Appendix S — Land: Surface Reflectance Model
Implementation status
| Model | DITL | PACE V3 | PACE V4 | Evaluation | Planned |
|---|---|---|---|---|---|
| — | — | — | x | — | — |
FastMAPOL NNs are trained from the RTSOS (Appendix B) with the land surface reflectance model represents both the angular dependence and polarization of reflected radiation from the land surface. The model combines:
- the Ross–Li kernel-driven bidirectional reflectance distribution function (BRDF);
- a polarized bidirectional reflectance distribution function (BPDF); and
- Fresnel reflection physics.
The resulting surface reflection is represented by a 4 × 4 Mueller matrix and is implemented in the PACE simulator for radiative transfer simulations.
Detailed geometry conventions and mathematical formulations of the individual kernels are provided in Appendix T.
S.1 Surface Reflectance Representation
The bidirectional reflectance distribution matrix \(\mathbf{R}\) converts the downwelling irradiance vector \(\mathbf{F}_s\) to the reflected radiance vector \(\mathbf{I}\):
\[ \mathbf{I}(\theta_v,\phi_v) = \frac{1}{\pi} \mathbf{R}(\theta_s,\theta_v,\phi_r) |\cos\theta_s| \mathbf{F}_s(\theta_s,\phi_s), \]
where \(\theta\) and \(\phi\) denote zenith and azimuth angles, respectively; subscripts \(s\) and \(v\) denote the incident and viewing directions; and
\[ \phi_r = \phi_v-\phi_s \]
is the relative azimuth angle.
The surface reflection matrix is represented as
\[ \mathbf{R}(\theta_s,\theta_v,\phi_r) = \left[ f_{\mathrm{iso}} + f_{\mathrm{vol}}K_{\mathrm{vol}} + f_{\mathrm{geo}}K_{\mathrm{geo}} \right]\mathbf{E} + B_{\mathrm{pol}}\mathbf{K}_{\mathrm{pol}}, \]
where
\[ \mathbf{E} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}. \]
The first term represents the scalar Ross–Li BRDF and consists of three components:
- \(f_{\mathrm{iso}}\): isotropic reflectance;
- \(f_{\mathrm{vol}}K_{\mathrm{vol}}\): volumetric scattering, primarily representing vegetation-canopy effects; and
- \(f_{\mathrm{geo}}K_{\mathrm{geo}}\): geometric-optical scattering associated with surface structure and shadowing.
The second term,
\[ B_{\mathrm{pol}}\mathbf{K}_{\mathrm{pol}}, \]
represents polarized surface reflection. The polarization kernel \(\mathbf{K}_{\mathrm{pol}}\) is based on Fresnel reflection following Maignan et al. (2009), while \(B_{\mathrm{pol}}\) controls its magnitude.
S.2 FastMAPOL Parameterization
For parameter sampling and neural-network training, FastMAPOL uses a scaled representation that reduces the number of independent surface parameters:
\[ \mathbf{R}(\theta_s,\theta_v,\phi_r) = f_{\mathrm{iso}}(\lambda) \left[ 1 + k_{\mathrm{vol}}K_{\mathrm{vol}} + k_{\mathrm{geo}}K_{\mathrm{geo}} \right]\mathbf{E} + B_{\mathrm{pol}}\mathbf{K}_{\mathrm{pol}}. \]
The corresponding Ross–Li coefficients are
\[ f_{\mathrm{vol}}(\lambda) = f_{\mathrm{iso}}(\lambda)k_{\mathrm{vol}}, \]
and
\[ f_{\mathrm{geo}}(\lambda) = f_{\mathrm{iso}}(\lambda)k_{\mathrm{geo}}. \]
In this representation, \(f_{\mathrm{iso}}(\lambda)\) is spectrally dependent, whereas \(k_{\mathrm{vol}}\) and \(k_{\mathrm{geo}}\) are assumed to be spectrally invariant.
The primary FastMAPOL surface parameters are:
| Parameter | Product variable | Description |
|---|---|---|
| \(f_{\mathrm{iso}}(\lambda)\) | land_fiso |
isotropic reflectance coefficient |
| \(k_{\mathrm{vol}}\) | land_kvol |
scaled volumetric-scattering coefficient |
| \(k_{\mathrm{geo}}\) | land_kgeo |
scaled geometric-scattering coefficient |
| \(B_{\mathrm{pol}}\) | land_bp |
polarization scaling parameter |
This parameterization allows the spectral magnitude of the land reflectance to vary through \(f_{\mathrm{iso}}(\lambda)\) while describing its angular dependence using the spectrally invariant \(k_{\mathrm{vol}}\) and \(k_{\mathrm{geo}}\) parameters.
S.3 White-Sky Albedo
The white-sky albedo (WSA), also referred to as bihemispherical reflectance, represents the surface albedo under completely diffuse illumination. For the RossThick–LiSparse-Reciprocal BRDF model, WSA can be derived directly from the three BRDF kernel coefficients (Strahler et al. 1999):
\[ \alpha_{\mathrm{WSA}} = f_{\mathrm{iso}} + 0.189184\,f_{\mathrm{vol}} - 1.377622\,f_{\mathrm{geo}}. \tag{S.1}\]
The numerical factors represent hemispherical integrals of the corresponding Ross–Li kernels.
Unlike black-sky albedo, which depends on solar zenith angle, white-sky albedo is integrated over all illumination directions and therefore does not depend on a specific solar geometry. In the FastMAPOL MAPOL_LAND product suite, WSA is reported as land_white_sky_albedo.
S.4 Polarized Surface Reflectance
The polarized component of land surface reflection is represented by a BPDF based on Fresnel reflection. The model uses the Fresnel Mueller matrix together with transformations between the incident meridian plane, scattering plane, and viewing meridian plane.
The current implementation assumes a surface refractive index of
\[ n = 1.5 + 0i. \]
For this refractive index, the normal-incidence Fresnel reflectance is approximately
\[ \left(\frac{n-1}{n+1}\right)^2 \approx 0.04. \]
The magnitude of the polarized contribution is controlled by the retrieved parameter \(B_{\mathrm{pol}}\).
The complete Fresnel formulation, Mueller matrix, polarization kernel, and reference-plane transformations are described in Appendix T.
For scalar radiative transfer simulations, only the \(R_{11}\) element of the surface reflection matrix is used. $$