Difference between tangent and sine-based calculations Error Spreadsheet   Robert Leverett
  Nov 26, 2004 10:27 PST 


      Our discussions on the difference between sin and tangent-based
calculations and Will Blozan's recent suggestion to compare what the
tangent-based height calculation would have been in our measurements by
treating the point shot with the laser to have been vertically over the
base led me to run comparative calculations on a sample of 1,330 trees
measured with laser and clinometer. From an original larger selection, I
eliminated those measurements in which the measurer would have been
closer to the trunk than a chain's distance (66 ft) or at an angle of to
the crown of 65 degrees or more. I considered these two situations to be
improbable measurements from the standpoint of an experienced user of
the tangent (% slope) method.

The results are as follows:

Differences between sin-based and tangent-based calculations for tree
height. Covers above eye calculations. Differences are in feet.

Diff       0-0.99  1-1.99  2-4.99  5-9.99  10-14.99  15-24.99  25-65.99  TOT
Count     123      131      333      368       167        138           70       1330
Pct Tot    9.2%   9.8%   25.0%  27.7%    12.6%    10.4%      5.3%
Cum Pct  9.2% 19.1%   44.1%   71.8%   84.4%    94.7%    100.0%

Max diff:      65.32
Ave diff:        8.30

     The average of the absolute value of the differences is 8.3 feet.
The max difference is a whopping 65.3 feet. These statistics plus the
ABOVE table of percentages tell most of the story. It is certainly
possible to reduce the error of the tangent-based calculations by
cross-triangulating the crown, but even that method has its limitation,
visibility of the same crown point from sufficiently separated spots
being the primary. Will Blozan and I did the crown cross-triangulating
for several years and described the method fully in: "Stalking the
Forest Monarchs - A Guide to Measuring Champion Trees".

      The next series of charts will look more closely at subsets of
these measurements. Please stay tuned. The above is enough for now.

      I'll send the Excel spreadsheet holding the 1330 measurements to
Ed in a couple of days.

RE: Difference between... MY REPLY   Will Blozan
  Nov 27, 2004 14:52 PST 


The numbers came out kind of distorted in my email. However, the errors are
very significant, especially given that in our ENTS measurements the true
top has already been identified- a task that can take hours with
cross-triangulation. Impressive and compelling!

I think we should present a synopsis of our findings to the website and
deliver it to certain parties for their "review". We have a strong case, one
that should be spread to the public. I am ready for ENTS to make more of an
impact in the tree related "playing fields". I also think a scientific push
can be made as well as in ecological mensuration (nest heights, canopy
layers, etc.) and in more utilitarian fields such as forestry.

Also, how accurate are waterfall heights in the East? I have heard some
seemingly outrageous heights claimed for waterfalls, and have often wondered
about the accuracy. I suspect I do not want to know...

RE: Difference between tangent and sine-based WILL IS ALMOST READY!   Will Blozan
  Nov 27, 2004 19:36 PST 

Bob, John, ENTS,

I have 10 trees of 10 species set up in EXCEL for calculations. I have
chosen 5 gymnosperms (e. hemlock, C. hemlock, red spruce, loblolly pine, and
white pine) and 5 angiosperms (tuliptree, white ash, red oak, sycamore, and
black birch).

To make my comparisons similar or identical to yours I need to know if you
have corrected for slope in the conventional height calculations. I have set
it up both ways, slope corrected and not, but I have some pretty serious
angles on some of the trees. I could select less steep angles, but the steep
tree measures are good to know as well, and are not undoable with cross

What would be good to know also is to "cross-triangulate" with the laser
from two 90 degree opposing angles (same top) to map the top relative to the
base, and then calculate the range of error over 360 degrees around the
tree. We would need to pick a "pointy topped" tree for this with a prominent

I have also looked at the average lean, both + and - from the observer, and
even with 10 samples the average is usually less than 2 feet. I find this
very interesting, and the greatest lean is actually in the e. hemlocks (~5
feet). Some of my red oaks have a 30' "lean"!

Also, I tend to shoot the base at the midslope on the side of the tree, not
the closest portion. This way, in my calculations, I do not need to correct
for the radius of the trunk.

More to come!