FAMOUS software, developed by F. Mignard (OCA, France) in the framework of the GAIA project (Mignard, 2005), carries out the frequency analysis of a time signal, with any kind of sampling. Usually time-series derived from observations are not regularly sampled (and even can be very irregularly sampled in the case of Geosciences) and the main feature of this application is to handle this sampling. This program detects the existing periods in the signal and estimates the associated amplitudes and phases. The amplitudes and phases of annual and semi-annual signals are estimated from each series using a non-linear least squares method called the Levenberg-Marquard method. The algorithm of this software decomposes a time-series y(t) as a Poisson series where it determines the frequencies η_k and the coefficients C_k(t) and S_k(t):

The first step of the algorithm consists of removing a polynomial trend from the time-series, adjusted by the least squares method. Then, a first least squares periodogram of the residual series is obtained in a set of frequencies. For each frequency, a constant and coefficients of sines and cosines are adjusted. So, the most powerful spectral line is retained and re-estimated by a non-linear optimisation adjustment to detect the next most significant spectral line. At each step, all frequencies, as well as the trend, are readjusted by the Levenberg-Marquard Method. The last step consists in analysing and filtering the frequencies to keep only the significant spectral lines, by means of frequency resolution and signal-to-noise ratio (S/N) test. The significance of the detected spectral lines is assessed in the spectral domain by dividing the square root of the amplitude by the median value of the noise computed once the line is removed (Collilieux et al., 2007). We adopt the value of 3.0 for analysing the time-series.

As results, the program gives; in addition to the cosine, sine terms and their standards deviations; the values of frequency η, amplitude A and phase φ, where the signal equation can be expressed by (Mignard, 2005):

According to the error propagation law, the standard deviations of the amplitude and phase are given as:

C_k, σ_C: Cosine term of k-th frequency and its standard deviation;

S_k, σ_S: Sinus term of k-th frequency and its standard deviation.

The Allan variance analysis was developed and is widely used for estimating the frequency stability of atomic clocks (Allan, 1966, 1987). This diagnosis tool was extended to geodetic data (Feissel-Verneir et al., 2007). By definition, the Allan variance of position residuals, for a given time interval, is computed by averaging the position residuals over that interval and computing the variance of differences between adjacent averaged values. This method allows one to characterise the statistical behaviour of time-series, in particular, to identify white noise (spectral density S independent of frequency f), flicker noise (S proportional to 1/f), and random walk noise (S proportional to 1/f²). Let us assume a time-series (X_j)_j=1,_N regular on a constant interval τ₀, for a given sampling time τ (with τ = M × τ₀), the Allan variance estimation is defined by:

The Allan variance can be expressed in function of the spectral density S, as:

A convenient and rigorous way to relate the Allan variance of a signal to its error spectrum is the interpretation of the Allan diagram, which gives the changes of the Allan variances for increasing values of τ, in logarithmic scales, corresponding to the following equation:

The noise type is measured by the slope of the Allan graph, which describes the log-log relationship of the Allan variance of the time series. The noise level is measured by the Allan deviation for a one-year sampling time of the non-linear, non-seasonal position time-series. These estimations are performed under a stationarity assumption (Feissel-Vernier et al., 2007), i.e., the scattering of the data is homogenous in time. Here, the autocorrelation function is used in order to check the stationarity of the time series.

In the context of this study, a white noise signature in the position residuals would point to random errors (Gaussian errors) affecting the measurements, while a flicker noise signature would point to perturbations that may have different origins, such as local tectonics, instrument defects, analysis consistency, etc. (Feissel-Vernier et al., 2007). Concerning the random walk, which is caused by uncorrected jumps in a time-series, it is not found in our analysis.

The station position time series are estimated with respect to the ITRF2005 mean position corrected from plate tectonics motion (ITRF2005 velocities), Earth solid tides, pole tide and oceanic loading effects. These time series must consequently show the atmospheric and/or hydrologic loading effects. Fig. 1 gives examples of time series of topocentric coordinate updates (N, E, U) for two stations McDonald (7080) and Yarragadee (7090), according to different solutions of combination. Table 4 exhibits the statistics of results obtained on coordinates updates of some ILRS stations. The contribution of the Starlette data on the determination of the estimated parameters of these stations is as follows:

• • increase of number of solutions compared to those obtained from the LA-1 and LA-1&LA-2 combinations;
• • improvement, of a few millimeter, of some stations coordinates quality; for example UP component of station 7090 and East component of station 7105; by adopting the LA-1&STAR solution compared to LA-1 one. However, a degradation of quality on coordinate updates is observed, of about 3 to 5 mm, which is mainly due to a noise induced by the forces effects on the low altitude orbit of Starlette.

