Precise modelling terms for PPP - Revision history

Gema.Cueto at 14:34, 15 June 2020

2020-06-15T14:34:21Z

← Older revision		Revision as of 14:34, 15 June 2020
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	:The satellite coordinates between epochs can be computed by polynomial interpolation.		:The satellite coordinates between epochs can be computed by polynomial interpolation.
	:As an example of polynomial interpolation, the [https://en.wikipedia.org/wiki/Lagrange_polynomial Lagrange method] is presented as follows. Given a table of values <math>(x_i,y_i), i=1,…,n,</math> the interpolated value <math>y≃P_n(x)</math> at a given x can be computed as (see [[Precise GNSS Satellite Coordinates Computation\|Precise GNSS Satellite Coordinates Computation]]):		:As an example of polynomial interpolation, the [https://en.wikipedia.org/wiki/Lagrange_polynomial Lagrange method] is presented as follows. Given a table of values <math>(x_i,y_i), i=1,…,n,</math> the interpolated value <math>y≃P_n(x)</math> at a given <math>x</math> can be computed as (see [[Precise GNSS Satellite Coordinates Computation\|Precise GNSS Satellite Coordinates Computation]]):

	::<math>		::<math>

Gema.Cueto at 13:44, 15 June 2020

2020-06-15T13:44:46Z

← Older revision		Revision as of 13:44, 15 June 2020
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	:The satellite coordinates between epochs can be computed by polynomial interpolation.		:The satellite coordinates between epochs can be computed by polynomial interpolation.
	:As an example of polynomial interpolation, the [https://en.wikipedia.org/wiki/Lagrange_polynomial Lagrange method] is presented as follows. Given a table of values <math>(x_i,y_i), i=1,…,n,</math> the interpolated value <math>y≃P_n(x)</math> at a given x can be computed as (see Precise GNSS Satellite Coordinates Computation):		:As an example of polynomial interpolation, the [https://en.wikipedia.org/wiki/Lagrange_polynomial Lagrange method] is presented as follows. Given a table of values <math>(x_i,y_i), i=1,…,n,</math> the interpolated value <math>y≃P_n(x)</math> at a given x can be computed as (see [[Precise GNSS Satellite Coordinates Computation\|Precise GNSS Satellite Coordinates Computation]]):

	::<math>		::<math>

Gema.Cueto at 13:38, 15 June 2020

2020-06-15T13:38:22Z

← Older revision		Revision as of 13:38, 15 June 2020
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	:where the [[Mapping of Niell\|mapping of Niell]] is used<ref name="Niell">5. Niell, A., 1996. Global mapping functions for the atmosphere delay at radio wavelengths. Journal of Geophysical Research. 101, pp. 3227-3246</ref>. The dry and wet tropospheric delays are given by		:where the [[Mapping of Niell\|mapping of Niell]] is used<ref name="Niell">Niell, A., 1996. Global mapping functions for the atmosphere delay at radio wavelengths. Journal of Geophysical Research. 101, pp. 3227-3246</ref>. The dry and wet tropospheric delays are given by

	::<math>		::<math>
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	: The ''carrier phase wind-up effect''<ref name="Wu">6. Wu, J.T., S,C. Wu, G.A. Hajj, W.I. Bertiger, and S.M. Lichten, 1993, Effects of antenna orientation on GPS carrier phase, Man. Geodetica 18, pp. 91-98</ref> due to the satellite motion is given by (7) (see [[Carrier Phase Wind-up Effect]])		: The ''carrier phase wind-up effect''<ref name="Wu">Wu, J.T., S,C. Wu, G.A. Hajj, W.I. Bertiger, and S.M. Lichten, 1993, Effects of antenna orientation on GPS carrier phase, Man. Geodetica 18, pp. 91-98</ref> due to the satellite motion is given by (7) (see [[Carrier Phase Wind-up Effect]])

