Angle error discussion Robert Leverett Apr 17, 2007 06:30 PDT
 Re: Angle error discussion Edward Frank Apr 17, 2007 12:41 PDT
 Bob, You can't ignore the tree height below eye level as the error is in the same direction as the top error and the errors tend to cancel out so the net error is minimal. So why should you be calculating the error for a 1 degree clinometer misalignment, when in actual application using the correct methodology these errors are not really applicable to height measurement errors? Unless you also add a section for the base measurement errors and then sum the two together I don't see this as a useful calculation, although it is impressive to look at Ed Frank
 Re: Angle error discussion dbhg-@comcast.net Apr 17, 2007 15:19 PDT
 Ed,     Some measurers sight trees from what they judge to be level ground and simply add their height to what they get from the slope reading, based on a 100-foot baseline. Will Blozan and I encountered this practice many years ago. In such cases, there isn't anything to offset or cancel out. We must remain sensitive to how measurements are actually carried out by the people who carry clinometers and tape measures with them. I would admit that doing all the math just to address a limited practice seems overkill, but there is more stuff on the way. I'll address your other point tomorrow. For now, gotta run. Bob
 Re: Angle error discussion Edward Frank Apr 17, 2007 17:20 PDT
 Bob, You write: A) "We can check the height error that results from a 1-degree error in the angle for both the sine and tangent methods."; B) "We can see from the differentials that the error made with the tangent method under this situation is greater than the corresponding error incurred with the sine method."; C) "Note that I intentionally ignored tree height below eye-level to simplify the above process. If the observer stands on level ground and simply adds his/her height at eye level then the above errors are propagated through the calculations as shown." Of course we must be sensitive to how measurements are being carried out in the field. I understand that some measurers 'guestimate' the base of the tree when using the tangent method. BUT that is not done when using the sine method. You are in effect comparing the errors of an inaccurate clinometer as it would often done in the field with the tangent method, with a "made up" scenario of a half done sine method that is not actually used. The second option used as part of the discussion is fiction. If you are comparing a real life situation with a fictitious one that does not exist. then the results are meaningless. Real life must be compared to real lfe to be meaningful. The only real comparison that can be made here is the amount of error introduced by the inaccurate clinometer using the tangent method with the actual height of the tree. A second tract could be written along the lines of what John Eicholz wrote about the inaccurate clinometer readings tending to cancel out when both the top sins and base sine are used in the calculation The math could be expanded upon and explained, But I seriously do not believe this argument you presented draws a meaningful comparison, even if the math is elegant. Ed Frank
 Re: Angle error discussion Don Bertolette Apr 17, 2007 19:34 PDT
 Ed/Bob- When attempting to measure a reasonably large object like a tree, say 100' tall, to a tenth of a foot, I'd think that the amount that your head pivots going from top to bottom of tree, is easily more than the tenth of a foot accuracy you're striving for...just pivoting my head (as if measuring my computer screen) as I look at top and bottom of monitor, describes a change in "eye height" of more than a tenth of a foot. -Don
 Over and under errors-continuing the discussion Robert Leverett Apr 18, 2007 06:50 PDT
 Re: Angle error discussion dbhg-@comcast.net Apr 18, 2007 08:51 PDT
 Don,    Head pivoting does change the triangles slightly by putting the spot at where a line from the top down through the clinometer, through the eye, and through the back of the head intersects the corresponding line coming from the base of the tree. We're talking about the impact of a few inches of distance, which translates to a tiny effect. I'll give examples in a future e-mail. Bob
 Back to Howard-Angle error discussion dbhg-@comcast.net Apr 18, 2007 09:09 PDT
