Angle
error discussion 
Robert
Leverett 
Apr
17, 2007 06:30 PDT 
ENTS,
The ENTS gathering at Cook Forest SP
draws near. I wish Monica and I
had more time to spend at Cook Forest this year, but we don't.
My focus
will be on my Saturday morning dendromorphometry presentation
and on
Will’s afternoon climb and modeling of the Seneca Hemlock. I
look
forward to the Saturday evening presentations. They have been
exceptionally good over the years.
Before proceeding, anyone who does not
like the mathematics of tree
measuring, best to stop at this point. The rest would be
torture.
In terms of the dendromorphometry,
I’ve been concentrating more on
the effect of angle and distance errors resulting from
tangentbased
versus sinebased measurement models. Our usual objection to
tangentbased models for measuring tree height is that they are
often
misapplied. The spot taken as the highest sprig and measured
anglewise
is seldom directly over the base, leading to crownpoint offset
errors.
But for the sake of argument, suppose the top of the tree is
directly
over the base, the trunk is vertical, and the measurer’s
apparatus
measures distances accurately. Further suppose that the
measurer’s
clinometer reads high by one degree, but that the clinometer
problem is
unknown to the measurer. We can check the height error that
results from
a 1degree error in the angle for both the sine and tangent
methods. We
can, of course, compute the errors directly, but a useful
approximation
of the error can be obtained by using the concept of
differentials from
differential calculus. We examine changes in a dependent
variable based
on small changes in an independent variable for a continuous
function of
one dependant and one or more independent variables.
Differentials have
some computational advantages that will be exploited in later
work. I
will introduce them first and derive the differentials for sine
and
tangentbased formulas where true distances to the target
(baseline and
hypotenuse) are fixed, but the angle to the crownpoint is
allowed to
vary. The derivation below shows the differential of height, H,
with
respect to a change in the angle, A.
D
= baseline distance
L
= hypotenuse distance to crownpoint
A
= angle to crownpoint
H
= Lsin(A) (Our
usual sine top
formula)
dH/dA
= d(Lsin(A))/dA (Derivative of H with
respect to A for sine formula)
dH/dA
= Lcos(A)dA/dA (Taking the derivative)
dH/dA
= Lcos(A) Simplifying:
dH
= Lcos(A)dA (Expressing result as a
differential)
Approaching the measurement from the
tangent perspective, we get:
H
= Dtan(A)
DH/dA
= d(Dtan(A))/dA (Derivative of H
with respect to A for tangent formula)
dH/dA
= Dsec^2(A)
dH/dA
= D/cos^2(A)
dH
= [D/cos^2(A)]dA
If there is no change in the angle A to the
crownpoint, then dH = 0.
But suppose we want to check the value of the differential H for
a
1degree change in A, say from 45 degrees to 46. The degree
represents
the amount that the clinometer is off, i.e. 1 degree over.
Suppose the baseline is 100 feet and the
crownpoint is on a
hypotenuse line that is exactly 45 degrees. Then the actual
hypotenuse
distance to the top is 141.2 feet. This is the distance that
will be
measured by the observer if the sine method is used and the
user’s
rangefinder is accurate to the inch. If the tangent method is
used, then
the measurer will get 100 feet for the baseline distance.
Let’s now
compute dH for the sine and tangent methods and compare them. We
use
radian measure for angles when we compute differentials.
45
degrees = 0.785398 radians
1
degree = 0.017453 radians
Sine Method:
dH
= Lcos(A)dA (height differential
for sine
measurement as derived above)
substituting: dH
= 141.2cos(0.785398)(0.017453)
dH
= 1.74 ft error in height
Tangent Method:
dH
= [D/cos^2(A)]dA (height differential for
tangent measurement)
dH
= [100/cos^2(0.785398)](0.017453)
dH
= 3.49
We can see from the differentials that
the error made with the
tangent method under this situation is greater than the
corresponding
error incurred with the sine method. I should emphasize that the
differentials are approximations. The actual difference in
height errors
(a longer calculation) differs little from the differential
approximations and is slightly more, not less. The question is:
Does the
direction of the difference in errors, i.e. tangent being more
than the
sine for a 45degree angle, hold for all angles or just some
angles?
We can show that the difference is
not only positive, but a
steadily increasing function value with increasing A. To show
this, we
let dHs represent the height differential for the sine formula
and dHt
represent height differential for the tangent formula.
Then:
dHs
= Lcos(A)dA
dHt
= (D/cos^2(A))dA
Z
= dHt – dHs = (D/cos^2(A))dA  Lcos(A)dA (Z
is defined
as the difference of differentials)
But, L = Dsec(A)
= D/cos(A)
So that Z = [DdA)/cos^2(A)] – [(Dcos(A)/cos(A))dA]
Simplifying:
Z
= DdA[(1/cos^2(A)) – 1 = DdA[(1 – cos^2(A))/cos^2(A)]
And remembering that sin^2(A) + cos^2(A) = 1, so that sin^2(A) =
1 –
cos^2(A)
Then: Z = DdA[sin^2(A)/cos^2(A)]
= Dtan^2(A)dA,
since
tan(A) = sin(A)/cos(A)
Z = Dtan^2(A)dA is a compact, and I believe potential useful,
formula
for expressing the difference in the differentials for the two
methods
of tree height determination. Holding the angle constant and
expanding
the baseline (and hypotenuse correspondingly) will be the next
development; then comes the two sources of error approached
through the
concept of a total differential expressed through partial
derivatives.
But that is for the next round of discussions. Let’s return to
the angle
situation.
Examining Dtan^2(A), we see that it is 0
when A =0 and increases as
A increases. So the differential of height for a small change in
A based
on the tangent formula is always larger than the corresponding
height
differential based on the sine formula. This, as far as I can
remember,
is a heretofore undiscussed reason for using the sine method.
The following table shows the height
differential for an angle
differential of 1 degree on a 100foot baseline calculated for
angles
ranging from 0 to 80 degrees at 10 degree increments. The
formatting of
the columns will probably be misaligned. So here are their
titles.
Column 1 = Angle in degrees
Column 2 = Angle
in radians
Column 3 = 1 degree angle
differential in radians
Column 4 = Baseline length in feet
Column 5 = Height difference (differential from tangent
calculation –
differential from sine calculation).
Col 1 Col 2 Col 3 Col 4 Col 5
5 0.0873 0.0175 100.00 0.01
10.00 0.1745 0.0175 100.00 0.05
20.00 0.3491 0.0175 100.00 0.23
30.00 0.5236 0.0175 100.00 0.58
40.00 0.6981 0.0175 100.00 1.23
45.00 0.7854 0.0175 100.00 1.75
50.00 0.8727 0.0175 100.00 2.48
60.00 1.0472 0.0175 100.00 5.24
65.00 1.1345 0.0175 100.00 8.03
70.00 1.2217 0.0175 100.00 13.17
80.00 1.3963 0.0175 100.00 56.14
Note that I intentionally ignored
tree height below eyelevel to
simplify the above process. If the observer stands on level
ground and
simply adds his/her height at eye level then the above errors
are
propagated through the calculations as shown. More elaborate and
complete development of the angle error effect will be reserved
for the
formal book on dendromorphometry.
I would call upon the other
mathematicians in Ents to check my
assumptions and the development of this line of analysis: John
Eichholz,
Howard Stoner, Lee Frelich, Don Bragg, Gary Beluzo, Tom Diggins,
Jess
Riddle, Paul Jost, John Knuerr, etc., etc. etc. It is in these
mathematical arguments that we swim or sink with our ENTS
methodology.
Bob
Robert T. Leverett
Cofounder, Eastern Native Tree Society

