PPP Fundamentals - Revision history

Forrest.Voight: added missing plus sign

2021-12-18T15:44:45Z

added missing plus sign

← Older revision		Revision as of 15:44, 18 December 2021
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	*<math> \rho</math> is the geometrical range between the satellite and the receiver, computed as a function of the satellite (<math>x_{Sat}</math>,<math>y_{Sat}</math>,<math>z_{Sat}</math>) and receiver (<math>x_{Rx}</math>,<math>y_{Rx}</math>,<math>z_{Rx}</math>) coordinates as:		*<math> \rho</math> is the geometrical range between the satellite and the receiver, computed as a function of the satellite (<math>x_{Sat}</math>,<math>y_{Sat}</math>,<math>z_{Sat}</math>) and receiver (<math>x_{Rx}</math>,<math>y_{Rx}</math>,<math>z_{Rx}</math>) coordinates as:
	::<math> \rho = \sqrt{(x_{Sat} - x_{Rx})^2 + (y_{Sat} - y_{Rx})^2 (z_{Sat} - z_{Rx})^2}</math>		::<math> \rho = \sqrt{(x_{Sat} - x_{Rx})^2 + (y_{Sat} - y_{Rx})^2 + (z_{Sat} - z_{Rx})^2}</math>

	*<math>b_{Rx}</math> is the receiver clock offset from the reference GNSS time;		*<math>b_{Rx}</math> is the receiver clock offset from the reference GNSS time;

Gema.Cueto: /* PPP Algorithm */

2020-06-15T10:36:05Z

PPP Algorithm

← Older revision		Revision as of 10:36, 15 June 2020
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	The above observations coming from all the satellites in view are processed together by adjustment and filtering algorithms that solve for the different unknowns, namely the receiver coordinates, the receiver clock, the Zenith Tropospheric Delay (ZTD) and the initial phase ambiguities.		The above observations coming from all the satellites in view are processed together by adjustment and filtering algorithms that solve for the different unknowns, namely the receiver coordinates, the receiver clock, the Zenith Tropospheric Delay (ZTD) and the initial phase ambiguities.

	Most [[PPP Standards\|implementations of PPP algorithms]] ~~use~~ a ~~sequential~~ filter ~~in which~~ the process noise for the coordinates is adjusted depending on the receiver dynamics, the time evolution of the clock is more or less unconstrained (white noise with a high sigma), and the process noise for the tropospheric delay is adjusted to standard tropospheric activity. ~~In the case of~~ phase ambiguities, they are considered as a constant per pass.		Most [[PPP Standards\|implementations of PPP algorithms]] are based on the sequential filtering of the available pseudorange and carrier-phase observables (e.g., by a least-square or a Kalman filter approach). In this case, the process noise for the coordinates is adjusted depending on the receiver dynamics, the time evolution of the clock is more or less unconstrained (white noise with a high sigma), and the process noise for the tropospheric delay is adjusted to standard tropospheric activity. Concerning phase ambiguities, they are considered as a constant per pass.

	Other implementations feature a batch algorithm instead, and therefore no process noise has to be modeled. In this case, the receiver clock offset is estimated at every measurement epoch, the coordinates are adjusted for all the observation interval (static mode) or per epoch (kinematic mode), the troposphere is estimated at regular fixed intervals and the ambiguities are also estimated per pass.		Other implementations feature a batch algorithm instead, and therefore no process noise has to be modeled. In this case, the receiver clock offset is estimated at every measurement epoch, the coordinates are adjusted for all the observation interval (static mode) or per epoch (kinematic mode), the troposphere is estimated at regular fixed intervals and the ambiguities are also estimated per pass. PPP algorithms based on batch adjustment are generally less efficient than those based on sequential adjustment, as they have to process a larger amount of data (especially for a large number of un-differenced observations).

