Posted: Thu May 24, 2012 8:23 pm
NTS,
I had lost track of John Eichholz's post of 2005 analyzing the implications of the 8.3-foot average difference between the trunk and crown baselines based on a sample of about 1,800 trees. The trees in that sample are mostly bigger, taller, older trees. Form the 8.3-foot average, John derived an average total crown-offset of 13 feet. I decided to reconstruct the derivation and include a diagram. They follow.
John extended the integration process over a range of 180 degrees. I show it for 90 degrees, and for the second quadrant. The results are the same for the 1st quadrant. In these kinds of problems, you have to express angles in radians as opposed to degrees. Thus the range of integration is show as 0 to π/2 instead of 0 to 90.
As the diagram indicates, this integration computes the average value of the y variable over the range of integration. At an angle of a = 0, the value of y = R. For a = π/2, y = 0. This is not exactly equal to the average value of D - L, but close enough. So y is a surrogate for D - L with the advantage that values of D and L do not have to be known.
The project that I am proposing that includes the spreadsheet in my previous post calls for us to add crown-offset determinations to our routine sin top-sin bottom measurements. This could be what opens the eyes of tape and clinometer users to the crown-offset problem and the need to see the problem as requiring two baselines. The idea may take time to catch on, but the logic of it is incontestable.
Bob
I had lost track of John Eichholz's post of 2005 analyzing the implications of the 8.3-foot average difference between the trunk and crown baselines based on a sample of about 1,800 trees. The trees in that sample are mostly bigger, taller, older trees. Form the 8.3-foot average, John derived an average total crown-offset of 13 feet. I decided to reconstruct the derivation and include a diagram. They follow.
John extended the integration process over a range of 180 degrees. I show it for 90 degrees, and for the second quadrant. The results are the same for the 1st quadrant. In these kinds of problems, you have to express angles in radians as opposed to degrees. Thus the range of integration is show as 0 to π/2 instead of 0 to 90.
As the diagram indicates, this integration computes the average value of the y variable over the range of integration. At an angle of a = 0, the value of y = R. For a = π/2, y = 0. This is not exactly equal to the average value of D - L, but close enough. So y is a surrogate for D - L with the advantage that values of D and L do not have to be known.
The project that I am proposing that includes the spreadsheet in my previous post calls for us to add crown-offset determinations to our routine sin top-sin bottom measurements. This could be what opens the eyes of tape and clinometer users to the crown-offset problem and the need to see the problem as requiring two baselines. The idea may take time to catch on, but the logic of it is incontestable.
Bob