REMOTE SENS. ENVIRON. 34:153-166 (1990) Modeling Bidirectional Reflectance of Forests and Woodlands Using Boolean Models and Geometric Optics Alan H. Strahler Center for Remote Sensing and Department of Geography, Boston University David L. B. Jupp Division of Water Resources, Commonwealth Scientific and Industrial Research Organization, Australia Principles of geometric optics and Boolean mod- els 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 in- struments. 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 collec- tion of discrete canopy envelopes with simple geo- metric shapes that are arranged on a contrasting background. The scene includes four components: sunlit canopy, shadowed canopy, sunlit back- ground, and shadowed background. The radiance or reflectance of the scene as a whole is modeled as the sum of the radiances or reflectances of the individual components as weighted by their areal proportions. The areal proportions of the compo- nents are determined by 1) principles of geometric optics as applied to the shapes of the canopy Address correspondence to Dr. A. H. Strahler, Center for Re- mote Sensing, Boston Univ., 725 Commonwealth Ave., Boston, MA 02215. Received 1 January 1990, revised 3 October 1990. 0034-4257/90 / $3.50 ©Elsevier Science Publishing Co. lnc., 1990 655 Avenue of the Americas, New York, NY 10010 envelopes and 2) Boolean models for random set overlap. These yield the expected proportions of the components as a function of angles of irradi- ance and exitance. At the leaf level, the canopy envelope can be treated as containing an assem- blage of leaves, and thus the radiance or re- flectance 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 direc- tions of irradiance and exitance, the model ac- counts 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 prin- ciples of geometric optics and Boolean modeling, they may easily be combined together in a single, two-stage model. Moreover, through further appli- cation of the mathematics of random sets, the aver- aging 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. 153