Angle error discussion   Robert Leverett
  Apr 17, 2007 06:30 PDT 


    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
Willís 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, Iíve been concentrating more on
the effect of angle and distance errors resulting from tangent-based
versus sine-based measurement models. Our usual objection to
tangent-based models for measuring tree height is that they are often
misapplied. The spot taken as the highest sprig and measured angle-wise
is seldom directly over the base, leading to crown-point 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 measurerís apparatus
measures distances accurately. Further suppose that the measurerís
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 1-degree 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
tangent-based formulas where true distances to the target (baseline and
hypotenuse) are fixed, but the angle to the crown-point 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 crown-point

                           A = angle to crown-point

                           H = Lsin(A)        (Our usual sine top

                           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

    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 crown-point, then dH = 0.
But suppose we want to check the value of the differential H for a
1-degree 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 crown-point 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 userís
rangefinder is accurate to the inch. If the tangent method is used, then
the measurer will get 100 feet for the baseline distance. Letís 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 45-degree 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.

            dHs = Lcos(A)dA

            dHt = (D/cos^2(A))dA

            Z = dHt Ė 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)] Ė [(Dcos(A)/cos(A))dA]

            Z = DdA[(1/cos^2(A)) Ė 1 = DdA[(1 Ė cos^2(A))/cos^2(A)]

And remembering that sin^2(A) + cos^2(A) = 1, so that sin^2(A) = 1 Ė

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. Letís return to the angle

    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 un-discussed reason for using the sine method.

    The following table shows the height differential for an angle
differential of 1 degree on a 100-foot baseline calculated for angles
ranging from 0 to 80 degrees at 10 degree increments. The formatting of
the columns will probably be mis-aligned. 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 Ė
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 eye-level 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.


Robert T. Leverett
Cofounder, Eastern Native Tree Society
Re: Angle error discussion   Edward Frank
  Apr 17, 2007 12:41 PDT 


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
  Apr 17, 2007 15:19 PDT 

    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 100-foot 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.

Re: Angle error discussion   Edward Frank
  Apr 17, 2007 17:20 PDT 

You write: A) "We can check the height error that results from a 1-degree 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 eye-level 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 

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.
Over and under errors-continuing the discussion    Robert Leverett
   Apr 18, 2007 06:50 PDT 


   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

    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. Letís 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 Ė at least theoretically. Suppose eye
level distance to the trunk is exactly 100 feet and we have measured
that distance accurately. Weíre 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 donít). 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

      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.
Letís 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 Ė 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
over-calculated by 3.55 feet at the top. Clearly, the 3.55-foot overage
at the top is not offset by the under-error 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
mid-point 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. Letís 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 mis-performing 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
Ė5 degrees and the crown angle to 55 degrees, the 1-degree 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 sin-based calculations, but not for
tangent-based ones. At 0 degrees base angle and 60 degrees to angle, the
1-degree 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 height-measuring techniques, looking for that which is not
obvious. The practicality of doing my approach will not always be
apparent, even to me, but thatís 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.
Heís a good Iowa farm boy, turned cowboy, turned mathematician. Dang,
Howard, I bet you had some confused cows.


Robert T. Leverett
Cofounder, Eastern Native Tree Society
Re: Angle error discussion
  Apr 18, 2007 08:51 PDT 

   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 e-mail.

Back to Howard-Angle error discussion
  Apr 18, 2007 09:09 PDT 

     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 out-of-calibration situations. Any thoughts?

Re: Over and under errors-continuing the discussion   Howard Stoner
  Apr 18, 2007 12:13 PDT 

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!


Back to Ed with a couple more thoughts   Robert Leverett
  Apr 18, 2007 12:29 PDT 


   You've probably read my response to your last e-mail 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 top-sin 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 e-mails 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.
Re: Back to Ed with a couple more thoughts   Edward Frank
  Apr 18, 2007 18:39 PDT 


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 top-only sine should
be skipped and you should move onto the next step without elaborating on
this portion of the comparison. In my opinion....

RE: Back to Ed with a couple more thoughts   Robert Leverett
  Apr 19, 2007 04:29 PDT 


   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.

Re: Back to Ed with a couple more thoughts   Edward Frank
  Apr 19, 2007 17:32 PDT 


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.

Angle errors revisited   Robert Leverett
  Apr 27, 2007 07:48 PDT 


      For folks not into the mathematics of tree measuring, best exit
this e-mail now. Because, the following discussion will either put you
to sleep or cause you to run screaming, pulling out gobs of hair as you


      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 entry-level tree measuring.

       Many discussions on the list have been devoted to the appropriate
mathematical model to apply in measuring a dimension, e.g. the
sine-based versus tangent-based model for computing tree height. If we
have a laser and a clinometer and can see the target, the sine-based
model is clearly the best, and if the sine model can be used, there is
no reason to employ the problematic tangent-based 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.


