By examining the behavior of the coefficients across the various top diameter, it was 

 found that the model 



TL = aiD 



where 



D = Top diameter inside bark, in inches 



m 



n = An sp-NF specific value of 1.0, 1,5, or 2.0 



was applicable. This model assumes that the top can be approximated by a conic form 

 (Gray 1956; Husch 1963; and Assman 1970). The power on d.b.h., n, can be thought of as 

 determining the diameter of the top cone projected to ground level. This is because 

 total tree height is measured as the distance from the ground level to the tip of the 

 tree and consequently the relationship 



Ik 

 D 



m 



H 



(8) 



will hold for similar triangles only if D is directly proportional to diameter at 

 ground level. 



With this background as a basis, the model for unmerchantable volume is therefore: 



(9) 



where 



= Gross cubic foot volume in the top and stump for a specified top diameter. 



Analysis of residuals indicated that weighting was necessary to homogenize variance. 

 A procedure described by Hann and McKinney (1975) was therefore used to obtain the 

 weight : 



H 



m 



- 1 









d3 h 



m 



(10) 



Weighted least squares regression was used on each sp-NF combination to obtain the 

 regression coefficients for n equal to 1.0, 1.5, and 2.0. The final regression equation 

 was that which minimized RMSQR. Further refinement for a value of n was not necessary 

 because of the small differences in the RMSQR values of the three values tested. The 

 final merchantable gross cubic foot volume equations were visually checked by plotting 

 the data points versus representative curves from the equations over diameter by height 

 classes . 



A word of caution concerning the interpretation of the terms in this model is 

 appropriate. While the logic behind the two components is clear, the final equations 

 cannot be divided into components because of multicollinearity between the two (Kmenta 

 1971). This can be seen by comparing the second component of the unmerchantable model 

 to the model derived separately for stump volume. The coefficients are in the same 

 proximity but are not equal. However, the presence of multicollinearity, so long as it 

 is not perfect, does not bias the estimation of unmerchantable volume (Kmenta 1971). 



34 