	==PPP Correction Models and Performance Driving Factors==		==PPP Correction Models and Performance Driving Factors==

Gema.Cueto: /* PPP Algorithm */

2020-06-15T10:32:26Z

PPP Algorithm

← Older revision		Revision as of 10:32, 15 June 2020
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	::<math>l_P = (f_A^2 l_{P,A} - f_B^2 l_{P,B}) / (f_A^2 - f_B^2)</math>		::<math>l_P = (f_A^2 l_{P,A} - f_B^2 l_{P,B}) / (f_A^2 - f_B^2)</math>

	*<math>l_\phi</math> is the ionosphere-free combination of <math>f_A</math> and <math>f_B</math> carrier-phase measurements <math>l(ϕ,A)</math> and <math>l(ϕ,B)</math>, respectively, as:		*<math>l_\phi</math> is the ionosphere-free combination of <math>f_A</math> and <math>f_B</math> carrier-phase measurements <math>l_{ϕ,A}</math> and <math>l_{ϕ,B}</math>, respectively, as:
	::<math>l_\phi = (f_A^2 l_{\phi,A} - f_B^2 l_{\phi,B}) / (f_A^2 - f_B^2)</math>		::<math>l_\phi = (f_A^2 l_{\phi,A} - f_B^2 l_{\phi,B}) / (f_A^2 - f_B^2)</math>

Gema.Cueto: /* PPP Algorithm */

2020-06-15T06:20:16Z

PPP Algorithm

← Older revision		Revision as of 06:20, 15 June 2020
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	==PPP Algorithm==		==PPP Algorithm==
	The PPP algorithm uses as input both code and phase observations from a dual-frequency receiver. These are then combined with precise satellite orbits/clocks and with additional very accurate error models, in order to calculate precise receiver coordinates and clock. The dual frequency observables are used un-differenced and combined into the so-called ionosphere-free combination, which allows for the removal of the first order effect of the ionosphere. The highlights of the algorithm are described next. Traditionally, L1 and L2 observables have been used for the PPP ionosphere-free combination in GPS. However, as modernized GPS and Galileo provide carrier-phase observables also in the L5/E5 frequency bands, the following mathematical details have been described for two generic carrier frequencies <math>f_A</math> and <math>f_B</math>. At a given epoch, and for a given satellite, the simplified observation equations are presented next:		The PPP algorithm uses as input both code and phase observations from a dual-frequency receiver. These are then combined with precise satellite orbits/clocks and with additional very accurate error models in order to calculate precise receiver coordinates and clock. The dual frequency observables are used un-differenced and combined into the so-called ionosphere-free combination, which allows for the removal of the first order effect of the ionosphere. The highlights of the algorithm are described next. Traditionally, L1 and L2 observables have been used for the PPP ionosphere-free combination in GPS. However, as modernized GPS and Galileo provide carrier-phase observables also in the L5/E5 frequency bands, the following mathematical details have been described for two generic carrier frequencies <math>f_A</math> and <math>f_B</math>. At a given epoch, and for a given satellite, the simplified observation equations are presented next:

