A system of nonlinear simultaneous equations for crown length and crown radius for the forest dynamics model SORTIE-ND

A system of equations was developed to predict crown length (CL) and crown radius (CRAD) for trees in structurally complex stands. The equations address two problems that often arise in crown allometry. First, relationships between the main stem and the crown are likely to change with intertree competition. Therefore, explicit measures of density were used along with main stem measurements as explanatory variables. Second, the physiological relationship between CL and CRAD is often overlooked when modeling crowns. This relationship is incorporated through the use of a simultaneous system of equations. Parameters were estimated using nonlinear three-stage least squares (N3SLS) in which first-stage equation estimates of CRAD are used to estimate CL and vice versa in the second and third stages of N3SLS. The equations were fitted and validated for four species: lodgepole pine (Pinus contorta Douglas ex Louden var. latifolia Engelm. ex S. Watson), hybrid spruce (Picea engelmannii Parry ex Engelm. × P. glauca (Moench) Voss), Douglas-fir (Pseudotsuga menziesii (Mirb.) Franco var. glauca (Beissn.) Franco), and trembling aspen (Populus tremuloides Michx.). The intent is to use the equations as alternatives to the crown equations in the spatially explicit forest growth model SORTIE-ND that use only main stem variables in estimating crowns over time.