NeQuick Ionospheric Model - Revision history

Gema.Cueto at 09:27, 16 April 2026

2026-04-16T09:27:17Z

← Older revision		Revision as of 09:27, 16 April 2026
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	In order to take into account both daily variation of the solar activity and the user’s local geomagnetic condition, the NeQuick-G model is driven by and computes its correction by use of the daily solar activity information, so-called effective ionization level (<i>Az</i>) expressed in solar flux unit (<i>10<sup>-22</sup>Wm<sup>-2</sup>Hz<sup>-1</sup></i>)		In order to take into account both daily variation of the solar activity and the user’s local geomagnetic condition, the NeQuick-G model is driven by and computes its correction by use of the daily solar activity information, so-called effective ionization level (<i>Az</i>) expressed in solar flux unit (<i>10<sup>-22</sup>Wm<sup>-2</sup>Hz<sup>-1</sup></i>)

	For Galileo single-frequency users, Galileo satellites broadcast three ionospheric coefficients in their navigation message that are used to compute the <i>Az</i> as follows [Galileo - Open Service SIS-ICD, ~~2016~~] <ref>[https://www.gsc-europa.eu/~~system~~/files/~~galileo_documents~~/~~Galileo-OS-SIS-ICD~~.pdf Galileo - Open Service - Signal-In-Space Interface Control Document (~~2016). “European GNSS (Galileo~~) ~~Open Service, Signal-In-Space Interface Control Document.” Issue 1.3, December 2016.~~ ]</ref>:		For Galileo single-frequency users, Galileo satellites broadcast three ionospheric coefficients in their navigation message that are used to compute the <i>Az</i> as follows [Galileo - Open Service SIS-ICD, 2025]<ref>[https://www.gsc-europa.eu/sites/default/files/sites/all/files/Galileo_OS_SIS_ICD_v2.2.pdf Galileo - Open Service - Signal-In-Space Interface Control Document (2025)]</ref>:

	::<math>		::<math>

Roberto.Prieto.Cerdeira at 21:28, 19 September 2017

2017-09-19T21:28:14Z

← Older revision		Revision as of 21:28, 19 September 2017
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	The coefficients <math>a_0</math>, <math>a_1</math>,<math>a_2</math> will be broadcasted to the users in the Galileo navigation message and updated at least once a day <ref group="footnotes"> These parameters are determined from measured TEC data obtained by the globally distributed Galileo Sensor Stations network.</ref>.		The coefficients <math>a_0</math>, <math>a_1</math>,<math>a_2</math> will be broadcasted to the users in the Galileo navigation message and updated at least once a day <ref group="footnotes"> These parameters are determined from measured TEC data obtained by the globally distributed Galileo Sensor Stations network.</ref>.

	::[[File: NeQuickIonoVTECmap.jpeg \|400px\|thumb\|right\|'''''Figure 1:''''' Global vertical Total Electron Content map generated with NeQuick G model at 13h UT for a day in ~~the month of~~ April ~~with an~~ Az=193. ]]		::[[File: NeQuickIonoVTECmap.jpeg \|400px\|thumb\|right\|'''''Figure 1:''''' Global vertical Total Electron Content map generated with NeQuick G model at 13h UT for a day in April and Az=193. ]]

	The Galileo single frequency receiver algorithm will be based in the following steps (the next paragraph is taken from [Arbesser-Rastburg, B., 2006]<ref name="Arbesser">[http://sidc.oma.be/esww3/presentations/Session4/Arbesser.pdf Arbesser-Rastburg, B., 2006. The Galileo Single Frequency Ionospheric Correction Algorithm. European Space Weather Week, Brussels]</ref>):		The Galileo single frequency receiver algorithm will be based in the following steps (the next paragraph is taken from [Arbesser-Rastburg, B., 2006]<ref name="Arbesser">[http://sidc.oma.be/esww3/presentations/Session4/Arbesser.pdf Arbesser-Rastburg, B., 2006. The Galileo Single Frequency Ionospheric Correction Algorithm. European Space Weather Week, Brussels]</ref>):

Roberto.Prieto.Cerdeira at 21:27, 19 September 2017

2017-09-19T21:27:12Z

← Older revision		Revision as of 21:27, 19 September 2017
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	The coefficients <math>a_0</math>, <math>a_1</math>,<math>a_2</math> will be broadcasted to the users in the Galileo navigation message and updated at least once a day <ref group="footnotes"> These parameters are determined from measured TEC data obtained by the globally distributed Galileo Sensor Stations network.</ref>.		The coefficients <math>a_0</math>, <math>a_1</math>,<math>a_2</math> will be broadcasted to the users in the Galileo navigation message and updated at least once a day <ref group="footnotes"> These parameters are determined from measured TEC data obtained by the globally distributed Galileo Sensor Stations network.</ref>.

