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|YearOfPublication=2011
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Performance of a positioning system is most usually measured in terms of positioning error ([[Accuracy|accuracy]]) and of the portion of the time it can be used for the intended application ([[Availability|availability]]). However, other performance parameters can become important for some [[GNSS Applications|applications]]. For instance, when a positioning system is used for air or maritime navigation, an unwarned large position error can seriously increase the risk of an accident, possibly causing damage of goods, injuries to people or even death. Such errors can occur without violating the [[Accuracy|accuracy]] specification, and this is why the civil aviation community has defined the concept of [[Integrity|integrity]] as a measure of the probability that such hazardous situations can take place. Also, since some flight phases (such as approach and landing) are of special criticality, as well as bounded in duration, it is important that the positioning system remains available without interruptions, very specially during such short periods, but actually all flight phases have certain needs at this respect. That is why the civil aviation community has defined a fourth performance parameter called [[Continuity|continuity]].


Accuracy is the degree of conformance between the estimated or measured position, velocity and/or time of a platform at a given time and its true position, velocity and/or time{{#tag:ref| This accuracy definition was adapted from the [http://www.navcen.uscg.gov/pdf/2008_Federal_Radionavigation_Plan.pdf 2008 US Federal Radionavigation Plan]<ref> [http://www.navcen.uscg.gov/pdf/2008_Federal_Radionavigation_Plan.pdf Federal Radionavigation Plan], DOT-VNTSC-RITA-08-02/DoD-4650.5, 2008</ref>|group="footnotes"}}.


==Performance Parameters==


The civil aviation community has put the greatest effort in the rationalisation and standardisation of positioning (navigation) performance parameters and requirements, thus specifying the so-called [[Wikipedia:Required navigation performance|Required Navigation Performance (RNP)]] that an airborne navigation system must accomplish<ref>Report of the Special Communications/Operations Divisional Meeting, [[Wikipedia:International Civil Aviation Organization|ICAO]] Doc. 9650, November 1995</ref>:
==Measuring Accuracy==


:'''[[Wikipedia:Required navigation performance|RNP]]:''' A statement of the navigation performance [[Accuracy|accuracy]], [[Integrity|integrity]], [[Continuity|continuity]] and [[Availability|availability]] necessary for operations within a defined airspace.
Although being very easily understood from a conceptual point of view, the way that accuracy is measured and what is measured is not always obvious. The accuracy concept is generally used to measure the accuracy of positioning but can be also be used to measure the accuracy of velocity and even the accuracy of timing. For positioning there are 3 variants depending on the number of dimensions being considered: one-dimensional accuracy (used for vertical accuracy), bidimensional accuracy (used for horizontal accuracy) and tridimensional accuracy (combining horizontal and vertical accuracy)


In literature and in system/product specifications it can be found measurements of accuracy such as CEP, rms, Percentile 67%, Percentile 95%, 1 sigma, 2 sigma. Some of these accuracy measures are averages while others are counts of distribution<ref>[http://www.gpsworld.com/gnss-system/receiver-design/gnss-accuracy-lies-damn-lies-and-statistics-4142 GNSS Accuracy: Lies, Damn Lies, and Statistics], GPS World, Frank van Diggelen, January 2007</ref>:
* '''x% Percentile (x%):''' Means that x% of the positions calculated have an error lower or equal to the accuracy value obtained. Typical used values are 50%, 67%, 75% and 95%. Having an accuracy of 5m (95%) means that in 95% of the time the positioning error will be equal or below 5m.


* '''Circular Error Probable (CEP):''' Percentile 50%. Means that 50% of the positions returned calculated have an error lower or equal to the accuracy value obtained.


