More Formulas

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TOPIC: Formula of Potential Value to Dale
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== 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 cross-sectional area yields the more familiar formula:

V = (H*PI)/12*[D1^2 + D3^2 + D1*D3]

Bob

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TOPIC: Convenient formula
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== 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

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TOPIC: From the Formula Factory
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== 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 point-based 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 cross-section
2. Frustum of a cone with an elliptical cross-section
3. Frustum of a paraboloid with a circular cross-section
4. Frustum of a paraboloid with an elliptical cross-section
5. Frustum of a neiloid with a circular cross-section
6. Frustum of a neiloid with an elliptical cross-section

Conifers lend themselves best to modeling by trunk sectioning. The broad-spreading 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 1/2-1/2 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 1/2-1/2 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 1/3-2/3 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 2/3-1/3 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 2/3-1/3 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 2/3-1/3 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 1/2-1/2 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 2/3-1/3 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 1/3-2/3 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 1/2-1/2 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 2/3-1/3 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 2/3-1/3 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 1/3-2/3 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 1/3-2/3 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 1/3-2/3 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 1/3-2/3 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 2/3-1/3 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 2/3-1/3 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 1/2-1/2 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 1/3-2/3 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 1/2-1/2 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 1/2-1/2 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 2/3-1/3 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 1/2-1/2 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 1/3-2/3 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 2/3-1/3 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 2/3-1/3 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 2/3-1/3 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 2/3-1/3 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 2/3-1/3 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 2/3-1/3 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 1/2-1/2 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 1/3-2/3 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 1/2-1/2 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 2/3-1/3 1/2-1/2 1/3-2/3 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 2/3-1/3 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 e-mail, 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)*[(h2-h1)*(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"

paraboloids, conoids and neiloids... :-)
xylometry sounds fun, but messy and non-repeatable

http://sres-associated.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[(H-h)/H]

If the geometric solid is that of a cone instead of a paraboloid, then,

r = R*[(H-h)/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*(H-h) + (1-f1)*(H-h)]

where 0 <= f1 <= 1 and represent the weight of the paraboloid and 1-f1 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