Clinometer Calibration - John Eicholz Edward Frank Dec 15, 2005 18:25 PST

John, ENTS,

I have been looking over various posts on Instrument Calibration. This
one gives a nice mathematical summary of the theoretical errors in
height due to potential clinometer error. The point I want to make here
is that if as you wrote:

 I think I can prove mathematically that the error in tree height that results from each degree of clinometer error is approximately between 1.75% and 1.9% of the horizontal distance to the trunk...Because the factor: (1/cos(@))*(Sin(@)-sin(@+e)) is nearly a constant! Its range is a smooth progression from 1.74% at 0 degrees to 1.9% at 80 degrees.

Doesn't that mean that if you are taking clinometer readings of both the
base and the top, that the errors would subtract from each other? So
therefore maximum error would be 1.9% - 1.74% = 0.16% of the baseline
distance to the tree? (So long as it was off less than 1 degree or so)?
Therefore the actual error generated by a clinometer error would be more
in the range of 2 inches per 100 feet baseline rather than the 1.5 to 2
feet cited in the discussion?

Ed Frank

John Eicholz-
wrote (See Discussion Thread):
 Bob, Howard, Lee, all, (Summary: Theoretical means are used to show plausible error rates of +/-3 feet for typical measurements, with 3 out of 4 falling within +/- 1.5 feet) I've been doing a little work on theoretical rates of error. Hold on now, its not that bad. I think I can prove mathematically that the error in tree height that results from each degree of clinometer error is approximately between 1.75% and 1.9% of the horizontal distance to the trunk. To show this, let DT be the true distance to the tip and let DB be the true distance to the point directly below the tip. Let @ be the true angle to the tip, and let e be the measurement error of the angle. Let H be the true height of the tip above horizontal, and let H' be the height calculated from the true distance to the tip times the sine of the measured angle. Then H - H' is the height error due to angle error. It is true that H = DT*sin(@). We have decided that H' = DT*sin(@+e). It is also true that DT = DB*1/cos(@). So we can then be sure that: H - H' = DB*(1/cos(@))*(Sin(@)-Sin(@+e)). Why bother with all this? Because the factor: (1/cos(@))*(Sin(@)-sin(@+e)) is nearly a constant! Its range is a smooth progression from 1.74% at 0 degrees to 1.9% at 80 degrees. Because of this, we can safely say that an upper bound of clinometer error is 2% of baseline per degree of error no matter what the angle. I think I'm always within +/-0.4 degrees with my Suunto. This translates to +/-0.8 feet per segment on a 100 foot baseline, or +/-1.6 feet overall. This clinometer error is pointing and leveling error, but the same logic applies to systematic error.