Bob and John,

I am trying to understand the mathematics of the calculations Bob made to get an average offset or 8.3 feet through the mathematical processing of 1800 tree survey data sets. Please bear with me.

Say you had a plane with a circle centered at the x-y intersection with a radius of Q. Then draw 100 radii lines evenly spaced around the circle. The deviation from the y axis could be calculated for each point as the absolute value of sin theta Q where sin theta was measured as the angle from the y axis. If you added all of these deviations from the y axis and then averaged them you would get an average deviation from the y-axis at all of these angles. As I envision it this number should be proportional to the actual radius of the circle. This proportion should be constant regardless of the radius of the circle. I am sure there is a simple and elegant integral to calculate out the average deviation from y in a circle centered on the y-axis, but my math isn't up to the task. [Could you enlighten me?] What this would mean is that by knowing the average of these deviations, the actual radius of the circle could be calculated.

In Bob's case of 1800 trees, the deviation from the y-axis (perpendicular to the line of sight with the z-axis running through the center of the base of the tree) can be easily calculated for every tree to see how far off the tops are in the x direction along the line of sight from the base of the tree. With 1800 samples, and a reasonable assumption that they the tops are randomly distributed in direction offset from the center of the base of the tree, the average deviation from the y-axis should be proportional to the average offset distance of the top in any direction from the true center of the base of the tree.

Essentially it means by knowing the offset in the x-direction you could calculate the actual average offset of the tops without knowing the amount of offset in the y direction. 

Does what I am saying make sense? Maybe this is what you have already done. How about a little help here guys.

Ed Frank 

Hi Ed,

I am glad to try and clarify the result of the calculation you wish to 
achieve. First let me say you have a clear grasp of the problem and it 
is a more interesting problem than it first appears. By solving it we 
can determine the theoretical average lean of the trees in Bob's 
database, even though this figure is not accurately calculated for each 
tree, by making a statistical assumption that the trees' lean is 
symmetric to the point of measurement. Quite an accomplishment of 
abstraction, it would appear. Thank you for the interesting problem.

Actually, the math is quite simple. The observed offset from vertical 
(as seen from the point of measurement) is proportional to the COSINE 
of the angle. Assume a radius of 1. The average offset would be the 
INTEGRAL of abs(cosine theta) around the circle, divided by the length 
of the circle. We can evaluate this as twice the integral of (cosine 
theta) from -pi/2 to pi/2, divided by 2*pi (the length of the path of 
integration). I apologize if this doesn't make sense, but 
mathematically it is accurate.

The integral of the cosine function is the sine function. We thus 
evaluate the integral as 2*[sin(pi/2) - sin(-pi/2)]/(2*pi). , or 0.6366 
and change.

Using an average observed offset of 8.3 feet yields a projected average 
true offset of 8.3/.6366, or about 13 feet.

I believe we can conclude by saying that the trees measured in Bob's 
database appear to have an average offset of 13 feet from bottom to 
measured tip.

John Eichholz

Bob and John,

The question comes to mind about how Bob calculated the offset and what assumptions were made. Bob if you could jump in and clarify the methodology. A first order approximation would be to calculate the horizontal distance to the base then calculate the horizontal distance to the top. 

Parallax could be safely ignored within the slight left or right offset we would be dealing with. 

There are a couple of issues that need to be factored into the calculations - 1) measuring errors with the laser rangefinder base distances, and 2) radius of the tree.

If the top is measured at a click-over point to obtain the greatest accuracy for the vertical aspect of the measurement, then it is likely that the distance measurement to the base is not at a clickover point. (If you move in or out to get a clickover then the calculations are really screwed.) Thus a reading of 25 yards means the distance to the base is somewhere between the reading of 25 and the clickover to 26. I see no reason that any one portion of this interval should be more likely to occur than any other. Therefore the average error over a large number of shots should be whatever distance of the 25.5 yards would turn out to be minus 25 yards. Therefore if the average rangefinder would clickover at exactly 25 yards, then the average distance for a 26 yard reading not at a clickover point would be 25.5 yards - 25 yards = 0.5 yards or a foot and a half. If the average clickover point (to 25 yards) for point was actually at 24.5 yards, then a large number of readings with random errors would actually average out to be a true 25 yards with an average erro of 0. In your estimation, at what distance do laser rangefinders actually change numbers? At approximately the correct number or at a lower value? That would be a distance that would need to be added to the base horizontal distance to correct for instrument error.

2) When you measure the distance to the base of the tree, you are measuring the distance to the front of the trunk not its center. That is being compared to the distance to top - essentially a distinct point. If the tree was 2 feet in diameter then the bottom distance would be short by 1 foot. To correct for this the cbh readings could be used to determine radius. The radius at the base is larger than at breast height so the cbh numbers would need to be corrected. Also the cbh is generally in inches rather than feet. If the base radius was 10% larger than at breast height, then the correction in feet would be = [cbh/(2)(3.14)(0.9)(12)] roughly about 1/68. For trees without any cbh a set number could be used to complete the calculation, A statement if cbh = null, then cbh = 1.2 (whatever number seems appropriate.).

These do not seem like major amounts of correction, but taken together- the laser is 1/2 yard off, the tree is 3 feet in diameter, then the total amount of correction would be 3 feet - a significant proportion of the total averaged offset of 8.3 feet. For the general average it doesn't matter because the values would be added to the values on one side of the y axis and subtracted from those on the other - provided there actually were equal numbers on each side of the axis. 

The reason for adding these corrections to the calculation would be so that a distribution of where the tops were located with respect to the center of the base of the tree could be made. 

The numbers from conifers could be compared to hardwoods for example...

