Hockey-Stick relation

Henyey-Greenstein function

$$p(g)=\frac{1+c}{2} \frac{1-b^{2}}{\left(1-2 b \cos (g)+b^{2}\right)^{3 / 2}}+\frac{1-c}{2} \frac{1-b^{2}}{\left(1+2 b \cos (g)+b^{2}\right)^{3 / 2}}$$

b (0 ≤ b ≤ 1) is the shape-controlling parameter, and c (−1 ≤ c ≤ 1) is the relative strength of backward and forward lobes.

b is the angular width of the backward (first term) or forward (second term) scattering lobe and c is the magnitude of the lobe.

c>0,后向；c<0 前向

b控制形状：b越大，侧向散射越弱（b=1例外）

Narrow lobes have b values close to 1, wide lobes have values << 1, and isotropically scattering particles have b = 0; 0 <= b < 1.

$p(g)=1$ 表示各向同性；$p(g)\neq 1$ 表示各项异性；$p(g) =0$ 表示完全吸收。

Normalized: ${1\over 4\pi} \int_0^\pi p(g)2\pi \sin g d g = 1$

Asymmetry factor:$<\cos \theta> = {1\over 4\pi} \int_0^\pi \cos\theta p(\theta)2\pi \sin \theta d \theta = -bc$

for a semi-infinite medium, the multiply scattered portion is much less sensitive to the particle phase function than the singly scattered fraction.

The brighter the surface, the more times the average photon is scattered before emerging from the surface, causing directional effects to be averaged out and the multiply scattered intensity distribution to closely approach the isotropic case.

Since the single-scattering term can be evaluated exactly for an arbitrary phase function and the multiply scattered term is relatively insensitive to phase function, a first-order expression for nonisotropic scatterers can be obtained by using the exact expression for the single-scattered term but retaining the isotropic solution for the multiply-scattered term.

多次散射项不像单次散射项那样对相函数明显，通常，表面越亮，多次散射的次数越多，其方向性就被平均掉了，所以一般多次散射项都当做各向同性处理。

hockey stick relation

$$c=3.29 \exp \left(-17.4 b^{2}\right)-0.908$$

other phase function（经验散射函数见Hapke2012，6.3节）

the Allen approxmation for Fraunhofer diffration
Legendre polynomial
Henyey-Greenstein function
Lambert sphere phase function
Lommel-Seeliger sphere phase function

• 单参数HG函数

$$p(g)=\frac{1-b^{2}}{\left(1+2 b \cos (g)+b^{2}\right)^{3 / 2}}$$

• $(1/4\pi)\int_{4\pi}p(g)d\Omega = 1$
• $\theta = \pi - g$
• 参数b为余弦不对称因子：$b = <\cos \theta>=-<\cos g>$
• b=0时，$p(g)=1$,各向同性

• 普通多项式（三阶）

$$f(g)=p_{1} g^{3}+p_{2} g^{2}+p_{3} g+p_{4}$$

• 普通多项式（四阶）

$$f(\alpha)=a_{0}+a_{1} \alpha+a_{2} \alpha^{2}+a_{3} \alpha^{3}+a_{4} \alpha^{4}$$

• 普通多项式（六阶）

$$f(\alpha, \lambda)=\mathrm{A}_{0}+\mathrm{A}_{1} \alpha+\mathrm{A}_{2} \mathrm{a}^{2}+\mathrm{A}_{3} \mathrm{\alpha}^{3}+\mathrm{A}_{4} \mathrm{\alpha}^{4}+\mathrm{A}_{5} \mathrm{\alpha}^{5}+\mathrm{A}_{6} \mathrm{\alpha}^{6}$$

• 普通多项式+指数项（指数项模拟背散射效应）

$$f(\alpha)=b_{0} e^{-b_{1} \alpha}+a_{0}+a_{1} \alpha+a_{2} \alpha^{2}+a_{3} \alpha^{3}+a_{4} \alpha^{4}$$

• 勒让德多项式

$$P(g)=1+b \times \cos (g)+c \times\left(1.5 \times \cos ^{2}(g)-0.5\right)$$

$P_1 = \cos \theta; P_2 = {1\over 2}(3\cos^2\theta -1)$

• 指数形式

$$f(\alpha)=e^{\beta \alpha+\gamma \alpha^{2}+\delta \alpha^{3}}$$

• 其他形式：

三阶多项式
一阶勒让德多项式
线性双项HG函数？

Rayleigh scattering

$$P(g) = {3\over 4} (1+\cos^2 g)$$

Are planetary regolith particles backscattering? (Hapke,1996)

Concepts

1. Continuous media(连续介质)
2. particulate media(离散介质): in which the scatters are separated by distance large compared to their size.
3. regolith: in which the particles are touching.