White Pine Volume Modeling  
  

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TOPIC: Rejuvenated White Pine Lists and Volume Modeling
http://groups.google.com/group/entstrees/browse_thread/thread/e37922c4f666f795?hl=en
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== 1 of 6 ==
Date: Tues, Nov 11 2008 2:26 pm
From: dbhguru@comcast.net

ENTS

The recent ENTS rendezvous in western Massachusetts has energized me to return to specialized big/tall tree lists. Beyond an interest in the knowledge encapsulated by the lists, my efforts are motivated by an awareness of informational gaps that need to be filled if the public is to understand and support our heritage big trees and stands of trees. In the case of the various white pine lists that Will Blozan, Dale Lutheringer, and I have conceived over the years, those lists are in recognition of the historic role of the white pine. This role especially applies to New England, but awareness of the history of the species has been diluted by a variety of factors, most of which work against preserving the impressive stands of white pine that we have remaining. In a nutshell, if people do not know what is significant or what will soon become significant, then preservation efforts will not likely be successful.
To look backward in time, Pinus strobus was THE tree species in colonial New England. White pines were coveted for ship masts by the English monarchy and were widely used for construction purposes by the colonists. However, for a period of decades, the great whites lost most of their value due to the white pine weevil and the white pine blister rust, but the species has largely survived those threats and is still the eastern United Statesís tallest species. The white pine is definitely New Englandís flagship species for stature. Without it, our site Rucker Indexes would suffer greatly. In my humble view, because of its historical importance and stature, the species deserves our respect beyond the mundane valuing of it for timber purposes. But for the public to value our heritage white pines, reliable information must be available on individual trees and stands of trees - information that has heretofore been very sparse.
To acknowledge what has been done, there are a few New England sites that have been preserved because of their large and/or old white pines. Sites that come immediately to mind include the Bradford and Tamworth Pines in New Hampshire, the Carlisle Pines in Massachusetts, the Cathedral and Gold Pines in Connecticut, and the Scott Fisher Memorial and Cambridge Pines in Vermont. Exemplary sites such as MTSF, Ice Glen, and the Bryant Homestead, all in Massachusetts, escaped notice until recently. Now thanks to the ENTS website, masterfully created and maintained by our webmaster Ed Frank, people who do Internet research can find dimension-based information and qualitative narratives on the white pine that put into context our beliefs about what may have grown in yesteryear as well as what is out there today. We are actively documenting white pine sites and gradually homing in on the genetic capabilities of the species across its geographical range. The latter mission has th
e greatest scientific value.
I believe it is in our interest to expand our white pine database and organize it for convenient web-based maintenance and for general public access, but this mission will take time and needs our collective input. For the present, we can construct a list of important trees. To this end, I recently proposed to Will a criteria for inclusion of white pines in a list that for the present is aimed predominantly at the Northeast. He tentatively agreed. We would like the input from others. Basically, a pine would be included in the list if it meets any of the following criteria:

1. Is 150 feet tall or more (maybe less in northern New England),
2. Is 12 feet in circumference or more,
3. Earns 1500 ENTS points or more (ENTS Pts = girth^2 * height/10),
4. Has a modeled trunk volume of 500 cubic feet or more.

These criteria are sufficiently strict in terms of what is growing now that the list, at least in the Northeast, isnít in danger of becoming so extensive as to give the idea that trees meeting any of the above criteria are everywhere common. That certainly is not the case and the point will need to be emphasized.
The most difficult of the above criteria to apply is #4, the modeled trunk volume, which is the subject of the remainder of this email. Fortunately, there are shortcuts to allow us to approximate volume based on the general formula:

V = F * A * H

where A = area of the base at a designated height (such as 4.5 feet),
H = full height of tree, and
F = the form factor that lies between 0.333 and 0.50.

For those who want to review the calculation of A, it can be done through any of the following formulas:

A = PI * R^2

A = PI * D^2/4

A = C^2/(4 * PI)

Where R is radius, D is diameter, and C is circumference (or girth).