 Howard,      It seems that issues with angles have long preoccupied us fellows in ways I would not choose to discuss on our tree list. But joking aside, I'd appreciate it if you would help me think through this latest approach to analyzing errors in tree measuring in terms of their effects - from the obvious to the not so obvious. Clinometer error has long plagued our measurements, at least mine, and this seems to be the same with that of a number of you. I've owned 6 Suunto clinometers and have experienced out of calibration situations or sticking at some point with all of them. John Eichholz has encountered significant errors with his instruments and has had to develop compensating methods.      It is easy to take the readings one gets from an instrument at face value. Point, shoot, calculate, record, and move on. But for us to be as good as we say we are, I think we need to spend more time investigating sources of error and coming up with simple techniques that we can routinely apply to compensate for out-of-calibration situations. Any thoughts? Bob
 Re: Over and under errors-continuing the discussion Howard Stoner Apr 18, 2007 12:13 PDT
 Bob, Roping a cow from a horse has some similarities to measuring the height of a tree. You need the distance to the cow and the angle the cow is below you. At that point the similarity ends. The distance between you and the cow and the angle the cow is below you must be accurately estimated multiple times (as you ride full speed on your horse) and in quick succession followed by a reflex response on the part of the arm to release the rope at exactly the right time. Try that on a tree! When cows get GM'ed(genetically modified) I suppose they will constantly raise and lower their head to avoid you ever getting an accurate reading on that angle! Howard
 Back to Ed with a couple more thoughts Robert Leverett Apr 18, 2007 12:29 PDT
 Ed,    You've probably read my response to your last e-mail by now and hopefully see where I'm going, but a couple of added points are still in order. You were quite right to be confused about my direction, since I didn't clearly state up front that I had set out to examine the efficacy of both the sin top-sin bottom and the tangent methods from the standpoint of angle error and was passing along material that I had just developed, i.e. the derivations.     The area of investigation covered in the last two e-mails is admittedly not needed for the Ents who use the superior sine method, but is meant for inclusion in the eventual dendromorphometry text, which I've begun to seriously work on. In the near future, I'll be exploring different topics with the rest of you and will be coming at them sometimes from unpredictable directions. Often, a topic that I will introduce will be irrelevant to current ENTS practices, but will make sense as material for the dendromorphometry text, i.e. I'll be thrashing out the topics relevant to tree measuring in general. However, I will preface the material with an explanation of why I'm passing it on to the rest of you, because the material will frequently be in snippet or tidbit form.         Bob
 Re: Back to Ed with a couple more thoughts Edward Frank Apr 18, 2007 18:39 PDT
 Bob, I can see where you are going, but also I think you are missing my point. I understand this a part of a larger argument comparing tangent and sine and clinometer errors. My point is that I don't think this line of reasoning should be pursued as the underlying assumption of one part of the exercise does not represent real life. The entire comparison of the respective errors of tangent method and just the top sine method does not represent anything meaningful and its discussion will only serve to confuse the issues involved. The entire argument would be better explained and understood without this particular sidetrack. The part about the top-only sine should be skipped and you should move onto the next step without elaborating on this portion of the comparison. In my opinion.... Ed
 RE: Back to Ed with a couple more thoughts Robert Leverett Apr 19, 2007 04:29 PDT
 Ed,    Your opinion is duly noted and respected as always. I did understand your point, and on thinking carefully about it, am inclined to agree with you. Actually, I intended to get to the full effect of tangent error from "angle shift" and compare it to the full effect of sine error from angle shift. I was doing it in pieces, but now see how that created an artificial comparison. Bob
 Re: Back to Ed with a couple more thoughts Edward Frank Apr 19, 2007 17:32 PDT
 Bob, I have expressed my viewpoint, but it is you who are writing the documents and if you feel you need to include a certain line of reasoning to make your arguments work as you envision them, then you should do what you feel is best. See you Saturday, maybe. Ed
 Angle errors revisited Robert Leverett Apr 27, 2007 07:48 PDT