	::<math>		::<math>

Gema.Cueto: /* PPP Correction Models and Performance Driving Factors */

2020-06-15T05:45:27Z

PPP Correction Models and Performance Driving Factors

← Older revision		Revision as of 05:45, 15 June 2020
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	In order to achieve the expected centimeter-level accuracy, PPP relies on very accurate error models that are not generally required when processing pseudorange observables for meter-level accuracy positioning. For instance, these models include an accurate estimation of the slant tropospheric delay <math>T_r</math>, which is expressed as a function of the ZTD (this parameter is actually estimated in PPP) through the use of accurate mapping functions such as the Vienna Mapping Function 1 (VMF1<ref name="Boehm">J. Boehm, et al: Troposphere mapping functions for GPS and very long baseline interferometry from European centre for medium-range weather forecasts operational analysis data, J. Geophys. Res. 111(B02406), 1–9 (2006)</ref>). In addition to that, higher-order ionospheric delay corrections need to be considered as well. The precise modeling of Earth dynamics, which causes variations of the static receiver coordinates with respect to the terrestrial reference frame, is normally based on the [http://www.iers.org/ International Earth Rotation and Reference Systems Service(IERS)] recommendations <ref name="IERS">[https://www.iers.org/IERS/EN/Home/home_node.html International Earth Rotation and Reference Systems Service (IERS)]</ref>. For the required PPP precision, the latter should also comprise the effect of solid Earth tides, ocean loading and Earth rotation. Furthermore, the modeling of the observables needs to include the offset between the antenna phase center and the satellite center of mass, as well as the so-called phase wind-up at the receiver. Finally, differential code biases corrections also need to be taken into consideration in the observation model (see [[PPP Standards]]).		In order to achieve the expected centimeter-level accuracy, PPP relies on very accurate error models that are not generally required when processing pseudorange observables for meter-level accuracy positioning. For instance, these models include an accurate estimation of the slant tropospheric delay <math>T_r</math>, which is expressed as a function of the ZTD (this parameter is actually estimated in PPP) through the use of accurate mapping functions such as the Vienna Mapping Function 1 (VMF1<ref name="Boehm">J. Boehm, et al: Troposphere mapping functions for GPS and very long baseline interferometry from European centre for medium-range weather forecasts operational analysis data, J. Geophys. Res. 111(B02406), 1–9 (2006)</ref>). In addition to that, higher-order ionospheric delay corrections need to be considered as well. The precise modeling of Earth dynamics, which causes variations of the static receiver coordinates with respect to the terrestrial reference frame, is normally based on the [http://www.iers.org/ International Earth Rotation and Reference Systems Service(IERS)] recommendations <ref name="IERS">[https://www.iers.org/IERS/EN/Home/home_node.html International Earth Rotation and Reference Systems Service (IERS)]</ref>. For the required PPP precision, the latter should also comprise the effect of solid Earth tides, ocean loading and Earth rotation. Furthermore, the modeling of the observables needs to include the offset between the antenna phase center and the satellite center of mass, as well as the so-called phase wind-up at the receiver. Finally, differential code biases corrections also need to be taken into consideration in the observation model (see [[PPP Standards]]).

	~~[[File:PPP_Performances.PNG\|400px\|left\|thumb]]~~

	The accuracy of the satellite clocks and orbits is one of the most important factors affecting the quality of the PPP Solution. Traditionally, the International GNSS Service (IGS) products were used due to their high accuracy, mainly for post-processed PPP solutions <ref name="IGS">[http://www.igs.org/ International GNSS Service (IGS)]</ref>. Nowadays, there exists a variety of providers achieving similar accuracy performance for both post-processed and real-time PPP implementations (see [[PPP Systems]]).		The accuracy of the satellite clocks and orbits is one of the most important factors affecting the quality of the PPP Solution. Traditionally, the International GNSS Service (IGS) products were used due to their high accuracy, mainly for post-processed PPP solutions <ref name="IGS">[http://www.igs.org/ International GNSS Service (IGS)]</ref>. Nowadays, there exists a variety of providers achieving similar accuracy performance for both post-processed and real-time PPP implementations (see [[PPP Systems]]).

Gema.Cueto: /* PPP Correction Models and Performance Driving Factors */

2020-06-15T05:44:57Z

PPP Correction Models and Performance Driving Factors

← Older revision		Revision as of 05:44, 15 June 2020
Line 51:		Line 51:
	==PPP Correction Models and Performance Driving Factors==		==PPP Correction Models and Performance Driving Factors==

