Measurements and Errors

1. Measurements and Errors

2. Definition of a Measurement • The application of a device or apparatus for the purpose of ascertaining an unknown quantity. • An observation made to determine an unknown quantity • (Usually read from a graduated scale on the device) • Excludes counting which can be exact

3. Kinds of Measurements • Direct (e.g. taped distance, angles measured by theodolite, …) • Indirect (e.g. coordinate inverse to determine distance, coordinate measurement by GPS • What about an EDM distance? Direct or indirect?

4. Characteristics of Measurements • No measurements are exact. • All measurements contain errors. • The true value of a quantity being measured is never known • The exact sizes of errors are unknown

5. Definition of Error • Difference between a measured quantity and its true value Where: ε = the error in the measurement y = the measured value μ = the true value

6. Error Sources • Instrumental errors • Natural errors • Personal errors

7. Instrumental Errors • Caused by imperfections in instrument construction or adjustment • Examples – imperfect spacing of graduations, nominally perpendicular axes not at exactly 90°, level bubbles or crosshairs misadjusted … • Fundamental principle – keep instrument in adjustment to the extent feasible, but use field procedures that assume misadjustment

8. Natural Errors • Errors caused by conditions in the environment that are not nominal • Examples – temperature different from standard when taping, atmospheric pressure variation, gravity variation, magnetic fields, wind

9. Personal Errors • Errors due to limitations in human senses or dexterity • Examples – ability to center a bubble, read a micrometer or vernier, steadiness of the hand, estimate between graduations, … • These factors may be influenced by conditions such as weather, insects, hazards, …

10. Some of the afore-mentioned errors (instrumental, natural, and personal) occur in a systematic manner and others behave with apparent randomness. They are therefore referred to as systematic and random errors.

11. Mistakes or Blunders • These are generally caused by carelessness • They are not classified as errors in the same sense as systematic or random errors • Examples – not setting the proper PPM correction in an EDM, misreading a scale, misidentifying a point, … • Mistakes need to be identified and eliminated • This is difficult when their effect is small

12. Systematic Errors • These follow physical laws and can be corrected as long as they are identified and the proper mathematical model is available • Lack of correction of a fundamental systematic error is often considered amistake • Temperature correction in taping is a typical example

13. Random Errors • These are the remaining errors which can not be avoided • They tend to be small and are equally likely to be positive as negative • They can be analyzed using the concepts of probability and statistics • They are sometimes referred to as accidental errors

14. Precision • Due to errors, repeated measurements will often vary • Precision is the degree to which measurements are consistent – measurements with a smaller variation are more precise • Good precision generally requires much skill • Precision is directly related to random error

15. Accuracy • Accuracy is the nearness to the true value • Since the true value is unknown, true accuracy is unknown • It is generally accepted practice to assess accuracy by comparison with measurements taken with superior equipment and procedures (the so-called test against a higher-accuracy standard)

16. Example Which is more precise? Which is more accurate?

17. Target Example • Accurate and precise • Accurate on average, but not precise • Precise but not accurate • Neither accurate nor precise Questions: Can one shot be precise? Can a group of shots be accurate?

18. Real-World Target for Measurements No bulls-eye

19. Redundant Measurements • Redundant measurements are those taken in excess of the minimum required • A prudent professional always takes redundant measurements • Mathematical conditions can be applied to redundant measurements • Examples – sum of angles of a plane triangle = 180°, sum of latitudes and departures in a plane traverse equal zero, averaging measurements of the length of a line

20. Benefits of Redundancy • Can apply least squares adjustment which is a mathematically superior method • Often disclose mistakes • Better results through averaging (adjustment) • Allows one to assign a plus/minus tolerance to the answer

21. Advantages of Least Squares Adjustment • Most rigorous of all adjustment procedures • Enables post-adjustment analysis • Gives most probable values • Can be used to perform survey design for a specified level of precision • Can handle any network configuration (not limited to traverse, for example)