	::<math>		::<math>

Gema.Cueto at 13:37, 15 June 2020

2020-06-15T13:37:35Z

← Older revision		Revision as of 13:37, 15 June 2020
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	:The satellite coordinates between epochs can be computed by polynomial interpolation.		:The satellite coordinates between epochs can be computed by polynomial interpolation.
	:As an example of polynomial interpolation, the [https://en.wikipedia.org/wiki/Lagrange_polynomial \| Lagrange method] is presented as follows. Given a table of values <math>(x_i,y_i), i=1,…,n,</math> the interpolated value <math>y≃P_n(x)</math> at a given x can be computed as (see Precise GNSS Satellite Coordinates Computation):		:As an example of polynomial interpolation, the [https://en.wikipedia.org/wiki/Lagrange_polynomial Lagrange method] is presented as follows. Given a table of values <math>(x_i,y_i), i=1,…,n,</math> the interpolated value <math>y≃P_n(x)</math> at a given x can be computed as (see Precise GNSS Satellite Coordinates Computation):

	::<math>		::<math>

Gema.Cueto at 13:37, 15 June 2020

2020-06-15T13:37:14Z

← Older revision		Revision as of 13:37, 15 June 2020
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	:The satellite coordinates between epochs can be computed by polynomial interpolation.		:The satellite coordinates between epochs can be computed by polynomial interpolation.
	:As an example of polynomial interpolation, the [[https://en.wikipedia.org/wiki/Lagrange_polynomial\|Lagrange method]] is presented as follows. Given a table of values <math>(x_i,y_i), i=1,…,n,</math> the interpolated value <math>y≃P_n(x)</math> at a given x can be computed as (see Precise GNSS Satellite Coordinates Computation):		:As an example of polynomial interpolation, the [https://en.wikipedia.org/wiki/Lagrange_polynomial \| Lagrange method] is presented as follows. Given a table of values <math>(x_i,y_i), i=1,…,n,</math> the interpolated value <math>y≃P_n(x)</math> at a given x can be computed as (see Precise GNSS Satellite Coordinates Computation):

	::<math>		::<math>

Gema.Cueto at 13:36, 15 June 2020

2020-06-15T13:36:41Z

← Older revision		Revision as of 13:36, 15 June 2020
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	:The satellite coordinates between epochs can be computed by polynomial interpolation.		:The satellite coordinates between epochs can be computed by polynomial interpolation.
	:As an example of polynomial interpolation, the Lagrange method is presented as follows. Given a table of values <math>(x_i,y_i), i=1,…,n,</math> the interpolated value <math>y≃P_n(x)</math> at a given x can be computed as (see Precise GNSS Satellite Coordinates Computation):		:As an example of polynomial interpolation, the [[https://en.wikipedia.org/wiki/Lagrange_polynomial\|Lagrange method]] is presented as follows. Given a table of values <math>(x_i,y_i), i=1,…,n,</math> the interpolated value <math>y≃P_n(x)</math> at a given x can be computed as (see Precise GNSS Satellite Coordinates Computation):

	::<math>		::<math>
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	:where the [[Mapping of Niell\|mapping of Niell]] is used. The dry and wet tropospheric delays are given by		:where the [[Mapping of Niell\|mapping of Niell]] is used<ref name="Niell">5. Niell, A., 1996. Global mapping functions for the atmosphere delay at radio wavelengths. Journal of Geophysical Research. 101, pp. 3227-3246</ref>. The dry and wet tropospheric delays are given by

	::<math>		::<math>
Line 86:		Line 86:


	: The ''carrier phase wind-up effect'' due to the satellite motion is given by (7) (see [[Carrier Phase Wind-up Effect]])		: The ''carrier phase wind-up effect''<ref name="Wu">6. Wu, J.T., S,C. Wu, G.A. Hajj, W.I. Bertiger, and S.M. Lichten, 1993, Effects of antenna orientation on GPS carrier phase, Man. Geodetica 18, pp. 91-98</ref> due to the satellite motion is given by (7) (see [[Carrier Phase Wind-up Effect]])

