Tree Diameter: Kalman Filter Observation
Trees are identified by their diameter. To obtain an estimate of the diameter of a tree a Kalman filter is
used. [3] proposes an equation for the observed diameter zD to be the average range, R’, in a tree cluster
multiplied by the subtended angle  (see Figure ?). Given the number of readings NR, the observation
equations (from [3]) are as follows:
zD = R’ 
R' =
1 • i∑n R' (i )
in - ii + 1 i = ii
π
β = (in - ii + 1) • NR
Is this an accurate measure of diameter? The answer to this question lies in the derivation provided below.
Given a set (cluster) of range values between an angle , associated with a tree, how can the diameter of
tree be computed? Assume the outer scans ii and in are tangent to the tree trunk (see Figure?).
Figure ? Derivation
The objective is to obtain the average height along the arc, y’, and to relate it back to average range R’.
Using the equation of a circle and solve for y,
y =
r 2 - x2
Thus, the average range can be computed from the radius r, the range of the midpoint m, and the average
height y’.
R’ = r – y’ + m
Because of symmetry, the integration interval to compute the average height y is [0,a], where
a = r cos 
1a
1a 2 2
∫y dx =
∫ r - x dx
a0
a0
1a 2 2
1 r cos θ 2 2
∫ r - x dx =
∫
r - x dx
a 0 r cos θ
r cos θ 0
1
1
- x r 2 - x2
∫
y' =
r 2 - x 2 dx =
r cos θ 0
r cos θ
2
r cos
x2
x
ff00
+ sin -1
2
r 0cos
- rcos θ r 2 (1- cos 2 θ rcos θ - 1
1
y' =
(
+
sin cos θ)
r cos θ
2
2