UTADA: Unified Theory of the Algebraic Differences Approaches---Derivation of Dynamic Site Equations from Yield-Site Relationships

Abstract

Dynamic-equation-based self-referencing models of the form: Y=f(y₀,t₀,t) describe changes in Y as a function of two variables: one longitudinal variable t, and one unobservable cross-sectional variable X. Traditionally, X is represented implicitly by its substitution of a snapshot value of Y, (y₀), at an arbitrary value of t, (t₀).  The unobservable variable X represents the environment potential, which cannot be directly measured or precisely defined due to its extreme complexity and variability. While the most elusive and difficult in handling, X is the most critical variable of the dynamic site equations due to its disproportionate impact on the modeled dynamics, yet, all traditional approaches to such modeling are predominantly based on a detailed analysis of primarily longitudinal relationships Y=u(t), which subsequently, to be helpful in practice, are modified into the self-referencing forms. All the former approaches devote little to no effort to explicitly model the cross-sectional relationships governed by the unobservable variable X.

The presented approach unifies the modeling efforts of defining yield and site relationships equally by focusing primarily on direct mathematical formulations describing the theory of their interaction. This approach considers the variable t only in the secondary analysis, adding it to the model through modifying the final model parameters. Despite the somewhat elusive nature of exploring the unobservable properties of the site, the new approach appears to be highly empowering by analyzing more direct yet more robust relationships between Y and X as opposed to those between Y and t.