	::<math>		::<math>

Gema.Cueto at 13:30, 15 June 2020

2020-06-15T13:30:18Z

← Older revision		Revision as of 13:30, 15 June 2020
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	The [[Precise Point Positioning\|Precise Point Positioning (PPP)]] technique allows achieving centimetre level accuracy for static positioning and decimetre level for kinematic, typically, after the best part of one hour. This high accuracy requires an accurate measurements modelling, where all model terms must be taken into account (up to the centimetre level or better).<ref name="Kouba">Kouba, Jan (et al.), Precise Point Positioning, Chapter 25, Handbook of Global Navigation Satellite Systems, 2017</ref><ref name="Hernandez">M. Hernandez-Pajares, J.M. Juan, J. Sanz, R. Orus, A. Garcia-Rigo, J. Feltens, A. Komjathy, S.C. Schaer, A. Krankowski: The IGS VTEC maps: A reliable source of ionospheric information since 1998, J. Geodesy 83(3/4), 263–275 (2009)</ref><ref name="King">M.A. King, Z. Altamimi, J. Boehm, M. Bos, R. Dach, P. Elosegui, F. Fund, M. Hernández-Pajares, D. Lavallee, P.J. Mendes Cerveira, N. Penna, R.E.M. Riva, P. Steigenberger, T. van Dam, L. Vittuari, S. Williams, P. Willis: Improved constraints on models of glacial isostatic adjustment: A review of the contribution of ground-based geodetic observations, Surveys Geophys. 31(5), 465–507 (2010)</ref>		The [[Precise Point Positioning\|Precise Point Positioning (PPP)]] technique allows achieving centimetre level accuracy for static positioning and decimetre level for kinematic, typically, after the best part of one hour. This high accuracy requires an accurate measurements modelling, where all model terms must be taken into account (up to the centimetre level or better).<ref name="Kouba">Kouba, Jan (et al.), Precise Point Positioning, Chapter 25, Handbook of Global Navigation Satellite Systems, 2017</ref><ref name="Hernandez">M. Hernandez-Pajares, J.M. Juan, J. Sanz, R. Orus, A. Garcia-Rigo, J. Feltens, A. Komjathy, S.C. Schaer, A. Krankowski: The IGS VTEC maps: A reliable source of ionospheric information since 1998, J. Geodesy 83(3/4), 263–275 (2009)</ref><ref name="King">M.A. King, Z. Altamimi, J. Boehm, M. Bos, R. Dach, P. Elosegui, F. Fund, M. Hernández-Pajares, D. Lavallee, P.J. Mendes Cerveira, N. Penna, R.E.M. Riva, P. Steigenberger, T. van Dam, L. Vittuari, S. Williams, P. Willis: Improved constraints on models of glacial isostatic adjustment: A review of the contribution of ground-based geodetic observations, Surveys Geophys. 31(5), 465–507 (2010)</ref>

	This modelling involves the following terms, among those considered in ~~the~~ [[Code Based Positioning (SPS)\|~~SPS~~]]:		This modelling involves the following terms, among those considered in [[Code Based Positioning (SPS)\|Code Based Positioning]]:


	*''' Precise satellite orbits and clocks:'''		*''' Precise satellite orbits and clocks:'''
	: ''The precise orbits and clocks files'' (see [[Precise GNSS Satellite Coordinates Computation]]) must be used instead of the broadcast ones used in conventional ~~code-based pseudorange positioning~~.		: ''The precise orbits and clocks files'' (see [[Precise GNSS Satellite Coordinates Computation\|Precise GNSS Satellite Coordinates Computation]]) must be used instead of the broadcast ones used in conventional [[Code Based Positioning (SPS)\|Code Based Positioning]].

