Piddling around in dendromorphometry as a cure for insomnia Robert Leverett Feb 20, 2007 10:51 PST
 ENTS, As the ENTS gathering at Cook Forest SP nears, Gary Beluzo and I have been polishing up our presentation on dendromorphometry - the art and science of measuring trees in the field. Gary has been my silent partner all along in dendromorphometry. He thought up the term. Will and Jess are also partners, but they are spending their time in the field these days doing critically important work. That leaves me to stay at home and piddle. In preparation for the lecture at Cook (It may put all of us to sleep, so bring an extra pillow), I've been searching for new formulas and processes that can aid the tree measurer in practicing his/her craft. Most of my forays into the world of trigonometry and geometry fail to produce anything of real value. But on occasion a formula pops out of an algebraic derivation that holds promise for use in the field. The objective is to make our workhorse formulas as computationally simple as possible. So, sometimes the result is just a more convenient form of a well-known formula. At other times, though, a truly novel approach emerges from considering a measurement problem realistically. As of late, I've been pre-occupied with trunk volume. What follows is the result of my piddling. Forest mensuration often treats the shapes of logs as truncated paraboloids. Slice the trunk vertically in two halves. The edges of a slice would follow parabolic arcs for part of the length of the trunk - perhaps the major part of the trunk, starting at about breast height. The top section of the tree is often treated more as conical in shape and the base more neiloid. So, taking the entire trunk, you might observe the base starting out as a neiloid shape (sides are concave), then changing to a quadratic parabaloid (sides are convex), to conical (sides are straight), and maybe back to paraboloid near the tip, but rarely so. Forest mensuration uses formulas for computing the volume of sections of a trunk. The sections are often 8 or 16 feet in length. While the ENTS area of interest isn't log volumes, the modeling of tree trunks, can use most of the same formulas. What might be an example? Let's say that the first 16-foot section of trunk starting at breast height from a 100-foot tall tree measures 4 feet for the bottom diameter and 3.4 feet for the top diameter taken at 20.5 feet (16+4.5). One foresty calculation called the Smalian method for log volume would return this calculation: 16*PI*[((3.4 + 4.0)/2)^2]/4, where PI = 3.141593.        This formula computes an average diameter from the two end diameters and then uses the result to compute an average cross-sectional area for the log. The average cross-sectional area is then multiplied by the trunk-length segment to get the volume of the trunk segment. The method assumes a paraboloid form that occupies half the cylinderical form based on the larger, or bottom, diameter. A slightly better way of applying this process is to take the actual diameter at 8 feet up from the starting point, i.e. the mid-point of the segment, which in this case would by at 12.5 feet up instead of averaging the diameters of the ends of the trunk segment. The averaging process can understate the volume, if the trunk is truly a paraboloid. The latter method is called the Huber method. However, the assumption for both these calculating methods is that the shape of the trunk segment is that of a parabaloid. Also, the cross-sectional form is considered to be uniformly circular. What if the form of the trunk segment is conical instead of that of a paraboloid? Then, the volume of the section, or frustum, would be calculated from (16*PI /12)*(4^2 + 3.4^2 + 4.0*3.4) - again, assuming a uniformly circular shape. The calcualted volumes from these two processes equate to 172.0 and 172.4 cubic feet respectively.          The actual trunk forms may not be strictly those of a parabaloid or a cone. Can some simple test be run to check the form assumptions? Were the 100-foot trunk a quadratic paraboloid, then the diameter, d, at any distance up the trunk, y, would be given by the relationship:     d = D*SQRT((H-y)/H), where H = total height and D = base diameter.         The equivalent formula for the cone is:      d = (D/H)*(H-y). Remember, y, is from the bottom up, instead of the apex down.      These are potentially useful formulas where sections being modeled are fairly long and curvatures are uncertain. A section of the trunk that fits neither the paraboloid or cone assumptions is the first few feet of the trunk. The base to DBH height is often in the shape of a neiloid (concave sides) because of the root flare. So if we need the volume of a neiloid frustum, we can use the formula:             V = H/4*(A1 + A2 + SQRT(A1*A2))            where V = volume                     H = length of frustum                     A1 = cross-sectional area of base of frustum                     A2 = cross-sectional area of top of frustum Note the formula for the volume of conical frustum is:             V = H/3*(A1 + A2 + SQRT(A1*A2)). The only difference is the factor 1/3 for the cone versus 1/4 for the neiloid. The factor for a paraboloid would be 1/2. A potentially usefful formula for the frustum of a cone that has an elliptical cross-sectional form is as follows: V = Pi*H/12*(D1*D2 + D3*D4 + SQRT(D1*D2*D3*D4)) where   V = colume H = length of frustum Pi = 3.141593 D1 = major axis of base of frustum D2 = minor axis of base of frustum D3 = major axis of top of frustum D4 = minor axis of top of frustum If the overall form is neiloid, then the formula becomes: V = Pi*H/16*(D1*D2 + D3*D4 + SQRT(D1*D2*D3*D4)) Ooh, I'm nodding off to sleep. Better stop now. ZZzzzzzzz Bob Robert T. Leverett Cofounder, Eastern Native Tree Society
