Two Methods of Deriving Advanced Dynamic Site Equations Demonstrated on Inland Douglas-fir Site Curves
Keywords:
Site index, dynamic equations, integrability, base-age invariance, state-space models, UTADA, inland Douglas-firAbstract
Two complementary methods are used to derive base-age-invariant dynamic site equations. Method 1 tests the compatibility of separately proposed age-growth and site-response rates. Compatible rates are integrated to recover a common surface, from which the unobserved site coordinate is eliminated. An inland Douglas-fir example reconstructs the combined-base model of Cieszewski (2001) using Monserud's published coefficients; a second example demonstrates forward construction and the resulting parameter restrictions. Six solved routes relate this construction to invertible maps and scalar differential equations. Method 2 applies the Unified Theory of the Algebraic Differences Approaches (UTADA; Cieszewski 2021), starting with a rational yield–site relationship and introducing age through its parameters. This family is a subclass of compatible surfaces. The method recovers the three-parameter Cieszewski (2001) equation and its four-parameter branch with equal site exponents and nonzero coefficients. With three global parameters and estimated reference heights, the former approximates Monserud's five-parameter Douglas-fir-series curves with root mean square error (RMSE) of approximately 0.15 ft over ages 10–200. This is a deterministic curve approximation, not independent empirical validation. Mathematical compatibility, site inversion, and biological domain restrictions are distinguished from the statistical requirements of fitting and validating the equations.
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