New Formula  (Sneak Preview) dbhg-@comcast.net Aug 21, 2007 15:16 PDT
 ENTS, A new ENTS formula is in the works. At this point, it is simple and may prove very useful to those of us woking with trunk volumes. Over the past several years of modeling, I've consistently seen that the computing the conical volume of single-stemmed white pines using the cone's base as the area just above the trunk flare (old root collar) and the full height of the tree over-shoots the volume on young and middle-aged pines and understates the volume on old growth trees. I've also observed that the conical volume using the area of the base of the cone as the area at breast height of the trunk along with the full tree height consistently understates the volume. In a sample of 30 modeled pines, the conical volume using breast height area understated the volume in 29 cases. By contrast, the turnk flare base yielded an overstatement in 21 of the 30 cases. For my Taking an average of the two bases yielded a volume that overstated the modeled volume in 12 of the 30 cases and understated 17 and equaled in 1. The average percent of the averaged volumes of the modeled volumes was 98 with standard deviation of 10. Yes, that's high, but if we introduce a form factor of 1.1 for old growth, columnar forms, and 0.95 for forms that taper extremely fast, and 1.00 for trees that appear just right - a judgment call in each case. Then the average of the ratios of the new volumes versus the modeled volumes remains at 98 and the standard deviation drops to 9. Changing the factors to 1.10, 0.90, and 1.00 yields an average ratio (turned into a percent) of 99 with the standard deviation remaining at 9. SOLD! The formula needed to estimate the volume of singe-stemmed pines is: V = H*F*(C1^2 + C2^2)/75.4 where: V = Trunk volume H = full tree height F = form factor (1.1, 1.0, 0.9) C1 = circumference at trunk flare height C2 = circumference at breast height. I'm sure Will, Jess, myself, and others interested in trunk modeling can refine this formula, but even as it stands, it is pretty good. Its algebraic derivation follows for those who wantit. D1 = diameter at trunk flare height D2 = diameter at breast height. C1 = circumference at trunk flare height C2 = circumference at breast height. A1 = cross-sectional area at trunk flare A2 = cross-sectional area at breast height V1 = trunk volume using cone base area as the area at trunk flare height V2 = trunk volume using cone base area as the area at breast height. H = full tree height F = form factor (1.1, 1.0, 0.9) V = averaged trunk volume V1 = H/3*A1 V2 = H/3*A2 V = (V1 + V2)/2 V = (H/3*A1 + H/3*A2)/2 V = H/6*(A1 + A2) A1 = PI*D1^2/4 = (PI*(C1/PI)^2)/4 = C1^2/(4*PI) A2 = C2^2/(4*PI) V = H/6*[C1^2/(4*PI) + C2^2/(4*PI)] V = H/(24*PI)*(C1^2 + C2^2) V = H*(C1^2 + C2^2)/75.4 where 24*PI = 75.4 Introducing the form factor F, we arrive at the final formula: V = H*F*(C1^2 + C2^2)/75.4. This formula does not replace a full modeling, but should give a pretty good approximation of trunk volume for single-stemmed white pines and I presume other conifers. I particularly like this formula because it does not require the use of either the RD1000 or Macroscope 25. The formula won't apply to most hardwoods. However, hardwoods such as young tuliptrees with a still symmetrical crown may fit well enough. My current objective is to develop an easy way of obtaining high and low estimates of trunk volume for single-stemmed white pines using simple calculations and avoiding the extra gadgets that some of us use. Once we have the simple method nailed down, we can then search for refinements. As a first cut, the simple formula presented above seems to fit the bill. Bob
 RE: Sneak Preview DON BERTOLETTE Aug 22, 2007 04:26 PDT
 Bob- Looks like a good first cut solution...you may recall that we've had conversations in the past over how badly USFS estimates of old-growth tree volumes are...that's because they have a uniform formula for the broad array of diameters/heights/form factors that they encounter across a broad geographic area. What you've done, is narrow the population's size diversity, and are now averaging species diversity  within the constrained population of old-growth trees...further refinement could be achieved through similar factors for specific species that typically fall outside the "old-growth un-normal norm"...cypress comes to mind as an obvious 'outlier' I noticed your use of the word 'judgement (call)'...while it can add confusion to the 'confidence level' when user judgement isn't of 'expert quality', constraining estimates to 'expert quality users' would likely improve quality of estimates. -DonB