			::[[File: NeQuickIonoVTECmap.jpeg \|400px\|thumb\|right\|'''''Figure 1:''''' Global vertical Total Electron Content map generated with NeQuick G model at 13h UT for a day in the month of April with an Az=193. ]]

	The Galileo single frequency receiver algorithm will be based in the following steps (the next paragraph is taken from [Arbesser-Rastburg, B., 2006]<ref name="Arbesser">[http://sidc.oma.be/esww3/presentations/Session4/Arbesser.pdf Arbesser-Rastburg, B., 2006. The Galileo Single Frequency Ionospheric Correction Algorithm. European Space Weather Week, Brussels]</ref>):		The Galileo single frequency receiver algorithm will be based in the following steps (the next paragraph is taken from [Arbesser-Rastburg, B., 2006]<ref name="Arbesser">[http://sidc.oma.be/esww3/presentations/Session4/Arbesser.pdf Arbesser-Rastburg, B., 2006. The Galileo Single Frequency Ionospheric Correction Algorithm. European Space Weather Week, Brussels]</ref>):

Roberto.Prieto.Cerdeira: New references

2017-09-19T20:21:41Z

New references

← Older revision		Revision as of 20:21, 19 September 2017
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	The coefficients <math>a_0</math>, <math>a_1</math>,<math>a_2</math> will be broadcasted to the users in the Galileo navigation message and updated at least once a day <ref group="footnotes"> These parameters are determined from measured TEC data obtained by the globally distributed Galileo Sensor Stations network.</ref>.		The coefficients <math>a_0</math>, <math>a_1</math>,<math>a_2</math> will be broadcasted to the users in the Galileo navigation message and updated at least once a day <ref group="footnotes"> These parameters are determined from measured TEC data obtained by the globally distributed Galileo Sensor Stations network.</ref>.

	The Galileo single frequency receiver algorithm will be based in the following steps (the next paragraph is taken from [Arbesser-Rastburg, B., 2006]<ref name="Arbesser">[http://sidc.oma.be/esww3/presentations/Session4/Arbesser.pdf Arbesser-Rastburg, B., 2006. The Galileo Single Frequency Ionospheric Correction Algorithm. ]</ref>):		The Galileo single frequency receiver algorithm will be based in the following steps (the next paragraph is taken from [Arbesser-Rastburg, B., 2006]<ref name="Arbesser">[http://sidc.oma.be/esww3/presentations/Session4/Arbesser.pdf Arbesser-Rastburg, B., 2006. The Galileo Single Frequency Ionospheric Correction Algorithm. European Space Weather Week, Brussels]</ref>):

	::'''1.''' <math>Az</math> is evaluated using <math>a_0</math>, <math>a_1</math>,<math>a_2</math> (from navigation message) and MODIP from the NeQuick model (which depends on estimated receiver position).		::'''1.''' <math>Az</math> is evaluated using <math>a_0</math>, <math>a_1</math>,<math>a_2</math> (from navigation message) and MODIP from the NeQuick model (which depends on estimated receiver position).
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	I_f = \frac{40.3\cdot 10^{16}}{f^2}TEC \qquad\mbox{(where f is in Hz)} \quad\mbox{(3)}</math>		I_f = \frac{40.3\cdot 10^{16}}{f^2}TEC \qquad\mbox{(where f is in Hz)} \quad\mbox{(3)}</math>

			An overview of the correction algorithm and initial performance results with broadcast parameters from Galileo satellites have been performed during [[Galileo IOV Satellites\|IOV]] in the period April 2013 to March 2014 during solar maximum of Solar Cycle 24 [Prieto-Cerdeira, Orús-Pérez et al, 2014] <ref>[http://gpsworld.com/innovation-the-european-way/ Prieto-Cerdeira, R., R. Orús-Pérez, E. Breeuwer, R. Lucas-Rodríguez and M. Falcone (2014). ‘Innovation: The European Way. Performance of the Galileo Single-Frequency Ionospheric Correction During In-Orbit Validation’. In: GPSworld 25.6, pp. 53–58. ]</ref>.