The advantage of the time series of station coordinates calculated in a homogeneous reference frame is to enable us to highlight residual signals compared to the a priori signals used in modelling (geophysical signals). In this framework, is carried out a frequency analysis on vertical component series of some stations by FAMOUS software. We have focused on this component because it is important for the geodynamical studies since it holds two-thirds amplitude of signals acting on the station motion (Coulot, 2005).

The results in terms of amplitude and phase of the signal in each UP series of station and according to combination solutions (LA-1, LA-1&LA-2 and LA-1&STAR) are given in Table 5. Annual and semi-annual signals, in vertical time series of stations (7080, 7840, 8834, 7110 and 7839), with amplitudes between 1 to 8 mm can be estimated. Since the effects of ocean loading were considered in the a priori model of orbit restitution, the estimated signals are probably related to residual loading effects as atmospheric and/or hydrologic loading effects, which typically have amplitudes of a few mm.

Fig. 2 gives an example of comparison between the UP periodic signals according to the different combinations and to the atmospheric loads model (ECMWF series), for 7080 and 7090 laser stations. These stations were chosen for their best quality of observational results and for their larger number of normal points of different satellites. The results revealed that there is coherence between the two signals for each station, with correlation of about 40 to 50%, in the case of the LA1&STAR combination. This explains that the UP variations of the two stations (7080 and 7090) are due to atmospheric loadings. The remains variations are come mainly from noise. It is interesting to indicate which type of noise affects the time series of coordinates in order to make diagnosis about the possible sources of errors in the estimation of these parameters. Table 6 provides the noise level and the noise type of some stations UP components. The white noise is the dominant noise in these series according to the different solutions, while a weak flicker noise combined with white noise is observed in case of LA-1 and LA-1&LA-2 solutions (Fig. 3). In addition, a periodic signal was detected in UP series of 7835 station. According to Table 4, the noise level is of about 3 to 7 mm.

Table 7 summarizes the statistics of the weekly series of Helmert transformation parameters (translation components and scale factor), between different combinations solutions and ITRF2005, and the daily series of pole motion parameters (pole coordinates and LOD).

5.2.1 Geocenter variations

Fig. 4 illustrates the translation time series according to the different Laser measurement combinations of LAGEOS-1&-2 and Starlette satellites. It can be seen that the solution of the combination (LA-1&STAR) is slightly larger compared to the reference solution (LA-1&LA-2). Indeed, in the case of the LA-1&STAR combination, the dispersion is of about 2.7 cm for TX, 2.8 cm for TY and 2.4 cm for TZ, with average RMS of about ±5 mm. However, in the case of the reference solution, it is only of about 2 cm, 1.9 cm and 1.6 cm, for TX, TY and TZ, respectively, with mean RMS of about ±4 mm (Table 7).

Fig. 5 shows a comparison between translation time series from (LA-1&STAR) and (LA-1&STARw) solutions. The latter is obtained by weighting the Starlette observations taking into account the corresponding standard deviations which are twice larger than those of LAGEOS performing the degree of freedom method (Coulot, 2005 for more details), in order to evaluate the effect of weighting on the results. It can be noted that the results are considerably equivalent for the parameters (TX and TY), and with a less degree, for the TZ component.

For phases, the agreements are less convincing. This is probably due to the choice of reference frame and physical models used in the computations (Coulot and Berio, 2006). The spectral behavior of time series of Geocenter motion, described by the method of the Allan variance, is shown in Fig. 6a. After removing the trend (estimated by linear regression) and the periodic components (annual and semi-annual terms), we applied the Allan variance on the resulting Geocenter motion time series. The dominant noise type in these series is white noise. However, the noise affecting the three Geocenter components of LA-1&STAR solution is slightly important, being about 2 mm. The solution (LA-1&LA-2) remains the least disturbed with a noise level of about 1 mm. On the other hand, Table 9 and Fig. 6b show that the weighting of Starlette measurements affected only the TZ component where the type of noise is a flicker noise, with the same order of 2 mm.

