On the free ocean tide loading provider we compute, for given location and ocean tide model, amplitudes $$A_j$$ and phase-lag $$\phi_j$$ of the loading effect for the 11 main tidal frequencies $$f_j$$ which are called harmonics. These frequencies are given in Table "11 main tidal frequencies".

11 main tidal frequencies
Species Harmonic frequency (cpd)
Long-period Ssa 0.0054758

Mm 0.0362916

Mf 0.0732022
Daily Q1 0.8932441

O1 0.9295357

P1 0.9972621

K1 1.0027379
Semi-diurnal N2 1.8959820

M2 1.9322736

S2 2.0000000

K2 2.0054758

For each partial tide $$j$$, the loading effect $$g_j$$ at time $$t$$ can be computed as

$g_j(t) = A_j \cos (a_j(t_0) + f_j \cdot(t-t_0) - \phi_j\tfrac{2\pi}{360})$

where $$A_j$$ and $$\phi_j$$ are given in the BLQ files. In oceanography a lag of a tide wave is reckoned positive while in earth tides (geodesy, astronomy) lags are reckoned negative. BLQ follows the oceanographic convention. The unit of the phase-lag is always degrees.

$$a_j(t_0)$$ is the phase of the astronomical angle of harmonic $$j$$ at time $$t_0$$. It can be computed using the with program ARG2.F available from the IERS. Actually, the $$\cos$$ is only used the for semi-diurnal and long-period tides while the $$\sin$$ function should be used for daily tides. However, this is taken care of in the computation of the angle $$a_j$$ which is shifted -90 degrees for diurnal tides. Another complication is that the tidal potential is for some harmonics (such as K1) shifted by 180 degrees. This is also taken into account in the computation of $$a_j$$.

Next, the direction we define as positive is given in Table 2 together with its unit.

positive directions of the amplitudes and its unit.
Displacement (m) gravity (nm/s$$^2$$) tilt (nrad)
Up Up North
West
East
South

Loading time series can be computed using the program ttimm. Another option the program hardisp developped by Duncan Agnew. Both programs use BLQ files as input. These time series can be used to remove the ocean tide loading effect from the observations.

For the analysis of tidal gravity time series, the Tsoft (van Camp, ROB) software packages is often used. A perl script olgte.pl can be used to adapt BLQ coefficients (with thanks to Hartmut Wziontek) to produce a table that Tsoft can read to subtract the loading effect from the observed gravity time series.

# Solid Earth tide

So far we have only focussed on ocean tide loading but normally one should also take the solid Earth tide into account. Tidal gravity analysis normally involves fitting various wavegroups to the observations. Each wavegroup consists out of various tidal harmonics, with amplitudes based on their value in the tidal potential, for which a single amplification factor and phase-lag is estimated. This exercise implicates a number of preconditions, ranging from the effective bandwidth assigned to wavegroups to site location parameters. This part of applications is beyond the scope of BLQ. See Boy et al. (2003).

For each harmonic the tidal potential can be computed as
$V_j(t,r,\theta,\lambda) = D\;K_j\;G_m(r,\theta) \cos(a_j(t_0) + f_j \cdot(t-t_0) + m\lambda \quad \quad (1)$
where $$D$$ = 2.627689 m2/s2 is Doodson's constant, and $$K_j$$ is the amplitude given in a harmonic expansion of the tidal potential such as the one of Tamura (1987) 1. Here we only allow positive values of $$K_j$$ since in the astronomical argument angle $$a_j$$ we already take care of a minus sign by shifting the angle with 180 degrees (and sine terms by shifting with $$\pm 90$$ degrees). $$\theta$$ is the latitude of the station and $$\lambda$$ its longitude. Thus, the angle $$a_j$$ corresponds to the phase of the tidal potential along the Greenwich meridian. At the station, the phase of the tidal potential is $$m\lambda$$ degrees ahead. Therefore, to refer the ocean tide loading phase-lag $$\phi$$ to the phase of the tidal potential at the station, one has to subtract $$m\lambda$$ from the phase-lag $$\phi$$.

