Tree Top Offset Project

[dbhguru]

Don, et al.,

       In the days prior to handheld electronic scientific calculators, we did calculations in the head, with pencil and paper, slide rule, and with the aid of tables. One could not have too many tables. They were a way of life. In forestry, I recognize that shortcuts were worked out to make the arithmetic simple for field measurements. In terms of how they were applied, I have no trouble believing that thousands upon thousands of western conifers were measured to an acceptable degree of accuracy. It was certainly the intent to do that. With large, eastern hardwoods, I expect the story has always been quite different. With the hardwoods, it isn’t so much a question of compensating for the lack of level on the trunk, but establishing the right baseline to the crown-point. It is, and always has been, a two-baseline problem. Moving forward or backward to adjust for a slope is fine for the base, but that doesn't work for the crown. Without a lot of work the measurer doesn't know what the crown-point offset is. Treating it as though it were zero has led to the eye-popping errors we've seen for the eastern broad-crowned hardwoods.

     With western conifers, a common level baseline to the trunk to serve for both the crown-point and base can be made to work, much, if not most of the time. With the hardwoods, at best it is a roll of the dice. However, there is a way to quantify these baseline measuring challenges. We can compute the crown-offset for the trees we measure as a standard part of our measuring protocol. On occasion, I set out to do this, but then I get lazy. However, I have a nifty Excel spreadsheet set up to handle crown-offset. If we all contribute, we could build a database that would quantify crown-offset for many species and age classes. To my knowledge other than what we in NTS have done, this kind of information is totally lacking in tree statistics. If many contribute, it will become second nature to collect the extra information. All that is required of the spreadsheet is:

1.        Direct crown-point distance
2.        Crown angle
3.        Direct base distance
4.        Base angle
5.        Azimuth of base
6.        Azimuth of crown-point

These 6 measurements are all that are needed for the spreadsheet. The first four are taken any way. So, we would be only adding two measurements, and fairly easy ones at that. The results for the crown-offset wouldn’t be extremely accurate, but sufficiently so to highlight the problem. An additional return would be the azimuth of the crown-point computed from the trunk, i.e. as if the measurer were standing with his/her back to the trunk and shooting toward the crown-point.  

Who is interested in undertaking this crown-point measuring project with me?



Bob

[edfrank]

NTS,

The project Bob is suggesting would be worthwhile.  We have previously talked about Tree Top Offset in many contexts.  In 2005 using Bob's existing dataset of about 1800 trees, we calculated an average offset for the tops of about 13 feet.

http://www.nativetreesociety.org/measure/crown/tree_top_offset.htm

There were some critical assumptions made in this analysis and procedural problems with some of the measurements which were not collected for the express purpose of measuring crown offset.  A final limitation of the original calculation is that it deals with the trees as an amalgam of all the measurements, so data for individual trees, and to a degree subsets of trees cannot extracted for the larger dataset in a meaningful manner.

Bob's proposal would allow the collection of data for individual trees so that the detail on this scale could be examined.  It can better demonstrate the problems with the tangent method of tree height measurement related to crown offset, and other aspects of tree form.  This project only requires two more measurements - azimuth to the top and azimuth to the base above those normally collected.

I would encourage active tree measurers to participate in the project.  It s important that for a particular trip, that all of the trees be measured including the azimuth readings to the top and base.  That assures that the data set will not be biased by preferential selection of those trees with greater offsets than average, whether deliberately or unconsciously. In any case it will eliminate any potential argument about bias in the data set.  

Edward Frank

The initial two posts in the 2005 thread are presented below:

Edward Frank
Sept 17, 2005

Bob and John,

I am trying to understand the mathematics of the calculations Bob made to get an average offset or 8.3 feet through the mathematical processing of 1800 tree survey data sets. Please bear with me.

Say you had a plane with a circle centered at the x-y intersection with a radius of Q. Then draw 100 radii lines evenly spaced around the circle. The deviation from the y axis could be calculated for each point as the absolute value of sin theta Q where sin theta was measured as the angle from the y axis. If you added all of these deviations from the y axis and then averaged them you would get an average deviation from the y-axis at all of these angles. As I envision it this number should be proportional to the actual radius of the circle. This proportion should be constant regardless of the radius of the circle. I am sure there is a simple and elegant integral to calculate out the average deviation from y in a circle centered on the y-axis, but my math isn't up to the task. [Could you enlighten me?] What this would mean is that by knowing the average of these deviations, the actual radius of the circle could be calculated.

In Bob's case of 1800 trees, the deviation from the y-axis (perpendicular to the line of sight with the z-axis running through the center of the base of the tree) can be easily calculated for every tree to see how far off the tops are in the x direction along the line of sight from the base of the tree. With 1800 samples, and a reasonable assumption that they the tops are randomly distributed in direction offset from the center of the base of the tree, the average deviation from the y-axis should be proportional to the average offset distance of the top in any direction from the true center of the base of the tree.

Essentially it means by knowing the offset in the x-direction you could calculate the actual average offset of the tops without knowing the amount of offset in the y direction.

Does what I am saying make sense? Maybe this is what you have already done. How about a little help here guys.

