Dave Abercrombie, abe@rahul.net

Bushnell makes a variety of laser rangefinders that measure distances to remote objects. These typically have a range of from 20 meters to 500 meters, with an advertised accuracy of +/-1 meter. I wanted to find out if these could be used to survey rural land, say for example, for mapping a stream through rugged terrain.

I've had previous experience with other methods for mapping a stream. For example, triangulation has the potential to be quite accurate. However, laying out a network of triangles and measuring all the necessary angles can be very time consuming, especially in a narrow gorge where the triangles must be small. Another unsatisfactory method is to use handheld consumer-grade GPS: the position error is just too large when you are interested in terrain features that are only 20 to 30 meters in size.

If instead, one could use a laser rangefinder for distance and a portable compass for direction, then a survey could be done fairly quickly (at least on flat terrain - more on elevation later). The purpose of this page is to evaluate this technique. I started by developing a mathematical model used for evaluating the feasibility off surveying with a laser rangefinder and compass. Since matematical analysis of the model seemed encouraging, I performed a field test on flat terrain. The field test showed that this technique was very efficient and reasonably accurate (about 1%). I am now planing a follow-up test on hilly terrain.

One way to use a laser rangefinder to survey would be to measure a series of straight line "legs" along the route of interest. Knowing the length and direction of each leg of the route would allow one to calculate the positions of all leg endpoints relative to the starting point. In rugged, forested terrain, each leg can not be very long, since vegetation or terrain will obstruct the visual line of sight. In areas I am familiar with, one can sometimes see about 30 meters or more, but often one can not see much more than about 20 meters (the minimum range of some rangefinders). The advertised accuracy of these rangefinders is one meter, which is a fairly large fraction of the length of each leg (one meter error is 4% of a typical 25 meter leg). It seemed that such relatively large errors might accumulate over the course of many legs, possible limiting the usefulness of laser rangefinders for this type of survey work

I decided to model these errors as a "random walk" and look at how these errors might accumulate over 10 or 20 legs. The advertising for these rangefinders do not clearly spell out what is meant by "one meter error". I decided to interpret the advertising claim as a standard deviation. I assumed that measurement errors would be normally distributed around a mean of zero with a standard deviation of one meter. In other words, the error would be less than one meter two-thirds of the time. I further assumed that this standard deviation of one meter applies independently to both the X and Y directions.

Of course, there will be errors also in direction as measured by a portable compass. If the compass has an error of about one degree, this would cause a position error of about 0.4 meters at a distance of 25 meters (the sine of one degree is 0.017). This is much less than the error of the rangefinder by itself, and can be ignored in this approximate random-walk model (it would contribute only about 9% of the total error).

I used a spreadsheet program to simulate a two-dimensional random walk with independent X and Y changes distributed normally with a standard deviation of one meter. Spreadsheet functions that were used included a random number generator and the inverse normal distribution function. Each X or Y change was calculated like this:

=NORMINV(RAND(), mean_error, std_dev_error)

I calculated the cumulative position after each of 20 legs. An example simulation run is shown below. The delta X and Y values were calculated using the above formula, and their cumulative sum give the X and Y coordinates after N legs. For the example random walk simulation run below, the cumulative error after 20 legs is just under seven meters.

N | delta X | delta Y | X | Y | radius |
---|---|---|---|---|---|

0 | -0.84 | -1.50 | 0.00 | 0.00 | |

1 | -0.83 | -0.67 | -0.84 | -1.50 | 1.72 |

2 | -1.03 | -0.21 | -1.66 | -2.17 | 2.74 |

3 | -0.76 | -0.12 | -2.70 | -2.38 | 3.60 |

4 | -0.26 | -0.91 | -3.46 | -2.50 | 4.27 |

5 | 1.16 | -0.52 | -3.72 | -3.41 | 5.04 |

6 | 0.10 | -1.82 | -2.56 | -3.93 | 4.69 |

7 | -1.87 | -1.16 | -2.46 | -5.74 | 6.25 |

8 | -0.16 | 0.01 | -4.33 | -6.91 | 8.15 |

9 | -0.90 | -1.04 | -4.49 | -6.90 | 8.23 |

10 | 1.18 | -0.38 | -5.39 | -7.94 | 9.59 |

11 | 0.64 | -0.82 | -4.21 | -8.31 | 9.32 |

12 | -0.03 | -0.56 | -3.57 | -9.13 | 9.80 |

13 | -1.66 | 1.06 | -3.60 | -9.69 | 10.33 |

14 | 0.79 | 0.19 | -5.26 | -8.63 | 10.10 |

15 | 1.40 | 1.49 | -4.46 | -8.43 | 9.54 |

16 | 1.74 | 1.72 | -3.07 | -6.94 | 7.59 |

17 | -0.88 | -0.43 | -1.32 | -5.22 | 5.39 |

18 | -0.21 | -0.61 | -2.21 | -5.66 | 6.07 |

19 | -0.12 | -0.07 | -2.42 | -6.26 | 6.72 |

20 | 0.90 | -0.43 | -2.54 | -6.33 | 6.82 |

Since X and Y errors are modelled as being random within a normal distribution, one simulation by itself is not too meaningful. So I ran the above simulation two dozen times and averaged the results at each value of N. The graph below illustrates the results of averaging over two dozen simulations. The average distance at each value of N (i.e., each "leg") is shown by a dot, and the pink line represents the best straight line through these points. It shows that after 10 legs, we'd expect a cumulative distance (i.e., error) of about 5 meters, rising to about 7 meters over 20 legs. This level of error seems like it might be acceptable, and was encouraging enough to warrant field tests.

