RE: Maximum Measurement Errors   
  Mar 2006 05:54 PST 

This is latest of a series of discussion on measurement errors related to laser rangefinders and clinometers posted to the discussion list and to the website.  

The earliest post from Nov 2003 contains a nice mathematical analysis of clinometer error by John Eicholz and discussions by Howard Stoner, Bob Leverett, and others.  The posts below continue that theme.

Ed Frank

Maximum Measurement Errors   Edward Frank
  Mar 15, 2006 08:33 PST 


Excellent report on the area. I think your measurements might be more
accurate than you think. There are numerous potential errors in the height
measurements: Picking the wrong point to define the base of the tree, errors
in reading the clinometer, out of calibration clinometer or laser
rangefinder. Many of these will only affect the measurements by a few
inches and tend to offset if multiple readings are taken of the height of a
particular tree. I plan to include a more detailed essay of the types and
magnitudes of all errors affecting the height measurements in a future post
to the discussion list and for the website.

One point I would like to address here concerns the accuracy of the laser
rangefinder readings. What does it mean when it says accurate to one yard?
The numbers that are shown when a particular distance is measured are
repeatable time after time. a particular point measured from x distance
will always be shown as being the same distance away. The readings are
repeatable. The repeatability of the readings means the instrument has a
high degree of precision. How well does that reflect the actual distance?
That is a question of accuracy. A rangefinder can be calibrated. If a
series of distances are measured with the rangefinder and compared to the
actual taped distances, the accuracy can be determined. Compare the
distance on the tape with the point the rangefinder changes from one number
reading to the next. The rangefinder may be off a matter of a few inches
from the values indicated on the tape. However if it is reading is off it
will always be off in the same direction by the same amount, and the same is
true for all distance in the same general range. So in this way you can
tell if 80 yards is really 80 yards plus 6 inches at the point the umbers
change. If the actual reading taken are written down in the field, they can
be later compared and corrected to the correct values using this calibration

In general the change over point will be within a few inches of the true
taped distance. For simplicity sake in the discussion, let us assume the
rangefinder is right on the money. If several measurements are taken of the
height of a single tree from several positions and distances, different
heights will be obtained. If all of the measurements were taken so that the
distances to the top of the tree was exactly at a point where the numbers
change from one reading to the next then the exact distance to the top of
the tree is known. Why are there different readings? There are different
readings because of variations in how well the clinometer readings were
taken, and other variables. The numbers for the tree height will generally
be within a foot or so of each other. How should the actual height be
determined? If the source of the errors are variations in the clinometer
readings, these errors tend to be random rather than systematic. Averaging
the values calculated for height will give the best approximation of the
true height, as the random errors will tend to offset.

Say your rangefinder has an accuracy of one yard, what does that mean? It
doesn't mean that the number is +/- 1 yard, it means that if the instrument
reads 80 yards, then the actual distance is somewhere between 80 yards and
81 yards.   -> 80 yards + 0 to 1 yard. It does not read short, the error is
in only 1 direction. (It actually means the precision of the instrument is
1 yard, not the accuracy). The accuracy of a laser that reads to 1 yard is
actually 1/2 yard. Whether 80 yards is really at 80 yards can be determined
by calibrating the instrument with a tape as indicated above) If the 80
yards for example is off by a few inches the same ideas here still work, but
are harder to explain so for simplicity I will still assume the measurements
at the change over point is actually the same as the distance. If several
readings are taken around the tree, but none of them are taken at the point
where the numbers change, how is the actual tree height best approximated?
Each laser reading will be from 0 to 1 yard short of the actual distance.
So starting out the heights determined will be short of the actual height.
None of the readings will over-estimate the height of the tree.

The next consideration is that the maximum error would not be 1 yard,
but using trig it would be:   sin(angle) x 1 yard = error in yards

In the case of a 45 degree angle, for example:
sin(45) x 1 yard = 0.707 yards.

So the actual error will be some value less than the maximum error.
This will be the maximum laser rangefinder distance error if a single
reading is taken.  If multiple readings are taken, different
heights will be found from different points around the tree, with differing
distances and angles. Unlike the example above where the measurements are
averaged, this is not the best approximation of the height of the tree.
(Any wildly short distances should be thrown out as you likely aren't
hitting the actual top.) The best approximation of the height among the
multiple readings is whichever height is the greatest. Since all of the
heights are measured short of the true height, the tallest would represent
the least amount the true height differs from the the measured height,
hence the best approximation.   By not measuring at change-over points any
other errors related to clinometer readings can not be averaged so that they
offset. The error for this height value will be +/- the sum of other errors
not related to the laser rangefinder accuracy.

