Combining pairs of signals and clock definition - Revision history

Claudia.Prajanu at 20:35, 23 September 2018

2018-09-23T20:35:11Z

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	The DCBs of the satellites are broadcast in the GNSS navigation messages as the TGD, (see [[Instrumental Delay]]). For instance, <math>TGD_{P1}=-\tilde{\alpha}_{1} K_{P21}^s</math> is broadcast in the legacy GPS message for the <math>P2-P1</math> code interfrequency bias (see [[GPS Signal Plan]]), and the DCBs between the codes on frequencies <math>E1,E5a</math>, and on frequencies <math>E5b,E1</math>, are broadcast in the Galileo F/NAV and I/NAV messages, respectively (see article [[GALILEO Signal Plan]]). IGS also provides DCBs for the GPS satellites and reference station receivers. These products match the IONEX TEC maps <ref group="footnotes"> ~~See, for instance, ftp://cddisa.gsfc.nasa.gov/gps/products/ionex.~~ Notice that <math>TGD_{P1_{[GPS,broadcast]}} \longrightarrow -\tilde{\alpha}_1\,DBC_{IONEX}</math>. On the other hand, they can be aligned to different references, and a global bias can appear.</ref> and the precise satellite orbits and clocks (see article [[Precise GNSS Satellite Coordinates Computation]]), which always refer to the code ionosphere-free combination <math>R_{_{PC}}</math>.		The DCBs of the satellites are broadcast in the GNSS navigation messages as the TGD, (see [[Instrumental Delay]]). For instance, <math>TGD_{P1}=-\tilde{\alpha}_{1} K_{P21}^s</math> is broadcast in the legacy GPS message for the <math>P2-P1</math> code interfrequency bias (see [[GPS Signal Plan]]), and the DCBs between the codes on frequencies <math>E1,E5a</math>, and on frequencies <math>E5b,E1</math>, are broadcast in the Galileo F/NAV and I/NAV messages, respectively (see article [[GALILEO Signal Plan]]). IGS also provides DCBs for the GPS satellites and reference station receivers. These products match the IONEX TEC maps <ref group="footnotes"> Notice that <math>TGD_{P1_{[GPS,broadcast]}} \longrightarrow -\tilde{\alpha}_1\,DBC_{IONEX}</math>. On the other hand, they can be aligned to different references, and a global bias can appear.</ref> and the precise satellite orbits and clocks (see article [[Precise GNSS Satellite Coordinates Computation]]), which always refer to the code ionosphere-free combination <math>R_{_{PC}}</math>.

Jaume.Sanz at 20:01, 4 February 2013

2013-02-04T20:01:01Z

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	</math>		</math>

	Once the <math>I+K_{21}</math> sum is computed, the DCBs (<math>K_{21}</math>) can be ~~only~~ decorrelated from the ionospheric refraction (<math>I</math>) using a geometrical description of the ionosphere, i.e., describing the satellite-receiver ray ionospheric sounding, due to the satellite motion along its path. In this way, the accuracy of the ionosphere and DCBs determinations mostly relies on the geometrical model used to describe the ionosphere (i.e., one layer model or two layers voxel model [Hernandez-Pajares et al., 1999]).		Once the <math>I+K_{21}</math> sum is computed, the DCBs (<math>K_{21}</math>) can be decorrelated from the ionospheric refraction (<math>I</math>) using a geometrical description of the ionosphere, i.e., describing the satellite-receiver ray ionospheric sounding, due to the satellite motion along its path. In this way, the accuracy of the ionosphere and DCBs determinations mostly relies on the geometrical model used to describe the ionosphere (i.e., one layer model or two layers voxel model [Hernandez-Pajares et al., 1999]).

	==References==		==References==

Jaume.Sanz at 09:52, 17 January 2013

2013-01-17T09:52:19Z

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	the following expressions are found (see article [[Combination of GNSS Measurements]]) :		the following expressions are found (see article [[Combination of GNSS Measurements]]) :

			Remark: The Antenna Phase Centre (APC) effect is neglected here for simplicity (see article [[Antenna Phase Centre]]).