      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 worst-case 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 doesnít 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 counter-intuitive, 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
sine-based 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 Hí. 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 HĒ. 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.

                                                HĒ = -L*sin(A).   (Note
the relationship to H)

       Since sin(A) is continuously positive for 0 <= A <= 90, the
product ĖL*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      

100                 0                  1                     1.75      

100               30                  1                     1.52       

100               60                  1                     0.87       

100               85                  1                     0.17       

     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

    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 over-rewards 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 e-mail, 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 out-of-calibration clinometer as A+1. The top
error, adding too much, does not exactly offset the bottom error, which
cheats the tree. Letís 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 isnít 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 straight-line 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 1-degree 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 Jakeís
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).


Robert T. Leverett
Cofounder, Eastern Native Tree Society
One more for the road   Robert Leverett
  Apr 27, 2007 10:11 PDT 


   One more e-mail for the road and then I'll mercifully disappear for
the weekend.

    It strikes me that proving that the angle error cancels for
sine-based 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 = laser-based hypotenuse distance to base at lower angle
           L2 = laser-based 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


           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 = (D2-D1)*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*(D2-D1),

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 crown-point offset of D2-D1
= 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 sine-based 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*(D2-D1) is sufficiently useful and insightful to qualify
for our list and become #15.


Robert T. Leverett
Cofounder, Eastern Native Tree Society
RTS   Robert Leverett
  May 03, 2007 06:49 PDT 


      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?


       Is there a simple relationship between the heights obtained from
correct distances but incorrect angles when using tangent versus
sine-based height determinations? Yes, there is. For any baseline
distance D, crown angle A, and angle error a, the ratio of the
tangent-based height to the sine-based 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
tangent-based height to the sine-based height as cos(50)/(cos(51.25) =
1.02943. For a 55 degree angle and 1.25 degree error, the ratio is

     The derivation of the RTS ratio follows.


    Ht = Dtan(A+a)   where D = baseline and angle to crown-point = A.
Here crown-point is directly over the base;

    Hs = Lsin(A+a)   where L = hypotenuse distance to crown-point for
angle A.

    L = D/cos(A), since we are dealing with a right triangle;

so that
    Hs = Dsin(A+a)/cos(A)

    RTS = Ht/Hs = [Dtan(A+a)]/[Dsin(A+a)/cos(A)].

By definition
    tan(A+a) = sin(A+a)/cos(A+a).    

    RTS = [Dsin(A+a)/cos(A+a)]/[Dsin(A+a)/cos(A)] and

    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 crown-point 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
sine-based height will be calculated as 151.61 feet. However, the
tangent-based 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
tangent-based height over the erroneous sine-based 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.?


Robert T. Leverett
Cofounder, Eastern Native Tree Society
RTSa, RTSd, and RTSo   Robert Leverett
  May 04, 2007 05:28 PDT 


   It seems like it has been a very, very long time since I've posted
any formulas and Iím experiencing formula withdrawal. So, I ask your
indulgence. We need the companion formula for the ratio of
tangent-based to sine-based 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
tangent-based calculations, though not by much at low angles Short
baselines and high angles give the highest ratios. For example, a
67-foot baseline, a 3-foot 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 1-degree 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

    The formula for the ratio of the tangent-based height to the
sine-based 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 equipment-related errors, but there is a crown-offset error? The
crown-offset only applies to the tangent-based 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 crown-offset 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.


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 


   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

RTSad   Robert Leverett
  May 07, 2007 06:02 PDT 


    As a reward for enduring the torture, we now have a ratio formula
for comparing a tangent-based to a sine-based height determination when
there is an error in the angle, but not the distance. The assumption is
that there is no crown-offset 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 crown-point (or base) and a is the error in
the angle. We note that distance to either the trunk or directly to the
crown-point 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 crown-point.

    I previously stated that the formula for the ratio of the
tangent-based to the sine-based 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 tangent-based height determinations to the
corresponding sine-based ones. From prior analysis, we saw that if the
errors are over, i.e. angle and/or distance is too much, then the
tangent-based height error is greater than the sine-based 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 tangent-based calculation
than for the sine-based one when there are over errors in both angle and
distance. Again, please keep in mind that there is no crown-point offset
error with the tangent calculation in this analysis.

    When the direction of error is to understate the angle, the
tangent-based calculation understates the computed height more than does
the sine-based calculation - the error in height goes the other
direction. Either way, angle over or under, the sine-based calculation
is closer to the actual height than is the tangent-based 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 ď*Ē 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 non-hierarchical. 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 crown-point.

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