refmod.hapke.roughness#

Surface roughness corrections based on Hapke’s model.

This module provides functions to calculate corrections for macroscopic surface roughness, a key component in photometric modeling as described by Hapke.

??? info “References”

Hapke (1984)

Module Contents#

refmod.hapke.roughness.__f_exp(x, y)#

Helper function for microscopic roughness calculation.

Calculates exp(-2 * y * x / pi).

Parameters:

  • x (npt.NDArray) – Input array.

  • y (float) – Factor, typically related to cot(roughness).

Returns:

Result of the exponential function.

Return type:

npt.NDArray

refmod.hapke.roughness.__f_exp_2(x, y)#

Helper function for microscopic roughness calculation.

Calculates exp(-(y^2 * x^2) / pi).

Parameters:

  • x (npt.NDArray) – Input array.

  • y (float) – Factor, typically related to cot(roughness), which is squared.

Returns:

Result of the exponential function.

Return type:

npt.NDArray

refmod.hapke.roughness.microscopic_roughness(roughness, incidence_direction, emission_direction, surface_orientation)#

Calculates the microscopic roughness factor for Hapke’s model.

This correction accounts for the effects of sub-resolution roughness on the observed reflectance.

Parameters:

  • roughness (float) – The mean slope angle of surface facets, in radians. A value of 0 means a smooth surface.

  • incidence_direction (npt.NDArray) – Incidence direction vector(s), shape (…, 3). Assumed to be normalized.

  • emission_direction (npt.NDArray) – Emission direction vector(s), shape (…, 3). Assumed to be normalized.

  • surface_orientation (npt.NDArray) – Surface normal vector(s), shape (…, 3). Assumed to be normalized.

Returns:

  • s (npt.NDArray) – The microscopic roughness factor, shape (…).

  • mu_0_prime (npt.NDArray) – The modified cosine of the incidence angle ($mu_0^{prime}$), accounting for roughness, shape (…).

  • mu_prime (npt.NDArray) – The modified cosine of the emission angle ($mu^{prime}$), accounting for roughness, shape (…).

Notes

The calculations are based on Hapke (1984).

  • The terms $mu_0^{prime}$ (mu_0_s0, mu_0_s) and $mu^{prime}$ (mu_s0, mu_s) are calculated based on different conditions for incidence angle i and emission angle e:

    • For prime-zero terms ($mu_0^{prime(0)}$, $mu^{prime(0)}$ used in mu_0_s0, mu_s0): See Hapke (1984, Eqs. 48, 49).

    • For $mu_0^{prime}$ and $mu^{prime}$ when $i < e$: See Hapke (1984, Eqs. 46, 47).

    • For $mu_0^{prime}$ and $mu^{prime}$ when $i ge e$: See Hapke (1984, Eqs. 50, 51).

  • Input vectors (incidence_direction, emission_direction, surface_orientation) are normalized internally.

  • If roughness is 0, s is 1, mu_0_prime is cos(i), and mu_prime is cos(e).

References

Hapke (1984)