RE: Maximum Measurement Errors Mar 2006 05:54 PST
 ENTS, This is latest of a series of discussion on measurement errors related to laser rangefinders and clinometers posted to the discussion list and to the website.   The earliest post from Nov 2003 contains a nice mathematical analysis of clinometer error by John Eicholz and discussions by Howard Stoner, Bob Leverett, and others.  The posts below continue that theme. Ed Frank
 Maximum Measurement Errors Edward Frank Mar 15, 2006 08:33 PST
 Max error considerations Robert Leverett Mar 15, 2006 12:29 PST
 Ed,    We guessed what you meant from the 0.707 value. ENTS,    Some sine values get quickly committed to memory. For instance:    sin(0) = 0.000    sin(30) = 0.500    sin(45) = 0.707    sin(60) = 0.866    sin(90) = 1.000    To keep an idea of the worst case scenarios for the equipment problems I have encountered with my own instruments, I keep handy tables such as the one following this narrative. The worst equipment problems that I have had are sticking clinometer that can read high or low by half a degree and laser that read long or short by one yard (this is a typical worse case scenario barring really major equipment problems). The table assumes the clinometer reads high by 0.r degrees and the laser shoots long by a yard. Some of the combinations in the table obviously don't make sense for trees. They are included for completeness. The table clearly reveals the danger of extremely long shots. It also shows that for very tall trees shot far enough back to see the top, with the laser-clinometer combination, an error of 3 to 3.5 feet can be made - least we get too complacent. But these are worst case scenarios for the the sin-sin method and include no partially offsetting measurement for any below eye-level component of the tree where the clinometer error of 0.5 degrees would be partially offset. The laser would still read long, but the error from the clinometer would go the other way. The table follows. Angle Ydg CalcHgt ActHgt Diff                 in Ft   in Ft   in Ft 0 150 3.95 0.00 3.953 15 150 121.06 116.47 4.590 30 150 229.91 225.00 4.915 45 150 323.10 318.20 4.904 60 150 394.27 389.71 4.560 75 150 438.57 434.67 3.904 90 150 452.98 450.00 2.983 0 125 3.30 0.00 3.299 15 125 101.02 97.06 3.959 30 125 191.85 187.50 4.350 45 125 269.61 265.17 4.444 60 125 328.99 324.76 4.235 75 125 365.96 362.22 3.738 90 125 377.99 375.00 2.986 0 100 2.64 0.00 2.644 15 100 80.97 77.65 3.328 30 100 153.78 150.00 3.784 45 100 216.11 212.13 3.983 60 100 263.72 259.81 3.910 75 100 293.35 289.78 3.571 90 100 302.99 300.00 2.988 0 75 1.99 0.00 1.990 15 75 60.93 58.23 2.696 30 75 115.72 112.50 3.219 45 75 162.62 159.10 3.522 60 75 198.44 194.86 3.585 75 75 220.74 217.33 3.404 90 75 227.99 225.00 2.991 0 50 1.34 0.00 1.335 15 50 40.89 38.82 2.065 30 50 77.65 75.00 2.653 45 50 109.13 106.07 3.061 60 50 133.16 129.90 3.261 75 50 148.13 144.89 3.238 90 50 152.99 150.00 2.994 0 25 0.68 0.00 0.681 15 25 20.84 19.41 1.433 30 25 39.59 37.50 2.088 45 25 55.63 53.03 2.601 60 25 67.89 64.95 2.936 75 25 75.52 72.44 3.071 90 25 78.00 75.00 2.997 0 15 0.42 0.00 0.419 15 15 12.83 11.65 1.181 30 15 24.36 22.50 1.862 45 15 34.24 31.82 2.416 60 15 41.78 38.97 2.806 75 15 46.47 43.47 3.004 90 15 48.00 45.00 2.998 Contrast the maximums with what can happen using the % slope method where errors can be in the 10's of feet. Bob
 Re: Max error considerations Edward Frank Mar 15, 2006 13:40 PST
 Bob, I did not repeat your entire post below. But there is a false assumption in your calculations concerning the clinometer errors. If the clinometer sticks when measuring the top, and doesn't when measuring the bottom the errors will b as you indicated. However this is really instrument failure - rather than instrument error. The false assumption is in the statement "partially offsetting measurement for any below eye-level component of the tree where the clinometer error of 0.5 degrees would be partially offset." The errors at the base would offset the errors at the top whether the base of the base of the tree were below, level, or above eye level. The offset would not be complete, because the distance to the top of the tree in most cases would be greater than the distance to the base (you could be looking down on the tree from a slope above), Thus the small error is applied over a greater distance than the same error is applied when shooting to the base of the tree. But it would still be almost completely offset. For example consider a basic 3-4-5 triangle for a tree 160 feet tall, measured from a distance 120 feet to the base on a level line, with the top of the tree 200 feet from the viewer. If the instrument misread the base of the tree to be at an angle of -1 degree, rather than 0 degrees, the calculated distance below the level line would be -2.09 feet. The top of the tree would be measured a similar amount low: 52.13 degrees, rather than the actual angle of 53.13 degrees. The calculated height of the tree above horizontal would be 157.88 feet. Adding the two numbers together gives you a total height of 159.97 feet instead of 160 feet. That is a net error of 0.03 feet with a clinometer that is reading off by a full degree.  Sticking is a completely different class of problem. If the instrument is sticking at one end of the readings and not the other, whether or not the base is above or below eye level, there will not be any offset of the error generated by the sticking instrument. That is simply a fixed error. Ed
