Height measurement Basics   Robert Leverett
  Nov 27, 2006 13:06 PST 

In the way of a quick general explanation of the best method for 
measuring tree height, it is called the sine top-sine bottom method. 
It is actually easier to use than our explanations of it. The question 
about angles gets to the heart of the issue. I'll give an example that may
help people to visualize the role of the angle in calculating tree height
when you can't stand directly under the high point and shoot straight up.

   Suppose you have positioned yourself under a tree next to the trunk
intending to shoot straight up to get the height of the tree, but you
just can't see through the foliage. So you proceed to back up from the
tree trunk to get a view. You find that you must back up a goodly
distance to be sure that you can distinguish the top twig. Let's say you
have backed away 60 feet from the trunk. Let's further say at that point
you can spot your quarry and you shoot the twig with your laser, getting
a distance from eye to twig of 105 linear feet. Now if you have a
clinometer, you can also get the angle to the twig, i.e. the angle of
the twig above eye level. Let's say you shoot the twig with the
clinometer and it reads 47 degrees above eye level. You now have the two
values needed to compute the height of the twig above eye level. The
calculations are as follows:

   Hgt above eye level = (105 ft) x sin(47 degrees) = 76.8 ft, where sin
is the trigonometric sine function of the angle. You can get the sine
value from a scientific calculator with sin, cos, and tan buttons.

    If you are standing on level ground, you can just add your own
height to eye level to the 76.8 ft. to get the total tree height.
However, without taking measurements, it is hard to judge if you are on
absolutely level ground. So it is better to repeat the above process to
get the height of your eye above the base of the tree. You have to stay
in exactly the same spot that you were in when you shot the twig. Let's
say that you shoot the base of the tree with the laser and you get 68
feet directly from eye to base. Let's say you then measure the angle
from your eye to the base and it turns out to be 7 degrees below eye
level. Then the height of your eye above the base is calculated as

Hgt of eye above base of tree = (68 ft.) x sin(7 degrees) = 8.3 ft.

The full height of the tree is 76.8 + 8.3 = 85.1 ft.

In the above calculations, notice that we did not make use of fact
that we were 60 feet from the trunk. It really doesn't matter what
distance you are from the trunk since the laser distances and angles to
crown twig and tree base allow you to compute the vertical components of
height, above and below eye level. Tree measurers stuck in the past
still think they have to shoot to the trunk and get the distance they
are from the trunk. If all they have is a tape measurer and a
clinometer, then indeed that is what they must do. That is to say, if
the only instrument one has is a clinometer and a tape measure, then one
has to take the error-prone approach of measuring height that is often
called the slope-distance method. The sins of that method is discussed
in the tree measuring guidelines put together by Will Blozan and
available on the ENTS website, courtesy of Ed Frank.

   We also discuss the height measurement errors to be expected from
using the clinometer only method. In extreme cases errors from the use
of a baseline (often 100 feet) and a clinometer shot to the crown of a
fairly tall, spreading hardwood can be over 50 feet.

As an additional point about angles, it is very seldom that you can
avoid use of angular as well as distance measures to be highly accurate
on tree heights, although the method you are using is going to be safer
than the clinometer and tape measure method. To get quick
approximations, all of us shoot directly upwards through the branches of
deciduous trees at times to get good approximations of height. The
biggest challenge is to find a spot where we're fairly confident that
we've found the top most twig, and if to do this we move laterally
outward, the angle comes necessarily into play. However, the sine of 85
degrees is 0.996, so being off the vertical by only 5 degrees hardly
effects accuracy. However, if you are at an angle of 70 degrees when you
shoot a crown point, then the sine is 0.94 and you can make a
significant error in using the laser distance from eye to twig as the
height of the tree. For instance, if your laser records a distance from
eye to twig of 100 feet and your eye is at an angle of 70 degrees
relatively to the twig, then the height of the twig above eye level is
actually (0.94)(100) = 94 feet.
    Is there a compensating way to use the laser only method? Averaging
crown shots by circling a tree is risky and error prone, since unless
the tree's crown comes to a point that is visible from different sides,
the average is of different crown points, none of which may be the
actual top and you are still left with the angle problem. Although if
the line from your eye to the twig stays at an angle of 85 degrees or
more, you can ignore the angle-induced error.

I'm happy to provide more examples. We can keep the dialog going. If 
one person has questions, I'm willing to bet others on the list do as well.

height measurement explanation    Edward Frank
   Nov 27, 2006 23:18 PST 

ENTS, Bob,

Many otherwise intelligent people have an aversion to math. One of the
difficulties is that people typically teaching or explaining math are those
to whom the math comes easily. Points that are intuitively obvious to math
people to the point they do not even think about it are not clear to others.
Steps like punching a formula into a calculator has been an sticking point
to some people. Most calculators use a pretty basic input format.

In Bob's example below there are four measurements that have been taken.
Angle to the tree top = 47 , distance to the top = 35 yards, angle to the
base = -7, and distance to base = 23 yards (I know Bob said 68 feet, but
that number does not display on the rangefinder). Write these numbers down
in your field book

How to calculate tree height:
1) First punch in the top angle [47],
2) press the sin button on the calculator [In this case the number that
appears will be 0.73135],
3] press the multiplication key [X],
4) type in the distance to the top in yards [35],
5) press the multiplication key [X],
6) type in the number 3 [3 feet per yard],
7) press the = sign. This will give you the height of the top above your
eye-level. [76.8]
8) Write this number down, it is the height of the tree top above eye

Next calculate the "height" of the base of the tree, using the same steps.
9) First punch in the base angle [7], Ignore at this step whether the
number is positive or negative.
10) press the sin button on the calculator [In this case the number that
appears will be 0.1219],
11] press the multiplication key [X],
12) type in the distance to the base in yards [23],
13) press the multiplication key [X],
14) type in the number 3 [3 feet per yard], this gives you 69 feet
15) press the = sign. This will give you the height of the top above your
eye-level. [8.4]
16) Write this number down, it is the height of the base of the tree above
or below eye level.

