SECULAR RATES OF THE ZONAL HARMONICS
We have determined the secular rates of the first I n zonal harmonics of the
geopotential. These secular rates are based on the direct computation of the I n timederivative, assuming linear dependence with time. In this method, all orbital elements
affected by the I n variations contribute to the information in the j n'
EOP parameters have been set to their IERS values. Coordinates have been solved for
all observing laser stations. Absolute velocities have been solved only for those stations
having observed a minimum of three years. Otherwise their NUVEL ··1 values have been
kept.
Several different solutions have been computed. (e.g., j2 alone; j2 and j3 ; j2 , j3 and
j4 ; j2 through j 5 ; 18.6-yr and SI tides solved, etc.) in order to evaluate the stability of
the estimated parameters. Table 2 gathers the results of a series of inversions. Quoted
uncertainties are formal errors, no external calibration of the results being performed. All
inversions give a values of j2 --3 x 10- 11 , in good agreement with recently published
solutions (see Table 3). While Lageos 1 alone does not allow us to separate j2 and j4,
the use of only two years of Lageos 2 data together with the eleven years of Lageos 1
data has permitted to solve for j4' A secular rate of -0.8 x 10- 11 is found for j4, with a
large uncertainty however. If the 18.6-yr ocean tide is also solved for, an unlikely high
value is obtained for j4 (see Table 2). But in this case, the correlation coefficient (as
given by the covariance matrix) between j4 and the 18.6-yr tide coefficients is quite high
so that we do not consider this solution as reliable. In comparison, solving or not for the
S 1 tide does not change the j 2 and j 4 solutions.
The reported value for j3 is - -1.75 x 10- 11 during 1984-1994. III no case we could
separate j3 and j 5, so that the j3 solution represents a lumped coefficient. The j3
solution is quite similar to the value of -1.86 x 10- 11 reported by Nerem et al. (1993) for
the years 1980-1989. These authors suggest that the j3 solution is polluted by the socalled 'Lageos 1 anomaly' appearing early 1989 (Eanes, 1991). This anomaly is seen in
the month to month j 3 time series, and produces an amplification of the annual variation.
It has been suggested that the 'Lageos 1 anomaly" is caused by some unmodeled forces in
the Lageos 1 orbit, perturbing principally the eccentricity, hence 1he odd I n and jn
solutions (Nerem et al., 1993). If we exclude the Lageos 1 data after 1989 from our
solution, a j3 solution of -0.15 ± 0.05 x 10- 11 is obtained, in good agreement with the
solution of Cheng et al. (1989) based on Starlette. Note however that recent multisatellites solutions of Nerem and Klosko (1994, 1995) give positive j3 values (2.3 x
10- 11 and 1.6 x 10- 11 ), i.e., of opposite sign compared to other j3 published values (see
Table 3). The cause of such a disagreement is unclear and further investigations are
needed to obtain reliable j 3 solutions.
144
We have determined the secular rates of the first I n zonal harmonics of the
geopotential. These secular rates are based on the direct computation of the I n timederivative, assuming linear dependence with time. In this method, all orbital elements
affected by the I n variations contribute to the information in the j n'
EOP parameters have been set to their IERS values. Coordinates have been solved for
all observing laser stations. Absolute velocities have been solved only for those stations
having observed a minimum of three years. Otherwise their NUVEL ··1 values have been
kept.
Several different solutions have been computed. (e.g., j2 alone; j2 and j3 ; j2 , j3 and
j4 ; j2 through j 5 ; 18.6-yr and SI tides solved, etc.) in order to evaluate the stability of
the estimated parameters. Table 2 gathers the results of a series of inversions. Quoted
uncertainties are formal errors, no external calibration of the results being performed. All
inversions give a values of j2 --3 x 10- 11 , in good agreement with recently published
solutions (see Table 3). While Lageos 1 alone does not allow us to separate j2 and j4,
the use of only two years of Lageos 2 data together with the eleven years of Lageos 1
data has permitted to solve for j4' A secular rate of -0.8 x 10- 11 is found for j4, with a
large uncertainty however. If the 18.6-yr ocean tide is also solved for, an unlikely high
value is obtained for j4 (see Table 2). But in this case, the correlation coefficient (as
given by the covariance matrix) between j4 and the 18.6-yr tide coefficients is quite high
so that we do not consider this solution as reliable. In comparison, solving or not for the
S 1 tide does not change the j 2 and j 4 solutions.
The reported value for j3 is - -1.75 x 10- 11 during 1984-1994. III no case we could
separate j3 and j 5, so that the j3 solution represents a lumped coefficient. The j3
solution is quite similar to the value of -1.86 x 10- 11 reported by Nerem et al. (1993) for
the years 1980-1989. These authors suggest that the j3 solution is polluted by the socalled 'Lageos 1 anomaly' appearing early 1989 (Eanes, 1991). This anomaly is seen in
the month to month j 3 time series, and produces an amplification of the annual variation.
It has been suggested that the 'Lageos 1 anomaly" is caused by some unmodeled forces in
the Lageos 1 orbit, perturbing principally the eccentricity, hence 1he odd I n and jn
solutions (Nerem et al., 1993). If we exclude the Lageos 1 data after 1989 from our
solution, a j3 solution of -0.15 ± 0.05 x 10- 11 is obtained, in good agreement with the
solution of Cheng et al. (1989) based on Starlette. Note however that recent multisatellites solutions of Nerem and Klosko (1994, 1995) give positive j3 values (2.3 x
10- 11 and 1.6 x 10- 11 ), i.e., of opposite sign compared to other j3 published values (see
Table 3). The cause of such a disagreement is unclear and further investigations are
needed to obtain reliable j 3 solutions.
144