OG Hemlock Modeling Bob Leverett
  Nov 21, 2007

TOPIC: More from the Formula Factory

== 1 of 1 ==
Date: Wed, Nov 14 2007 6:50 pm
From: dbhguru


Right. Will started the whole thing by sending me a spreadsheet that consisted of columns A - F, showing the taper of the listed hemlocks from their breast-high diameters to their diameters at 95.5 feet. Will was getting a feel for how quickly or gradually the giant hemlocks tapered in terms of how much diameter they had lost going from 4.5 to 95.5 feet. I added the remaining columns in order to predict what the taper would be were the whole form of the tree a paraboloid, a cone, half way between a paraboloid and a cone (or conoid as it is sometimes called), and 2/3rd paraboloid and 1/3 cone. The percentage columns show how much the the predicted deviates from the actual. The way I developed the predictive equations, I employed radius, thus the "r" columns. I then convert to circumference, thus the "c" columns. I frequently bounce between circumference, diameter, and radius without much thought given. I probably eed to be more consistent for the sake of the rest of you.

The R, H, h1, and h2 columns figure into the frustum formulas for 100% paraboloid and cone and the hybrids. The direction of the analysis suggests two things: (1) the huge hemlocks tend toward the paraboloid form, and (2) there are mavericks among them that don't behave themselves.


TOPIC: OG Hemlock modeling

   The following table shows the results of testing 38 modeled OG hemlocks against standard shapes. All trees are ones you've climbed and modeled. The notes in the table explain how well modeled hemlocks fit the paraboloid, cone, 50% paraboloid and 50% cone, 67% paraboloid and 33% cone, and 33% paraboloid and 67% cone. More to come.
Results of OG hemlock modeling                  
Sample Size % Fit Parabola Abs(%) Parabola % Fit Cone Abs(%) Cone Para -Cone   % 50-50 Abs(%)   50-50 Para - Cone   % 67-33 Abs(%)   67-33 Para - Cone  % 33-67 Abs(%)   33-67  
38 13.4 18.5 -32.5 32.5 9.6 16.1 -1.9 14.4 -17.2 20.4  
Notes: Objective is to predict circumference at a designated height using a variety of predictive models.
    utilizing the following formula. The designated height is 100 feet.  
  c = C*[f1*SQRT((H-h)/H) + (1-f1)*(H-h)/H]  
  c = the predicted circumference for the mix of paraboloid and cone  
  based on proportions f1 and (1-f1)  
  C = circumference at 4.5 feet  
  H = total height  
  h  = height at designated point where c is calculated.  
  % fit is calculated by:    [(Predicted - Actual)/Actual]*100  
  The best fit is 67% paraboloid and 33% cone  
  The 2nd best fit is 50% paraboliid abd 50% cone  
  The worst fit is the cone. It under-predicts in 100% of the cases.  
  The paraboloid over-predicts in 27 of 38 cases.  
  The 67-33 over-predicts in 15 of 38 cases.  
  The 50-50 over-predicts in 6 cases, meaning that it under-predicts   
     in 32 cases. This means that the correct mix needs more paraboloid