==============================================================================
TOPIC: Formula of Potential Value to Dale
http://groups.google.com/group/entstrees/browse_thread/thread/8d14ca1f8059bdf7?hl=en
==============================================================================
== 1 of 1 ==
Date: Sat, Oct 27 2007 1:36 am
From: dbhguru
Dale,
Thanks for the spreadsheet you sent off list. I'm glad we have the
Seneca Pine nailed down in terms of volume. Its 965 cubes is
impressive. BTW, if you plan to do more modelings with your
Macroscope 25, you might want to use the formula for a frustum of an
ellipse as listed below  if you decide to measure a diameter at a
particular height and then check for ellipticality by measuring the
diameter at the same height, but at a 90 degrees rotation on the
trunk.
Let D1 = major axis of upper ellipse of the frustum
D2 = minor axis of upper ellipse of the frustum
D3 = major axis of lower ellipse of the frustum
D4 = minor axis of lower ellipse of the frustum
H = height of frustum
V = volume of frustum
PI = 3.141593
V = (H*PI)/12*[D1*D2 + D3*D4 + SQRT(D1D2D3D4)]
Note that this formula is a little more involved than the equivalent
for a circle. In terms of the above definitions, the conical frustum
based on a circular crosssectional area yields the more familiar
formula:
V = (H*PI)/12*[D1^2 + D3^2 + D1*D3]
Bob
==============================================================================
TOPIC: Convenient formula
http://groups.google.com/group/entstrees/browse_thread/thread/ad916649629f116c?hl=en
==============================================================================
== 1 of 1 ==
Date: Thurs, Nov 1 2007 4:42 pm
From: dbhguru
ENTS,
I'm fond of developing single formulas where possible to use in a
process as opposed to implementing a long series of disjoint steps.
Since determining trunk volume has steadily grown in importance for
a group of us, from time to time, it is instructive to revisit the
modeling tools currently at our disposal.
For old growth Eastern Hemlocks, a fairly typical shape is a very
slowly tapering trunk up to where limbs become numerous and that a
sharp change in taper to a point. The overall shape appears to fit a
neiloid up to about breast height and then a frustum of a cone up to
the point of rapid taper and then a cone to the top. Figuring the
volume of the neiloid section is a challenge because of the roots.
However, the top two sections can be combined into a single formula
using the following definitions.
If:
d1 = diameter of base of lower frustum
d2 = diameter of top of lower frustum & base of upper cone
h1 = height of lower frustum
h2 = height of top cone
pi = 3.141593
H = h1 + h2
V = combined volume of the two solids
Then:
V = (pi/12) * [H * d2^2 + h1 * d2*(d1 + d2)],
The chief value of this formula is a small gain in reducing the
number of arithmetic operations. If the underlying model doesn't fit
the situation, then more frustums have to be added. Then, there is
no convenient formula to use to combine 3 or more frustums into one
formula. It can certainly be done, but the formula is unweildy and
needs to be broken down into discreet steps..
Bob
==============================================================================
TOPIC: From the Formula Factory
http://groups.google.com/group/entstrees/browse_thread/thread/aaa56531101e321f?hl=en
==============================================================================
== 1 of 4 ==
Date: Sun, Nov 11 2007 7:52 am
From: dbhguru
ENTS,
Folks, hold your hats, this submission is going to be a toad
strangler. Here goes.
We periodically broach the subject of "bigness" as applied
to the forms of trees. We investigate measures of physical size and
we consider psychological size, i.e the perception of size as it
relates not only to actual dimensions, but also to shape, especially
symmetry. In terms of just physical size, American Forests tackled
the concept of bigness decades ago and after much analysis settled
on a simple formula that incorporate the dimensions of height,
girth, and crown spread. All of us who measure are all familiar with
the formula:, but I'll repeat for those who don't measure.
Let C = circumference in inches at 4.5 feet from base
H = full height of tree in feet
S = average crown spread in feet
P = points
Then
P = C + H + (1/4)*S.
The American Forests recipe was promulgated through the state
champion tree programs and has become the standard. Although, I'm
stretching a bit, it appears to me that for the most part big tree
aficianados ceased to think creatively about the concept of bigness
and just applied the pointbased American Forests formula with an
occasional Hmm. Probably those who thought about overall comparative
measures of tree size saw the flaws in the formula, but given the
wide variety of tree shapes, accepted that compromises had to be
made and that that had been adequtely done in the American Forests
champion tree formula. So imperfect as it was, the formula was it
used to as the determiner of tree bigness. Whichever tree of a
particular species scored the highest on the American Forests
formula was considered to be the biggest of its kind. Maybe in the
distant bacground there were objections, but nothing noteworthy.
When ENTS came on the scene questioning everything, little by
little, we move toward alternative measures of tree bigness. We
