Sept 17, 2005
Bob and John,
I am trying to understand the mathematics of the calculations Bob made to get an average offset or 8.3 feet through the mathematical processing of 1800 tree survey data sets. Please bear with me.
Say you had a plane with a circle centered at the x-y intersection with a radius of Q. Then draw 100 radii lines evenly spaced around the circle. The deviation from the y axis could be calculated for each point as the absolute value of sin theta Q where sin theta was measured as the angle from the y axis. If you added all of these deviations from the y axis and then averaged them you would get an average deviation from the y-axis at all of these angles. As I envision it this number should be proportional to the actual radius of the circle. This proportion should be constant regardless of the radius of the circle. I am sure there is a simple and elegant integral to calculate out the average deviation from y in a circle centered on the y-axis, but my math isn't up to the task. [Could you enlighten me?] What this would mean is that by knowing the average of these deviations, the actual radius of the circle could be calculated.
In Bob's case of 1800 trees, the deviation from the y-axis (perpendicular to the line of sight with the z-axis running through the center of the base of the tree) can be easily calculated for every tree to see how far off the tops are in the x direction along the line of sight from the base of the tree. With 1800 samples, and a reasonable assumption that they the tops are randomly distributed in direction offset from the center of the base of the tree, the average deviation from the y-axis should be proportional to the average offset distance of the top in any direction from the true center of the base of the tree.
Essentially it means by knowing the offset in the x-direction you could calculate the actual average offset of the tops without knowing the amount of offset in the y direction.
Does what I am saying make sense? Maybe this is what you have already done. How about a little help here guys.
Sept 17, 2005
I am glad to try and clarify the result of the calculation you wish to achieve. First let me say you have a clear grasp of the problem and it is a more interesting problem than it first appears. By solving it we can determine the theoretical average lean of the trees in Bob's database, even though this figure is not accurately calculated for each tree, by making a statistical assumption that the trees' lean is symmetric to the point of measurement. Quite an accomplishment of abstraction, it would appear. Thank you for the interesting problem.
Actually, the math is quite simple. The observed offset from vertical (as seen from the point of measurement) is proportional to the COSINE of the angle. Assume a radius of 1. The average offset would be the INTEGRAL of abs(cosine theta) around the circle, divided by the length of the circle. We can evaluate this as twice the integral of (cosine theta) from -pi/2 to pi/2, divided by 2*pi (the length of the path of integration). I apologize if this doesn't make sense, but
mathematically it is accurate.
The integral of the cosine function is the sine function. We thus evaluate the integral as 2*[sin(pi/2) - sin(-pi/2)]/(2*pi). , or 0.6366 and change.
Using an average observed offset of 8.3 feet yields a projected average true offset of 8.3/.6366, or about 13 feet.
I believe we can conclude by saying that the trees measured in Bob's database appear to have an average offset of 13 feet from bottom to measured tip.
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