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.