Re:
Angle error discussion 
Edward
Frank 
Apr
17, 2007 12:41 PDT 
Bob,
You can't ignore the tree height below eye level as the error is
in the same
direction as the top error and the errors tend to cancel out so
the net
error is minimal. So why should you be calculating the error for
a 1 degree
clinometer misalignment, when in actual application using the
correct
methodology these errors are not really applicable to height
measurement
errors? Unless you also add a section for the base measurement
errors and
then sum the two together I don't see this as a useful
calculation, although
it is impressive to look at
Ed Frank

Re:
Angle error discussion 
dbhg@comcast.net 
Apr
17, 2007 15:19 PDT 
Ed,
Some measurers sight trees from what
they judge to be level ground and simply add their height to
what they get from the slope reading, based on a 100foot
baseline. Will Blozan and I encountered this practice many years
ago. In such cases, there isn't anything to offset or cancel
out. We must remain sensitive to how measurements are actually
carried out by the people who carry clinometers and tape
measures with them. I would admit that doing all the math just
to address a limited practice seems overkill, but there is more
stuff on the way. I'll address your other point tomorrow. For
now, gotta run.
Bob

Re:
Angle error discussion 
Edward
Frank 
Apr
17, 2007 17:20 PDT 
Bob,
You write: A) "We can check the height error that results
from a 1degree error in the angle for both the sine and tangent
methods."; B) "We can see from the differentials that
the error made with the tangent method under this situation is
greater than the corresponding error incurred with the sine
method."; C) "Note that I intentionally ignored tree
height below eyelevel to simplify the above process. If the
observer stands on level ground and simply adds his/her height
at eye level then the above errors are propagated through the
calculations as shown."
Of course we must be sensitive to how measurements are being
carried out in the field. I understand that some measurers 'guestimate'
the base of the tree when using the tangent method. BUT that is
not done when using the sine method. You are in effect comparing
the errors of an inaccurate clinometer as it would often done in
the field with the tangent method, with a "made up"
scenario of a half done sine method that is not actually used.
The second option used as part of the discussion is fiction. If
you are comparing a real life situation with a fictitious one
that does not exist. then the results are meaningless. Real life
must be compared to real lfe to be meaningful. The only real
comparison that can be made here is the amount of error
introduced by the inaccurate clinometer using the tangent method
with the actual height of the tree.
A second tract could be written along the lines of what John
Eicholz wrote about the inaccurate clinometer readings tending
to cancel out when both the top sins and base sine are used in
the calculation The math could be expanded upon and explained,
But I seriously do not believe this argument you presented draws
a meaningful comparison, even if the math is elegant.
Ed Frank

Re:
Angle error discussion 
Don
Bertolette 
Apr
17, 2007 19:34 PDT 
Ed/Bob
When attempting to measure a reasonably large object like a
tree, say 100'
tall, to a tenth of a foot, I'd think that the amount that your
head pivots
going from top to bottom of tree, is easily more than the tenth
of a foot
accuracy you're striving for...just pivoting my head (as if
measuring my
computer screen) as I look at top and bottom of monitor,
describes a change
in "eye height" of more than a tenth of a foot.
Don

