Re: Tangent errors pot pouri

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#1)  Tangent errors pot pouri

Postby dbhguru » Sun Jan 24, 2016 1:16 pm

Hi Folks,

  I am pleased to report that the AF National Cadre is gaining steam. We now have 27 members and should soon add a 28th. A new unofficial responsibility of the Cadre that looms large on the horizon is assistance in training state people. While the Cadre must use the Sine Method for measuring tree height, state people will likely continue using the Tangent Method for several years to come. The best we can do is to train them on how to avoid making large errors, and to understand the method at a deeper level. For NTS members who either are Cadre or are not, this is a tall order. This post is intended to aid in the training.  

Baseline for Tangent Method

  The first hurdle is to get tangent measurers to understand what constitutes the correct baseline for a tangent-based measurement, especially for the crown. Simply stated, the correct baseline for the tree top is the level line from the eye to the point that is vertically below the point being measured. What will come as the first shock is that the correct baseline has nothing to do with the trunk except in cases where the top is vertically above the point on the trunk that is level with the eye. In other words if a tape is stretched level from the eye to the trunk and the point where the tape touches the trunk is vertically beneath the top being measured, then the trunk does plays a convenient role in establishing the baseline. As we know, this works reasonably well for lots of conifers, especially plantation trees, but often fails for broad-crowned hardwoods, such as we often measure for championship status.

  So, what is the disconnect? Clinometer users are taught (or assume) that the instrument was designed to measure tree height and that if they used it as instructed, they will get tree height. However, what the clinometer is actually designed to do is convert an angle in the vertical plane to the vertical leg of a right triangle based on an assumed horizontal distance. If the horizontal distance is a surveyor's chain length of 66 feet or a multiple thereof, all the measurer need do is read height off of a scale as if the baseline is 66 feet, or multiply by 2 if the baseline is 132 feet. If the assumed clinometer baseline is 100 feet, the clinometer scale reads the tree's height. No calculation needed for any distance. But all the clinometer scale is doing is returning the height leg of a right triangle for the angle measured as a expressed as a percentage of the baseline.

 The critical assumption is that a right triangle has been formed by the baseline, the line from the end of the baseline up to the top, and the line from the top back to the eye (hypotenuse). Either it does or it doesn't, but clinometer manufacturers don't tell measurers what to do if the tree being measured does not correspond to the one shown in their diagram, i.e. with the top positioned vertically over the base. Cadre trainers have to get these points across to trainees.

Methods for Establishing the Right Baseline

  Once the trainee understands the requirement to establish the correct baseline, the trainer can cover cross-traingulation as one method of pinpointing the spot on the ground vertically under the top. A second method is to assume a position in which the level line of sight to the trunk is 90 degrees to the vertical plane containing the end of the baseline and the top. These methods are illustrated in the AF guidelines. However, the trainer should also alert the trainee to situations where either or both methods can't be used. For example, when one changes one's location, the visibility of a point in the crown may be lost.

Understanding the Impact of Distance and Angle Errors

 Even if the trainee understands what constitutes a correct baseline for the top being measured, there is the possibility of angle and baseline distance errors. How do these kinds of errors translate to errors in height? Are there rules or guidelines to be invoked that could bring down an error in either distance, angle, or both? Here is where matters get complicated. There are guidelines to follow, but used blindly, they can lead you in the wrong direction.

 If we consider an error in baseline distance with no angle error, the rule is that the lower the angle to the target, meaning a longer baseline, the less the error in height. It's that simple, or is it? Well, as you get farther back, unless the ground slopes downward, going back, the more of the crown use see - but less distinctly. It will always be a situation of tradeoffs.

 If we consider an error in angle only, no baseline error, then the minimum error in height will be at 45 degrees - exactly 45 degrees. At higher or lower angles, the height error goes up.

 If we have errors in both angle and distance, and they are both in the same direction, i.e. over or under, then the minimum error in height will be at an angle to the target of less than 45 degrees. It could be as low as 30 degrees.

  If we have errors in both angle and distance, and they are in opposite direction, i.e.one over and one under, then the minimum error in height will vary enormously. It's Katie bar the door, For example, if the distance error is -2 feet and the angle error is +1.5 degrees, would you have guessed that an angle to the target of 50.6 degrees minimizes the height error? Now, let's switch the sign of the errors to +2 feet and -1.5 degrees. Where's the sweet spot? It is still at 50.6 degrees. If our error had been +1 foot and still -1.5 degrees, the sweet spot would be at 49.1 degrees. Would that have been obvious?

  Assuming the worst distance error would be +/-3 feet and angle error +/-1.5 degrees, we find the sweet spots at 53.5 for (-3,1.5) and 55.4 for (3,-1.5). So much for a simple all inclusive rule. It helps to play what-if games using an Excel spreadsheet. I think some of the combinations will elicit amusing responses from some of you.

 The problems associated with high angles to the target often manifest themselves in conflicting ways. You have greater visibility form being closer, but look upward at a steep angle lessens your chances of seeing the top.In addition, your instrument may be less reliable at high angles. Something to think of.

  The attached spreadsheet allows you to play what-if games. I hope I haven't made any formula errors. Catching my goofs isn't a strength of mine. It would be greatly appreciated if some of you could check the logics. Remember that in Excel, angles used in functions must be expressed in radians as opposed to degrees. The functions Degrees() and Radians() allows you to convert back and forth.

  Note that, the spreadsheet starts by you supplying an assumed tree height in cell A14. Starting at 25 feet, distances are incremented by 10 feet for each new row. The resulting angle from eye to top is calculated. You supply distance and angle errors in cells E14 and F14 respectively. Column H shows the height calculated from the original height and distance plus the errors in distance and angle.

  Note that included in the last 3 columns are height error calculations via the total differential approach. Basically, the lesson here is that if the error in a measurement is a substantial percentage of its base value, the total differential is not a good method. That point has never been in question, but I don't think I have stressed it previously.

   I'll go through the spreadsheet again in a couple of days looking for errors. But here it is now.

Bob
Attachments
TangentErrorAlaCart.xlsx
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Robert T. Leverett
Co-founder and Executive Director
Native Native Tree Society
Co-founder and President
Friends of Mohawk Trail State Forest

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