 ENTS,       For folks not into the mathematics of tree measuring, best exit this e-mail now. Because, the following discussion will either put you to sleep or cause you to run screaming, pulling out gobs of hair as you go. THINKING BACK:       On our list, we frequently discuss the common sources of measurement error when calculating tree height. In terms of our instruments, the errors are in the angle, the distance, or both. We may make an angle error because our clinometer is out of calibration, because we misread it, because it sticks, because we measure the angle to the wrong object, or some combination of the above. Similarly, our distance may be off because our laser is out of calibration, or because we hit the wrong target. We may get accurate angle and distance measurements to the target that we select, but apply the wrong mathematical model to calculate the intended dimension. Anticipating these sources of error and attempting to eliminate or compensate for them is part of entry-level tree measuring.        Many discussions on the list have been devoted to the appropriate mathematical model to apply in measuring a dimension, e.g. the sine-based versus tangent-based model for computing tree height. If we have a laser and a clinometer and can see the target, the sine-based model is clearly the best, and if the sine model can be used, there is no reason to employ the problematic tangent-based model. We may teach it to others so they will understand the full range of measurement methods that they are likely to encounter, but we we employ the sine method. So, the discussion below assumes we are using the sine method. IMPACT OF ANGLE ERRORS:       In investigating errors in the angle, or distance, or both, we want to know how bad things can get. But after examining some extremes and getting a sense of the worst-case scenarios, where do we go? One line of investigation is to explore what happens when we hold one variable, e.g. hypotenuse distance, constant and look at the pattern of the height change as the angle changes. We can then apply that knowledge to investigate small angle errors. By holding the hypotenuse distance constant and checking height changes from angle changes, it doesn’t take much experimentation before we see that we make the greatest errors in height in response to changes in the lower angles, i.e. angles closer to 0. This may seem counter-intuitive, but it is true.       We can get a better feel for not only the magnitude of the height error from an angle error, but the rate at which the height changes as the angle changes by small amounts. To investigate changes in the rate of increase or decrease, we appeal to a concept in calculus called the derivative of a function. Without much explanation, basically the derivative of a function measures the instantaneous rate of change in one variable that is induced by changes in another. Appealing to the sine-based formula H = L*sin(A), the derivative of H with respect to A is denoted by dH/dA and dH/dA = L*cos(A). An alternative notation of the derivative is H’. The derivative measures the instantaneous rate at which H changes as A changes for any value of A tested where the derivative exists. We can deduce the rate of height change by recognizing that cos(A) steadily decreases as A increases. Therefore, since L is constant, L*cos(A) is greatest at small angles. In fact, at 0 degrees, L*cos(A) = L and at 90 degrees, L*cos(A) = 0. So the rate of change drops from L to 0 as the angle goes from 0 to 90 degrees.       There is another approach that we can use. We can take something called a second derivative, which we will denote here as H”. The second derivative measures how the rate of height, itself, changes as the angle changes. If the rate of height change decreases as A increases, we get a negative second derivative. The following formula gives the second derivative for H.                                                 H” = -L*sin(A).   (Note the relationship to H)        Since sin(A) is continuously positive for 0 <= A <= 90, the product –L*sin(A) stays negative throughout the angle range. This simply means that the rate at which H changes as A increases is a continuously decreasing value. We can see this effect clearly in the table below: Hypotenuse    Angle   Change                 Change        2nd Distance                     in Angle                 in Height       Derivative 100                 0                  1                     1.75           -1.75 100               30                  1                     1.52           -50.00 100               60                  1                     0.87        -86.60 100               85                  1                     0.17        -99.62      As we see in the table, the change in height (as opposed to the absolute height) drops as the angle increases for a constant hypotenuse distance. This fact is reflected in the negative second derivative. Of course, the absolute height increases as the angle increases, but at a diminishing rate as reflected by the negative value of the second derivative.     