Pole Tide - Revision history

Jaume.Sanz at 07:33, 5 April 2013

2013-04-05T07:33:09Z

← Older revision		Revision as of 07:33, 5 April 2013
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	Taking the earth's angular rotation <math>\omega_E= 7.29\cdot 10^{-5}</math>, the earth's equatorial radius <math>R_e=6378\cdot 10^3\, m</math> and the gravitational acceleration <math>g=9.8\, m/sec^2</math>, it follows:		Taking the earth's angular rotation <math>\omega_E= 7.29\cdot 10^{-5} rad/s</math>, the earth's equatorial radius <math>R_e=6378\cdot 10^3\, m</math> and the gravitational acceleration <math>g=9.8\, m/sec^2</math>, it follows:

Jaume.Sanz at 08:51, 17 February 2013

2013-02-17T08:51:41Z

← Older revision		Revision as of 08:51, 17 February 2013
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	The instantaneous earth rotation axis shifts inside a square of about <math>20</math> meters in relation to a point with fixed coordinates on the earth (i.e., Chandler wobble with a period of <math>14</math> months). This entails a varying elastic response of the earth's crust. This has an effect smaller than <math>2.5</math> centimetres in vertical and <math>0.7</math> centimetres in horizontal, but must be taken into account if the observations are carried out over periods longer than two months.		The instantaneous earth rotation axis shifts inside a square of about <math>20</math> meters in relation to a point with fixed coordinates on the earth (i.e., Chandler wobble with a period of <math>14</math> months). This entails a varying elastic response of the earth's crust. This has an effect smaller than <math>2.5</math> centimetres in vertical and <math>0.7</math> centimetres in horizontal, but must be taken into account if the observations are carried out over periods longer than two months.

	From the IERS Conventions [Denis et al., 2004] <ref> [Denis et al., 2004] Denis, D., McCarthy and Petit, G., 2004. IERS Conventions (2003). IERS Technical Note 32.. IERS Convention Center., Frankfurt am Main.</ref>, pages 83-84, the following expression <ref group="footnotes"> Notice the use of latitude <math>\displaystyle \varphi</math> in equations (1 and 2), instead of the co-latitude <math>\displaystyle \theta</math> used in the IERS equations.</ref> can be derived for the displacement at a point of ~~geographic~~ latitude <math>\displaystyle \varphi</math> and longitude <math>\displaystyle \lambda</math>:		From the IERS Conventions [Denis et al., 2004] <ref> [Denis et al., 2004] Denis, D., McCarthy and Petit, G., 2004. IERS Conventions (2003). IERS Technical Note 32.. IERS Convention Center., Frankfurt am Main.</ref>, pages 83-84, the following expression <ref group="footnotes"> Notice the use of latitude <math>\displaystyle \varphi</math> in equations (1 and 2), instead of the co-latitude <math>\displaystyle \theta</math> used in the IERS equations.</ref> can be derived for the displacement at a point of geocentric latitude <math>\displaystyle \varphi</math> and longitude <math>\displaystyle \lambda</math>:


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	The displacement <math>\displaystyle \delta</math> is given in the radial, longitude and latitude <math>(\hat{\mathbf r}, \hat{\boldsymbol \lambda}, \hat{\boldsymbol \varphi})</math> vectors, ~~i.e~~., the ~~Up East North~~ (~~UEN) local system, and~~, ~~thence~~, ~~can be transformed to the TRS reference system (see [[Reference Frames in GNSS]]~~) ~~applying the following latitude and longitude rotations indicated in figure 1:~~		The displacement <math>\displaystyle \delta</math> is given in the radial, longitude and latitude <math>(\hat{\mathbf r}, \hat{\boldsymbol \lambda}, \hat{\boldsymbol \varphi})</math> vectors (positive upwards, eastwards and northwards, respectively).
			Thus, the displacement vector in the (x, y, z) ECEF Cartesian coordinates is given by