 Re: Dry Creek, NC Edward Frank Mar 15, 2006 14:02 PST
 Josh, The value of .8 yards long seems to me to be higher than the normal range of offset to be expected straight out of the box. Perhaps the company could recalibrate it for you if it is till under warranty. What you need to determine is how far the instrument is off for the ranges you are typically measuring tree heights. You need to make sure of the calibration in these intervals which are longer than the 50 meters you have information about. People commonly make a mistake when using bathroom scales. They adjust the settings to read exactly zero when empty. That doesn't matter. What they need to do is to adjust the bathroom scale to match the scale at the doctors office when the person is standing on them. Thus all of the values close to your weight will be calibrated to read as accurately as possible. Who cares if it is accurate in a weight range you are not measuring, especially if that calibration would compromise the accuracy of the values in the range you are measuring. Ed ----- Original Message ----- From: "Josh Kelly"  Sent: Wednesday, March 15, 2006 3:47 PM Subject: Re: Dry Creek, NC Ed, Thanks for the education about range finders. I have figured out the click over on my range finder out to 50 meters. I haven't figured it out beyond that. I also know that my range finder shoots .8 yards long. A major source of uncertainty for me is that my range finder only delivers even numbers, still, I am impressed so far with its precision. Josh
 RE: Max error considerations Robert Leverett Mar 16, 2006 05:54 PST
 Ed, As usual, you caught my verbal shortcut. Instrument failure is the better choice of description. What I should have said was error due to the clinometer reading - whether the source of the error is a mis-reading due to bad eye sight, a sticking clinometer where the direction of the sticking is not consistent, or a clinometer that is off in its calibration by a consistent amount in one direction or the other. Having owned 5 clinometers over the past 15 years, I've experienced the range of problems with them. One of my clinometers is out of calibration by about a degree. It reads high. As you correctly note, the errors from a clinometer being consistently off almost cancel. However, I've encountered situations where the out of calibration clinometer would stick - unpredictably. It gives the user reason to continuously be checking for angle errors. Bob
 Max error considerations - clinometer Edward Frank Mar 16, 2006 07:52 PST
 RE: Max error considerations - clinometer Robert Leverett Mar 16, 2006 08:56 PST
 Ed,    Excellent discussion of the sources of error that can be associated with the clinometer.    For a clinometer in good working condition (the operative assumption), it is my experience that one can learn to read it accurately to within a quarter of a degree. That doesn't happen starting off, though. But using the methods you describe, most readers can eventually become that accurate. However for analysis purposes, the half degree is a safe maximum error for angle reading.     For those shaky with the math, consider the calculation:      Error in Height = distance x sine(error in angle)     The greater the distance and/or the greater the error in the angle, the greater the height error attributable to the error in the angle. For any given error in the angle, if you double the distance, you double the height error. The relationship is linear. However, if you double the angle error for a given distance, you don't exactly double the height error, although the difference is inconsequential for most purposes. Here the doubling is in the error of the angle, not the base reading of the angle. Just want to make that point clear. Bob
 Re: Max error considerations - clinometer Edward Frank Mar 16, 2006 10:25 PST
 Bob, The angle you are looking up makes a big difference. What you say is true for errors at around 0 degrees, but if you are looking up at an angle of 60 degrees, the error is half that value. The simplification may be adequate for people with a math phobia but with just a slightly more complicated equation a better estimate can be obtained: Error height = distance x sin(angle) - distance x sin(angle + error in angle) for 30 degrees, at 70 yards, with an error of +0.5 degrees for example: Error height = 210 x sin(30) - 210 x sin(30.5)                    = 105 - 106.583                    = -1.583 feet The basic formula you suggested: [Error in Height = distance x sine(error in angle)] would give an error of 1.83 feet. This is not a big difference between the two formulas, but the difference becomes greater with an increasingly steeper angle. A more complete formula would include the calculations for both the base of the tree and the top of the tree: Error height = [distance to base x sin(angle at base) - distance to base x sin(angle to base + error in angle)] + [distance to top x sin(angle to top) - distance to top x sin(angle to top + error in angle)] I would be interested in your insight on what techniques will let you make better estimates, problems in the field, how to tell if your instrument is sticking, and other problems associated with making inclination measurements not covered by the basic number crunching I posted. Ed