adopted ENTS points as circcumference x height. The system never
really went anywhere and with good reason. Later, ENTS, throughWill
Blozan, added a new measure of tree bigness through his TDI system
which utilizes the three dimensions employed by American Forests,
but can utilize more or less. Will's system can be viewed either as
a replacement for the American Forests formula or as a companion to
it.
Most of us in ENTS recognize that neither system is sufficient to
completely tie down the concept of bigness, and Will would be the
first to acknowledge this. If I had to choose between the two
systems, I'd take TDI. If I were limited to only one system, I would
be an unhappy camper because capturing tree bigness must address the
range of tree shapes and the extreme range of shapes was forcefully
brought to our attentionby our friend Larry Tuccei Jr.. Courtesy of
Larry's awesome live oak discoveries and his generous sharing of
their images, our awareness of the complexity of tree bigness has
taken a leap forward. The bad news is that satisfactory
quantification of tree bigness through a simple formula has not
surrendered to our collective wit. Nor is it likely to at any time
in the near future. We have to come at the problem from multiple
directions.
Starting all the way back at square one, we know that in the
public's eye big trunks will generally trump great height as a
measure of tree bigness. Large spreading crowns vie with large
trunks as the most prominent measure of bigness to John Q. Citizen.
The spread of a large crown certainly creates the illusion of
bigness. I say illusion because much of the crown is empty space. If
great girth and height are combined then the combination trumps just
great girth. But what if the girth is great only near the base of
the tree and only a few feet up the trunk narrows down. We need to
somehow be able to capture the trunks full behavior and incorporate
it in our comparisons.
Well, boldly accepting that challenge, some of us have forged ahead.
Will Blozan, Jess Riddle, and myself, have set upon the numerically
intensive task of determining trunk and limb volume as the best
single measure of tree bigness. We believe that trunk and limb
volume are indispensible to deciding which among competing trees is
the bigger. If volume is then combined with the three standard
dimensions, then maybe we can crack the nut. Volume can be
incorporated into Will's TDI system as a 4th comparative measure of
bigness. Volume can then be converted into mass, if we want to head
in that direction. Either way, calculating trunk and limb volume is
our operative challenge.
Wrapping a tape around a tree at a predetermined height is something
anyone can do, but determining trunk volume is a very different
matter. Readers of the list, who have an interest in volume
calculations, well understand the importance we place on sectioning
the trunk and volume modeling each section. Adding up the volumes of
all sections gives our best estimates of the volume of the trunk to
the extent that the sections have been modeled properly.
Basically, there are 6 volume models for the sections. They are
listed below:
1. Frustum of a cone with a circular crosssection
2. Frustum of a cone with an elliptical crosssection
3. Frustum of a paraboloid with a circular crosssection
4. Frustum of a paraboloid with an elliptical crosssection
5. Frustum of a neiloid with a circular crosssection
6. Frustum of a neiloid with an elliptical crosssection
Conifers lend themselves best to modeling by trunk sectioning. The
broadspreading hardwoods create the most work. I can't even imagine
the work that those live oaks will require. So, I've been spending
time developing techniques for modeling eastern conifers the quick
and easy, but less accurate, way. My purpose is to provide members
with simpler approximating techniques. Those with reticles can do
the full Monty. One way is to divide the tree up into two sections.
Take the tree at 4.5 feet above base and apply either the neiloid or
cone frustum formula. Then apply the cone of parabaloid formula to
the upper section. An even simpler method is to consider the form of
a conifer to fall between that of a cone and paraboloid.We might
consider different mixes, such as 2/3 cone and 1/3 paraboloid, or
half and half, or 1/3 cone and 2/3 paraboloid. The formulas for the
associated solids are listed below. PI = 3.14 1593. allthe formulas.
V = (7/72)*H*PI*D^2 for 2/3 cone and 1/3 paraboloid
V = (5/48)*H*PI*D^2 for 1/2 cone and 1/2 paraboloid
V = (1/9)*H*PI*D^2 for 1/3 cone and 2/3 paraboloid
A test of the fit of each formula was applied to
50 trees in my database that have been
modeled. The following table shows the
results.
From the above data,
Species 
Hgt 
Diam 
Modeled
Vol 
Cone 
Paraboloid 
2/3Cone
and 1/3Para 
1/2Cone
and 1/2Para 
1/3Cone
and 2/3Para 
Cone
Fac 
Par
Fac 
2/3C
1/3P 
1/2C
1/2P 
1/3C
2/3P 
OG 
Best
fit 
WP 
154.40 
4.11 
954.00 
681.55 
1022.32 
795.14 
851.93 
908.73 
0.71 
1.07 
0.83 
0.89 
0.95 
YES 
Para 
WP 
173.20 
3.98 
921.00 
717.86 
1076.78 
837.50 
897.32 
957.14 
0.78 
1.17 
0.91 
0.97 
1.04 
YES 
1/21/2 
WP 
150.30 
4.40 
821.00 
761.79 
1142.68 
888.75 
952.23 
1015.71 
0.93 
1.39 
1.08 
1.16 
1.24 
YES 
Cone 
WP 
160.20 
3.98 
812.00 
663.97 
995.96 
774.64 
829.97 
885.30 
0.82 
1.23 
0.95 
1.02 
1.09 