Assuming the angle to the base of a tree is shallow and steep toward the top, does this mean that if we make an angle error at the base of a tree of say 1 degree and make the same angle error in magnitude and direction at the top of the tree, that the two errors will NOT offset one another because the impact of the error for small angles is greater? No, not necessarily. They WILL offset each other IF the sine method is used AND the top is vertically over the base because the hypotenuse stretches as the angle increases. The greater hypotenuse distance to the top creates the offsetting error. The error at the bottom cheats the tree of height by the same amount that it over-rewards the tree at the top. Ed Frank has frequently mentioned the offsetting errors from a constant angle error. In the configuration of top directly over base, the offsetting error effect is another reason to use the sine method. As was shown in a prior e-mail, the errors do NOT offset one another when using the tangent method.     Now, suppose that the crown top is horizontally closer to us than the base, a frequently encountered situation in the field. Let us further assume that the closer horizontal distance leads to a hypotenuse distance that exactly equals the base distance. This assumption is to keep things simple. We define the differential dHa, where dHa defines an angle change, as                                                        dHa = L*cos(A)*dA      Since L and dA do not change in value, the value of dHa is dependent on the term cos(A), which decreases as A increases. This means that the error in height made at the top as approximated by the differential dHa is less than the error at the bottom. Here we assume that the base angle is 0, but measured as 1 degree and the top angle is A, but is measured by our out-of-calibration clinometer as A+1. The top error, adding too much, does not exactly offset the bottom error, which cheats the tree. Let’s take an example from the Jake Swamp Tree data.      What happens frequently in the field is that the base angle is less than the crown angle and the base distance is closer than the crown distance so that a constant angle error in bottom and top angle measurements do not exactly offset one another when computing height, but often the difference isn’t great. The last measurement of the Jake Swamp white pine in MTSF was:                    Crown Distance: 76 Yds                       Base Distance:    66 Yds                          Crown Angle:      34.20 Degrees                     Base Angle:       -11.80 Degrees      The straight-line eye to target difference between the two distances is 30 feet and the eye level to point difference in the angles is 22.4 degrees (not total angle). The impact of the height error at the base from a 1-degree angle error would be to rob Jake of 3.38 feet. The corresponding error at the crown would add 3.29 feet too much to Jake’s height. The difference of 0.09 feet is hardly worth worrying about. But, the reason the error is so little is that the horizontal offset of the high point of the Jake tree from the base is only 5.24 feet, making the measurement closer to case #1 (top directly over base) instead of case #2 (top significantly removed from base in horizontal direction). Bob Robert T. Leverett Cofounder, Eastern Native Tree Society
 One more for the road Robert Leverett Apr 27, 2007 10:11 PDT
 ENTS,    One more e-mail for the road and then I'll mercifully disappear for the weekend.     It strikes me that proving that the angle error cancels for sine-based calculations when dealing with a right triangle tree configuration ought to be something that we formally do. So here goes.      