Appendix T — Land: Mathematical Formulation

Implementation status

Model	DITL	PACE V3	PACE V4	Evaluation	Planned
—	—	—	x	—	—

This chapter provides the detailed mathematical formulation of the land surface reflectance model summarized in Appendix S. It describes the geometry conventions, Ross volumetric kernel, Li geometric-optical kernel, Fresnel reflection matrix, and polarized surface reflection kernel used in the RTSOS (Appendix B) and trained into FastMAPOL NN.

T.1 RTSOS Geometry Convention

The model follows the RTSOS light-ray convention, in which the zenith and azimuth angles are defined with respect to the direction of light propagation.

Let \(\theta_s\) and \(\theta_v\) denote the solar and viewing zenith angles, respectively, and \(\phi_s\) and \(\phi_v\) the corresponding azimuth angles. The relative azimuth angle is defined as

\[ \phi_r = \phi_v-\phi_s. \]

The RTSOS geometry convention is summarized below:

Quantity	RTSOS definition
Solar zenith	\(\theta_s > 90^\circ\)
Solar azimuth	\(\phi_s=0\)
Viewing zenith	\(\theta_v < 90^\circ\)
Relative azimuth	\(\phi_r=\phi_v-\phi_s\)

Under this convention, the relative azimuth corresponds to the following scattering geometries:

Condition	Scattering geometry
\(\phi_r=0\)	forward scattering
\(\phi_r=\pi\)	backscattering

T.1.1 Geometry Variables

For convenience, define the zenith-angle cosines as

\[ \mu_s = \cos\theta_s, \qquad \mu_v = \cos\theta_v. \]

The corresponding sine terms are

\[ \sin\theta_s = \sqrt{1-\mu_s^2}, \qquad \sin\theta_v = \sqrt{1-\mu_v^2}. \]

T.1.2 Scattering and Phase Angles

The scattering angle \(\Theta\) between the illumination and viewing directions is

\[ \cos\Theta = \mu_s\mu_v + \sqrt{1-\mu_s^2} \sqrt{1-\mu_v^2} \cos\phi_r. \]

The corresponding phase angle used in the BRDF formulation is

\[ \Theta_{\mathrm{phase}} = \pi-\Theta. \]

T.2 Ross Volumetric Kernel

The Ross volumetric kernel represents volumetric scattering within vegetation canopies (Roujean et al. 1992; Strahler et al. 1999; Wanner et al. 1995).

The RossThick volumetric kernel is

\[ K_{\mathrm{vol}} = \frac{ \left( \frac{\pi}{2}-\Theta_{\mathrm{phase}} \right) \cos\Theta_{\mathrm{phase}} + \sin\Theta_{\mathrm{phase}} } { |\mu_s|+|\mu_v| } - \frac{\pi}{4}. \]

The volumetric contribution to the surface BRDF is

\[ f_{\mathrm{vol}}K_{\mathrm{vol}}. \]

T.3 Li Geometric-Optical Kernel

The Li geometric-optical kernel represents shadowing and geometric effects associated with three-dimensional surface structures. RTSOS uses the Li-Sparse-Reciprocal formulation.

T.3.1 Projected Zenith Angles

Define the tangent terms

\[ |\tan\theta_s| = \frac{\sqrt{1-\mu_s^2}}{|\mu_s|}, \]

and

\[ |\tan\theta_v| = \frac{\sqrt{1-\mu_v^2}}{|\mu_v|}. \]

The projected zenith angles are

\[ \theta'_s = \tan^{-1} \left( \frac{B}{R}|\tan\theta_s| \right), \]

and

\[ \theta'_v = \tan^{-1} \left( \frac{B}{R}|\tan\theta_v| \right), \]

where \(B/R\) is the vertical-to-horizontal crown-shape ratio.

T.3.2 Projected Phase Angle

The projected phase angle \(\xi\) is defined by

\[ \cos\xi = \cos\theta'_s\cos\theta'_v + \sin\theta'_s\sin\theta'_v \cos(\phi_r+\pi). \]

T.3.3 Projected Distance

The projected distance is

\[ D = \sqrt{ \tan^2\theta'_s + \tan^2\theta'_v - 2\tan\theta'_s\tan\theta'_v \cos(\phi_r+\pi) }. \]

Define

\[ \sec\theta'_s = \frac{1}{\cos\theta'_s}, \]

and

\[ \sec\theta'_v = \frac{1}{\cos\theta'_v}. \]

T.3.4 Overlap Function

The auxiliary angle \(t\) is determined from

\[ \cos t = \frac{H}{B} \frac{ \sqrt{ D^2 + \left[ \tan\theta'_s \tan\theta'_v \sin(\phi_r+\pi) \right]^2 } }{ \sec\theta'_s+\sec\theta'_v }. \]

If

\[ |\cos t|>1, \]

the overlap function is set to

\[ O=0. \]

Otherwise,

\[ O = \frac{1}{\pi} (t-\sin t\cos t) (\sec\theta'_s+\sec\theta'_v). \]

T.3.5 Li Geometric Kernel

The final geometric-optical kernel is

\[ K_{\mathrm{geo}} = O - \sec\theta'_s - \sec\theta'_v + \frac{1}{2} (1+\cos\xi) \sec\theta'_s \sec\theta'_v. \]

The geometric contribution to the surface BRDF is

\[ f_{\mathrm{geo}}K_{\mathrm{geo}}. \]

T.4 Polarized Surface Reflectance

The polarized bidirectional reflectance distribution function is based on Fresnel reflection following Maignan et al. (2009).

