International Mass Loading Service: Tidal Ocean Loading


The Earth as a whole responses to external forces as an elastic body while the ocean response far from the shore takes the shape of equipotential surface. For this reason we observe ocean tides: relative redistribution of ocean with respect to the lithosphere. Due to viscosity response the ocean response deviates from the equipotential surface, especially in shallow seas. Ocean mass re-distribution due to tides causes crustal deformation called ocean tidal loading.

Tides can be represented as a sum of quasi-harmonic constituents that are characterized by frequency, angular acceleration, phase, sine, and cosine amplitudes. The last three parameters depend on site coordinates.

Tidal constituents consist of short-periodic tides with frequencies around the diurnal frequency and its harmonics, long-periodic (or zonal) with periods from 18.6 years to 5 days, and several pole tide constituents. The pole tide is caused by changes in the centrifugal potential due to polar motion. Other tides are caused by gravitational attraction of the Moon and the Sun. The vertical displacements typically have peak to peak variations around 1 cm, although there are areas where they can be as large as 10 cm.

Service for the tidal ocean loading

On 2014.06.17 the service of tidal atmospheric pressure loading was established by Leonid Petrov. The service is based on the spherical harmonics transform of the bottom pressure exerted by the ocean tide according to several tidal models. In general, tides are split into
  1. gravity tides described by the tidal model that are based on assimilation of satellite altimeter data and the underlying numerical model that is a result of solving hydrodynamics equations. Gravity tides consist of a number of quasi-harmonics constituents: A_c cos ( P + Ω t + 1/2 α t2 ) + A_s sin ( P + Ω t + 1/2 α t2 )   where α is the angular acceleration.

  2. long periodic tides described by the so-called equilibrium tidal model that assumes ocean water for these constituents follows the equipotential surface with correction for total ocean mass conservation and self-loading, i.e. considering crustal deformation of the ocean bottom. Deviation of long-periodic tides from the equilibrium is not large. Some gravity tides model include several (but not all) long periodic models.

  3. pole tides described by the equilibrium tidal response and a harmonic model of polar motion. Pole motion and therefore, changes in the centrifugal potential over [1984, 2014] are approximated as a sum of three harmonic component with periods 365.24 days (annual), 431.18 days, and 445.41 days (Chandler period) with rms of residuals 0.029 mas. Therefore, pole tides can be considered using the same formalism as gravity tides.

The service provides


  1. Darwin, G.H., On variations in the vertical due to elasticity of the Earth's surface, Phil. Mag., Ser. 5, col. 14, N. 90, 409--427, 1882.
  2. Farrell, W.E, Deformation of the Earth by Surface Loads, Rev. Geophys. and Spac. Phys., vol. 10(3), pp. 751--797, 1972.


This work was supported by NASA
Earth Surface & Interior program, grant NNX12AQ29G.

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This page was prepared by Leonid Petrov ()
Last update: 2021.07.19_08:01:58