	::<math>		::<math>
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	::[[File:Pole_Tide_UEN2CTS.png\|none\|thumb\|400px\|'''''Figure 1:''''' Transformation from UEN <math>(\hat{\mathbf r}, \hat{\boldsymbol \lambda}, \hat{\boldsymbol \varphi})</math> to TRS <math>(x,y,x)</math> coordinates.]]		::[[File:Pole_Tide_UEN2CTS.png\|none\|thumb\|400px\|'''''Figure 1:''''' Transformation from UEN <math>(\hat{\mathbf r}, \hat{\boldsymbol \lambda}, \hat{\boldsymbol \varphi})</math> to TRS <math>(x,y,x)</math> coordinates.]]

			where <math> {{\mathbf R}_3(-\lambda )\cdot {\mathbf R}_2(\varphi )} </math> are the rotations in latitude (1) and longitude (2) indicated in figure 1 (see [[Reference Frames in GNSS]]).

Jaume.Sanz at 12:20, 13 January 2013

2013-01-13T12:20:19Z

← Older revision		Revision as of 12:20, 13 January 2013
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	Taking the earth's angular rotation <math>\omega_E= 7.29\cdot 10^{-5}</math>, the earth's equatorial radius <math>R_e=6378\, km</math> and the gravitational acceleration <math>g=9.8\, m/sec^2</math>, it follows:		Taking the earth's angular rotation <math>\omega_E= 7.29\cdot 10^{-5}</math>, the earth's equatorial radius <math>R_e=6378\cdot 10^3\, m</math> and the gravitational acceleration <math>g=9.8\, m/sec^2</math>, it follows:

Jaume.Sanz at 10:17, 13 January 2013

2013-01-13T10:17:44Z

Carlos.Lopez at 11:35, 23 February 2012

2012-02-23T11:35:03Z

← Older revision		Revision as of 11:35, 23 February 2012
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	\|Level=Intermediate		\|Level=Intermediate
	\|YearOfPublication=2011		\|YearOfPublication=2011
	~~\|Logo=gAGE~~
	\|Title={{PAGENAME}}		\|Title={{PAGENAME}}
	}}		}}

Jaume.Sanz at 10:37, 8 February 2012

2012-02-08T10:37:34Z

← Older revision		Revision as of 10:37, 8 February 2012
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	The displacement <math>\displaystyle \delta</math> is given in the radial, longitude and latitude <math>(\hat{\mathbf r}, \hat{\boldsymbol \lambda}, \hat{\boldsymbol \varphi})</math> vectors, i.e., the North~~-East-Up~~ (~~NEU~~) local system, and, thence, can be transformed to the TRS reference system (see [[Reference Frames in GNSS]]) applying the following latitude and longitude rotations indicated in figure 1:		The displacement <math>\displaystyle \delta</math> is given in the radial, longitude and latitude <math>(\hat{\mathbf r}, \hat{\boldsymbol \lambda}, \hat{\boldsymbol \varphi})</math> vectors, i.e., the Up East North (UEN) local system, and, thence, can be transformed to the TRS reference system (see [[Reference Frames in GNSS]]) applying the following latitude and longitude rotations indicated in figure 1:


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	::[[File:Pole_Tide_UEN2CTS.png\|none\|thumb\|400px\|'''''Figure 1:''''' Transformation from ~~NEU~~ <math>(\hat{\mathbf r}, \hat{\boldsymbol \lambda}, \hat{\boldsymbol \varphi})</math> to TRS <math>(x,y,x)</math> coordinates.]]		::[[File:Pole_Tide_UEN2CTS.png\|none\|thumb\|400px\|'''''Figure 1:''''' Transformation from UEN <math>(\hat{\mathbf r}, \hat{\boldsymbol \lambda}, \hat{\boldsymbol \varphi})</math> to TRS <math>(x,y,x)</math> coordinates.]]

