Limb Length Again

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Limb Length Again 1/2/09 9:50 AM



            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.  



Re: [ENTS] Limb Length Again Edward Frank 1/2/09 1:55 PM
Looking at your diagram there are a coupe of practical notes that must be made.  First the distance and angle measurements must be taken from a point in the plane of the parabola, i.e.. under the limb. 
Looking at the illustration I see that angle 1 (a1) is in a different direction than are a2 and a3.  Is this accounted for in your calculations?  Need it be listed as an angle > 90 degrees.  In the diagram, must you be to the left of point d2.  How do you deal with the different points being on the opposite side of the person making the measurements?
"The most beautiful thing we can experience is the mysterious. 
It is the source of all true art and all science." - Albert Einstein
Re: [ENTS] Re: Limb Length Again 1/2/09 2:55 PM
     You are absolutely correct. The measurer must be within the plane of the parabola and the eye of the measurer is the origin of the Cartesian coordinate system that includes the limb. In this case, points above the origin yield positive Y values, but points on opposite sides of the origin have signs in terms of the X values. This is a point I didn't address adequately in the description. I should explain the sign change. Every thing works out when these conventions are followed. It is a darn good thing that you are willing to run interference and interpret for me. Thanks again.
     As a fianl comment, the system only works for limbs that lie within the same vertical plane.
RE: [ENTS] Limb Length Again DON BERTOLETTE 1/2/09 11:53 PM
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!

Subject: [ENTS] Limb Length Again
Date: Fri, 2 Jan 2009 17:50:47 +0000
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Re: [ENTS] Re: Limb Length Again 1/3/09 6:16 AM
      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 res earch on limb length and volume modeling?
RE: [ENTS] Re: Limb Length Again DON BERTOLETTE 1/3/09 4:50 PM
On the first, my description would be limited to that within reach...the use of laser hypsometers and such to constrain limb flare measurements would at best be problematic, but at worst, better than most alternatives.  One thing for sure...short of full immersion/displacement measurements, all volume determinations would be estimates.

On the second, thanks for the kind words, but I'd be more inclined to look to remote sensing as a solution.  I've not lately revisited the LIDAR research on determining tree/forest volume, but will soon be in contact with a friend who will be employing it for remote Alaskan forests.  I realize that it isn't yet a solution for all applications, but clearly advances are being made.

Subject: [ENTS] Re: Limb Length Again
Date: Sat, 3 Jan 2009 14:16:50 +0000
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Re: [ENTS] Re: Limb Length Again Edward Frank 1/3/09 5:09 PM
Where is the beginning of the limb?  People have been avoiding this questions since we first discussed about whether for limb length it should be considered to be at the surface of the trunk or at the center of the tree. 
The trunk of the tree  for a short section may be thought of as a cylinder.  When a limb sprouts from the trunk, a section of wood bulges outward from this idealized trunk shape forming a root collar.  This root collar forms a buttress at the base of the limb.  The limb itself extends outward from the center to upper portion of the root collar.  I propose for measurement purposes that:
"The base or start of the limb is considered to be where the center of the limb would intersect the regular surface of the trunk."
As with the definition of the base of the tree on a slope, this definition would require a minor bit of extrapolation, but it would eliminate any variation in measurement points caused by different sized limb collars, and as a practical matter would mark the functional change between trunk and limb structures.
Edward Frank
"The most beautiful thing we can experience is the mysterious. 
It is the source of all true art and all science." - Albert Einstein
Re: [ENTS] Re: Limb Length Again 1/3/09 5:21 PM
  To quote you: "The base or start of the limb is considered to be where the center of the limb would intersect the regular surface of the trunk." Your definition is basically the one I employ now, even if sloppily.
RE: [ENTS] Re: Limb Length Again DON BERTOLETTE 1/3/09 9:24 PM
It probably doesn't matter much as long as the method is explained, as all volume determinations will be an estimate that has an inherent error (inherent in that the branch flare around the bole is generally irregular at best).


Subject: [ENTS] Re: Limb Length Again
Date: Sun, 4 Jan 2009 01:21:03 +0000

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