For forest-grown pines in the age-class of 150 years or more, the F factor will commonly be between 0.38 and 0.44. Stocky old-growth outlier pines may achieve a factor between 0.45 and 0.47. I doubt any pine will be 0.5, which is the factor that determines the paraboloid shape. The overall shape of a trunk of a mature pine characteristically begins as a neiloid (F=0.25), change to paraboloid (F=0.5), and then into a cone (F=0.333), but a single F value can be used to calculate the volume of a pine. Determining the value of F for a particular tree is our challenge.
For initial inclusion of a pine in the list, the F factor can be estimated if a more exacting determination canít be made such as through use of the Macroscope 25/45. I acknowledge that a lot of work remains to be done on figuring out how to derive the F factor for a pine to get a quick volume approximation, but at this juncture, we are not helpless. We have a good head start.
From data weíve collected thus far, it is safe to assume all but the highly columnar pines will have a trunk volume that is less than the calculated value achieved by taking the cross-sectional area just above the root collar, the full height of the tree, and an F value of 0.333. By contrast, single-trunk old-growth specimens are proving to have volumes very close to the calculated volume using the base set at the root collar, with some trees requiring an F value as high as 0.35 or even 0.37.
At the least, we presently have a strategy for homing in on the trunk volume by first surrounding it with high and low volume estimates. We then can refine the estimate by choosing an F value that appears to match the trunk. Judgment is involved, but with experience, we can eventually reduce the error to an acceptable level without being forced to fully model a tree. A fact obvious to me now that was not several years ago is that relatively few of us in ENTS are driven to calculate trunk volumes through full trunk modeling and do other kinds of tree modeling that is numerically intensive. So, if the few of us want volume-based lists created by more than just ourselves, we are going to have to come up with handy ways of calculating volumes that involve a minimum of calculations and that means refining the application of the F value. Today we are a lot closer to doing that than we were a couple of years ago.
At this point, I feel confident in saying that for young pines, a straightforward trunk volume calculation using DBH, full height, and an F between 0.333 and 0.35 will give a close approximation for the junior class of pines. For old-growth specimens with straight trunks, DRH (R=root), full height, and an F between 0.30 and 0.40 will do the job in most cases. F values as low as 0.30 can be required where root flare is extreme and the tree tapers fast. For white pines of intermediate age, the volume calculation is equally challenging and we can approach it in several ways. Using BH (breast height), F can vary between 0.35 and 0.40. Using RH (root collar height), F will be between 0.30 and 0.36. Occasionally, the trunk will be so stocky that the F value for RH needs to be as high as 0.37, but that will not often occur on intermediate-aged pines. In general, the volume of a single-trunk, intermediate-aged white pine will usually lie between the two methods just described a
nd often near the midway point. Letís now examine some specific trees.

Jake Swamp Pine

The average of the two volumes (CBH and CRH with F = 0.333) for the Jake Swamp tree is 574 cubes. This uses a CBH of 10.4 feet, which is what Willís model uses, as opposed to my more liberal 10.5 feet, Willís full modeling of Jake on November 1st yielded 573 cubes. This is an amazing match, and it is not accidental. The shape of Jake falls between that of young and old pines. Let us formalize these volume calculations.

Let ABH = area of trunk at breast height,
ARH = area of trunk at root collar height,
H = full height of tree,
F = form factor,
VEI = volume estimate of intermediate-aged pine.

VEI = F*H*(ABH + ARH)/2
Where F = 0.333

Tecumseh Pine

Letís try the VEI formula on the Tecumseh Pine, an older, stockier tree, but not yet truly old growth. The circumference at the root collar is 13.24 feet and at breast height is 11.9 feet. The full height of the tree is 163 feet as determined by Will on his November 2nd climb.

VEI = 693 cubes. Compared to the modeled volume of 779 cubes this is significantly low. However, the Tecumseh Tree is a stocky pine. Consequently, its volume can be better approximated by: VE = F*H*ARH, which yields 788 cubes and that is much closer. The F value for RH needed to reach the modeled volume is 0.35675, which falls between or 0.333 and 0.37 parameters.
So, the volume estimation process works pretty well provided adjustments are made to the F value and the base area is calculated as either the lower, upper, or mid-point of collar to breast height span. I stress that the choices are dependent on the overall form of the tree.