	Notice that, as with the [[Klobuchar Ionospheric Model\|Klobuchar]] model, the ionospheric corrections computed by the NeQuick G can be used for any GNSS signal (GPS, GLONASS, Galileo…) simply by setting the corresponding frequency in the equation (3).		Notice that, as with the [[Klobuchar Ionospheric Model\|Klobuchar]] model, the ionospheric corrections computed by the NeQuick G can be used for any GNSS signal (GPS, GLONASS, Galileo…) simply by setting the corresponding frequency in the equation (3).

Roberto.Prieto.Cerdeira: Differences between NeQuick G Galileo version and climatological NeQuick

2017-09-19T19:24:34Z

Differences between NeQuick G Galileo version and climatological NeQuick

← Older revision		Revision as of 19:24, 19 September 2017
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	{{Article Infobox2		{{Article Infobox2
	\|Category=Fundamentals		\|Category=Fundamentals
	\|Authors=J. Sanz Subirana, J.M. Juan Zornoza and M. Hernández-Pajares, Technical University of Catalonia, Spain.		\|Authors=J. Sanz Subirana, J.M. Juan Zornoza and M. Hernández-Pajares, Technical University of Catalonia, Spain. [Updated 2017]: R. Prieto-Cerdeira, R. Orús Pérez, European Space Agency, ESA/ESTEC.
	\|Level=Intermediate		\|Level=Intermediate
	\|YearOfPublication=2011		\|YearOfPublication=2011, updated 2017
	\|Title={{PAGENAME}}		\|Title={{PAGENAME}}
	}}		}}
	~~The~~ NeQuick G is the ionospheric model ~~proposed to be used~~ by the Galileo ~~single frequency receiver~~ to compute ionospheric corrections. It is based on the original profiler proposed by [Di Giovanni and Radicella, 1990]<ref> ~~[Di Giovanni and Radicella, 1990]~~ Di Giovanni, G. and Radicella, S. M., 1990. An analytical model of the electron density pro_le in the ionosphere. Advances in Space Research. 10(11), pp. 27-30.</ref>.		<b>NeQuick-G</b> is the ionospheric model adopted by the Galileo system in order to help to compute the ionospheric delay corrections for its single-frequency users [Galileo Open Service-Ionospheric Correction Algorithm, 2016] <ref>[http://www.gsc-europa.eu/system/files/galileo_documents/Galileo_Ionospheric_Model.pdf European Union (2016). European GNSS (Galileo) Open Service-Ionospheric Correction Algorithm for Galileo Single Frequency Users. 1.2.] </ref>. The NeQuick-G model is an adaptation for real-time users of the ITU-R NeQuick ionospheric electron density model [ITU-R, 2013] <ref> ITU-R (2013). Ionospheric propagation data and prediction methods required for the design of satellite services and systems. Recommendation ITU-R P. 531-12. </ref> based on the original profiler proposed by [Di Giovanni and Radicella, 1990]<ref> Di Giovanni, G. and Radicella, S. M., 1990. An analytical model of the electron density pro_le in the ionosphere. Advances in Space Research. 10(11), pp. 27-30.</ref> [Hochegger, Nava et al., 2000]<ref>Hochegger, G., B. Nava, S. Radicella and R. Leitinger (2000). “A family of ionospheric models for different uses”. In: Physics and Chemistry of the Earth, Part C: Solar, Terrestrial & Planetary Science 25.4, pp. 307–310. issn: 1464-1917. doi: 10.1016/ S1464-1917(00)00022-2. </ref>.NeQuick a three-dimensional and time dependent ionospheric electron density model based on an empirical climatological representation of the ionosphere, which predicts monthly mean electron density from analytical profiles, depending on solar-activity-related input values: sun spot number or solar flux, month, geographic latitude and longitude, height and Universal Time (UT).

