mathrm{j}\omega t}+\hat{\mathbf {E}}^{*}\left( \mathbf {r}\right) \times \hat{\mathbf {H}}^{*}\left( \mathbf {r}\right) e^{2\mathrm{j}\omega t}\right] }\\ {+\dfrac{1}{4}\left[ \hat{\mathbf {E}}^{*}\left( \mathbf {r}\right) \times \hat{\mathbf {H}}\left( \mathbf {r}\right) +\hat{\mathbf {E}}\left( \mathbf {r}\right) \times \hat{\mathbf {H}}^{*}\left( \mathbf {r}\right) \right] .} \end{array} \end{aligned}$$

(3.269)

Then, taking into account that

$$\begin{aligned} \mathbf {S}=\mathrm {Re}\{\hat{\mathbf {S}}\}, \end{aligned}$$

(3.270)

$$\begin{aligned} \mathrm {Re}\{\hat{\mathbf {E}}^{*}\left( \mathbf {r}\right) \times \hat{\mathbf {H}}\left( \mathbf {r}\right) \}=\mathrm {Re}\{\hat{\mathbf {E}}\left( \mathbf {r}\right) \times \hat{\mathbf {H}}^{*}\left( \mathbf {r}\right) \} \end{aligned}$$

(3.271)

and

$$\begin{aligned} \left\langle e^{\pm 2\mathrm{j}\omega t}\right\rangle =0, \end{aligned}$$

(3.272)

we obtain the expression for the time-averaged Poynting vector in the form of Eq. (3.44).

Appendix 2: Computations of Legendre Polynomials

The associated Legendre polynomials $P_{l}^{m}\left( \mu \right)$ (sometimes referred to as Ferrers’ functions) can be computed recurrently. The base of the recurrence is given by two formulas for m = l and l = m + 1, respectively,

$$\begin{aligned} P_{m}^{m}\left( \mu \right) =(-1)^{m}(2m-1)!!(1-\mu ^{2})^{m/2} \end{aligned}$$

(3.273)

and

$$\begin{aligned} P_{m+1}^{m}(\mu )=\mu (2m+1)P_{m}^{m}\left( \mu \right) \end{aligned}$$

(3.274)

with !! the double factorial (n!! equals to the product of all the integers from 1 up to n that have the same parity (odd or even) as n — it should not be confused with the factorial iterated twice, which would be written as (n!)!). Polynomials $P_{l}^{m}\left( \mu \right)$ for $l\ge m+2$ are computed by using three-term recurrence relation with respect to the degree (for m = 0 known is Bonnet’s recursion formula):

$$\begin{aligned} (2l+1)\mu P_{l}^{m}(\mu )=(l+m)P_{l-1}^{m}(\mu )+(l-m+1)P_{l+1}^{m}(\mu ). \end{aligned}$$

(3.275)

For validation of the recurrent formula implementation, it is convenient to use analytical forms of the first few associated Legendre polynomials are listed in Table 3.4.

Note that polynomials $P_{l}^{m}\left( \mu \right)$ satisfy the following normalization condition

$$\begin{aligned} \int _{-1}^{1}P_{l}^{m}(\mu )P_{l'}^{m}(\mu )d\mu =\frac{2}{2l+1}\frac{(l+m)!}{(l-m)!}\delta _{ll'} \end{aligned}$$

(3.276)

and the maximum value of $P_{l}^{m}\left( \mu \right)$ rapidly increases with m. To avoid overflow during the recurrent process, it is recommended to compute normalized Legendre polynomials $\bar{P}_{l}^{m}\left( \mu \right)$ defined as

$$\begin{aligned} \bar{P}_{l}^{m}(\mu )=\sqrt{\left( 2l+1\right) \frac{(l-m)!}{(l+m)!}}P_{l}^{m}(\mu ) \end{aligned}$$

(3.277)

instead of $P_{l}^{m}\left( \mu \right)$. For $\bar{P}_{l}^{m}(\mu )$ the following normalization condition holds:

$$\begin{aligned} \int _{-1}^{1}\bar{P}_{l}^{m}(\mu )\bar{P}_{l'}^{m}(\mu )d\mu =2\delta _{ll'}. \end{aligned}$$

(3.278)

Table 3.4 The first few associated Legendre polynomials

Appendix 3: Computations of Generalized Spherical Functions

The generalized spherical functions $R_{l}^{m}\left( \mu \right)$ and $T_{l}^{m}\left( \mu \right)$ can be expressed as follows [67, 109,110,111]:

$$\begin{aligned} R_{l}^{m}\left( \mu \right) =-\frac{1}{2}(i)^{m}\sqrt{\frac{(l+m)!}{(l-m)!}}\left\{ P_{m,2}^{l}\left( \mu \right) +P_{m,-2}^{l}\left( \mu \right) \right\} , \end{aligned}$$

(3.279)

$$\begin{aligned} T_{l}^{m}\left( \mu \right) =-\frac{1}{2}(i)^{m}\sqrt{\frac{(l+m)!}{(l-m)!}}\left\{ P_{m,2}^{l}\left( \mu \right) -R_{m,-2}^{l}\left( \mu \right) \right\} , \end{aligned}$$