Jaume.Sanz at 13:06, 6 February 2012

2012-02-06T13:06:43Z

← Older revision		Revision as of 13:06, 6 February 2012
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	The displacement <math>\displaystyle \delta</math> is given in the radial, longitude and latitude <math>(\hat{\mathbf r}, \hat{\boldsymbol \lambda}, \hat{\boldsymbol \varphi})</math> vectors, i.e., the East~~, North,~~ Up (~~ENU~~) local system, and, thence, can be transformed to the TRS reference system (see [[Reference Frames in GNSS]]) applying the following latitude and longitude rotations indicated in figure 1:		The displacement <math>\displaystyle \delta</math> is given in the radial, longitude and latitude <math>(\hat{\mathbf r}, \hat{\boldsymbol \lambda}, \hat{\boldsymbol \varphi})</math> vectors, i.e., the North-East-Up (NEU) local system, and, thence, can be transformed to the TRS reference system (see [[Reference Frames in GNSS]]) applying the following latitude and longitude rotations indicated in figure 1:


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	::[[File:Pole_Tide_UEN2CTS.png\|none\|thumb\|400px\|'''''Figure 1:''''' Transformation from ~~ENU~~ <math>(\hat{\mathbf r}, \hat{\boldsymbol \lambda}, \hat{\boldsymbol \varphi})</math> to TRS <math>(x,y,x)</math> coordinates.]]		::[[File:Pole_Tide_UEN2CTS.png\|none\|thumb\|400px\|'''''Figure 1:''''' Transformation from NEU <math>(\hat{\mathbf r}, \hat{\boldsymbol \lambda}, \hat{\boldsymbol \varphi})</math> to TRS <math>(x,y,x)</math> coordinates.]]

Jaume.Sanz at 13:04, 6 February 2012

2012-02-06T13:04:54Z

← Older revision		Revision as of 13:04, 6 February 2012
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	The instantaneous earth rotation axis shifts inside a square of about <math>20</math> meters in relation to a point with fixed coordinates on the earth (i.e., Chandler wobble with a period of <math>14</math> months). This entails a varying elastic response of the earth's crust. This has an effect smaller than <math>2.5</math> centimetres in vertical and <math>0.7</math> centimetres in horizontal, but must be taken into account if the observations are carried out over periods longer than two months.		The instantaneous earth rotation axis shifts inside a square of about <math>20</math> meters in relation to a point with fixed coordinates on the earth (i.e., Chandler wobble with a period of <math>14</math> months). This entails a varying elastic response of the earth's crust. This has an effect smaller than <math>2.5</math> centimetres in vertical and <math>0.7</math> centimetres in horizontal, but must be taken into account if the observations are carried out over periods longer than two months.

	From the IERS Conventions [Denis et al., 2004] <ref> [Denis et al., 2004] Denis, D., McCarthy and Petit, G., 2004. IERS Conventions (2003). IERS Technical Note 32.. IERS Convention Center., Frankfurt am Main.</ref>, pages 83-84, the following expression <ref group="footnotes"> Notice the use of latitude <math>\varphi</math> in equations (1 and 2), instead of the co-latitude <math>\theta</math> used in the IERS equations.</ref> can be derived for the displacement at a point of geographic latitude <math>\displaystyle \varphi</math> and longitude <math>\displaystyle \lambda</math>:		From the IERS Conventions [Denis et al., 2004] <ref> [Denis et al., 2004] Denis, D., McCarthy and Petit, G., 2004. IERS Conventions (2003). IERS Technical Note 32.. IERS Convention Center., Frankfurt am Main.</ref>, pages 83-84, the following expression <ref group="footnotes"> Notice the use of latitude <math>\displaystyle \varphi</math> in equations (1 and 2), instead of the co-latitude <math>\displaystyle \theta</math> used in the IERS equations.</ref> can be derived for the displacement at a point of geographic latitude <math>\displaystyle \varphi</math> and longitude <math>\displaystyle \lambda</math>:


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	::[[File:Pole_Tide_UEN2CTS.png\|none\|thumb\|400px\|'''''Figure 1:''''' Transformation from ~~UEN~~ <math>(\hat{\mathbf r}, \hat{\boldsymbol \lambda}, \hat{\boldsymbol \varphi})</math> to TRS <math>(x,y,x)</math> coordinates.]]		::[[File:Pole_Tide_UEN2CTS.png\|none\|thumb\|400px\|'''''Figure 1:''''' Transformation from ENU <math>(\hat{\mathbf r}, \hat{\boldsymbol \lambda}, \hat{\boldsymbol \varphi})</math> to TRS <math>(x,y,x)</math> coordinates.]]