	~~The GSA~~ has ~~recently published~~ a ~~new document on~~ the ~~NeQuick Ionospheric Model~~, ~~titled “European GNSS (Galileo) Open Service Ionospheric Correction Algorithm for Galileo Single Frequency Users~~,” and ~~downloadable~~, ~~containing detailed description~~ of ~~the algorithm~~ and ~~results from years of intense research~~. ~~<ref name="GAL_NEQUICK">[http://www.gsc-europa.eu/system/files/galileo_documents/Galileo_Ionospheric_Model.pdf Galileo Ionospheric Model]</ref>~~		NeQuick model has been adapted for real-time Galileo single-frequency ionospheric corrections (for convenience referred to as NeQuick-G) in order to derive real-time predictions based on a single input parameter, the Effective ionisation level, Az, which is determined using three coefficients broadcast in the navigation message. As a three-dimensional and time-dependent model, it has the capability to provide vertical or slant ionospheric delay correction by integrating the predicted electron density along the line of sight vector between satellite and receiver.

			In order to take into account both daily variation of the solar activity and the user’s local geomagnetic condition, the NeQuick-G model is driven by and computes its correction by use of the daily solar activity information, so-called effective ionization level (<i>Az</i>) expressed in solar flux unit (<i>10<sup>-22</sup>Wm<sup>-2</sup>Hz<sup>-1</sup></i>)

	~~NeQuick is a tridimensional and time~~-~~dependent~~ ionospheric ~~electron density model, which provides electron density~~ in ~~the ionosphere as a function of the position and time. Thence, it allows~~ to compute ~~ionospheric delays (TEC or STEC),~~ as ~~the integrated electron density along any ray path.~~		For Galileo single-frequency users, Galileo satellites broadcast three ionospheric coefficients in their navigation message that are used to compute the <i>Az</i> as follows [Galileo - Open Service SIS-ICD, 2016] <ref>[https://www.gsc-europa.eu/system/files/galileo_documents/Galileo-OS-SIS-ICD.pdf Galileo - Open Service - Signal-In-Space Interface Control Document (2016). “European GNSS (Galileo) Open Service, Signal-In-Space Interface Control Document.” Issue 1.3, December 2016. ]</ref>:


	~~NeQuick FORTRAN 77 code was accepted by the International Telecommunication Union Radiocommunication Sector (ITU~~-~~R) in 2000 and revised in 2002. It is freely available from~~ [~~http~~://www.~~itu~~.~~int~~/~~dms_pub~~/~~itu-r~~/~~oth~~/~~0A/04/R0A040000180001ZIPE~~.~~zip the ITU]. It is referred to either as version 1 or ITU~~-~~R. This package includes a comprehensive description of the implementation as well as numerical integration subroutines allowing to compute Vertical and Slant TEC.~~


	~~The input parameters of the model are the position~~ (~~longitude, latitude and height~~)~~, the epoch~~ (~~month and UT~~) ~~and the solar activity (either expressed <ref group="footnotes"> The F10.7 index is a measure of the solar activity~~, i.e., the flux level generated by the sun at at the earth's orbit at a 10.7 cm wavelength. It has been found to correlate well with the sunspot number (Rz). The sunspot number is defined from counts of the number of individual sunspots as well as the number of sunspot groups. The F10.7 index can be measured relatively easily and quickly and has replaced the sunspot number as an index of solar activity for many purposes. The F10.7 and the smoothed sunspot number R12 (12-~~months moving average) are related by <math>\mbox{R12}=\sqrt{167273~~.~~0+1123~~.6\,~~(\mbox{F10.7}-63.7)}-408~~.~~99</math>. [ITU-R recommendation~~].</ref> ~~as F10.7 or R12). Other internal parameters are the foF2 and M(3000)F2 values, which can be defined according to the ITU-R, amongst other options depending on the purpose.~~


	The NeQuick model running in the Galileo single frequency receivers will be driven by the ''Effective Ionisation Level'', <math>Az</math> parameter (replacing the solar flux), that is a function of the receiver location.

	::<math>		::<math>
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	being <math>I</math> the true magnetic inclination<ref group="footnotes"><math>I</math> is <math>0^o</math> at the magnetic equator and <math>90^o</math> at the magnetic poles.</ref>, or ''dip'' in the ionosphere (usually at 300 km), and <math>\Phi</math> the geographic latitude of the receiver [Rawer, 1963]<ref> ~~[Rawer, 1963]~~ Rawer, K., 1963. Propagation of decameter waves (HF-band), in Meteorological and Astronomical Inuences on Radio Wave Propagation. Ed. Landmark, B. Pergamon Press, New York.</ref>.		being <math>I</math> the true magnetic inclination<ref group="footnotes"><math>I</math> is <math>0^o</math> at the magnetic equator and <math>90^o</math> at the magnetic poles.</ref>, or ''dip'' in the ionosphere (usually at 300 km), and <math>\Phi</math> the geographic latitude of the receiver [Rawer, 1963]<ref> Rawer, K., 1963. Propagation of decameter waves (HF-band), in Meteorological and Astronomical Inuences on Radio Wave Propagation. Ed. Landmark, B. Pergamon Press, New York.</ref>.