 Back to Don Bertolette dbhg-@comcast.net Aug 22, 2007 10:47 PDT
 Don,    Yes, you're quite right. I saw no other way to get a handle on volume calculations via a simple process than to narrow the field of investigation. I expect most of the eastern conifers, certainly the pines, spruces, and firs, will lend themselves to the outlined modeling process. I'd love to perfect it more in the coming months and then see how well it applies to the ponderosas and other Rocky Mountain conifers next summer. Lots more to do though in the interim. Bob
 New Formula dbhg-@comcast.net Aug 23, 2007 19:36 PDT
 Don, Don, Ed, Will, Jess, et. al.      Today I modeled two more old pines in Childs Memorial Park, Northampton. My sample now consists of 32 pines. Using the formula:      V = H*F*(C1^2 + C2^2)/75.4     Where V = trunk volume                H = full tree height                F = a choice from the set { 0.90, 1.00, 1.10)                C1 = girth at trunk/root flare (TF)                C2 = girth at breast height (BH)     and modeling the volumes using the Macroscope 25, I got the following results. Tree                      1             2 Vol-TF             481.6         409.2     Vol-BH            287.3         307.5 Vol-Avg           384.4         355.3 Vol-Adj Avg    346.0         390.9 Vol-Modeled   350.0         371.3 Adj Avg/MV      0.99           1.05            I applied the factor 0.9 to the first tree and 1.1 to the second. The first tree has a large trunk flare that leads to much too high of a volume. The second confirms to the old growth white pine model, so I used 1.1.     The ratio of the adjusted average to modeled volume (MV) for all 32 sample points is a surprising 0.99 with a standard deviation of 0.09. I can't beat the average ratio value, but the standard deviation could be tighter. However, it is what it is. I could cheat and bring down the standard deviation for the smaple by including a lot of younger trees. But my purpose is to investigate the range of values we can expect using plenty of trees in all ages classes. So far, I'm satisfied with the results and believe the formula has value.     Tomorrow, I'll model a few younger pines and present the results. Oh yes, the stats on the modeled pines are (120.0, 9.5) and (120.8, 9.7). Despite their closeness in the two common measures, the first has a girth at trunk flare height of 12.3 feet. The second has a girth at trunk flare of 11.3 feet. Bob
 New Formula Continued dbhg-@comcast.net Aug 24, 2007 12:10 PDT
 New Formula Cont' dbhg-@comcast.net Aug 25, 2007 20:05 PDT
 Will, Jess, Don, Don, Ed, Lee, et. al.:    I am up to 41 white pines modeled for testing the new trunk volume fomula. With the inclusion of 3 more pines, the ratio of AATV/MV stands at 0.992 and the standard deviation is 0.089. The unadjusted ATV/MV is 0.982 and the standard deviation is 0.102. Two of the added pines are old growth and one is mature. Adding young pines would improve the ratio and standard deviation, but that is not the purpose. I expect the long run ratio to stabalize around 0.98 with a standard deviation of probably 0.095. We'll see.    In an experiment, I applied the formula to the big Dunbar Brook hemlock and the AATV fell well short of that modeled. The old growth hemlock form will require an F value of 1.2, if not more. Applying the formula to old growth hemlocks is going to be a challenge.    Well about an hour ago, I went and did it. I ordered the Macroscope 45. I wanted a backup instrument to the indispensible Macroscope 25 and elected to upgrade to the 45. The telescope version works the same for both models, but the microscope feature has a magnification factor of 45 for the Macroscope 45 instead of 25. One can't have too many gizmos.    I'm anxious to purchase one of those extremely accurate laser rangefinders that Paul Jost is tracking. The TruPulse 200 is useful, but falls well short of shooting the holes like the Nikon Prostaff 440 does. It is the real work horse of the laser clan.    While scanning the Ben Meadows catalog, I saw new entries in the one instrument to do everything competiton, but all their tree height routines are the flawed tangent method. So, for now, I'll stick with the simpler instruments and do the math through the ENTS formula package. Bob
 RE: Back to Don again dbhg-@comcast.net Aug 26, 2007 05:56 PDT
