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'''Dilution of precision''' ('''DOP'''), or '''geometric dilution of precision''' ('''GDOP'''), is a term used in [[satellite navigation]] and [[geomatics engineering]] to specify the additional multiplicative effect of navigation satellite geometry on positional measurement precision. | |||
[[Image:Geometric Dilution Of Precision.svg|right|thumb|Understanding the Geometric Dilution of Precision (GDOP) with a simple example. In '''A''' someone has measured the distance to two landmarks, and plotted their point as the intersection of two circles with the measured radius. In '''B''' the measurement has some error bounds, and their true location will lie anywhere in the green area. In '''C''' the measurement error is the same, but the error on their position has grown considerably due to the arrangement of the landmarks.]] | |||
[[Image:Bad gdop.png|right|thumb|Navigation satellites with poor geometry for Geometric Dilution of Precision (GDOP).]] | |||
[[Image:Good gdop.png|right|thumb|Navigation satellites with good geometry for Geometric Dilution of Precision (GDOP).]] | |||
== | ==Introduction== | ||
The concept of dilution of precision (DOP) originated with users of the [[LORAN|Loran-C navigation system]].<ref>{{cite web | |||
| url = http://gauss.gge.unb.ca/papers.pdf/gpsworld.may99.pdf | |||
| title = Dilution of Precision | |||
| author = Richard B. Langley | |||
| work = GPS World | |||
| date = May 1999 | |||
| accessdate = 2011-10-12 | |||
}}</ref> The idea of Geometric DOP is to state how errors in the measurement will affect the final state estimation. This can be defined as:<ref>{{cite book | |||
| last1 = Dudek | |||
| first1 = Gregory | |||
| authorlink1 = Gregory Dudek | |||
| last2 = Jenkin | |||
| first2 = Michael | |||
| title = Computational Principles of Mobile Robotics | |||
| publisher = [[Cambridge University Press]] | |||
| year = 2000 | |||
| isbn = 0-521-56876-5 | |||
}} | |||
</ref> | |||
<math>GDOP=\frac{\Delta ( {\rm Output \ Location} )}{\Delta ( {\rm Measured \ Data} )}</math> | |||
Conceptually you can imagine errors on a measurement resulting in the <math>\Delta ( {\rm Measured \ Data} )</math> term changing. Ideally small changes in the measured data will not result in large changes in output location, as such a result would indicate the solution is very sensitive to errors. The interpretation of this formula is shown in the figure to the right, showing two possible scenarios with acceptable and poor GDOP. | |||
More recently, the term has come into much wider usage with the development and adoption of GPS. Neglecting ionospheric <ref>{{cite web | |||
| url = http://www.insidegnss.com/node/1579 | |||
| title = GNSS and Ionospheric Scintillation: How to Survive the Next Solar Maximum | |||
| author = Paul Kintner, Cornell University | |||
| coauthors = Todd Humphreys, University of Texas-Austin; Joanna Hinks, Cornell University | |||
| work = Inside GNSS | |||
| date = July/August 2009 | |||
| accessdate = 2011-10-12 | |||
}}</ref> and tropospheric<ref>[http://www.trimble.com/gps/howgps-error.shtml GPS errors (Trimble tutorial)]</ref> effects, the signal from navigation satellites has a fixed precision. Therefore, the relative satellite-receiver geometry plays a major role in determining the precision of estimated positions and times. Due to the relative geometry of any given satellite to a receiver, the precision in the [[pseudorange]] of the satellite translates to a corresponding component in each of the four dimensions of position measured by the receiver (i.e., <math>x</math>, <math>y</math>, <math>z</math>, and <math>t</math>). The precision of multiple satellites in view of a receiver combine according to the relative position of the satellites to determine the level of precision in each dimension of the receiver measurement. When visible navigation satellites are close together in the sky, the geometry is said to be weak and the DOP value is high; when far apart, the geometry is strong and the DOP value is low. Consider two overlapping rings, or [[annulus (mathematics)|annuli]], of