4 Computer experiments

4.1 Simulated signal

  We will now present three different numerical simulations. The first one shows the effect of a correlation on the power spectrum and its consequences on the measurement of frequencies. The second permits to verify the convergence of the second step of the proposed algorithm for the estimation of the covariance matrix Q, and the third one estimates the performance of the whole algorithm in recovering the parameters.

• The first simulation aims to analyse the effect of modes correlation on the PSD when L=2. The poles of the AR models have both modulus 0.9 and angle $\pm 2\pi 0.25$, $\pm 2\pi 0.35$. The power of each excitation equals Q₁₁=Q₂₂=2 and the three cases: Q₁₂=0, r₁₂=-3/4, r₁₂=3/4 have been studied. The three PSD calculated using (15) are given in Fig. 1. The effect of the cross-correlation is clearly visible on the plot but it must be emphasized that this effect disappears when the two modes move away from each other. As previously noticed, the existence of a correlation will affect the measured value of the rotational splitting. This result confirms the insight given by the parametric approach for the cross-correlation estimation.
• The purpose of the next simulation is to study the convergence of $\hat{Q}^{[k]}$ and the dependence on the initial values. The poles of the AR models have modulus 0.9 and angle $\pm 2\pi. 0.15$, $\pm 2\pi. 0.35$ and are assumed to be known. N=200 samples of the process y_n have been generated with Q₁₁=Q₂₂=2 and Q₁₂=1. Figure 2 represents 10 iterations of (15) when the initial condition equals $\hat{Q}^{[0]}=\alpha I$, $\alpha=0.5$, 1, 1.5, 2, 2.5 and 3. The smoothing has been performed with R=0.01. This simulation confirms the correct convergence of the algorithm.

  The Cramér Rao bounds [Stoica & Moses 1997], i.e. the asymptotic variance of the estimated value of Q, have been computed for this simulation, see Appendix A. They equal 0.0968 for Q₁₁, 0.1185 for Q₁₂ and 0.0968 for Q₂₂.

• The last simulation of this section evaluates the overall performance of the algorithm. The poles of the model are $0.9\exp(\pm j 2\pi 0.2)$, $0.87\exp(\pm j 2\pi 0.35)$. The excitation covariance is Q₁₁=2, Q₂₂=1, Q₁₂=0, R=0.1 and N=500. 110 independent realisations of the signal (3) have been generated using these parameters. For each realisation, the poles have been estimated using an autocorrelation matrix of order M=20 and the result has been used for the estimation of the excitation covariance. The EM algorithm has been initialised with $\hat{Q}^{[0]}=1.5 I$ and stopped after 100 iterations. Note that for the GOLF signal (next paragraph), the algorithm is performed with 1000 iterations.

  The mean and standard deviation of $\hat{Q}$ and $\hat{r}_{12}$ estimated from the 110 computed simulations are:

  $$\textrm{mean}[\hat{Q}]=\left( \begin{array} {rr} 1.922 & 0.010 \\ 0.010 & 0.977\end{array}\right),$$

  $$\textrm {std}[\hat{Q}]=\left( \begin{array} {rr} 0.228 & 0.272 \\ 0.272 & 0.162\end{array}\right),$$

  $$\textrm {mean}[\hat{r}_{12}]=-0.006,\; \textrm {std}[\hat{r}_{12}]=0.198.$$

    

  Figure 1: Effect of Q on the PSD. Q11=Q22=2. Continuous line (-): r12=0, dashed line ($-\,-$): r12=3/4, dashed-dotted line ($-\cdot$): r12=-3/4

    

  Figure 2: Convergence of $\hat{Q}^{[k]}$ for different initial values

    

  Figure 3: Continuous line: Theoretical PSD of the simulated signal. Dot-dashed line: Average of the estimated PSD of 110 simulations

  It can be seen that the algorithm does not introduce any bias in the estimation of the covariance matrix Q. Even if these estimates do not asymptotically achieve the Cramér-Rao lower bound due to the fact that the dynamical parameters are not estimated by the maximum likelihood method, these results confirm the good performance of the proposed method for the estimation of correlations. Moreover, this performance could be increased by a larger number of iterations. Figure 3 shows that the averaged PSD of the simulations is close to the original one. It has been checked also that the estimates of Q do not depend substantially on the estimates of F.