The four parameters on which [[Wikipedia:Required navigation performance|RNP]] specification is based are summarised hereafter<ref>Annex 10 (Aeronautical Telecommunications) To The Convention On International Civil Aviation, Volume I – Radio Navigation Aids, International Standards And Recommended Practices (SARPs), [[Wikipedia:International Civil Aviation Organization|ICAO]] Doc. AN10-1, 6<sup>th</sup> Edition, July 2006</ref>:
* '''Root Mean Square Error (rms):''' The square root of the average of the squared error. This measurement is an average but assuming that the error follows a normal distribution (which is close but not exactly true) it will correspond to the percentile 68% in one-dimensional distributions (e.g. vertical error or timing error) and percentile 63% for bidimensional distributions (e.g. horizontal error). For the horizontal error this measurement is also referred as drms and can have variants such as 2rms or 2drms (2 times rms)


:'''[[Accuracy]]{{#tag:ref| This [[Accuracy|accuracy]] definition has been taken from the [http://www.navcen.uscg.gov/pdf/2008_Federal_Radionavigation_Plan.pdf 2008 US Federal Radionavigation Plan]<ref> [http://www.navcen.uscg.gov/pdf/2008_Federal_Radionavigation_Plan.pdf Federal Radionavigation Plan], DOT-VNTSC-RITA-08-02/DoD-4650.5, 2008</ref>|group="footnotes"}}:''' is the degree of conformance between the estimated or measured position and/or velocity of a platform at a given time and its true position or velocity
* '''x sigma:''' 1 sigma corresponds to one standard deviation and x sigma corresponds to x times 1 sigma. Assuming normal distributions 1 sigma corresponds to Percentile 68% in one-dimensional distributions and Percentile 39% for bidimensional distributions.




:'''[[Availability]]:''' is the portion of time during which the system is to be used for navigation during which reliable navigation information is presented to the crew, autopilot, or other system managing the flight of the aircraft
Less used that the previous measurements are the:
* '''Mean Error:''' Average error. Corresponds Percentile 68% in one-dimensional distributions and Percentile 54% for bidimensional distributions.


* '''Standard Deviation:''' Standard deviation of the error. Same as one sigma. Corresponds Percentile 58% in one-dimensional distributions and Percentile 39% for bidimensional distributions.


:'''[[Continuity]]:''' is the capability of the system to perform its function without unscheduled interruptions during the intended operation
Although the mean error and standard deviation are as less used as accuracy measurements, assuming normal distributions its use is as legitimate as the other measurements usually used.


==Relationship between Accuracy Measurements==


:'''[[Integrity]]:''' is a measure of the trust which can be placed in the correctness of the information supplied by the total system. Integrity includes the ability of a system to provide timely and valid warnings to the user (alerts) when the system must not be used for the intended operation (or phase of flight)
Assuming normal distributions these accuracy measurements can be converted between themselves. There is a correspondence between sigmas and percentiles. This correspondence can be used to convert between accuracy measurements since an accuracy of 1m (1 sigma) corresponds to 2m (2 sigma) , 3m (3 sigma) and xm (x sigma).




==Uses of GNSS Perfomance Parameters==
For one-dimensional distributions:
{| class="wikitable"
|-
! Sigma
! Percentile
|-
| 0,67
| 0,5 (CEP)
|-
| 0,80
| 0,58 (mean error)
|-
| 1
| 0,6827 (rms and std deviation)
|-
| 1,15
| 0,75
|-
| 1,96
| 0,95
|-
| 2
| 0,9545
|-
| 2,33
| 0,98
|-
| 2,57
| 0,99
|-
| 3
| 0,9973
|-
| 4
| 0,999936
|-
| 5
| 0,99999942
|-
| 6
| 0,999999998
|}


The performance parameters described above measure the performance of the GNSS system. These performance parameters can be used in different ways, with different purposes and in different conditions.