Let   D = base of right triangle (top is directly over bottom)            A1 = lower angle (to base)            A2 = upper angle (to top)            L1 = laser-based hypotenuse distance to base at lower angle            L2 = laser-based hypotenuse distance to crown at upper angle            H1 = height of base above(below) eye level for angle A1            H2 = height of top above eye level for angle A2            a = angle error for both A1 and A2            dA = a   (treating a as a differential) Now,       H1 = L1*sin(A1) and        H2 = L2*sin(A2) But        L1 = D/cos(A1)             (top is directly above base in this development) And        L2 = D/cos(A2)             (same logic) The differentials are:            dH1 = L1*cos(A1)*dA            dH2 = L2*cos(A2)*dA Substituting for L1 and L2:            dH2 - dH1 = L2*cos(A2)*dA - L1*cos(A1)*dA            dH2 - dH1 = [D/Cos(A2)]*cos(A2)*dA - [D/Cos(A1)]*cos(A1)*dA Simplyfing:            dH2 - dH1 = D*dA - D*dA = 0. The cosines obviously cancel, giving us the result we seek, i.e. a difference of 0. So now, we can all sleep at night...... Uh, but what if the top is not directly over the base so that D does not remain the same as we move from angle A1 for the base to A2 for the top. Under this assumption,            D1 = base for A1            D2 = base for A2 Then dH2 - dH1 = D2*dA - D1*dA = (D2 - D1) * dA If we define Eh as the error in height incurred from a constant error in angle, then               Eh = (D2 - D1) *dA where dA is the error in the angle expressed in radians. This last result is useful, but we can improve on it. The radian equivalent of a 1 degree angle is 0.017453. So a handy formula for a constant angle error of 1 degree for situations where there is a crown point offset unequal to zero is               Eh = (D2-D1)*0.17453. But why stop at 1 degree. We can take this process one step farther. If we define k as some multiple of an angle error of 1 degree (k could be 0.25, 0.5, 1.5, or 2.0, etc.). The formula becomes              Eh = 0.017453*k*(D2-D1), which gives us the error in height for an angle error k expressed in degrees, recognizing that k*0.17453 = the angle error in radians.            As an example, suppose we have a crown-point offset of D2-D1 = 25 and k = 2, an extreme case of angle error. Then              Eh = 0.017453*2*25 = 0.87. An offset of 25 is fairly large and a consistant angle error of 2 degrees is very large. So, it is reasonable to assume that the error in height due to a consistant angle error for sine-based calculations will almost always be under a foot and usually well under a foot. We can really go to sleep now.    Unless someone finds an error in the above construction, I do think Eh = 0.017453*k*(D2-D1) is sufficiently useful and insightful to qualify for our list and become #15. Bob Robert T. Leverett Cofounder, Eastern Native Tree Society
 RTS Robert Leverett May 03, 2007 06:49 PDT
 ENTS,       One more round of the error analysis and I'll let the subject rest for a while. However, as a rationale for going deeper into the subject, ENTS needs to fully explore all the sources of errors in our calculations, not merely to be aware of them or qualitatively describe them, but to quantitatively evaluate them. Obviously, we all recognize that if we make angle and distance errors, we can expect errors in our calculated results, but what are the sources with the greatest impacts? When do errors offset one another, etc.? Do equipment errors propagate through the different measuring techniques (sine versus tangent) for a dimension in the same way? TANGENT/SINE:        Is there a simple relationship between the heights obtained from correct distances but incorrect angles when using tangent versus sine-based height determinations? Yes, there is. For any baseline distance D, crown angle A, and angle error a, the ratio of the tangent-based height to the sine-based height (RTS) is given by:        RTS = cos(A)/cos(A+a).       This means that the distance measurement isn't involved when only the angle varies. As an example, suppose the angle is 50 degrees and the angle error is 1.25 degrees. Then the RTS formula gives the ratio of the tangent-based height to the sine-based height as cos(50)/(cos(51.25) = 1.02943. For a 55 degree angle and 1.25 degree error, the ratio is 1.03241.      The derivation of the RTS ratio follows. Let:                    Ht = Dtan(A+a)   where D = baseline and angle to crown-point = A. Here crown-point is directly over the base;     Hs = Lsin(A+a)   where L = hypotenuse distance to crown-point for angle A. Now,     L = D/cos(A), since we are dealing with a right triangle; so that     Hs = Dsin(A+a)/cos(A) And     RTS = Ht/Hs = [Dtan(A+a)]/[Dsin(A+a)/cos(A)]. By definition     tan(A+a) = sin(A+a)/cos(A+a).     Substituting         RTS = [Dsin(A+a)/cos(A+a)]/[Dsin(A+a)/cos(A)] and Simplifying,         RTS = cos(A)/cos(A+a).      At 65 degrees and an error of 1.25 degrees, RTS = 1.04934     A tree 70 feet away with a crown-point directly over the base at 65 degrees is 150.12 feet in height. The hypotenuse distance is 165.6 feet. Suppose the angle is measured as 66.25 instead of 65. Then the erroneous sine-based height will be calculated as 151.61 feet. However, the tangent-based height will be calculated as 159.1. The ratio of 159.1 to 151.61 is 1.04934. Now, RTS = cos(A)/cos(A+a) = cos(65)/cos(66.25) = 1.04934. Therefore, RTS expresses the ratio of the erroneous tangent-based height over the erroneous sine-based equivalent. I believe that RTS has practical significance, at least for educational purposes, and is easy to apply. Are there any supporters of this point Detractors? Will, Lee, Jess, Paul, Ed, etc.? Bob Robert T. Leverett Cofounder, Eastern Native Tree Society
