Formulas March 2008

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TOPIC: A useful formula
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== 1 of 1 ==
Date: Fri, Mar 7 2008 2:43 pm
From: dbhguru@comcast.net

ENTS,

All of us who measure trees with some degree of seriousness encounter those inescapable places that are cluttered by underbrush. We may be able to see the base of a tree we want to measure with our eye, but can't get the laser to penetrate the brush. We may be short on time, the terrain may be rough, etc. to preclude extra time being taken with a tree. So want to shoot and move on. If we can spot a point not to high on the trunk, in case of a lean where we can get a clean laser shot, we're in business.

Let

H = height of tree from eye level to the base
a = angle from eye level to point on trunk that we can shoot
D = laser distance from eye to point on trunk that can be clearly seen
( can be above or below eye level)
b = angle between eye and base of tree.

Then

H = D * cos(a) * tan(b)

This formula gives us the component of height between the level of our eye and the base of the tree. The component of height above eye level is measured in the ordinary way using the sine top method. The above calculation works only if the tree does not have a substantial lean or if it does that we can use close enough to the base that the error introduced is insignificant. However, in situations where it can be applied, it solves the problem of clearing away the brush to get a clear shot.
I've applied in above method in a two step process many times, but had not boiled it down to a single formula.

Bob

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TOPIC: New formula test
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== 1 of 1 ==
Date: Mon, Mar 10 2008 2:08 pm
From: dbhguru@comcast.net

Beth, Ed, et al:
I did a very quick test this afternoon of the cos-tan bottom method compared to the sine bottom. The table below shows the results.

 CosTan Bottom Sin Bottom Diff Abs(Diff) -3.01 -3.00 -0.01 0.01 -8.08 -8.50 0.42 0.42 -10.24 -10.00 -0.24 0.24 -16.15 -16.00 -0.15 0.15 4.76 5.00 -0.24 0.24 -3.67 -3.50 -0.17 0.17 -21.05 -21.00 -0.05 0.05 16.80 17.00 -0.20 0.20 -21.10 -21.00 -0.10 0.10 -19.00 -18.50 -0.50 0.50 Average 0.21

Average 0.21

With care I could have probably gotten the difference to as low as 0.1 feet. In deciding to do the test, there wasn't any expectation of a problem, but I thought potential users of the method would want to know what experimental difference could be expected from the two methods.
To repeat the formula and its purpose.
Let

H = height of tree from eye level to the base
a = angle from eye level to point on trunk that we can see and shoot without obstruction
D = laser distance from eye to the point on trunk that can be clearly seen
( can be above or below eye level)
b = angle between eye and base of tree.

Then

H = D * cos(a) * tan(b)

This formula gives us the component of height between the level of our eye and the base of the tree. The component of height above eye level is measured in the ordinary way using the sine top method. The above calculation works only if the tree does not have a substantial lean or if it does that we can use close enough to the base that the error introduced is insignificant. However, in situations where it can be applied, it solves the problem of clearing away the brush to get a clear shot.
Bob

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TOPIC: Three more formulas
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== 1 of 3 ==
Date: Sat, Mar 15 2008 8:48 am
From: dbhguru@comcast.net

ENTS,

There is a multitude of measuring situations that present special challenges to those of us who measure trees in a forest environment and want to achieve the requisite ENTS level of accuracy. Two situations are covered below.

We've all encountered in-forest conditions where clutter around the base of a tree prevents a clean laser shot and it isn't practical for us to worm our way to the base, e.g. across a stream or ravine, too much rhodo, we're pooped, etc. However, as covered in a previous e-mail, if we can spot the base with our eye and measure the angle from eye to base (even though we can't hit the base with the laser) and then measure the distance and angle to a point not to high up on the trunk, we can compute the component of height from base to eye level using the formula I previously gave. Our assumption is the the tree's lean is negligible. But, the byproducts of computation don't end here. An associated problem is to compute the straight-line distance from eye to the base under the above conditions. This distance can be computed as follows:

Let

D = straight-line distance from eye to visible spot on trunk
a = angle between eye and spot on trunk
b = angle between eye and base of tree
S = straight-line distance from eye to base.

Then

S= D * cos(a)/cos(b). [Remembering that the height component is D * cos(a)*tan(b)]

While knowing S isn't necessary to computing the component of height between the base and the eye (our primary objective), it is good to be able to completely delineate the triangles we are constructing to measure the components of height. Computing S is a way of filling all the squares.
A significantly more challenging situation faces us when we want, from a distance, to determine the angle of lean toward or away from the measurer of a tree trunk. Let's assume that we can't bank on shooting the distance directly to the point on the trunk either to eye level or to the base. Those areas of the trunk may be obscured. However, if we can shoot to two points on the trunk (one above and one below eye level) that are sufficiently separated from one another to create an angle of several degrees, we can establish with acceptable accuracy the degree of a lean with the following formula. Note that the symbol "|" in the formula denotes absolute value. That is, |x| means the absolute value of x. Also please note that a set of six formulas is needed to calculate trunk lean under all conditions (e.g. both angles are positive, both are negative, one positive and one negative, combined with a trunk lean toward the observer or away from the observer).