$$G_m(r,\theta)$$ is called the Geodetic function (Bartels, 1957) and depends on the height of the station (radius $$r$$) and its latitude $$\theta$$. Furthermore, it depends on the species $$m$$. For long period tides $$m=0$$, diurnal tides $$m=1$$ and for semi-diurnal tides $$m=2$$. Their equations are:
$G_0(r,\theta) = \left(\frac{r}{R}\right)^2 (1- 3 \sin^2\theta) \quad \quad (2a)$
$G_1(r,\theta) = \left(\frac{r}{R}\right)^2 \sin 2\theta \quad \quad \quad (2b)$
$G_2(r,\theta) = \left(\frac{r}{R}\right)^2 \cos^2\theta \quad \quad \quad (2c)$

where $$R$$ is the mean radius of the Earth, 6371 km.

Since the solid Earth is elastic, it deforms due to the tidal forcing of the Moon and Sun. The amount of deformation is proportional to the forcing and expressed with the use of Love numbers $$h_2$$ and $$l_2$$ (second degree). The vertical displacement $$u$$ of the solid Earth tide is:
$u = h_2 \frac{V}{g}$

For this component the ocean tide loading the solid Earth has the same phase as the tidal potential. Note that Geodetic function $$G_m(r,\theta)$$ can change sign depending on the latitude of the station and the species $$m$$. This can cause a 180 degree phase difference with the angular argument of the tidal potential. The horizontal East solid Earth tide is: $v_E = \frac{l_2}{g\cos\theta} \frac{\partial V}{\partial \lambda}$

Since $$d\cos(\psi+\lambda)/d\lambda=-\sin(\psi+\lambda)=cos(\psi+\lambda+\tfrac{\pi}{2})$$, the solid Earth tide of the displacement towards the east is 90 degrees ahead of the tidal potential. Again, depending on the latitude of the station this can also be -90 degrees. The horizontal North solid Earth tide is: $v_N = \frac{l_2}{g} \frac{\partial V}{\partial \theta}$

This only affects the Geodetic function $$G_m(r,\theta)$$ and thus causes variations of 180 degrees in the phase of the solid Earth tide. These various situations of the phase of the solid Earth tide (referred to the longitude of the station) are shown in the figure below.

For gravity and tilt the situation is very similar although other combination of Love numbers is involved. Let us write the local solid Earth tide phase as: $a_j(t_0) + f_j \cdot(t-t_0) + \nu\tfrac{\pi}{2}$

where $$j$$ again enumerates the tide waves, m = 0, 1, 2 for long-period, diurnal, and semi-diurnal, respectively. The value of $$\nu$$ is a table element given in the Table "Phase relations" and can be used to derive the phase of the solid Earth tide for gravity, East and North tilt for a few latitude values of the station and for the three species ($$m=0$$, 1 and 2).

Phase relations
Latitude -60 -10 10 60
m 0 1 2 0 1 2 0 1 2 0 1 2
gravity 2 2 2 2 2 2 2 2 2 2 2 2
east tilt * 1 1 * 1 1 * -1 1 * -1 1
north tilt 2 2 2 2 0 2 0 2 0 0 0 0
_____
1 However, note that a particular harmonic tide development's amplitudes may not necessarily relate to the Geodetic functions in a 1:1 fashion; Tamura e.g. used Associated Legendre functions, whence conversion factors would appear in equations (2) .

# References

Agnew, D.C., Earth Tides, Chap. 3-06 in: G. Schubert (ed.) Treatise on Geophysics, 2nd ed., , 2015 1st ed. PDF

Bartels, J., Gezeitenkräfte, Handbuch der Physik, Vol. XLVIII, Geophysik II, Springer-Verlag, Berlin, 1957.

Boy, J.-P., M. Llubes, J. Hinderer, and N. Florsch, A comparison of tidal ocean loading models using superconducting gravimeter data, J. Geophys. Res., 108(B4), 2193, doi:10.1029/2002JB002050, 2003.

Tamura, Y., A harmonic development of the tide-generating potential, Bulletin Inform. Marées Terres., 99, 6813–6855, 1987.