Hapke Model

Hapke’s photometric model（地大师姐推荐文章(Sato, Robinson et al. 2014)）：也见（Hapke书9.4节（9.47）式）

$$I / F=L S\left(i_{e}, e_{e}\right) K \frac{w}{4}\left[p(g)\left(1+B_{S 0} B_{S}(g)\right)+M\left(i_{e}, e_{e}\right)\right]\left[1+B_{C 0} B_{C}(g)\right] S(i, e, \psi)$$

注：$I/F$为radiance factor，而reflectance factor(REFF)为${I/F\over \pi}$,这里的I/F就是一种表示方法（RADF），不要分开看。

1. LS is the Lommel-Seeliger function,$i_e$ and $e_e$ are effective angle of incidence and emission respectively
2. K is porosity factor,$\phi$,[0,1]
3. w is single scattering albedo,[0,1]
4. p(g) is phase function:b,c；The probability that a photon will be scattered in the direction g is given by p(g)。b,[0,1];c,p(g)>0;
5. $B_{S0}$ is amplitude of Shadow Hiding Opposition Effect (SHOE),Bs,[0,1]
6. $B_S(g)$ is SHOE function,$h_S$
7. $M(i_e, e_e)$ is Isotropic Multiple-Scattering Approximation (IMSA)
8. $B_{C0}$ is amplitude of Coherent Backscatter Opposition Effect (CBOE)，[0,1]
9. $B_C(g)$ is CBOE function,$h_C$
10. S(i,e,𝜓) is shadowing function, and 𝜓 is azimuthal angle between the planes of incidence and emergence；The term $\bar{\theta}_p$ is the shadowing function and corrects for macroscopic roughness of the surface and depends on mean slope angle 𝜃.

1. Henyey-Greenstein double-lobed single particle phase 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）

$p(g)=1$ 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。

• ‘hockey-stick” diagram

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

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

Hapke2012,p121

1. The SHOE function BS(g) is given by

$$B_{S}(g)=1 /\left[1+\tan (g / 2) / h_{S}\right]$$

$h_S$ is the angular width parameter of SHOE,

1. The IMSA function $M(i_e, e_e)$ is given by

$$M\left(i_{e}, e_{e}\right)=H\left(\frac{\cos \left(i_{e}\right)}{K}, w\right) H\left(\frac{\cos \left(e_{e}\right)}{K}, w\right)-1$$

where H(x,w) is Ambartsumian-Chandrasekhar H function, which is approximated by

$$H(x, w) \simeq\left\{1-w x\left[r_{0}+\frac{1-2 r_{0} x}{2} \ln \left(\frac{1+x}{x}\right)\right]\right\}^{-1}$$

简化版：

$$H(x)={1+2x\over 1+2\gamma x}$$

$r_0$ is diffusive reflectance, given by

$$r_{0}=\frac{1-\sqrt{1-w}}{1+\sqrt{1-w}}$$

1. The CBOE function $B_C(g)$ is given by

$$B_{C}(g)=\frac{1+\frac{1-\exp \left[-\tan (g / 2) / h_{C}\right]}{\tan (g / 2) / h_{C}}}{2\left[1+\tan (g / 2) / h_{C}\right]^{2}}$$

where $h_C$ is angular width of CBOE

• $h_C = \lambda/4\pi\Lambda$;$\Lambda$ is the transport mean free path in the medium.

1. The porosity factor K is

$$K=\frac{-\ln \left(1-1.209 \phi^{2 / 3}\right)}{1.209 \phi^{2 / 3}}$$

where 𝜙 is filling factor that is equivalent to 1− porosity. The K monotonically increases with the filling factor 𝜙 (= 1− porosity)

孔隙度因子K不同于孔隙度（porosity）

1. The $i_e, e_e,$(注：下标e表示effective) and the shadowing function S(i, e,𝜓) are the functions of $\bar{\theta}_p$ (the mean slope angle), the effective value of the photometric roughness.宏观粗糙度。

公式很复杂，待补充

拟合数据得到如上九个参数，这些参数中有些与物性有关

简化的5参数Hapke模型(地大师姐)：SHOE-only.

$$I / F={\cos i \over \cos i+\cos e}\frac{w}{4}\left[p(g)\left(1+{B_{S 0} \over 1+\tan(g/2)/h_S}\right)+H(\cos i)H(\cos e)-1\right]$$

参数范围：0≤b≤1，-1≤c≤1，0≤w≤1，$0\leq B_{S0} \leq 2$，$0\leq h_S \leq 1$

或者，CBOE-only

在不同的应用场景下，SHOE与CBOE对opposition effects 的贡献不一样，因此，按需选择。得到5参数的hapke模型

$$\begin{aligned} \begin{array}{l} \text {IOF} / L S=\frac{w}{4}\left[p(g)+H\left(\mu_{0}, w\right) H(\mu, w)-1\right]\left[1+B_{C 0} B_{C}\left(g, h_{c}\right)\right] S(i, e, \theta) \\ L S=\mu_{0} /\left(\mu_{0}+\mu\right) \end{array} \end{aligned}$$

简化hapke模型（林洪磊）:仅3个参数，w,b,c

不考虑冲效应，但考虑多次散射，因此得到3参数的hapke模型，

K= 1，因为月壤最上层的填充率未知

忽略SHOE和COBE，因为冲效应仅仅发生在小相角处，而第十月昼的相角大于50°

$$r={\cos i \over \cos i+\cos e}\frac{w}{4\pi}\left[p(g)+H(\cos i)H(\cos e)-1\right]$$

若继续不考虑多次散射，则，hapke模型简化为LS模型

Hapke,2012:

• Single scattering:the Lommel-Seeliger law（p197)

$r = {\varpi \over 4\pi}{\mu_0 \over \mu_0 +\mu}f(g)$

• the bidirectional reflectance of a sparse medium of isotropic scatters(p197)

二流近似

$r = {\varpi \over 4 \pi}{\mu_0\over \mu_0 +\mu}{1+2\mu_0 \over 1+2\gamma\mu_0}{1+2\mu \over 1+2\gamma\mu}$

变分法

$r = {\varpi \over 4 \pi}{\mu_0\over \mu_0 +\mu}H(\mu_0)H(\mu)$

• the isotropic multiple-scattering approximation(IMSA)

$I_{SS} = J{\varpi \over 4\pi}{\mu_0 \over \mu_0 +\mu}p(g)$

$I_{MS} = J{\varpi \over 4 \pi}{\mu_0\over \mu_0 +\mu}[H(\mu_0)H(\mu)-1]$

$r = {\varpi \over 4 \pi}{\mu_0\over \mu_0 +\mu}[p(g)+H(\mu_0)H(\mu)-1]$

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。

相函数有多种形式：

• HG函数；

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

b = −0.17, c = 0.70。

• 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
• b>0时，

• 双参数HG函数

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

• 第一项为后向项，第二项为前向项；

• 三参数HG函数：$b_1,b_2,c$

• 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？

基于得到的相函数做光度校正：

• 比值法
• 解单次反照率：因为多次散射项中包含单次反照率，因此不能直接作比。

以上所有相函数都可用比值法，但HG函数和勒让德多项式来源于Hapke模型，因此还可以基于hapke模型作光度校正。