Midpoint Procedure   Robert Leverett
  Apr 10, 2006 07:58 PDT 
ENTS,

     I spent my Sunday with PhD candidate Nancy Rich from Antioch
Graduate School collecting data in the Trees of Peace area of MTSF.
We've undertaken a new project to gather at least 50 sample points for
white pines. We will measure their diameters at the midpoint of their
full heights. The midpoint diameters will then be compared to what would
be obtained from treating the white pine form as a regular cone with
base at just above the root flare. We have defined the process to use
for locating the midpoints of the pine sample and getting measurements
at the midpoints with a monocular such as the Macroscope 25. The process
follows:

Definitions:

Let

      d1 = hypotenuse distance from eye to base of trunk
      d2 = hypotenuse distance from eye to point at half way up trunk
(midpoint of trunk)
      d3 = hypotenuse distance from eye to top of trunk.
      d4= straight line distance from eye to point on trunk at eye
level
      a1 = angle from eye to base of trunk
      a2 = angle from eye to point on trunk at midpoint
      a3 = angle from eye to top of trunk
      h1 = trunk length below eye level
      h2 = trunk length from eye level to midpoint
      h3 = trunk length from eye level to top of trunk
      h4 = trunk length from bottom of trunk to midpoint, i.e. half of
total trunk length
      H = total trunk length
      MD = diameter at midpoint as calculated using monocular
      CD = diameter at midpoint, assuming conical shape to the trunk
      BD = base diameter of tree (just above root flare)
      mm = recticle value from monocular expressed in millimeters
      F = factor to use with monocular to calculate diameter from mm
reading. For the Macroscope 25, it is 75 as determined by Will and Jess.

   
Formulas:

      (1) h1 = d1(sin(a1))   

      (2) d4 = d1cos(a1)   

      (3) h3 = d3sin(a3)

      (4) H = h1 + h3     Steps (1), (3), and (4) are common to all
Ents using the sin-sin method.

      (5) h4 = H/2    

      (6) h2 = h4 - h1

      (7) a2 =arctan( h2 / d4) arctan is the inverse tangent function.
It returns the angle corresponding to a calculated tangent value. The
familiar formula is tan(a2) = h2/d4.

      (8) Spot the midpoint on trunk at angle a2. From eye-level go up
the trunk a2 degrees using clinometer and observe where the point is on
the trunk.

      (9) Use laser to measure the distance from the eye to the
midpoint. This is the distance d2.

      (10) MD = (mm)d2/F.

      Assuming the trunk is in the shape of a perfect cone, the diameter
at the midpoint of the trunk based on a conical shpe would be:

      (11) CD = (h4/H)BD

      The objective of the new project is to develop a data set based on
the differences MD-CD. We would calculate the differences as a percent
of the CD values. Perhaps develop a regression model. From what the eye
perceives, young pine trunks can form almost perfect cones. This conical
shape continues well into maturity. The Jake Swamp pine measures 167.3
feet tall and has a base diameter of 3.61 feet just above the root
flare. There is between 1.25 and 1.5 feet of vertical root flare on the
mid-slope sides of the tree. At the midpoint of the trunk, Nancy and I
measured the diameter at 1.96 feet. A perfect cone would have given 1.8
feet. The difference is 8.9%. Treating the Jake tree like a perfect cone
from base to top, the volume would be 571 cubic feet. The last full
modeling we did for the Jake tree puts its volume at 563 cubes. That
represents a variance of only 1.3%. BTW, this is pretty much what the
eye sees. Previous modelings of the Grandfather pine in MSF yielded
volumes from 930 to 1004 cubic feet. The conic volume via the above
method yields 1066 cubic feet. The jury is still out on whether the
Grandfather tree makes 1000 cubes. It is close and there is no doubt
that when limbs are included, it makes the 1000-cube club.   

      Refinements of methods such as the above 11-step process above
will be included in the ENTS book on Dendromorphometry.

