Don,
Many of the limbs we would be measuring for length or volume are far
beyond the reach of a ground-based measurer. However, for length measurements,
it is a matter of shooting from the tip to the straight bole of the tree above
or beneath the limb flare and calculating the horizontal distance between. This
would give the horizontal limb length that includes the limb flare. In terms of
volume measurements as opposed to simple length, the method you describe sounds
good for a limb that is near ground level and horizontally extended.
On your other point, remembering past conversations with Dr. Alan Gordon,
there has been no shortage of high-end mathematical modeling of trunks, limbs,
etc. by people sitting behind computers manipulating advanced mathematcis and
graphics packages. However, Alan is really sour on these efforts. In Alan's
judgement, the modeling has extremely limited applicabilty. One runs into all
sorts of implicit assumptions with higher degree polynomials generally applied.
The challenge is to find or develop tools that individual Ents will use if they
want to measure limb length. At this point, the spreadsheet solution seems to
be the simple answer to limb length measurement according to the proposed
definitions I previously offered. A wider search may eventually provide us with
better tools. You have proven yourself a good researcher - far better and more
patient than I am. Would you be willing to do some research on limb length and
volume modeling?
Bob
-------------- Original message --------------
From: DON BERTOLETTE <[email protected]>
Bob-
A quick thought, late at night, about your 'where to start the limb'
quandry...not that it makes your formula/estimation task easier. Earlier I
had weighed in on a biologic basis of identifying where a limb starts, and
still think there's logic to it. But in terms of measuring the volume of the
limb that extends beyond the bole, it seems relatively simple to envision the
bole continuing from the cylinder/frustrum measured immediately below the limb
flare, to a point immediately above the limb flare. Given an otherwise simple
bole (ie, no other limbs or items that affect the bole) it would be a matter of
sliding a dtape/ctape up the bole until the limb flare causes the
diameter/circumference to increase (where it would without limb otherwise
decrease)...conversely, going to a point above the limb and measuring the
diameter/circumference of the bole downward until major change is noted (where
flare rapidly 'enlarges' diameter/circumference).
I can't image there's not a known shape for that limb flare and known formuli
to quantify volume!
-DonRB
From: [email protected]
To: [email protected]
Subject: [ENTS] Limb Length Again
Date: Fri, 2 Jan 2009 17:50:47 +0000
ENTS,
Perhaps it is a good time for us to circle the wagons and discuss
where we want to go in 2009 in terms of our tree measurement mission. To this
point we have concentrated on the use of sine-based mathematics to measure tree
height, Rucker height indexing, trunk volume determinations, use of TDI as a
method to crown champion trees, and support of the champion tree programs of
the states and American Forests. Pursuit of other measurement and measurement
methodologies has been sporadic. Advances in our methodologies have been
limited principally to volume determinations.
Recently we started to think seriously about methods for computing
limb length and that led to several methods ranging from simple to
mathematically involved. In a prior email, yours truly proposed some
definitions for consideration. The following is a slight refinement of those
definitions for limb length.
Lh = Horizontal length (horizontal distance from start to end of limb)
using 2 measurement points. This is the shadow length of a limb looking
directly down from above.
Ls = Straight line distance from start to end of limb (slope distance)
using 2 measurement points, one at the tip and the other at the base of the
limb.
Lp = Parabolic arc length of limb using 3 measurement points, one at the
tip, one at the base, and one at or near the midpoint of the limb. This method
requires the derivation of the parabola that passes through the tree points.
The model is y = ax2 + bx + c.
Lr = Length based on a bivariate curvilinear regression model using
multiple measurement points. The curvilinear equation will be one of three
forms:
y = ax2 + bx + c
y = ax3 + bx2 + cx + d
y = axb + c
Lc= Length based on division of the limb into segments with each segment
measured using one of the previous methods. This is a composite length.
