ENTS, Bob,
First let me apologize for the two empty posts that appeared from.me regarding
this matter. I was having some computer problems, and it appears that the
problem was simply my mouse going bad and was multi-clicking everything.
Mathematical discussions are hard to articulate and I am never satisfied with
the explanations and arguments that I present. Bob, this is mostly directed at
you, please forgive my referring to you in the third person. as I am not sure
how else to state it. In Bob's original post he provided these three
formulas:
Summary
ABH = area of trunk at breast height,
ARH = area of trunk at root collar height,
H = full height of tree,
F = form factor,
There is lots more to come on this topic along with lists of pines based on
the proposed criteria, but to summarize. As a first cut, if the pine is young
use:
VEY = 0.333 * ABH * H.
If the tree is a stocky old-growth specimen, use:
VEO = 0.333 * ARH * H
If the tree is intermediate in form and age, use:
VEI = 0.333 * H * (ABH + ARH)/2
For a particular tree, as more measurements are taken, the F value can
adjusted to better fit the observed form.
These represent the general equations for volume calculations in white pines in
his data sets. The first formula: VEY = 0.333 * ABH * H. the volume is equal
to the form factor for a cone (0.33) x (the cross-sectional area at breast
height) x (tree height). Bob reports: Young white pines hold to a conical
shape with surprising consistency and the conical volume using BH comes pretty
close to a more thoroughly modeled form. The number of pines I have modeled to
arrive at this conclusion numbers around 150. I believe this a a reasonable
formulation that is supported by a large data set. I have no complaints on
this matter.
In the second formula presented Bob says: If the tree is a stocky old-growth
specimen, use: VEO = 0.333 * ARH * H This is the formula I am trying to
discuss. In the initial characterization the tree is described as a "stocky
old-growth." It is not conical in shape, yet in the formula Bob has chosen to
use the F or shape factor for a cone as the first parameter. Since this tree
is larger in volume than a cone, if you keep the F or shape factor for the
cone, one of the other parameters must be increased to represent the increased
volume of the tree. The height is pretty straight forward, so it was left
alone. In the formula, instead of changing the F value Bob has opted to change
the cross-sectional area used in the tree. He is using the cross-sectional
area of the tree at the root collar instead of at breast height. This number
will be bigger than the cross-sectional area at breast height, therefore will
generate a bigger number for volume. We know the volume of the stocky
old-growth tree is larger, this manipulation generates a bigger volume number,
therefore there will be a positive correlation between the two. This is
exactly what the formulas say....
The problem is that by changing the cross-sectional area instead of the F or
shape factor for the tree, the volume modeled is not the same shape as the
tree. The lower portions of the trunk will be exaggerated in volume, while the
upper portions will be under-estimated. The hope is that the amount of
exaggeration at the bottom is exactly the same as the amount the top is
under-estimated, thus yielding a good volume for the tree. For example a cone
with a cross-sectional area of its base three times larger than that of a
cylinder of the same height, will have exactly the same volume as the cylinder,
even though they have a different shape. What I don't see is why the
cross-section at the top of the root collar should be that balancing point
where the bottom exaggerations and the upper under-estimates match. There
surely is some place where they do match, but why should it be this
cross-sectional area of the tree at the root collar?
The third formula suggests that some trees are intermediate in shape between
the conical young trees and the stocky old-growth trees. Sure I can buy that,
but the formula states that the value for the intermediate form will be a
cross-sectional area somewhere between that at breast height and that at the
root collar. Why should breast height and root collar be the boundaries?
Breast height is reasonable as it can be demonstrated by field measurements,
but as far as I can see the root collar cross-section is an arbitrary value
that may be higher or lower than is appropriate for a given set of trees. If
you are just picking out a cross-sectional area from a given range, knowing
that the number you pick out does not actually represent the shape of the tree
but a hoped for balance point of errors, would it not be better to pick out
from a range a value that actually represented the shape of the tree? A range
of F shape factors? Bob says: For a particular tree, as more measurements are
taken, the F value can adjusted to better fit the observed form. Would this be
an adjustment to F as used in the first formula, or does he mean an adjustment
as a fudge factor to the F in the second equation, and therefore not really
representing the overall form of the tree at all?
What I am suggesting is that instead of picking the cross-sectional area at the
root collar and hoping it is somewhere close to the theoretical balance point
between lower exaggerations and upper under-estimates, it would be better to
find some repeatable protocol for estimating the F or shape factor for a given
species of trees in an area, and keep the basic cross-sectional are at breast
height as a constant in the formula. That way you are changing the true
variable in the formula, shape of the trunk, to fit the tree being examined,
instead of just picking a value at the root collar and hoping it will be OK.
This would be a simpler way to deal with the different shapes of trees and a
sounder approach in my opinion. I anticipate that there will be a different F
shape factor for each of the different classes of trees being examined and that
a pattern will emerge. I am not sure, and don't believe that this same
pattern will be observable if volume is calculated by manipulating the
cross-sectional area. Another advantage of developing a protocol for assigning
an F value would be that trees with unusually wide basal flairs or those who
have had their tops broken out will tend to fall into separate ranges of F,
rather than being intermixed as might be the case using formula 2.
Ed Frank
"Two roads diverged in a yellow wood, And sorry I could not travel both. "
Robert Frost (1874–1963). Mountain Interval. 1920.
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