Let

D1 = distance to uppermost of the two points on the trunk we will shoot
D2 = distance to lowermost of two points
a = angle to upper point
b = angle to lower point
c = angle of lean from the vertical

If the cinditions D2*cos(b) > D1*cos(a), a>0, b<0 are met, then

c = 90 - arctan[( |D1*|sin(a)| + D2*|sin(b)|| )/( |D2*|cos(b)| - D1*|cos(a)||)]

If D2*cos(b) < D1*cos(a), a>0, b<0 then

c = arctan[( |D1*|sin(a)| + D2*|sin(b)|| )/( |D2*|cos(b)| - D1*|cos(a)||)]

The above formulas for c cover 2 of the 6 cases. The others will follow in other communications. Note that all these formulas and their derivations will be included in the publication on dendromorphometry that is in the works.

Finally, the complexity of these formulas suggests the development of spreadsheets for the actual calculations.

Bob

== 2 of 3 ==
Date: Sat, Mar 15 2008 11:58 am
From: Larry

Bob, You are the Man! Good to see you feeling better! Your formulas
are way cool. I need to get back to the Volume project, the stuff you
set up on excel is Awesome, way cool stuff. I have to get a new
laptop and I will begin sending you results on the project. Its 77
down here today with lots of flowers starting to bloom. Again, I'm
glad to see you able to get back in the forest, doing the stuff you
love so much. Bob, I think we need another one of your Asethetic
stories if you get time. Larry

== 3 of 3 ==
Date: Sat, Mar 15 2008 12:15 pm
From: dbhguru@comcast.net

Larry,

Thanks very much. I think Monica and I are going for a third trip to Mohawk this Sunday, if the weather holds. Gosh, does it ever feel good to even think about getting out. The power of those marvelous Mohawk trees is not to be denied.

On the story side, as I plan Monica's and my June-July trip to Idaho, I've been thinking about old haunts we intend to visit along the way and the one that always seems to emerge on top is the Cloud Peak Wilderness Area in the Bighorn Mountains of north-central Wyoming- my ultimate mountain Mecca. Lots of stories to tell there. I'll put a couple together.

Bob

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TOPIC: Three more formulas
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== 1 of 1 ==
Date: Sun, Mar 16 2008 3:02 pm
From: dbhguru@comcast.net

ENTS,

I've been able to boil the trunk lean fomulas down to just two after working through the 6 scenarios.

Let

D1 = distance to uppermost of the two points on the trunk we will shoot
D2 = distance to lowermost of two points
a = angle to upper point
b = angle to lower point
c = angle of lean from the vertical

If a >0 and b < 0 then

c = atan([|D1*|cos(a)| - D2*|cos(b)||]/[|D1*|sin(a)| +D2*|sin(b)||])

If a >0 and b >0 or
a<0 and b<0 then

c = atan([|D1*|cos(a)| - D2*|cos(b)||]/[|D1*|sin(a)| -2*|sin(b)||])

Please take note of the "| |" notation. The bars identify the absolute value of what is enclosed between them. Also note that relative to the observer, the trunk can have a forward-backward and a lateral lean. The above formulas ignore the lateral lean and assume the observer is in alignment with the trunk so that a vertical plane that includes the measurer's eye would also include all of the trunk between the two measurement points. A slight lateral lean won't change the calculations much. Taking both components of lean into account involves much more calculating and will be the subject of later communications.

Bob

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== 1 of 1 ==
Date: Wed, Mar 19 2008 12:32 pm
From: dbhguru@comcast.net

ENTS,

Sheet1 of the attached Excel workbook (TrunkSlopeDiagram.xls) shows the formula for calculating trunk lean from the vertical as determined from a distance. Note that absolute values are taken to avoid negative angles in the resulting addition. I may have gone overboard with the absolute value function, but what the heck. The diagram provided on the Sheet1 deals with the case where angle a >0 (above eye level) and angle b <0 (below eye level). If both angles are either above or below eye level, the formula changes in that the factors in the denominator are subtracted instead of added. This is indicated by the +/- sign in the denominator of the last shown formula.

Sheet2 is a handy dandy calculator for tree lean. Remember, the assumption is that we cannot get to the tree, so we are attempting to determine the trunk lean as an angle from the vertical taking two points on the trunk and measuring them from a distance. We also assume that the area of the trunk at eye level may be obscured also. So the location of the two points is arbitrary.