For direction measurements, I used a Brunton pocket transit mounted on a non-magnetic tripod. For distance measurements, I used a Bushnell Yardage Pro 500. For the rangefinder target, I used a four-inch diameter plastic ball, covered with aluminum foil, and suspended about six feet high on an aluminum tripod. I also used a handheld Suunto MC-1 compass to supplement direction measurements with the Brunton.

The Brunton compass gave readings in degrees East or West of North or South. In other words, it was graduated from zero to 90 degrees in four separate quadrants, with a reading of zero at North and South, and a reading of 90 at East and West. For example, a reading of "45 degrees East of South" is equivalent to Southeast. The Suunto, on the other hand, read in two-degree increments starting at zero for North, 90 for East, 180 for South, and so on.

This field test was done on flat terrain so that I could ignore elevation changes. My plan was to survey a closed loop of about ten legs, each about 25 meters long. At each point I would measure the distance and direction to the next point. Simple trigonometry would be able to convert the measured length and direction of each leg to the equivalent changes in X and Y coordinates. If all measurements were completely accurate, the the sum of all of the changes in X would be zero, as would the sum of all changes in Y. The actual sum of all changes in X and Y would provide a measure of the accuracy of this survey method. Because of this simple design, I could arbitrarily choose my starting point as the origin of the coordinate system, and I would also be able to ignore magnetic declination.

I started by placing the compass at a well-marked location. I placed the target tripod at a distance estimated visually to be about 25 meters in a direction that was convenient. I would return to the compass and read the direction to the target with both compasses. I would mentally convert from on compass's coordinate system into the other compass's system to avoid gross errors in readings. I would then measure the distance using the laser rangefinder. Observations were recorded with ink in a bound notebook. I would then move the compass tripod to be directly under the target tripod, and then move the target tripod another 25 meters or so. I repeated these measurements and movements until I had traced out a loop that returned to the well-marked starting position.

The following table provides both the raw observational data as well as the calculations. Comparing the readings from the two compasses, one can see that they seem to have their magnetic declination corrections set to different values. The Brunton was much easier to use and read, and it was graduated in single degrees, rather than two-degree increments of the Suunto. For these reasons, I decided to use only the Brunton observations as the basis of my calculations.

After 11 legs totalling 280 meters, the X-Y closure error was only two meters.

Suunto azimuth (degrees) | Brunton direction (degrees) | Brunton quadrant | Range finder meters | Sine of direction (X, East) | Cosine of direction (Y, North) | Sign of X based on quadrant | Sign of Y based on quadrant | delta X (meters) | delta Y (meters) |
---|---|---|---|---|---|---|---|---|---|

255 | 79 | W of S | 21 | 0.982 | 0.191 | -1 | -1 | -20.61 | -4.01 |

250 | 77 | W of S | 21 | 0.974 | 0.225 | -1 | -1 | -20.46 | -4.72 |

324 | 34 | W of N | 25 | 0.559 | 0.829 | -1 | 1 | -13.98 | 20.73 |

279 | 80 | W of N | 28 | 0.985 | 0.174 | -1 | 1 | -27.57 | 4.86 |

294 | 62 | W of N | 26 | 0.883 | 0.469 | -1 | 1 | -22.96 | 12.21 |

198 | 21 | W of S | 25 | 0.358 | 0.934 | -1 | -1 | -8.96 | -23.34 |

134 | 45 | E of S | 29 | 0.707 | 0.707 | 1 | -1 | 20.51 | -20.51 |

123 | 55 | E of S | 23 | 0.819 | 0.574 | 1 | -1 | 18.84 | -13.19 |

74 | 78 | E of N | 25 | 0.978 | 0.208 | 1 | 1 | 24.45 | 5.20 |

75 | 75 | E of N | 29 | 0.966 | 0.259 | 1 | 1 | 28.01 | 7.51 |

58 | 60 | E of N | 28 | 0.866 | 0.500 | 1 | 1 | 24.25 | 14.00 |

280 meterstotal |
X, Y error (distance from starting point) | 1.51 | -1.27 | ||||||

radius error (meters) |
2.0 |

The following map depicts the location of each point. Note how close the start and end points appear (on the right side of image).

I conclude that this technique of surveying with a laser rangefinder is very efficient and reasonably accurate. There are many projects, especially in rural areas, where this level of accuracy, say about 1%, is sufficient. And the method is very quick. It took only about seven to ten minutes per leg. The equipment is very portable, and easily carried by one person in rugged terrain.

Being used to thinking of azimuth as ranging from zero to 360, increasing clockwise from North, I was at first uncomfortable with the four-quadrant, zero to 90 degree scale of the Brunton. However, the Brunton's coordinate system turned out to be much easier to calculate, especially if using a slide rule or printed trigonometry tables.

Copyright Dave Abercrombie, 2002