	In order to achieve the expected centimeter-level accuracy, PPP relies on very accurate error models that are not generally required when processing pseudorange observables for meter-level accuracy positioning. For instance, these models include an accurate estimation of the slant tropospheric delay <math>T_r</math>, which is expressed as a function of the ZTD (this parameter is actually estimated in PPP) through the use of accurate mapping functions such as the Vienna Mapping Function 1 (VMF1, <ref name="Boehm">J. Boehm, et al: Troposphere mapping functions for GPS and very long baseline interferometry from European centre for medium-range weather forecasts operational analysis data, J. Geophys. Res. 111(B02406), 1–9 (2006)</ref>). In addition to that, higher-order ionospheric delay corrections need to be considered as well. The precise modeling of Earth dynamics, which causes variations of the static receiver coordinates with respect to the terrestrial reference frame, is normally based on the [http://www.iers.org/ International Earth Rotation and Reference Systems Service(IERS)] recommendations <ref name="IERS">[https://www.iers.org/IERS/EN/Home/home_node.html International Earth Rotation and Reference Systems Service (IERS)]</ref>. For the required PPP precision, the latter should also comprise the effect of solid Earth tides, ocean loading and Earth rotation. Furthermore, the modeling of the observables needs to include the offset between the antenna phase center and the satellite center of mass, as well as the so-called phase wind-up at the receiver. Finally, differential code biases corrections also need to be taken into consideration in the observation model (see [[PPP Standards]]).		In order to achieve the expected centimeter-level accuracy, PPP relies on very accurate error models that are not generally required when processing pseudorange observables for meter-level accuracy positioning. For instance, these models include an accurate estimation of the slant tropospheric delay <math>T_r</math>, which is expressed as a function of the ZTD (this parameter is actually estimated in PPP) through the use of accurate mapping functions such as the Vienna Mapping Function 1 (VMF1<ref name="Boehm">J. Boehm, et al: Troposphere mapping functions for GPS and very long baseline interferometry from European centre for medium-range weather forecasts operational analysis data, J. Geophys. Res. 111(B02406), 1–9 (2006)</ref>). In addition to that, higher-order ionospheric delay corrections need to be considered as well. The precise modeling of Earth dynamics, which causes variations of the static receiver coordinates with respect to the terrestrial reference frame, is normally based on the [http://www.iers.org/ International Earth Rotation and Reference Systems Service(IERS)] recommendations <ref name="IERS">[https://www.iers.org/IERS/EN/Home/home_node.html International Earth Rotation and Reference Systems Service (IERS)]</ref>. For the required PPP precision, the latter should also comprise the effect of solid Earth tides, ocean loading and Earth rotation. Furthermore, the modeling of the observables needs to include the offset between the antenna phase center and the satellite center of mass, as well as the so-called phase wind-up at the receiver. Finally, differential code biases corrections also need to be taken into consideration in the observation model (see [[PPP Standards]]).

	[[File:PPP_Performances.PNG\|400px\|left\|thumb]]		[[File:PPP_Performances.PNG\|400px\|left\|thumb]]

Gema.Cueto: /* PPP Algorithm */

2020-06-15T05:43:51Z

PPP Algorithm

← Older revision		Revision as of 05:43, 15 June 2020
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	::<math>l_\phi = (f_A^2 l_{\phi,A} - f_B^2 l_{\phi,B}) / (f_A^2 - f_B^2)</math>		::<math>l_\phi = (f_A^2 l_{\phi,A} - f_B^2 l_{\phi,B}) / (f_A^2 - f_B^2)</math>

	*<math> \rho</math> is the geometrical range between the satellite and the receiver, computed as a function of the satellite (<math>x_{Sat}</math>,<math>y_{Sat}</math>,<math>z_{Sat}</math>) and receiver (<math>~~x_Rx~~</math>,<math>~~y_Rx~~</math>,<math>~~z_Rx~~</math>) coordinates as:		*<math> \rho</math> is the geometrical range between the satellite and the receiver, computed as a function of the satellite (<math>x_{Sat}</math>,<math>y_{Sat}</math>,<math>z_{Sat}</math>) and receiver (<math>x_{Rx}</math>,<math>y_{Rx}</math>,<math>z_{Rx}</math>) coordinates as:
	::<math> \rho = \sqrt{(x_{Sat} - x_{Rx})^2 + (y_{Sat} - y_{Rx})^2 (z_{Sat} - z_{Rx})^2}</math>		::<math> \rho = \sqrt{(x_{Sat} - x_{Rx})^2 + (y_{Sat} - y_{Rx})^2 (z_{Sat} - z_{Rx})^2}</math>