So anyway, the number you are generating are likely closer than you
indicate. Consistency indicates that the measurements you are getting are
not filled with major busts and that big systematic errors also are absent.
It is bad to over-estimate the accuracy of your height readings, but it is
also bad to underestimate the accuracy of the readings. I am sure if I have
made any grossly wrong assumptions here, our mathematicians will point them
out to me.

Ed Frank

----- Original Message -----
From: "Josh Kelly"
Sent: Tuesday, March 14, 2006 6:59 PM
Subject: Dry Creek, NC

All measurements listed
below were made with laser and clinometer using the sine method. I caution
readers that while my measurements are consistent, I still do not consider
them that accurate: I rate them only to within +/- 3 ft. of the actual
height of the tree.
Max error considerations   Robert Leverett
  Mar 15, 2006 12:29 PST 


   We guessed what you meant from the 0.707 value.


   Some sine values get quickly committed to memory. For instance:

   sin(0) = 0.000
   sin(30) = 0.500
   sin(45) = 0.707
   sin(60) = 0.866
   sin(90) = 1.000

   To keep an idea of the worst case scenarios for the equipment
problems I have encountered with my own instruments, I keep handy tables
such as the one following this narrative. The worst equipment problems
that I have had are sticking clinometer that can read high or low by
half a degree and laser that read long or short by one yard (this is a
typical worse case scenario barring really major equipment problems).
The table assumes the clinometer reads high by 0.r degrees and the laser
shoots long by a yard. Some of the combinations in the table obviously
don't make sense for trees. They are included for completeness. The
table clearly reveals the danger of extremely long shots. It also shows
that for very tall trees shot far enough back to see the top, with the
laser-clinometer combination, an error of 3 to 3.5 feet can be made -
least we get too complacent. But these are worst case scenarios for the
the sin-sin method and include no partially offsetting measurement for
any below eye-level component of the tree where the clinometer error of
0.5 degrees would be partially offset. The laser would still read long,
but the error from the clinometer would go the other way. The table

Angle Ydg CalcHgt ActHgt Diff
                in Ft   in Ft   in Ft

0 150 3.95 0.00 3.953
15 150 121.06 116.47 4.590
30 150 229.91 225.00 4.915
45 150 323.10 318.20 4.904
60 150 394.27 389.71 4.560
75 150 438.57 434.67 3.904
90 150 452.98 450.00 2.983

0 125 3.30 0.00 3.299
15 125 101.02 97.06 3.959
30 125 191.85 187.50 4.350
45 125 269.61 265.17 4.444
60 125 328.99 324.76 4.235
75 125 365.96 362.22 3.738
90 125 377.99 375.00 2.986

0 100 2.64 0.00 2.644
15 100 80.97 77.65 3.328
30 100 153.78 150.00 3.784
45 100 216.11 212.13 3.983
60 100 263.72 259.81 3.910
75 100 293.35 289.78 3.571
90 100 302.99 300.00 2.988

0 75 1.99 0.00 1.990
15 75 60.93 58.23 2.696
30 75 115.72 112.50 3.219
45 75 162.62 159.10 3.522
60 75 198.44 194.86 3.585
75 75 220.74 217.33 3.404
90 75 227.99 225.00 2.991

0 50 1.34 0.00 1.335
15 50 40.89 38.82 2.065
30 50 77.65 75.00 2.653
45 50 109.13 106.07 3.061
60 50 133.16 129.90 3.261
75 50 148.13 144.89 3.238
90 50 152.99 150.00 2.994

0 25 0.68 0.00 0.681
15 25 20.84 19.41 1.433
30 25 39.59 37.50 2.088
45 25 55.63 53.03 2.601
60 25 67.89 64.95 2.936
75 25 75.52 72.44 3.071
90 25 78.00 75.00 2.997

0 15 0.42 0.00 0.419
15 15 12.83 11.65 1.181
30 15 24.36 22.50 1.862
45 15 34.24 31.82 2.416
60 15 41.78 38.97 2.806
75 15 46.47 43.47 3.004
90 15 48.00 45.00 2.998

Contrast the maximums with what can happen using the % slope method
where errors can be in the 10's of feet.