Over
and under errorscontinuing the discussion 
Robert
Leverett 
Apr
18, 2007 06:50 PDT 
ENTS,
Ed, you are welcome to think through my posts
critically, and comment
as you see fit. That's never a problem for me, but please bear
with me
on this. There is more to come and it IS relevant, at least in
some
context.
The big problem for you, Ed, I think is
decoding my style. I usually
choose to develop tree measurement material incrementally,
focusing on
certain tidbits rather than swallowing the meal whole and then
regurgitating it out in a logical appearing order for the
convenience of
the reader. You can never be quite sure where I am going with
something.
I acknowledge that this can be confusing for anyone unaware of
my style.
But, hey, it is the southerner in me. Despite living in New
England
since 1975, I remain laid back. I eventually get there, just not
as
quickly as folks up in these parts would like. Now, back to the
subject
at hand.
From time to time, we
discuss the problems encountered when
equipment is out of calibration and measures incorrectly, such
as a
clinometer that consistently reads high or low or a laser
rangefinder
that reads long or short by a yard or meter over part or all its
range.
One question that needs to be answered early on is whether or
not
certain classes of errors compensate for one another. For
example, if we
have a clinometer that consistently reads high or low by a set
amount
over its full angle range, does the height error we incur at one
end of
the tree compensate for the error at the other end. If the
clinometer
reads high, we cheat the tree at the base measurement and give
it too
much at the top. Do the errors compensate? Well, the answer
turns out to
depend on our measurement technique we use. Let’s take an
example.
Suppose the base of a
tree is 1 degree below eye level and the
top is 45 degrees above eye level. Suppose the top of the tree
is
directly over the base so that the tangent method can be used
just as
effectively as the sine method – at least theoretically.
Suppose eye
level distance to the trunk is exactly 100 feet and we have
measured
that distance accurately. We’re set to go. But let us say that
our
clinometer reads consistently high by a degree (Yes, we could
compensate, if we knew, but we don’t). Now, based on the
information
given, the tree's height is 101.75 feet. That is a fact. One
hundred
feet of the full height are above eye level and 1.75 feet are
below eye
level.
To measure the tree, we can
theoretically use either the tangent
or sine method, since the top of the tree is directly over the
base.
Let’s say we use the tangent method because we don't have a
laser
rangefinder with us. Shooting the base of the tree, the
clinometer will
read 0 degrees instead of the actual 1 and the measurer will
start off
his/her height determination by being off by 1.75 feet –
cheating the
tree. Shooting the top of the tree, the measurer's clinometer
will read
46 degrees and the measurer will compute the above eye level
height as
103.55 feet or 3.55 feet too much, since the actual above eye
level
height is exactly 100 feet. The measurer thinks the tree is
103.55 feet
tall, while it is actually 101.75. The difference is 1.8 feet,
which
results from the tree getting shorted by 1.75 feet at the base
and
overcalculated by 3.55 feet at the top. Clearly, the 3.55foot
overage
at the top is not offset by the undererror at the bottom. The
difference is 1.8 feet. So, for this application of the tangent
method,
the errors do not offset one another. Now, were eye level at the
midpoint of the tree, then the errors would offset one another,
but we
seldom measure from such a vantage point. And the taller the
tree and
the shallower the base angle, the greater the differential error
that
will be made with the tangent method. Let’s take another
example.
If top of the vertical
tree is at 55 degrees instead of 45 and
the base is at say 5 and the eye level distance to the trunk is
still
100 feet and the clinometer still reads a degree high, then the
bottom
error becomes 1.77 feet and the top error becomes 5.44 feet. The
difference is 3.69 feet and that is definitely significant.
Okay, we
have demonstrated another problem with the tangent method along
the
limited lines of investigation of a misperforming clinometer.
What about the sine method? What if we
apply the sine method to the
first example? If our distances are measured correctly with our
rangefinder, but we goof on the angles because of our clinometer
problem, we shortchange the tree at the base by 1.75 feet and we
add
1.75 feet too much at the top. The difference is 0 feet, which
is your
point Ed. No argument. I worked through that exercise several
years ago.
The bottom angle error does offset the top error  when using
the sine
method. What about the second situation? If we change the base
angle to
–5 degrees and the crown angle to 55 degrees, the 1degree
problem with
the clinometer is again a wash.
So long as the eye level distance
to a vertical tree remains the
same and its top is directly over its base, consistent over or
under
angle errors are offsetting for sinbased calculations, but not
for
tangentbased ones. At 0 degrees base angle and 60 degrees to
angle, the
1degree upward shift of the clinometer angle leads to an
aggregate
error of 5.45 feet in the tree height determination using the
tangent
method. This is what I was aiming to get at. I was just taking
my time.
What I am doing is investigating every
nook and cranny of the use of
these tree heightmeasuring techniques, looking for that which
is not
obvious. The practicality of doing my approach will not always
be
apparent, even to me, but that’s what you get when you get a
southerner
stuck up here in Yankee Land, yearning for one good breakfast of
smoked
ham, grits, and fresh eggs cooked to order. Hey, Howard
understands.
He’s a good Iowa farm boy, turned cowboy, turned
mathematician. Dang,
Howard, I bet you had some confused cows.
Bob
Robert T. Leverett
Cofounder, Eastern Native Tree Society