Bob

Robert T. Leverett
Cofounder, Eastern Native Tree Society
Midpoint Procedure   Robert Leverett
  Apr 10, 2006 
Will, (individual email)

For what it is worth, the attachment provides the step by step process that I used this past weekend to compute diameter at the midpoint of the trunk of the Jake Swamp and Frank Decontie trees. Nothing special about the process. It's straightforward stuff, but I thought it was time to begin formalizing these new procedures that we're using in preparation for "Dendromorphometry". I'll follow this spreadsheet up with a more generalized version that computes the diameter at any specified height, not just the midpoint. On Sunday, I did calculate positions on the trunks of the named pines other than at their midpoints, but I will be concentrating on just the midpoint for a while. 

From the small amount of modeling that I've done so far with the Macroscope, it looks like the conical representation of the trunk of a single-stemmed white pine that is not too old can be done for many, many trees and that these representations can serve as checks on what we do as more detailed volume determinations. bTW, I did do the calculation for Monica's pine yesterday evening just before the light failed. The base diameter of Monica's pine is 2.83 feet. So its projected diameter at the midpoint is 1.42 feet. My calculated value in the lest than perfect light was 1.31 feet. Not bad. As I mentioned over the phone, I expect the conical volume based on the base of the tree being just above the root collar to often slightly overshoot the volume that we come up with by segmentation. At least this is what I expect for young to middle-aged forest-grown trees. I think the older ones will go the other way, but perhaps only slightly. 

It seems to me that one of the primary values of our planned book will be the compilation of procedures that can be applied in a series of simple steps for folks who are not comfortable thinking through the mathematics on their own. So, every ENTS method would have its procedure like a military manual with checklists galore. Thoughts, anyone? 

Bob

 
Midpoint Procedure - Improved Worksheet   Robert Leverett
  Apr 13, 2006 
From: "Leverett, Bob" 
13 Apr 2006 14:34:19.0966 

Folks,

The attached Excel workbook is a slight improvement over prior ones for computing diameters at the midpoint of the trunk and comparing the results to an assumed conical form. The first sheet of the pair presents the computational model. The cells having red entries are for the raw data. One column can be either - the distance from eye to the point on the trunk for which the diameter will be measured. I did this because one can quickly shoot the point with the laser, but it can be calculated from other data, i.e. the secant of the angle to the proportion point times the level distance to the trunk, assuming insignificant lean. We all recognize that the computational approach helps when intervening branches make it difficult to get an accurate trunk bounce at the proportion point. The second sheet provides a data enty format for recording the raw data, but includes the whole process and shows the calculations that need to be made, if the measurer wants to do the calculations in the field. 

I believe that the assumption of a conical relationship as defined from the base just above the root falre to tip of crown is going to hold fairly well at the mid-point of the trunk at least for white pines under 200 years of age. I wonder about ponderosas, loblollies, etc. Of course hemlocks need to be considered sooner or later. The Excel model allows for an proportion of trunk. I'm presently going only to the midpoint.

After I get enough trees modeled at the midpoint to see if the conical form fits, I'll look at going to other spots on the trunk. I don't have the time now to do the whole tree at multiple points along the trunk. I'm grabbing a tree or two at the most when I get off work. By limiting myself to the midpoint, I can add data points quickly and see if I'm spinning my wheels. From the data collected thus far, it looks like I'm not. 

Ideally, we will eventually find a combination of diameters that box in the total trunk volume as determined by complete modelings so that we can first quickly make calculations on a new tree to see if we want to go the whole way. I would think that we could get by with 4 diameters for sure, but possibly 3. I'd be surprised if just the base and midpoint do it.

I'm sure that computer modeling is the more powerful and efficient way to go, but at this point, one would need to spend increasingly many hours in front of the computer instead of out in the woods. That would suck big time.

I have a hard time quality checking my own work, so there may be notational errors in the spreadsheet. I don't think the actual formulas in the cells have any errors. 

Bob