New measurement method   Robert Leverett
  Jan 11, 2007 06:51 PST 

ENTS,

     As I mentioned in a prior e-mail, at the April 2007 ENTS rendezvous
at Cook Forest, I will be giving an update presentation on our ENTS
measurement methods to include new measurement techniques. As an example
of what will be included, what follows is a new formula for measuring
tree height above eye level using nothing but a clinometer and tape
measure. The advantage of the method is that it avoids the pitfall of
the top being measured not being directly over the base. However, there
is an assumption that absolutely must be met. The measurer is on a patch
of level ground that allows forward and/or backward movement. The method
works as follows.

    From a vantage point, either the angle to the crown point or the
slope percent is determined (left scale angle, right scale slope percent
on the Suunto clinometer with those scales). The measurer then moves
directly forward or backward a known distance to a second point where
the crown point is still visible. The crown point, measurer's original
position, and measurer's new position all fall in the same line. The
measurer's eye level remains unchanged. The measurer takes the angle or
slope percentage of the crown point from the new position. The measurer
takes the distance between the two points of measurement. This is a
level distance.

    So we end up with three measurements: two angles or two slope
percentages from the eye to the crown point and the linear distance
between the two points of measurement. If the assumptions are fulfilled,
the following formula calculates the height of the crown point above eye
level.

Let:
    D = level distance between the two measurement points

         A = angle of eye to crown point from nearer location   

    B = angle of eye to crown point from more distant location

    H = height of crown point above eye level

Then:

    H   = [(D)tan(A)tan(B)] / [tan(A) - tan(B)]


    The algebraic derivation of the above formula along with others will
be handed out at the Cook Forest rendezvous.

     Note that the reading of the right scale of the Suunto clinometer
with a percent scale is just the tangent of the angle expressed as a
percent. The measurer need only move the decimal place two positions to
the left to get the tangent. Consequently, no trigonometry tables need
to be accessed. The calculations can be done by the simplest
calculators.

    The dangers of carelessly applying this technique are the same as
the simple baseline to trunk and slope angle to crown. If the measurer
is indeed on level ground then adding height to eye level gives the full
height of the crown point above the tree's base. However, if that is not
the case, then the procedure for shooting the crown can be applied to
shooting the base. I suspect that many would-be measurers will find the
procedure to already be too calculation intensive, but it avoids the
pitfall of assuming the crown point is directly over the base - the most
common source of tree height calculation errors.

    Note that no assumption has to be made as to how distant the trunk
is. There is no baseline from measurer to trunk. So this method can be
applied where getting to the base of the trunk is difficult such as when
there is an intervening stream. Of course, the base of the trunk has to
be visible.

     A caution is immediately in order for the users of the percent
slope scale of a clinometer. It cannot be read to a sufficient number of
decimal places to replace actually calculating the tangent of the angle.
However, the process, as described above, still provides a better height
approximation than ignoring the horizontal offset of the crown-point
from the tree's base. As is frequently pointed out, measurers armed with
only clinometer and tape measurer usually apply the vertical telephone
pole in the level parking lot model that is the basis for the
height-measuring instructions accompanying clinometers and hypsometers.

     The fundamental lesson to be learned in the application of all
these measurement techniques is that the measurer MUST understand the
assumptions made behind the model and verify that those assumptions are
fulfilled. Ignorance is not bliss. Our ENTS workshops will increasingly
focus on different models, the assumptions behind the models, and the
magnitude of the errors introduced when assumptions are not fulfilled.

    This new method in no way replaces the much preferred sine top-sine
bottom method, but should be useful as an interim method for folks who
possess a clinometer and tape measure and who are saving their pennies
for a laser rangefinder.

Bob

Robert T. Leverett
Cofounder, Eastern Native Tree Society
Re: New measurement method   Jess Riddle
  Jan 11, 2007 19:57 PST 

Bob,

Seeing a new strategy for measuring trees proposed is exciting.
When I saw how you had the triangles set up, I had to try and derive
the formula myself. What I came up with produces the same results as
the formula you provided, but is much messier to use. In checking the
formulas, it appears to me that the distances involved with applying
this technique are realistic for practical application. Shifting 15
or 20' will provide a reasonably large angle difference to work with
for a 100' tall tree.