Re:
Angle error discussion 
dbhg@comcast.net 
Apr
18, 2007 08:51 PDT 
Don,
Head pivoting does change the triangles
slightly by putting the spot at where a line from the top down
through the clinometer, through the eye, and through the back of
the head intersects the corresponding line coming from the base
of the tree. We're talking about the impact of a few inches of
distance, which translates to a tiny effect. I'll give examples
in a future email.
Bob

Back
to HowardAngle error discussion 
dbhg@comcast.net 
Apr
18, 2007 09:09 PDT 
Howard,
It seems that issues with angles
have long preoccupied us fellows in ways I would not choose to
discuss on our tree list. But joking aside, I'd appreciate it if
you would help me think through this latest approach to
analyzing errors in tree measuring in terms of their effects 
from the obvious to the not so obvious. Clinometer error has
long plagued our measurements, at least mine, and this seems to
be the same with that of a number of you. I've owned 6 Suunto
clinometers and have experienced out of calibration situations
or sticking at some point with all of them. John Eichholz has
encountered significant errors with his instruments and has had
to develop compensating methods.
It is easy to take the readings
one gets from an instrument at face value. Point, shoot,
calculate, record, and move on. But for us to be as good as we
say we are, I think we need to spend more time investigating
sources of error and coming up with simple techniques that we
can routinely apply to compensate for outofcalibration
situations. Any thoughts?
Bob

Re:
Over and under errorscontinuing the discussion 
Howard
Stoner 
Apr
18, 2007 12:13 PDT 
Bob,
Roping a cow from a horse has some similarities to measuring the
height of a tree.
You need the distance to the cow and the angle the cow is below
you. At that point
the similarity ends. The distance between you and the cow and
the angle
the cow is below you must be accurately estimated multiple times
(as you
ride full speed on your horse) and in quick succession followed
by a
reflex response on the part of the arm to release the rope at
exactly the right time.
Try that on a tree!
When cows get GM'ed(genetically modified) I suppose they will
constantly raise and lower
their head to avoid you ever getting an accurate reading on that
angle!
Howard

Back
to Ed with a couple more thoughts 
Robert
Leverett 
Apr
18, 2007 12:29 PDT 
Ed,
You've probably read my response to your last
email by now and
hopefully see where I'm going, but a couple of added points are
still in
order. You were quite right to be confused about my direction,
since I
didn't clearly state up front that I had set out to examine the
efficacy
of both the sin topsin bottom and the tangent methods from the
standpoint of angle error and was passing along material that I
had just
developed, i.e. the derivations.
The area of investigation covered in the
last two emails is
admittedly not needed for the Ents who use the superior sine
method, but
is meant for inclusion in the eventual dendromorphometry text,
which
I've begun to seriously work on. In the near future, I'll be
exploring
different topics with the rest of you and will be coming at them
sometimes from unpredictable directions. Often, a topic that I
will
introduce will be irrelevant to current ENTS practices, but will
make
sense as material for the dendromorphometry text, i.e. I'll be
thrashing
out the topics relevant to tree measuring in general. However, I
will
preface the material with an explanation of why I'm passing it
on to the
rest of you, because the material will frequently be in snippet
or
tidbit form.
Bob

Re:
Back to Ed with a couple more thoughts 
Edward
Frank 
Apr
18, 2007 18:39 PDT 
Bob,
I can see where you are going, but also I think you are missing
my point. I
understand this a part of a larger argument comparing tangent
and sine and
clinometer errors. My point is that I don't think this line of
reasoning
should be pursued as the underlying assumption of one part of
the exercise
does not represent real life. The entire comparison of the
respective
errors of tangent method and just the top sine method does not
represent
anything meaningful and its discussion will only serve to
confuse the issues
involved. The entire argument would be better explained and
understood
without this particular sidetrack. The part about the toponly
sine should
be skipped and you should move onto the next step without
elaborating on
this portion of the comparison. In my opinion....
Ed

RE:
Back to Ed with a couple more thoughts 
Robert
Leverett 
Apr
19, 2007 04:29 PDT 
Ed,
Your opinion is duly noted and respected as
always. I did understand
your point, and on thinking carefully about it, am inclined to
agree
with you. Actually, I intended to get to the full effect of
tangent
error from "angle shift" and compare it to the full
effect of sine error
from angle shift. I was doing it in pieces, but now see how that
created
an artificial comparison.
Bob

Re:
Back to Ed with a couple more thoughts 
Edward
Frank 
Apr
19, 2007 17:32 PDT 
Bob,
I have expressed my viewpoint, but it is you who are writing the
documents
and if you feel you need to include a certain line of reasoning
to make your
arguments work as you envision them, then you should do what you
feel is
best. See you Saturday, maybe.
Ed