A sample of actual measurements I took from my patio follows. The first measurement was a test only to see what the resulted looked like.

A sample of actual measurements I took from my patio follows. The first measurement was a test only to see what the resulted looked like.

 D5 h1 D3 h2 Species D1 a D2 b abs(D1*abs(cos(a))) abs(D1*abs(sin(a))) abs(D2*abs(cos(b))) abs(D2*abs(sin(b))) c Test 106.42 18.00 100.00 -6.00 101.21 33.43 99.45 10.47 2.29 NRO 80.50 9.70 81.50 -13.60 79.35 13.63 79.21 19.35 0.23 BB 48.50 4.00 50.50 -17.50 48.38 3.39 48.16 15.42 0.67 HM 41.50 4.50 44.00 -17.60 41.37 3.26 41.94 13.52 -1.94 WP 89.00 -5.00 91.50 -13.90 88.66 7.77 88.82 22.20 0.63 WO 54.50 -0.50 57.00 -17.60 54.50 0.48 54.33 17.51 -0.56 BB 45.50 4.00 48.00 -18.50 45.39 3.18 45.52 15.50 -0.40 NRO 78.50 -0.60 78.50 -9.90 78.50 0.82 77.33 13.56 -5.22

You can see in the above table, only one lean is significant, the Northern Red Oak at -0.522 degrees. The tree is on an embankment and leans away from the location I from where I shot it. It leans toward the downhill side. Folks with programable calculators, might want to program in the formula. However, it might not be easy to be able to take in the addition/subtraction option in the denominator, i.e. if a>0 and b <0, then the factors in the denominator are added, else subtracted. If D5>D3, then the tree leans away, otherwise, it leans toward the measurer.

The practicality and applicability of the above method is admittedly limited and not apt to find much usage except where the measuring environment is cluttered, the tree has limited accessibility, and the steaks to get it right are high.

Bob

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TOPIC: Possible new spreadsheet for Larry
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== 1 of 2 ==
Date: Wed, Mar 19 2008 5:16 pm
From: dbhguru@comcast.net

Larry,

Uh oh, I've been at it again. What happens when a long limb of one of the Live Oaks is a curve as opposed to being straight? The attached spreadsheet may be a way to get a better length calculation. It fits a parabola to limb curve. Three measurements are required: (x0,y0), (x1,y1), and (x2,y2). The diagram shows where they would be taken. The rest is automatic. The example shows a limb that has a horizontal length of 65 feet. What do you think.

Bob

ParabolicLimbModeling.xls

== 2 of 2 ==
Date: Wed, Mar 19 2008 5:21 pm
From: "Dale Luthringer"

Bob,

This definitely would come in handy to model the Pinchot Sycamore.

Dale

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TOPIC: Possible new spreadsheet for Larry
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== 1 of 1 ==
Date: Thurs, Mar 20 2008 3:33 am
From: dbhguru@comcast.net

Dale,

Yes, I think you're right. I had that in mind as well as the Granby Oak. Are there any trees out your way that we could test it on?

Bob

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TOPIC: Possible new spreadsheet for Larry
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== 2 of 4 ==
Date: Thurs, Mar 20 2008 8:48 am
From: Larry

Bob, Way cool, looks like it will help alot. I'd guess 50% of all
Live Oak limbs are curved. Your skills on excel are Awesome! Lots of
flowers blooming down this way. Larry

== 3 of 4 ==
Date: Thurs, Mar 20 2008 12:12 pm
From: dbhguru@comcast.net

Larry,

Thanks. As a general observation, my long term goal is to establish a set of measurement methods that requires the fewest number of field measurements to be taken by the measurer and then automate each method in a popular product like Excel. In pursuit of this personal objective, I'm also available to my fellow and lady Ents to create customized spreadsheets for any of you who might wish them.
BTW, we can adopt other curve forms besides the parabola to model limb length, but so far I see no reason to do that. Also, if we want to take more measurments along the length of the limb, a best fit parabola can be determined via curvilinear regression analysis. The regression process can be built into a spreadsheet although there are sophisticated statistics programs that do all that. The problem is that they invariably force onto the user far more fiddling around with complex software than the user may be willing to commit to. I think it is better to have a simple set of tools that can be individually applied unless you are a real heavyweight. There comes a point where a piecemeal approach won't work and you have to commit to sophisticated software, but that greatly limits the number of users. So, I'm trying to fill in with an in between solution to what otherwise tends to be all or nothing.
As a final comment, In response to your request, I've started on a write-up of an aesthetic look at a favored nature site of mine. I should get the first installement out in about three days.

Bob

== 4 of 4 ==
Date: Thurs, Mar 20 2008 7:19 pm
From: Larry

Bob,