Jaume.Sanz at 13:01, 6 February 2012

2012-02-06T13:01:16Z

← Older revision		Revision as of 13:01, 6 February 2012
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	The displacement <math>\displaystyle \delta</math> is given in the radial, longitude and latitude <math>(\hat{\mathbf r}, \hat{\boldsymbol \lambda}, \hat{\boldsymbol \varphi})</math> vectors, i.e., the ~~UEN~~ local system, and, thence, can be transformed to the TRS reference system (see [[Reference Frames in GNSS]]) applying the following latitude and longitude rotations indicated in figure 1:		The displacement <math>\displaystyle \delta</math> is given in the radial, longitude and latitude <math>(\hat{\mathbf r}, \hat{\boldsymbol \lambda}, \hat{\boldsymbol \varphi})</math> vectors, i.e., the East, North, Up (ENU) local system, and, thence, can be transformed to the TRS reference system (see [[Reference Frames in GNSS]]) applying the following latitude and longitude rotations indicated in figure 1:

Jaume.Sanz at 12:57, 6 February 2012

2012-02-06T12:57:11Z

← Older revision		Revision as of 12:57, 6 February 2012
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	where <math>(m_1,m_2)</math> are the displacements from the 1903.0 CIO, pole position, and <math>h= 0.6027</math>, <math>l=0.0836</math> are the Love numbers.		where (<math>\displaystyle m_1</math>,<math>\displaystyle m_2</math>) are the displacements from the 1903.0 CIO, pole position, and <math>\displaystyle h= 0.6027</math>, <math>\displaystyle l=0.0836</math> are the Love numbers.


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	where <math>(m_1,m_2)</math> are the displacements (given in seconds of arc). Pole displacements can be found at ftp://hpiers.obspm.fr/iers/eop/eop.others.		where (<math>\displaystyle m_1</math>,<math>\displaystyle m_2</math>) are the displacements (given in seconds of arc). Pole displacements can be found at ftp://hpiers.obspm.fr/iers/eop/eop.others.


	The displacement <math>\delta</math> is given in the radial, longitude and latitude <math>(\hat{\mathbf r}, \hat{\boldsymbol \lambda}, \hat{\boldsymbol \varphi})</math> vectors, i.e., the UEN local system, and, thence, can be transformed to the TRS reference system (see [[Reference Frames in GNSS]]) applying the following latitude and longitude rotations indicated in figure 1:		The displacement <math>\displaystyle \delta</math> is given in the radial, longitude and latitude <math>(\hat{\mathbf r}, \hat{\boldsymbol \lambda}, \hat{\boldsymbol \varphi})</math> vectors, i.e., the UEN local system, and, thence, can be transformed to the TRS reference system (see [[Reference Frames in GNSS]]) applying the following latitude and longitude rotations indicated in figure 1:

← Older revision		Revision as of 10:17, 13 January 2013
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	::<math>		::<math>
	\begin{array}{ll}		\begin{array}{ll}
	\delta_{\hat{\mathbf r}}=&-\frac{\omega^2_E \, ~~R^2~~}{2\,g}\,h \, \sin 2\varphi \left ( m_1 \cos \lambda + m_2 \sin \lambda \right ) \,\hat {\mathbf r}\qquad \\[0.3cm]		\delta_{\hat{\mathbf r}}=&-\frac{\omega^2_E \, R_e}{2\,g}\,h \, \sin 2\varphi \left ( m_1 \cos \lambda + m_2 \sin \lambda \right ) \,\hat {\mathbf r}\qquad \\[0.3cm]
	\delta_{\hat{\mathbf \lambda}}= & -\frac{\omega^2_E \, ~~R^2~~}{g}\, l \, \sin \varphi \left (-m_1 \sin \lambda + m_2 \cos \lambda \right ) \, \hat{\mathbf \lambda} \qquad \\[0.3cm]		\delta_{\hat{\mathbf \lambda}}= & -\frac{\omega^2_E \, R_e}{g}\, l \, \sin \varphi \left (-m_1 \sin \lambda + m_2 \cos \lambda \right ) \, \hat{\mathbf \lambda} \qquad \\[0.3cm]
	\delta_{\hat{\mathbf \varphi}}= & -\frac{\omega^2_E \, ~~R^2~~}{g}\, l\, \cos 2\varphi \left (\, m_1 \cos \lambda + m_2 \sin \lambda \right )\, \hat{\mathbf \varphi}\qquad		\delta_{\hat{\mathbf \varphi}}= & -\frac{\omega^2_E \, R_e}{g}\, l\, \cos 2\varphi \left (\, m_1 \cos \lambda + m_2 \sin \lambda \right )\, \hat{\mathbf \varphi}\qquad
	\end{array} \qquad\mbox{(1)}</math>		\end{array} \qquad\mbox{(1)}</math>