Angle
errors revisited 
Robert
Leverett 
Apr
27, 2007 07:48 PDT 
ENTS,
For folks not into the
mathematics of tree measuring, best exit
this email now. Because, the following discussion will either
put you
to sleep or cause you to run screaming, pulling out gobs of hair
as you
go.
THINKING BACK:
On our list, we frequently
discuss the common sources of
measurement error when calculating tree height. In terms of our
instruments, the errors are in the angle, the distance, or both.
We may
make an angle error because our clinometer is out of
calibration,
because we misread it, because it sticks, because we measure the
angle
to the wrong object, or some combination of the above.
Similarly, our
distance may be off because our laser is out of calibration, or
because
we hit the wrong target. We may get accurate angle and distance
measurements to the target that we select, but apply the wrong
mathematical model to calculate the intended dimension.
Anticipating
these sources of error and attempting to eliminate or compensate
for
them is part of entrylevel tree measuring.
Many discussions on
the list have been devoted to the appropriate
mathematical model to apply in measuring a dimension, e.g. the
sinebased versus tangentbased model for computing tree height.
If we
have a laser and a clinometer and can see the target, the
sinebased
model is clearly the best, and if the sine model can be used,
there is
no reason to employ the problematic tangentbased model. We may
teach it
to others so they will understand the full range of measurement
methods
that they are likely to encounter, but we we employ the sine
method. So,
the discussion below assumes we are using the sine method.
IMPACT OF ANGLE ERRORS:
In investigating errors in
the angle, or distance, or both, we
want to know how bad things can get. But after examining some
extremes
and getting a sense of the worstcase scenarios, where do we go?
One
line of investigation is to explore what happens when we hold
one
variable, e.g. hypotenuse distance, constant and look at the
pattern of
the height change as the angle changes. We can then apply that
knowledge
to investigate small angle errors. By holding the hypotenuse
distance
constant and checking height changes from angle changes, it
doesn’t take
much experimentation before we see that we make the greatest
errors in
height in response to changes in the lower angles, i.e. angles
closer to
0. This may seem counterintuitive, but it is true.
We can get a better feel for
not only the magnitude of the height
error from an angle error, but the rate at which the height
changes as
the angle changes by small amounts. To investigate changes in
the rate
of increase or decrease, we appeal to a concept in calculus
called the
derivative of a function. Without much explanation, basically
the
derivative of a function measures the instantaneous rate of
change in
one variable that is induced by changes in another. Appealing to
the
sinebased formula H = L*sin(A), the derivative of H with
respect to A
is denoted by dH/dA and dH/dA = L*cos(A). An alternative
notation of the
derivative is H’. The derivative measures the instantaneous
rate at
which H changes as A changes for any value of A tested where the
derivative exists. We can deduce the rate of height change by
recognizing that cos(A) steadily decreases as A increases.
Therefore,
since L is constant, L*cos(A) is greatest at small angles. In
fact, at 0
degrees, L*cos(A) = L and at 90 degrees, L*cos(A) = 0. So the
rate of
change drops from L to 0 as the angle goes from 0 to 90 degrees.
There is another approach
that we can use. We can take something
called a second derivative, which we will denote here as H”.
The second
derivative measures how the rate of height, itself, changes as
the angle
changes. If the rate of height change decreases as A increases,
we get a
negative second derivative. The following formula gives the
second
derivative for H.
H”
= L*sin(A). (Note
the relationship to H)
Since sin(A) is
continuously positive for 0 <= A <= 90, the
product –L*sin(A) stays negative throughout the angle range.
This simply
means that the rate at which H changes as A increases is a
continuously
decreasing value. We can see this effect clearly in the table
below:
Hypotenuse Angle Change Change 2nd
Distance in
Angle in
Height
Derivative
100 0 1 1.75
1.75
100 30 1 1.52
50.00
100 60 1 0.87
86.60
100 85 1 0.17
99.62
As we see in the table, the change
in height (as opposed to the
absolute height) drops as the angle increases for a constant
hypotenuse
distance. This fact is reflected in the negative second
derivative. Of
course, the absolute height increases as the angle increases,
but at a
diminishing rate as reflected by the negative value of the
second
derivative.
Assuming the angle to the base of a tree
is shallow and steep toward
the top, does this mean that if we make an angle error at the
base of a
tree of say 1 degree and make the same angle error in magnitude
and
direction at the top of the tree, that the two errors will NOT
offset
one another because the impact of the error for small angles is
greater?
No, not necessarily. They WILL offset each other IF the sine
method is
used AND the top is vertically over the base because the
hypotenuse
stretches as the angle increases. The greater hypotenuse
distance to the
top creates the offsetting error. The error at the bottom cheats
the
tree of height by the same amount that it overrewards the tree
at the
top. Ed Frank has frequently mentioned the offsetting errors
from a
constant angle error. In the configuration of top directly over
base,
the offsetting error effect is another reason to use the sine
method. As
was shown in a prior email, the errors do NOT offset one
another when
using the tangent method.
Now, suppose that the crown top is
horizontally closer to us than
the base, a frequently encountered situation in the field. Let
us
further assume that the closer horizontal distance leads to a
hypotenuse
distance that exactly equals the base distance. This assumption
is to
keep things simple. We define the differential dHa, where dHa
defines an
angle change, as
dHa
= L*cos(A)*dA
Since L and dA do not change in
value, the value of dHa is
dependent on the term cos(A), which decreases as A increases.
This means
that the error in height made at the top as approximated by the
differential dHa is less than the error at the bottom. Here we
assume
that the base angle is 0, but measured as 1 degree and the top
angle is
A, but is measured by our outofcalibration clinometer as A+1.
The top
error, adding too much, does not exactly offset the bottom
error, which
cheats the tree. Let’s take an example from the Jake Swamp
Tree data.
What happens frequently in the
field is that the base angle is less
than the crown angle and the base distance is closer than the
crown
distance so that a constant angle error in bottom and top angle
measurements do not exactly offset one another when computing
height,
but often the difference isn’t great. The last measurement of
the Jake
Swamp white pine in MTSF was:
Crown Distance: 76 Yds
Base Distance: 66
Yds
Crown Angle: 34.20
Degrees
Base Angle: 11.80
Degrees
The straightline eye to target
difference between the two
distances is 30 feet and the eye level to point difference in
the angles
is 22.4 degrees (not total angle). The impact of the height
error at the
base from a 1degree angle error would be to rob Jake of 3.38
feet. The
corresponding error at the crown would add 3.29 feet too much to
Jake’s
height. The difference of 0.09 feet is hardly worth worrying
about. But,
the reason the error is so little is that the horizontal offset
of the
high point of the Jake tree from the base is only 5.24 feet,
making the
measurement closer to case #1 (top directly over base) instead
of case
#2 (top significantly removed from base in horizontal
direction).
Bob
Robert T. Leverett
Cofounder, Eastern Native Tree Society

