Modeling bidirectional reflectance of forests and woodlands using boolean models and geometric optics

Abstract

Principles of geometric optics and Boolean models for random sets in a three-dimensional space provide the mathematical basis for a model of the bidirectional radiance or reflectance of a forest or woodland as remotely sensed by radiometric instruments. The model may be defined at two levels: whole-canopy and individual-leaf. At the whole-canopy level, the forest scene is treated as a collection of discrete canopy envelopes with simple geometric shapes that are arranged on a contrasting background. The scene includes four components: sunlit canopy, shadowed canopy, sunlit background, and shadowed background. The radiance or reflectance of the scene as a whole is modeled as the sun of the radiances or reflectances of the individual components as weighted by their areal proportions. The areal proportions of the components are determined by 1) principles of geometric optics as applied to the shapes of the canopy envelopes and 2) Boolean models for random set overlap. These yield the expected proportions of the components as a function of angles of irradiance and exitance. At the leaf level, the canopy envelope can be tretaed as containing an assemblage of leaves, and thus the radiance or reflectance is a function of the areal proportions of sunlit leaf, shadowed leaf, sunlit background, and shadowed background. Because the proportions of scene components are dependent upon the directions of irradiance and exitance, the model accounts for the “hotspot” that is well known in leaf and tree canopies. Because both whole-canopy and individual-leaf models are driven by the same principles of geometric optics and Boolean modeling, they may easily be combined together in a single, two-stage model. Moreover, through further application of the mathematics of random sets, the averaging and variance that occurs when a scene is imaged by a sensor with a finite field of view may be accommodated. In addition, the models are capable of inversion, yielding estimates of size, shape, and spacing of crowns and/or leaves from directional and spatial statistics of remotely sensed radiances.