different centres. If they overlap at right angles, the greatest extent of the overlap is much smaller than if they overlap in near parallel. Thus a low DOP value represents a better positional precision due to the wider angular separation between the satellites used to calculate a unit's position. Other factors that can increase the effective DOP are obstructions such as nearby mountains or buildings. DOP can be expressed as a number of separate measurements. HDOP, VDOP, PDOP, and TDOP are respectively Horizontal, Vertical, Positional (3D), and Time Dilution of Precision. They follow mathematically from the positions of the usable satellites. Signal receivers allow the display of these positions (''skyplot'') as well as the DOP values. | |||
The term can also be applied to other location systems that employ several geographical spaced sites. It can occur in electronic-counter-counter-measures (electronic warfare) when computing the location of enemy emitters (radar jammers and radio communications devices). Using such an [[interferometry]] technique can provide certain geometric layout where there are degrees of freedom that cannot be accounted for due to inadequate configurations. | |||
The effect of geometry of the satellites on position error is called geometric dilution of precision and it is roughly interpreted as ratio of position error to the range error. Imagine that a [[square pyramid]] is formed by lines joining four satellites with the receiver at the tip of the pyramid. The larger the volume of the pyramid, the better (lower) the value of GDOP; the smaller its volume, the worse (higher) the value of GDOP will be. Similarly, the greater the number of satellites, the better the value of GDOP. | |||
==Meaning of DOP Values== | |||
{| class="wikitable" | |||
|- | |||
! DOP Value | |||
! Rating | |||
! Description | |||
|- | |||
| 1 | |||
| Ideal | |||
| This is the highest possible confidence level to be used for applications demanding the highest possible precision at all times. | |||
|- | |||
| 1-2 | |||
| Excellent | |||
| At this confidence level, positional measurements are considered accurate enough to meet all but the most sensitive applications. | |||
|- | |||
| 2-5 | |||
| Good | |||
| Represents a level that marks the minimum appropriate for making business decisions. Positional measurements could be used to make reliable in-route navigation suggestions to the user. | |||
|- | |||
| 5-10 | |||
| Moderate | |||
| Positional measurements could be used for calculations, but the fix quality could still be improved. A more open view of the sky is recommended. | |||
|- | |||
| 10-20 | |||
| Fair | |||
| Represents a low confidence level. Positional measurements should be discarded or used only to indicate a very rough estimate of the current location. | |||
|- | |||
| >20 | |||
| Poor | |||
| At this level, measurements are inaccurate by as much as 300 meters with a 6 meter accurate device (50 DOP × 6 meters) and should be discarded. | |||
|} | |||
The DOP factors are functions of the diagonal elements of the [[covariance matrix]] of the parameters, expressed either in a global or a local geodetic frame. | |||
==Computation of DOP Values== | |||
As a first step in computing DOP, consider the unit vectors from the receiver to satellite i: <math>\scriptstyle \left(\frac {(x_i-x)} {R_i}, \frac {(y_i-y)} {R_i}, \frac {(z_i-z)} {R_i}\right)</math> where <math>\scriptstyle R_i= \sqrt{(x_i- x)^2 + (y_i-y)^2 + (z_i-z)^2}</math> and where <math>\scriptstyle\ x,\ y</math> and <math>\scriptstyle \ z</math> denote the position of the receiver and <math>\scriptstyle \ x_i, y_i</math> and <math>\scriptstyle \ z_i</math> denote the position of satellite i. Formulate the matrix, A, as: | |||
:<math>A = | |||
\begin{bmatrix} | |||
\frac {(x_1- x)} {R_1} & \frac {(y_1-y)} {R_1} & \frac {(z_1-z)} {R_1} & -1 \\ | |||
\frac {(x_2- x)} {R_2} & \frac {(y_2-y)} {R_2} & \frac {(z_2-z)} {R_2} & -1 \\ | |||
\frac {(x_3- x)} {R_3} & \frac {(y_3-y)} {R_3} & \frac {(z_3-z)} {R_3} & -1 \\ | |||
\frac {(x_4- x)} {R_4} & \frac {(y_4-y)} {R_4} & \frac {(z_4-z)} {R_4} & -1 | |||
\end{bmatrix} | |||
