Measurements and Errors
description
Transcript of Measurements and Errors
![Page 1: Measurements and Errors](https://reader035.fdocuments.in/reader035/viewer/2022081505/56815cbf550346895dcac428/html5/thumbnails/1.jpg)
Measurements and Errors
![Page 2: Measurements and Errors](https://reader035.fdocuments.in/reader035/viewer/2022081505/56815cbf550346895dcac428/html5/thumbnails/2.jpg)
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
![Page 3: Measurements and Errors](https://reader035.fdocuments.in/reader035/viewer/2022081505/56815cbf550346895dcac428/html5/thumbnails/3.jpg)
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?
![Page 4: Measurements and Errors](https://reader035.fdocuments.in/reader035/viewer/2022081505/56815cbf550346895dcac428/html5/thumbnails/4.jpg)
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
![Page 5: Measurements and Errors](https://reader035.fdocuments.in/reader035/viewer/2022081505/56815cbf550346895dcac428/html5/thumbnails/5.jpg)
Definition of Error
• Difference between a measured quantity and its true value
y
Where:
ε = the error in the measurement
y = the measured value
μ = the true value
![Page 6: Measurements and Errors](https://reader035.fdocuments.in/reader035/viewer/2022081505/56815cbf550346895dcac428/html5/thumbnails/6.jpg)
Error Sources
• Instrumental errors• Natural errors• Personal errors
![Page 7: Measurements and Errors](https://reader035.fdocuments.in/reader035/viewer/2022081505/56815cbf550346895dcac428/html5/thumbnails/7.jpg)
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
![Page 8: Measurements and Errors](https://reader035.fdocuments.in/reader035/viewer/2022081505/56815cbf550346895dcac428/html5/thumbnails/8.jpg)
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
![Page 9: Measurements and Errors](https://reader035.fdocuments.in/reader035/viewer/2022081505/56815cbf550346895dcac428/html5/thumbnails/9.jpg)
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, …
![Page 10: Measurements and Errors](https://reader035.fdocuments.in/reader035/viewer/2022081505/56815cbf550346895dcac428/html5/thumbnails/10.jpg)
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.
![Page 11: Measurements and Errors](https://reader035.fdocuments.in/reader035/viewer/2022081505/56815cbf550346895dcac428/html5/thumbnails/11.jpg)
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
![Page 12: Measurements and Errors](https://reader035.fdocuments.in/reader035/viewer/2022081505/56815cbf550346895dcac428/html5/thumbnails/12.jpg)
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 a mistake
• Temperature correction in taping is a typical example
![Page 13: Measurements and Errors](https://reader035.fdocuments.in/reader035/viewer/2022081505/56815cbf550346895dcac428/html5/thumbnails/13.jpg)
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
![Page 14: Measurements and Errors](https://reader035.fdocuments.in/reader035/viewer/2022081505/56815cbf550346895dcac428/html5/thumbnails/14.jpg)
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
![Page 15: Measurements and Errors](https://reader035.fdocuments.in/reader035/viewer/2022081505/56815cbf550346895dcac428/html5/thumbnails/15.jpg)
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)
![Page 16: Measurements and Errors](https://reader035.fdocuments.in/reader035/viewer/2022081505/56815cbf550346895dcac428/html5/thumbnails/16.jpg)
ExampleObservation pacing taping EDM
1 571 567.17 567.133
2 563 567.08 567.124
3 566 567.12 567.129
4 588 567.38 567.165
5 557 567.01 567.144
average 569 567.15 567.133
Which is more precise? Which is more accurate?
![Page 17: Measurements and Errors](https://reader035.fdocuments.in/reader035/viewer/2022081505/56815cbf550346895dcac428/html5/thumbnails/17.jpg)
Target Example(a) Accurate and precise
(b) Accurate on average, but not precise
(c) Precise but not accurate
(d) Neither accurate nor precise
Questions:
Can one shot be precise?
Can a group of shots be accurate?
![Page 18: Measurements and Errors](https://reader035.fdocuments.in/reader035/viewer/2022081505/56815cbf550346895dcac428/html5/thumbnails/18.jpg)
Real-World Target for Measurements
No bulls-eye
![Page 19: Measurements and Errors](https://reader035.fdocuments.in/reader035/viewer/2022081505/56815cbf550346895dcac428/html5/thumbnails/19.jpg)
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
![Page 20: Measurements and Errors](https://reader035.fdocuments.in/reader035/viewer/2022081505/56815cbf550346895dcac428/html5/thumbnails/20.jpg)
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
![Page 21: Measurements and Errors](https://reader035.fdocuments.in/reader035/viewer/2022081505/56815cbf550346895dcac428/html5/thumbnails/21.jpg)
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)