Table 9

Bruit des séries temporelles des paramètres de transformation et du facteur d’échelle. La pente du graphe log-log de la variance d’Allan désigne le type du bruit et NL est le niveau de bruit.

	TX (cm)	TY (cm)	TZ (cm)	D (ppb)
Solution	Slope	NL	Slope	NL	Slope	NL	Slope	NL
LA-1	–0.9	0.16 ± 0.01	–0.8	0.16 ± 0.01	–0.7	0.16 ± 0.01	–0.6	0.28 ± 0.02
LA-1&LA-2	–0.7	0.14 ± 0.01	–0.7	0.12 ± 0.004	–0.9	0.12 ± 0.01	–0.6	0.22 ± 0.01
LA-1 &STAR	–0.7	0.19 ± 0.02	–0.8	0.17 ± 0.01	–0.7	0.19 ± 0.01	–0.8	0.32 ± 0.04
LA-1 &STARw	–0.8	0.19 ± 0.01	–0.8	0.19 ± 0.01	–0.5	0.22 ± 0.01	–0.8	0.46 ± 0.08

5.2.2 Scale factor variations

Scale factor variations of the reference frame are affected by errors of the stations’ vertical component determination (Coulot, 2005). Thus, the range biases and errors on the radial components due to residual orbital errors, which limit the accuracy of the vertical components, affect the scale factor variations. On the other hand, the network effect (Collilieux and Altamimi, 2008) is another factor that may also affect the scale factor. Fig. 7 shows the time series of the scale factor variations, corresponding to different combinations.

Fig. 8 shows the graph of the Allan variance of time series of the scale factor. It can be seen that the dominant noise for combinations using measurements of Starlette is a white noise of about 2 to 3 mm level. For LAGEOS solutions, one notices, in addition to white noise, the presence of flicker noise with a level of 1 to 2 mm (Table 9).

Fig. 9 presents the time series residuals of the pole coordinates (X_p, Y_p), and of the LOD. These times series were processed, for different solutions, with respect to the standard solution EOPC04 of IERS, and expressed according to a coherent reference frame with ITRF2005. From Table 2, the standard deviations of the (LA-1&-2) solution are, respectively, of about 0.14 mas (∼4 mm), and 0.01 ms (∼5 mm) for the pole coordinates and time.

According to the LA-1&STAR solution, the RMS are slightly large, in the range of 0.20 to 0.23 mas (or, 6–7 mm) for pole coordinates, and 0.02 ms (or 10 mm) for LOD. In addition, the estimation of pole parameters is satisfactory for the SLR technique and the obtained values are coherent with published values of IERS, which are at a level of 200–300 μas (Gambis, 2004).

Fig. 10 shows an agreement between both time series of the LA1&STAR solutions, in the case of weighted and un-weighted Starlette observations. In fact, in terms of RMS, the two series are at a level of 5 to 6 mm for pole coordinates and of about 10 mm for LOD; Table 2. The EOP solutions obtained by SLR technique appear to be inconsistent with EOPC04 at level of 5 to 10 mm (∼200 to 300 μas) (Gambis, 2004).

Table 10 depicts the results of frequency analysis of pole parameters series, using FAMOUS software. The periodic signals are decomposed, with respect to three periods: inter-annual, annual and short periods (from few days to few months < 100 days). The choice of this decomposition is based on the periods of the geophysical phenomena causing the variations in the pole motion. These phenomena are mainly due to the redistribution of masses in the Earth, oceans and atmosphere (Frède, 1999).

The results of different combinations are very close, because the maximum amplitudes of different signals of polar motion do not exceed 74 μas (or 2.2 mm). As can be observed, the residual effects of geophysical phenomena, mentioned above, on the polar motion, are weak and of about few millimeters. For LOD, the variations are induced by the friction of the winds and ocean currents on the solid Earth surface (Frède, 1999). Similarly, variations of this parameter can be decomposed into three terms. The estimated amplitudes of these terms range from about 4 to 6 μs (approximately 2–3 mm) for LAGEOS combinations and are around 7 to 14 μs (or 4–7 mm) for Starlette and LAGEOS-1 combinations. Generally, these values remain very small because they describe the residual signals of the geophysical phenomena.

According to Table 11, all the combinations are characterised by a flicker noise, with levels of about 3 mm and 4 mm in polar motion, 9.5 mm and 14.5 mm in LOD, for LA-1&LA-2 and LA-1&Star, respectively.