Improved model for lower basal wedge volume

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#1)  Improved model for lower basal wedge volume

Postby M.W.Taylor » Thu Sep 14, 2017 8:35 pm

Attached is a new model I am evaluating for the section of wood on trees below high side ground level and the ground interface. This section of tree is a frustum with a slant base.   For some of the big redwoods this piece of wood is substantial, 5,000 cubic feet or more sometimes. It should be accurately modeled in order to estimate the total volume of the tree. Up until now I was using a 1/2 wedge model, which does not really reflect the shape of the trunk form.   The formula used is that for a "slanted cone". The formula and derivation can be found at:

https://math.stackexchange.com/question ... -a-frustum

see attachment for further details.  I had to rework the formula a little to work for a slanted frustum.
Attachments
slant frustum.jpg
new analytical model with exact solution
slanted wedge.jpg
lower part of tree below high side ground and ground interface
slanted wedge.jpg (6.31 KiB) Viewed 104 times

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John Harvey, mdvaden, Will Blozan
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#2)  Re: Improved model for lower basal wedge volume

Postby MarkGraham » Thu Sep 14, 2017 11:41 pm

A 5,000 cubic foot wedge, that is a big tree on a steep slope.

The approach seems reasonable, some of it is beyond my understanding.  

It is very tempting, based on the diagram, to calculate the wedge volume as the difference between two cones.
The first cone, the big one, involves H for radius and then 2r + d for height.  Then divide that by 2 since it is a half cone.
The the second cone, the smaller one, involves d for radius and H for height.  Then divide that by 2 since it is a half cone.
Then subtract second cone from first cone.

Trying for a 16 foot diameter redwood, on a 45 degree slope, this is about 1000 cu ft.
Trying for a 27 foot diameter redwood, on a 45 degree slope, this is about 4500 cu ft.

Tree volume is a real bear to calculate, the trunk is one thing.  But how much wood is above the ground, including the limbs, gosh, it makes my head hurt.
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#3)  Re: Improved model for lower basal wedge volume

Postby M.W.Taylor » Fri Sep 15, 2017 1:04 am

Mark, The method you describe is the old way. I think this model will more accurately reflect the true shape of the lower wedge.  At some point I will compare both models to a true point cloud map of the lower wedge of the same tree, that has been volumized.
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#4)  Re: Improved model for lower basal wedge volume

Postby dbhguru » Fri Sep 15, 2017 8:51 am

Michael,

 Kudos to you for working on this problem. I look forward to going through the derivation (well, not too forward). Anyway, I know this is a problem you've been interested in for a long time. Great job!

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