Clinometer Errors and Calibration

General discussions of measurement techniques and the results of testing of techniques and equipment.

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Karlheinz
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Re: Clinometer Errors and Calibration

Post by Karlheinz » Mon Jan 21, 2013 8:36 am

I would like to come back to the origin of the discussion.
I made calculations of my own: What is the effect of +1° angular error on reading of height?
Tabelle-en.jpg
Distance 50m-100m.jpg
This calculation raises the following hypothesis:
condition is:
- vertical measuring body (tree) and given measuring position on horizontal baseline
- clinometer with angular error constant over the entire range (e.g. +1 °)


1) Height error is exactly constant, no matter what the angle
For example, measurement error in the case of height = baseline (45°) is constant at 1.75% of height (related to +1° error), no matter what angle to measuring point.

2) If determining the height as difference between two measurement points (sine-based), angle measurement error will be compensated completely

Please refute my hypothesis or render the mathematical proof!

I conclude:
Clinometer calibration is of no importance, if you take the two measuring points of the sine-based measurement
by the same clinometer instrument.


With best regards
Karlheinz

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KoutaR
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Re: Clinometer Errors and Calibration

Post by KoutaR » Mon Jan 21, 2013 4:57 pm

Karlheinz,

Only a minor note: The height error is not exactly constant, though so small that other error sources greatly exceed it. If you add one decimal, for example, in the case of 50 m baseline the difference (and the height measuring error) is 0.873m - 0.865m = 0.008m = 8 mm. I would not have believed the error is so small without calculating it!

In the case of the baseline = 0 m, the angle to the top is not 45 degrees but 90 degrees, though it has no infuence on the result.

Kouta

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edfrank
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Re: Clinometer Errors and Calibration

Post by edfrank » Mon Jan 21, 2013 5:19 pm

I ask Bob to do a nice excel table and graph as he is better at it than I am, but basically these are the points to be made:

If you consider a tree as a vertical pole for the purposes of calculating clinometer errors, and consider that all measurements are taken from the same point, and that the amount of clinometer error in degrees is equal and in the same direction at all angles, then the following statements are true:

1) The distance to the tree along the hypotenuse as measured by the laser rangefinder will vary with the angle, with the greatest distance being at the steepest angles.
2) No matter what the length of the hypotenuse, the baseline distance from the measurer to the tree will be the same at all angles.
3) If the clinometer reads incorrectly by one degree, and that reading is one degree higher that true, then the readings at all angles will be the correct angle @ + one degree.
4) The actual height at any measuring point will be tan@ x baseline
5) The indicated height will be tan(@ +1) x baseline
6) The height error at any angle therefore will be (tan(@+1) – tan@) x baseline

What you will see is that since tan isn’t a linear function that the value (tan(@+1) – tan@) will not be equal at all angles.

The net result is that the errors at the top and bottom tend to offset, but not perfectly and at some angle combinations the offset is worse than others. This makes a difference if you are measuring at those angles, if you are trying to measure different portions of the tree from different positions and adding them together, or if you are using a pole or similar device to measure the distance from eye level to the base of the tree.

This is a generalized equation for all trees and all error amounts.

m = top angle
n = bottom angel
e = clinometer error
ht = hypotenuse for top triangle
hb = hypotenuse for bottom triangle

[[tan(m) – tan(m-e)] x [cos(m-e) x ht]] – [[tan(n) – tan(n-e)] x [cos(n-e) x hb]] = error


Ed
"I love science and it pains me to think that so many are terrified of the subject or feel that choosing science means you cannot also choose compassion, or the arts, or be awe by nature. Science is not meant to cure us of mystery, but to reinvent and revigorate it." by Robert M. Sapolsky

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Karlheinz
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Re: Clinometer Errors and Calibration

Post by Karlheinz » Tue Jan 22, 2013 4:28 am

My consideration is:
If one adjusts the indicator of the Clinometer by +1°, the measured distance by the laser remain unchanged. Only the built-in calculation routine of the Clinometer calculates the vertical component incorrectly by the factor (sin (angle +1 °) - sin angle). When lowering the laser beam this factor gets greater, but the measured distance gets smaller. Both compensates completely, the amount of the height measurement error remains constant.
I empirically found out this by specification of numbers for different angles. I know that mathematical proof has yet to be delivered.

Karlheinz

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edfrank
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Re: Clinometer Errors and Calibration

Post by edfrank » Tue Jan 22, 2013 11:38 am

Karlheinz,

Yes I made an error in my post. The angles at the top and bottom tend to cancel out. There is some minor differences, but not enough to be significant in the total calculations.

Ed
"I love science and it pains me to think that so many are terrified of the subject or feel that choosing science means you cannot also choose compassion, or the arts, or be awe by nature. Science is not meant to cure us of mystery, but to reinvent and revigorate it." by Robert M. Sapolsky

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dbhguru
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Re: Clinometer Errors and Calibration

Post by dbhguru » Wed Jan 23, 2013 12:49 pm

NTS,

The interaction of angle and distance variables with the measuring method (tangent versus sine) can lead to results that are not always intuitive. Some general principles can be given, but then there are "behaviors" that don't seem intuitive, at least to me. I've attached an Excel workbook with two spreadsheets to hopefully shed some light on this area of NTS analysis. The first spreadsheet follows tangent errors based on a 100-foot baseline to both top and base over a fixed range of top to base angle of 60 degrees, i.e. top angle - base angle stays constant at 60 degrees. We impose a fixed angle error of +1 degree, i.e. we're over a degree at the top and at the base. We then follow the impact as we swing from +30 degrees top to -30 degrees base to +85 degrees top and +25 degrees bottom. As we see the height error is close to canceling out where the top angle is +30 degrees and the base angle is -30 degrees. Then the tangent method starts to fall apart.