1/21/2 
WP 
140.60 
4.60 
789.00 
778.88 
1168.32 
908.69 
973.60 
1038.50 
0.99 
1.48 
1.15 
1.23 
1.32 
YES 
Cone 
HM 
116.00 
3.98 
757.00 
480.78 
721.17 
560.91 
600.98 
641.04 
0.64 
0.95 
0.74 
0.79 
0.85 
YES 
Para 
WP 
121.30 
4.00 
713.00 
508.10 
762.15 
592.78 
635.13 
677.47 
0.71 
1.07 
0.83 
0.89 
0.95 
YES 
1/32/3 
WP 
163.30 
3.76 
695.00 
603.14 
904.71 
703.66 
753.93 
804.19 
0.87 
1.30 
1.01 
1.08 
1.16 

2/31/3 
WP 
161.80 
3.72 
679.00 
587.52 
881.27 
685.43 
734.39 
783.35 
0.87 
1.30 
1.01 
1.08 
1.15 

2/31/3 
WP 
147.00 
3.72 
632.00 
533.77 
800.66 
622.74 
667.22 
711.70 
0.84 
1.27 
0.99 
1.06 
1.13 

2/31/3 
WP 
157.30 
3.47 
604.00 
495.74 
743.60 
578.36 
619.67 
660.98 
0.82 
1.23 
0.96 
1.03 
1.09 

1/21/2 
WP 
169.10 
3.31 
570.00 
485.15 
727.73 
566.01 
606.44 
646.87 
0.85 
1.28 
0.99 
1.06 
1.13 

2/31/3 
WP 
183.10 
3.51 
569.00 
590.57 
885.85 
689.00 
738.21 
787.43 
1.04 
1.56 
1.21 
1.30 
1.38 
YES 
Cone 
WP 
141.90 
3.31 
548.00 
407.12 
610.67 
474.97 
508.89 
542.82 
0.74 
1.11 
0.87 
0.93 
0.99 

1/32/3 
WP 
157.00 
3.25 
547.00 
434.15 
651.22 
506.50 
542.68 
578.86 
0.79 
1.19 
0.93 
0.99 
1.06 

1/21/2 
WP 
125.00 
3.31 
520.00 
358.63 
537.94 
418.40 
448.29 
478.17 
0.69 
1.03 
0.80 
0.86 
0.92 

Para 
WP 
156.60 
3.50 
516.00 
502.22 
753.33 
585.93 
627.78 
669.63 
0.97 
1.46 
1.14 
1.22 
1.30 