T.4.1 Specular Incidence Angle

Using the scattering angle \(\Theta\) defined above, the incidence angle corresponding to reflection from a locally flat surface is

\[ \theta_1 = \frac{\pi-\Theta}{2}. \]

This angle is used to evaluate the Fresnel reflection coefficients.

T.5 Fresnel Reflection

Let the complex refractive index be

\[ n=n_r+i n_i. \]

The amplitude reflection coefficient for parallel polarization is

\[ r_p = \frac{ n^2\cos\theta_1 - \sqrt{n^2-\sin^2\theta_1} }{ n^2\cos\theta_1 + \sqrt{n^2-\sin^2\theta_1} }, \]

and that for perpendicular polarization is

\[ r_s = \frac{ \cos\theta_1 - \sqrt{n^2-\sin^2\theta_1} }{ \cos\theta_1 + \sqrt{n^2-\sin^2\theta_1} }. \]

The current implementation assumes

\[ n=1.5+0i. \]

At normal incidence, the Fresnel reflectance can therefore be approximated as

\[ \left(\frac{n-1}{n+1}\right)^2 \approx 0.04. \]

The Brewster-angle behavior is associated with the maximum polarization contrast represented by

\[ \frac{|A-B|}{A+B}. \]

T.6 Fresnel Mueller Matrix

Define

\[ A=\frac{1}{2}|r_p|^2, \]

\[ B=\frac{1}{2}|r_s|^2, \]

and

\[ \gamma=r_p^*r_s, \]

where \(r_p^*\) denotes the complex conjugate of \(r_p\).

The Fresnel reflection Mueller matrix is

\[ \mathbf{F} = \begin{bmatrix} A+B & A-B & 0 & 0 \\ A-B & A+B & 0 & 0 \\ 0 & 0 & \operatorname{Re}(\gamma) & -\operatorname{Im}(\gamma) \\ 0 & 0 & \operatorname{Im}(\gamma) & \operatorname{Re}(\gamma) \end{bmatrix}. \]

The element \(F_{11}=A+B\) represents the scalar Fresnel reflectance.

T.7 Polarization Kernel

The polarization kernel follows the formulation of Maignan et al. (2009):

\[ \mathbf{K}_{\mathrm{pol}} = \frac{ e^{-\tan\theta_1} e^{-v_{\mathrm{fac}}} }{ 4(|\mu_s|+|\mu_v|) } \mathbf{L}(\pi-i_2) \mathbf{F}(n,\theta_1) \mathbf{L}(i_1), \]

where \(\mathbf{L}(i_1)\) rotates the Stokes vector from the incident meridian plane to the scattering plane, and \(\mathbf{L}(\pi-i_2)\) rotates it from the scattering plane to the viewing meridian plane. The quantities \(i_1\) and \(i_2\) are the corresponding Stokes reference-plane rotation angles.

The empirical factor used in the current implementation is

\[ v_{\mathrm{fac}}=0.1. \]

The magnitude of the polarized surface contribution is controlled by \(B_{\mathrm{pol}}\):

\[ \mathbf{R}_{\mathrm{pol}} = B_{\mathrm{pol}}\mathbf{K}_{\mathrm{pol}}. \]

The NDVI-dependent scaling included in the original BPDF formulation is absorbed into \(B_{\mathrm{pol}}\) in the RTSOS implementation.

T.8 Complete Surface Reflection Matrix

Combining the scalar Ross–Li BRDF and polarized BPDF components gives

\[ \mathbf{R} = \left[ f_{\mathrm{iso}} + f_{\mathrm{vol}}K_{\mathrm{vol}} + f_{\mathrm{geo}}K_{\mathrm{geo}} \right]\mathbf{E} + B_{\mathrm{pol}}\mathbf{K}_{\mathrm{pol}}. \]

Equivalently, using the scaled parameterization,

\[ \mathbf{R} = f_{\mathrm{iso}}(\lambda) \left[ 1 + k_{\mathrm{vol}}K_{\mathrm{vol}} + k_{\mathrm{geo}}K_{\mathrm{geo}} \right]\mathbf{E} + B_{\mathrm{pol}}\mathbf{K}_{\mathrm{pol}}. \]

For scalar radiative transfer simulations, only the \(R_{11}\) element is used.

Maignan et al. 2009. Polarized reflectances of natural surfaces: Spaceborne measurements and analytical modeling. Remote Sensing of Environment, 113, (12), 2642–2650. https://doi.org/https://doi.org/10.1016/j.rse.2009.07.022.

Roujean et al. 1992. A bidirectional reflectance model of the earth’s surface for the correction of remote sensing data. Journal of Geophysical Research: Atmospheres, 97, (D18), 20455–20468. https://doi.org/https://doi.org/10.1029/92JD01411.

Strahler et al. 1999. MODIS BRDF/Albedo Product: Algorithm Theoretical Basis Document. 53.

Wanner et al. 1995. On the derivation of kernels for kernel-driven models of bidirectional reflectance. Journal of Geophysical Research: Atmospheres, 100, (D10), 21077–21089. https://doi.org/https://doi.org/10.1029/95JD02371.