 Don,    C1 is at the point of trunk/root flare, which is usually from 1.0 to 2.5 feet from ground contact point, at least for eastern trees. C2 is at breast height. I realize that the C1 point is often debatable - a weakness in the method, but not a significant flaw for the vast majority of white pines. It remains to be seen how well the method will hold up for other conifers. For spruce, I think it will be even more accurate, but probably less so for hemlock. Bob -------------- Original message -------------- From: DON BERTOLETTE ; Bob- You said "I'd say it is safe to conclude that the breast high determination consistently underestimates the trunk volume." By the volume amount due to the flare? Are C1 and C2 circumferences at base and breast height? Seems 'calculable'... -DonB
 Onward marching volumes dbhg-@comcast.net Aug 26, 2007 10:30 PDT
 ENTS,     I just finished modeling Monica's tuliptree. I wanted to see how well the AATV formula fit that slender, graceful tree. I believed the match would be close. The rest of you can be the judges. Here are the numbers. Monica's Tuliptree CRH       7.95' (Girth at the trunk flare) CBH       6.60' (Girth at 4.5 ft above base) Height     123.0' (full height) V1        206.2 (volume using the cone's base as the area computed at the trunk flare point) V2         142.1 (volume using the cone's base as the area computed at 4.5 feet above base)     VC        180.2 (volume modeled with the reticle) AATV    174.2 Note: AATV = 1.00*123*(7.95^2+6.6^2)/75.4 AATV/VC = 0.97    This obviously is a very good trunk volume estimation for Monica's tuliptree. Her tree is arrow straight, has a gradual, even taper and a noticeable, but not exaggerated trunk flare. So, I set the F value to 1.00.     Adding Monica's tree to the sample, I now have 42 modeled trees. My contention continues to be that the formula fits young forest-grown trees very well. Old trees are the rascals. No surprise there, but I think for trees in the 1.0 to 5.0-ft diameter range, the AATV formula will prove valuable. For example, the huge Grandfather white pine in Monroe SF has an AATV value of 976 cubes. Past modeled volumes run from 930 to 1020 cubes, with the most likely value being around 950. Why the variation? Well, it is very difficult to see the trunk in the crown region from the ground. As soon as my toe heals sufficiently and the insect population crashes, it is of the Dunbar Brook I go to re-model the Grandfather pine. I expect to spend the entire day, circling the tree to get more and more points. Hopefully, when Will comes in October, we'll put the issue to rest with a climb. Bob
 RE: New Formula Cont' DON BERTOLETTE Aug 28, 2007 13:37 PDT
 Bob/et al- As I read your post below, realizing how accurate your procedures have become, I couldn't help but think that there probably needs to be different levels of measuring devices for different levels of accuracy needed. Conceptually, I would suggest there are three levels...that of 1)approximation, 2)estimation, and 3)exacting (for the lack of a better word.    Respectively, these would be purposely for 1) a measurement triage, taken with lightweight field gear [as defined by absence of gear belt/overloaded day- or fanny-pack...;>] such as a clinometer and rag tape; 2) a measure of candidacy for superlative listing, taken with extensive/expensive digital field gear [defined as the array needed to induce field duct tape repairs due to equipment overload and repeated face plants...;>]; and  3) an exacting, undeniably accurate and precise measure that stands up to the highest scrutiny, and only needed for trees of such superlative dimension that they are likely to be eligible for state or national tree champion status. This third level would necessarily have to be precise (capable of replication by independent measurement) and field going (though not necessarily awkward, but probably so...:>}.  I'm thinking something that has precise control of vertical and horizontal axis, like a 'Total Station' or such.  There's just too much 'loss of control' in handheld electronics, at the  level of accuracy that ENTS procedures can muster. Stand to the side of anyone measuring the angle to the top and bottom, and you'll see more 'movement' vertically than the accuracy striven for (tenths of a foot). -DonB
 Re: New Formula Cont' Edward Frank Aug 28, 2007 17:17 PDT
 Don, Bob, et al, The fourth level of accuracy would be called excruciating.... Ed Frank
 Back to Don dbhg-@comcast.net Aug 30, 2007 22:18 PDT