Cone 
WP 
130.00 
3.70 
507.00 
465.92 
698.89 
543.58 
582.41 
621.23 
0.92 
1.38 
1.07 
1.15 
1.23 
YES 
2/31/3 
WP 
151.00 
3.40 
503.00 
456.99 
685.48 
533.15 
571.23 
609.32 
0.91 
1.36 
1.06 
1.14 
1.21 

2/31/3 
WP 
120.00 
3.00 
482.00 
282.74 
424.12 
329.87 
353.43 
376.99 
0.59 
0.88 
0.68 
0.73 
0.78 

Para 
WP 
118.00 
3.10 
465.00 
296.88 
445.31 
346.35 
371.09 
395.83 
0.64 
0.96 
0.74 
0.80 
0.85 

Para 
WP 
146.10 
2.90 
445.00 
321.67 
482.51 
375.29 
402.09 
428.90 
0.72 
1.08 
0.84 
0.90 
0.96 

1/32/3 
WP 
120.80 
3.10 
409.20 
303.92 
455.88 
354.57 
379.90 
405.23 
0.74 
1.11 
0.87 
0.93 
0.99 

1/32/3 
WP 
107.00 
3.20 
395.00 
286.85 
430.27 
334.66 
358.56 
382.46 
0.73 
1.09 
0.85 
0.91 
0.97 

1/32/3 
WP 
122.90 
3.30 
372.00 
350.39 
525.58 
408.79 
437.98 
467.18 
0.94 
1.41 
1.10 
1.18 
1.26 
YES 
Cone 
WP 
110.50 
3.10 
361.10 
278.01 
417.01 
324.34 
347.51 
370.67 
0.77 
1.15 
0.90 
0.96 
1.03 

1/32/3 
WP 
126.80 
2.80 
309.10 
260.26 
390.39 
303.63 
325.32 
347.01 
0.84 
1.26 
0.98 
1.05 
1.12 

2/31/3 
WP 
115.30 
2.90 
288.90 
253.86 
380.79 
296.17 
317.32 
338.48 
0.88 
1.32 
1.03 
1.10 
1.17 

2/31/3 
WP 
107.30 
2.80 
284.00 
220.23 
330.35 
256.94 
275.29 
293.65 
0.78 
1.16 
0.90 
0.97 
1.03 

1/21/2 
WP 
105.00 
2.70 
275.00 
200.39 
300.59 
233.79 
250.49 
267.19 
0.73 
1.09 
0.85 
0.91 
0.97 

1/32/3 
WP 
112.70 
3.10 
272.00 
283.54 
425.31 
330.80 
354.43 
378.05 
1.04 
1.56 
1.22 
1.30 
1.39 

Cone 
HM 
127.10 
2.56 
271.00 
218.07 
327.10 
254.41 
272.59 
290.76 
0.80 
1.21 
0.94 
1.01 
1.07 

1/21/2 
WP 
141.10 
2.60 
256.30 
249.71 
374.57 
291.33 
312.14 
332.95 
0.97 
1.46 
1.14 
1.22 
1.30 

Cone 
WP 
120.70 
2.50 
244.90 
197.49 
296.24 
230.41 
246.87 
263.33 
0.81 
1.21 
0.94 
1.01 
1.08 

1/21/2 
WP 
120.00 
2.60 
239.00 
212.37 
318.56 
247.77 
265.46 
283.16 
0.89 
1.33 
1.04 
1.11 
1.18 

2/31/3 
HM 
104.50 
2.67 
237.30 
195.03 
292.55 
227.54 
243.79 
260.04 
0.82 
1.23 
0.96 
1.03 
1.10 

1/21/2 
WP 
138.40 
2.60 
232.00 
244.94 
367.40 
285.76 
306.17 
326.58 
1.06 
1.58 
1.23 
1.32 
1.41 

Cone 
WP 
121.20 
2.40 
220.90 
182.77 
274.15 
213.23 
228.46 
243.69 
0.83 
1.24 
0.97 
1.03 
1.10 