Saheda Pine

As the last example, the Saheda Pine was modeled in 2007 by Will. It is comparably aged pine to Tecumseh Pine, but less stocky in its upper portion. Its measurements in 2007 were CBH = 11.8 feet, Height = 163.6 feet. Its girth at root flair, as determined by Will was CRH = 13.3 feet. Will modeled Saheda at 695 cubes. The RH volume is 767 cubes and its BH volume is 604 cubes. The average of the two is 685 cubes, which is close to the modeled volume of 695 cubes. The averaging method works for Saheda. The more slender upper portions of the Saheda Pine virtually guarantee that the RH volume will over-estimate the modeled volume. The greater age of the pine guarantees that the BH volume will under-estimate the modeled volume. The F value needed to produce the modeled volume using BH is 0.3835. The F value needed at RH is 0.302. This latter value is necessitated because of the slender double trunk near the top of Saheda in combination with the root flare.

Summary

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.

Bob


== 2 of 6 ==
Date: Tues, Nov 11 2008 3:05 pm
From: "Edward Frank"


Bob,

An interesting article. I especially liked the background summary and the concept for your specialized list. looking at the volume formulas you suggested as a rough estimate, I have some questions. In the final set of equations the basic difference between the three formulas is based upon the difference in area of a section at breast height versus the area of the trunk at the root collar. I am sure you have run statistics on the numbers and these produce meaningful results on a first pass. What bothers me about the process is that the entire trunk is being characterized by the differences in the tree diameter at breast height and at the root collar. Is this relatively tiny fraction of the total volume of the tree really an adequate basis for projecting the volume of the entire tree? I really have my doubts in a broad sampling that it does. First we really don't seem to know why some trees have more of a flared base than others. Across species it seems, based upon observations only and not any modeling, that species that grow on a more unstable substrate have a larger flair at the base of their trunk. Does this observation stand up to analysis - I don't know. Does it also apply within a single species - would it apply to pine trees - again would think so, but I don't actually know, but I think it should be considered. So if the amount of flair between breast height and the root collar is not dependant on overall trunk shape, but upon some other factor, such as the nature of the substrate, then it would not serve as a good indication of overall volume.

The second question that comes to mind is that you are characterizing young trees as having one shape, old growth trees as having another shape, and also an intermediate category. I wonder if these generalizations are valid over a larger sampling size. Is is age that affects the shape, is it the size of the tree, is it the history of suppression and rapid growth, is it dependant on age or the history of a particular tree?

In the final formulas you present you essentially are adjusting the cross-sectional area used in a basic formula by considering whether to use the breast height area, the root collar area, or something in between. I would feel a more appropriate approach would be to keep the position of the cross-section are at breast height and adjust the Form Factor between the suggested ranges, 0.33 to 0.50 based upon visual observation of the taper of the trunk, whether or not the tree has excessive flair extending above breast height, and whether or not the tree appears to have been topped. Your formulas may provide a good estimate, I am just wondering if a different approach that just included a form factor might yield better results, as the role of trunk basal flair in overall trunk volume is unclear (at least to me.)

Ed Frank

"Two roads diverged in a yellow wood, And sorry I could not travel both. "
Robert Frost (1874Ė1963). Mountain Interval. 1920.



== 3 of 6 ==
Date: Tues, Nov 11 2008 4:15 pm
From: dbhguru@comcast.net


Ed,

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.
Often the pines in a stand will show great consistency of form. Increasing sample size dramatically doesn't add much new information. As pines age and the root structure develops, the volume implied by using the area calculated at RH overcomes any change in trunk shape toward the paraboloid form.
I will have to do a lot more work before I'd take any formula to the bank, but the formulas and range of F values boxes in the true volumes pretty well. It is a start.
In terms of what species fall within the formulas and F values, well, I'd be reluctant to go beyond the white pine at this point. The formulas probably don't apply to the hemlock, at least not without changing the F value range.
The use of one predominant shape for young trees, another for old-growth pines, and a third for intermediate age trees follows from the data I have, but there are lots of exceptions. I'll discuss them in future communications. The age criteria is just a starting point. Overall shape or form really drives the volume, but pines change shape over time along the lines indicated by the formulas. More to come on this topic.

