A LOG model of the following form was fitted to arrive at a preliminary estimate 

 of variance as a function of estimated cubic volume: 



LOG((Y - Y)^) = a + h LOG(Y) 



where: 



2 



(y - J) = estimated variance, residuals squared 



Y = field measured cubic volume 



Y = estimated cubic volume from previous regression. 

 Substituting V for (7 - J)^ and taking the antilog yielded: 



V = a?. 



Therefore, the estimated weight was: 



Since a is a constant, it was dropped from the weight without affecting the relationship 

 between the weights over estimated cubic volume. Dropping the constant yielded: 



After the initial estimate of the weight was determined, the model was fitted using 

 weighted least squares techniques. From the weighted model, a new estimate of variance 

 and corresponding weight was obtained by again fitting a LOG model. Having obtained a 

 new b value for the estimated weight, a new weighted regression was computed. The 

 fourth iteration of this process yielded weights and regression coefficients essentially 

 the same as the third iteration; this indicated the weights had stabilized and an 

 appropriate weight had been found. The final weight used for the weighted regression 

 was : 



W = 



1 



where: 



Y = estimated cubic volume from previous iteration 

 i> - 1.43 for pinyon and 1.40 for juniper 



Next another plot of the residuals was made over estimated cubic volume. The 

 residuals were weighted by the square root of the weight used for the final weighted 

 regression. The plot showed that the residuals: 



1. Appeared in a horizontal band 



2. Appeared balanced overall 



3. Appeared unbalanced close to the origin 



4. Appeared unbalanced for large estimated cubic volumes 



5. For juniper only, had a conspicuous absence of positive residuals at 15 ft 

 (0.42 m ) offset by an absence of negative residuals at 25 ft (0.71 m ). 



Items 1 and 2 indicate appropriate weights were used. Items 3 and 4 indicate an 

 intercept value should be included in the model. Item 5 indicates that two segments of 

 the juniper model do not fit the data trends, but the lack of fit in one segment is off- 

 set by a corresponding lack of fit in another. Another examination of the means used 

 to derive the original model revealed that to correct for the lack of fit in the two 

 areas would result in an unrealistic model form. Therefore, the original model form 

 was retained. 



4 