(3.280)

where for $l\ge \max (m,2)$,

$$\begin{aligned} \begin{array}{l} {P_{m,n}^{l}\left( \mu \right) =A_{m,n}^{l}\left( 1-\mu \right) ^{-(n-m)/2}(1+\mu )^{-(n+m)/2}}\\ {\times \frac{d^{l-n}}{d\mu ^{l-n}}\left[ \left( 1-\mu \right) ^{l-m}\left( 1+\mu \right) ^{l+m}\right] ,} \end{array} \end{aligned}$$

(3.281)

with

$$\begin{aligned} A_{m,n}^{l}=\frac{\left( -1\right) ^{l-m}\left( i\right) ^{n-m}}{2^{l}\left( l-m\right) !}\sqrt{\frac{(l-m)!(l+n)!}{(l+m)!(l-n)!}}. \end{aligned}$$

(3.282)

For $P_{m,n}^{l}$ the recursion formula was derived in [109]:

$$\begin{aligned} e_{m,n}^{l}P_{m,n}^{l+1}\left( \mu \right) =(2l+1)\mu P_{m,n}^{l}\left( \mu \right) -f_{m,n}^{l}P_{m,n}^{l-1}\left( \mu \right) -\frac{mn\left( 2l+1\right) }{l(l+1)}P_{m,n}^{l}\left( \mu \right) \end{aligned}$$

(3.283)

with

$$\begin{aligned} e_{m,n}^{l}=\frac{1}{l+1}\sqrt{(l+m+1)(l-m+1)(l+n+1)(l-n+1)} \end{aligned}$$

(3.284)

and

$$\begin{aligned} f_{m,n}^{l}=\frac{1}{l}\sqrt{(l+m)(l-m)(l+n)(l-n)}. \end{aligned}$$

(3.285)

Given Eqs. (3.279)-(3.285), the recurrence formulas for $R_{l}^{m}$ and $T_{l}^{m}$ can be derived for l > max(m, 2) and $m\ge 0$:

$$\begin{aligned} \begin{array}{l} {\frac{l+1-m}{l+1}\sqrt{(l+3)(l-1)}R_{l+1}^{m}\left( \mu \right) =}\\ {(2l+1)\mu R_{l}^{m}\left( \mu \right) -\frac{l+m}{l}\sqrt{(l+2)(l-2)}R_{l-1}^{m}\left( \mu \right) -\frac{2m(2l+1)}{l(l+1)}T_{l}^{m}\left( \mu \right) ,} \end{array} \end{aligned}$$

(3.286)

$$\begin{aligned} \begin{array}{l} {\frac{l+1-m}{l+1}\sqrt{(l+3)(l-1)}T_{l+1}^{m}(\mu )=}\\ {(2l+1)\mu T_{l}^{m}(\mu )-\frac{l+m}{l}\sqrt{(l+2)(l-2)}T_{l-1}^{m}(\mu )-\frac{2m(2l+1)}{l(l+1)}R_{l}^{m}(\mu ).} \end{array} \end{aligned}$$

(3.287)

The base of recurrence computation is

$$\begin{aligned} R_{j}^{j}\left( \mu \right) =\frac{(2j)!}{2^{j}j!}\sqrt{\frac{j(j-1)}{(j+1)(j+2)}}\sqrt{1-\mu ^{2}}\frac{1+\mu ^{2}}{1-\mu ^{2}}, \end{aligned}$$

(3.288)

$$\begin{aligned} T_{j}^{j}\left( \mu \right) =\frac{(2j)!}{2^{j}j!}\sqrt{\frac{j(j-1)}{(j+1)(j+2)}}\sqrt{1-\mu ^{2}}\frac{2\mu }{1-\mu ^{2}}, \end{aligned}$$

(3.289)

while

$$\begin{aligned} R_{l}^{m}\left( \mu \right) =T_{l}^{m}\left( \mu \right) =0,\mathrm{\;\;}l<m. \end{aligned}$$

(3.290)

In particular, it follows:

$$\begin{aligned} R_{2}^{2}\left( \mu \right) =\frac{\sqrt{6}}{2}\left( 1+\mu ^{2}\right) , \end{aligned}$$

(3.291)

$$\begin{aligned} T_{2}^{2}\left( \mu \right) =\sqrt{6}\mu . \end{aligned}$$

(3.292)

For m = 1, the initial values are

$$\begin{aligned} R_{2}^{1}\left( \mu \right) =-\frac{1}{2}\mu \sqrt{6}\sqrt{1-\mu ^{2}}, \end{aligned}$$

(3.293)

$$\begin{aligned} T_{2}^{1}\left( \mu \right) =-\frac{1}{2}\sqrt{6}\sqrt{1-\mu ^{2}}. \end{aligned}$$

(3.294)

Finally, for m = 0, we have

$$\begin{aligned} R_{2}^{0}\left( \mu \right) =\frac{\sqrt{6}}{4}(1-\mu ^{2}), \end{aligned}$$

(3.295)

$$\begin{aligned} T_{2}^{0}\left( \mu \right) =0. \end{aligned}$$

(3.296)

Note that $R_{l}^{m}$ and $T_{l}^{m}$ satisfy the following normalization condition:

$$\begin{aligned} \int _{-1}^{+1}\left[ R_{l}^{m}\left( \mu \right) R_{r}^{m}\left( \mu \right) +T_{l}^{m}\left( \mu \right) T_{r}^{m}\left( \mu \right) \right] d\mu =1, \end{aligned}$$

(3.297)

where l, r $\ge$ max(m, 2).