	::[[File: Comb_Pairs_Sign_Fig_1.png\|none\|640px]]		::[[File: Comb_Pairs_Sign_Fig_1.png\|none\|640px]]

Jaume.Sanz at 11:11, 13 January 2013

2013-01-13T11:11:38Z

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	Notice that, as with the code instrumental delay, the carrier phase bias <math>b</math> can be split in two different terms, associated to the receiver and to the satellite. That is:		Notice that, as with the code instrumental delay, the carrier phase bias <math>b</math> can be split in two different terms, associated to the receiver and to the satellite. That is:
	<math>b_1=b_{1,r}-b_1^s</math>.		<math>b_1=b_{1,r}-b_1^s</math>.

			'''Comment:''' At first glance it could be surprising to see the ‘code DCBs’ (i.e. <math>K_{21}</math> ) included in the equation of carrier phase measurements, joining the ionospheric term. Nevertheless, as will be shown next, it provides closed expressions for the different combinations of measurements, where the ionospheric term always joins the DCB as ‘<math>I +K_{21}</math> ’, with a frequency- dependent scale factor, see the equations below. In this way, combinations removing the ionosphere, such as MW, will also be free from DCBs. On the other hand, as the carrier measurement contains an unknown bias, it can be redefined without loss of generality to include such a term. See remarks and additional comments below.

Jaume.Sanz at 10:50, 13 January 2013

2013-01-13T10:50:28Z

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	:* Notice that the wind-up term does not appear in the wide-lane carrier phase combination. This is because the signals are subtracted in cycles, see equation (2). That is, <math>\~~Phi_C~~=\frac{f_1 \Phi_1 -f_2 \Phi_2}{f_1 -f_2}=c(\frac{\~~Phi_1}{\lambda_1}~~-\~~frac{\Phi_2~~}{~~\lambda_2~~}~~)=c(\phi_1-\phi_2)~~</math>, and both signals are affected by the same fraction of cycle by the wind-up.		:* Notice that the wind-up term does not appear in the wide-lane carrier phase combination. This is because the signals are subtracted in cycles, see equation (2). That is, <math>\Phi_W=\frac{f_1 \Phi_1 -f_2 \Phi_2}{f_1 -f_2}=c\,\frac{\phi_1-\phi_2}{f_1-f_2}</math>, and both signals are affected by the same fraction of cycle by the wind-up.

	:* The GRAPHIC combination [Yunck, 1993] <ref>[Yunck, 1993] Yunck, T., 1993. Coping with the atmosphere and ionosphere in precise satellite and ground positioning. Environmental E_ects on Spacecraft Trajectories and Positioning, AGU Monograph.</ref> provides a ionosphere-free single frequency measurement with reduced noise (half than the code noise, see [[Receiver noise]]), but it contains the unknown ambiguity of the carrier phase. This combination is used, for instance, for GPS single-frequency orbit determination for Low Earth Orbiting (LEO) satellites, see for instance, [Montenbruck and Ramos-Bosch, 2007] <ref>[Montenbruck and Ramos-Bosch, 2007] Montenbruck, O. and Ramos-Bosch, P., 2007. Precision real-time navigation of LEO satellites using global positioning system measurements. GPS Solutions. 12(3), pp. 187-198.</ref>, [Hernandez-Pajares et al., 1999] <ref> [Hernandez-Pajares et al., 1999] Hernandez-Pajares, M., Juan, J., Sanz, J. and		:* The GRAPHIC combination [Yunck, 1993] <ref>[Yunck, 1993] Yunck, T., 1993. Coping with the atmosphere and ionosphere in precise satellite and ground positioning. Environmental E_ects on Spacecraft Trajectories and Positioning, AGU Monograph.</ref> provides a ionosphere-free single frequency measurement with reduced noise (half than the code noise, see [[Receiver noise]]), but it contains the unknown ambiguity of the carrier phase. This combination is used, for instance, for GPS single-frequency orbit determination for Low Earth Orbiting (LEO) satellites, see for instance, [Montenbruck and Ramos-Bosch, 2007] <ref>[Montenbruck and Ramos-Bosch, 2007] Montenbruck, O. and Ramos-Bosch, P., 2007. Precision real-time navigation of LEO satellites using global positioning system measurements. GPS Solutions. 12(3), pp. 187-198.</ref>, [Hernandez-Pajares et al., 1999] <ref> [Hernandez-Pajares et al., 1999] Hernandez-Pajares, M., Juan, J., Sanz, J. and