One
more for the road 
Robert
Leverett 
Apr
27, 2007 10:11 PDT 
ENTS,
One more email for the road and then I'll
mercifully disappear for
the weekend.
It strikes me that proving that the
angle error cancels for
sinebased calculations when dealing with a right triangle tree
configuration ought to be something that we formally do. So here
goes.
Let D = base of
right triangle (top is directly over bottom)
A1
= lower angle (to base)
A2
= upper angle (to top)
L1
= laserbased hypotenuse distance to base at lower angle
L2
= laserbased hypotenuse distance to crown at upper angle
H1
= height of base above(below) eye level for angle A1
H2
= height of top above eye level for angle A2
a
= angle error for both A1 and A2
dA
= a (treating a as a differential)
Now, H1 = L1*sin(A1)
and H2 =
L2*sin(A2)
But L1 =
D/cos(A1) (top
is directly above base in
this development)
And L2 =
D/cos(A2) (same
logic)
The differentials are:
dH1
= L1*cos(A1)*dA
dH2
= L2*cos(A2)*dA
Substituting for L1 and L2:
dH2
 dH1 = L2*cos(A2)*dA  L1*cos(A1)*dA
dH2
 dH1 = [D/Cos(A2)]*cos(A2)*dA  [D/Cos(A1)]*cos(A1)*dA
Simplyfing:
dH2
 dH1 = D*dA  D*dA = 0. The cosines obviously cancel,
giving us the result we seek, i.e. a difference of 0.
So now, we can all sleep at night...... Uh, but what if the top
is not
directly over the base so that D does not remain the same as we
move
from angle A1 for the base to A2 for the top. Under this
assumption,
D1
= base for A1
D2
= base for A2
Then dH2  dH1 = D2*dA  D1*dA = (D2  D1) * dA
If we define Eh as the error in height incurred from a constant
error
in angle, then
Eh
= (D2  D1) *dA
where dA is the error in the angle expressed in radians.
This last result is useful, but we can improve on it. The radian
equivalent of a 1 degree angle is 0.017453. So a handy formula
for a
constant angle error of 1 degree for situations where there is a
crown
point offset unequal to zero is
Eh
= (D2D1)*0.17453.
But why stop at 1 degree. We can take this process one step
farther.
If we define k as some multiple of an angle error of 1 degree (k
could
be 0.25, 0.5, 1.5, or 2.0, etc.). The formula becomes
Eh
= 0.017453*k*(D2D1),
which gives us the error in height for an angle error k
expressed in
degrees, recognizing that k*0.17453 = the angle error in
radians.
As
an example, suppose we have a crownpoint offset of D2D1
= 25 and k = 2, an extreme case of angle error. Then
Eh
= 0.017453*2*25 = 0.87.
An offset of 25 is fairly large and a consistant angle error of
2
degrees is very large. So, it is reasonable to assume that the
error in
height due to a consistant angle error for sinebased
calculations will
almost always be under a foot and usually well under a foot. We
can
really go to sleep now.
Unless someone finds an error in the above
construction, I do think
Eh = 0.017453*k*(D2D1) is sufficiently useful and insightful to
qualify
for our list and become #15.
Bob
Robert T. Leverett
Cofounder, Eastern Native Tree Society