1/32/3 
WP 
132.40 
2.10 
216.00 
152.86 
229.29 
178.34 
191.08 
203.81 
0.71 
1.06 
0.83 
0.88 
0.94 

2/31/3 
WP 
110.60 
2.60 
201.40 
195.74 
293.60 
228.36 
244.67 
260.98 
0.97 
1.46 
1.13 
1.21 
1.30 

Cone 
WP 
125.70 
2.30 
199.70 
174.08 
261.13 
203.10 
217.61 
232.11 
0.87 
1.31 
1.02 
1.09 
1.16 

2/31/3 
WP 
113.40 
2.20 
164.80 
143.69 
215.54 
167.64 
179.61 
191.59 
0.87 
1.31 
1.02 
1.09 
1.16 

2/31/3 
WP 
114.70 
2.18 
158.60 
142.71 
214.06 
166.49 
178.38 
190.28 
0.90 
1.35 
1.05 
1.12 
1.20 

2/31/3 
WP 
108.00 
2.20 
152.00 
136.85 
205.27 
159.66 
171.06 
182.46 
0.90 
1.35 
1.05 
1.13 
1.20 

2/31/3 
WP 
118.40 
2.02 
144.70 
126.48 
189.72 
147.56 
158.10 
168.64 
0.87 
1.31 
1.02 
1.09 
1.17 

2/31/3 
WP 
84.00 
2.00 
108.60 
87.96 
131.95 
102.63 
109.96 
117.29 
0.81 
1.21 
0.94 
1.01 
1.08 

1/21/2 
WP 
101.30 
1.63 
91.40 
70.46 
105.69 
82.21 
88.08 
93.95 
0.77 
1.16 
0.90 
0.96 
1.03 

1/32/3 
WP 
123.50 
1.50 
77.80 
72.75 
109.12 
84.87 
90.93 
97.00 
0.94 
1.40 
1.09 
1.17 
1.25 

Cone 
WP 
106.60 
1.48 
75.10 
61.13 
91.69 
71.32 
76.41 
81.51 
0.81 
1.22 
0.95 
1.02 
1.09 