Re: Max error considerations    Edward Frank
   Mar 15, 2006 13:40 PST 


I did not repeat your entire post below. But there is a false assumption in
your calculations concerning the clinometer errors. If the clinometer
sticks when measuring the top, and doesn't when measuring the bottom the
errors will b as you indicated. However this is really instrument failure -
rather than instrument error. The false assumption is in the statement
"partially offsetting measurement for any below eye-level component of the
tree where the clinometer error of 0.5 degrees would be partially offset."
The errors at the base would offset the errors at the top whether the base
of the base of the tree were below, level, or above eye level. The offset
would not be complete, because the distance to the top of the tree in most
cases would be greater than the distance to the base (you could be looking
down on the tree from a slope above), Thus the small error is applied over a
greater distance than the same error is applied when shooting to the base of
the tree. But it would still be almost completely offset.

For example consider a basic 3-4-5 triangle for a tree 160 feet tall,
measured from a distance 120 feet to the base on a level line, with the top
of the tree 200 feet from the viewer. If the instrument misread the base of
the tree to be at an angle of -1 degree, rather than 0 degrees, the
calculated distance below the level line would be -2.09 feet. The top of
the tree would be measured a similar amount low: 52.13 degrees, rather than
the actual angle of 53.13 degrees. The calculated height of the tree above
horizontal would be 157.88 feet. Adding the two numbers together gives you
a total height of 159.97 feet instead of 160 feet. That is a net error of
0.03 feet with a clinometer that is reading off by a full degree. 

Sticking is a completely different class of problem.
If the instrument is sticking at one end of the readings and not the other,
whether or not the base is above or below eye level, there will not be any
offset of the error generated by the sticking instrument. That is simply a
fixed error.


Re: Dry Creek, NC   Edward Frank
  Mar 15, 2006 14:02 PST 


The value of .8 yards long seems to me to be higher than the normal range of
offset to be expected straight out of the box. Perhaps the company could
recalibrate it for you if it is till under warranty. What you need to
determine is how far the instrument is off for the ranges you are typically
measuring tree heights. You need to make sure of the calibration in these
intervals which are longer than the 50 meters you have information about.
People commonly make a mistake when using bathroom scales. They adjust the
settings to read exactly zero when empty. That doesn't matter. What they
need to do is to adjust the bathroom scale to match the scale at the doctors
office when the person is standing on them. Thus all of the values close to
your weight will be calibrated to read as accurately as possible. Who cares
if it is accurate in a weight range you are not measuring, especially if
that calibration would compromise the accuracy of the values in the range
you are measuring.


----- Original Message -----
From: "Josh Kelly" 
Sent: Wednesday, March 15, 2006 3:47 PM
Subject: Re: Dry Creek, NC


Thanks for the education about range finders. I have figured out the click
over on my range finder out to 50 meters. I haven't figured it out beyond
that. I also know that my range finder shoots .8 yards long. A major
source of uncertainty for me is that my range finder only delivers even
numbers, still, I am impressed so far with its precision.


RE: Max error considerations   Robert Leverett
  Mar 16, 2006 05:54 PST 


As usual, you caught my verbal shortcut. Instrument failure is the
better choice of description. What I should have said was error due to
the clinometer reading - whether the source of the error is a
mis-reading due to bad eye sight, a sticking clinometer where the
direction of the sticking is not consistent, or a clinometer that is off
in its calibration by a consistent amount in one direction or the other.

Having owned 5 clinometers over the past 15 years, I've experienced
the range of problems with them. One of my clinometers is out of
calibration by about a degree. It reads high. As you correctly note, the
errors from a clinometer being consistently off almost cancel.

However, I've encountered situations where the out of calibration
clinometer would stick - unpredictably. It gives the user reason to
continuously be checking for angle errors.

