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 _________________________________________________________________ Send e-mail faster without improving your typing skills. http://windowslive.com/online/hotmail?ocid=TXT_TAGLM_WL_hotmail_acq_speed_122008 --~--~---------~--~----~------------~-------~--~----~ Eastern Native Tree Society http://www.nativetreesociety.org You are subscribed to the Google Groups "ENTSTrees" group. To post to this group, send email to [email protected] To unsubscribe send email to [email protected] For more options, visit this group at http://groups.google.com/group/entstrees?hl=en -~----------~----~----~----~------~----~------~--~---