</math> | |||
The first three elements of each row of ''A'' are the components of a unit vector from the receiver to the indicated satellite. If the elements in the fourth column are ''c'' which denotes the [[speed of light]] then the <math>\sigma_{t}</math> factor (time dilution) is always 1. If the elements in the fourth column are ''-1'' then the <math>\sigma_{t}</math> factor is calculated properly.<ref>http://www.colorado.edu/geography/gcraft/notes/gps/gif/gdop.gif</ref> Formulate the matrix, ''Q'', as: | |||
:<math>Q = \left (A^T A \right )^{-1} | |||
</math> | |||
This computation is in accordance with [http://www.gmat.unsw.edu.au/snap/gps/gps_survey/chap1/142.htm Section 1.4.2 of ''Principles of Satellite Positioning''] where the weighting matrix, ''P'', has been set to the identity matrix. | |||
The elements of ''Q'' are designated as: | |||
:<math>Q = | |||
\begin{bmatrix} | |||
\sigma_x^2 & \sigma_{xy} & \sigma_{xz} & \sigma_{xt} \\ | |||
\sigma_{xy} & \sigma_{y}^2 & \sigma_{yz} & \sigma_{yt} \\ | |||
\sigma_{xz} & \sigma_{yz} & \sigma_{z}^2 & \sigma_{zt} \\ | |||
\sigma_{xt} & \sigma_{yt} & \sigma_{zt} & \sigma_{t}^2 | |||
\end{bmatrix} | |||
</math> | |||
PDOP, TDOP and GDOP are given by: | |||
<math>\begin{align} | |||
PDOP &= \sqrt{\sigma_x^2 + \sigma_y^2 + \sigma_z^2}\\ | |||
TDOP &= \sqrt{\sigma_{t}^2}\\ | |||
GDOP &= \sqrt{PDOP^2 + TDOP^2}\\ | |||
\end{align}</math> | |||
in agreement with [http://www.gmat.unsw.edu.au/snap/gps/gps_survey/chap1/149.htm Section 1.4.9 of ''Principles of Satellite Positioning'']. | |||
The horizontal dilution of precision, <math>\scriptstyle HDOP = \sqrt{\sigma_x^2 + \sigma_y^2}</math>, and the vertical dilution of precision, <math>\scriptstyle \ VDOP = \sqrt{\sigma_{z}^2} </math>, are both dependent on the coordinate system used. To correspond to the local horizon plane and the local vertical, ''x'', ''y'', and ''z'' should denote positions in either a north, east, down coordinate system or a south, east, up coordinate system. | |||
== References == | |||
=== Notes === | |||
{{reflist}} | |||
=== General === | |||
{{refbegin}} | |||
* [http://gauss.gge.unb.ca/papers.pdf/gpsworld.may99.pdf DOP Factors] | |||
* [http://www.developerfusion.co.uk/show/4652/2/ Writing Your Own GPS Applications: Part 2 - Causes of Precision Error] | |||
* [http://www.colorado.edu/geography/gcraft/notes/gps/gif/gdop.gif manually calculating GDOP] | |||
* [http://users.erols.com/dlwilson/gpshdop.htm HDOP AND GPS HORIZONTAL POSITION ERRORS] | |||
{{refend}} | |||
== External links == | |||
* Article on DOP and Trimble's program: [http://freegeographytools.com/2007/determining-local-gps-satellite-geometry-effects-on-position-accuracy Determining Local GPS Satellite Geometry Effects On Position Accuracy]. | |||
* Notes & [[GIF]] image on manually calculating GDOP: [http://www.colorado.edu/geography/gcraft/notes/gps/gps.html#Gdop Geographer's Craft] | |||
* GPS Errors & Estimating Your Receiver's Accuracy: [http://www.edu-observatory.org/gps/gps_accuracy.html Sam Wormley's GPS Accuracy Web Page] | |||
{{DEFAULTSORT:Dilution Of Precision (Gps)}} | |||
[[Category:Global Positioning System]] | |||
[[Category:Satellite navigation systems]] |
Revision as of 16:48, 13 March 2013
Dilution of precision (DOP), or geometric dilution of precision (GDOP), is a term used in satellite navigation and geomatics engineering to specify the additional multiplicative effect of navigation satellite geometry on positional measurement precision.
Introduction
The concept of dilution of precision (DOP) originated with users of the Loran-C navigation system.[1] The idea of Geometric DOP is to state how errors in the measurement will affect the final state estimation. This can be defined as:[2]
Conceptually you can imagine errors on a measurement resulting in the term changing. Ideally small changes in the measured data will not result in large changes in output location, as such a result would indicate the solution is very sensitive to errors. The interpretation of this formula is shown in the figure to the right, showing two possible scenarios with acceptable and poor GDOP.