Carlos.Lopez at 11:26, 23 February 2012

2012-02-23T11:26:08Z

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	\|Level=Advanced		\|Level=Advanced
	\|YearOfPublication=2011		\|YearOfPublication=2011
	~~\|Logo=gAGE~~
	\|Title={{PAGENAME}}		\|Title={{PAGENAME}}
	}}		}}

Carlos.Lopez at 14:41, 30 January 2012

2012-01-30T14:41:07Z

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	Thanks to the (satellite and receiver) clock re-definition (from expression 3), the ionosphere free combination of code measurements is free from DCBs (either for satellite or receivers). But this is not the case for the carrier phase measurements, which still contain the real bias <math>b_C=b_{C,rec}+b_C^{sat}</math> because the clocks have been referred to the code measurements.		Thanks to the (satellite and receiver) clock re-definition (from expression 3), the ionosphere free combination of code measurements is free from DCBs (either for satellite or receivers). But this is not the case for the carrier phase measurements, which still contain the real bias <math>b_C=b_{C,rec}+b_C^{sat}</math> because the clocks have been referred to the code measurements.

	Nevertheless, this carrier phase bias <math>b_C</math> is not a problem for carrier-phase based positioning techniques like the Precise Point Positioning (PPP) presented in article [[Code and Carrier ~~based positioning~~ (PPP)]], because it is assimilated in the unknown ambiguity when it is estimated by the [[Kalman Filter]] as a real number, i.e., floating the ambiguities.		Nevertheless, this carrier phase bias <math>b_C</math> is not a problem for carrier-phase based positioning techniques like the Precise Point Positioning (PPP) presented in article [[Code and Carrier Based Positioning (PPP)]], because it is assimilated in the unknown ambiguity when it is estimated by the [[Kalman Filter]] as a real number, i.e., floating the ambiguities.

	It neither affects the ambiguity fixing in differential positioning techniques like Real Time Kinematics (RTK), because any remaining bias cancels out when forming the double differences between satellites and receivers <math>\nabla \Delta</math>, remaining only the integer part of the ambiguities (<math>\nabla \Delta B_i= \nabla \Delta N_i</math>, <math>i=1,2</math>), see article [[Carrier Phase Ambiguity Fixing\| Double differenced ambiguity fixing]]). Nevertheless, this bias (fractional part of the ambiguities) must be considered if the ambiguities are tried to be fixed in undifferenced mode, i.e., [[Carrier Phase Ambiguity Fixing\|Undifferenced Ambiguity Fixing]].		It neither affects the ambiguity fixing in differential positioning techniques like Real Time Kinematics (RTK), because any remaining bias cancels out when forming the double differences between satellites and receivers <math>\nabla \Delta</math>, remaining only the integer part of the ambiguities (<math>\nabla \Delta B_i= \nabla \Delta N_i</math>, <math>i=1,2</math>), see article [[Carrier Phase Ambiguity Fixing\| Double differenced ambiguity fixing]]). Nevertheless, this bias (fractional part of the ambiguities) must be considered if the ambiguities are tried to be fixed in undifferenced mode, i.e., [[Carrier Phase Ambiguity Fixing\|Undifferenced Ambiguity Fixing]].