One of the uses for the GNSS Performance Paramenters is to define requirements that and GNSS System should have in specific conditions. All GNSS systems had  such requirements during the specification phase of the development of the system. These requirements are one of the drivers of the system design and development. During the operational phase these requirements are kept and sometimes updated as operational requirements against which actual performance will be verified. Such [[GALILEO Performances|performance]] requirements for [[GALILEO|Galileo]] can be found in the [http://ec.europa.eu/dgs/energy_transport/galileo/doc/galileo_hld_v3_23_09_02.pdf Galileo Mission High Level Definition] coming from the Galileo System Requirement Document<ref> [http://ec.europa.eu/dgs/energy_transport/galileo/doc/galileo_hld_v3_23_09_02.pdf Galileo Mission High Level Definition], v3, September 2002.</ref><ref> Galileo System Requirement Document, ESA-APPNS-REQ-00011, issue 3.0, June 20 2003</ref>. The GPS Performance requirements can be found in the [http://www.gps.gov/technical/ps/ GPS Performance Requirements Documents]<ref> [http://www.pnt.gov/public/docs/2008/spsps2008.pdf GPS Standard Positioning Service (SPS) Performance Standard], 4<sup>th</sup> Edition, September 2008.</ref><ref> [http://www.pnt.gov/public/docs/2007/ppsps2007.pdf GPS Precision Positioning Service (PPS) Performance Standard], 1<sup>st</sup> Edition, February 2007.</ref>.
For bidimensional distributions (Rayleigh distribution) :
{| class="wikitable"
|-
! Sigma
! Percentile
|-
| 1
| 0,394 (std deviation)
|-
| 1,18
| 0,5 (CEP)
|-
| 1,25
| 0,544 (mean error)
|-
| 1,414
| 0,632 (rms)
|-
| 1,67
| 0,75
|-
| 2
| 0,865
|-
| 2,45
| 0,95
|-
| 2,818
| 0,982 (2rms)
|-
| 3
| 0,989
|-
| 3,03
| 0,99
|-
| 4
| 0,9997
|-
| 5
| 0,999997
|-
| 6
| 0,999999985
|}


In a similar way this kind of performance requirements can be used also to define the performance that an [[GNSS Applications|application]] will need from a GNSS system in order for a the application to be able to perform its functionality adequately.


In the same way that these parameters are used for the definition of requirements of a GNSS system they are use to evaluate its actual performance under typical and well defined conditions. The actual [[GPS Perfomance|performance of GPS]] is monitored by several entities such as the [[Wikipedia:Federal Aviation Administration|US Federal Aviation Adminstration]] in its GPS Performance Analysis Reports><ref> [http://www.nstb.tc.faa.gov/reports/PAN72_0111.pdf FAA Global Positioning System (GPS) Standard Positioning Service (SPS) Performance Analysis Report], Report 72, January 2011.</ref>.


In the same way that Global Navigation Space Systems ([[GALILEO Perfomances|GALILEO]], [[GPS Performances|GPS]], [[GLONASS Performances|GLONASS]], [[COMPASS Performances|COMPASS]]) are evaluated in terms of performance using these parameters also the increase in performance given by Space Based Augmentations Systems ([[EGNOS Perfomances|EGNOS]], [[WAAS Performances|WAAS]], [[MSAS Performances|MSAS]]) and of other [[GNSS Augmentations]] are measured using these parameters.
Besides being used to evaluate the performance of the GNSS systems and it augmentation systems and tecnologies these performance paramenters are used to measure the perfomance of the receivers that use these systems. This performance evaluation can be made for generic conditions (such as open sky locations) or for the the evaluation of the receiver on specific conditions since the receiver performance can be influenced by local factors
The performance of GNSS systems can depend on several local aspects such as:
* '''Location of the user''' - Different locations yield a different performances since different satellites will be visible at different position in the sky.
* '''Time of day''' - Despite the fact that different views of constelation will be visible at different times of day, some errors (such as the errors caused by [[Ionospheric Delay]]) have different impacts during different times of the day.
* '''Surrounding environment''' - Surrounding buildings and vegetation can lead to masking of the sky leading to less visible satellites, signal attenuation effects comming from tree fooliage and multi-path effects comming from the reflection of the GNSS signal on buildings or other landmarks.