 Re: Piddling around in dendromorphometry as a cure for insomnia Edward Frank Feb 20, 2007 15:10 PST
 Bob, Reading your post a few things bother me. The calculations of the volumes of different shapes of trunk segments seem fine. The trunk may vary in shape in a continuum from cylindrical, to concave outward, to conical, to concave inward with varying degrees of concavity. So the question becomes, how can you tell what shape a particular trunk segment represents? How to do that is the what needs to be determined. If you look at some of the latest numbers Will compiled for tree volumes you can see a variety of shapes represented. If a cylinder equal to the diameter and height of the tree has a value of 100%, then trees measured in his Tsuga Search Project have varied in percentages from 52.3% to 34.8% for intact, single trunked trees. The Sag Branch Tulip represents 66% of the ideal volume. So I don't see how any specific formula, even if idealized to position on the trunk bottom to top, would be able to represent the actual variation in trunk shape and volumes. The only way I can see that is practical is to measure relatively short segments of the trunk to minimize the variations from ideal. I am not sure that characterizing these smaller segments as parabolic curves versus simple conical segments would increase the accuracy of the calculations to a significant degree. I do like the formula for the elliptical cross-section conical. To what degree will there be a difference between using this formula for a section that is only moderately elliptical compared to treating it as a circular cone of the diameter equal to the average of the two axis of the ellipse? Ed Frank
 Re: Piddling around in dendromorphometry as a cure for insomnia Edward Frank Feb 20, 2007 19:13 PST
 Bob, In the example below from the middle of your post, you are talking about the Huber method in which you are measuring the diameter half-way between the ends to determine a better average with the assumption the trunk shape is paraboloid. If you had that measurement, then I think you would be better off computing the two smaller sections separately and adding them together. If you are doing ground based or climb based measurements you could determine the shape of a segment by comparing how well it fit one shape pattern or the other by progressively examining the measured diameter with the diameter measurements taken at the next height above and below it respectively. Ed Frank
 RE: Piddling around in dendromorphometry as a cure for insomnia Robert Leverett Feb 21, 2007 05:15 PST
 Ed,    I am in total agreement with you about the utility (or lack there-of) of volume formulas applied to a large section of trunk. The irregularities are simply too many and assumptions about a parabolic outline are usually a stretch. I present the formulas more for completeness of the subject than with any expectation of their wide spread use by ENTS. The subject is periodically worth investigating and I'd like to be able show others that we consider traditional forestry models and search for legitimate applications for them so that it doesn't appear that we're trying to re-invent the wheel.    The traditional models tend to work where large numbers of logs are involved. The law of averages takes over. But applying one of the models to a long section of trunk can lead to errors beyond what we in ENTS are willing to accept. In the end, that's why Will climbs the trees.     I should have issued a caveat to the e-mail, explaining the risks of assuming a regular geometrical shape over a long segment of trunk. Still, some of these shortcuts can give us approximations, so we should still have them as part of our repertory - just used with caution. Bob
 RE: Piddling around in dendromorphometry as a cure for insomnia Robert Leverett Feb 21, 2007 05:22 PST
 Ed,    Yes, if you have the the 3 measurements, you would be better of using them all. Again, I was attempting to show (without explaining it) how the process traditionally works. There has been a ton of work done of trunk shape and taper and basically the conclusion is that across the broad sweep of species and forms, there is too much irreglarity to attempt a one size fits all approach. When giving lectures, I will be careful to state the caveats and assumptions. Bob
 Re: Piddling around in dendromorphometry as a cure for insomnia Edward Frank Feb 21, 2007 14:54 PST
