The AMSA (Anisotropic Multiple Scattering Approximation) Model#

The AMSA model, an acronym for Anisotropic Multiple Scattering Approximation, represents an advanced formulation of Hapke’s theory designed to more accurately account for the effects of anisotropic single-particle scattering on the multiple-scattering term. The definitive form of this model, which also incorporates a sophisticated treatment of opposition effects, is detailed by Hapke [2001].

The functions refmod.hapke.amsa.amsa and refmod.hapke.amsa.amsa_derivative in this library implement this comprehensive and powerful model.

AMSA Reflectance Equation#

The final expression for the AMSA model, as given by Hapke [2001] (Eq. 38), combines the single-scattering term, the anisotropic multiple-scattering term, and both major opposition effects. When combined with the macroscopic roughness correction, the full equation is:

\[ r(i, e, g) = \frac{w}{4\pi} \frac{\mu_{0e}}{\mu_{0e} + \mu_e} \left[ p(g) B_{SH}(g) + M(\mu_{0e}, \mu_e) \right] B_{CB}(g) \cdot S(i, e, g, \bar{\theta}) \]

The components are broken down as follows:

  1. Single-Scattering Term (\(p(g) B_{SH}(g)\))

    • \(p(g)\) is the single-particle phase function (e.g., a Legendre polynomial expansion).

    • \(B_{SH}(g)\) is the Shadow-Hiding Opposition Effect (SHOE), which multiplies only the single-scattering term. It is given by (Eq. 28, 29): $\( B*{SH}(g) = 1 + \frac{B*{S0}}{1 + \frac{1}{h_S} \tan(g/2)} \)$

  2. Anisotropic Multiple-Scattering Term (\(M(\mu_{0e}, \mu_e)\))

    • This is the core improvement of AMSA, replacing the simpler \(H(\mu_0)H(\mu)-1\) term from IMSA. It is defined by Hapke [2001] (Eq. 17): $\( M(\mu_0, \mu) = P(\mu_0)[H(\mu) - 1] + P(\mu)[H(\mu_0) - 1] + \bar{P}[H(\mu) - 1][H(\mu_0) - 1] \)$

    • The functions \(P(\mu_0)\), \(P(\mu)\), and \(\bar{P}\) are averaged phase functions defined in terms of Legendre coefficients.

    • The H-functions used here should be the more accurate “level 2” approximations (Eq. 13 in the paper; h_function_2 in refmod).

  3. Coherent Backscatter Opposition Effect (\(B_{CB}(g)\))

    • This term multiplies the entire reflectance (both single and multiple scattering components). It is defined by (Eq. 32): $\( B*{CB}(g) = 1 + B*{C0} \cdot B_C(g) \)\( where \)B_C(g)$ is a complex function modeling the coherent backscatter peak.

  4. Macroscopic Roughness (\(S\))

    • The standard roughness correction [Hapke, 1984] is applied to the final reflectance. The effective angles \(\mu_{0e}\) and \(\mu_e\) are used as inputs to the main scattering function.

Derivative Function#

The presence of refmod.hapke.amsa.amsa_derivative is highly significant. This function calculates \(\partial r / \partial w\) (the derivative of the AMSA reflectance with respect to the single-scattering albedo \(w\)). This capability is crucial for model inversion and sensitivity analysis, making the AMSA implementation in refmod particularly powerful for quantitative analysis of remote sensing data.