 RE: Max error considerations - clinometer Robert Leverett Mar 16, 2006 13:06 PST
 Ed,    Some of our readers might like to see the absolute amount of height error that a 0.5 degree clinometer error has as both the distance and angle to the target increase. The angle error in the examples is on the high side. Dist-to-target   Angle   Angle-error    Height-error 50                  0         0.5           0.44 50                 30         0.5           0.38 50                 45         0.5           0.31 50                 60         0.5           0.22         100                 0         0.5           0.87 100                30         0.5           0.75 100                45         0.5           0.61 100                60         0.5           0.43 150                 0         0.5           1.31 150                30         0.5           1.13 150                45         0.5           0.92 150                60         0.5           0.65 200                 0         0.5           1.75 200                30         0.5           1.51 200                45         0.5           1.23 200                60         0.5           0.87 250                 0         0.5           2.18 250                30         0.5           1.88 250                45         0.5           1.54 250                60         0.5           1.08    Practically speaking this says the largest error that is likely to be made as a consequence of misreading the clinometer or the clinometer consistently reading off by as much as 0.5 degrees is a little over 2 feet. However, if the laser shoots long by a yard and the clinometer reads high by 0.5 degrees, then the combination produces an error of 2.21 feet at 0 degrees and 250 feet, but 3.69 feet at 60 degrees and 250 feet. We're obviously not measuring an eastern tree in the latter case, at least none in this era.    If we assume that 120 yards is the longest distance we'll shoot and 33.8 degrees is the highest angle we'll shoot at that distance (tree is about 200 feet-dream on Bob), we make an error of 2.6 feet from the clinometer alone, 1.7 feet from the laser alone, and 4.3 feet from the combination of clinometer and laser. This is an extreme distance from which to measure a tree. If we move closer and shoot the same tree at a distance of say 240 feet, our combined clinometer-laser error drops to 3.4 feet. So shooting a very tall tree at 80 yards with a clinometer reading that's off by 0.5 degrees and with a laser that's off by a yard results in a height error just a little over 3 feet.    The lesson in all this is that at distance of 80 yards or less, unless equipment is a real problem, one has to work hard to make an error in the height measurement of 3 feet. I learned that a long time ago and began boldly proclaiming that most of our measurements would be within 1.5 feet of the actual height of what was being measured.       Ed, I'll address your questions tomorrow. Bob
 RE: Max error considerations - clinometer John Eichholz Mar 16, 2006 18:06 PST
 ENTS, I think recently my clinometer error is closer to +/- 0.2 degree. This comes with long practice, and with careful calibration of the individual quirks of my clinometer using a "peg test". The test involves mounting the clinometer on a tripod, placing thin black tape at judicious places on the door frames, and generally acting like a nut. I only do it at home alone. Whenever I want the most accurate readings, I use a small handheld tripod which I brace on the trunk of a conveniently located tree. Since there is no motion I have very little reading error. These readings are repeatable to within the limits of my vision, so I believe they are quite accurate. I know the math is complicated, but my post of 11/10/03 and I think a later post shows that height error due to angle error in measurements taken of a vertical object from a fixed point are almost equal independent of the angle. This is because as the angle increases, the error rate decreases but the distance increases, and the math shows these changes are in a near lock step offset. So, a height reading from a fixed point with a consistent clinometer error (either always too high or always too low) using an angle reading at both the top and the bottom, should nearly cancel. I guess that is good news. One interesting detail I have observed while calibrating the rangefinder is that it is much more accurate with a sky background than with a terrestrial background. I notice this with narrow objects, such as the edge of my picnic table, less so with tree trunks. This is likely due to overlapping reflections from more distant objects. The readings are always longer. With a reflector, the readings are as accurate as those with a sky background. Since we almost always have a sky background at the tip, I recommend calibrating against a sky background. I have used a thin stick projecting above my roof, with a long tape trailing to the ground. John