Adjusting a linear trend with known starting point

A compositional linear trend is determined by an initial composition a = [a1, …, aD], and a unitary composition p = [p1, …, pD] that defines the direction of the trend ( von Eynatten et al., 2003). In this BMS 961 study, we used the geometric mean of the corestone samples as a, and, as discussed below, used non-central principal component analysis to define p.
Any composition y on the linear trend defined by a and p is determined by a scalar t (t ∈ ℜ) such thatequation(1)y=a⊕t⊗p=Ca1p1t,…,aDpDt.
In Eq. (1), C is the closure operator and the symbols ⊕ and ⊗ represent perturbation and power transformation respectively. The scalar t is a measure of the change in composition relative to the original state as a result of some process. As such, t is a measure of the degree of intensity to which the process has been active ( von Eynatten, 2004 and von Eynatten et al., 2003). In terms of processes operating in the regolith, t represents a translation invariant measure of the weathering intensity factor as defined in A, CN, K subspaces ( von Eynatten, 2004 and von Eynatten et al., 2003).