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Astron. Astrophys. 333, 362-368 (1998)

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2. The Monte Carlo approach to p-modes parameters uncertainties

IRIS is a non-imaging atomic resonance spectrophotometer making a full-disk measurement of the integrated solar velocity, and producing a discrete one-dimensional time-series [FORMULA]. The existence of gaps in the IRIS network data, as well as in any other interrupted solar observation, alters the spectral density [FORMULA] of the time series mainly by the existence of sidelobes of the peaks which represent the solar eigenmodes. Our team has investigated two options: gap filling techniques allow to a certain extent to get rid of these spurious peaks (Pantel 1996). However interesting, this technique works best with 75% to 80% duty cycles, which we did not yet achieve. The other possibility is to take into account those sidelobes in the analysis of the power spectra. To the first order, they are Lorentz shaped peaks located at multiples of [FORMULA] Hz around each p-mode peak, and they impact significantly on the fitting strategy leading to the extraction of the mode parameters from the power spectrum. They also influence the precision of the result to an amount which, so far, has been poorly known. To compute the statistical uncertainty on p-mode parameters, we have used a Monte-Carlo (MC) approach, which can be described as a 3 stage process,

  • We perform a maximum likelihood (ML) fit in a frequency range including a [FORMULA] or [FORMULA] group in order to extract a set [FORMULA] of initial parameters.
  • we generate many realizations using random perturbations of the real and imaginary part of the Fourier transform, given by the initial values [FORMULA].
  • we perform a ML fit on each simulation in order to determine the noisy parameters. Then we model the probability density function of each p-mode parameter (frequency, width and amplitude) and evaluate its statistical uncertainty.

2.1. Initial estimation of the parameters

A typical [FORMULA] mode is modeled by a Lorentz function depending on the line frequency [FORMULA], the linewidth [FORMULA], the mode amplitude [FORMULA] and the background noise B like:

[EQUATION]

To handle cases with more than a singlet mode, we introduced a composite profile [FORMULA] where [FORMULA] is a variable size vector of parameters which can describe just as well an [FORMULA] mode, [FORMULA], or a group of 2 peaks like [FORMULA] and [FORMULA], [FORMULA], where the parameter Sdls-ratio adjust the ratio between the amplitude of a peak and its nearest sidelobe. This is just an example and we refer the reader to our previous paper (Gelly et al. 1997), as to the detailed fitting strategy. A Gauss-Newton algorithm is then used to minimize (within bounds) the expression of the maximum likelihood:

[EQUATION]

where [FORMULA] is the spectral density of the solar signal at the discrete frequencies [FORMULA]. This gives the first estimate of the vector of parameters [FORMULA]. As to the initial "guess parameters" needed by the optimization to work, they have no particular value refering to an "a-priori" knowledge of the result. The minimisation converges toward a global minimum in most cases, and two different maximum likelihood fits (with very different guess parameters) will give parameters [FORMULA] matching [FORMULA] 0.2 [FORMULA].

2.2. Spectrum modelisation

The distribution function of helioseismic spectral density is a [FORMULA] deriving from the gaussian distribution of the imaginary and real parts of the Fourier transform of [FORMULA]. Consequently we have used two independent random gaussian variables [FORMULA] and [FORMULA], having a zero mean value and an unit variance to simulate a given frequency range of the Fourier transform [FORMULA]:

[EQUATION]

To be realistic this [FORMULA] must be convolved with the complex window in order to show the daily sidelobes. This last operation introduces additional difficulties:

  • it increases the widths of the peaks. Therefore the initial 'width' parameter, which was estimated on the true spectrum, must be modified in order to restitute the true width of the peaks after this convolution. We perform two Monte Carlo computations: the first for evaluating the increase of the width and the second to give the error bars computed with the modified initial parameter.
  • to take care of the power redistribution of the peaks in the sidelobes we used an amplitude normalized window function
  • finally the background noise is added in the form of a complex gaussian random perturbation only after the convolution.

One realisation of the spectral density is obtained by taking the squared modulus of the Fourier transform. Fig. 1 shows the different stages of the spectrum modelling of a [FORMULA] group for the 1990 dataset ([FORMULA], [FORMULA] and [FORMULA], [FORMULA]).

2.3. Error-bars estimation

We compute 400 Monte Carlo perturbed spectral densities in order to ensure the convergence of the mean toward the initial parameters. The error bars are then given by modeling the probability density function (e.g. the histogram) of the values taken from the many realizations of the parameters.

2.3.1. Frequency

Due to the symmetrical Lorentzian modelling of the p-modes, the frequency PDF shows a symmetric distribution around the mean value, which is fitted by a gaussian distribution Eq. (4) (Gelly et al., 1997). Fig. 2a shows the frequency PDF of [FORMULA], [FORMULA] mode for the year 1990.

[EQUATION]

The frequency uncertainty is given by [FORMULA].

[FIGURE] Fig. 2. Probability density function of frequency (a) and width (b) of the n=19, l=1 mode for the year 1990 dataset in the MC simulation. The superimposed fits give [FORMULA] =2828.25 [FORMULA] 0.15 mHz and [FORMULA] =0.864 [FORMULA] 0.138 µHz.

2.3.2. Amplitude and width

The amplitude and width PDF's are not symmetric because of the [FORMULA] power spectrum distribution and are best modeled by :

[EQUATION]

K being a normalization constant and c being an adjustable shape parameter. The statistical uncertainties [FORMULA] and [FORMULA] are then derived numerically from the modeled PDF.

Fig. 2b shows the width probability density function of [FORMULA], [FORMULA] for the year 1990. It is worth noticing that the width and amplitude PDF have the same characteristics as the distribution function for time averaged energies of stochastically excited solar p-modes (Kumar, 1988). This is just the effect of gaussian random perturbation in the MC simulation of the real and imaginary part of the Fourier transform, that corresponds to the hypothesis of the stochastic nature of the excitation. Fig. 3a-c shows three normalized PDF of the amplitudes, computed by MC simulations, as a function of [FORMULA] (where T is the duration of observation and [FORMULA] is proportional to the inverse of the lifetime of the mode), and the corresponding modeled distribution given by Eq. (5). For low values of [FORMULA], the PDF are well modeled by a Boltzmann law ([FORMULA] =0) and by a gaussian law (high values of [FORMULA]) for large values of [FORMULA].


[FIGURE] Fig. 3a-c. Variations of the amplitude probability density function as a function of [FORMULA], in agreement with the stochastic excitation of solar p-modes (Kumar, 1988).

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© European Southern Observatory (ESO) 1998

Online publication: April 15, 1998
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