The one part of your explanation that I didn't follow regarded
calculating the tangents instead of reading them directly off the
clinometer. How does using a calculator to find the tangents increase
the resolution of the clinometer? If you calculate the tangents from
the angle in degrees read off the clinometer, aren't you just
'rounding' to some irrational number that you obtained after rounding
the degrees?

This technique's ability to preclude projecting heights to an
illusionary top (produced by the high point not being over the base),
and doing so without complicated calculations, give it great
potentially. Certainly, the requirement for level ground will
restrict the use, but many state big tree programs could still improve
a significant proportion of their data if they used this method
instead of the standard tangent method where applicable.

Jess
Back to Jess   Robert Leverett
  Jan 15, 2007 05:46 PST 

Jess,

   You are absolutely correct. Good thinking. I thought about the
rounding error situation after I submitted the e-mail. Having all those
extra digits from a tangent table based on an approximate clinometer
reading is of no real value to accuracy. Two heads are definitely better
than one, especially if one of the heads is yours.

   I had intuitively settled on 20 feet as the baseline between the two
angles, but had not done any calculations to investigate sensitivity. I
have a process in the mill for dealing with the situation where the
baseline between the two measurement points is not level. The math is
messy and I doubt that anyone would want to adopt it, but I'll submit it
nonetheless.

Bob
Dendromorphometry extended   Robert Leverett
  Jan 19, 2007 05:23 PST 

ENTS,

   Not all ENTS-developed formulas are destined for high use in the
field, but nonetheless form part of our "fleshing out" of the new
discipline of dendromorphometry.

   I recently gave a formula for calculating the height of a crown-point
above eye level that requires only a clinometer and tape measure. The
assumptions that must be fulfilled for the formula to be used are: two
clinomter angle measurements are made of the same crown-point from
locations that are on the same level. The crown-point, eye position of
the first measurement point, and eye position of the second measurement
point all lie in the same vertical plane.

If:

   a1 = clinometer angle to crown-point from nearer measurement point
   a2 = clinometer angle to crown-point from more distant measurement
point
   D = level distance between two measurement points (eye position to
eye position)
   H = height of crown-point above eye level

then:

   H = [(D)tan(a1)tan(a2)]/[tan(a1) - tan(a2)]

   Since, the right scale of a clinometer with a degrees and percent
slope scale is just the tangent of the angle x 100, one can use the
above formula without trigonometry tables. That makes it attractive to
people who want to avoid trigonometry. Jess Riddle independently tested
the formula and stated that a baseline of 15 or 20 feet should be
sufficient to get sufficient angle differentiation to allow the formula
to do its work.

   The toughest condition to be met is the requirement that both eye
positions be at the same level. In mountainous terrain, this condition
often cannot be met will keeping the two eye positions and the
crown-point all in the same vertical plane. One might be able to move
laterally at the same level, but that would violate a key requirement
that the three points lie in the same vertical plane.

   Suppose that the second observation point is not at the same level as
the first. Can we still solve the problem? Suppose the second, more
distant eye position is at a higher elevation. In this case more
triangles that must be constructed to model the situation and the
calculations become more involved, as can be seen below.

If:

   a1 = clinometer angle to crown-point from nearer measurement point
   a2 = clinometer angle from eye-level at second measurement point down
to eye level at first measurement point (note that a2 here is not the
same as a2 in the first formula)
   a3 = clinometer angle to crown-point from more distant measurement
point (at the hiher elevation)
   d5 = straight-line distance from the eye at first measuremet point to
second the eye at the second measurement point. The line d5 is sloped.   
   

   Remember that the crown-point, first eye position, and second eye
position all lie in the same vertical plane. If the assumptions are met,
then the height of the crown-point above the first eye position is:

   h = [tan(a1){d5cos(a2)tan(a3) + d5sin(a2)}] / [tan(a1) - tan(a3)]

   This toad strangler of a formula is not likely to take the measuring
world by storm, but it can be used by those who want as many tools in
their toolkit as they can get.

   Both formulas with accompanying diagrams will be thoroughly explained
at the April ENTS event at Cook Forest.