	The coefficients <math>a_0</math>, <math>a_1</math>,<math>a_2</math> will be broadcasted to the users in the Galileo navigation message and updated at least once a day <ref group="footnotes"> These parameters will be determined from measured TEC data obtained during the last 24h by the globally distributed Galileo Sensor Stations network.</ref>. Ionospheric Disturbance Flag alerts for five predefined regions (see Figure 1) will be also broadcasted to aware the users when the ionospheric correction coming from the Galileo broadcast message might not meet the specified performance. These disturbance flags will be transmitted continuously and updated with the update rate of the Navigation Message (every <math>100</math> minutes).

	::[[File: NeQuick_5zones.png \|none\|thumb\|400px\|'''''Figure 1:''''' The 5 Regions defined for the Disturbance Flags (in degrees MODIP) -90 to -60, -60 to -30, -30 to 30, 30-60, 60-90. (courtesy of [Arbesser-Rastburg, B., 2006]<ref name="Arbesser"> [Arbesser-Rastburg, B., 2006] Arbesser-Rastburg, B., 2006. The Galileo Single Frequency Ionospheric Correction Algorithm. http://sidc.oma.be/esww3/presentations/Session4/Arbesser.pdf</ref>).]]

			The coefficients <math>a_0</math>, <math>a_1</math>,<math>a_2</math> will be broadcasted to the users in the Galileo navigation message and updated at least once a day <ref group="footnotes"> These parameters are determined from measured TEC data obtained by the globally distributed Galileo Sensor Stations network.</ref>.

	The Galileo single frequency receiver algorithm will be based in the following steps (the next paragraph is taken from [Arbesser-Rastburg, B., 2006]<ref name="Arbesser"/>):		The Galileo single frequency receiver algorithm will be based in the following steps (the next paragraph is taken from [Arbesser-Rastburg, B., 2006]<ref name="Arbesser">[http://sidc.oma.be/esww3/presentations/Session4/Arbesser.pdf Arbesser-Rastburg, B., 2006. The Galileo Single Frequency Ionospheric Correction Algorithm. ]</ref>):

	::'''1.''' <math>Az</math> is evaluated using <math>a_0</math>, <math>a_1</math>,<math>a_2</math> (from navigation message) and MODIP from the NeQuick model (which depends on estimated receiver position).		::'''1.''' <math>Az</math> is evaluated using <math>a_0</math>, <math>a_1</math>,<math>a_2</math> (from navigation message) and MODIP from the NeQuick model (which depends on estimated receiver position).
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	Notice that, as with the [[Klobuchar Ionospheric Model\|Klobuchar]] model, the ionospheric corrections computed by the NeQuick can be used for any GNSS signal (GPS, GLONASS, Galileo…) simply by setting the corresponding frequency in the equation (3).		Notice that, as with the [[Klobuchar Ionospheric Model\|Klobuchar]] model, the ionospheric corrections computed by the NeQuick G can be used for any GNSS signal (GPS, GLONASS, Galileo…) simply by setting the corresponding frequency in the equation (3).

Roberto.Prieto.Cerdeira: Differences between NeQuick G Galileo version and climatological NeQuick

2017-09-19T18:32:18Z

Differences between NeQuick G Galileo version and climatological NeQuick

← Older revision		Revision as of 18:32, 19 September 2017
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	\|Title={{PAGENAME}}		\|Title={{PAGENAME}}
	}}		}}
	The NeQuick is the ionospheric model proposed to be used by the Galileo single frequency receiver to compute ionospheric corrections. It is based on the original profiler proposed by [Di Giovanni and Radicella, 1990]<ref> [Di Giovanni and Radicella, 1990] Di Giovanni, G. and Radicella, S. M., 1990. An analytical model of the electron density pro_le in the ionosphere. Advances in Space Research. 10(11), pp. 27-30.</ref>.		The NeQuick G is the ionospheric model proposed to be used by the Galileo single frequency receiver to compute ionospheric corrections. It is based on the original profiler proposed by [Di Giovanni and Radicella, 1990]<ref> [Di Giovanni and Radicella, 1990] Di Giovanni, G. and Radicella, S. M., 1990. An analytical model of the electron density pro_le in the ionosphere. Advances in Space Research. 10(11), pp. 27-30.</ref>.

