Exercise 4 Allometric Equations



Problem: You need allometric relations established for a species for which you have no direct information, but the site index and fraction of light that penetrates through the canopy at ages 20, 70, and 150 years and yield tables that relate diameter to stem volume.


Solution: Use published volume tables to create allometric relations between stem diameter and tree mass, and determine appropriate relation between stem diameter and growth in foliage through simulations with 3-PG.



  1. Open spreadsheet ‘Wind River Ps me’.  Observe Volume Table-derived data for site III Douglas-fir. Note formula displayed on graph shows allometric relationship between stem diameter and tree mass (with wood density assumed constant at 400 k/m3). This relationship is similar for all sites, but trees obtain larger diameters on better sites, so the equation can be extended.


  1. Note gray table that displays measured values of Leaf Area   Index, m2/m2, as well as modeled and measured values of stand basal area, m2/ha.

The LAI values were obtained using Beer’s Law and measured values of photosynthetically active radiation absorbed  (fPAR) by a pure canopy of Douglas-fir, where

fPAR = 1-e –(k)(LAI);  Example, If k = 0.5, LAI =6, then e-3.0 = 0.05 of PAR penetrates through canopy, fPAR = 1-0.05 = 0.95.


  1. The fraction of foliage growth to stem growth decreases as tree diameter increases.3-PG is configured to compare partitioning ratios between foliage and stems when trees are 2 cm (pFS2) and 20 cm in diameter (pFS20).

Use the following sequence of values and select the pair that best fit measured LAI and Stem Volume increment from Year 20-Year 160.

pFS2            pFS20                             pFS2            pFS20

1.0                                  0.8                                 1.3                   0.7

1.5                                  1.0                                 1.2                   0.6

1.3                 0.8                                 1.1                   0.5


  1. Open spreadsheet labeled ‘Conversions’ and insert best pair of values under ‘pFS2 and pFS20’ shown with background in yellow and numbers appearing in red.


E.    Record conventional form of allometric equation that predicts Foliage Mass, (kg) 

as a function of stem diameter (b) in cm = (aF)*(bnF). Record derived parameters here: 

       Foliage Mass, kg =___________________.