By contrast, the sine method yield good results at all but the most extreme combinations and even then, it isn't bad. What is NOT obvious is that with the sine method, the largest errors occur at the lowest angles, not the highest ones, which is the case with the tangent method. So the implication is that you want steeper angles for the sine method. This implies that one wants to get closer to the target. Ideally, you are positioned well above the base of the tree, so that the unsigned values of the angles to top and bottom are closer to one another.

For most of the trees that we measure and for the typical range of clinometer errors we can expect, calibration isn't a consideration. However, in actual field conditions, we often must mix the methods, e.g. sine top and tangent bottom. We may be able to see the base of a tree with our eye and thus get a clinometer reading, but not be able to shoot through the clutter with the laser. Since lean won't likely be much of an issue upon the trunk where we can get a laser shot, the preferred combination would be high angle to crown for the sine part of the calculation and low angle to the base for the tangent part of the calculation.

The interaction of angle, distance, and mixed methods along with errors creates a complex stew with unpredictable behaviors. In the second spreadsheet, we're looking at the impact on sine-based calculations of a fixed clinometer error and repositioning to a closer/or more distant vantage point on the same level base line. We're only looking here at shots to the crown. We position ourselves 100 feet horizontal distance from the target and compute the height error associated with a 1-degree clinometer error. Then we move to a more distant spot level with the first and shoot the target and compare the errors we get at both locations. We try another scenario. What we keep fixed in the new trial is the distance between the two measuring points. Notice that the difference between errors remains fixed for that particular difference between measuring locations. I've filled up the spreadsheet with trials that speak for themselves. More discussion of this type of analysis to come. I's put out there as food for thought at this point.

What should be apparent to all is that the very simple rules of the road accompanying most tree measuring guides don't offer a clue as to how all the variables interact.

Bob
Robert T. Leverett
Co-founder, Native Native Tree Society
Co-founder and President
Friends of Mohawk Trail State Forest
Co-founder, National Cadre

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pdbrandt
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Re: Clinometer Errors and Calibration

Post by pdbrandt » Wed Jan 23, 2013 1:03 pm

Thanks Ed, Kouta, Bob, Karl et al,

Your examples have helped me to understand why clinometer error tends to negate itself when measuring a tree -- especially when you are upslope from the base of the tree.
Patrick

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Karlheinz
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Re: Clinometer Errors and Calibration

Post by Karlheinz » Wed Jan 23, 2013 6:18 pm

Referring to my posting #11), after precise calculations, I realized that my hypothesis isn't perfectly accurate. I must admit, there are minor differences, as Ed, Bob, Kouta and others already have stated, however in this case in the range of 1 cm, absolutely negligible.

The sketch with corresponding formulas shows how I calculated the height error:
Skizze delta alpha.jpg
k is the height error in meters caused by a fixed angle error of +1 degree

There are three equations:

1) s=sqrt(h²+b²)
2) h/s=sin(α)
α=arcsin(h/s)
3) (h+k)/s=sin(α+1)

Thereof derived:

k=s*(sin(α+1)-sin(α))
k=s*sin(arcsin(h/s)+1)-h
k=sqrt(h²+b²)*sin(arcsin(h/sqrt(h²+b²))+1)-h

Inserted a few of concrete numbers for constant baseline and different heights:

h = 0 m, b = 50 m
k = 0.87262 m
-------------------------
h = 25 m, b = 50 m
k = 0.86881 m
-------------------------
h = b = 50 m
k = 0.86501 m
-------------------------
h = 100 m, b = 50 m
k = 0.85739 m

Greetings, Karlheinz

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dbhguru
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Re: Clinometer Errors and Calibration

Post by dbhguru » Wed Jan 23, 2013 7:08 pm

Karl,

You've confirmed for yourself a key value of the sine method. I look forward to other interactions with you in exploring the impact of errors in both angle and distance for different methods of measurement. You may wish to read some of the past posts in Measurement and Dendromorphometry. A formula that I have posted in the past that invokes differential calculus gives the approximate error in height from any combination of angle, distances, and errors thereto. I'll revisit the topic on a future post. We are really happy to have you aboard.

Bob
Robert T. Leverett
Co-founder, Native Native Tree Society
Co-founder and President
Friends of Mohawk Trail State Forest
Co-founder, National Cadre

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edfrank
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Re: Clinometer Errors and Calibration

Post by edfrank » Wed Jan 23, 2013 9:33 pm

Karlheinz,

Thank you for contributing to this discussion. This is what we need, more voices to keep everybody from getting off track in these discussions. The problem with my initial post was that the clinometer error not only affected the height values, but also changes the effective baseline for the tangent calculations. This is something I overlooked in my post, but was apparent after looking at your calculations.

Edward Frank
"I love science and it pains me to think that so many are terrified of the subject or feel that choosing science means you cannot also choose compassion, or the arts, or be awe by nature. Science is not meant to cure us of mystery, but to reinvent and revigorate it." by Robert M. Sapolsky

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