Gema.Cueto: /* PPP Algorithm */

2020-06-15T05:43:10Z

PPP Algorithm

← Older revision		Revision as of 05:43, 15 June 2020
Line 26:		Line 26:
	::<math>l_\phi = (f_A^2 l_{\phi,A} - f_B^2 l_{\phi,B}) / (f_A^2 - f_B^2)</math>		::<math>l_\phi = (f_A^2 l_{\phi,A} - f_B^2 l_{\phi,B}) / (f_A^2 - f_B^2)</math>

	*<math> \rho</math> is the geometrical range between the satellite and the receiver, computed as a function of the satellite (<math>x_{Sat}</math>,<math>y_{Sat}</math>,<math>z_{Sat}</math>) and receiver (x_Rx,y_Rx,z_Rx) coordinates as:		*<math> \rho</math> is the geometrical range between the satellite and the receiver, computed as a function of the satellite (<math>x_{Sat}</math>,<math>y_{Sat}</math>,<math>z_{Sat}</math>) and receiver (<math>x_Rx</math>,<math>y_Rx</math>,<math>z_Rx</math>) coordinates as:
	::<math> \rho = \sqrt{(x_{Sat} - x_{Rx})^2 + (y_{Sat} - y_{Rx})^2 (z_{Sat} - z_{Rx})^2}</math>		::<math> \rho = \sqrt{(x_{Sat} - x_{Rx})^2 + (y_{Sat} - y_{Rx})^2 (z_{Sat} - z_{Rx})^2}</math>

Gema.Cueto: /* PPP Algorithm */

2020-06-15T05:41:46Z

PPP Algorithm

← Older revision		Revision as of 05:41, 15 June 2020
Line 23:		Line 23:
	::<math>l_P = (f_A^2 l_{P,A} - f_B^2 l_{P,B}) / (f_A^2 - f_B^2)</math>		::<math>l_P = (f_A^2 l_{P,A} - f_B^2 l_{P,B}) / (f_A^2 - f_B^2)</math>

	*<math>l_\phi</math> is the ionosphere-free combination of <math>f_A</math> and <math>f_B</math> carrier-phase measurements <math>l(_ϕ,A)</math> and <math>l_(ϕ,B)</math>, respectively, as:		*<math>l_\phi</math> is the ionosphere-free combination of <math>f_A</math> and <math>f_B</math> carrier-phase measurements <math>l(ϕ,A)</math> and <math>l(ϕ,B)</math>, respectively, as:
	::<math>l_\phi = (f_A^2 l_{\phi,A} - f_B^2 l_{\phi,B}) / (f_A^2 - f_B^2)</math>		::<math>l_\phi = (f_A^2 l_{\phi,A} - f_B^2 l_{\phi,B}) / (f_A^2 - f_B^2)</math>

Gema.Cueto: /* PPP Algorithm */

2020-06-15T05:40:57Z

PPP Algorithm

← Older revision		Revision as of 05:40, 15 June 2020
Line 23:		Line 23:
	::<math>l_P = (f_A^2 l_{P,A} - f_B^2 l_{P,B}) / (f_A^2 - f_B^2)</math>		::<math>l_P = (f_A^2 l_{P,A} - f_B^2 l_{P,B}) / (f_A^2 - f_B^2)</math>

	*<math>l_\phi</math> is the ionosphere-free combination of <math>f_A</math> and <math>f_B</math> carrier-phase measurements <math>l_(ϕ,A)</math> and <math>l_(ϕ,B)</math>, respectively, as:		*<math>l_\phi</math> is the ionosphere-free combination of <math>f_A</math> and <math>f_B</math> carrier-phase measurements <math>l(_ϕ,A)</math> and <math>l_(ϕ,B)</math>, respectively, as:
	::<math>l_\phi = (f_A^2 l_{\phi,A} - f_B^2 l_{\phi,B}) / (f_A^2 - f_B^2)</math>		::<math>l_\phi = (f_A^2 l_{\phi,A} - f_B^2 l_{\phi,B}) / (f_A^2 - f_B^2)</math>