RTS 
Robert
Leverett 
May
03, 2007 06:49 PDT 
ENTS,
One more round of the error
analysis and I'll let the subject rest
for a while. However, as a rationale for going deeper into the
subject,
ENTS needs to fully explore all the sources of errors in our
calculations, not merely to be aware of them or qualitatively
describe
them, but to quantitatively evaluate them. Obviously, we all
recognize
that if we make angle and distance errors, we can expect errors
in our
calculated results, but what are the sources with the greatest
impacts?
When do errors offset one another, etc.? Do equipment errors
propagate
through the different measuring techniques (sine versus tangent)
for a
dimension in the same way?
TANGENT/SINE:
Is there a simple
relationship between the heights obtained from
correct distances but incorrect angles when using tangent versus
sinebased height determinations? Yes, there is. For any
baseline
distance D, crown angle A, and angle error a, the ratio of the
tangentbased height to the sinebased height (RTS) is given by:
RTS = cos(A)/cos(A+a).
This means that the distance
measurement isn't involved when only
the angle varies. As an example, suppose the angle is 50 degrees
and the
angle error is 1.25 degrees. Then the RTS formula gives the
ratio of the
tangentbased height to the sinebased height as
cos(50)/(cos(51.25) =
1.02943. For a 55 degree angle and 1.25 degree error, the ratio
is
1.03241.
The derivation of the RTS ratio
follows.
Let:
Ht = Dtan(A+a) where D
= baseline and angle to crownpoint = A.
Here crownpoint is directly over the base;
Hs = Lsin(A+a) where L
= hypotenuse distance to crownpoint for
angle A.
Now,
L = D/cos(A), since we are dealing with
a right triangle;
so that
Hs = Dsin(A+a)/cos(A)
And
RTS = Ht/Hs = [Dtan(A+a)]/[Dsin(A+a)/cos(A)].
By definition
tan(A+a) = sin(A+a)/cos(A+a).
Substituting
RTS = [Dsin(A+a)/cos(A+a)]/[Dsin(A+a)/cos(A)]
and
Simplifying,
RTS = cos(A)/cos(A+a).
At 65 degrees and an error of 1.25
degrees, RTS = 1.04934
A tree 70 feet away with a crownpoint
directly over the base at 65
degrees is 150.12 feet in height. The hypotenuse distance is
165.6 feet.
Suppose the angle is measured as 66.25 instead of 65. Then the
erroneous
sinebased height will be calculated as 151.61 feet. However,
the
tangentbased height will be calculated as 159.1. The ratio of
159.1 to
151.61 is 1.04934. Now, RTS = cos(A)/cos(A+a) =
cos(65)/cos(66.25) =
1.04934. Therefore, RTS expresses the ratio of the erroneous
tangentbased height over the erroneous sinebased equivalent. I
believe
that RTS has practical significance, at least for educational
purposes,
and is easy to apply. Are there any supporters of this point
Detractors?
Will, Lee, Jess, Paul, Ed, etc.?
Bob
Robert T. Leverett
Cofounder, Eastern Native Tree Society

RTSa,
RTSd, and RTSo 
Robert
Leverett 
May
04, 2007 05:28 PDT 
ENTS,
It seems like it has been a very, very long
time since I've posted
any formulas and I’m experiencing formula withdrawal. So, I
ask your
indulgence. We need the companion formula for the ratio of
tangentbased to sinebased height determination that
incorporates a
distance error, but no angle error. The formula is presented
below.
Everyone will be spared the algebraic derivation.
D = baseline distance
d = error in distance (baseline and
hypotenuse)
A = angle
RTS = (D+d)/(D+d*cos(A))
At 0 degrees the ratio is 0 and at 90
degrees it is (D+d)/D. For a
given d, the ratio changes as the angle increases to a maximum
of 1+d/D.
Since the situations where the angles are 0 or 90 degrees are of
no
practical significance, the ratio is seen to always be greater
than 1,
meaning that the height error made from a distance error is
greater for
tangentbased calculations, though not by much at low angles
Short
baselines and high angles give the highest ratios. For example,
a
67foot baseline, a 3foot baseline error of too much, and an
angle of
65 degrees lead to a ratio of 1.0254. However, switching from
distance
to degrees, a 1degree error at 65 degrees leads to a ratio of
1.0390
regardless of the baseline distance. The inescapable conclusion
is that
for modest distance or angle errors, the angle error is the more
serious.
The formula for the ratio of the
tangentbased height to the
sinebased height where there are errors in both the angle and
distance
is cumbersome and won't be presented. The picture of effects is
clear
from the two formulas.
One final formula to cure my formula
withdrawal, what if there are
no equipmentrelated errors, but there is a crownoffset error?
The
crownoffset only applies to the tangentbased calculation, but
we can
still compute the ratio of the height determinations from the
two
methods. Well, the formula is:
RTS = (D+d)/D*cos(A).
To distinguish between the different RTS
situations, we could settle
on the following definitions:
RTSa = RTS associated with an angle
error,
RTSd = RTS associated with a distance
error,
RTSo = RTS associated with a
crownoffset error.
These formulas in conjunction with the
total differential dH
provides us with a means of investigating the impact of errors
from
angle, distance, and method.
Bob
Robert T. Leverett
Cofounder, Eastern Native Tree Society

Re:
Angle errors revisited 
Edward
Frank 
May
04, 2007 08:17 PDT 
ENTS and Bob,
I want to thank Bob for taking on the giant task of trying to
better define
all of the aspects of tree measuring in terms of mathematics. I
am sure I
could not do all of the many equations Bob has posted. We
may argue about
details, but the important aspect at this point is to get them
out there. I
am sorry I missed Bob's presentation at Cook last weekend. The
idea that
the top error and the bottom angle error offset was noted by
John Eicholz in
a post several years ago. The offset is not perfect but the
errors are both
in the same direction and to a large degree cancel each other,
even when
using large errors such as two degrees in Bob's example. Smaller
angle
errors have smaller net errors and the sum of the offsets are
even closer to
zero when using the top/bottom sine methods. I just wanted to
point out the
idea was not mine originally.
Ed Frank