4.2 GOLF signal

The proposed estimation technique has been applied to GOLF data. The data set corresponds to one and half year of continuous observations. The signal corresponding to each analysed frequency band has been demodulated and decimated by direct and inverse Fourier transforms of a 581 000 point GOLF signal sampled every 80 seconds.

We analysed two frequency ranges containing l=1 modes: the n=11 and n=12. The first band is [1746, 1752]$~\mu$Hz. The corresponding decimated signal contains N=503 points. The poles estimated using an autocorrelation matrix of order M=10 are $0.941\exp(\pm j2\pi 0.198)$ and $0.942\exp(\pm j2\pi 0.275)$. The estimated value of R is 2.1. The iterations of (15) have been initialised with $\hat{Q}^{[0]}=0.5I$.

The estimated value of Q after 1000 iterations ($\Vert\hat{Q}^{[1000]}-\hat{Q}^{[999]}\Vert _2\approx 10^{-6}$) is:

$$\hat{Q}=\left( \begin{array} {rr} 0.180 & -0.100 \\ -0.100 & 0.193\end{array}\right)$$

and $\hat{r}_{12}=-0.533$. Figures 4a, 4b represent the second and fourth components of the smoothed stated vector at the last iteration, i.e. the estimated signal corresponding to each mode. Figure 4c, represents the corresponding GOLF signal its estimated smoothed version.

  

Figure 4: Band [1746, 1752]$~\mu$Hz. a) and b) reconstructed signal of the first and second mode. c) GOLF signal and smoothed signal

The second band is [1882, 1888] $\mu$Hz. The corresponding decimated signal contains N=559 points. The poles estimated using an autocorrelation matrix of order M=10 are $0.922\exp(\pm j2\pi 0.212)$ and $0.925\exp(\pm j2\pi 0.278)$. The estimated value of R is 2. The iterations of (15) have been initialised with $\hat{Q}^{[0]}=0.6I$. The estimated value of Q after 1000 iterations:

$$\hat{Q}=\left( \begin{array} {rr} 0.490 & -0.108 \\ -0.108 & 0.570\end{array}\right)$$

and $\hat{r}_{12}=-0.204$.

Finally we analysed the triplet l=2 in the band [1942, 1950] $\mu$Hz, with N=745. The poles are estimated using an autocorrelation matrix of order M=20. In order to take into account the a priori information on the width of the modes, the moduli of the estimated poles have been replaced by the mean of the three estimated moduli, that is all the three modes are forced to have approximately the same width. The corresponding values are: $0.924\exp(\pm j2\pi 0.188)$,$0.925\exp(\pm j2\pi 0.228)$ and $0.929\exp(\pm j2\pi 0.295)$.The estimated value of R is 2.4. The iterations of (15) have been initialised with $\hat{Q}^{[0]}=0.3I$. The estimated value of Q after 1000 iterations is:

$$\hat{Q}=\left( \begin{array} {rrr} 0.225 & 0.095 & -0.135 \\... ...95 & 0.169 & -0.156\\ -0.135 & -0.156 & 0.265\end{array}\right)$$

and $\hat{r}_{12}=0.489$, $\hat{r}_{13}=-0.554$ $\hat{r}_{23}=-0.737$.

In Figs. 5, 6, 7 the estimated PSD are compared to the PSD assuming uncorrelation of the modes and fitting Lorentzian to the periodogram by maximum likelihood criterion, [Fierry-Fraillon et al. 1998]. 2

  

Figure 5: Estimated PSD. Continuous line (-): proposed method, dashed line ($-\,-$): fit of Lorentzian, dotted line ($\cdots$) periodogram

  

Figure 6: Estimated PSD. Continuous line (-): proposed method, dashed line ($-\,-$): fit of Lorentzian, dotted line ($\cdots$) periodogram

  

Figure 7: Estimated PSD. Continuous line (-): proposed method, dashed line ($-\,-$): fit of Lorentzian, dotted line ($\cdots$) periodogram