More recently, the term has come into much wider usage with the development and adoption of GPS. Neglecting ionospheric [3] and tropospheric[4] effects, the signal from navigation satellites has a fixed precision. Therefore, the relative satellite-receiver geometry plays a major role in determining the precision of estimated positions and times. Due to the relative geometry of any given satellite to a receiver, the precision in the pseudorange of the satellite translates to a corresponding component in each of the four dimensions of position measured by the receiver (i.e., , , , and ). The precision of multiple satellites in view of a receiver combine according to the relative position of the satellites to determine the level of precision in each dimension of the receiver measurement. When visible navigation satellites are close together in the sky, the geometry is said to be weak and the DOP value is high; when far apart, the geometry is strong and the DOP value is low. Consider two overlapping rings, or annuli, of different centres. If they overlap at right angles, the greatest extent of the overlap is much smaller than if they overlap in near parallel. Thus a low DOP value represents a better positional precision due to the wider angular separation between the satellites used to calculate a unit's position. Other factors that can increase the effective DOP are obstructions such as nearby mountains or buildings. DOP can be expressed as a number of separate measurements. HDOP, VDOP, PDOP, and TDOP are respectively Horizontal, Vertical, Positional (3D), and Time Dilution of Precision. They follow mathematically from the positions of the usable satellites. Signal receivers allow the display of these positions (skyplot) as well as the DOP values.
The term can also be applied to other location systems that employ several geographical spaced sites. It can occur in electronic-counter-counter-measures (electronic warfare) when computing the location of enemy emitters (radar jammers and radio communications devices). Using such an interferometry technique can provide certain geometric layout where there are degrees of freedom that cannot be accounted for due to inadequate configurations.
The effect of geometry of the satellites on position error is called geometric dilution of precision and it is roughly interpreted as ratio of position error to the range error. Imagine that a square pyramid is formed by lines joining four satellites with the receiver at the tip of the pyramid. The larger the volume of the pyramid, the better (lower) the value of GDOP; the smaller its volume, the worse (higher) the value of GDOP will be. Similarly, the greater the number of satellites, the better the value of GDOP.
Meaning of DOP Values
DOP Value | Rating | Description |
---|---|---|
1 | Ideal | This is the highest possible confidence level to be used for applications demanding the highest possible precision at all times. |
1-2 | Excellent | At this confidence level, positional measurements are considered accurate enough to meet all but the most sensitive applications. |
2-5 | Good | Represents a level that marks the minimum appropriate for making business decisions. Positional measurements could be used to make reliable in-route navigation suggestions to the user. |
5-10 | Moderate | Positional measurements could be used for calculations, but the fix quality could still be improved. A more open view of the sky is recommended. |
10-20 | Fair | Represents a low confidence level. Positional measurements should be discarded or used only to indicate a very rough estimate of the current location. |
>20 | Poor | At this level, measurements are inaccurate by as much as 300 meters with a 6 meter accurate device (50 DOP × 6 meters) and should be discarded. |
The DOP factors are functions of the diagonal elements of the covariance matrix of the parameters, expressed either in a global or a local geodetic frame.
Computation of DOP Values
As a first step in computing DOP, consider the unit vectors from the receiver to satellite i: where and where and denote the position of the receiver and and denote the position of satellite i. Formulate the matrix, A, as:
The first three elements of each row of A are the components of a unit vector from the receiver to the indicated satellite. If the elements in the fourth column are c which denotes the speed of light then the factor (time dilution) is always 1. If the elements in the fourth column are -1 then the factor is calculated properly.[5] Formulate the matrix, Q, as:
This computation is in accordance with Section 1.4.2 of Principles of Satellite Positioning where the weighting matrix, P, has been set to the identity matrix.
The elements of Q are designated as:
PDOP, TDOP and GDOP are given by:
in agreement with Section 1.4.9 of Principles of Satellite Positioning.
The horizontal dilution of precision, , and the vertical dilution of precision, , are both dependent on the coordinate system used. To correspond to the local horizon plane and the local vertical, x, y, and z should denote positions in either a north, east, down coordinate system or a south, east, up coordinate system.
References
Notes
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
General
- DOP Factors
- Writing Your Own GPS Applications: Part 2 - Causes of Precision Error
- manually calculating GDOP
- HDOP AND GPS HORIZONTAL POSITION ERRORS
External links
- Article on DOP and Trimble's program: Determining Local GPS Satellite Geometry Effects On Position Accuracy.
- Notes & GIF image on manually calculating GDOP: Geographer's Craft
- GPS Errors & Estimating Your Receiver's Accuracy: Sam Wormley's GPS Accuracy Web Page
- ↑ Template:Cite web
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ Template:Cite web
- ↑ GPS errors (Trimble tutorial)
- ↑ http://www.colorado.edu/geography/gcraft/notes/gps/gif/gdop.gif