In almost every case the top of the tree will be above eye level, so it will
be a positive number.
17) If the base of the tree is below eye level [negative angle], add the
two numbers [from steps 8 and 16] together to get the height of the tree.
18) If the base of the tree is above eye level, subtract the base height
[step 16] from the top height [step 8] to obtain the height of the tree.

If you think about it, it makes sense. If you know the top of the tree is
so many feet above eye level, then if the base is below eye level, you must
add the two together to get the height of the tree. If the base is above
eye level, then the tree starts out at some point above eye level and that
base height must be subtracted from the height of the top above eye level.

In the method above, I said to ignore when doing the basic number
calculations whether or not the base angle was positive or not. If you
calculate the sin x distance of a negative angle you will get a negative
number. In a rigorous mathematical sense you, if you do use the sign + or -
for the base height, you subtract the base height from the top height. If
the base angle is positive, then like the list above [18] you subtract the
base height for the top height to get a true height of the tree. If the
angle is negative, then you are subtracting a negative height from the top
height. Subtracting a negative number is the same as adding the two heights
together with both being considered positive as listed in step 17 above.

If you are using a calculator with the odd rpn notation, or are programming
it with a formula that requires you enter whether the top angle and base
angle are each positive or negative in a single long function with both
trigonometric, and arithmetic functions, then you probably don't need the
discussion above anyway.

Ed Frank

Re: height measurement explanation    Edward Frank
   Nov 28, 2006 08:49 PST 


Another potential sticking point for the beginner in understanding how the
laser/clinometer height measurements with regard to the sine function. I
like to understand the process itself rather than just going through the

Essentially when you measure a tree using our laser clinometer methods you
are creating a giant triangle with one square corner. I am sure everyone
has heard of Pythagorean's theorum: a(squared) + b(squared) = c(squared).
The far side of the triangle is a vertical line extending down from the top
to eye level. Think of this as a plumb-bob line dropping down from the top
of the tree. This represents the A side - A for altitude. The bottom of
the triangle is a level line from the measurer's eye to a point directly
below the top of tree. This is the B side - B for bottom or base. The A
line meets the B line at an angle of 90 degrees, forming a square corner or
a right angle. The third side is the hypotenuse or C-side. This is the
distance to the top of the tree from your eye that you "C" when you measure
it with the laser rangefinder.

A useful property of a right triangle is that for every angle, there is a
unique ratio between the lengths of the sides of the triangle. This ratio
is the same no matter how large or small the triangle. If you know one of
the comer angles, in addition the square corner, and the length of one of
the sides, you can calculate the length of the other two sides. With the
clinometer you are measuring this a second angle, and with the rangefinder
you are measuring the length of side C. Therefore you can calculate both
the height of the top above eye level (Side A), and how far the top is
displaced horizontally from your position (Side B). The height (Side A) is
equal to sin(angle) x distance C. The horizontal displacement distance
(Side B) is equal to the cos(angle) x distance C.

In our measurements we want to measure the length C as accurately as
possible. The best reading is at the point where the rangefinder changes
readings from one number to the next higher number. It always changes at
the same distance. You can increase the accuracy of the measurement by
checking the distance the number with a tape stretched out along the ground.
If the "click-over point' is for example three inches farther than what the
tape says it should be, then you can apply a correction to the readings you
are taking - in effect calibrating the rangefinder. Most of the time the
rangefinder is pretty much right on.

When measuring a tree there really are two triangles. One triangle
represents the height of the tree above eye level. The second triangle is
the "height" of the base of the tree above or below eye level. By adding
the two numbers together, the total height of a tree can be calculated. It
is generally true that a clickover point at the top will not also be a
click-over point for the base of the tree. The best thing to do is to
carefully move in toward or out a short distance from the base to find a
point that is a click over point while trying to stay on the same horizontal
plane - trying to stay level.. As the rangefinders read in increments of 1
yard or 1/2 yard. This adjustment should be no more than 1/2 yard. This
will assure that the best height measurement is obtained. It will mess up
the horizontal distance some, but it really isn't important for our uses,
except as a point of conversation.

You can see how these functions work by trying a couple of examples. Say
you are measuring a point 60 feet directly above your head. You know
already that the vertical height is 60 feet, and the horizontal displacement
(it is directly above you) is 0. What do the formulas say? Using the
formula sin(angle) x distance = height sin(90) = 1, therefore 1 x 60 = 60 =
height. The cosine function works the same cos(angle) x distance =
horizontal displacement    cos(90) = 0, therefore )x 60 = 0 = horizontal

What about a point 60 feet away on level with your eye - inclination =0?
sin(0) = 0, therefore 0 x 60 = 0 = height. cos(0) =1, therefore 1 x 60 = 60
= horizontal displacement.

At an angle of 45 degrees the horizontal and vertical displacement are the
same. sin(45) = 0.707 = cos(45), therefore the displacement is equal to
0.707 x 60 = 42.43 feet height and 42.43 feet horizontally.

Unless you are shooting straight up, the height of the tree top above your
head will always be less than the distance you measure. I have memorized
several approximations that lets me judge the height of the tree before
punching the number into the calculator - sort of on the fly surveying of a
series of trees.

45 degrees = 0.7 x distance
65 degrees = 0.9 x distance
70 degrees = 0.94 x distance
80 degrees = 0.98 x distance

Ed Frank