There are a number of assumptions and estimates being made with these corrections. 1) The average calibration correction for the laser rangefinder is whatever. 2) The base of the tree is a certain percentage larger than the tree at breast height. 3) The general assumption of circularity of the tree trunk. 4) The bottom of the tree was measured from the same point as the top.

The errors associated with making these assumptions are much much smaller than if these corrections were not made at all.

Again I am just brainstorming and trying to figure out what the numbers mean.

Ed Frank

Lee, Ed, John, Will, Jess, Dale:

The attachment speaks for itself --- I hope. Suggestions? Improvements?

Bob

Database Developer and Systems Analyst
Information Technologies

Attachment: Offset Diagram

Bob,

I got the spreadsheet. I will think about it. My biggest concern is still the calculation of AD, as I explained in a note a couple days ago. 

1) If not at a clickover point then the rangefinder should on average be reading 1/2 yard short along line AE;
2) The radius of the trunk should be added to the length of AE. 

Both would increase the length of the line AE and hence AD. The offset GD would be increased for tops in front of the plane of the trunk base and decreased for those beyond the plane of the trunk.

Ed

Ed, 

As a user of the rangefinder back to 1996, I learned to routinely scan the trunk from side to side. I look for a clickover point between the middle and the sides of the trunk, since what we are really doing is going for the center of the tree. Additionally, I work the problem of laser rangefinder clickover for both the crown and the base. One must be conscious of elevation changes shifting forward or backward by up to 3 feet searching for the other clickover, but the elevation change is usually half that. Since I'm seldom on a slope of more than 45 degrees, the vertical correction is usually less than half a foot. In future e-mail discussions, we may want to emphasize these points to the rest of the list, although beyond about 12 of us, I doubt that the others have much interest. For the trunk distance, if I find that I can't get a clickover between the center and the side, I move forward or backward until I do get a clickover. If the clickover is at the center of the bole, I can now use the RD 1000 relascope/dendrometer to get the add-on to the pith. This extra step isn't particularly important for small trees, but trees such as the big tuliptrees in the Smokies that are 4 to 7 feet thick must be handled much more carefully. 

I would add to what you say below - just for clarification. The impact on height determinations using the tangent method (without crown point cross-triangulation) would be to overstate height where AG is less than AD and understate it where AG is greater than AD. In the first case, the baseline is overstated, and in the second case, understated. Interestingly, there are cases where AG = AD, even when GD is of significant magnitude. Tangent users get away with sloppy technique where AG and AD act as radii of a circle through G and D. On occasion, I've been with measurers who lucked out when their results matched mine. They proudly proclaimed their methods of equal efficacy and wondered what all my fuss was about in championing hypotenuse-driven measurements to the actual crown point. None had a clue as to the assumptions each was implicitly making and it proved pointless for me to try to educate them. 

Bob

Bob,

If you measure the position of the top, and then move and measure the position of the base these two numbers are no longer directly comparable. The horizontal offset will change as you move, even if the height measurements are not affected significantly. If you move toward the trunk by 2 feet to get to a clickover point then the relative position of the measured top and the measured base will also be offset by the same amount - 2 feet. The offset can only be measured if the top and bottom positions are measured from the same point. Movement adds or subtracts the amount of lateral movement toward or away from the trunk to the calculated offset position. 

If you move to a base clickover point, that would offset the non-clickover point correction I suggested, but add another correction that would depend on whether you tended to move forward or backward to obtain clickover more often. From a measurement perspective, and I understand the point of the measurements was to obtain height not top offset, the procedure would be better to not move to a click-over point, but accept the average error that convention would entail. This error is a statistically random error that can be averaged over the dataset- the other is a subjective error that is not random and might be applied differently between different people in similar situations.

I need to think about this some more. Your second paragraph is right on the money. It is exactly what it means. To elaborate half of the errors would be greater than the average offset errors. (and half the errors would be less than the average offset errors) I think that the tops measured are more likely to be on the facing side of the tree than on the opposite side of the tree, but for calculation purposes we used to calculate actual offset, this front to back asymmetry is not important. [It could be calculated from the dataset]. Therefore the likelihood of a tree being measured too high, even if the actual top is found would be greater than the chance of it being underestimated. If the measurer is not hitting the top, then almost definitely it would be high. The laser lets you identify the top more clearly than other methods because it generates a number, so the chance of not picking a top, or a correct top is much much greater when just guessing without using a laser. Most of the bigger errors are the result of mis-identifying the top, lesser errors are a direct result of the top not being directly over the base, even if it is identified correctly.


Ed

Ed:

Yes, of course. You are correct. I changed horses in the middle of the stream as I began reviewing my method for getting ever more accurate heights. Switching to a system where offsets are considered does change the rules. Ed, be damned if I know how we got along before you joined us. I hope others appreciate your contributions as much as I do. I know Will Blozan, Dale Luthringer, and the main measuring crew does.

Your second paragraph is so very important. Oddly foresters more consistently resist this logic than others. The mistakes of top identification are indeed on the measurer's side of the tree far more frequently than on the opposite side where high tops are often not even visible. It is the steep angle of a jutting limb high within the crown that creates the illusion of a top and leads to measurement of false tops. Those who mechanically set base lines of 100 feet so they won't have to do any math with their clinometer are the ones who mis-measure by 50 feet and more. 

I'm always surprised that more professional foresters don't buy themselves cheap laser rangefinders and make good uses of them. Don Bertolette once told me that as a group foresters tend to be weak in conceptualizing mathematical ideas. They do fine running numbers in an add/subtract mode, but are seldom strong as abstract thinkers. Since Don holds a masters in forestry, I gave considerable weight to what he said. He said made that statement years ago and in the intervening time, I've confirmed what he said over and over. 

Bob

Bob,

I guess the tradeoff is more accurate heights, versus better offsets. Which is more important - I would vote for heights.

Ed