Bob


Robert T. Leverett
Cofounder, Eastern Native Tree Society
Refinements on tangent-based calculations   Robert Leverett
  Jan 29, 2007 05:26 PST 

ENTS,

   What follows below is a more complete treatment than was previously
given for measuring the height of a crown-point above eye level using a
clinometer and a baseline between two measurement points that are
aligned with the crown-point. The procedure and formulas that are
presented below are not meant as substitutes for sine-based measuring,
but the tangent-based procedure can fill the accuracy gap that
ordinarily accompanies the typical use of a clinometer.

    First a quick review. If the measurer is able to see top of the tree
and the top is directly over the base, then a baseline from eye to trunk
and the slope % to the crown top is all one needs to compute height
above eye level. A similar procedure can be used for height below eye
level. The slope percent converted to the equivalent decimal value times
the level baseline distance from eye to trunk gives the height of the
tree above eye level. This is the standard clinometer-baseline procedure
for instruments that have a percent slope scale. The two scales for a
clinometer that make the most sense are degrees and percent slope. Of
the two, degrees is the most important.

    But what if the crown-point is not directly over the base? What can
be done to get an accurate height measurement if one does not know where
a plumb line from the crown-point to the ground would fall? Some timber
specialists develop a good eye and can make educated guesses, but that
is risky, especially for broad-crowned hardwoods. There is the
crown-point cross-triangulation method as described in our guidelines,
but that process is difficult to implement without two long tapes, an
assistant, and continuous visibility of the crown-point being measured
as one moves from one location to another. However, there is another
procedure that one can apply that uses just the clinometer and tape.

   The measurer positions himself/herself at a spot where the target is
visible and shoots the angle to the target (or percent slope). The
measurer then moves back to a second vantage point and takes a second
reading with the clinometer. The height of the crown spot above the eye
position at the first vantage points is calculated by using any of the
three formulas below.

DEFINTIONS:

d5 = straight line distance between positions of the eye at first and
second vantage points. This is the baseline. Note that it does not go
from measurer to the trunk, which is the traditional baseline.

a1 = angle to crown spot an closer vantage point

a2 = angle to crown spot at farther vantage point

a3 = angle between eye positions at the two vantage points
(You may nned a pole to keep track of the locations.)

FORMULAS FOR THE THREE SITUATIONS:

(1). Baseline between two vantage points is level

h = [(d5){tan(a1)tan(a2)}] / [tan(a1) - tan(a2)]

(2). Baseline is not level, more distant point at higher elevation

h = [(d5)tan(a1){sin(a2)+cos(a2)tan(a3)}] / [tan(a1)-tan(a3)]

(3). Baseline is not level, more distant point at lower elevation

h = [(d5)tan(a1){sin(a2)-cos(a2)tan(a3)}] / [tan(a3)-tan(a1)]

    I used a combination of brackets, braces, and parentheses to make
the formulas clearer.

    It's apparent that cases (2) and (3) lead to awkward calculations. I
doubt many people will want to use these formulas. However, case (1) is
more straightforward, and again, please note that the distance from the
measurer's position to the tree does not have to be determined. This may
come as a surprise. Also note that the crown-point, eye position #1, and
eye position #2 all must lie in the same vertical plane. Also, please
bear in mind that the tangent of angles a1, a2, and a3 can be determined
directly from the right scale of a Suunto clinometer with a degrees and
percent slope scale. You simply divide the percent slope read from the
right scale by 100 to get the tangent of the angle. Slope, defined as
the rise divided by the run IS the tangent of the angle. The challenge
in applying this technique is find a place to get a level baseline. This
new procedure is especially useful when the measurer cannot get to the
trunk of the tree (across water, surrounded by briars, poison ivy,
etc.).

DISCUSSION ABOUT THE PROCEDURE:

    I highly doubt that the above procedure is going to take the
measuring world by storm. Those who have a laser rangefinder,
clinometer, and scientific calculator can apply the much superior sine
top-sine bottom method. Those who don’t need high accuracy, but do need
great efficiency in the forest, will likely not employ a technique that
is calculation-intensive. Those who nourish delusions about tree form
aren’t reachable. However, there is a third group of intrepid souls who
want to measure the best way, but are operating on a razor thin budget.
They are saving their pennies for the needed equipment, but in the
interim, they want to be out in the forest measuring trees. The above
procedure could be of help.

Bob


Robert T. Leverett
Cofounder, Eastern Native Tree Society