Missing Data: Location of Dead Trees

The values representing a tree's angle and distance from the center of its FIA plot are missing for a total of 4006 trees. 3539 of the 4006 trees died between the 4th and 5th measurement cycles, and 356 died between the 5th and 6th cycles. Thirty-eight of the trees are alive and have dbh measurements for all three time periods.

Previously, these trees were assigned uniformly random positions within the square 22.7 m. by 22.7 m. reduced plots. (Reduced plots are described in section x.x.) Since the reduced plots have 64% the area of the original circular plots, 36% of the (dead) trees that have been placed in the reduced square would normally have been omitted from the data. This means there are 157% the expected number of tree stumps located on the reduced plots.

A possible solution is to assign these trees random distances and angles from the centers of the initial circular FIA plots, before the plots are reduced to smaller, square plots. Care must be taken, however, to assure that the spacing of the trees is even. Whereas the initial method of assigning uniform random values to the longitude and latitude of a tree assures even spacing, a uniform random value assigned to the radius (distance from the center of the circular FIA plot) would cause an uneven distribution of trees, causing a much higher density of trees in the center of the plot.

To solve this, a uniform random number can be assigned to the angle $\theta$ of the tree, and a random number with the following population density function can be assigned to the distance rfrom the center of the plot, where the radius of the plot is r₀.

$$f(r)= {{2 \pi r} \over {\pi r_0^2}}$$