The IMSA (Isotropic Multiple Scattering Approximation) Model#

The term IMSA in the Hapke framework historically stands for Isotropic Multiple Scattering Approximation. It refers to a foundational version of the theory where the multiple-scattering component of light is treated as isotropic, even if the single-particle phase function itself is anisotropic [Hapke, 2001].

The name “Inversion of Multiple Scattering and Absorption,” which may be suggested by a module’s name, can describe the intended application of this particular variant, which is well-suited for inverting reflectance data to derive physical parameters.

The function refmod.hapke.imsa.imsa implements a model consistent with this classic IMSA formulation, augmented with terms for opposition effects and macroscopic roughness.

IMSA Reflectance Equation#

The fundamental IMSA equation, as given by Hapke [2001] (Eq. 1), is:

\[ r(i, e, g) = \frac{w}{4\pi} \frac{\mu_0}{\mu_0 + \mu} \left[ p(g) + H(\mu_0)H(\mu) - 1 \right] \]

This base equation is then typically modified with a simplified opposition effect (like SHOE) and a standard macroscopic roughness correction.

Key Features#

  • Multiple Scattering: Treated as isotropic, using the standard product of Chandrasekhar’s H-functions, \(H(\mu_0)H(\mu)\). Simpler H-function approximations (like Eq. 2 in Hapke [2001]) are often associated with this model.

  • Phase Function (\(p(g)\)): The refmod implementation allows for flexible, user-supplied callable phase functions.

  • Simplicity for Inversion: Due to its relative simplicity and fewer free parameters compared to the full AMSA model, the IMSA model is well-suited for use in inversion routines that fit observational data to derive physical parameters.

Note on refmod Normalization

An important implementation detail in refmod.hapke.imsa.imsa is an additional division by \(4\pi\) (refl /= 4*np.pi). This is a significant deviation from the published formula (Eq. 1 in Hapke [2001]) and should be treated with care when comparing refmod outputs to results from other standard Hapke models.