 Don,     I think you are on to something. By recognizing and formalizing the 3 levels of measurement, we would be making a definitive statement about accuracy while making room for those to participate who may be just starting out and intimidated by all the expensive equipment and/or math required to go to the extremes. We would be emphasizing the lesser level of accuracy of Type I measurements and drawing attention to them without rejecting them outright - although Type I measurements would not be ENTS-certified. Let's continue this thread.     With respect to equipment, LaserTech is delivering a TruePulse 360 to my doorstep this morning at 9:00AM. Way cool! Then the testing will begin. Bob
 Re: New Formula Cont' dbhg-@comcast.net Aug 30, 2007 22:42 PDT
 Ed, Don, et al:     Or exhilarating. The 4th stage is where we call in Will to climb the tree. Then depending on how the climb goes, it is excruciating or exhilarating - maybe both. For us landlubbers remaining on the ground, probably just fascinating. Bob
 Comparisons and modeling dbhg-@comcast.net Sep 12, 2007 01:39 PDT
 Re: Comparisons and modeling Dean Hedin Sep 12, 2007 20:56 PDT
 I don't like F. It seems to make the rest of the formula almost meaningless if you can just multiply a constant to everything to make it come out nice. In my line a work we call that a "Fudge Factor". I'll be honest, I haven't measured many trees, but I don't need that experience not to like F. I understand the difficulty of the problem. You would like to get a very good estimate of volume from a minimal set of measurements. Such a problem may not have a simple solution (or any). I know how I would find out. I presume you have a set of data that consist of the "simple measures" along with data of tree volumes measured in some careful manner and are relatively accurate. I would then write a genetic algorithm that tries different "formula combinations" and then runs these across the data set as a test for fitness of the formula. This process is repeated many times over with the "formula combinations" crossbred until a formula of required accuracy is found (or not). It's a brute force method. In other words, let the computer find the best formula (or let it tell you it can't find one). I would think that one could get good estimate of tree volume for an "isolated" deciduous tree by merely taking a high resolution digital picture of the tree against a clear background (like the sky) in the winter, when the leaves have fallen. So long as a scale is indicated in the image an algorithm could then count up the dark pixels in the image and then estimate the volume.
 Modeling dbhg-@comcast.net Sep 17, 2007 17:43 PDT
 Dale, Will, Jess, Howard, et al:         On Friday I went to MTSF for a brief period, but long enough to model the Mirror tree, a handsome white pine that stands 156.6 feet tall and has an 11.0-foot girth. It models to 533.0 cubes. The AATV formula yields 516.5 cubes for the Mirror Tree. The difference of 3.3% in volume resulting from use of the formula is acceptable as an estimate of the tree's volume. This morning, I modeled a neighbor's pine, a gorgeous tree 8.1 feet in girth and 120.0 feet tall. My neighbor's white pine yields 232.0 cubes and computes on the formula to 239.1. The difference is 3.1 percent. This afternoon, I modeled a pine on Monica's land. It models to 237.1 cubes. The unadjusted formula gives 244.7 and the adjusted formula gives 232.5.       I have now modeled 46 pines and applied the formula to the same. The volume yielded by the formula averages 98.3% of modeled volume with a standard deviation of 9.6%. Using the adjusted volumes by applying the shape factor to the situations that I've previously described, the formula yields 99.2% of modeled volume with a standard deviation of 8.6%. The very old trees and trees on steep slopes are the ones that predictably produce the high standard deviation. Nonetheless, The formula is proving its worth as a tool for ENTS to use. The beauty of it is that it gives us a good estimate of volume for single-trunk eastern conifers with very little measuring. However, as it under-estimates the volume of old growth forms, the Seneca Pine's volume would be under-estimated by between 10% and 20% unless the shape factor is applied.      Dale, will you have any time to experiment with the formula in the coming months? Similarly, Will, Jess, and Howard, will you be able to give it a fair test on conifers in your region? I am especially interested in how the formula would work if tweaked it with one extra measurement taken at halfway up the trunk? A measurement at that point would negate the need for the subjectively applied shape factor, but would require the reticle and finding the midway point of the trunk. That would deter its wide spread usage, but allow us to catalog many more trees. Bob
