Deriving compatible taper functions from volume ratio equations based on upper-stem height

Abstract

A very common model for prediction of tree stem volumes to upper-stem height or diameter limits is the use of a merchantable to total volume ratio function multiplied by a total stem volume function. Many users of these prediction systems also desire taper equations that can predict heights to upper-stem diameters. While taper equations compatible with volume ratio equations have been used for many years, compatible taper equations from volume ratio equations that are functions of upper-stem height have been used infrequently. Yet many studies have indicated that height-based ratio equations perform well and frequently have statistics of fit that are comparable with diameter-based volume ratio equations. Compatible taper equations derived from height-based ratio equations are presented here. The methodology that uses height-based merchantable to total volume ratios does not require the solution of a differential equation after differentiating the height-based volume ratio, as is necessary when using the method of deriving taper equations from diameter-based merchantable to total volume ratios. This could be an advantage depending on the complexity of the ratio function. Example taper equations fitted to loblolly pine (Pinus taeda L.) data from the southeastern USA and the state of Oklahoma, USA, indicate good fit to these data, whether fitted directly to taper data or implicitly by using parameters fitted to volume ratio data.

Taper and Volume Systems Based on Ratio Equations for Pinus pseudostrobus Lindl. in Mexico

“…The procedure for fitting systems based on volume ratio equations implies that the taper equation is derived from the ratio equation, thus guaranteeing compatibility of the components [11]. This methodology allowed simultaneous fitting of the components of Vs, Vt and Vb.…”

“…Vm (m 3 ) is the cumulative volume up to height h (m) above the ground, Vs (m 3 ) is total stem volume, and p is the ratio between commercial height and total height (p = h H ) [10,11]. Compatible taper and volume systems included five equations or components: Vm, d, Vs, Vt and Vb.…”

“…The Vm equations were developed from volume ratio functions, which describe changes in the ratio of cumulative volume to total stem volume over relative stem height [10], whereas the d equations correspond to the derivation of the expression of cumulative volume [11]. The systems were named S1, S2, S3, S4, S5 and S6 ( Table 2).…”

“…The equation of commercial height (h) is given by solving the d equation. For more details, review the study conducted on these types of equations [10,11]. Table 2.…”

“…Hence, compatible taper equations were developed; they were derived from volume ratio functions based on relative tree height [11]. The compatible taper equations derived from ratio functions based on relative diameters have been used for decades, but those based on relative height have been infrequently used [10,11].…”