## Introduction

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) variations 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.

Theoretically, the 3D loading displacement along the vertical, east, and northern directions DU, DE, and DN can be computed in two steps: firstly, expanding the field of the surface pressure anomaly into spherical harmonics of infinite degree/order, scaling them at a certain function that depends only on the degree of the expansion

,

where Δ P is the surface pressure anomaly, M is the land-sea mask (a number within 0 and 1 that is the share of land in the cell if the atmospheric pressure loading or land water storage loading computed and or the share of ocean if the ocean loading is computed), ρ is the mean Earth's density and g0 is the equatorial gravity acceleration, and h′, l′ are loading Love numbers of degree n; and, second, performing inverse spherical harmonics transform:

Since in practice we cannot operate with spherical harmonics transform of infinite degree/order, we have to use spherical harmonics transform truncated to degree/order m denoted as S(m). Loading computation with truncated spherical harmonics transform involves four steps: 1) spherical harmonics transform of the pressure anomaly with band-limited, weighted land-sea mask truncated at degree/order t=2699, 2) scaling the first copy with Lvn = h′/(2n+1) and second copy with Lhn = h′/(2n+1); 3) inverse spherical harmonics transform truncated at degree/order t=2699; 4) computation of sampling correction for the vertical component δU (since the sampling correction for the horizontal components is always less than 0.01mm it is neglected), using the so-called sampling correction land-sea mask that depends on two truncation parameters: tf=21599 and t=2699:

,

where the bandlimited land-sea mask Mb(t) and the sampling correction matrix Cs(tf,t) are computed as

,

where W_n(t) is the windowing function. Specifically, Blackmann window is used:

,

where α = 7938/18608, β = 9240/18608, γ = 1430/18608. The sampling correction significantly reduces (although not eliminate entirely) artifacts related to truncation of the spherical harmonics transform to degree/order 2699. The sampling correction δu is added to Dui.

The algorithm outlined above provides computation of the the 3D field of loading displacements at the regular longitude/latitude grid with resolution 2′ × 2′ (3.7 × 3.7 cos φ   km). This resolution is sufficient for computation of mass loading at the specific location using interpolation with B-splines.

## Surface pressure models

Mass loading is a substantially non-local effect. It is not sufficient to know mass variations 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, a number of jumps, secular rate, sine and cosine variations at annual, semi-annual, diurnal, semi-diurnal, ter-diurnal, 4-diurnal bands is computed. (Ter-diurnal and 4-diurnal terms are not estimated if the pressure fields has time step 6 hours). The contribution of the surface pressure due to the secular rate, annual and semi-annual bands is retained, while the contribution of the mean value, jumps, and harmonic variations at all other bands is removed. Therefore, the loading time series do not have these variations. The harmonic loading variations at these frequencies are computed separately. The sum of loading time series and loading harmonics variations should be used in data reduction.

The land-water storage models have several jumps related to changes in the data analysis. The sudden changes occur for every cell in the grid. The epochs of these jumps are identified. The magnitude of these jumps is evaluated together with the gridded mean surface pressure and the harmonic model. The output mass loading field does not have these jumps.

## What is provided

The following data products are provided for each geophysical model of the surface pressure variations:

1. The time series of 3D displacements caused by mass loading at the regular longitudinal/latitudinal 2′ × 2′ grid with harmonics variations in diurnal, semi-diurnal, ter-diurnal, and 4-diurnal bands removed.
2. The coefficients of harmonic variations of 3D displacements at diurnal, semi-diurnal, ter-diurnal, and 4-diurnal bands caused by mass loading at the regular longitudinal/latitudinal 2′ × 2′ grid.
3. The time series of 3D displacements caused by mass loading for 1272 space geodesy sites with harmonics variations in diurnal, semi-diurnal, ter-diurnal, and 4-diurnal bands removed.
4. The coefficients of harmonic variations of 3D displacements at diurnal, semi-diurnal, tear-diurnal, and 4-diurnal bands caused by mass loading at the regular longitudinal/latitudinal 2′ × 2′ 1272 space geodesy sites.
5. The time series of integrated mean mass loading height. The integration is performed over the entire Earth, as well as over the land only, over the ocean only, over the regions with latitude less by module 66°.
6. The time series of the Stokes coefficients of the contribution of surface pressure variations to gravity truncated to degree/order 64.
7. The time series of the expansion of 3D loading displacements into 2D B-spline basis.
8. The coefficients of expansion of harmonic variations of 3D displacements at diurnal, semi-diurnal, tear-diurnal, and 4-diurnal bands caused by mass loading into the 2D Bspline basis.
9. The online Internet service that computes the mass loading for the user-supplied list of stations
10. The online Internet service that provides the list of files of the specified mass loading for the specified date range.