 RE: Modeling DON BERTOLETTE Sep 18, 2007 01:32 PDT
 Bob- I'm thinking that you may end up with a formula adjustment factor (FAF) that may be effective within species, ie, o-g white pines would have xx.x FAF, whereas hemlocks might likely have a somewhat different one, etc., etc.    As well as age, growth habit (steep slopes, creeping soils, etc.), and other peculiarities affecting base. -Don
 RE: Modeling dbhg-@comcast.net Sep 18, 2007 07:30 PDT
 Dale,    The formula is as follows:     H = full tree height,     C1 = Circumference at root flare     C2 = Circumference at 4.5 feet     F = shape factor     V = volume     V = H*F*(C1^2 + C2^2)/75.4     If the difference between C1 and C2 is over 2 feet, you can set F to either 0.95 if the taper is slow or 0.9 if the taper is fast. If the taper is extremely slow as with an OG tree, F could be as high as 1.2 for a normal root flare. Big root flare and extremely slow taper would cancel one another, if you get the idea. You be the judge. Young white pines with out a pronounced root flare have F=1.00. Bob     -------------- Original message -------------- From: Dale Luthringer ; Bob, I’d like to try it out on a couple of pines, and maybe compare it a couple of hemlocks as well. Can you please send me the formula again? Sorry, I’m just starting to get my head above water over here after the busy summer season. Dale
 Back to Dale dbhg-@comcast.net Sep 18, 2007 11:05 PDT
 Dale,      We will need to experiment with the form factor to cut down on the element of judgement. Making it species specific as Don Bertolette suggests will be required. Hemlocks will most likely have a wider range for the form factor.       It will be interesting to see if we can get this approach to really work and with John Eichholz coming back on board in the not too distant future, we'll have a good team to work on the process. As it stands now, the formula does work for a limited range of white pine shapes, but that's all I can say for sure.      Now as to Anthony's Mohawk haircut, well even though he doesn't think he would look handsome in a nice new Mohawk, I think he would be mobbed by the ladies. He'd owe us big time. What do you think? Bob
 RE: Modeling dbhg-@comcast.net Sep 18, 2007 14:33 PDT
 Don,    Yes, that's the direction all this seems to be pointed. I think the formula will apply well to spruce in Alaska - hint, hint. Bob
 #49 and counting dbhg-@comcast.net Sep 20, 2007 12:34 PDT
 Will, Dale, Howard, Jess, Lee, Don, et al:     I just returned from modeling white pine #49 - a very mature tree growing near upper Broad Brook. Its stats are as follows:     Girth at Trunk Flare: 10.3 '     Girth at 4.5 feet:          9.1'     Girth at 6 feet:             8.8'      Total Hgt:                115.3'      The modeled volume with reticle is 289.5 cubes      The unadjusted formula volume is 288.9 cubes.      What can I say? I didn't apply an adjustment because the taper is normal and the trunk flare is under 1.5 feet. This tree is a dream match to the fromula, but of course, other trees aren't. The unadjusted percentage of the average of the formula-calculated volumes to the equivalent averaged modeled volume stands at 98.7% with a standard deviation of 9.7%. The comparable stats for the adjusted formula are 99.3% and 8.4% respectively. Although, I'll continue modeling pines, I think the case for that species has been made. It's mainly a question of tewaking the adjustment factor. Anyone have thoughts on the best way to proceed? Bob
 #51, counting and adding species dbhg-@comcast.net Sep 20, 2007 12:45 PDT
 ENTS, After the success with #49, I added two northern red oaks. One I had previously modeled but had excluded from the data. The other is an black-scarlet hybrid in Monica's front yard. The tree's vital stats are:     Girth at 1.0 ft = 7.6'     Girth at 4.5 ft = 6.3'      Height = 84.0 feet     The modeled volume is 107.2 cubes. The main trunk holds true to near the top branches, so the overall form is right for formula application. The unadjusted formula volume is 108.6 cubes. Wow! Two tight matches in one day of different species. This is starting to get way cool. Bob
 #53 dbhg-@comcast.net Sep 21, 2007 13:00 PDT
 ENTS,    I just modeled a slender white oak across the street from Monica's house. Its stats are:      Height: 95.9 ft      CFH:   6.59 ft      CBH:   4.58 ft     Its modeled volume is 88.5 cubes and the formula volume computes to 82.0 cubes. The 6.5-cube difference seems a lot, given the prior close matches, but not unexpected.     I am going to turn my attention to hemlocks now and see if the younger ones conform as well as the younger white pines. Don Bertolette,     Don, any chance of applying the formula to some of the straight young to mature spruce in your neck of the woods? My current belief is that spruce should be most compliant. Bob