Bob



== 6 of 6 ==
Date: Tues, Nov 11 2008 9:40 pm
From: "Edward Frank"


Bob,

What I am thinking is that really what you are looking at is a change in shape of the trunk in different trees. In your summary formulas you are mimicking the change in shape by substituting the cross sectional area of the trunk at either breast height, root collar height, or somewhere in between into a formula while the shape factor is represented as a constant (0.33). Then you suggest that the F factor should be modified to fit other situations. By substituting cross-sectional area for shape, you are approaching the same values, the trend of the calculations are going in the same direction,as would be obtained by adjusting the F shape factor. So there should be a statistical correlation in what you are doing but that doesn't mean it is an optimal solution. I think the logic of the substitution is flawed as I don't see any direct reasonable relationship to the amount of basal flair to total tree trunk volume. The correlation is coincidental because it is going in the same direction as the tree shape parameter is going. If this parameter must be modified further by changing the F shape factor in certain cases, then what is gained?

You are creating an artificial variable (changing the cross-sectional area) that is in the same direction as the actual variable - shape, so that you have two variables instead of one. To me it seems much more reasonable to keep the cross-sectional area a constant at one height - say breast height in the formulas - and to manipulate the one true variable - the F shape factor - in the general formula. One variable that is real, instead of two - one fake (cross-sectional area) variable that approaches the same volume as the trunk and second fake variable (an F factor-like factor that really doesn't represent anything measurable in the tree) that is essentially a fudge factor to make the first fake variable better match the actual trunk volume. It is like you are saying in your summary formulas - the tree really isn't this shape, but if we adjust the basal area in this way, and throw in a fudge factor when needed the resulting generated form will be somewhat similar in volume to the actual trunk.

What I would suggest is to instead develop a standard Protocol to assign a tree specific F shape depending on the general shape of the trunk. It likely will fall into broad groups like you suggest, I believe you are on the button in that regard. However the methodology is more sound when you are manipulating the formula based upon variations trunk shape, rather than generating an artificial variable value by changing the basal cross section.

Ed Frank

"Two roads diverged in a yellow wood, And sorry I could not travel both. "
Robert Frost (1874Ė1963). Mountain Interval. 1920.


== 2 of 2 ==
Date: Tues, Nov 11 2008 11:27 pm
From: "Edward Frank"


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


from Don Bertolette, Nov 12, 2008

Bob-
I wondered if your tree library had any of the following references on eastern white pine volume data :
 
PART B. DIAGNOSTIC CRITERIA
VOLUME TABLE/EQUATIONS

  • Form-Class Volume Tables (2nd Edition). 1948. Department of Mines and Resources, Mines, Forests, and Scientific Services Branch, Dominion Forest Service, Ottawa, Canada.
SITE INDEX EQUATIONS
  • Beck, Donald E. 1971. Polymorphic Site Index Curves for White pine in the Southern Appalachians. Dept. of Agriculture Forest Service, Southeastern Forest Experiment Station, Asheville, North Carolina.
YIELD TABLES
  • Normal
    • Leak, William B. 1970. Yields of Eastern white pine in New England Related to Age, Site, and Stocking. USDA Forest Service Research Paper, Northeastern Forest Experiment Station, Upper Darby, PA.
  • Empirical
    • Leak, William B. 1970. Yields of Eastern white pine in New England Related to Age, Site, and Stocking. USDA Forest Service Research Paper, Northeastern Forest Experiment Station, Upper Darby, PA.
  • Variable Density
    • Leak, William B. 1970. Yields of Eastern white pine in New England Related to Age, Site, and Stocking. USDA Forest Service Research Paper, Northeastern Forest Experiment Station, Upper Darby, PA.