Ed,
I see what you are getting at. I'm not sure how you got the 2.62
feet
and the subsequent errors, though. Actually, the situation is
dependent
on you assumptions. If you are starting with a given distance
measurement to the top that is different from starting with a
given
distance horizontally. I have set up a spreadsheet with the
latter
assumption. This addresses the situation which I feel is most
practical:
You are standing a set distance from the tree and you want to
know how
much height error you will have if you don't read your
clinometer
accurately. (You might regret getting me started, but that is a
different thing.)
Actually, the error is nearly constant in this scenario,
regardless of
the angle. I have tried to make the spreadsheet as clear as I
could. If
you (or anyone) comes up with different numbers it is because we
are
making different assumptions. Not wrong, just different. I would
like to
continue with this if there are scenarios that people want to
explore.
As Colby said in one of the posts in this series (Nov 6, 2003),
" I
finally decided that I had only paid $199 for the rangefinder,
and it was
not realistic to expect absolute accuracy. If it was supposed to
be
accurate to a yard, and it was consistent to several inches,
that wasn't
bad. Calibrating for all sorts of possible deviations would be a
nuisance
and might add as many (or more) errors than it corrected. "
The
instruments are only "so" accurate, so don't try to
split hairs. But, I
still think an awareness of the source and magnitude of errors
as they
propagate in our calculations is useful.
Also see many earlier posts in November, 2003 on this subject.
John
Eichholz
Error xls file analy20041124b.xls
File as a html webpage
(noninteractive)
p.s. if you don't have Excel, here are the formulas:
2004 11 24
Analysis of height error due solely to angle measurement error
using the
sine*distance (ENTS) method.
Scenario: You are at a fixed distance from a tree. (DB) You
accurately
measure the distance to the top. (DT)
You measure the angle to the top, or any other point on the
tree, but
there is an error in your measurement (e).
Since we are focusing on angle error, we assume the tree is
vertical.
The only difference would be a change in the baseline. The
"error as a
percent of baseline" is still the same.
DB is the true horizontal distance to the point directly beneath
the top.
DT is the true distance to the top.
a is the true angle to the top.
e is the angle measurement error.
H is the true height.
H' is the height calculated using the erroneous angle
measurement.
Facts: DT = DB/cos(a); H = DT*sin(a); H' = DT*sin(a+e)
Formula used: HH' = DT*sin(a)  DT*sin(a+e) =
(DB/cos(a))*(sin(a)sin(a+e))
My conclusion: At a given baseline, the height error due to
angle
measurement error is nearly the same no matter what the angle.
Also, at 50 yards horizontal, the height error due to a 0.4
degree
clinometer error is about 1 foot.
Enter the angle error in b6. Enter the baseline distance to the
tree in b7.
angle error (degrees)>> 0.4 e
true baseline distance to point below top (yards)>> 50 DB
Here are a series of angles and the height error that results
a a+e DT H H' HH' (HH')/H
true angle true angle + error (yards) true height (feet) calc
height
(feet) height error (feet) hgt error percent of baseline
10 9.6 50.77 26.45 25.40 1.0478 0.70%
5 4.6 50.19 13.12 12.08 1.0475 0.70%
0 0.4 50.00 0.00 1.05 1.0472 0.70%
5 5.4 50.19 13.12 14.17 1.0469 0.70%
10 10.4 50.77 26.45 27.50 1.0465 0.70%
15 15.4 51.76 40.19 41.24 1.0462 0.70%
20 20.4 53.21 54.60 55.64 1.0459 0.70%
25 25.4 55.17 69.95 70.99 1.0455 0.70%
30 30.4 57.74 86.60 87.65 1.0451 0.70%
35 35.4 61.04 105.03 106.08 1.0446 0.70%
40 40.4 65.27 125.86 126.91 1.0441 0.70%
45 45.4 70.71 150.00 151.04 1.0435 0.70%
50 50.4 77.79 178.76 179.81 1.0428 0.70%
55 55.4 87.17 214.22 215.26 1.0420 0.69%
60 60.4 100.00 259.81 260.85 1.0409 0.69%
65 65.4 118.31 321.68 322.72 1.0394 0.69%
70 70.4 146.19 412.12 413.16 1.0371 0.69%
75 75.4 193.19 559.81 560.84 1.0335 0.69%
80 80.4 287.94 850.69 851.72 1.0265 0.68%
85 85.4 573.69 1714.51 1715.51 1.0054 0.67%
90 90.4 816227613880954000.00 2448682841642860000.00
2448623168969300000.00 59672673563136.0000 39781782375424.00%