Jaume.Sanz at 11:31, 16 January 2012

2012-01-16T11:31:05Z

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	the following expressions are found (see ~~the~~ article [[Combination of GNSS Measurements]]) :		the following expressions are found (see article [[Combination of GNSS Measurements]]) :

	::[[File: Comb_Pairs_Sign_Fig_1.png\|none\|640px]]		::[[File: Comb_Pairs_Sign_Fig_1.png\|none\|640px]]

Jaume.Sanz at 11:30, 16 January 2012

2012-01-16T11:30:21Z

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	{{Article Infobox2		{{Article Infobox2
	\|Category=Fundamentals		\|Category=Fundamentals
	~~\|Title={{PAGENAME}}~~		\|Authors=J. Sanz Subirana, J.M. Juan Zornoza and M. Hernández-Pajares, Technical University of Catalonia, Spain.
	\|Authors= J. Sanz Subirana, JM. Juan Zornoza and M. ~~Hernandez~~-Pajares, University of ~~Catalunia~~, Spain.
	\|Level=Advanced		\|Level=Advanced
	\|YearOfPublication=2011		\|YearOfPublication=2011
	\|Logo=gAGE		\|Logo=gAGE
			\|Title={{PAGENAME}}
	}}		}}

	Before starting to apply the equations to the code and phase measurements, some arrangement will be done to the expressions		Before starting to apply the equations to the code and phase measurements, some arrangement will be done to the expressions

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	in the article [[Combination of GNSS Measurements]]~~, the following expressions are found~~:		the following expressions are found (see the article [[Combination of GNSS Measurements]]) :

	::[[File: Comb_Pairs_Sign_Fig_1.png\|none\|640px]]		::[[File: Comb_Pairs_Sign_Fig_1.png\|none\|640px]]

Carlos.Lopez at 13:21, 27 October 2011

2011-10-27T13:21:43Z

← Older revision		Revision as of 13:21, 27 October 2011
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	The DCBs of the satellites are broadcast in the GNSS navigation messages as the TGD, (see [[Instrumental Delay]]). For instance, <math>TGD_{P1}=-\tilde{\alpha}_{1} K_{P21}^s</math> is broadcast in the legacy GPS message for the <math>P2-P1</math> code interfrequency bias (see [[GPS Signal Plan]]), and the DCBs between the codes on frequencies <math>E1,E5a</math>, and on frequencies <math>E5b,E1</math>, are broadcast in the Galileo F/NAV and I/NAV messages, respectively (see article [[~~Galileo~~ Signal Plan]]). IGS also provides DCBs for the GPS satellites and reference station receivers. These products match the IONEX TEC maps <ref group="footnotes"> See, for instance, ftp://cddisa.gsfc.nasa.gov/gps/products/ionex. Notice that <math>TGD_{P1_{[GPS,broadcast]}} \longrightarrow -\tilde{\alpha}_1\,DBC_{IONEX}</math>. On the other hand, they can be aligned to different references, and a global bias can appear.</ref> and the precise satellite orbits and clocks (see article [[Precise GNSS Satellite Coordinates Computation]]), which always refer to the code ionosphere-free combination <math>R_{_{PC}}</math>.		The DCBs of the satellites are broadcast in the GNSS navigation messages as the TGD, (see [[Instrumental Delay]]). For instance, <math>TGD_{P1}=-\tilde{\alpha}_{1} K_{P21}^s</math> is broadcast in the legacy GPS message for the <math>P2-P1</math> code interfrequency bias (see [[GPS Signal Plan]]), and the DCBs between the codes on frequencies <math>E1,E5a</math>, and on frequencies <math>E5b,E1</math>, are broadcast in the Galileo F/NAV and I/NAV messages, respectively (see article [[GALILEO Signal Plan]]). IGS also provides DCBs for the GPS satellites and reference station receivers. These products match the IONEX TEC maps <ref group="footnotes"> See, for instance, ftp://cddisa.gsfc.nasa.gov/gps/products/ionex. Notice that <math>TGD_{P1_{[GPS,broadcast]}} \longrightarrow -\tilde{\alpha}_1\,DBC_{IONEX}</math>. On the other hand, they can be aligned to different references, and a global bias can appear.</ref> and the precise satellite orbits and clocks (see article [[Precise GNSS Satellite Coordinates Computation]]), which always refer to the code ionosphere-free combination <math>R_{_{PC}}</math>.