This is a post written after Frouin and Pelletier (2015)'s tutorial, with the goal of giving it a reproducible and shareable platform for experimentation of the concepts laid out. A lot of the text describing the model was lifted off verbatim from the paper refereced above.

Problem

The inputs available are:

• $L_{toa}$: radiance measured by satellite ocean color sensor at a given band
• $F_s$: extraterrestrial solar irradiance (corrected for earth-sun distance)
• $\theta_s$: sun zenith angle

Expressing $L_{toa}$ in terms of bidirectional reflectance yields: $$\rho_{toa} = \pi \frac{L_{toa}}{F_s cos(\theta_s)}$$

Typically, $\rho_{toa}$ is modeled as:

$$\rho_{toa} = T_g \bigg[ \rho_{mol} +\rho_{aer} +\rho_{mol-g} +\rho_{aer-g} +\rho_{mol-aer} + \rho_gt_a + \frac{T_a\rho_f}{1 - S_a\rho_f} + \frac{T_a\rho_w}{1-S_a\rho_w} \bigg] $$

where,

• $T_g$: gaseous transmittance; accounts for absorption of photons by
  • nitrous oxide
  • ozone
  • oxygen
  • water vapor
• $\rho_{mol}$: molecular reflectance
  • accounts for multiple scattering of photons by molecules
• $\rho_{aer}$: aerosol reflectance
  • accounts for multiple scattering of photons by molecules
• $\rho_{mol-g}$:
  • accounts for interactions b/w molecules and photons reflected by a wavy surface
• $\rho_{aer-g}$:
  • accounts for interactions b/w aerosols and photons reflected by a wavy surface
• $\rho_{mol-aer}$:
  • accounts for coupling b/w scattering by molecules and scattering/absorption by aerosols
• $\rho_g$: sun glint reflectance
• $t_a$: direct transmittance
  • along the paths sun-to-surface & surface-to-sensor
• $T_a$: total (direct + diffuse) transmittance
  • along the paths sun-to-surface & surface-to-sensor
• $\rho_f$: backscattering of photons by whitecaps
• $S_a$: spherical albedo of atmosphere
  • accounts for successive photon interactions with the surface, the atmosphere, and the surface again
• $\rho_w$: water reflectance
  • accounts for photons backscattered by the water body