	The GSA has recently published a new document on the NeQuick Ionospheric Model, titled “European GNSS (Galileo) Open Service Ionospheric Correction Algorithm for Galileo Single Frequency Users,” and downloadable, containing detailed description of the algorithm and results from years of intense research. <ref name="GAL_NEQUICK">[http://www.gsc-europa.eu/system/files/galileo_documents/Galileo_Ionospheric_Model.pdf Galileo Ionospheric Model]</ref>		The GSA has recently published a new document on the NeQuick Ionospheric Model, titled “European GNSS (Galileo) Open Service Ionospheric Correction Algorithm for Galileo Single Frequency Users,” and downloadable, containing detailed description of the algorithm and results from years of intense research. <ref name="GAL_NEQUICK">[http://www.gsc-europa.eu/system/files/galileo_documents/Galileo_Ionospheric_Model.pdf Galileo Ionospheric Model]</ref>

Paulo.Pombal at 14:12, 1 June 2015

2015-06-01T14:12:30Z

← Older revision		Revision as of 14:12, 1 June 2015
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	}}		}}
	The NeQuick is the ionospheric model proposed to be used by the Galileo single frequency receiver to compute ionospheric corrections. It is based on the original profiler proposed by [Di Giovanni and Radicella, 1990]<ref> [Di Giovanni and Radicella, 1990] Di Giovanni, G. and Radicella, S. M., 1990. An analytical model of the electron density pro_le in the ionosphere. Advances in Space Research. 10(11), pp. 27-30.</ref>.		The NeQuick is the ionospheric model proposed to be used by the Galileo single frequency receiver to compute ionospheric corrections. It is based on the original profiler proposed by [Di Giovanni and Radicella, 1990]<ref> [Di Giovanni and Radicella, 1990] Di Giovanni, G. and Radicella, S. M., 1990. An analytical model of the electron density pro_le in the ionosphere. Advances in Space Research. 10(11), pp. 27-30.</ref>.

			The GSA has recently published a new document on the NeQuick Ionospheric Model, titled “European GNSS (Galileo) Open Service Ionospheric Correction Algorithm for Galileo Single Frequency Users,” and downloadable, containing detailed description of the algorithm and results from years of intense research. <ref name="GAL_NEQUICK">[http://www.gsc-europa.eu/system/files/galileo_documents/Galileo_Ionospheric_Model.pdf Galileo Ionospheric Model]</ref>

Carlos.Lopez at 11:34, 23 February 2012

2012-02-23T11:34:25Z

← Older revision		Revision as of 11:34, 23 February 2012
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	\|Level=Intermediate		\|Level=Intermediate
	\|YearOfPublication=2011		\|YearOfPublication=2011
	~~\|Logo=gAGE~~
	\|Title={{PAGENAME}}		\|Title={{PAGENAME}}
	}}		}}

Jaume.Sanz at 10:05, 21 February 2012

2012-02-21T10:05:12Z

← Older revision		Revision as of 10:05, 21 February 2012
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	::::<math>		::::<math>
	I_f = \frac{40.3\cdot 10^{16}}{f^2}TEC \qquad\mbox{(~~meters,~~ where f is in Hz)} \quad\mbox{(3)}</math>		I_f = \frac{40.3\cdot 10^{16}}{f^2}TEC \qquad\mbox{(where f is in Hz)} \quad\mbox{(3)}</math>

Jaume.Sanz at 10:04, 21 February 2012

2012-02-21T10:04:48Z

← Older revision		Revision as of 10:04, 21 February 2012
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	::::<math>		::::<math>
	I_f = \frac{40.3\cdot 10^{16}}{f^2}TEC \qquad\mbox{(meters)} \quad\mbox{(3)}</math> ~~(where f is in Hz)~~		I_f = \frac{40.3\cdot 10^{16}}{f^2}TEC \qquad\mbox{(meters, where f is in Hz)} \quad\mbox{(3)}</math>