Ed Frank

John Eicholz
Sept 17, 2005

Hi Ed,

I am glad to try and clarify the result of the calculation you wish to achieve. First let me say you have a clear grasp of the problem and it is a more interesting problem than it first appears. By solving it we can determine the theoretical average lean of the trees in Bob's database, even though this figure is not accurately calculated for each tree, by making a statistical assumption that the trees' lean is symmetric to the point of measurement. Quite an accomplishment of abstraction, it would appear. Thank you for the interesting problem.

Actually, the math is quite simple. The observed offset from vertical (as seen from the point of measurement) is proportional to the COSINE of the angle. Assume a radius of 1. The average offset would be the INTEGRAL of abs(cosine theta) around the circle, divided by the length of the circle. We can evaluate this as twice the integral of (cosine theta) from -pi/2 to pi/2, divided by 2*pi (the length of the path of integration). I apologize if this doesn't make sense, but
mathematically it is accurate.

The integral of the cosine function is the sine function. We thus evaluate the integral as 2*[sin(pi/2) - sin(-pi/2)]/(2*pi). , or 0.6366 and change.

Using an average observed offset of 8.3 feet yields a projected average true offset of 8.3/.6366, or about 13 feet.

I believe we can conclude by saying that the trees measured in Bob's database appear to have an average offset of 13 feet from bottom to measured tip.

John Eichholz

[Don]

Bob-
Of course I'd be interested, having thought that what we refer to as crown-point offset would be a valuable bit of information for researchers...essentially represents the trees primary vector (as the twig is bent, so grows the tree).  While I'm more used to the coniferous shape, deciduous trees show significant competition between upper branches for apical dominance.
Individual tree crown-point (tcp) offset information is interesting enough of its own, but the array of tcp vectors across a stand and their responses to aspect and topography intrigue me.
Where's your spreadsheet?
-Don

[dbhguru]

Don,

  It is attached with three trees included as samples from my neighbor's yard. I'm happy to modify the spreadsheet to make it better if you want other features. If we can develop a real database of trees measured with crown-offsets, we will be breaking new ground - I think.

Bob

[dbhguru]

NTS,

   I had lost track of John Eichholz's post of 2005 analyzing the implications of the 8.3-foot average difference between the trunk and crown baselines based on a sample of about 1,800 trees. The trees in that sample are mostly bigger, taller, older trees. Form the 8.3-foot average, John derived an average total crown-offset of 13 feet. I decided to reconstruct the derivation and include a diagram. They follow.

               

                       

                                       

               

  John extended the integration process over a range of 180 degrees. I show it for 90 degrees, and for the second quadrant. The results are the same for the 1st quadrant. In these kinds of problems, you have to express angles in radians as opposed to degrees. Thus the range of integration is show as 0 to π/2 instead of 0 to 90.

  As the diagram indicates, this integration computes the average value of the y variable over the range of integration. At an angle of a = 0, the value of y = R. For a = π/2, y = 0. This is not exactly equal to the average value of D - L, but close enough. So y is a surrogate for D - L with the advantage that values of D and L do not have to be known.

  The project that I am proposing that includes the spreadsheet in my previous post calls for us to add crown-offset determinations to our routine sin top-sin bottom measurements. This could be what opens the eyes of tape and clinometer users to the crown-offset problem and the need to see the problem as requiring two baselines. The idea may take time to catch on, but the logic of it is incontestable.

Bob

[edfrank]

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

.

[dbhguru]

Ed,

  Excellent point. I have little doubt that what you suggest is true. That is especially the case for a lot of the early measurements I took when higher angles figured in more prominently in my thinking. As the years have passed my sense of where to look for a top of a tree keyed to species has improved greatly. So, my early data would have been greatly skewed toward the 2nd and 3rd quadrants. I'm anxious to get to cracking on an ever more careful documentation of crown-offset, stratifying by species and age class. This is the dawning of a new era of NTS tree measuring. With the spreadsheet that I presented, there is no messy calculations that the measurer has to perform. They're all done in the spreadsheet.

Bob

[dbhguru]

NTS,

  A useful byproduct of the analytical process that we've been discussing for anyone who has a fairly large dataset consisting of randomly distributed positions relative to the crown-offset line is the following formula:


                                                                           C = (A x π)/2

  where A is the average of the (D - L) values and C = average crown-offset. D = level baseline to trunk, and L = level baseline to crown-point. This formula has no real value for small datasets, but for large ones, it may prove useful.

Bob

[dbhguru]

NTS,

  One last tidbit. In the Excel spreadsheet posted to allow measurers compute the crown-offset distance and direction, angle a5 allows us to know whether the crown-point is between the measurer and trunk or behind it in terms of horizontal distances. If a5 is less than 90 degrees, the crown-point lies in a band that is between the measurer and trunk. If greater than 90 degrees, the crown-point is behind.

Bob

[dbhguru]

(Copied from another thread)

Don,

 What you have said makes sense. We are behind you. Let us know of anything we can do to help, e.g. collect data to meet an experimental design that would satisfy the requirements. Consider us your troops. To this end I recently posted an Excel spreadsheet that computes crown-offset. It requires two additional measurements (azimuths to trunk and crown), Then the spreadsheet does the rest. If we all contributed data through this spreadsheet, would it provide you with information of value?

I've attached the spreadsheet for your review. I'd be pleased to modify it in any way that would make it more useful.

               

                       

CrownOffsetAmountAndDirection.xlsx

                                               

(74.46 KiB) Downloaded 43 times

               

               

Robert T. Leverett