Max error considerations - clinometer   Edward Frank
  Mar 16, 2006 07:52 PST 


The consideration of how much error is caused by misread clinometer readings
is not as straight forward as those dealing with the laser range finder.
There are several considerations. 1) Calibration Errors. A calibration
error means the instrument is consistently reading high by some amount or
consistently reading low by some amount. In a previous post I showed that
calibration errors of 1 degree did not result in any significant error in a
typical distances we are using to measure trees. The errors at the base of
the tree are offset by the errors at the top of the tree with a net result
of total height errors of a few hundredths of a foot. 2) Sticking
instrument. Bob Leverett has reported many times that he has a clinometer
that sticks slightly leading to errors of a half a degree or so. This is an
instrument malfunction error. An analysis of this type of error is
difficult, because the amount and direction of sporadic error can not be
predicted. The only way to deal this this type of error is to catch it in
the field when doing the measurement. I do not believe it is a common
problem, although Bob might disagree, and this type of error is outside the
bounds of a reading error analysis. It is an different class of error. 3)
How accurately can the user read the instrument? John Eicholz (Nov 10,
2003)suggested he was getting measurements +/- 0.4 degrees consistently.
Other people read the instrument to a smaller estimate and suggest that they
are getting consistent readings to a smaller error value. For error
discussions a value of +/- 0.5 degrees is a reasonable starting point. The
amount of height error resulting from the inclination being off increases
with greater distance from the target, and decreases with an increasing
angle. This suggests the best strategy would be to measure the tree height
from as close as possible while still being able to see the actual top of
the tree. Here are some numbers for 0.5 degree errors as measured from a
distance of 300 feet. The distance is the same for each point because the
error is in the clinometer reading to the same point on the tree.

0.0 = 0
0.05 = 2.62
error = 2.62

30.0 = 150
30.5 = 152.26
error = 2.26

60.0 = 259.81
60.5 = 261.11
error = 1.30

80.0 = 295.44
80.5 = 295.89
error = 0.45

Errors may occur at either the base or top of the treee. If the direction
of the error is opposite at each end of the tree - ie. both errors make the
tree taller, or both errors make the tree shorter, the total error is
additive. If one error makes the tree shorter and the other makes the tree
taller, the error at the base will be partially offset by the error at the
top. The amount of error at the base and top may be different, and the
distance to the target from the observer to the top of the tree and the base
of the tree will be different, so the amount of error will vary from instance
to instance. The total amount of error from clinometer reading error can
easily exceed one yard, so in many cases, most of the error in measuring
tree height is from inaccurate clinometer readings, rather than from laser
rangefinder errors (which do not increase with distance). These errors can
be dealt with in a couple ways. The first is to take extra care when making
the inclination readings. With care and practice the amount of error in the
reading can be reduced. The instrument could be braced to reduce shake.
Bracing the instrument on a solid object is often impractical, but there are
techniques of holding the clinometer that can be taken from photography to
help reduce the shake. Measurements can be taken with the arm braced
tightly on the reader's chest and with the breath held. The other way to
reduce the net error in clinometer readings is to take multiple readings
from different places. Clinometer reading errors tend to be random. If the
tree heights are measured from different points, with the distance to the
target taken at changeover points (clickover points where the numbers on the
rangefinder changes), this will essentially eliminate distance errors.
Averaging the resulting calculated heights would tend to offset the positive
and negative errors from the clinometer readings and achieve a more accurate
height value. Field conditions may make some of these suggestions difficult
to implement.

Ed Frank
RE: Max error considerations - clinometer   Robert Leverett
  Mar 16, 2006 08:56 PST 


   Excellent discussion of the sources of error that can be associated
with the clinometer.

   For a clinometer in good working condition (the operative
assumption), it is my experience that one can learn to read it
accurately to within a quarter of a degree. That doesn't happen starting
off, though. But using the methods you describe, most readers can
eventually become that accurate. However for analysis purposes, the half
degree is a safe maximum error for angle reading.

    For those shaky with the math, consider the calculation:

     Error in Height = distance x sine(error in angle)

    The greater the distance and/or the greater the error in the angle,
the greater the height error attributable to the error in the angle. For
any given error in the angle, if you double the distance, you double the
height error. The relationship is linear. However, if you double the
angle error for a given distance, you don't exactly double the height
error, although the difference is inconsequential for most purposes.
Here the doubling is in the error of the angle, not the base reading of
the angle. Just want to make that point clear.


Re: Max error considerations - clinometer    Edward Frank
   Mar 16, 2006 10:25 PST 


The angle you are looking up makes a big difference. What you say is true
for errors at around 0 degrees, but if you are looking up at an angle of 60
degrees, the error is half that value. The simplification may be adequate
for people with a math phobia but with just a slightly more complicated
equation a better estimate can be obtained:

Error height = distance x sin(angle) - distance x sin(angle + error in

for 30 degrees, at 70 yards, with an error of +0.5 degrees for example:

Error height = 210 x sin(30) - 210 x sin(30.5)
                   = 105 - 106.583
                   = -1.583 feet

The basic formula you suggested: [Error in Height = distance x sine(error
in angle)] would give an error of 1.83 feet. This is not a big difference
between the two formulas, but the difference becomes greater with an
increasingly steeper angle.

