Simplified Equation For Lower Basal Wedge - Improved Model

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#1)  Simplified Equation For Lower Basal Wedge - Improved Model

Postby M.W.Taylor » Thu Sep 21, 2017 8:41 pm

Attached is a simplified version of the equation for the lower slanted basal frustum below high side ground level and ground perimeter. As demonstrated in the graphic, using the 1/2 frustum model is an over estimate, especially for trees on a steep slope.  Bob Leverett came up with a proof that shows this upper wedge is always smaller than the lower wedge. Maybe he'll post that here ?

For example, we have a lower trunk of a tree with high side of ground level radius, as measured by tape wrap, at 5 feet ("r" in the attached formula"). The height differential of the low side of base and high side is 3 feet ("H" in the attached formula) and the trunk off-set from high side to low side in the horizontal direction is 1 foot ("d" in the attached formula).  This would give a volume for the lower slant frustum of 123.51 cubic feet.  If you model the lower basal wedge using the 1/2 frustum model,  you get 142.94 cubic feet. This shows the old model gives an over-estimate by around 15% for this example. For trees on steeper slopes, this over-estimate becomes even greater.
Attachments
slanted frustum-V2.jpg
new simplified model of the lower slant frustum
frustums.jpg
the 1/2 frustum model breaks down as the the slope becomes greater and greater
Last edited by M.W.Taylor on Fri Sep 22, 2017 2:38 am, edited 1 time in total.
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#2)  Re: Simplified Equation For Lower Basal Wedge - Improved Mod

Postby mdvaden » Thu Sep 21, 2017 10:19 pm

What's extra interesting, is how many of the big trees, say along the coast, are not just pure conical shaped below upper grade, but can curve outward or even bulge.

It's as if the Bull Creek Flats trees have shapes more fitting to your diagram and equations, but the the slopes where that would be very useful are around Redwood National Park where a good number of trees have bulgy bases.

Sure enjoy your math equations though. Way over my understanding, but definitely detailed and thought through.
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#3)  Re: Simplified Equation For Lower Basal Wedge - Improved Mod

Postby M.W.Taylor » Fri Sep 22, 2017 2:40 am

mdvaden wrote:What's extra interesting, is how many of the big trees, say along the coast, are not just pure conical shaped below upper grade, but can curve outward or even bulge.

It's as if the Bull Creek Flats trees have shapes more fitting to your diagram and equations, but the the slopes where that would be very useful are around Redwood National Park where a good number of trees have bulgy bases.

Sure enjoy your math equations though. Way over my understanding, but definitely detailed and thought through.



Mario,  If the base of the tree has a large bulge, the only way to get an accurate volume measurement is to create a point cloud and measure its volume as a free form.
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