Back
to Ed: RTSa, RTSd, and RTSo 
Robert
Leverett 
May
04, 2007 09:55 PDT 
Ed,
Thanks. It is a labor of love, and until
recently, it was mostly
going over plowed ground. But more recently I started pushing
myself to
revisit mathematical concepts that I haven't thought about
literally for
years, searching for potential applications to our ENTS work.
I'll incorporate all this error analysis in
the upcoming guide to
Dendromorphometry, which all of you will get an ample
opportunity to
edit.
Bob

RTSad 
Robert
Leverett 
May
07, 2007 06:02 PDT 
ENTS,
As a reward for enduring the torture, we
now have a ratio formula
for comparing a tangentbased to a sinebased height
determination when
there is an error in the angle, but not the distance. The
assumption is
that there is no crownoffset error, so we are dealing with the
same
right triangle. We want to see how the two methods for
calculating
height compare to one another when there is an angle error. The
ratio
formula is
RTSa = cos(A)/cos(A+a)
where A = angle to the crownpoint (or
base) and a is the error in
the angle. We note that distance to either the trunk or directly
to the
crownpoint is not involved.
We also developed a corresponding ratio
formula when there is an
error in the distance, but not angle. The formula is
RTSd = (D+d)/(D+d*cos(A))
where D = eye level distance to
trunk, d = error in distance, and
A=angle to crownpoint.
I previously stated that the formula for
the ratio of the
tangentbased to the sinebased height determination when there
are
errors in both distance and angle is too cumbersome, but I
realized that
there is a simple way of expressing the relationship. If we
define
RTSad as the ratio when there are errors in both distance and
angle
then, RTSad can be expressed simply as:
RTSad = RTSa*RTSd.
That is, RTSad is just the product of
the separate ratios RTSa and
RTSd. This wasn't intuitive to me. That's why I missed it. It
still
isn't intuitive, but it is the case. So, substituting the
formulas for
RTSa and RTSd into RTSad, we get:
RTSad = [cos(A)/cos(A+a)]*[(D+d)/(D+d*cos(A))]
The ratio formulas RTSa, RTSd, and RTSad give
us three potentially
useful tools for comparing tangentbased height determinations
to the
corresponding sinebased ones. From prior analysis, we saw that
if the
errors are over, i.e. angle and/or distance is too much, then
the
tangentbased height error is greater than the sinebased height
error
and that fact is reflected in the ratio formula being greater
than 1.
RTSad incorporates the individual ratio formulas RTSa and RTSd
as
factors. This means that the height error from the combined
angle and
distance errors will be even greater for the tangentbased
calculation
than for the sinebased one when there are over errors in both
angle and
distance. Again, please keep in mind that there is no
crownpoint offset
error with the tangent calculation in this analysis.
When the direction of error is to
understate the angle, the
tangentbased calculation understates the computed height more
than does
the sinebased calculation  the error in height goes the other
direction. Either way, angle over or under, the sinebased
calculation
is closer to the actual height than is the tangentbased
calculation.
This is not an effect that the makers of hypsometers who follow
convention are going to want to hear about. There are probably
offsetting impacts on the composite ratio when one error is over
and the
other is under. This will be my next area of investigation.
For those who are interested, the derivation
of the RTSad product
formula follows (Beth, please don't look at this stuff, it will
make you
catatonic!). I've numbered the steps for easy reference. I note
that for
mathematical expressions, having to use Topica's text editor is
a real
pain. It is much easier when one can use proper mathematical
notation.
However, to partially compensate for the crappy editor, I've
used the
grouping symbols "[", "]", "{",
and "}" as well as parentheses and the
computer symbol “*” for multiplication to hopefully make the
expressions
a little easier to follow. Brackets, braces, and parentheses are
all
algebraic grouping symbols that have the same purpose. As
grouping
symbols, they are nonhierarchical. By judiciously using them
instead of
only parentheses, matching left and right grouping symbols
becomes a
little easier. Well, here goes.
1. RTSa = cos(A)/cos(A+a) by previous development
2. RTSd = (D+d)/(D+d*cos(A)) by previous development
3. RTSad = [(D+d)*tan(A+a)]/[(L+d)*sin(A+a)] by definition
4. However, tan(A+a) = sin(A+a)/cos(A+a) and L = D/cos(A),
where L = hypotenuse distance from eye to
crownpoint.
5. Substituting, we get
RTSad = [(D+d)*sin(A+a)/cos(A+a)]/[{d+D/cos(A)}*sin(A+a)]
6. Canceling the sin(A+a) factors in the numerator and
denominator, we
simplify the equation to
RTSad = [(D+d)/cos(A+a)]/[d + D/cos(A)]
7. The right side can be rewritten to give
RTSad = [(D+d)/cos(A+a)]/[(d*cos(A)+D)/cos(A)]
8. Simplifying the fractions we get
RTSad = [cos(A)*(D+d)/cos(A+a)]/[{d*cos(A)+D}]
9. Rearranging terms,
RTSad = [cos(A)*(D+d)/[cos(A+a)*{D+d*cos(A)}]
10. Which can be further rewritten as
RTSad = [cos(A)/cos(A+a)]/[(D+d)/{D+d*cos(A)}]
In this last form, we see the roles of
RSTa and RTSd. Substituting
them, gives us last equation:
11. RTSad = RTSa * RTSd, which is the result we were seeking.
Robert T. Leverett
Cofounder, Eastern Native Tree Society

