Errors in Survey Measurements

Random v. Systematic

Presentation by Tom Arneson for the 2011 MSPS Conference

1

Classifications of Survey Errors

Mistakes or Blunders

Systematic errors

Random errors

2

Mistakes or Blunders

These are not really errors, under error theory, but they

are the one of the main reasons why surveyors purchase

errors and omissions insurance.

Some examples are:

Transposing digits i.e. writing 144.03 instead of 114.03

Recording the wrong point number for a traverse point.

Not recording a change in prism pole height.

Blunders that cause large errors in the final survey are

usually detected by normal survey checks, however small

blunders may remain undetected.

3

Systematic Errors

Systematic errors, or biases errors, result from the

physical properties of the measuring system.

Systematic errors are constant under constant measuring

conditions and change as conditions change. A classical

example is the change in length of a tape as the

temperature changes.

4

Correcting Systematic Errors

5

Because systematic errors are caused by the physics of the

measurement system, they can be mathematically

modeled and corrections computed to offset these

errors. For example temperature correction for a steel

tape:

Where k is a constant:, (6.45x10-6 for degrees Fahrenheit) ;

Tm is the temperature of the tape; Ts is the standard

temperature; and L is the uncorrected length measured. If the

temperature is above standard, then the tape is too long, and

a measured distance will be too short.

Total Station EDM Corrections for

Systematic Errors

6

The EDM (Electronic Distance Measurement) part of a total station measures distances using light waves. The velocity of light in air varies according to the air density. If the operator enters air temperature and pressure, the systematic error caused by this variation is corrected by most total stations.

The atmospheric pressure reported by The National Weather Service and used in weather reports is not an actual pressure. It is corrected to sea level pressure, so that a stated pressure means the same thing, as far as weather trends go, in areas of different elevations. For example an pressure of 30.10 inches of mercury (inHg) at Boston, near sea level, corresponds to an actual pressure of 24.79 inHg at Denver, which is about 5000 feet elevation.

If the survey not at sea level, using sea level pressure introduces a systematic error. This error, of course, increases with elevation. Ignoring the correction at an elevation of 900 feet causes a distance error of about 10 ppm (parts per million).

Correcting Sea Level Pressure

7

The following table shows actual pressure at selected

elevations when the sea level pressure is 29.92 inHg, this

pressure is also known as one Standard Atmosphere.

Elevation (feet) Pressure (inHg) Correction

(inHg)

0 29.92 0.00

200 29.70 -0.22

400 29.49 -0.43

600 29.28 -0.64

800 29.06 -0.86

1000 28.85 -1.07

1200 28.64 -1.28

1400 28.44 -1.48

1600 28.23 -1.69

1800 28.02 -1.90

2000 27.82 -2.10

8

The elevation of the Twin Cities area is around 900 feet

(274 m). For an easy to use correction: Subtract 1.0 inHg

from the announced air pressure and enter that value

into the total station.

Example:

Announced pressure: 29.2 inHg

Correction 900 feet: -1.0 inHg

=====

Enter: 28.2 inHg on the total station.

Random Errors

Remain after mistakes are prevented or eliminated and

measurements corrected for systematic errors

Random errors are as likely to be positive as negative

Small random errors are much more likely than large

ones.

Study of random errors began in the 18th Century

9

Carl Friedrich Gauss

Gauss was a German mathematician and scientist who

contributed significantly to many fields, including number

theory, statistics, analysis, differential geometry, geodesy,

geophysics, electrostatics, astronomy, and optics.

Although others also worked on random error theory,

Carl Gauss got much of the credit, to the extent that

Germany honored him on the Ten-Mark note. (This note

is obsolete since the advent of the common European

currency, the Euro.)

10

Ten Mark Bill honoring Carl Gauss

11

Reverse of 10-Mark bill showing a

triangulation diagram and a sextant

12

Notation and equations

The lower case Greek letter σ, sigma denotes the standard

deviation of a series

Upper case sigma, ∑ denotes the sum of a series

A lower case x denotes a series of measurements

A Lower case x with an over-score denotes the mean of a

series

13

Useful Statistical Equations

Mean:

Residuals:

14

Equations for Standard Deviations

Standard Deviation:

Standard Deviation of

the mean:

15

Normal (Gaussian) Distribution

Equation:

Graph and equation

shown on 10-Mark bill:

16

The probability of ±1σ is 68%

17

The probability of ±2σ is 95%

18

The probability of ±3σ is 99.7%

19

Reducing Random Errors

Make repeated measurements – Making 4 times as

many measurements will reduce the random error by

half.

Include redundant measurements in the survey network

and use a Least Squares Adjustment with proper

weights for measured values.

20

References

Elementary Surveying, An Introduction to Geomatics, Charles D.

Ghilani and Paul R Wolf

Adjustment Computations: Spatial Data Analysis, Charles D.

Ghilani and Paul R Wolf

21

Web Links

22

Normal Distribution Calculator/Table

wikipedia Gaussian_function

http://en.wikipedia.org/wiki/Normal_distribution

http://www.cs.princeton.edu/introcs/11gaussian/

http://www.physics.ohio-ate.edu/~gan/teaching/spring04/Chapter3.pdf

simulation of binary events

wikipedia Carl_Friedrich_Gauss

http://davidmlane.com/hyperstat/z_table.html

http://en.wikipedia.org/wiki/Systematic_error#Systematic_versus_random_error

http://en.wikipedia.org/wiki/Observational_error

Book Fooled by Randomness

http://www.netmba.com/statistics/distribution/normal/

http://en.wikipedia.org/wiki/Abraham_de_Moivre

http://www.york.ac.uk/depts/maths/histstat/demoivre.pdf