The sheer amount of measuring and calculating involved in
determining volumes and limb length simply overpowers the level of interest
that most Ents have. This simple stating of the obvious is not meant as a
criticism of others. It is just an acknowledgement of reality. So if the few of
us who are obsessed with measurement methodology want others among us to become
more involved, we have to provide the calculating tools needed to avoid
burdensome calculations. To this end, I’ve taken on the job of trying to
provide those tools and have chosen the form of Excel workbooks. There are
better routes, but they involve specialized software. For instance, a
statistical package named Minitab provides a wealth of statistical analysis
techniques to include regression analysis. But one must feel confident using
that kind of software and be able to purchase it. In contrast, most computer
users have Excel. So by default, it becomes the tool of choice.
In a past communication, I attached an Excel Workbook that includes
3 spreadsheets. The first shows the model of a limb and the hypothetical
position of a measurer taking 3 measurements. The formulas for the calculations
for the 3 points are shown. The spreadsheet is entitled Diagram. The second
spreadsheet fits a parabola to the 3 points by computing the coefficients of
the parabola, designated as a, b, and c. The length of the parabolic arc that
encompasses the three points is designated as s. Its evaluation requires
integral calculus. The definite integral that has to be evaluated is shown.
However, there is a numeric process for evaluating definite integrals called
Simpson’s Rule or Method. I’ve programmed in the method to calculate s
automatically. The spreadsheet is named Parabola.
The latest spreadsheet, added last night is entitled Regression and
fits a parabola to up to 10 points. The method of calculating the points is
just an extension of the 3-point system of the prior spreadsheets. The third
spreadsheet calculates the length of the parabolic arc that is based on a
parabola derived through bivariate curvilinear regression analysis. The
regression-based parabola assumes at least 4 points have been determined along
the surface of the limb and have been fitted into a Cartesian coordinate
system. Up to 10 points are allowed. I could have built in an allowance for
more, but doubt that limb length calculations will often use more than 4 or 5
points.
I have included the equations for a, b, c, and s in the Parabola
and Regression spreadsheets. The user need not concern himself/herself with the
equations. They are included for documentation purposes. In the case of the
equations, they are not always in their simplest forms, algebraically speaking.
That results from how I went about deriving them. However, since all the
calculating is performed by Excel, the user doesn’t need to be concerned with
computational efficiency.
The next step is to add the cubic equation to the mix of tools. The
cubic equation has the form y = ax3+bx2+cx+d. Its derivation is a bear, either
as an exact fit to 4 points or to more in a least squares regression sense. I
would argue for the inclusion of the cubic model because it incorporates a
point of inflection where the curve goes from concave to convex or vice versa.
This corresponds to the architecture of many limb segments that we see. So from
the standpoint of fitting a curve to the smallest number of measurement points
to follow limb curvature, the cubic model adds an important tool to our
inventory. I will begin working on a spreadsheet for the cubic model and
eventually add it to the workbook with the parabolas.
In the above discussion, I have not addressed the challenge of
deciding where to place the start of a limb. That is likely to be an ongoing
discussion. I admit to being a little sloppy on point of origin determinations
because the error associated with one system or another is manageable. I prefer
to work on the determinations that address sources of significant error and
leave the fine-tuning to the end. To employ the reverse process seems to me to
lead to the straining at a gnat and swallowing a camel syndrome.
In the attachment, I’ve cleaned things up a bit. Nothing more. Just
to reinforce the point that the user need only enter raw measurements into the
green cells. Excel does the rest. In justifying the effort that I have put into
the spreadsheet approach, use of spreadsheets seems to make sense only if they
are used and that won’t happen if mathematical expressions have to be created
by the user. In the limb length workbook, the user only needs to determine the
coordinates of the points on the limb with laser and clinometer. As a final
comment, I again acknowledge that the limb lies completely in a vertical plane
(no lateral twisting and turning). Where this condition is violated, the limb
must be broken up into segments, and of course, this approach will often be
necessitated by a variety of factors and consid eration, visibility being a big
one.
Bob
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