	where (<math>\displaystyle m_1</math>,<math>\displaystyle m_2</math>) are the displacements from the 1903.0 CIO, pole position, and <math>\displaystyle h= 0.6027</math>, <math>\displaystyle l=0.0836</math> are the Love numbers.		where (<math>\displaystyle m_1</math>,<math>\displaystyle m_2</math>) are the displacements (in meters) from the 1903.0 CIO, pole position, and <math>\displaystyle h= 0.6027</math>, <math>\displaystyle l=0.0836</math> are the Love numbers.


	Taking the earth's angular rotation <math>\omega_E= 7.29\cdot 10^{-5}</math>, the earth's equatorial radius <math>R=6378\, km</math> and the gravitational acceleration <math>g=9.8\, m/sec^2</math>, it follows:		Taking the earth's angular rotation <math>\omega_E= 7.29\cdot 10^{-5}</math>, the earth's equatorial radius <math>R_e=6378\, km</math> and the gravitational acceleration <math>g=9.8\, m/sec^2</math>, it follows:


	::<math>		::<math>
	\begin{array}{ll}		\begin{array}{ll}
	\delta_{\hat{\mathbf r}}=&-32 \, \sin 2\varphi \left ( ~~m_1~~ \cos \lambda + ~~m_2~~ \sin \lambda \right ) \, \hat {\mathbf r}\qquad \mbox{(mm)} \\[0.3cm]		\delta_{\hat{\mathbf r}}=&-32 \, \sin 2\varphi \left ( x_1 \cos \lambda + x_2 \sin \lambda \right ) \, \hat {\mathbf r}\qquad \mbox{(mm)} \\[0.3cm]
	\delta_{\hat{\mathbf \lambda}}= & -9 \, \sin \varphi \left (-~~m_1~~ \sin \lambda + ~~m_2~~ \cos \lambda \right ) \, \hat{\mathbf \lambda} \qquad \mbox{(mm)} \\[0.3cm]		\delta_{\hat{\mathbf \lambda}}= & -9 \, \sin \varphi \left (-x_1 \sin \lambda + x_2 \cos \lambda \right ) \, \hat{\mathbf \lambda} \qquad \mbox{(mm)} \\[0.3cm]
	\delta_{\hat{\mathbf \varphi}}= & -9 \, \cos 2\varphi \left (\, ~~m_1~~ \cos \lambda + ~~m_2~~ \sin \lambda \right )\, \hat{\mathbf \varphi}\qquad \mbox{(mm)}		\delta_{\hat{\mathbf \varphi}}= & -9 \, \cos 2\varphi \left (\, x_1 \cos \lambda + x_2 \sin \lambda \right )\, \hat{\mathbf \varphi}\qquad \mbox{(mm)}
	\end{array} \qquad\mbox{(2)}</math>		\end{array} \qquad\mbox{(2)}</math>


	where (<math>\displaystyle ~~m_1~~</math>,<math>\displaystyle ~~m_2~~</math>) are the displacements (given in seconds of arc). Pole displacements can be found at ftp://hpiers.obspm.fr/iers/eop/eop.others.		where (<math>\displaystyle x_1</math>,<math>\displaystyle x_2</math>) are the displacements (given in seconds of arc). Pole displacements can be found at ftp://hpiers.obspm.fr/iers/eop/eop.others.