==Notes==
==Notes==

Revision as of 16:46, 28 March 2011


FundamentalsFundamentals
Title GNSS Performances
Author(s) Rui Barradas Pereira
Level Basic
Year of Publication 2011
Logo GMV.png


Accuracy is the degree of conformance between the estimated or measured position, velocity and/or time of a platform at a given time and its true position, velocity and/or time[footnotes 1].


Measuring Accuracy

Although being very easily understood from a conceptual point of view, the way that accuracy is measured and what is measured is not always obvious. The accuracy concept is generally used to measure the accuracy of positioning but can be also be used to measure the accuracy of velocity and even the accuracy of timing. For positioning there are 3 variants depending on the number of dimensions being considered: one-dimensional accuracy (used for vertical accuracy), bidimensional accuracy (used for horizontal accuracy) and tridimensional accuracy (combining horizontal and vertical accuracy)

In literature and in system/product specifications it can be found measurements of accuracy such as CEP, rms, Percentile 67%, Percentile 95%, 1 sigma, 2 sigma. Some of these accuracy measures are averages while others are counts of distribution[2]:

  • x% Percentile (x%): Means that x% of the positions calculated have an error lower or equal to the accuracy value obtained. Typical used values are 50%, 67%, 75% and 95%. Having an accuracy of 5m (95%) means that in 95% of the time the positioning error will be equal or below 5m.
  • Circular Error Probable (CEP): Percentile 50%. Means that 50% of the positions returned calculated have an error lower or equal to the accuracy value obtained.
  • Root Mean Square Error (rms): The square root of the average of the squared error. This measurement is an average but assuming that the error follows a normal distribution (which is close but not exactly true) it will correspond to the percentile 68% in one-dimensional distributions (e.g. vertical error or timing error) and percentile 63% for bidimensional distributions (e.g. horizontal error). For the horizontal error this measurement is also referred as drms and can have variants such as 2rms or 2drms (2 times rms)
  • x sigma: 1 sigma corresponds to one standard deviation and x sigma corresponds to x times 1 sigma. Assuming normal distributions 1 sigma corresponds to Percentile 68% in one-dimensional distributions and Percentile 39% for bidimensional distributions.


Less used that the previous measurements are the:

  • Mean Error: Average error. Corresponds Percentile 68% in one-dimensional distributions and Percentile 54% for bidimensional distributions.
  • Standard Deviation: Standard deviation of the error. Same as one sigma. Corresponds Percentile 58% in one-dimensional distributions and Percentile 39% for bidimensional distributions.

Although the mean error and standard deviation are as less used as accuracy measurements, assuming normal distributions its use is as legitimate as the other measurements usually used.

Relationship between Accuracy Measurements

Assuming normal distributions these accuracy measurements can be converted between themselves. There is a correspondence between sigmas and percentiles. This correspondence can be used to convert between accuracy measurements since an accuracy of 1m (1 sigma) corresponds to 2m (2 sigma) , 3m (3 sigma) and xm (x sigma).


For one-dimensional distributions:

Sigma Percentile
0,67 0,5 (CEP)
0,80 0,58 (mean error)
1 0,6827 (rms and std deviation)
1,15 0,75
1,96 0,95
2 0,9545
2,33 0,98
2,57 0,99
3 0,9973
4 0,999936
5 0,99999942
6 0,999999998


For bidimensional distributions (Rayleigh distribution) :

Sigma Percentile
1 0,394 (std deviation)
1,18 0,5 (CEP)
1,25 0,544 (mean error)
1,414 0,632 (rms)
1,67 0,75
2 0,865
2,45 0,95
2,818 0,982 (2rms)
3 0,989
3,03 0,99
4 0,9997
5 0,999997
6 0,999999985



Notes

  1. ^ This accuracy definition was adapted from the 2008 US Federal Radionavigation Plan[1]

References

  1. ^ Federal Radionavigation Plan, DOT-VNTSC-RITA-08-02/DoD-4650.5, 2008
  2. ^ GNSS Accuracy: Lies, Damn Lies, and Statistics, GPS World, Frank van Diggelen, January 2007
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