 Back to Don dbhg-@comcast.net Sep 22, 2007 13:48 PDT
 Don,    #54 was modeled an hour ago - a fine black oak in a neighbors yard. The modeled volume was 159.8 cubes. The formula gave 158.9. Since the species is an oak there is a lot of branching, as you would expect. So, what the formula means or produces when applied to such a form, is unclear. The close of numbers match is very interesting. More on this later. Bob
 RE: Back to Don DON BERTOLETTE Sep 23, 2007 11:49 PDT
 Bob- I would guess that boles with branching may have more hidden mass/volume than those without, the way that many trees will buttress their limbs for strength... -Don
 #54 and #55 dbhg-@comcast.net Sep 25, 2007 13:23 PDT
 ENTS,     Yesterday, I modeled Baaby's Hemlock on Petticoat Hill in Williamsburg, MA ((yes, that's two A's in Baaby). The stats for the tree are:      CRH: 9.8 '      CBH: 8.1'      Height: 127.1'      The modeled volume of Baaby's Hemlock is 277.3 cubes. The unadjusted formula volume is 271.1. The difference represents 2.2% of modeled volume, which is quite acceptable. The F value in the formula was left at 1.00, because the fast taper of the tree appears to be offset by the slightly high trunk flare (1.7 feet more girth than at 4.5 feet).      The primary reason I’ve modeled Baaby's Hemlock is to establish an ENTS record of significant hemlocks in New England before they succumb to the adelgid. Baaby's Hemlock has lost a lot of foliage and may be past the point of no return. It is significant because its 127.1-foot height makes it the tallest measured so far in the eastern Berkshires. Its age appears to be between 130 and 180 years, probably in the 140-150 range.       Baaby’s Hemlock becomes #54 modeled and compared to the calculated volume via the trunk volume formula, which continues to perform well when applied to single-trunk conifers up to 200 years in age and not growing on extremely steep slopes or in excessively wet areas, where trunk flare becomes exaggerated.       Today, Gary Beluzo and I searched for tuliptrees and I modeled another hemlock along Broad Brook. Stats are:      CRH = 9.35’      CBH = 8.2’       Height = 117.0’       Modeled Volume = 229.1 cubes       Adjusted Formula Volume = 228.0 cubes      Unadjusted Formula Volume = 240.0 cubes       The tree had a normal root flare, but a very fast taper, so F was set to 0.95.       The reason the formula works well is that the volume of the trunk of eastern conifers usually falls between two conical forms. The upper limit form is the cone formed by using its basal area equal to that of the tree’s cross-sectional trunk area taken at the trunk flare point. The lower limit is the conical form with the cone’s basal area equal to the tree’s cross-sectional area at 4.5 feet up. In both cases, the cone height equals the full height of the tree. Trees that taper extremely fast or extremely slow and trees that have an extreme trunk/root flare can fall outside the above range. A modest compensating factor can be applied that will capture about 3/4ths to 4/5ths of these trees, but some trees will fall out side the range of +/- 10%. We can eventually derive factors to capture most of eastern conifers and perhaps a lot in the West, at least the Rocky Mountain West.        To repeat the formula:        C1 = circumference at trunk/root flare        C2 = circumference at 4.5 feet above base        H   = full height of tree        F = trunk form adjustment factor        V = trunk volume        V = F*H*(C1^2+C2^2)/75.4       The adjusted formula volume averages 99.33% of the modeled volume with a standard deviation of 7.94%. This is after 55 modelings. Bob
 New formula dbhg-@comcast.net Sep 25, 2007 16:37 PDT
 Will, Dale, Jess, Don, et al:            I modeled another white pine today - a slender one on Broad Brook that exhibits the trunk form that works well in the formula. The modeled volume came out to be 147.3 cubes and the formula, unadjusted, yielded 144.7 cubes. The difference of 2.6 cubes is pretty minimal. The Broad Brook pine becomes number 48 modeled with reticle to test the formula. I won't be content until I have 100 pines modeled. My guess is that the unadjusted volume via the formula will be about 98% of modeled volume with a standard deviation of around 10%. The adjusted version will be about 99.5% and 8.5%. Bob