ENTS,

    Problem #5 is presented in the two attachments. The Word document presents the problem with the solution and summary comments. The spreadsheet is a handy dandy tool to compute the answer to problems of the type described automatically . As with other spreadsheets, the green cells are for user input.

    With this submission, the bank now contains 5 solved problems. I am willing to continue presenting problems if at least a few of you want them, but before sending more problems, I would appreciate a show of hands from those of you who want t he process to continue. I'd like to test the demand for this kind of on-line help. If just a few of you, say 5 or 6 , want the process to continue , I'll be happy to oblige .

Bob

Problem 5 xls

Problem #5: A measurer using a tape and clinometer wants to limit the error in calculating tree height attributable to the horizontal offset of the crown-point relative to the trunk. The measurer has been told that if the crown-point, trunk, and measurer’s eye are all in the same vertical plane and then the measurer circles around 90 degrees from that vertical plane, so that the new vertical plane that includes the trunk and measurer’s eye is at 90 degrees relative to the first plane, then the error in the height calculation will be reduced to a minimum. However, the measurer knows that circumstances do not always permit the measurer to circle a full 90 degrees and still retain visibility of the crown-point. How can the measurer develop a mathematical process to compute the crown-point offset error for different combinations of crown height, distance to trunk, and angle between the vertical planes.

Solution: The following diagram shows the variables and their relationships. The left diagram is a top down view. P₁ is a point on the trunk, P₂ is the crown-point, and P₃ is the measurer’s eye. The right view shows the height of the crown-point relative to the measurer.

Definitions:

            a = angle between direction of crown-point offset and measurer’s eye. Trunk is

                  the vertex of the triangle

            b = angle of crown-point being measured relative to measurer’s eye.

            D = distance from measurer to trunk.

            d = horizontal offset distance from trunk to crown-point.

            n = distance from P₂ to the line D which runs from P₁ and P₃. The line n

                  is at a right angle to D. The lines n and D are horizontal.

            m = distance to P1 from intersection of n and D. The lines n, m, and d form a

                      right triangle.

            D₁ = distance from end of m to P_3. D = D₁ + m.

            S = horizontal distance from P₃ to P₂. 

            H = height of crown-point above measurer’s eye.

            E = error in height from using D as baseline instead of S

            H_O = S - D

            From these definitions we can solve the problem through the following sequence of steps:

            With this sequence of step, we can determine the error from initial values of a,b,d, and D. The following table illustrates the procedure.

            In the table, a, d, D, and b are the knowns. The values of other variables are computed. The measurer can analyze the error for different value sets. From the example, the 90 degree scenario gives almost the exact value. If the measurer were close to a very tall tree with a big horizontal offset, the error would increase dramatically. Errors will generally be under 10 feet for the most exaggerated situations. However, errors in the over 10-foot range can occur.

Summary Discussion: This problem illustrates how a known source of error with clinometer and tape can be minimized without employing crown-point cross-triangulation, a method if done correctly will eliminate the horizontal offset error entirely. In actual field practice, visibility of different points in the crown from different vantage points often eliminates crown-point cross-triangulation as a viable field technique for accurately measuring the height of a tree.