 RTSa, RTSd, and RTSo Robert Leverett May 04, 2007 05:28 PDT
 ENTS,    It seems like it has been a very, very long time since I've posted any formulas and I’m experiencing formula withdrawal. So, I ask your indulgence. We need the companion formula for the ratio of tangent-based to sine-based height determination that incorporates a distance error, but no angle error. The formula is presented below. Everyone will be spared the algebraic derivation.     D = baseline distance     d = error in distance (baseline and hypotenuse)     A = angle     RTS = (D+d)/(D+d*cos(A))         At 0 degrees the ratio is 0 and at 90 degrees it is (D+d)/D. For a given d, the ratio changes as the angle increases to a maximum of 1+d/D. Since the situations where the angles are 0 or 90 degrees are of no practical significance, the ratio is seen to always be greater than 1, meaning that the height error made from a distance error is greater for tangent-based calculations, though not by much at low angles Short baselines and high angles give the highest ratios. For example, a 67-foot baseline, a 3-foot baseline error of too much, and an angle of 65 degrees lead to a ratio of 1.0254. However, switching from distance to degrees, a 1-degree error at 65 degrees leads to a ratio of 1.0390 regardless of the baseline distance. The inescapable conclusion is that for modest distance or angle errors, the angle error is the more serious.     The formula for the ratio of the tangent-based height to the sine-based height where there are errors in both the angle and distance is cumbersome and won't be presented. The picture of effects is clear from the two formulas.     One final formula to cure my formula withdrawal, what if there are no equipment-related errors, but there is a crown-offset error? The crown-offset only applies to the tangent-based calculation, but we can still compute the ratio of the height determinations from the two methods. Well, the formula is:      RTS = (D+d)/D*cos(A).     To distinguish between the different RTS situations, we could settle on the following definitions:     RTSa = RTS associated with an angle error,     RTSd = RTS associated with a distance error,     RTSo = RTS associated with a crown-offset error.     These formulas in conjunction with the total differential dH provides us with a means of investigating the impact of errors from angle, distance, and method. Bob Robert T. Leverett Cofounder, Eastern Native Tree Society
 Re: Angle errors revisited Edward Frank May 04, 2007 08:17 PDT
 ENTS and Bob, I want to thank Bob for taking on the giant task of trying to better define all of the aspects of tree measuring in terms of mathematics. I am sure I could not do all of the many equations Bob has posted.   We may argue about details, but the important aspect at this point is to get them out there. I am sorry I missed Bob's presentation at Cook last weekend. The idea that the top error and the bottom angle error offset was noted by John Eicholz in a post several years ago. The offset is not perfect but the errors are both in the same direction and to a large degree cancel each other, even when using large errors such as two degrees in Bob's example. Smaller angle errors have smaller net errors and the sum of the offsets are even closer to zero when using the top/bottom sine methods. I just wanted to point out the idea was not mine originally. Ed Frank
 Back to Ed: RTSa, RTSd, and RTSo Robert Leverett May 04, 2007 09:55 PDT
 Ed,    Thanks. It is a labor of love, and until recently, it was mostly going over plowed ground. But more recently I started pushing myself to revisit mathematical concepts that I haven't thought about literally for years, searching for potential applications to our ENTS work.    I'll incorporate all this error analysis in the upcoming guide to Dendromorphometry, which all of you will get an ample opportunity to edit. Bob
 RTSad Robert Leverett May 07, 2007 06:02 PDT