The Earth as a whole responses to external forces as an elastic body. Putting an additional mass on the Earth's surface causes a crust deformation. Changes of loading mass result in variable displacements of Earth's surface. If not properly accounted, such variations distort precise geodetic measurements. Loading displacements have a typical magnitude of 2–5 mm, but may reach 60 mm in extreme cases.
Mass loading has four constituents: 1) variations of air mass that results in a change of surface air pressure; 2) variations in ocean level due to lunar and solar tides; 3) variations in ocean level due to wind and atmospheric pressure; 4) variatiosn in soil moisture. These four loadings are to be taken into account in processing space geodetic data when precision better 1 cm is required.
Assuming the Earth's response to the pressure of loading mass is linear, i.e. deformation is proportional to stress, as it was shown by Augustine Love, the spherical harmonics of degree m and order n of the displacement field d are proportional to the spherical harmonics of the pressure field P of the same order, the same degree: Ηmn(d) = αn Ηmn(P) where the coefficients α, proportional to dimensionless parameters called Love numbers, depend on the distribution of density and elastic property of the Earth.
Computation of mass loading is done in two steps. First the anomaly of the pressure field ΔP(t,φ,λ) is expanded into spherical harmonics and scaled by numbers that depend on the degree of the transform:
Here ρ is the mean Earth's density, g is the equatorial gravity acceleration, h'n is the vertical Love number, and l'n is the horizontal Love number (sometimes referred to as the Shida number). Here we presented the acting pressure field as a product of the continuous pressure and the land/sea mask Lφ,λ. In a case of the atmosphere pressure and land water storage loadings the value of the land/sea mask is the share of land in a cell, and in the case of ocean loading, the value of the land/sea mask is the share of the ocean in an elementary cell. Ymn(φ,λ) is the spherical harmonic function of degree m and order n. While the output of the numerical model that presents the pressure field is band-limited, the land/sea mask is not. Therefore, one needs to expand the product of the pressure field and the land/sea mask to rather high degree/order, at least 1023, or even higher.
The second step is to perform the inverse vector spherical harmonic transform of the scaled spherical harmonics Vmn and Hmn. The result is the displacement vector D with components in vertical (U), east (E) and north (N) directions:
Mass loading is a substantially non-local effect. It is not sufficient to know mass verations in the point of interest. In order to compute mass loading displacements, the mass change over entire Earth should be known.
The pressure field is derived from the output of numerical models of the atmosphere, ocean or hydrology. The average global pressure field over the specified time range is computed and subtracted, and the result, called pressure anomaly ΔP is used for the spherical harmonics transform. The mean loading displacements computed by averaging over exactly the same interval of time is below one micron (due to rounding errors it is not exactly zero). However, the mass loading averaged over any other interval is not zero due to inter-annual variations.
It is desirable for some applications to separate harmonic loading variations from aharhmonic. Therefore, the model that includes the mean, secular rate, sine and cosine variations at annual, semi-annual, diurnal, semi-diurnal, ter-diurnal, 4-diurnal bands is computed. The contribution of the surface pressure due to the secular rate, annual and semi-annual bands is retained, while the contribution of the harmonics variations at all other bands (i.e. with diurnal and sub-diurnal frequencies) is removed. Therefore, the loading time series do not have these variations. The harmonic loading variations are computed separately. The sum of loading time series and loading harmonics variations should be used in data reduction.