 Don, As I noted in the second post regarding Bob's musings, we could calculate the "shape" of each tree segment by looking at three successive diameters, then moving upward one measurement at a time. We could show that some sections are nearly cylindrical, while others narrow sharply. It would be interesting to see how the form of a measured tree changed in terms of shape functions from top to bottom as well as showing the shape changes graphically. For those trees for which we have volumes, we could also calculate the degree of change that best matches the actual measured volume of the tree of a given base circumference and height. I am not sure what any of this would get you, but it could be done without hideous calculations. Maybe it would be a try it and see what you get experiment? Ed
 Re: Piddling around in dendromorphometry as a cure for insomnia Don Bertolette Feb 21, 2007 21:51 PST
 Ed- In the perfect world, the shapes are continuous and follow some kind of tree oriented Fibonacci Code that you and Bob seek. From my experience measuring thousands of mundane, ordinary trees they'll pretty well fit into a normal curve, but it's the outliers that usually become difficult to estimate volume...they'll have sweep or crook, oblong bases, vertical ribs, spirals, swollen butts, j-hooked bases responding to soil creep/erosion, juvenile wood accumulations, and all manner of distortions. It's way technical, but I like my friend's efforts to measure virtual volume displacement...using high resolution LIDAR to quantify the trees volume...I think that he/they may arrive at a fairly high level of accuracy/precision, albeit contrived, technological, and disparate to all the reasons we want to go out into the woods for anyway... -Don
 Re: Piddling around in dendromorphometry as a cure for insomnia Fores-@aol.com Feb 22, 2007 14:09 PST
 Bob: You have actually hit on one of the greatest fears of older foresters...emergence of the computer simulation based Power Point generation of resource managers. It seems that the physical aspects of the forestry profession (or almost any other job that requires strenuous bodily movement)is a major turnoff to today's youth and there is growing concern of a future shortage of trained, qualified and productive technical help. I do not see the efforts of ENTS as a group of entity being out done by some future generation of foresters. Russ
 Re: Piddling around in dendromorphometry Edward Frank Feb 22, 2007 18:44 PST
 Back to Ed Robert Leverett Feb 23, 2007 05:07 PST
 Ed,    Your comments always provide food for thought. Please continue doing just what you are doing.    Basically, I expect to run through a couple dozen formulas/processes to find one of real practical significance to us. So, a lot of this stuff will drop by the wayside. One project that I have been thinking of is the developing of a list (yep, another list) of the most useful formulas to ENTS that utilize the equipment that we use (laser rangefinder, clinometer, scientific calculator, tape measure, DBH tape, prism, GPS Receiver, etc.). We could describe the situations where each formula is of most use. It would be a way of organizing the wealth of material that flows through these e-mails. It could also be part of the dendromorphometry book that is in the works. Bob
 Re: Piddling around in dendromorphometry as a cure for insomnia Don Bertolette Feb 23, 2007 19:43 PST
 Ed- As I tried to visualize your 'shaping', work by Bill Wilson at UMASS came to mind, as he worked with modelling tree growth...he was 'branching' out of Chaos theory, Mandelbrot...the idea being that 'fractal iterations' when constrained by our 'tree branching' rules could be used to model tree growth. Most of the Mandelbrot sets I've seen (they're fascinating) have been two dimensional...I suspect that taking it to the 3rd dimension becomes computationally intense... -DonB
 RE: Piddling around in dendromorphometry as a cure for insomnia Steve Galehouse Feb 23, 2007 20:13 PST
 Don, ENTS- Could these "fractal iterations" of branches then be extended to foliage size and shape: degree of serration, degree of lobing, compound(ed)ness, etc.? Perhaps the ultimate measure of size of a tree should be the amount of photosynthetic square footage it presents to to the sun, rather than its "support system"---but then we would have to factor in the efficiency of the foliage--that would really be computationally intense! Steve Galehouse
 Re: Piddling around in dendromorphometry Don Bertolette Feb 23, 2007 20:26 PST
 Ed- Your use of the words 'perfect cylinder' and the proportion of it, reminded me of what was going on in my head when I was discussing the LIDAR approach...LIDAR (Laser Imaging Detection and Ranging) can do a pretty good job of measuring 'virtual volume', as it sorts out the 'backscatter'...folks are working with software that effectively is like submerging the tree in a large graduated cylinder to obtain 'volume', by 'displacement'. Re 'not being a numbers guy', you had me fooled! I'm REALLY not a numbers guy!! You both go a long ways towards bringing these conceptualizations to a level where us 'lay people' can grasp at least some level of it. -Don