1/21/2 
WP 
99.60 
1.40 
55.40 
51.11 
76.66 
59.63 
63.88 
68.14 
0.92 
1.38 
1.08 
1.15 
1.23 

Cone 
















Avg 








0.84 
1.25 
0.98 
1.05 
1.12 


Over/Under 







0.16 
0.25 
0.02 
0.05 
0.12 


Std
Dev 








0.11 
0.16 
0.13 
0.13 
0.14 


















Best
Fit Summary 













Cone 
Para 
2/31/3 
1/21/2 
1/32/3 
Total 










11 
5 
15 
10 
9 
50 










Examining the above table, the formula V =
(7/72)*H*PI*D^2 fits best. This formula is based on 2/3 cone and 1/3
paraboloid and is the best fit in 15 out of 50 times,but has a
standard deviation of 0.13, so misses can be substantial. Old growth
specimens are unruly as to be expected. If a tree's shape is very
conical, then the cone formula should be used. If the tree is
extremely columnar, then the paraboloid formula should be used. If
it isn't clear, then the 2/31/3 can be justified based on the above
table. Let commonsense rule.
The above formulas are meant for quick and dirty use in volume
modeling. They apply to the whole trunk. And consequently, they have
a built in flaw, especially for old growth forms that often show two
or three different trunk forms. A
They don't replace the need for a more rigorous modeling effort,
which requires sectioning. If sections are short, then it is
appropriate to apply the volume formula for a frustum of a cone:
V = (H*PI/12)*[D1^2 + D2^2 + D1*D2]\
If the frustum is longer and has noticeably convex sides, then we
can apply the formula for a frustum of a paraboloid, which is:
V = (H*PI/8)*[D1^2 + D2^2].
If curvature is slight, then the form of the frustum may lie between
a cone and a paraboloid. The formula for a frustum that lies half
way between cone and paraboloid is:
V = (H*PI/48)*[5*D1^2 + 5*D2^2 + 2*D1*D2]
Since these intermediate forms are a judgement calls, I do not
extend the frustum formulas to include (2/3)*C + (1/3)*P and (1/3)*C
+ (12/3)*P. As explained, the formula implemnets (1/2)C + (1/2)P.
Well, I've probably confused everyone enough, so I'll shut it down
here and continue the thread in another email, where I will show
that you can modeled a tree trunk with a minimum of diameter
measurements, but still take changes in shape into account. The
process will never equal the reticle process, but it can get one
into the ball park.
Bob
== 2 of 4 ==
Date: Sun, Nov 11 2007 7:42 pm
From: dbhguru
ENTS,
The formula factory is at it again. Thinking abstractly about trunk
modeling on the cheap, the alternative to dividing the trunk into
multiple segments and modeling each one as a frustum of a cone or
some other geometric solid is to postulate a geometric solid that
takes in most or all the trunk. Evidence is then sought to justify
the choice of solid. For instance, the huge eastern hemlocks of the
Smokies often appear columnar, but they do taper, just slowly.
However, their actual form for the first 100 feet or so may be close
to the paraboloid. If that is the case then we might be able to come
close to the actual volume of the first 100 feet by computing the
volume of a frustum of the parabola. But how much volume is in a
section that starts at height h1 and goes through h2 of a paraboloid
that extends from the base to the tip? If the tree has height H and
radius R at the base or at the point where the paraboloid shape is
assumed t o start, then the equation of the parabola t
hat represents the shape of the paraboloid sliced vertically in half
is:
h = H  (H/R^2)*r^2
Using Intergral Calculus, we can develop a formula that represents
the volume of a frustum of the paraboloid. The derivation can be
supplied if anyone is interested. Otherwise, the formula is:
V = (PI*R^2/H)*[(h2h1)*(H(h2+h1)/2)]
We can use this formula to test our hypothesis of the paraboloid
shape between h1 and h2. We can use the Macroscope 25/45 to volume
model to a high degree of accuracy the volume of the trunk between
heights h1 and h2 and compare the result to the volume from the
above formula betwen h1 and h2.
We are at the beginning of this type of analysis. Much more to come.
Bob
== 3 of 4 ==
Date: Sun, Nov 11 2007 8:10 pm
From: "Lawrence J. Winship"
You may already know about this web site  nice equations for
paraboloids, conoids and neiloids... :)
xylometry sounds fun, but messy and nonrepeatable
http://sresassociated.anu.edu.au/mensuration/volume.htm
== 4 of 4 ==
Date: Sun, Nov 11 2007 9:58 pm
From: dbhguru
Larry,
I'd visited the site a few times before, but had then lost track of
it. Thanks for the URL. I'll peruse the site again.
Actually, as I recall, their approach seemed a little fanciful to
me. My experience is that the more complex the form of the form or
volume equation the less useful it is, the less it can be
productively applied to individual trees. Complex exponential and
logarithmic equations are more academic exercises than intended for
field use. Nonetheless, all approaches go into the hopper as grist
to stimulate the neurons.
Hey, Larry, when can we get together so that I can show you the
TruPulse 360 and discuss a crown mapping project? Also, I'd love to
present my PowerPoint presentation on Dendromorphometry to one of
your classes. Any possibility of that?
Bob
ENTS,
Imagine a paraboloid shape circumscribed about
the trunk of a tree in such a way that the bases of the paraboloid
and tree coincide and the tip of the trunk concides with the
vertex of the paraboloid. Assume the following definitions:
R = Radius of base of paraboloid
H = Height of paraboloid
r,h = point on the parabola generated by slicing
the paraboloid vertically. The point will lie on the trunk if
the paraboloid and trunk match.
For any height h, the corresponding radius r is
computed by:
r = R*SQRT[(Hh)/H]
If the geometric solid is that of a
cone instead of a paraboloid, then,
r = R*[(Hh)/H]
Now, we can also compute r where r is intermediate
between the cone and the parabloid by using:
r = (R/H)*[f1*SQRT(H*(Hh) + (1f1)*(Hh)]
where 0 <= f1 <= 1 and represent the
weight of the paraboloid and 1f1 is the weight of the cone. If
the two solids are weighted equally, then f1 = 0.50.
We can use this last formula to
investigate the taper of the southern Appalachian hemlocks. Will
has listed the circumferences at base and at 100 feet. The
following table analyzes taper of 7 hemlocks. Scroll to the right
to see rightmost columns
