Posted: Thu May 24, 2012 10:00 pm
Bob, NTS,
There is another level of data that can be culled from the original dataset, or similar dataset without specific crown offset measurements. Please excuse my crude hand drawn diagram. The horizontal lines represent an offset in the direction of the measurer of 0 feet, 10 feet, 20 feet, and 30 feet. The half circles represent the actual limb offset from the base of the tree in terms of 10 feet, 20 feet, and 30 feet.
As can be seen from diagram trees with an offset of between 20 and 30 feet can have an apparent offset in the direction of the measurer in any of the offset ranges calculated using the cosine(theta) x distance formula. But you can figure out if the actual tree top directions in the sample set were truly distributed randomly about the base point or it the simple act of managing to find the top through the brush gave some preference to a particular offset direction. I think those tops pointed toward the measurer might have been sampled more frequently than those off toward the side because they were easier to find.
If the tree tops are randomly oriented there should be an equal number of tree tops in each equal area of the band between the 20 foot radius circle and the 30 foot radius. We can calculate the area of each of the segments marked A through I on the diagram. We know that every top that with a measured offset toward the measurer that was greater than 20 feet was located in the section marked A. Since we know the area of A, we can calculate how many tree tops are found in this section per area. If the tops are randomly distributed in the sample set, then since we know the area of sections B and C we can calculate how many of the tree tops showing an offset toward the measurer within the 10 to 20 can be accounted for by tree tops with actual offsets of between 20 and 30 feet. Therefore, if you subtract that number of tree tops from the total for the 10 to 20 foot range, what is left over is the number of tree tops in section F, which represents trees with actual offsets between 10 and 20 feet. We know the area of F and therefore can calculate the density of tree tops for section F.
Applying the same process we can calculate the number of trees in sections D and E, using the density in section A, and calculate the number of trees in areas G and H, using the density in section F. What is left over from the total number of trees with offsets toward the observer is the number of trees in section I. The area of section I can be calculated, and the density of trees within section I.
The distribution of crown offset densities should logically form a nice bell curve. If it doesn’t, then there is some preferential bias in the sampling process. You might be able to tease out the actual distribution by assuming a directional bias toward the observer and manipulating the densities until you go a good match along both the x and y axis.
In the real data set, numbers greater than 30 feet could be excluded to simplify the process. It might be better to do 5 foot segments rather than 10 feet, but the smaller the range examined the more calculations are required. 1800 samples or a comparably large set should be robust enough to support smaller scale subsampling.
Another worthwhile subsampling would be to distinguish between conifers and deciduous, and break out individual species where there were sufficient sample numbers.
Edward Frank
.
There is another level of data that can be culled from the original dataset, or similar dataset without specific crown offset measurements. Please excuse my crude hand drawn diagram. The horizontal lines represent an offset in the direction of the measurer of 0 feet, 10 feet, 20 feet, and 30 feet. The half circles represent the actual limb offset from the base of the tree in terms of 10 feet, 20 feet, and 30 feet.
As can be seen from diagram trees with an offset of between 20 and 30 feet can have an apparent offset in the direction of the measurer in any of the offset ranges calculated using the cosine(theta) x distance formula. But you can figure out if the actual tree top directions in the sample set were truly distributed randomly about the base point or it the simple act of managing to find the top through the brush gave some preference to a particular offset direction. I think those tops pointed toward the measurer might have been sampled more frequently than those off toward the side because they were easier to find.
If the tree tops are randomly oriented there should be an equal number of tree tops in each equal area of the band between the 20 foot radius circle and the 30 foot radius. We can calculate the area of each of the segments marked A through I on the diagram. We know that every top that with a measured offset toward the measurer that was greater than 20 feet was located in the section marked A. Since we know the area of A, we can calculate how many tree tops are found in this section per area. If the tops are randomly distributed in the sample set, then since we know the area of sections B and C we can calculate how many of the tree tops showing an offset toward the measurer within the 10 to 20 can be accounted for by tree tops with actual offsets of between 20 and 30 feet. Therefore, if you subtract that number of tree tops from the total for the 10 to 20 foot range, what is left over is the number of tree tops in section F, which represents trees with actual offsets between 10 and 20 feet. We know the area of F and therefore can calculate the density of tree tops for section F.
Applying the same process we can calculate the number of trees in sections D and E, using the density in section A, and calculate the number of trees in areas G and H, using the density in section F. What is left over from the total number of trees with offsets toward the observer is the number of trees in section I. The area of section I can be calculated, and the density of trees within section I.
The distribution of crown offset densities should logically form a nice bell curve. If it doesn’t, then there is some preferential bias in the sampling process. You might be able to tease out the actual distribution by assuming a directional bias toward the observer and manipulating the densities until you go a good match along both the x and y axis.
In the real data set, numbers greater than 30 feet could be excluded to simplify the process. It might be better to do 5 foot segments rather than 10 feet, but the smaller the range examined the more calculations are required. 1800 samples or a comparably large set should be robust enough to support smaller scale subsampling.
Another worthwhile subsampling would be to distinguish between conifers and deciduous, and break out individual species where there were sufficient sample numbers.
Edward Frank
.