A more complete formula would include the calculations for both the base of
the tree and the top of the tree:

Error height = [distance to base x sin(angle at base) - distance to base x
sin(angle to base + error in angle)] + [distance to top x sin(angle to
top) -
distance to top x sin(angle to top + error in angle)]

I would be interested in your insight on what techniques will let you make
better estimates, problems in the field, how to
tell if your instrument is sticking, and other problems associated with
making inclination measurements not covered by the basic number crunching I


RE: Max error considerations - clinometer    Robert Leverett
   Mar 16, 2006 13:06 PST 


   Some of our readers might like to see the absolute amount of height
error that a 0.5 degree clinometer error has as both the distance and
angle to the target increase. The angle error in the examples is on the
high side.

Dist-to-target   Angle   Angle-error    Height-error

50                  0         0.5           0.44
50                 30         0.5           0.38
50                 45         0.5           0.31
50                 60         0.5           0.22        

100                 0         0.5           0.87
100                30         0.5           0.75
100                45         0.5           0.61
100                60         0.5           0.43

150                 0         0.5           1.31
150                30         0.5           1.13
150                45         0.5           0.92
150                60         0.5           0.65

200                 0         0.5           1.75
200                30         0.5           1.51
200                45         0.5           1.23
200                60         0.5           0.87

250                 0         0.5           2.18
250                30         0.5           1.88
250                45         0.5           1.54
250                60         0.5           1.08

   Practically speaking this says the largest error that is likely to be
made as a consequence of misreading the clinometer or the clinometer
consistently reading off by as much as 0.5 degrees is a little over 2
feet. However, if the laser shoots long by a yard and the clinometer
reads high by 0.5 degrees, then the combination produces an error of
2.21 feet at 0 degrees and 250 feet, but 3.69 feet at 60 degrees and 250
feet. We're obviously not measuring an eastern tree in the latter case,
at least none in this era.

   If we assume that 120 yards is the longest distance we'll shoot and
33.8 degrees is the highest angle we'll shoot at that distance (tree is
about 200 feet-dream on Bob), we make an error of 2.6 feet from the
clinometer alone, 1.7 feet from the laser alone, and 4.3 feet from the
combination of clinometer and laser. This is an extreme distance from
which to measure a tree. If we move closer and shoot the same tree at a
distance of say 240 feet, our combined clinometer-laser error drops to
3.4 feet. So shooting a very tall tree at 80 yards with a clinometer
reading that's off by 0.5 degrees and with a laser that's off by a yard
results in a height error just a little over 3 feet.

   The lesson in all this is that at distance of 80 yards or less,
unless equipment is a real problem, one has to work hard to make an
error in the height measurement of 3 feet. I learned that a long time
ago and began boldly proclaiming that most of our measurements would be
within 1.5 feet of the actual height of what was being measured.   

   Ed, I'll address your questions tomorrow.


RE: Max error considerations - clinometer   John Eichholz
  Mar 16, 2006 18:06 PST 


I think recently my clinometer error is closer to +/- 0.2 degree. This
comes with long practice, and with careful calibration of the individual
quirks of my clinometer using a "peg test". The test involves mounting
the clinometer on a tripod, placing thin black tape at judicious places
on the door frames, and generally acting like a nut. I only do it at
home alone. Whenever I want the most accurate readings, I use a small
handheld tripod which I brace on the trunk of a conveniently located
tree. Since there is no motion I have very little reading error. These
readings are repeatable to within the limits of my vision, so I believe
they are quite accurate.

I know the math is complicated, but my post of 11/10/03 and I think a
later post shows that height error due to angle error in measurements
taken of a vertical object from a fixed point are almost equal
independent of the angle. This is because as the angle increases, the
error rate decreases but the distance increases, and the math shows
these changes are in a near lock step offset. So, a height reading from
a fixed point with a consistent clinometer error (either always too high
or always too low) using an angle reading at both the top and the
bottom, should nearly cancel. I guess that is good news.

One interesting detail I have observed while calibrating the rangefinder
is that it is much more accurate with a sky background than with a
terrestrial background. I notice this with narrow objects, such as the
edge of my picnic table, less so with tree trunks. This is likely due
to overlapping reflections from more distant objects. The readings are
always longer. With a reflector, the readings are as accurate as those
with a sky background. Since we almost always have a sky background at
the tip, I recommend calibrating against a sky background. I have used
a thin stick projecting above my roof, with a long tape trailing to the