Solid Tides - Revision history

Natalia.Castrillo at 07:45, 20 August 2025

2025-08-20T07:45:32Z

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 ==Rotational ellipsoid model for solid Earth tide with high precision==
-A new [https://rdcu.be/elmWR model] for the solid Earth tide was published in 2024<ref name="Solid Earth Tide high precision"> Yang, Y., Zhang, Y., Liu, Q. et al. A rotational ellipsoid model for solid Earth tide with high precision. Sci Rep 14, 28527 (2024). https://doi.org/10.1038/s41598-024-79898-8 </ref>, presenting an updated approach for modeling Earth's tidal deformations with improved accuracy, and integrating recent geophysical data and refined computational methods.
+A new [https://rdcu.be/elmWR model] for the solid Earth tide was published in 2024<ref name="Solid Earth Tide high precision"> Yang, Y., Zhang, Y., Liu, Q. et al. A rotational ellipsoid model for solid Earth tide with high precision. Sci Rep 14, 28527 (2024). https://doi.org/10.1038/s41598-024-79898-8 </ref>, presenting a rotating ellipse for computing Earth's tidal deformations with improved accuracy, and integrating recent geophysical data and refined computational methods. The vertical displacement of a reference point relative to the Earth's center at a time ''t'' can be written as:
 ==Notes==

Natalia.Castrillo at 14:51, 9 May 2025

2025-05-09T14:51:06Z

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	~~Solid~~ tides comprise the Earth's crust movement (and thus the receiver location coordinates variation) due to gravitational attracting forces produced by external bodies, mainly the sun and the moon. The Solid Tides produce vertical and horizontal displacements that can be expressed by spherical harmonics expansion (<math>\displaystyle m</math>,<math>\displaystyle n</math>), characterised by the Love and Shida numbers <math>\displaystyle h_{mn}</math>, <math>\displaystyle l_{mn}</math>.		Earth solid tides comprise the Earth's crust movement (and thus the receiver location coordinates variation) due to gravitational attracting forces produced by external bodies, mainly the sun and the moon. The Solid Tides produce vertical and horizontal displacements that can be expressed by spherical harmonics expansion (<math>\displaystyle m</math>,<math>\displaystyle n</math>), characterised by the Love and Shida numbers <math>\displaystyle h_{mn}</math>, <math>\displaystyle l_{mn}</math>.

			An accurate modeling of Earth solid tides is essential for ensuring that GNSS-derived positions are reliable, especially for scientific and engineering applications.

	A simplified model for the tide displacement, but with few ~~millimetres~~ of accuracy, is given by the following expression (from IERS Conventions degree 2 tides displacement model --in-phase corrections--, see [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>, page 79):
			==IERS solid Earth tide model==

			A simplified model for the tide displacement, but with few millimeters of accuracy, is given by the following expression (from IERS Conventions degree 2 tides displacement model --in-phase corrections--, see [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>, page 79):

	:<math>		:<math>
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			==Rotational ellipsoid model for solid Earth tide with high precision==

			A new [https://rdcu.be/elmWR model] for the solid Earth tide was published in 2024<ref name="Solid Earth Tide high precision"> Yang, Y., Zhang, Y., Liu, Q. et al. A rotational ellipsoid model for solid Earth tide with high precision. Sci Rep 14, 28527 (2024). https://doi.org/10.1038/s41598-024-79898-8 </ref>, presenting an updated approach for modeling Earth's tidal deformations with improved accuracy, and integrating recent geophysical data and refined computational methods.

	==Notes==		==Notes==

Carlos.Lopez at 11:41, 23 February 2012

2012-02-23T11:41:36Z

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Jaume.Sanz at 10:27, 8 February 2012

2012-02-08T10:27:00Z

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	Finally, it must be taken into account that the previous equations provide the correction to obtain the coordinates relative to the "conventional tide free". To obtain the position relative to the "mean tide" the following vector must be added <ref group="footnotes"> According to the Resolution of 18th IAG General Assembly, 1983 (see [Denis et al., 2004] pages 9-10, and 83, the (degree 2 zonal) tidal potential contains a time independent (i.e., permanent) part, which is included in the geoid definition. Thence, the "mean tide" position is obtained after removing this part to the tidal displacement. This is done, by adding the vector (4) to the tidal displacement computed from equation (1).</ref>:		Finally, it must be taken into account that the previous equations provide the correction to obtain the coordinates relative to the "conventional tide free". To obtain the position relative to the "mean tide" the following vector must be added <ref group="footnotes"> According to the Resolution of 18th IAG General Assembly, 1983 (see [Denis et al., 2004] pages 9-10, and 83, the (degree 2 zonal) tidal potential contains a time independent (i.e., permanent) part, which is included in the geoid definition. Thence, the "mean tide" position is obtained after removing this part to the tidal displacement. This is done, by adding the vector in equation (4) to the tidal displacement computed from equation (1).</ref>:

	::<math>		::<math>

Jaume.Sanz at 12:30, 6 February 2012

2012-02-06T12:30:56Z

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	A small correction of latitude (<math>\varphi</math>) dependence can be considered in the <math>\displaystyle h_2</math> and <math>\displaystyle l_2</math> values of previous equation:		A small correction of latitude (<math>\varphi</math>) dependence can be considered in the <math>\displaystyle h_2</math> and <math>\displaystyle l_2</math> values of previous equation (1) according to the expression:

	~~::<math> N=nint\left[\frac{\Delta \Phi_{previo}-\delta \Phi}{2\pi}\right] \qquad~~
	~~\zeta=\hat{k}\cdot \left~~(~~\vec{D}^{\prime}\times\vec{D}\right~~)~~</math>~~

	according to the expression:

	::<math>		::<math>

Jaume.Sanz at 12:29, 6 February 2012

2012-02-06T12:29:28Z

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	Notice that the radial component of this part can amount for about <math>-12</math> centimetres at the poles and about <math>+6</math> centimetres at the equator.		Notice that the radial component of this part can amount for about <math>-12</math> centimetres at the poles and about <math>+6</math> centimetres at the equator.

	An alternative set of equations describing a similar model implemented in GIPSY-OASIS II can be found in [Webb and Zumberge, 1993] <ref> [Webb and Zumberge, 1993] Webb, F. and Zumberge, J., 1993. An Introduction to GIPSY/OASIS-II. Jet Propulsion Laboratory, JPL, 4800 Oak Grove Drive, Pasadena, CA 91109.</ref>		An alternative set of equations describing a similar model implemented in GIPSY-OASIS II can be found in [Webb and Zumberge, 1993] <ref> [Webb and Zumberge, 1993] Webb, F. and Zumberge, J., 1993. An Introduction to GIPSY/OASIS-II. Jet Propulsion Laboratory, JPL, 4800 Oak Grove Drive, Pasadena, CA 91109.</ref>.

Jaume.Sanz at 12:25, 6 February 2012

2012-02-06T12:25:25Z

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	A simplified model for the tide displacement, but with few millimetres of accuracy, is given by the following expression (from IERS Conventions degree 2 tides displacement model --in-phase corrections--, see [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>, page 79):		A simplified model for the tide displacement, but with few millimetres of accuracy, is given by the following expression (from IERS Conventions degree 2 tides displacement model --in-phase corrections--, see [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>, page 79):

	:<math>		:<math>

Jaume.Sanz at 12:22, 6 February 2012

2012-02-06T12:22:07Z

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	::{\|		::{\|
	\|+align="bottom"\|''Figure 1: Solid Tides: Range and position domain effect ''		\|+align="bottom"\|''Figure 1: Solid Tides: Range and position domain effect.''
	\| [[File:Solid_Tides_Ex2b2.png\|none\|thumb\|400px\|frameless]]		\| [[File:Solid_Tides_Ex2b2.png\|none\|thumb\|400px\|frameless]]
	\| [[File: Solid_Tides_Ex2b3.png\|none\|thumb\|400px\|frameless]]		\| [[File: Solid_Tides_Ex2b3.png\|none\|thumb\|400px\|frameless]]
	\|-		\|-
	\| [[File:Solid_Tides_Ex2b1.png\|none\|thumb\|400px\|frameless]]		\| [[File:Solid_Tides_Ex2b1.png\|none\|thumb\|400px\|frameless]]
	\| '' First row shows the horizontal (left) and vertical (right) positioning error using (blue) or not using (red) the solid tides correction (from equations (1) to (3). The effect on range of solid tides displacement is shown in the second row at left.''		\| '' First row shows the horizontal (left) and vertical (right) positioning error using (blue) or not using (red) the solid tides correction (from equations (1) to (3)). The effect on range of solid tides displacement is shown in the second row at left.''
	\|}		\|}

Jaume.Sanz at 12:20, 6 February 2012

2012-02-06T12:20:10Z

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	Figure 1 illustrates the effect of the Solid Tides correction. The navigation solution computed using the solid tides correction is shown in blue and the solution without using this correction in red. The effect of solid tides displacement on range is shown in the second row at left.		Figure 1 illustrates an example of the effect of the Solid Tides correction. The navigation solution computed using the solid tides correction is shown in blue and the solution without using this correction in red. The effect of solid tides displacement on range is shown in the second row at left.

Jaume.Sanz at 12:17, 6 February 2012

2012-02-06T12:17:18Z

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	where <math>h_3=0.292</math> and <math>l_3=0.015</math>. Only the moon's contribution (<math>j=2</math>) need to be computed, because the contribution of sun (<math>j=3</math>) is negligible. Nevertheless, also the moon's contribution to this degree 3 tide to the radial displacement does not exceed the <math>1.7</math> millimetres in radial and <math>0.02</math> millimetres in transversal components.		where <math>\displaystyle h_3=0.292</math> and <math>\displaystyle l_3=0.015</math>. Only the moon's contribution (<math>j=2</math>) need to be computed, because the contribution of sun (<math>j=3</math>) is negligible. Nevertheless, also the moon's contribution to this degree 3 tide to the radial displacement does not exceed the <math>1.7</math> millimetres in radial and <math>0.02</math> millimetres in transversal components.