	:The satellite coordinates between epochs can be computed by polynomial interpolation.		:The satellite coordinates between epochs can be computed by polynomial interpolation.
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	: ''Relativistic effects''(see <ref name=Ashby">Ashby, N., 2003. Relativity in the Global Positioning System</ref> for a very good review of relativistic effects on GPS): The gravitational path range correction (3) (see [[Relativistic Path Range Effect]]), can be added among the satellite clock correction due to the orbit eccentricity (4) considered in ~~the~~ [[Code Based Positioning (SPS)\|~~SPS~~]] (see [[Relativistic Clock Correction]]).		: ''Relativistic effects''(see <ref name=Ashby">Ashby, N., 2003. Relativity in the Global Positioning System</ref> for a very good review of relativistic effects on GPS): The gravitational path range correction (3) (see [[Relativistic Path Range Effect]]), can be added among the satellite clock correction due to the orbit eccentricity (4) considered in [[Code Based Positioning (SPS)\|Code Based Positioning]] (see [[Relativistic Clock Correction]]).

	::<math>		::<math>
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	* '''Antenna biases and orientation:'''		* '''Antenna biases and orientation:'''
	: The ''satellite and receiver antenna phase centres'' can be found in the IGS ANTEX files, after the GPS week ~~<math>~~1400~~</math>~~ (see [[Antenna Phase Centre]]).		: The ''satellite and receiver antenna phase centres'' can be found in the IGS ANTEX files, after the GPS week 1400 (see [[Antenna Phase Centre]]).

Gema.Cueto: /* Notes */

2020-06-15T11:56:02Z

Notes

@@ Line 119: / Line 119: @@
-==Notes==
+==Related Articles==
 <references group="footnotes"/>
 ==References==

Gema.Cueto at 11:53, 15 June 2020

2020-06-15T11:53:03Z

← Older revision		Revision as of 11:53, 15 June 2020
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	:The satellite coordinates between epochs can be computed by polynomial interpolation.		:The satellite coordinates between epochs can be computed by polynomial interpolation.
	As an example of polynomial interpolation, the Lagrange method is presented as follows. Given a table of values <math>(x_i,y_i), i=1,…,n,</math> the interpolated value <math>y≃P_n(x)</math> at a given x can be computed as (see Precise GNSS Satellite Coordinates Computation):		:As an example of polynomial interpolation, the Lagrange method is presented as follows. Given a table of values <math>(x_i,y_i), i=1,…,n,</math> the interpolated value <math>y≃P_n(x)</math> at a given x can be computed as (see Precise GNSS Satellite Coordinates Computation):

	::<math>		::<math>

Gema.Cueto at 11:52, 15 June 2020

2020-06-15T11:52:04Z

← Older revision		Revision as of 11:52, 15 June 2020
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	: ''The precise orbits and clocks files'' (see [[Precise GNSS Satellite Coordinates Computation]]) must be used instead of the broadcast ones used in conventional code-based pseudorange positioning.		: ''The precise orbits and clocks files'' (see [[Precise GNSS Satellite Coordinates Computation]]) must be used instead of the broadcast ones used in conventional code-based pseudorange positioning.

	The satellite coordinates between epochs can be computed by polynomial interpolation.		:The satellite coordinates between epochs can be computed by polynomial interpolation.
	As an example of polynomial interpolation, the Lagrange method is presented as follows. Given a table of values <math>(x_i,y_i), i=1,…,n,</math> the interpolated value <math>y≃P_n(x)</math> at a given x can be computed as (see Precise GNSS Satellite Coordinates Computation):		As an example of polynomial interpolation, the Lagrange method is presented as follows. Given a table of values <math>(x_i,y_i), i=1,…,n,</math> the interpolated value <math>y≃P_n(x)</math> at a given x can be computed as (see Precise GNSS Satellite Coordinates Computation):

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	Notice that, the orbits are referred to the satellite mass centre, and thence, the [[Satellite Antenna Phase Centre]] offset must be applied to compute the <math>{\boldsymbol \Delta}_{APC}</math> vector offset (see [[Satellite Antenna Phase Centre]]).		:Notice that, the orbits are referred to the satellite mass centre, and thence, the [[Satellite Antenna Phase Centre]] offset must be applied to compute the <math>{\boldsymbol \Delta}_{APC}</math> vector offset (see [[Satellite Antenna Phase Centre]]).

	::<math>		::<math>