Eigenvalues and eigenfunctions of the two-stream radiative transfer problem in vegetation

Abstract:

The eigenvalue problem of the radiative transfer operator has exceptional importance for understanding the basics of vegetation-radiation interaction and allows efficient and robust methods to estimate the reflectance, transmittance and absorption of shortwave radiation in plant canopies. The eigenvalue problem for the general integro-differential radiative transfer equation is complex and difficult to interpret in terms of the effects observed in the real world, e.g., in the field of passive optical remote sensing. Therefore, we analysed the eigenvalue problem in a simple the two-stream approximation, often applied in remote sensing of homogeneous plant canopies, with known analytical solutions.

We derived the equation to calculate the eigenvalues for this important case and solved it numerically, as no analytical solution of the eigenvalue problem exists. We found up to an infinite number of eigenvalues and the respective eigenfunctions. The eigenfunctions are defined by a trigonometric or hyperbolic sine function for the downward radiation component, and by a sum of sine and cosine components for the upward direction. The eigenvalues are determined by the leaf area index, leaf reflectance to transmittance ratio and leaf orientation. Surprisingly, the first eigenvalue keeps changing even at large leaf area index values, where most other characteristics of the radiation field tend to saturate.

The results provide a simple approximation to calculate an important spectral invariant, photon recollision probability (p-value), related to the first eigenvalue, and the directionality of vegetation scattering. Eigenvalues and the p-value are mostly structural characteristics of vegetation with a minor dependence on optical properties of canopy elements. The results obtained help us to better understand the properties of radiative transfer in vegetation and parameterize it for remote sensing applications.