 Re: Piddling around in dendromorphometry Don Bertolette Feb 23, 2007 20:26 PST
 Steve- Cutting and pasting an example from http://www.simulistics.com/examples/catalogue/modeldescription.php?Id=branching1 Description This is a demonstration of Simile's ability to handle fractal models. L-system (Lindenmayer system) models are based on the idea that a complex structure can be obtained by repeatedly applying quite simple rules to a simple initial stucture. Thus, for example, the branching structure of a mature tree can be arrived at by applying simple rules to a seedling: rules such as "replace a bud by an extension to the branch, a side shoot, and two buds". In the Simile implementation, we use a population submodel to hold the set of branch segments on the tree, and an association submodel to tell which segment is connectedto which other segment. As we iterate through time, each segment gives rise to two daughter segemnts, and thus the structure develops. Each segment is angled with respect to its parent in 3D: thus, the resulting structure is in 3D, and could be viewed as such in an appropriate 3D rendering environment. Currently, the model is not available for download. Contact r.muetzelfeldt@ed.ac.uk for more information. I'm reminded of some inked drawings from the younger Leverett, of old growth that aren't that dis-similar from this. -Don ---------------------------------------------------------------------- And lastly for the night, the following narrative that ties the Fibonacci sequence (if you saw or read DaVinci Code, you may recall it playing a 'role' in the movie) to tree branching, in a more mathematical approach... Fibonacci is the well known sequence of numbers that describes phyllotaxis - the geometric distribution of petals in circular patterns in plants. The sequence runs 1,2,3,5,8,13 and is constructed by taking the sum of the previous 2 numbers and then adding that number to the sequence and so on. We can use recursion to calculate a Fibonacci number for a given n'th degree in the sequence. function fibonacci( n ) { if ( n == 0 || n == 1) return n; else return fibonacci( n - 1 ) + fibonacci( n - 2 ); } We can then grab a section of the sequence using this function to distribute movieclips to produce patterns such as those appreciated on a pinecone. Check out Gabriel Mulzer's Fibonacci Flowers and while your there have a look at the Recursive Patterns. -Don --------------------------------------------------- Steve- An article that expresses the reasonably long history at attempting to model branching, with some interesting images, at... http://algorithmicbotany.org/papers/abop/abop-ch2.lowquality.pdf -Don
 Fractals Edward Frank Feb 24, 2007 19:02 PST
 Don, Regarding fractals, there are a number of natural systems that have characteristics similar to the mathematical concept of fractals. A fractal is an object or quantity that displays self-similarity.   An object is said to be self-similar if it looks "roughly" the same on any scale. The object need not exhibit exactly the same structure at all scales, but the same "type" of structures must appear on all scales. When I was working in cave systems in the Bahamas and Puerto Rico, one mathematician suggested that the semi-circular rooms had fractal characteristics. Whether looking at the interlocking pattern of rooms in a cave, the semi-circular cut-outs on the edges of the room, to the smaller scale semi-circular cut-outs in these smaller areas, to even finer scale ones in these areas the pattern remained the same. If you looked at a diagram of any of these different scale features they would be indistinguishable. In natural systems there are limits to the extent to which the fractal array will continue. These are limited by the processes that forms them. For example if the fractals were driven by tidal processes, then they would tend to not be larger than the range of the tides. The lower end is similarly limited. It is an area I wanted to investigate further, but funding was cut, and I do not have sufficient math to do the work myself. I know there are fractal functions that resemble trees, but I am not convinced that there is any relationship between the two forms. And even if trees were found to have branching in a fractal pattern across a limited range (mathematical fractals are unlimited) I don't know of how value looking at mathematical generated fractals would be of any use in understanding trees. Maybe they could grow them for movies scenes or something. Looking a real system that has a fractal pattern within a specific limited range of scales is useful, because by identifying the limits of the pattern, you may be able to identify the factor and processes that are responsible for this limiting boundary value. Ed Frank
 Re: Fractals Don Bertolette Feb 24, 2007 19:48 PST
 Ed- Yes, I too was fascinated with fractals for awhile...as I revisited them, I was intrigued by the medical applications (neurons, dendritic pathways, etc.). -DonB
 RE: Fractals Pamela Briggs Feb 26, 2007 08:24 PST
 Here are two of my favorite fractals sites: http://home.att.net/~Paul.N.Lee/FotD/FotD.html http://beautifulandes.com/ Pamela